Cold molecules: theory, experiment, applications

COLD MOLECULES Theory, Experiment, Applications © 2009 by Taylor and Francis Group, LLC COLD MOLECULES Theory, Exper...

Author: Roman Krems | Bretislav Friedrich | William C Stwalley

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COLD MOLECULES Theory, Experiment, Applications

© 2009 by Taylor and Francis Group, LLC

COLD MOLECULES Theory, Experiment, Applications Edited by

Roman V. Krems William C. Stwalley Bretislav Friedrich

CRC Press Taylor & Francis Croup Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informs business


CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4200-5903-8 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Cold molecules: theory, experiment, applications / [edited by] Roman Krems, Bretislav Friedrich, and William C. Stwalley. p. cm. Includes bibliographical references and index. ISBN 978-1-4200-5903-8 (hard back: alk. paper) 1. Collisions (Nuclear physics) 2. Low temperatures. 3. Quantum solids. 4. Quantum liquids. 5. Cold gases. 6. Molecular dynamics. I. Krems, Roman. II. Friedrich, Bretislav. III. Stwalley, William C, 1942- IV. Title. QC794.6.C6C635 2009 530.4-dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com


2009015987

Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi A Guided Tour of the Monograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix Part I

Cold Collisions

Chapter 1

Theory of Cold Atomic and Molecular Collisions . . . . . . . . . . . . . . . . . 3 Jeremy M. Hutson

Chapter 2

Electric Dipoles at Ultralow Temperatures . . . . . . . . . . . . . . . . . . . . . . 39 John L. Bohn

Chapter 3

Inelastic Collisions and Chemical Reactions of Molecules at Ultracold Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Goulven Quéméner, Naduvalath Balakrishnan, and Alexander Dalgarno

Chapter 4

Effects of External Electromagnetic Fields on Collisions of Molecules at Low Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Timur V. Tscherbul and Roman V. Krems

Part II

Photoassociation

Chapter 5

Ultracold Molecule Formation by Photoassociation . . . . . . . . . . . . . 169 William C. Stwalley, Phillip L. Gould, and Edward E. Eyler

Chapter 6

Molecular States Near a Collision Threshold . . . . . . . . . . . . . . . . . . . 221 Paul S. Julienne

Chapter 7

Prospects for Control of Ultracold Molecule Formation via Photoassociation with Chirped Laser Pulses . . . . . . . . . . . . . . . . . . . 245 Eliane Luc-Koenig and Françoise Masnou-Seeuws

Chapter 8

Adiabatic Raman Photoassociation with Shaped Laser Pulses . . . . 291 Evgeny A. Shapiro and Moshe Shapiro v


vi

Contents

Part III

Few- and Many-Body Physics

Chapter 9

Ultracold Feshbach Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Francesca Ferlaino, Steven Knoop, and Rudolf Grimm

Chapter 10

Molecular Regimes in Ultracold Fermi Gases . . . . . . . . . . . . . . . . . 355 Dmitry S. Petrov, Christophe Salomon, and Georgy V. Shlyapnikov

Chapter 11

Theory of Ultracold Feshbach Molecules . . . . . . . . . . . . . . . . . . . . . 399 Thomas M. Hanna, Hugo Martay, and Thorsten Köhler

Chapter 12

Condensed Matter Physics with Cold Polar Molecules . . . . . . . . . 421 Guido Pupillo, Andrea Micheli, Hans-Peter Büchler, and Peter Zoller

Part IV

Cooling and Trapping

Chapter 13

Cooling, Trap Loading, and Beam Production Using a Cryogenic Helium Buffer Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Wesley C. Campbell and John M. Doyle

Chapter 14

Slowing, Trapping, and Storing of Polar Molecules by Means of Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 Sebastiaan Y.T. van de Meerakker, Hendrick L. Bethlem, and Gerard Meijer

Part V

Tests of Fundamental Laws

Chapter 15

Preparation and Manipulation of Molecules for Fundamental Physics Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 Michael R. Tarbutt, Jony J. Hudson, Ben E. Sauer, and Edward A. Hinds

Chapter 16

Variation of Fundamental Constants as Revealed by Molecules: Astrophysical Observations and Laboratory Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 Victor V. Flambaum and Mikhail G. Kozlov

Part VI

Quantum Computing

Chapter 17

Quantum Information Processing with Ultracold Polar Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 Susanne F. Yelin, Dave DeMille, and Robin Côté


vii

Contents

Part VII Chapter 18

Cold Molecular Ions Sympathetically Cooled Molecular Ions: From Principles to First Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 Bernhard Roth and Stephan Schiller


Foreword This volume offers an exhilarating journey to a compelling frontier of molecular physics. It is a special pleasure to applaud the authors and editors. Intrepid explorers of the frontier, they are also adept guides for both recruits and curious visitors. The result is a lucid and lively evangelical survey of seminal results, experimental and theoretical methods, prospects and challenges. My benediction will simply touch on some kinships, past or future, with more civilized fields. Any physical chemist of my vintage is delighted to see molecules now ardently embraced by physicists. The oft-quoted dictum: “A diatomic molecule has one atom too many” has been disavowed. The new converts to molecular physics are emulating illustrious ancestors from early decades of the 20th century, who found that molecules presented challenging questions for the then fledgling quantum theory. Today, much of the “hot” appeal of molecules in the “cold” ( E, which is separated from the classically allowed region by the outer turning point. In general, of course, it is possible for there to be more than one classically allowed region and additional classical turning points. This is particularly common in cold molecule applications when the collision energy is lower than the height of a centrifugal barrier in VL (r). For a scattering calculation, it is sufficient to solve Equation 1.15 numerically by propagating ψL (r) outwards, starting at a point rmin deep in the inner classically forbidden region. In the classically forbidden region, the wavefunction curves away from zero according to Equation 1.15. Thus a solution that starts from zero will increase approximately exponentially until it reaches the inner turning point, then start to oscillate. The usual procedure is to propagate the solution out to a distance rmax and then to match the wavefunction and its derivative to Equation 1.17 to obtain the phase shift δL . This is then repeated for sufficient values of L to obtain convergence of the summations in the expressions for cross-sections. The propagation can be carried out using any of a variety of techniques for solving second-order differential equations, such as the renormalized Numerov method of Johnson [12]. For bound states the problem is a little more difficult, because solutions of Equation 1.15 that satisfy the boundary conditions Equation 1.23 do not exist at all energies. In general, if a solution of the differential equation that satisfies the boundary condition at one end of the range is calculated at an arbitrary energy, it will not satisfy the boundary condition at the other end of the range. Furthermore, propagating a solution of a differential equation into a classically forbidden region is a notoriously unstable procedure, because any exponentially increasing component quickly dominates the (usually desired) exponentially decreasing component, and any information about the exponentially decreasing component becomes lost in the numerical errors. This is not a problem when propagating a solution out of a classically forbidden region, because the exponentially increasing solution is then the one that is wanted. The approach usually adopted is to guess an energy Etrial and propagate a solution ψ+ (r) outwards from rmin to a point rmid in the classically allowed region. A second solution ψ− (r) is then propagated inwards to meet it from a point rmax , far into the outer classically forbidden region. The two solutions are arbitrarily normalized, so their values can always be made to agree at the matching point by renormalizing them. However, for a finite potential VL (r), the derivative ψ (r) of the wavefunction must be continuous, as well as its value ψ(r). The criterion for Etrial to be an eigenvalue is therefore [ψ+ ] (rmid )/ψ+ (rmid ) = [ψ− ] (rmid )/ψ− (rmid ), (1.34) where the superscripts + and − indicate outwards and inwards solutions originating at short and long range, respectively. The matching function [ψ+ ] /ψ+ − [ψ− ] /ψ− is thus a function of energy that is zero when E is an eigenvalue; it would be possible to © 2009 by Taylor and Francis Group, LLC

Theory of Cold Atomic and Molecular Collisions

15

converge on an eigenvalue simply by using one of the standard numerical procedures for finding a zero of a function, such as the secant method, to find a zero of the difference of the normalized derivatives. However, in the one-dimensional case, it is common to use Numerov propagation with an explicit energy correction formula due to Cooley [13], which gives quadratic convergence in the energy. This algorithm usually converges to ±10−6 cm−1 in less than 10 iterations, so that it offers a very efficient method for finding eigenvalues and eigenfunctions of one-dimensional Schrödinger equations.

1.3.2

MULTICHANNEL SCATTERING

The discussion of quantum scattering so far has concentrated on collisions between unstructured particles with only a single internal state. However, most systems of interest involve structured particles. The Hamiltonian for the colliding pair may generally be written 2 (1.35) − ∇ 2 + Hˆ int (τ) + V (r, τ), 2μ where τ denotes all coordinates except the interparticle distance r and Hˆ int (τ) represents the internal Hamiltonians of the two particles. The intermolecular potential V (r, τ) is now a function of τ as well as r. The wavefunction for a pair of colliding particles may be written in the form

Ψ(r, τ) = r −1 φi (τ)ψi (r), (1.36) i

where the channel functions {φi (τ)} form a basis set in the τ coordinates. The total wavefunction Ψ(r, τ) thus has components in each channel i. For both bound states and scattering, the wavefunction must be regular at the origin, and when V (r) 0 as r → 0 the short-range boundary conditions are ψi (r) → 0

as

r → 0.

(1.37)

The particular sets of channel functions needed are different for each type of scattering problem, and it would be quite impossible to enumerate all the possibilities, so just two relatively simple examples will be given here: atom–diatom scattering and collisions of metastable helium atoms. 1.3.2.1 Atom–Diatom Scattering Consider collisions between a structureless atom A and a closed-shell diatomic molecule BC, which is allowed to rotate but not vibrate. Nuclear spins are neglected. The orientation of the diatomic molecule is described by a unit vector rˆBC with spherical polar coordinates (θBC , φBC ). The corresponding rotational wavefunctions are spherical harmonics Yjmj (ˆrBC ) = Yjmj (θBC , φBC ). The position of the atom with respect to the center of mass of the diatom is given by spherical polar coordinates (r, rÂ ) = (r, θA , φA ) with corresponding spherical harmonics YLmL (ˆrA ). © 2009 by Taylor and Francis Group, LLC

16

Cold Molecules: Theory, Experiment, Applications

It is possible to expand the total wavefunction as in Equation 1.36 using simple products of spherical harmonics, φi (τ) = Yjmj (ˆrBC )YLmL (ˆrA ).

(1.38)

In this representation, the index i represents the set of quantum numbers { j, mj , L, mL }. This expansion is suitable under some circumstances, such as calculations on collisions in applied electric and magnetic fields. However, for field-free scattering it is inefficient because it fails to take advantage of the fact that the total angular momentum J is a good quantum number. It is therefore usual to use the Arthurs–Dalgarno total angular momentum representation [14], with basis functions that are eigenfunctions of Jˆ 2 and JˆZ as well as ˆj2 and Lˆ 2 . These are formed as linear combinations of the simple product functions,

JMJ φi (τ) = YjL = jmj LmL |JMJ Yjmj (ˆrBC )YLmL (ˆrA ), (1.39) mj mL

where jmj LmL |JMJ is a Clebsch–Gordan vector coupling coefficient. The index i thus represents the set of quantum numbers { j, L, J, MJ }. The Hamiltonian of the colliding pair is diagonal in the quantum numbers J and MJ (and independent of MJ ). It is therefore adequate to solve equations for each value of J separately, which results in very large savings in computer time. The computational efficiency may be further increased by taking advantage of other symmetries such as total parity. For each value of J, the channel basis set contains a set of functions with values of j and L that can couple together to give a resultant J. There are matrix elements involving the anisotropic part of the interaction potential that are off-diagonal in j and L. A common approach is to include all monomer rotational functions with j limited by a maximum value jmax , and for each j to include all allowed values of L of the desired parity (−1) j+L that satisfy the triangle condition | j − L| ≤ J ≤ j + L. Such calculations are often referred to as close-coupling calculations. The value of jmax required depends on the anisotropy of the interaction potential and it is of course always necessary to check convergence of the results with respect to jmax . The total angular momentum J plays the part of a partial wave quantum number. As for collisions of structureless particles, integral cross-sections involve sums over partial waves and differential cross-sections involve double sums, with interference between different partial waves playing an important role. 1.3.2.2

Scattering of Metastable Helium Atoms

States of individual atoms are usually described by quantum numbers L, S, and Ja for the electronic orbital, spin, and total angular momenta, respectively. However, in scattering and bound-state problems involving pairs of atoms or molecules it is common to use lower-case letters for quantum numbers of individual collision partners and reserve capital letters for quantities that refer to the collision system (or complex) as a whole. Thus, in this subsection we will use l and s for the quantum numbers of a single helium atom and reserve L and S for the end-over-end angular momentum of the atomic pair and the total spin, respectively. © 2009 by Taylor and Francis Group, LLC

17


The helium atom has a metastable 3 S1 excited state with quantum numbers a = ja = 1 and a radiative lifetime of 8000 s. To carry out quantum scattering calculations on collisions of two such atoms, we need channel functions to handle the two spin quantum numbers and the mechanical angular momentum L. Once again it would be possible to use simple product functions as channel functions, |s1 ms1 |s2 ms2 |LML ,

(1.40)

but in the absence of external fields it is more efficient to use a total angular momentum representation. There is more than one way to construct such functions from three angular momenta, but the obvious one in this case is to couple s1 and s2 to give a total S and then couple S to L to give a resultant J, |s1 s2 SMS =

s1 ms1 s2 ms2 |SMS |s1 ms1 |s2 ms2 ;

ms1 ms2

|SL JMJ =

SMS LML |JMJ |SMS |LML .

(1.41)

MS ML

Once again, the resulting equations are diagonal in J and independent of MJ . Additional savings may be made by constructing functions that are symmetric with respect to exchange of identical atoms. In the case of metastable 4 He, the functions that are antisymmetric with respect to exchange may be discarded because the atoms are composite bosons.

1.3.3

COUPLED EQUATIONS

Substituting the expansion (1.36) into the total Schrödinger equation and projecting onto a single channel function φj (τ) yields a set of coupled equations,

2 d2 − − E ψ (r) = − Wji (r)ψi (r), j 2μ dr 2

(1.42)

i

where i and j are collective indices that each include all the quantum numbers needed to describe a channel. The coupling matrix W(r) generally contains off-diagonal as well as diagonal terms, Lˆ 2 ∗ Wji (r) = φj (τ) Hint (τ) + V (r, τ) + φi (τ) dτ. (1.43) 2μr 2 The coupling matrix W(r) does not become zero at long range, because different channels correspond to different energies of the separated monomers. Indeed, in some channel basis sets, W(r) may not even be diagonal at long range. However, it is almost always best to transform into a representation in which W(r) is diagonal, r→∞

Wii (r) ∼ wi δij , © 2009 by Taylor and Francis Group, LLC

(1.44)

18


before applying scattering boundary conditions. The energies wi , which correspond to levels of (pairs of) free monomers, are often referred to as thresholds because they represent energies at which pairs of monomers can be separated into the states concerned. The channels in a coupled-channel calculation at total energy E are usually divided into open channels, which correspond to energetically accessible monomer levels (E ≥ wi ), and closed channels, which correspond to energetically inaccessible monomer levels (E < wi ). The channel wavenumber ki for channel i is defined through the equation 2 ki2 = |E − wi | = |Ekin,i |, 2μ

(1.45)

where Ekin,i is the kinetic energy for channel i. In quantum scattering, we envisage a situation where there is an incoming wave in a single channel j, corresponding to the incident particles, and outgoing waves in all open channels, corresponding to scattered particles. The corresponding unnormalized wavefunction is asymptotically r→∞ −1

Ψ(r, τ) = r

−1/2 −ikj r+iLj π/2 φj (τ)kj e

+

−1/2 iki r−iLi π/2 Sji φi (τ)ki e

,

i

(1.46) where the sum runs over open channels only. There is a separate solution corresponding to each possible incoming channel, and the solution is characterized at long range by the S-matrix with elements Sji . The S-matrix is an Nopen × Nopen complex symmetric matrix, where Nopen is the number of open channels. It is unitary, that is, SS† = I, where S† indicates the Hermitian conjugate and I is a unit matrix. If the physical problem is factorized into separate sets of coupled equations for different symmetries (such as total angular momentum or parity), there is a separate S-matrix for each symmetry. All properties that correspond to completed collisions, such as elastic and inelastic integral and differential cross-sections, can be written in terms of S-matrices. There are usually several (or many) channels i, with different values of the orbital angular momentum Li , corresponding to each pair of internal states α of the colliding atoms or molecules. We indicate the set of channels corresponding to a particular set of monomer quantum numbers α as i ∈ α. The integral cross-section for molecules to undergo a transition from states α before the collision and to states β after the collision is π

σαβ = 2 gi |δij − Sij |2 , (1.47) kα i∈α j∈β

where the degeneracy factor gi is a function of the channel quantum numbers that depends on the particular channel basis set in use. In practice, the coupled equations usually factorize into sets with different symmetries (such as total angular © 2009 by Taylor and Francis Group, LLC

19


momentum), each with its own S-matrix, so that there is an additional sum over symmetries. 1.3.3.1

Scattering Calculations

If the basis set contains N channel functions of a particular symmetry, a single solution of the coupled equations 1.42 is a vector ψ(r) made up of N radial strength functions ψi (r) for i = 1 to N. However, at any energy there are N linearly independent solution vectors that satisfy the boundary conditions (1.37) at short range, and it is usually not possible to select a single one of them that is physically relevant until longrange boundary conditions are applied. In computational terms it is therefore usually necessary to solve for all N solutions simultaneously to obtain an N × N wavefunction matrix Ψ(r) with elements ψij (r). The radial strength functions ψij (r) are in general complex. However, it is almost always possible to transform the problem to work with real functions and to express the complex solutions, when needed, in terms of these. For both open and closed channels, the boundary conditions require that Ψ(r) → 0 at short range. However, at long range the scattering boundary conditions corresponding to real solutions are r→∞

Ψ(r) = J(r) + N(r)K,

(1.48)

where the S-matrix is related to the open-open part K oo of the real symmetric K-matrix by S = (1 + iK oo )−1 (1 − iK oo ).

(1.49)

The matrices J(r) and N(r) are diagonal matrices with elements for open channels given by [15] 1/2

jLj (kj r)

1/2

nLj (kj r),

[J(r)]ij = δij rkj [N(r)]ij = δij rkj

(1.50)

where jL and nL are spherical Bessel functions, and for closed channels given by [J(r)]ij = δij (kj r)−1/2 ILj +1/2 (kj r) [N(r)]ij = δij (kj r)−1/2 KLj +1/2 (kj r),

(1.51)

where IL and KL are modified spherical Bessel functions of the first and third kinds [6]. (Note that the Riccati–Bessel functions used by Johnson [15] are krjL (kr) and krnL (kr) in the notation of Ref. [6].) The use of Bessel functions in place of sines and cosines takes into account the effect of the centrifugal terms in W(r) and makes it possible to apply the boundary conditions at much shorter range. 1.3.3.2

Numerical Methods for Scattering

As in the single-channel case, the coupled-channel equations are usually solved by numerical propagation. One approach is to start a solution matrix Ψ(r) at a distance © 2009 by Taylor and Francis Group, LLC

20


rmin , far enough into the classically forbidden region (in all channels) to assume that Ψ(rmin ) = 0. The solution is propagated outwards to rmax , which is chosen to be at long enough range that the effects of the potential V (r, τ) are negligible outside rmax . The wavefunction matrix and its derivative are then matched to the boundary conditions Equation 1.48 to obtain the real symmetric matrix K. This is then transformed using Equation 1.49 to obtain the physically meaningful S-matrix. There are many different propagators that have been developed to take account of the special properties of Equation 1.42, and a detailed discussion is beyond the scope of this chapter. Among those commonly used in cold molecule collisions are the renormalized Numerov approach [16,17] and several different log-derivative propagators [15,18,19]. The latter actually propagate the log-derivative matrix, defined as Y(r) =

dΨ [Ψ(r)]−1 , dr

(1.52)

in place of the wavefunction itself. The K matrix can be obtained from the logderivative matrix at rmax without requiring the wavefunction matrix explicitly [15]. The log-derivative propagators have good step-size convergence properties and do not suffer from stability problems arising from the presence of deeply closed channels. The airy-based log-derivative propagator of Alexander and Manolopoulos [19] has particular advantages for cold collisions because it allows very large propagation steps at long range. A variety of program packages are available to construct and solve sets of coupled equations for molecular quantum scattering calculations. These include the MOLSCAT package of Hutson and Green [20] and the HIBRIDON package of Alexander et al. [21]. 1.3.3.3

Decoupling Approximations

The computer time taken to solve a set of coupled equations is generally proportional to N 3 , where N is the number of channel basis functions in the expansion of Equation 1.36. Even for atom–diatom scattering, the size of the basis set required for close-coupling calculations can be prohibitively large, especially for systems with small rotational constants or strongly anisotropic potential energy surfaces. For more complicated systems, such as molecule–molecule scattering or for problems involving open-shell or nonlinear molecules, the problem is substantially worse. There are thus a number of decoupling approximations that factorize the coupled equations into smaller sets that can be solved separately. Approaches that have been popular for collisions at higher energies include the CS (coupled states or centrifugal sudden) approximation and the IOS (infinite-order sudden) approximation. A detailed discussion of these is beyond the scope of this chapter, but there is a good (if old) review by Kouri [22]. Decoupling approximations can save enormous amounts of computer time and can often provide approximate solutions for otherwise intractable scattering problems. However, they necessarily involve approximations, and must be treated with great care for cold molecule interactions. In particular, the CS and IOS approximations in their usual form both approximate the end-over-end angular momentum © 2009 by Taylor and Francis Group, LLC

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operator Lˆ 2 and thus affect centrifugal barriers, which can be extremely important in the ultracold regime. Nevertheless, carefully crafted decoupling approximations can be of great value. 1.3.3.4

Bound States

Provided the interaction potential is sufficiently attractive (which it almost always is), the multichannel Schrödinger equation supports bound states at energies En below the lowest threshold. The corresponding wavefunction for state n is Ψn (r, τ) = r −1

φi (τ)ψin (r).

(1.53)

i

The boundary conditions for bound-state problems are ψin (r) → 0

as

r→0

or ∞

(1.54)

in all channels i. Multichannel bound-state problems are much more complicated than singlechannel problems. Although to a first approximation it may be true that each bound state is supported primarily by a single channel with a well-defined potential well, this is not generally the case. The coupling between channels provided by W(r) mixes and shifts the levels in a nontrivial way that is outside the scope of this chapter but has been explored extensively in the context of the spectroscopy of Van der Waals complexes [23]. 1.3.3.5

Numerical Methods for Bound States

There are two general approaches to the calculation of bound states of multichannel systems: coupled-channel methods and radial basis set methods. Coupled-channel methods [24] operate by direct numerical solution of Equation 1.42, propagating either a wavefunction matrix or a derived quantity such as the log-derivative matrix (Eqution 1.52) across a grid of r values. As in the singlechannel case, the propagation is carried out outwards from a distance rmin at very short range and inwards from a distance rmax at long range to a matching point rmid in the classically allowed region. Solutions that satisfy bound-state boundary conditions (Equation 1.54) exist only for certain values of the energy E, so an extra layer of calculation is needed to locate the energies (eigenvalues). In the multichannel case, the desired wavefunction is a column vector ψn (r) whose elements are the channel wavefunctions ψin (r). In order to start propagating a solution to the coupled equations, it is necessary to know not only the initial values of the functions ψin (r) at rmin and rmax (which are given by the boundary conditions) but also their derivatives (which are not). It is therefore once again necessary to propagate an N × N wavefunction matrix Ψ(r), made up of a complete set of N vectors that spans the space of all possible initial derivatives. The particular wavefunction vector that is continuous (and has a continuous derivative) at the matching point is not identified until after converging on an energy eigenvalue. © 2009 by Taylor and Francis Group, LLC

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Because the columns of Ψ(r) span the space of all possible initial derivatives, any wavefunction that satisfies the boundary conditions can be expressed as a linear combination of them. The true wavefunction vector ψn (r) can thus be expressed as ψn (r) = Ψ+ (r)C + , −

−

ψn (r) = Ψ (r)C ,

R ≤ rmid ; R ≥ rmid ,

(1.55)

where Ψ+ (r) and Ψ− (r) are the wavefunction matrices propagated from short and long range and C + and C − are r-independent column vectors that must be found. For an acceptable wavefunction, both ψn and its derivative must match at R = rmid , ψn (rmid ) = Ψ+ (rmid )C + = Ψ− (rmid )C − ; ψn (rmid ) = [Ψ+ ] (rmid )C + = [Ψ− ] (rmid )C − ,

(1.56)

where the primes indicate radial differentiation. Gordon [25] combined these two equations into the single equation + − + − + Ψ Ψ [Ψ ] [Ψ ] C − C − = 0, (1.57) where the matrix on the left-hand side is evaluated at R = rmid . It is a matrix of order 2N × 2N, and is a function of the energy at which the wavefunctions are calculated. A nontrivial solution of Equation 1.57 exists only if the determinant of the matrix on the left is zero, and this is true only if the energy used is an eigenvalue En of the coupled equations. In Gordon’s method, eigenvalues of the coupled equations are located by solving the coupled equations at a series of trial energies E and searching for energies at which the determinant is zero. It is then straightforward to find the transformation vectors C + and C − . Propagating wavefunctions explicitly in the presence of deeply closed channels is notoriously numerically unstable. It is much more satisfactory to use a propagator that is stable in the presence of closed channels, such as one of the log-derivative propagators described above. The multichannel matching condition can be expressed very simply in terms of the log-derivative matrix Y(r) of Equation 1.52. If E is an eigenvalue of the coupled equations, there must exist a wavefunction vector − ψn (rmid ) = ψ+ n (rmid ) = ψn (rmid ) for which − [ψ+ n ] (rmid ) = [ψn ] (rmid ),

(1.58)

Y + (rmid )ψn (rmid ) = Y − (rmid )ψn (rmid ),

(1.59)

[Y + (rmid ) − Y − (rmid )]ψn (rmid ) = 0.

(1.60)

so that

or equivalently

Thus the wavefunction ψn (rmid ) is an eigenvector of the matching matrix Y + (rmid ) − Y − (rmid ) with eigenvalue zero. Eigenvalues En of the coupled equations can be located © 2009 by Taylor and Francis Group, LLC


23

by searching for energies at which the log-derivative matching matrix has a zero eigenvalue, using standard methods for finding zeroes of a function, such as the secant method. The propagator methods described here are implemented in the widely distributed BOUND package [26]. However, most bound-state problems of interest in cold molecule studies, such as those involving open-shell species and applied magnetic fields [27,28] require extensions to the distributed code. A quite different approach for calculating bound states is to expand the radial functions ψin (r) in a basis set of functions {χj (r)}, so that the full wavefunction is expanded

cijn φi (τ)χj (r). (1.61) Ψn (r, τ) = ij

The approach is then simply to calculate the complete Hamiltonian matrix in the product representation and diagonalize it to find the eigenvalues and eigenvectors. This approach is generally preferred for low-lying (deeply bound) states, because in that case quite compact radial basis sets can be used. Methods of this general type have been extensively reviewed in the context of the vibration-rotation spectroscopy of Van der Waals complexes and other molecules that exhibit wide-amplitude motion [29–31]. General-purpose programs are available for closed-shell triatomic and tetraatomic molecules [32,33]. However, these methods present some difficulties for the near-dissociation states that are important for cold molecule studies. The general problem is that quite large basis sets of radial basis functions {χj (r)} are often needed to describe both the short-range and long-range parts of the potential. For N channels and Nr radial basis functions, the full Hamiltonian matrix is of dimension NNr and the time taken to diagonalize it is proportional to N 3 Nr3 . By contrast, the time taken by propagation methods is proportional to N 3 but is linear in the number of propagation steps. There has nevertheless been a considerable amount of work on methods to calculate near-dissociation bound states using product basis sets [34,35].

1.3.4

QUASIBOUND STATES AND SCATTERING RESONANCES

True bound states can exist only at energies below the lowest threshold of the same symmetry as the state concerned. However, quasibound states can exist at energies above threshold, as shown in Figure 1.3. These are (relatively) long-lived states that can be seen in spectroscopy in much the same way as true bound states, but are coupled to a continuum and decay (dissociate) spontaneously. As a quasibound state has a finite lifetime τ, it has a width in energy ΓE rather than a precisely defined eigenvalue, ΓE = /τ.

(1.62)

Quasibound states also produce sharp features in collision properties as a function of energy, and in this context they are usually referred to as scattering resonances. There are two different types of scattering resonance that can occur, as shown schematically in Figure 1.3. A shape resonance corresponds to a state that is confined © 2009 by Taylor and Francis Group, LLC

Energy

24


Quasibound states (Feshbach resonances) Centrifugal barrier Quasibound state (shape resonance) Dissociation energy

Bound states

r

FIGURE 1.3 The different types of bound and quasibound states and scattering resonances. For the purpose of illustration only two potential curves are shown, corresponding to the lowest channel and one excited channel.

behind a barrier in a centrifugally corrected potential (Equation 1.16). A shape resonance can exist even in single-channel scattering and decays by tunneling through the barrier. Conversely, a Feshbach resonance [36] (sometimes called a compound state resonance) corresponds to a state that is supported by an attractive well in channels that are energetically closed at the energy of interest. It decays by curve-crossing to the open channel. If there is only one open channel, a resonance (of either type) is characterized by a sharp change in the phase shift δL , which increases sharply by π across the width of the resonance. The phase shift follows a Breit–Wigner form as a function of energy,

ΓE −1 , (1.63) δL (E) = δbg + tan 2(Eres − E) where δbg is a slowly varying background term, Eres is the resonance position, and ΓE is its width (in energy space). In general the parameters δbg , Eres and ΓE are weak functions of energy, but this is often neglected for reasonably narrow resonances. The 1 × 1 S-matrix for single-channel scattering is given by SL (E) = e2iδL (E) .

(1.64)

As δL (E) is real, SL (E) describes a circle of radius 1 in the complex plane across a resonance. © 2009 by Taylor and Francis Group, LLC


25

When there is more than one open channel, the complex symmetric scattering matrix S has elements Sij . The diagonal S-matrix element in incoming channel i has magnitude |Sii | ≤ 1 and may be written in terms of a complex phase shift δi (E) with a positive imaginary part [37], Sii (E) = e2iδi (E) .

(1.65)

In the multichannel case, the quantity that follows the Breit–Wigner form of Equation 1.63 is the S-matrix eigenphase sum [38,39], which is the sum of the phases obtained from the eigenvalues of the S-matrix. The eigenphases and the eigenphase sum are real, unlike the phases δi obtained from individual diagonal elements, because the S-matrix is unitary, so that all its eigenvalues have modulus 1. Across a resonance, the individual S-matrix elements describe circles in the complex plane [40], Sij (E) = Sbg,ij −

igEi gEj , E − Eres + iΓE /2

(1.66)

where gEi is complex. The radius of the circle in Sij is |gEi gEj |/ΓE , which is usually less than 1. The partial width for channel i is commonly defined as a real quantity, ΓEi = |gEi |2 . For a narrow resonance, the total width is just the sum of the partial widths, ΓE =

ΓEi .

(1.67)

i

The physical interpretation of the partial width is that, when the quasibound state corresponding to the resonance decays, the fractional population produced in product channel i is ΓEi /ΓE .

1.3.5

LOW-ENERGY SCATTERING

At low energies, scattering is commonly dominated by one or a few partial waves and the cross-sections σ(k) may be decomposed into partial wave contributions σL (k) from different incoming partial waves L. The limiting low-energy behavior for different values of L is given by the Wigner threshold laws [41]. For elastic cross-sections, tan δL (k) ∼ k 2L+1

and

σel,L ∼ k 4L ,

(1.68)

where k is the wavenumber in the incoming channel. For a potential that behaves as V (r) = −Cn r −n at long range (with n > 2), there is an additional L-independent term that dominates at high L [42], tan δL (k) ∼ k n−2 © 2009 by Taylor and Francis Group, LLC

and

σel,L ∼ k 2n−6 .

(1.69)

26


For inelastic (deexcitation) cross-sections, σinel,L ∼ k 2L−1 .

(1.70)

For the purpose of the Wigner laws, a transition is classed as inelastic if the kinetic energy released is large compared to the centrifugal barrier in the outgoing channels. At low energies, the complex phase shift defined by Equation 1.65 can be expressed in terms of a complex energy-dependent scattering length, a(k) = α(k) − iβ(k) [43], defined by analogy with Equation 1.26 as − tan δi (k) 1 a(k) = = k ik

1 − Sii (k) . 1 + Sii (k)

(1.71)

This expression is to be preferred to the common approximation, a(k) ≈

−Im Sii , ik

(1.72)

because Equation 1.71 remains valid close to resonance when a(k) is large and |1 − Sii | is not close to zero. For an incoming channel with L = 0 (s-wave scattering), the scattering length a(k) becomes constant at limitingly low energy. The elastic and total inelastic cross-sections are exactly [44] σel (k) =

4π|a|2 1 + k 2 |a|2 + 2kβ

(1.73)

and tot σinel (k) =

1.3.6

4πβ . k(1 + k 2 |a|2 + 2kβ)

(1.74)

COLLISIONS IN EXTERNAL FIELDS

Collisions in applied electric and magnetic fields are important for several reasons. Most importantly, applied fields provide ways to control ultracold gases, and for atomic systems this control has already led to the observation of a wide range of new phenomena including molecule formation and transitions to superfluid phases [45]. There is little doubt that applied fields will provide the key to controlling the much richer behavior expected for ultracold molecules. A subsidiary issue is that cold molecules are often manipulated or trapped using applied electric or magnetic fields. Such methods generally rely on the molecules remaining in a particular quantum state, and are disrupted if they undergo transitions to different states. In addition, inelastic collisions that convert internal energy into kinetic energy can eject both collision partners from the trap if the kinetic energy released is greater than the trap depth. The most important effect of an applied electric or magnetic field is that the energy levels of the colliding monomers split and shift by amounts that can be quite large © 2009 by Taylor and Francis Group, LLC


27

compared to the kinetic energy. The shifts in the positions of scattering thresholds have profound effects on collision properties, as will be seen below. Another effect of an applied field is that it breaks the isotropic character of space, so that the total angular momentum is no longer a good quantum number. This presents computational problems because much greater numbers of channels are coupled together. Many problems that are computationally tractable in the absence of fields become intractable when fields are applied. Despite the proliferation of channels and the need for more complicated Hamiltonians, the formal theory of atomic and molecular collisions is not much changed in the presence of fields. Applied fields do reduce symmetry, so that collisions no longer take place in an isotropic environment: total angular momentum is no longer conserved, and differential cross-sections need not be cylindrical symmetrical in the center-of-mass frame. However, in computational terms, it is still necessary to solve sets of coupled differential equations subject to bound-state and scattering boundary conditions. Perhaps the most important effect is that applied fields provide access to resonance phenomena that do not occur in field-free scattering. In particular, they provide access to zero-energy Feshbach resonances, which are at the heart of most approaches to controlling ultracold gases.

1.3.6.1

Basis Sets without Total Angular Momentum

As described in Section 1.3.2 above, field-free scattering calculations are usually carried out in total angular momentum representations. The equations arising from each value of the total angular momentum are independent and may be solved separately. In the presence of an applied field, this symmetry is lost and there is much less advantage in working in a total angular momentum representation. Several cases may be envisaged: • In the presence of a magnetic field with no electric field, the quantities that are conserved are the projection Mtot of the total angular momentum onto the field axis and the total parity [46–48]. • In the presence of an electric field but no magnetic field, Mtot is again preserved but the total parity is lost. Pairs of states with ±|Mtot | are degenerate with one another. States with Mtot = 0 separate into blocks that are even and odd with respect to reflection in a plane containing the field axis. • In the presence of collinear electric and magnetic fields, only Mtot is conserved. • In the presence of electric and magnetic fields that are not collinear, there is no symmetry at all and even Mtot is lost [49]. As in field-free calculations, there are very many basis sets that may be used in different circumstances. A totally uncoupled representation is often convenient because the matrix elements needed are usually quite simple. For example, for collisions of a molecule in a 3 Σ state with an unstructured atom we need basis functions to handle the molecular rotation n, electron spin s, and end-over-end angular momentum © 2009 by Taylor and Francis Group, LLC

28


L. In this case the totally uncoupled representation is made up of simple product functions φi (τ) = |nmn |sms |LML .

(1.75)

However, other basis functions sometimes describe the real eigenfunctions more compactly and may be useful in arriving at physical interpretations. In this case the functions that describe the diatomic molecule most efficiently at low fields are obtained by coupling n and s to give a resultant j, which does not appear in the basis set of Equation 1.75. 1.3.6.2

Spin Relaxation

Paramagnetic molecules are often confined in magnetic traps, with a magnetic field that is zero at the center of the trap and increases in all directions away from it. Such an arrangement traps molecules in low-field-seeking states, whose energy increases away from the center, but not those in high-field-seeking states. However, because the lowest-energy state in a field is always high-field-seeking, it is always possible for molecules to be transferred from trapped states to untrapped states by inelastic collisions. The rate of this process limits the time for which molecules can be trapped. Calculations on the collisions of cold molecules in fields are discussed in more detail in this book in the chapter by Tscherbul and Krems (Chapter 4). 1.3.6.3

Zero-Energy Feshbach Resonances

Vibrational energy levels are quite closely packed near thresholds. In cold molecule studies, it is common to have several closely spaced thresholds, corresponding to different nuclear hyperfine states or other small energy splittings of the colliding monomers. Because the bound states of the two-body system usually have different Stark and Zeeman effects from the constituent monomers, near-dissociation bound states may often be tuned across thresholds with applied electric and magnetic fields. An example of this is shown in Figure 1.4. At the point where the bound state and the threshold exactly coincide, there is a zero-energy Feshbach resonance. Consider the case of a zero-energy Feshbach resonance as a function of magnetic field B. At constant Ekin , the phase shift follows a form similar to Equation 1.63,

ΓB , (1.76) δ(B) = δbg + tan−1 2(Bres − B) where Bres is the field at which Eres = E = Ethresh + Ekin . An example of this is shown in Figure 1.5. The width ΓB is a signed quantity given by ΓB = ΓE /Δμ, where the magnetic moment difference Δμ is the rate at which the energy Ethresh of the open-channel threshold tunes with respect to the resonance energy, Δμ =

dEres dEthresh − . dB dB

(1.77)

ΓB is thus negative if the bound state tunes upwards through the energy of interest, as in the case in Figure 1.5. © 2009 by Taylor and Francis Group, LLC

29


0

(1, 1)–1

(mfin1 mfin2) (–1, 1) (0, 0) (0, 1)

(1, 1)–2

Energy/h (GHz)

–1

(1, 2)–4 (1, 2)–4

(1, 1)

–2

–3

(1, 1)–3 (2, 2)–5 (2, 2)–5 (2, 2)–5

–4 (f1, f2) v ¢ 0

200

400 600 Magnetic field (G)

800

1000

1200

FIGURE 1.4 Tuning of molecular levels (solid lines) and atomic thresholds (dashed lines) for 87 Rb as a function of magnetic field. Feshbach resonances occur at the points marked with 2 filled circles, where a molecular state crosses a threshold. (From Marte, A. et al., Phys. Rev. Lett., 89, 283202, 2002. With permission. Copyright 2002, the American Physical Society.)

When there is only one open channel, the S-matrix element S = e2iδ(B) describes a circle of radius 1 in the complex plane. In the ultracold regime, the background phase shift δbg goes to zero as k → 0 according to Equation 1.26 (with aL (k) replaced by abg , which is finite and becomes constant as k → 0). However, the resonant term still exists. The scattering length passes through a pole when δ(B) = n + corresponding to S = −1. The scattering length follows the formula [51],

π,

a(B) = abg

ΔB 1− . B − Bres

1 2

(1.78)

An example of this is shown in Figure 1.6. The elastic cross-section given by Equation 1.73 thus shows a sharp peak of height 4π/k 2 at resonance. The two widths ΓB and ΔB are related by ΓB = −2abg kΔB .

(1.79)

At limitingly low energy, ΓB is proportional to k while ΔB is constant [52]. For multichannel scattering, the situation is more complicated. As a function of magnetic field, the scattering length passes through a pole only if δ0 (B) passes through © 2009 by Taylor and Francis Group, LLC

30


10.0

Eigenphase sum/p

9.8

9.6

9.4

9.2

9.0 7168.7553

7168.7550

7168.7556

7168.7559

Magnetic field (G)

FIGURE 1.5 Phase shift for elastic 3 He–NH collisions in the vicinity of an elastic Feshbach resonance at a kinetic energy of 10−6 K. (From González-Martínez, M.L. and Hutson, J.M., Phys. Rev. A, 75, 022702, 2007. With permission. Copyright 2007, the American Physical Society.)

n + 21 π, corresponding to S00 = −1. If there is any inelastic scattering, the radius of the resonant circle in S00 is proportional to k at low energy and this does not occur. The formula followed by the complex scattering length is [53] a(B) = abg +

ares , 2(B − Bres )/Γinel B +i

(1.80)

where ares is a resonant scattering length that characterizes the strength of the resonance. When ares is small, the scattering length shows only small peaks or oscillations across a resonance. Both ares and the background term abg can in general be complex and are independent of kinetic energy at low energy. When ares is complex, the total inelastic cross-section given by Equation 1.74 can show troughs as well as peaks in the vicinity of a resonance.

1.4

REACTIVE SCATTERING

Reactive scattering, in which the products of a collision are chemically different from the reactants, is formally similar to inelastic scattering. However, there are complications that arise from the fact that a basis set that efficiently describes the reactants is usually inefficient to describe the products and vice versa. Even the coordinate system to be used requires some care: for example, Jacobi coordinates © 2009 by Taylor and Francis Group, LLC

31

Theory of Cold Atomic and Molecular Collisions 3 × 104 Ekin = 1.0 mK

Re (scattering length) (Å)

2 × 104

1 × 104

0 × 100

–1 × 104

–2 × 104

–3 × 104 7168.7550

7168.7553

7168.7556

7168.7559

Magnetic field (G)

FIGURE 1.6 Scattering length for elastic 3 He–NH collisions in the vicinity of an elastic Feshbach resonance, from calculations at a kinetic energy of 10−6 K. (From GonzálezMartínez, M.L. and Hutson, J.M., Phys. Rev. A, 75, 022702, 2007. With permission. Copyright 2007, the American Physical Society.)

are usually inappropriate because the reduced mass is different in the reactant and product coordinate systems. The approach most commonly used to circumvent these difficulties, at least for triatomic systems, is to express the problem in hyperspherical coordinates, with the relative positions described by a single radial coordinate ρ (the hyperradius) and two hyperangles. The different atomic arrangements simply correspond to different values (or ranges of values) of the hyperangles, and a single basis set is capable of describing all arrangements. The time-independent scattering problem again reduces to a set of coupled differential equations that can be solved by propagation as in the inelastic case. Several variants of hyperspherical coordinates have been used, and a general comparison has been given by Pack and Parker [54]. There are a variety of channel basis sets that have been used by different authors: in some approaches the basis functions are solutions of a fixed-ρ Schrödinger equation that is solved at each value of the hyperradius [54–56], and in others the basis functions represent the reactants and products [57]. At long range (in the hyperradius ρ), the solutions are transformed back into the basis sets appropriate for the reactants or products individually and matched onto Bessel functions to give the scattering S-matrix as described in Section 1.3.3.1. These methods, together with time-dependent (wavepacket) approaches for reactive scattering, have been recently reviewed by Hu and Schatz [58]. © 2009 by Taylor and Francis Group, LLC

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The time-independent procedure works well for atom–diatom reactions involving light atoms that proceed on a single potential energy surface. A general-purpose program to solve the time-independent scattering problem in this case is available [57]. However, applying the theory outside this narrow range of systems is not yet routine. Low-energy reactive scattering calculations have been carried out for single-surface systems as heavy as K + K2 [59]. Calculations have also been carried out at higher energy for light systems such as F + H2 [60], where the reaction occurs on several surfaces coupled by spin–orbit interactions. However, heavy-atom systems involving multiple surfaces are not yet tractable, even for triatomic systems. Methods for performing full-dimensional quantum scattering calculations on four-atom and larger systems have been developed [61,62], but are mostly based on time-dependent wavepacket approaches that are not easily applicable to ultracold collisions. There has so far been relatively little work on reactive collisions in the presence of external fields, but Tscherbul and Krems [63] have described the general theory required in the presence of an electric field and have carried out pilot calculations on the LiF + H ↔ Li + FH reactions.

1.4.1

LANGEVIN MODEL FOR BARRIERLESS REACTIONS

There is a considerable class of chemical reactions for which the potential energy surfaces are barrierless with deep wells at short range. These include many ion–molecule reactions as well as atom exchange reactions in atom–molecule and molecule– molecule collisions involving alkali metal dimers. In the ultracold domain where only one or a few partial waves contribute to collision cross-sections, such reactions still require a full quantum treatment. However, for strongly exothermic barrierless reactions at slightly higher temperatures (several mK) it is a reasonable approximation that all collisions that sample the short-range region of the potential surface lead to reaction. This is the basis of the Langevin capture model [64]. The effective potential VL (r) for a partial wave with L > 0, Equation 1.16, is governed at long range by the centrifugal and dispersion terms, VL (R) =

2 L(L + 1) C6 − 6, 2μR2 R

(1.81)

where C6 is the atom–molecule or molecule–molecule dispersion coefficient. There is thus a centrifugal barrier at a distance

RLmax with height

VLmax

=

6μC6 = 2 L(L + 1)

2 L(L + 1) μ

1/4 (1.82)

3/2 (54C6 )−1/2 .

(1.83)

At collision energies below the centrifugal barrier in the incoming channel, the partial cross-sections for each L follow Wigner laws given by Equation 1.70. If there are © 2009 by Taylor and Francis Group, LLC

33


many open channels, however, the inelastic probabilities above the centrifugal barrier come close to their maximum possible value of 1. The cross-sections then vary as E −1 because of the k −2 factor in the expression for the cross-section. At collision energies high enough that many partial waves are involved, the total inelastic cross-section and rate coefficient become 1/3 C6 capture σinel (E) = 3π ; 4E (1.84) 1/3 1/2 1/3 3πC6 E 1/6 C6 2E capture = . kinel (E) = 3π 4E μ 21/6 μ1/2 For atom–molecule and molecule–molecule reactions, the centrifugal barriers for low partial waves are typically a few mK or less. For barrierless reactions, therefore, Langevin behavior sets in at temperatures between 1 and 100 mK. Above this temperature, the details of the short-range potential become unimportant. An example of this is shown in Figure 1.7, which shows inelastic collision rates for Li + Li2 for boson and fermion dimers initially in v = 1 and 2 [65]. The full quantum result approaches the Langevin value at collision energies above about 10 mK.

Rate coefficient (cm3/s–1)

10−9

Bosons: vi = 1 Bosons: vi = 2 Fermions: vi = 1 Fermions: vi = 1

10−10

10−6

Langevin model

10−5

10−4

10−3 Collision energy (K)

10−2

10−1

FIGURE 1.7 Total inelastic rate coefficients for collisions of Li with Li2 (v = 1 and 2, with j = 0 for boson dimers and j = 1 for fermion dimers). The dotted line shows the result of Langevin capture theory. (From Critaš, M.T. et al., Phys. Rev. Lett., 94, 033201, 2005. With permission.)


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22. Kouri, D.J., Rotational excitation II: Approximate methods, in Bernstein, R.B., ed., Atom–Molecule Collision Theory: A Guide for the Experimentalist, Plenum Press, New York, 1979, p. 301–358. 23. Hutson, J.M., An introduction to the dynamics of Van der Waals molecules, in Advances in Molecular Vibrations and Collision Dynamics, Vol. 1A, JAI Press, Greenwich, Connecticut, 1991, p. 1–45. 24. Hutson, J.M., Coupled-channel methods for solving the bound-state Schrödinger equation, Comput. Phys. Commun., 84, 1–18, 1994. 25. Gordon, R.G., A new method for constructing wavefunctions for bound states and scattering, J. Chem. Phys., 51, 14, 1969. 26. Hutson, J.M., BOUND computer program, version 5. Distributed by Collaborative Computational Project No. 6, UK Engineering and Physical Sciences Research Council, 1993. 27. González-Martínez, M.L. and Hutson, J.M., Ultracold atom–molecule collisions and bound states in magnetic fields: zero-energy Feshbach resonances in He–NH (3 Σ− ), Phys. Rev. A, 75, 022702, 2007. 28. Hutson, J.M., Tiesinga, E., and Julienne, P.S., Avoided crossings between bound states of ultracold Cesium dimers, Phys. Rev. A, 78, 052703, 2008. 29. Tennyson, J., The calculation of the vibration-rotation energies of triatomic molecules using scattering coordinates, Comput. Phys. Rep., 4, 1–36, 1986. 30. Baˇcić, Z. and Light, J.C., Theoretical methods for rovibrational states of floppy molecules, Annu. Rev. Phys. Chem., 40, 469–498, 1989. 31. Carrington, T., Methods for calculating vibrational energy levels, Can. J. Chem.–Rev. Can. Chim., 82, 900–914, 2004. 32. Tennyson, J., Kostin, M.A., Barletta, P., Harris, G.J., Polyansky, O.L., Ramanlal, J., and Zobov, N.F., DVR3D: a program suite for the calculation of rotation-vibration spectra of triatomic molecules, Comput. Phys. Commun., 163, 85–116, 2004. 33. Kozin, I.N., Law, M.M., Tennyson, J., and Hutson, J.M., New vibration-rotation code for tetraatomic molecules exhibiting wide-amplitude motion: WAVR4, Comput. Phys. Commun., 163, 117–131, 2004. 34. Tiesinga, E., Williams, C.J., and Julienne, P.S., Photoassociative spectroscopy of highly excited vibrational levels of alkali-metal dimers: Green-function approach for eigenvalue solvers, Phys. Rev. A, 57, 4257–4267, 1998. 35. Mussa, H.Y. and Tennyson, J., Bound and quasi-bound rotation-vibrational states using massively parallel computers, Comput. Phys. Commun., 128, 434–445, 2000. 36. Feshbach, H., Unified theory of nuclear reactions, Ann. Phys., 5, 357–390, 1958. 37. Mott, N.F. and Massey, H.S.W., The Theory of Atomic Collisions, 3rd edn., Clarendon Press, Oxford, 1965, p. 380. 38. Hazi, A.U., Behavior of the eigenphase sum near a resonance, Phys. Rev. A, 19, 920–922, 1979. 39. Ashton, C.J., Child, M.S., and Hutson, J.M., Rotational predissociation of the Ar–HCl Van der Waals complex—close-coupled scattering calculations, J. Chem. Phys., 78, 4025, 1983. 40. Taylor, J.R., Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Wiley, New York, 1972, p. 411–412. 41. Wigner, E.P., On the behavior of cross sections near thresholds, Phys. Rev., 73, 1002–1009, 1948. 42. Sadeghpour, H.R., Bohn, J.L., Cavagnero, M.J., Esry, B.D., Fabrikant, I.I., Macek, J.H., and Rau, A.R.P., Collisions near threshold in atomic and molecular physics, J. Phys. B–At. Mol. Opt. Phys., 33, R93–R140, 2000.


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43. Balakrishnan, N., Kharchenko, V., Forrey, R.C., and Dalgarno, A., Complex scattering lengths in multi-channel atom–molecule collisions, Chem. Phys. Lett., 280, 5–9, 1997. 44. Cvitaš, M.T., Soldán, P., Hutson, J.M., Honvault, P., and Launay, J.M., Interactions and dynamics in Li + Li2 ultracold collisions, J. Chem. Phys., 127, 074302, 2007. 45. Pethick, C.J. and Smith, H., Bose–Einstein Condensation in Dilute Gases, Cambridge University Press, 2002. 46. Volpi, A. and Bohn, J.L., Magnetic-field effects in ultracold molecular collisions, Phys. Rev. A, 65, 052712, 2002. 47. Krems, R.V. and Dalgarno, A., Quantum-mechanical theory of atom–molecule and molecular collisions in a magnetic field: Spin depolarization, J. Chem. Phys., 120, 2296–2307, 2004. 48. Krems, R.V. and Dalgarno, A., Collisions of atoms and molecules in external magnetic fields, in Fundamental World of Quantum Chemistry, Brändas, E.J., and Kryachko, E.S., Eds, Vol. 3, Kluwer Academic, 2004, p. 273–294. 49. Tscherbul, T.V. and Krems, R.V., Controlling electronic spin relaxation of cold molecules with electric fields, Phys. Rev. Lett., 97, 083201, 2006. 50. Marte, A., Volz, T., Schuster, J., Durr, S., Rempe, G., van Kempen, E.G.M., and Verhaar, B.J., Feshbach resonances in rubidium 87: Precision measurement and analysis, Phys. Rev. Lett., 89, 283202, 2002. 51. Moerdijk, A.J., Verhaar, B.J., and Axelsson, A., Resonances in ultracold collisions of Li-6, Li-7, and Na-23, Phys. Rev. A, 51, 4852–4861, 1995. 52. Timmermans, E., Tommasini, P., Hussein, M., and Kerman, A., Feshbach resonances in atomic Bose–Einstein condensates, Phys. Rep., 315, 199–230, 1999. 53. Hutson, J.M., Feshbach resonances in the presence of inelastic scattering: threshold behavior and suppression of poles in scattering lengths, New J. Phys., 9, 152, 2007. Note that there is a typographical error in Equation 22 of this paper: the last term on the right-hand side should read −βres instead of +βres . 54. Pack, R.T, and Parker, G.A., Quantum reactive scattering in 3 dimensions using hyperspherical (APH) coordinates—theory, J. Chem. Phys., 87, 3888–3921, 1987. 55. Schatz, G.C., Quantum reactive scattering using hyperspherical coordinates: results for H + H2 and Cl + HCl, Chem. Phys. Lett., 150, 92–98, 1988. 56. Launay, J.M. and LeDourneuf, M., Hyperspherical close-coupling calculation of integral, cross-sections for the reaction H + H2 → H2 + H, Chem. Phys. Lett., 163, 178, 1989. 57. Skouteris, D., Castillo, J.F., and Manolopoulos, D.E., ABC: a quantum reactive scattering program, Comput. Phys. Commun., 133, 128–135, 2000. 58. Hu, W. and Schatz, G.C., Theories of reactive scattering, J. Chem. Phys., 125, 132301, 2006. 59. Quéméner, G., Honvault, P., Launay, J.M., Soldán, P., Potter, D.E., and Hutson, J.M., Ultracold quantum dynamics: Spin-polarized k + k2 collisions with three identical bosons or fermions, Phys. Rev. A, 71, 032722, 2005. 60. Alexander, M.H., Manolopoulos, D.E., and Werner, H.-J., An investigation of the F + H2 reaction based on a full ab initio description of the open-shell character of the F(2 P) atom, J. Chem. Phys., 113, 11084–11000, 2000. 61. Zhang, D.H. and Light, J.C., Quantum state-to-state reaction probabilities for the H + H2 O → H2 + OH reaction in six dimensions, J. Chem. Phys., 105, 1291–1294, 1996. 62. Meyer, H.D. and Worth, G.A., Quantum molecular dynamics: propagating wavepackets and density operators using the multiconfiguration time-dependent Hartree method, Theor. Chem. Acc., 109, 251–267, 2003.



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63. Tscherbul, T.V. and Krems, R.V., Quantum theory of chemical reactions in the presence of electromagnetic fields, J. Chem. Phys., 129, 034112, 2008. 64. Levine, R.D. and Bernstein, R.B., Molecular Reaction Dynamics and Chemical Reactivity, Oxford University Press, 1987. 65. Cvitaš, M.T., Soldán, P., Hutson, J.M., Honvault, P., and Launay, J.M., Ultracold Li + Li2 collisions: Bosonic and fermionic cases, Phys. Rev. Lett., 94, 033201, 2005.


Electric Dipoles at 2 Ultralow Temperatures John L. Bohn CONTENTS 2.1 2.2 2.3

General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review of Classical Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum-Mechanical Dipoles in Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Rotating Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Molecules with Lambda-Doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Field due to a Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Example: j = 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Example: j = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Interaction of Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Potential Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Adiabatic Potential Energy Surfaces in Two Dimensions . . . . . . . . . . 2.5.3 Example: j = 1/2 Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Adiabatic Potential Energy Curves in One Dimension: Partial Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Asymptotic Form of the Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

39 40 42 42 45 48 51 53 55 57 58 61 62 65 66 67 67

GENERAL REMARKS

Any object with a net positive charge on one end and a net negative charge on the other end possesses an electric dipole moment. In ordinary classical electromagnetism this dipole moment is a vector quantity that can point in any direction, and is subject to electrical forces that are fairly straightforward to formulate mathematically. However, for a quantum mechanical object like an atom or molecule, the strength and orientation of the object’s dipole moment can depend strongly on the object’s quantum mechanical state. This is a subject that becomes relevant in low-temperature molecular samples, where an ensemble of molecules can be prepared in a single internal state, as described in Chapters 5, 9, 13, 14, and 15. In such a case, the mathematical description becomes more elaborate, and indeed the dipole–dipole interaction need not take the classical 39 © 2009 by Taylor and Francis Group, LLC

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form given in textbooks. The description of this interaction is the subject of this chapter. We approach this task in three steps: first, we introduce the ideas of how dipoles arise in quantum mechanical objects; second, we present a formalism within which to describe these dipoles; and third, we give examples of the formalism that illustrate some of the basic physics that emerges. The discussion will explore the possible energy states of the dipole, the field generated by the dipole, and the interaction of the dipole with another dipole. We restrict the discussion to a particular “minimal realistic model,” so that the most important physics is incorporated, but the arithmetic is not overwhelming. Although we discuss molecular dipoles in several contexts, our main focus is on polar molecules that possess a Λ doublet in their ground state. These molecules are the most likely, among diatomic molecules at least, to exhibit their dipolar character at moderate laboratory field strengths. Λ-doubled molecules have another peculiar feature, namely, their ground states possess a degeneracy even in an electric field. This means that there is more than one way for such a molecule to align with the field; the two possibilities are characterized by different angular momentum quantum numbers. This degeneracy leads to novel properties of both the orientation of a single molecule’s dipole moment and the interaction between dipoles. In the examples we present, we focus on revealing these novel features. We assume the reader has a good background in undergraduate quantum mechanics and electrostatics. In particular, the ideas of matrix mechanics, Dirac notation, and time-independent perturbation theory are used frequently. In addition, the reader should have a passing familiarity with electric dipoles and their interactions with fields and with each other. Finally, we will draw heavily on the mathematical theory of angular momentum as applied to quantum mechanics in more detail in the classic treatise of Brink and Satchler [1]. When necessary, details of the structure of diatomic molecules have been drawn from Brown and Carrington’s recent authoritative text [2].

2.2

REVIEW OF CLASSICAL DIPOLES

The behavior of a polar molecule is largely determined by its response to electric fields. Classically, an electric dipole appears when a molecule has a little bit of positive charge displaced a distance from a little bit of negative charge. The dipole moment is then a vector quantity that characterizes the direction and magnitude of this displacement: μ =

qξ rξ ,

ξ

where the ξ-th charge qξ is displaced rξ from a particular origin. Because we are interested in forces exerted on molecules, we will take this origin to be the center of mass of the molecule. (Defining μ = 0 would instead identify the center of charge of the molecule—quite a different thing!) By convention, the dipole moment vector points away from the negative charges, and toward the positive charges, inside the molecule. © 2009 by Taylor and Francis Group, LLC

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Electric Dipoles at Ultralow Temperatures

A molecule has many charges in it, and they are distributed in a complex way, as governed by the quantum-mechanical state of the molecule. In general there is much more information about the electrostatic properties of the molecule than is contained in its dipole moment. However, at distances far from the molecule (as compared to the molecule’s size), these details do not matter. The forces that one molecule exerts on another in this limit are strongly dominated by the dipole moments of the two molecules. In this limit, too, the details of the dipole moment’s origin are irrelevant, and we consider the molecule to be a “point dipole,” with a dipole moment characterized by a magnitude μ and a direction μ. ˆ We consider in this chapter only electrically neutral molecules, so that the Coulomb force between molecules is absent. its energy depends on the relative If a dipole μ is immersed in an electric field E, orientation of the field and the dipole, via Eel = −μ · E. This follows simply from the fact that the positive charges will be pulled in the direction of the field, while the negative charges are pulled the other way. Thus a dipole pointing in the same direction as the field is lower in energy than a dipole pointing in exactly the opposite direction. In classical electrostatics, the energy can continuously vary between these two extreme limits. As an object containing charge, a dipole generates an electric field, which is given, r ). For a point as usual, by the gradient of an electrostatic potential, Emolecule = −∇Φ( dipole the potential Φ is given by Φ(r ) =

μ · rˆ , r2

(2.1)

where r = r rˆ denotes the point in space, relative to the dipole, at which the field is to be evaluated [3]. The dot product in Equation 2.1 gives the field Φ a strong angular dependence. For this reason, it is convenient to use spherical coordinates to describe the physics of dipoles, because they explicitly record directions. Setting rˆ = (θ, φ) and μ ˆ = (α, β) in spherical coordinates, the dipole potential becomes Φ=

μ (cos α cos θ + sin α sin θ cos(β − φ)). r2

(2.2)

For the most familiar case of a dipole aligned along the positive z-axis (α = 0), this yields the familiar result Φ = μ cos θ/r 2 . This potential is maximal along the dipole’s axis (θ = 0 or π), and vanishes in the direction perpendicular to the dipole (θ = π/2). From the results above we can evaluate the interaction potential between two dipoles. One of the dipoles generates an electric field, which acts on the other. Taking the scalar product of one dipole moment with the gradient of the dipole potential Equation 2.2 due to the other, we obtain [3] ) = Vd (R


ˆ μ ˆ 2 − 3(μ 1 · R)( 2 · R) μ 1 · μ , 3 R

(2.3)

42


= RRˆ is the relative coordinate of the dipoles. This result is general for any where R orientation of each dipole, and for any relative position of the pair of dipoles. In a special case where both dipoles are aligned along the positive z-axis, and where the vector connecting the centers of mass of the two dipoles makes an angle θ with this axis, the dipole–dipole interaction takes a simpler form: 1 − 3 cos2 θ . (2.4) R3 Note that the angle θ as used here has a different meaning from the one in Equation 2.2. We will use θ in both contexts throughout this chapter, hopefully without causing undue confusion. The form 2.4 of the interaction is useful for illustrating the most basic fact of the dipole–dipole interaction: if the two dipoles line up in a head-to-tail orientation (θ = 0 or π), then Vd < 0 and they attract one another; if they lie side by side (θ = π/2), then Vd > 0 and they repel one another. (This expression ignores a contact potential that must be associated to a point dipole to conserve lines of electric flux [3]. However, real molecules are not point dipoles, and the electrostatic potential differs greatly from this dipolar form at length scales inside the molecule, scales that do not concern us here.) Our main goal in this chapter is to investigate how these classical results change when the dipoles belong to molecules that are governed by quantum mechanics. In Section 2.3 we evaluate the energy of a dipole exposed to an external field. In Section 2.4 we consider the field produced by a quantum-mechanical dipole. In Section 2.5 we address the interaction between two dipolar molecules. Vd (r ) = μ1 μ2

2.3

QUANTUM-MECHANICAL DIPOLES IN FIELDS

Whereas the classical energy of a dipole in a field can take a continuum of values between its minimum and maximum, this is no longer the case for a quantummechanical molecule. In this section we will establish the energy spectrum of a polar molecule in an electric field, building from a set of simple examples. To start, we will define the laboratory z-axis to coincide with the direction of an externally applied electric field, so that E = E zˆ . In this case, the projection of the total angular momentum on the z-axis is a conserved quantity.

2.3.1 ATOMS Our main focus in this chapter will be on electrically polarizable dipolar molecules. But before discussing this in detail, we first consider the simpler case of an electrically polarizable atom, namely, hydrogen. This will introduce both the basic physics ideas, and the angular momentum techniques that we will use. In this case a negatively charged electron separated a distance r from a positively charged proton forms a dipole moment μ = −er . Because dipoles require us to consider directions, it is useful to cast the unit vector rˆ into its spherical components [1]: x √ y z ±i = C10 (θφ). = ∓ 2C1±1 (θφ), r r r © 2009 by Taylor and Francis Group, LLC

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Here the C’s are reduced spherical harmonics, related to the usual spherical harmonics by [1] 4π Ckq = Ykq , 2k + 1 and given explicitly for k = 1 by 1 C1±1 (θφ) = ∓ √ sin θe±iφ 2 C10 (θφ) = cos θ.

(2.5)

In general, it is convenient to represent interaction potentials in terms of the functions Ckq (because they do not carry extra factors of 4π), and to use the functions Ykq as wavefunctions in angular degrees of freedom (because they are already properly normalized, by Ylm |Yl m = δll δmm ). Integrals involving the reduced spherical harmonics are conveniently related to the 3-j symbols of angular momentum theory, for example:

d(cos θ) dφCk1 q1 (θφ)Ck2 q2 (θφ)Ck3 q3 (θφ) = 4π

k1 q1

k2 q2

k3 q3

k1 0

k2 0

k3 . 0

The 3-j symbols, in parentheses, are related to the Clebsch–Gordan coefficients. They are widely tabulated and easily computed for applications. In terms of these functions, the Hamiltonian for the atom–field interaction is Hel = −(−er ) · E = ezE = er cos(θ)E = erEC10 (θφ).

(2.6)

The possible energies for a dipole in a field are given by the eigenvalues of the Hamiltonian 2.6. To evaluate these energies in quantum mechanics, we identify the usual basis set of hydrogenic wavefunctions, |nlm, where we ignore spin for this simple illustration: r, θ, φ | nlm = fnl (r)Ylm (θ, φ). The matrix elements between any two hydrogenic states are nlm | −μ · E | n l m = erE

∗ d(cos θ) dφYlm C10 Yl m

= erE(−1)m (2l + 1)(2l + 1) l 1 l l 1 l × . 0 0 0 −m 0 m

(2.7)

Here er = r 2 drfnl (r)rfn l is an effective magnitude of the dipole moment, which can be analytically evaluated for hydrogen [4]. © 2009 by Taylor and Francis Group, LLC

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Some important physics is embodied in Equation 2.7. First, the electric field defines an axis of rotational symmetry (here the z-axis). On general grounds, we therefore expect that the projection of the total angular momentum of the molecule onto this axis is a constant. And indeed, this is built into the 3-j symbols: because the sum of all m quantum numbers in a 3-j symbol should add to zero, Equation 2.7 asserts that m = m , and the electric field cannot couple two different m’s together. A second feature embodied in Equation 2.7 is the action of parity. The hydrogenic wavefunctions have a definite parity, that is, they either change sign, or else remain invariant, upon converting from a coordinate system (x, y, z) to a coordinate system (−x, −y, −z). The sign of the parity-changed wavefunction is given by (−1)l . Thus an s-state (l = 0) has even parity, while a p-state (l = 1) has odd parity. For an electric field pointing in a particular direction, the Hamiltonian 2.6 itself has odd parity, and thus serves to change the parity of the atom. For example, it can couple the s and p states to each other, but not to themselves. This is expressed in Equation 2.7 by the first 3-j symbol, whose symmetry properties require that l + 1 + l = even. This seemingly innocuous statement is the fundamental fact of electric dipole moments of atoms and molecules. It says that, for example the 1s ground state of hydrogen, with l = l = 0, does not, by itself, respond to an electric field at all. Rather, it requires an admixture of a p state to develop a dipole moment. (These remarks are not strictly true. The ground state of hydrogen already has a small admixture of odd-parity states, due to the parity-violating part of the electroweak force. This effect is far too small to concern us here, however.) To evaluate the influence of an electric field on hydrogen, therefore, we must consider at least the nearest state of opposite parity, which is the 2p state. These two states are separated in energy by an amount E1s2p . Considering only these two states, and ignoring any spin structure, the atom-plus-field Hamiltonian is represented by a simple 2 × 2 matrix: −E1s2p /2 μE H= . (2.8) μE E1s2p /2 where the dipole matrix element is given by the convenient shorthand μ = 1s, m = √ 0 | ez | 2p, m = 0 = 128 2ea0 /243 [4]. Of course there are many more p states that the 1s state is coupled to. Plus, all states are further complicated by the spin of the electron and (in hydrogen) the nucleus. Matrix elements for all these can be constructed, and the full matrix diagonalized to approximate the energies to any desired degree of accuracy. However, we are interested here in the qualitative features of dipoles, and so limit ourselves to Equation 2.8. The Stark energies are thus given approximately by E± = ± (μE)2 + (E1s2p /2)2 .

(2.9)

This expression illustrates the basic physics of the quantum mechanical dipole. First, there are necessarily two states (or more) involved. One state decreases in energy as the field is turned on, representing the “normal” case where the electron moves to negative z and the electric dipole moment aligns with the field. The other state, © 2009 by Taylor and Francis Group, LLC


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however, increases in energy with increasing field and represents the dipole moment antialigning with the field. Classically, it is of course possible to align the dipole against the field in a state of unstable equilibrium. Similarly, in quantum mechanics this is a legitimate energy eigenstate, and the dipole will remain antialigned with the field in the absence of perturbations. A second observation about the energies of Equation 2.9 is that the energy is a quadratic function of E at low fields, and only becomes linearly proportional to E at higher fields. Thus the permanent dipole moment of the atom, defined by the zero-field limit ∂E− μpermanent ≡ lim , E →0 ∂E vanishes. The atom, in an energy eigenstate in zero field, has no permanent electric dipole moment. This makes sense because, in zero field, the electron’s position is randomly varied about the atom, lying as much on one side of the nucleus as on the opposite side. The transition from quadratic to linear Stark effect is an example of a competition between two opposing tendencies.At low field, the dominant energy scale is the energy splitting E1s2p between opposite parity states. At higher field values, the interaction energy with the electric field becomes stronger, and the dipole is aligned. The value of the field where this transition occurs is found roughly by setting these energies equal to find a “critical” electric field: Ecritical = E1s2p /2μ. For atomic hydrogen, this field is on the order of 109 V/cm. However, at this field it is already a bad approximation to ignore that fact that there are both 2p1/2 and 2p3/2 states, as well as higher-lying p states, and further coupling between p, d, etc., states. We will not pursue this subject further here.

2.3.2

ROTATING MOLECULES

With these basics in mind, we can move on to molecules. We focus here on diatomic, heteronuclear molecules, although the principles are more general. We will consider only electric fields so small that the electrons cannot be polarized in the sense of the previous section; thus we consider only a single electronic state. However, the charge separation between the two atoms produces an electric dipole moment μ in the rotating frame of the molecule. We assume that the molecule is a rigid rotor and we will not consider explicitly the vibrational motion of the molecule, focusing instead solely on the molecular rotation. (More precisely, we consider μ to incorporate an averaging over the vibrational coordinate of the molecule, much as the electron–proton distance r was averaged over for the hydrogen atom in the previous section.) As a mathematical preliminary, we note the following. To deal with molecules, we are required to transform freely between the laboratory reference frame and the bodyfixed frame that rotates with the molecule. The rotation from the lab frame (x, y, z) © 2009 by Taylor and Francis Group, LLC

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to the body frame (x , y , z ) is governed by a set of Euler angles (α, β, γ). The first two angles α = φ, β = θ coincide with the spherical coordinates (θ, φ) of the body frame’s z axis. By convention, we take the positive z direction to be parallel to the dipole moment μ. The third Euler angle γ serves to orient the x axis in a desired orientation within the body frame; it is thus the azimuthal angle of rotation about the molecular axis. Consider a given angular momentum state | jm referred to the lab frame. This state is only a state of good m in the lab frame, in general. In the body frame, which points in some other direction, the same state will be a linear superposition of different m’s, which we denote in the body frame as ω’s to distinguish them. Moreover, this linear superposition will be a function of the Euler angles, with a transformation that is conventionally denoted by the letter D: D(αβγ) | jω =

| jm jm | D(α, β, γ) | jω

m

≡

j

| jmDmω (α, β, γ).

m

This last line defines the Wigner rotation matrices, whose properties are widely j tabulated. For each j, Dmω is a unitary transformation matrix; note that a rotation can only change m-type quantum numbers, not the total angular momentum j. One of the more useful properties of the D matrices, for us, is dα d cos(β) dγ = 8π2

j1 m1

j

j

j

Dm11 ω1 (αβγ)Dm22 ω2 (αβγ)Dm33 ω3 (αβγ) j2 m2

j3 m3

j1 ω1

j2 ω2

j3 . ω3

(2.10)

Because the dipole is aligned along the molecular axis, and because the molecular axis is tilted at an angle β with respect to the field, and because the field defines the z-axis, the dipole moment is defined by its magnitude μ times a unit vector with polar coordinates (β, α). The Hamiltonian for the molecule–field interaction is given by Hel = −μ · E = −μEC10 (βα) = −μEDq0 (αβγ). j∗

(2.11)

For use below, we have taken the liberty of rewriting C10 as a D-function; since the second index of D is zero, this function does not actually depend on γ, so introducing this variable is not as drastic as it seems. To evaluate energies in quantum mechanics we need to choose a basis set and take matrix elements. The Wigner rotation matrices are the quantum-mechanical eigenfunctions of the rigid rotor. With normalization, these wavefunctions are αβγ | nmn λn =


2n + 1 n∗ D . 8π2 mn λn


47

As we did for hydrogen, we here ignore spin. Thus n is the quantum number of rotation of the atoms about their center of mass, mn is the projection of this angular momentum in the lab frame, and λn is its projection in the body frame. In this basis, the matrix elements of the Stark interaction are computed using Equation 2.10 to yield · E | n mn λ = −μE(−1)mn −λn (2n + 1)(2n + 1) nmn λn | −μ n 1 n n 1 n × . (2.12) −mn 0 mn −λn 0 λn In an important special case, the molecule is in a Σ state, meaning that the electronic angular momentum projection λn = 0. In this case, Equation 2.12 reduces to the same expression as that for hydrogen, apart from the radial integral. This is as it should be: in both objects, there is simply a positive charge at one end and a negative charge at the other. It does not matter if one of these is an electron, rather than an atom. More generally, however, when λn = 0 there will be a complicating effect of lambda-doubling, which we will discuss in the next section. Thus the physics of the rotating dipole is much the same as that of the hydrogen atom. Equation 2.12 also asserts that, for a Σ state with λn = 0, the electric field interaction vanishes unless n and n have opposite parity. For such a state, the parity is related to the parity of n itself. Thus, for the ground state of a 1 Σ molecule with n = 0, the electric field only has an effect by mixing this state with the next rotational state with n = 1. These states are split by an energy Erot = 2Be , where Be is the rotational constant of the molecule. (In zero field, the state with rotational quantum number n has energy Be n(n + 1).) We can formulate a simple 2 × 2 matrix describing this situation, as we did for hydrogen: −Erot /2 −μE H= , (2.13) −μE +Erot /2 where the dipole matrix element is given by the convenient shorthand notation μ = nmn 0 | μq=0 | n mn 0. There is one such matrix for each value of mn . Of course there are many more rotational states that these states are coupled to. Plus, all states would further be complicated by the spins (if any) of the electrons and nuclei. The matrix 2.13 can be diagonalized just as Equation 2.8 was above, and the same physical conclusions apply. Namely, the molecule in a given rotational state has no permanent electric dipole moment, even though there is a separation of charges in the body frame of the molecule. Second, the Stark effect is quadratic for low fields, and linear only at higher fields, with the transition occurring at a “critical field” Ecrit = Erot /2μ.

(2.14)

To take an example, the NH molecule possesses a 3 Σ ground state. For this state, ignoring spin, the critical field is of the order 7 × 106 V/cm. This is far smaller than the field required to polarize electrons in an atom or molecule, but still large for laboratory-strength electric fields. Diatomic molecules with smaller rotational constants, such as LiF, would have correspondingly smaller critical fields. In any event, by the time the critical field is applied, it is already a bad approximation to © 2009 by Taylor and Francis Group, LLC

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ignore coupling to the other rotational states of the molecule, which must be included for an accurate treatment. We do not consider this topic further here.

2.3.3

MOLECULES WITH LAMBDA-DOUBLING

As we have made clear in the previous two subsections, the effect of an electric field on a quantum-mechanical object is to couple states of opposite parity. For a molecule in a Π or Δ state, there are often two such parity states that are much closer together in energy than the rotational spacing. The two states are said to be the components of a “Λ-doublet.” Because they are close together in energy, these two states can then be mixed at much smaller fields than are required to mix rotational levels. The physics underlying the lambda doublet is rather complex, and we refer the reader to the literature for details [2,5]. However, in broad terms, the argument is something like this: a Π state has an electronic angular momentum projection of magnitude 1 about the molecular axis. This angular momentum comes in two projections, for the two senses of rotation about the axis, and these projections are nominally degenerate in energy. The rotation of the molecule, however, can break the degeneracy between these levels, and (it so happens) the resulting nondegenerate eigenfunctions are also eigenfunctions of parity. The main point is that the resulting energy splitting is usually quite small, and these parity states can be mixed in fields much smaller than those required to mix rotational states. To this end, we modify the rigid-rotor wavefunction of the molecule to incorporate the electronic angular momentum: αβγ | jmω =

2j + 1 j∗ Dmω (αβγ). 8π2

(2.15)

Here j is the total (rotation-plus-electronic) angular momentum of the molecule, and m and ω are the projections of j on the laboratory and body-fixed axes, respectively. Using the total j angular momentum, rather than just the molecular rotation n, marks the use of a “Hund’s case a” representation, rather than the Hund’s case b that was implicit in the previous section [2]. In this basis the matrix element of the electric field Hamiltonian 2.11 becomes jmω | −μ · E | j m ω = −μE(−1)

m−ω

(2j

+ 1)(2j

j + 1) −m

1 0

j m

j −ω

1 0

j . (2.16) ω

In Equation 2.16, the 3-j symbols imply conservation laws. The first asserts that m = m is conserved, as we already knew. The second 3-j symbol adds to this the fact that ω = ω . This is the statement that the electric field cannot exert a torque around the axis of the dipole moment itself. Moreover, in the present model we assert that j = j , because the next higher-lying j level is far away in energy, and only weakly mixes with the ground state j. With this approximation, the 3-j symbols have simple © 2009 by Taylor and Francis Group, LLC

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algebraic expressions, and we can simplify the matrix element: jmω | −μ · E | jmω = −μE

mω . j( j + 1)

The physical content of this expression is illustrated in Figure 2.1. Notice that both m and ω can have a sign, and that whether the energy is positive or negative depends on both signs. An essential point is that there is not a unique state representing the dipole aligned with the field. Rather, there are two such states, distinguished by different angular momentum quantum numbers but possessing the same energy. To distinguish these in the following we will refer to the two states in Figures 2.1a and b as molecules of type | a and type | b, respectively. Likewise, for molecules nominally antialigned with the field, we will refer to types | c and | d, corresponding to the two states in Figures 2.1c and d. The existence of these degeneracies will lead to novel phenomena in these kinds of molecules, as we discuss below. As to the lambda doubling, it is, as we have asserted, diagonal in a basis where parity is a good quantum number. In terms of the basis (Equation 2.15), wavefunctions of well-defined parity are given by the linear combinations 1 | jmω ¯ = √ [| jmω ¯ + | jm − ω]. ¯ 2

(a)

m>0

j

(2.17)

(b) w>0 w>0 m0

j

(d) w0 j

m 0, ω > 0 (a), or (ii) j aligns against the field and m < 0, ω < 0 (b). Similar remarks apply to dipoles that antialign with the field (c,d).


50


Here we define ω ¯ =| ω |, the absolute value of ω. For a given value of m, the linear combinations of ±ω ¯ in Equation 2.17 are distinguished by the parity quantum number = ±1. It is straightforward to show that in the parity basis Equation 2.17 the Λ-doubling is off-diagonal. The net result is that for each value of m, the Hamiltonian for our lambda-doubled molecule can be represented as a 2 × 2 matrix, similar to the ones above: −Q −Δ/2 H= , −Δ/2 Q where Δ is the lambda doubling energy, that is, the energy difference between the two parity states, and |m|ω ¯ Q ≡ μE j( j + 1) is a manifestly positive quantity. This is the Hamiltonian we will treat in the remainder of this chapter. The difference here from the previous subsections is that the basis is now that of Equation 2.15, which diagonalizes the electric field interaction rather than the zero-field Hamiltonian. This change reflects our emphasis on molecules in strong fields where their dipole moments are made manifest. The zero-field Λ-doubling Hamiltonian is considered, for the most part, to be a perturbation. In general, for field interactions that are not infinitely larger than the lambda doublet, the energy eigenstates are superpositions of the strong-field states | jm ± ω. ¯ For each value of m (which, remember, is conserved by the field), the mixing of +ω ¯ and −ω ¯ states is conveniently represented by a mixing angle that we denote by δm : ¯ + sin δm | jm − ω | jmω ¯ = + = cos δm | jmω ¯ ¯ + cos δm | jm − ω. ¯ | jmω ¯ = − = − sin δm | jmω

(2.18)

Explicitly, the mixing angle as a function of field is given by tan δ|m| =

Δ/2 = − tan δ−|m| , Q + Q 1 + η2

in terms of the energy Q and the dimensionless parameter ηm =

Δ . 2Q

Notice that with this definition, δm is positive when m is positive, and δ−m = −δm . The energies of these states are conveniently summarized by the expression Emω ¯

mω ¯ = −μE 1 + η2 , j( j + 1)

m = 0.

This is a very compact way of writing the results that will facilitate writing the expressions below. Notice that the intuition afforded by Figure 2.1 is still intact in © 2009 by Taylor and Francis Group, LLC

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these energies, but by replacing the sign of for the sign of ω. Thus states with m > 0 have negative energy, while states with m < 0 have positive energy. These expressions are somewhat problematic in the limit where the field vanishes, which sets Q = 0. However, this limit exists and it is easily shown that δ| m| = π/4 in this limit, while the energies become −μEsign(m)Δ/2. Likewise, we also have Q = 0 whenever m = 0, for any value of the field strength. Again, we can still write the eigenstates in the form of 2.18, provided that we set δ0 =

π , 4

and understand that the corresponding energies, independent of field, are simply E0ω ¯ = −Δ/2. Like the dipoles considered above, this model has a quadratic Stark effect at low energies, rolling over to linear at electric fields exceeding the critical field given by setting Q = Δ/2. This criterion gives a critical field of Ecrit =

Δj( j + 1) . 2μ| m | ω ¯

(2.19)

To take an example, consider the ground state of OH, which has j = ω ¯ = 3/2, μ = 1.7 Db, and a Λ-doublet splitting of 0.06 cm−1 . In its m = 3/2 ground state, its critical field is ∼1600 V/cm. Again, we have explicitly ignored the spin of the electron. The parity states are therefore easily mixed in fields that are both small enough to easily obtain in the laboratory, and small enough that no second-order coupling to rotational or electronic states needs to be considered. Keeping the relatively small number of molecular states is therefore a reasonable approximation, and highly desirable as it simplifies our discussion. (In OH there is also the ω ¯ = 1/2 state to consider, but it is also far away in energy compared to the Λ-doublet, and we ignore it. It does play a role in the fine structure of OH, however, and this should be included in a quantitative model of OH.)

2.4 THE FIELD DUE TO A DIPOLE Once polarized, each of the Λ-doubled molecules discussed above is itself the source of an electric field. The field due to a molecule is given above by Equation 2.1. In what follows, it is convenient to cast this potential in terms of the spherical tensors defined above: Φ(r ) =

μ · rˆ μ

= 2 (−1)q C1q (αβ)C1−q (θφ). 2 r r q

(2.20)

That this form is correct can be verified by simple substitution, using the definitions of Equation 2.5 and comparing to Equation 2.2. It seems at first unnecessarily complicated to write Equation 2.20 in this way. However, the effort required to do so © 2009 by Taylor and Francis Group, LLC

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will be rewarded when we need to evaluate the potential for quantum-mechanical dipoles below. For a classical dipole, defining the direction (αβ) of its dipole moment would immediately specify the electrostatic potential it generates according to Equation 2.20. However, in quantum mechanics the potential will result from suitably averaging the orientation of μ over the distribution of (αβ), weighted by probabilities that are dictated by the molecule’s wavefunction. To evaluate this, we need to evaluate matrix elements of Equation 2.20 in the basis of energy eigenstates 2.18. This is easily done using Equation 2.10, along with formulas that simplify the 3-j symbols. The result is

jmω | C1q

⎧ ( j + m)( j − m ) ⎪ ⎪ ⎪ , q = +1 − ⎪ ⎨ 2 ω | jm ω = δωω m, q=0 , j( j + 1) ⎪ ⎪ ⎪ ⎪ ⎩+ ( j − m)( j + m ) , q = −1 2

(2.21)

where the quantum numbers m and ω are signed quantities. Also note that this integration is over the molecular degrees of freedom (αβ) in Equation 2.20. This still leaves the angular dependence on (θφ), which characterizes the field in space around the dipole. From expression 2.21 it is clear that this matrix element changes sign upon either (1) reversing the sign of both m and m or (2) changing the sign of ω. Moreover, jmω | C1q | jm − ω = 0, because ω is conserved by the electric field. Using these observations, we can readily compute the matrix elements of Φ in the dressed basis 2.18. Generally they take the form ¯ = μ jmω ¯ | C1q | jm ω ¯ (−1)q jmω ¯ | Φ(r ) | jm ω

C1−q (θφ) . r2

(2.22)

The matrix elements in front of Equation 2.22 represent the quantum-mechanical manifestation of the dipole’s orientation. These matrix elements follow from the above definition of the eigenstates (Equation 2.18). Explicitly, jmω, ¯ | C1q | jm ω, ¯ = cos(δm + δm ) jmω ¯ | C1q | jm ω ¯ jmω, ¯ −| C1q | jm ω, ¯ + = jmω, ¯ + | C1q | jm ω, ¯ −

(2.23)

= − sin(δm + δm ) jmω ¯ | C1q | jm ω. ¯ In all these expressions, the value of q is set by angular momentum conservation to q = m − m . This description, while complete within our model, nevertheless remains somewhat opaque. Let us therefore specialize it to the case of a particular energy eigenstate | jmω. ¯ In this state, the averaged electrostatic potential of the dipole is

mω cos 2δm Φ(r ) ≡ jmω ¯ | Φ(r ) | jmω ¯ = μ j( j + 1) © 2009 by Taylor and Francis Group, LLC

cos θ . r2

(2.24)


53

Here the factor in parentheses is a quantum-mechanical correction to the magnitude of the dipole moment. The factor cos 2δm expresses the degree of polarization: in a strong field, δm = 0 and the dipole is at maximum strength, whereas in zero field δm = π/4 and the dipole vanishes. Notice that for a given value of m, the potential generated by the states with = ± differ by a sign. this is appropriate, because these states correspond to dipoles pointing in opposite directions (Figure 2.1). The off-diagonal matrix elements in Equation 2.22 are also important, for two reasons. First, it may be desirable to create superpositions of different energy eigenstates, and computing these matrix elements requires the off-diagonal elements in Equation 2.22, as we will see shortly. Second, when two dipoles interact with each other, one will experience the electric field due to the other, and this field need not lie parallel to the z-axis. Hence, the m quantum number of an individual dipole is no longer conserved, and elements of Equation 2.22 with q = 0 are required.

2.4.1

EXAMPLE: j = 1/2

To illustrate these abstract points, we consider here the simplest molecular state with a Λ doublet: a molecule with j = 1/2, which has ω ¯ = 1/2, and consists of four internal states in our model. Based on the discussion above, we tabulate the matrix elements between these states in Table 2.1. Because we have j = 1/2, we suppress the index j in this section. For concreteness, we focus on a type | a molecule, as defined in Figure 2.1. This molecule aligns with the field and produces an electrostatic potential (Equation 2.24). However, if the molecule is prepared in a state that is a superposition of this state with another, a different electrostatic potential can result. We first note that combining | a with | b produces nothing new, because both states generate the same potential. An alternative superposition combines state | a with state | c. In this case the two states have the same value of m, but are nevertheless nondegenerate. We define iω0 t 1 −iω0 t 1 | ψac = Ae ω, ¯ + + Be ω, ¯ − , 2 2 for arbitrary complex numbers A and B with | A |2 +| B |2 = 1. Because the two states are nondegenerate, it is necessary to include the explicit time-dependent phase factors, where ω0 = | Emω ¯ |/, and 2ω0 is the energy difference between the states. As usual in quantum mechanics, these phases will beat against one another to make the observables time dependent. Now some algebra identifies the mean value of the electrostatic potential, averaged over state | ψac , as μ | A |2 −| B |2 cos 2δ1/2 ψac | Φ(r ) | ψac = 3 cos θ − 2| AB | sin 2δ1/2 cos(2ω0 t − δ) . (2.25) r2 This potential has the usual cos θ angular dependence, meaning that the dipole remains aligned along the field’s axis. However, the magnitude, and even the sign, of the dipole © 2009 by Taylor and Francis Group, LLC

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TABLE 2.1 ¯ | C1q | jm ω ¯ for a j = 1/2 Molecule Matrix Elements jm ω

1 ω+ ¯ 2 1 ω− ¯ 2 1 ¯ − ω+ 2 1 ¯ − ω− 2

1 ω+ ¯ 2

1 ω− ¯ 2

1 cos 2δ1/2 3

1 − sin 2δ1/2 3

1 − sin 2δ1/2 3 √ 2 3

1 − cos 2δ1/2 3

0

2 3

0

1 − cos 2δ1/2 3

1 − sin 2δ1/2 3

1 − sin 2δ1/2 3

1 cos 2δ1/2 3

0

√ −

2 3

1 − ω+ ¯ 2

1 − ω− ¯ 2

√ −

2 3

0 √

To obtain matrix elements of the electrostatic potential Φ(r ), these matrix elements should be multiplied by (−1)q C1−q (θφ)μ/r 2 , where q = m − m .

change over time. The first term in square brackets in Equation 2.25 gives a constant, dc component to the dipole moment, which depends on the population imbalance | A |2 −| B |2 between the two states. The second term adds to this an oscillating component with angular frequency 2ω0 . The leftover phase δ is an irrelevant offset, and comes from the phase of A∗ B, that is, the relative phase of the two components at time t = 0. It is therefore possible to construct a superposition of states of the dipole, such that the effective dipole moment of the molecule bobs up and down in time. The amount that the dipole bobs, relative to the constant component, can be controlled by the relative population in the two states. Moreover, the degree of polarization of the molecule plays a significant role. For a fully polarized molecule, when δ1/2 = 0, only the dc portion of the dipole persists, although even it can vanish if there is equal population in the two states, “dipole up” and “dipole down.” As another example, we consider the superposition of | a with | d. Now the two states have different values of m as well as different energies: iω0 t 1 −iω0 t 1 ¯ + + Be ¯ + . | ψad = Ae 2 ω, − 2 ω, The dipole potential this superposition generates is cos θ μ ψad | Φ(r ) | ψad = cos 2δ1/2 | A |2 −| B |2 3 r2 2μ sin θ Re A∗ Be−i(φ+2ω0 t) + . 3 r2 This expression can be put in a useful and interesting form if we parametrize the coefficients A and B as α α A = cos e−iβ/2 B = sin eiβ/2 . 2 2 © 2009 by Taylor and Francis Group, LLC


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This way of writing A and B seems arbitrary, but it is not. It is the same parametrization that is used in constructing the Bloch sphere, which is a powerful tool in the analysis of any two-level system [6]. This parametrization leads to the following expression for the potential: ψad | Φ(r ) | ψad =

1μ cos 2δ cos α cos θ + sin α sin θ cos((β − 2ω t) − φ) . 1/2 0 3 r2 (2.26)

The interpretation of this result is clear upon comparing it to the classical expression 2.2. First consider that the molecule is perfectly polarized, so that cos 2δ1/2 = 1. Then Equation 2.26 represents the potential due to a dipole whose polar coordinates are (α, β − ωt). That is, this dipole makes (on average) an angle α with respect to the field, and it precesses about the field with an angular frequency 2ω0 . Interestingly, even in this strong field limit where the field nominally aligns the dipole along z, quantum mechanics allows the dipole to point in quite a different direction. As the field relaxes, the z-component reduces, but this dipole still has a component precessing about the field.

2.4.2

EXAMPLE: j = 1

We also consider a molecule with spin j = 1. Here there are in principle three mixing angles, δ1 , δ0 , and δ−1 . However, as noted above we have δ−1 = −δ1 and δ0 = π/4, so that the entire electric field dependence of these matrix elements is incorporated in the single parameter δ1 . In this notation, the matrix elements of the electrostatic potential for a j = 1 molecule are given in Table 2.2. Similar remarks apply to the spin-1 case as applied to the spin-1/2 case. If the molecule is in an eigenstate, say | +1ω, ¯ −, then the expectation value of the dipole points along the field axis, and its distribution has the usual cos θ dependence. In an eigenstate with m = 0, however, the expectation value of the dipole vanishes altogether. As before, the molecule can also be in a superposition state. No matter how complicated this superposition is, the expectation value of the dipole must instantaneously point in some direction, since the only available angular dependence resides in the C1q functions, which yield only dipoles. In other words, no superposition can generate the field pattern of a quadrupole moment, for example. Where this dipole points, and how its orientation evolves with time, however, can be nontrivial. For example, a superposition of | +1ω, ¯ + and | +1ω, ¯ − can bob up and down, just like the analogous superposition for j = 1/2. However, for j = 1 molecules additional superpositions are possible. For example, consider the combination | ψ3 = Aeiω0 t | +1ω, ¯ + + BeiωΔ t | 0ω, ¯ + + Ceiω0 t | −1ω, ¯ −, where ωΔ = Δ/2 is a shorthand notation for half the lambda doubling energy. Let us further assume for convenience that A, B, and C are all real. Then the expectation © 2009 by Taylor and Francis Group, LLC

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TABLE 2.2

¯ | C1q | jm ω ¯ for a j = 1 Molecule Matrix Elements of jmω ¯ | +1ω−

¯ | 0ω+

¯ | 0ω−

¯ | −1ω+

¯ | −1ω−

+1ω+ ¯ |

1 cos 2δ1 2

1 − sin 2δ1 2

1 − cos(δ1 + π/4) 2

1 sin(δ1 + π/4) 2

0

0

+1ω− ¯ |

1 − sin 2δ1 2

1 − cos 2δ1 2

1 sin(δ1 + π/4) 2

1 cos(δ1 + π/4) 2

0

0

0ω+ ¯ |

1 cos(δ1 + π/4) 2

1 − sin(δ1 + π/4) 2

0

0

1 − cos(−δ1 + π/4) 2

1 sin(−δ1 + π/4) 2

0ω− ¯ |

1 − sin(δ1 + π/4) 2

1 − cos(δ1 + π/4) 2

0

0

1 sin(−δ1 + π/4) 2

1 cos(−δ1 + π/4) 2

0

0

1 cos(−δ1 + π/4) 2

1 − sin(−δ1 + π/4) 2

1 − cos 2δ1 2

1 − sin 2δ1 2

−1ω+ ¯ |

1 1 1 1 cos 2δ1 − sin(−δ1 + π/4) − cos(−δ1 + π/4) − sin 2δ1 2 2 2 2 q 2 To obtain the matrix elements of the electrostatic potential Φ(r ), these matrix elements should be multiplied by (−1) C1−q (θφ)μ/r , where q = m − m . −1ω− ¯ |

0


0


¯ | +1ω+

57


value of the electrostatic potential is ψ3 | Φ(r ) | ψ3 =

cos θ μ μ sin θ cos 2δ1 A2 + C 2 + √ cos(δ1 + π/4)B 2 2 r2 r 2 × [A cos((ωΔ − ω0 )t − φ) − C cos(−(ωΔ − ω0 )t − φ)].

By analogy with remarks in the previous section, this represents a dipole with a constant component along the z-axis, which depends on both the strength of the field and on | A |2 + | C |2 , the total population in the ±m states. It also has a component in the (x–y)-plane (given by the sin θ dependence), orthogonal to the field’s direction. In the case where C = 0, this component would precess around the field axis with a frequency ωΔ − ω0 , in a clockwise direction as viewed from the +z direction. On the other hand, if A = 0, this component would rotate at this frequency but in a counterclockwise direction. If both components are present and A = C, then the result will be, not a rotation, but an oscillation of this component from, say, +x to −x, in much the same way that linearly polarized light in a superposition of left- and right-circularly polarized components. More generally, if A = C, then the tip of the dipole moment will trace out an elliptical path. However, in the limit of zero field, ω0 reduces to ωΔ and these time-dependent effects go away.

2.5

INTERACTION OF DIPOLES

Having thus carefully treated individual dipoles and their quantum-mechanical matrix elements, we are now in a position to do the same for the dipole–dipole interaction between two molecules. This interaction depends on the orientation of each dipole, μ 1 . This interaction has the form (Equation 2.3): and μ 2 , and on their relative location, R ˆ μ ˆ 2 − 3(μ μ 1 · μ 1 · R)( 2 · R) R3 √ 6μ1 μ2

=− (−1)q [μ1 ⊗ μ2 ]2q C2−q (θφ). R3 q

) = Vd (R

(2.27)

In going from the first line to the second, we assume that the intermolecular axis = (R, θ, φ). The makes an angle θ with respect to the laboratory z-axis, so that R angles θ and φ thus stand for something slightly different than in the previous section. The second line in Equation 2.27 also rewrites the interaction in a compact tensor notation that is useful for the calculations we are about to do. Here √

1 1 q 2 [μ1 ⊗ μ2 ]2q = 5 C1q1 (β1 α1 )C1q2 (β2 α2 ) (−1) q −q1 −q2 q1 q2

denotes the second-rank tensor composed of the two first-rank tensors (i.e., vectors) C1q1 (β1 α1 ) and C1q2 (β2 α2 ) that give the orientation of the molecular axes [1]. Equation 2.27 highlights the important point that the orientations of the dipoles are © 2009 by Taylor and Francis Group, LLC

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intimately tied to the relative motion of the dipoles: if a molecule changes its internal state and sheds angular momentum, that angular momentum may appear in the orbital motion of the molecules around each other.

2.5.1

POTENTIAL MATRIX ELEMENTS

Equation 2.27 is a perfectly reasonable way of writing the classical dipole–dipole interaction. Quantum mechanically, however, we are interested in molecules that are in particular quantum states | jmω, ¯ , rather than molecules whose dipoles point in particular directions (α, β). We must therefore construct matrix elements of the interaction potential 2.27 in the basis we have described in Section 2.3.3. Writing the interaction in the form above has the advantage that each term in the sum factors into three pieces: one depending on the coordinates of molecule 1, another depending on the coordinates of molecule 2, and a third depending on the relative coordinates (θ, φ). This makes it easier to evaluate the Hamiltonian in a given basis. For two molecules we consider the basis functions α1 β1 γ1 | jm1 ω, ¯ 1 α2 β2 γ2 | jm2 ω, ¯ 2 ,

(2.28)

as defined above. In this basis, matrix elements of the interaction become jm1 ω, ¯ 1 ; jm2 ω, ¯ 2 | Vd (θφ) | jm1 ω, ¯ 1 ; jm2 ω, ¯ 2 √ 2 1 1 = − 30μ1 μ2 q −q1 −q2 × jm1 ω, ¯ 1 | C1q1 | jm1 ω, ¯ 1 jm2 ω, ¯ 2 | C1q2 | jm2 ω, ¯ 2

C2−q (θφ) (2.29) R3

¯ are evaluated in Equawhere matrix elements of the form jmω, ¯ | C1q | jm ω, tion 2.23. Conservation of angular momentum projection constrains the values of the summation indices, so that q1 = m1 − m1 , q2 = m2 − m2 , and q = q1 + q2 = (m1 + m2 ) − (m1 + m2 ). To make this model concrete, we report here the values of the second-rank reduced spherical harmonics [1]: 1 (3 cos2 θ − 1) 2 1/2 3 =∓ cos θ sin θe±iφ 2 1/2 3 = sin2 θe±2iφ . 8

C20 = C2±1 C2±2

We also tabulate the relevant 3-j symbols in Table 2.3. Viewed roughly as a collision process, we can think of two molecules approaching each other with angular momenta m1 and m2 , scattering, and departing with angular © 2009 by Taylor and Francis Group, LLC

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TABLE 2.3 The 3-j Symbols Needed to Construct the Matrix Elements in Equation 2.29

q

q1

q2

0 0 1 2

0 1 1 1

0 −1 0 1

2 q

1 1 −q1 −q2 √ 2/15 √ 1/ 30 √ −1/ 10 √ 1/ 5

Source: Brink, D.M. and Satchler, G.R., Angular Momentum, 3rd ed., Oxford University Press, 1993. Note that these symbols remain invariant under interchanging the indices q1 and q2 , as well as under simultaneously changing the signs of q1 , q2 , and q.

momenta m1 and m2 , in which case q is the angular momentum transferred to the relative angular momentum of the pair of molecules. Remarkably, apart from a numerical factor that can be easily calculated, the part of the quantum-mechanical dipole–dipole interaction corresponding to angular momentum transfer q has an angular dependence given simply by the multipole term C2−q . Suppose that the molecules, when far apart, are in the well-defined states of 2.28. Then the diagonal matrix element of the dipole–dipole potential evaluates to (1 − 3 cos2 θ) m1 1 ω ¯ m 2 2 ω ¯ cos 2δm1 cos 2δm2 μ2 . (2.30) μ1 j( j + 1) j( j + 1) R3 This has exactly the form of the interaction for classical, polarized dipoles, as in Equation 2.4. The difference is that each dipole μ1 , μ2 , is replaced by a quantumcorrected version (in large parentheses). It is no coincidence that this is the same quantum-corrected dipole moment that appeared in expression 2.24 for the field due to a single dipole. When both dipoles are aligned with the field, we have m1 1 > 0 and m2 2 > 0 (e.g., both molecules are of type | a), and the interaction has the angular dependence ∝ (1 − 3 cos2 θ). On the other hand, when one dipole is aligned with the field and the other is against (e.g., one molecule is of type | a and the other is of type | c), then the opposite sign occurs – just as we would expect from classical intuition. More generally, at finite electric field, or at finite values of R, the molecules do not remain in the separated-molecule eigenstates 2.28, because they exert torques on one another. The interaction among several different internal molecular states makes the scattering of two molecules a “multichannel problem,” the formulation and solution of which is described in Chapters 5, 9, 13, 14, and 15. However, a good way to visualize the action of the dipole–dipole potential on the molecules is to construct an adiabatic , the relative surface. To do so, we diagonalize the interaction at a fixed value of R location of the two molecules. Before doing this, we must consider the quantum statistics of the molecules. If the two molecules under consideration are identical bosons or identical fermions (with © 2009 by Taylor and Francis Group, LLC

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μ1 = μ2 = μ), then the total two-molecule wavefunction must account for this fact. This total wavefunction is α1 β1 γ1 | jm1 ω ¯ 1 α2 β2 γ2 | jm2 ω ¯ 2 Fjω;m ¯ 1 1 m2 2 (R, θ, φ). This wavefunction is either symmetric or antisymmetric under the exchange of the two particles, which is accomplished by swapping the internal states of the molecules, while simultaneously exchanging their center-of-mass coordinates, that to −R : is, by mapping R (α1 β1 γ1 ) ↔ (α2 β2 γ2 ) R→R

(2.31)

θ→π−θ φ → π + φ.

For the molecule’s internal coordinates, a wavefunction with definite exchange symmetry is given by α1 β1 γ1 | jm1 ω ¯ 1 α2 β2 γ2 | jm2 ω ¯ 2 s =

1

2(1 + δm1 m2 δ1 2 ) ¯ 1 α2 β2 γ2 | jm2 ω ¯ 2 × α1 β1 γ1 | jm1 ω

+ sα2 β2 γ2 | jm1 ω ¯ 1 α1 β1 γ1 | jm2 ω ¯ 2 . (2.32)

The new index s = ±1 denotes whether the combination 2.32 is even or odd under the → −R interchange. If s = +1, then F must be symmetric under the transformation R for bosons, and odd under this transformation for identical fermions. If s = −1, the reverse must hold. We now have the tools required to consider the form of the dipole–dipole interaction beyond the “pure” dipolar form 2.30. The details of this analysis will depend on the Schrödinger equation to be solved. The Schrödinger equation reads

2 2 − ∇ + Vd + HS Ψ = EΨ. 2mr

Here HS stands for the threshold Hamiltonian that includes Λ-doubling and electric field interactions, and is assumed to be diagonal in the basis 2.32; and mr is the reduced mass of the pair of molecules. In the usual way, we expand the total wavefunction Ψ as 1

Ψ(R, θ, φ) = Fi (R, θ, φ) | i , R i

where the index i stands for the collective set of quantum numbers { jω; ¯ m1 1 m2 2 s}. © 2009 by Taylor and Francis Group, LLC

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Inserting this expansion into the Schrödinger equation and projecting onto the ket i | leads to the following set of coupled equations: 2 ∂ ∂2 1 ∂ 1 ∂2 − Fi + 2 sin θ + 2mr ∂R2 ∂θ R sin θ ∂θ R2 sin2 θ ∂φ2

+ i | Vd | i Fi + i | HS | iFi = EFi . i

If we keep N channels i, then this represents a set of N coupled differential equations. We can, in principle, solve these subject to physical boundary conditions for any bound or scattering problem at hand. For visualization, however, we will find it convenient to reduce these equations to fewer than three independent variables. We carry out this task in the following subsections.

2.5.2 ADIABATIC POTENTIAL ENERGY SURFACES IN TWO DIMENSIONS Applying an electric field in the zˆ -direction establishes zˆ as an axis of cylindrical symmetry for the two-body interaction. The angle φ determines the relative orientation of the two molecules about this axis, thus the interaction cannot depend on this angle. To handle this, we include an additional factor in our basis set, 1 | ml = √ exp(iml φ). 2π We then expand the total wavefunction as ΨMtot (R, θ, φ) =

1

F (R, θ)| ml | i . R i ml i ml

In each term of this expression the quantum numbers must satisfy the conservation requirement for fixed total angular momentum projection, Mtot = m1 + m2 + ml . In addition, applying exchange symmetry to each term requires that Fi,ml (R, π − θ) = s(−1)ml Fi,ml (R, θ) for bosons, and s(−1)ml +1 Fi,ml (R, θ) for fermions. Inserting this expansion into the Schrödinger equation yields a slightly different set of coupled equations: 2 − 2mr +

1 ∂ ∂2 ∂ Fi,ml + sin θ ∂θ ∂R2 R2 sin2 θ ∂θ

2 ml2 2mr

R2 sin2 θ

Fiml +

i ml

i | Vd2D | i Fi m + i | HS | iFiml = EFiml . (2.33) l

This substitution has the effect of replacing the differential form of the azimuthal kinetic energy, ∝ ∂ 2 /∂φ2 , by an effective centrifugal potential ∝ ml2 /R2 sin2 θ. In © 2009 by Taylor and Francis Group, LLC

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addition, the matrix elements of the dipolar potential Vd2D are slightly different from those of Vd . Recall that the (θ, φ)-dependent part of the matrix element 2.29 is proportional to C2−q (θ, φ), which we will write explicitly as C2−q (θφ) ≡ C2−q (θ) exp(−iqφ). This equation explicitly defines a new function C2−q (θ) that is a function of θ alone, and that is proportional to an associated Legendre polynomial. The matrix element of the potential now includes the following integral:

1 1 dφ √ e−iml φ C2−q (θ)e−iqφ √ eiml φ 2π 2π C2−q (θ) = dφei(Mtot −Mtot )φ 2π

ml | C2−q | ml =

C2−q (θ), = δMtot Mtot

which establishes the conservation of the projection of total angular momentum by the dipole–dipole interaction. Therefore, matrix elements of Vd2D in this representation are identical to those in of Vd in Equation 2.29 except that the factor exp(−iqφ) is . replaced by δMtot Mtot With these matrix elements in hand, we can construct solutions to the coupled differential equations 2.33. However, to understand the character of the potential surface, it is useful to construct adiabatic potential energy surfaces. This means that, for a fixed relative position of the molecules (R, θ), we find the energy spectrum of Equations 2.33 by diagonalizing the Hamiltonian Vc2D + Vd2D + HS , where Vc2D is a short-hand notation for the centrifugal potential discussed above. This approximation is common throughout atomic and molecular physics, and amounts to defining a single surface that comes as close as possible to representing what is, ultimately, multichannel dynamics.

2.5.3

EXAMPLE: j = 1/2 MOLECULES

Analytic results for the adiabatic surfaces are rather difficult to obtain. Consider the simplest realization of our model, a molecule with spin j = 1/2. In this case each molecule has four internal states (two values of m and two values of ), so that the two-molecule basis comprises 16 elements. Dividing these according to exchange symmetry of the molecules’ internal coordinates, there are ten channels within the manifold of s = +1 channels, and six within the s = −1 manifold. These are the cases we will discuss in the following, although the same qualitative features also appear in higher-j molecules. As the simplest illustration of the influence of internal structure on the dipolar interaction, we will focus on the lowest-energy adiabatic surface, and show how it differs from the “pure dipolar” result (Equation 2.30) as the molecules approach one another. The physics underlying this difference arises from the fact that the dipole–dipole interaction becomes stronger as the molecules get closer together, and at some point this © 2009 by Taylor and Francis Group, LLC

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interaction is stronger than the action of the external field that holds their orientation fixed in the lab. The intermolecular distance at which this happens can be approximately calculated by setting the two interactions equal, μ2 /R03 = (μE)2 + (Δ/2)2 , yielding a characteristic distance R0 =

μ2 (μE)2 + (Δ/2)2

1/3 .

When R R0 , the electric field interaction is dominant, the dipoles are aligned, and the interaction is given by Equation 2.30. When R becomes comparable to, or less than, R0 , then the dipoles tend to align in a head-to-tail orientation to minimize their energy, regardless of their relative location. Before proceeding, it is instructive to point out how large the scale R0 can be for realistic molecules. For the OH molecule considered above, with μ = 1.7 D and Δ = 0.06 cm−1 , the molecule can be polarized in a field of E ≈ 1600 V/cm. At this field, the characteristic radius is approximately R0 ≈ 120a0 (where a0 = 0.053 nm is the Bohr radius), far larger than the scale of the molecules themselves. Therefore, while the dipole–dipole interaction is by far the largest interaction energy at large R, over a significant range of R this potential does not take the usual dipolar form. To take an even more extreme case, the molecule NiH has a ground state of 2 Δ symmetry with j = 5/2 [2]. Because it is a Δ, rather than a Π, state, its Λ doublet is far smaller, probably on the order of ∼10−5 cm−1 . This translates into a critical field of E ≈ 0.5 V/cm, and a characteristic radius at this field R0 ≈ 2000a0 ≈ 0.09 μm. This length is approaching a nonnegligible fraction of the interparticle distance in a Bose–Einstein condensed sample of such molecules (assuming a density of 1014 cm−3 , this spacing is of order 0.2 μm). Deviations from the simple dipolar behavior may thus influence the macroscopic properties of a quantum degenerate dipolar gas (see Chapter 12). As an example, we present in Figure 2.2 sections of the lowest-energy adiabatic potential energy surfaces for a fictitious j = 1/2 molecule whose mass, dipole moment, and Λ doublet are equal to those of OH. These were calculated in a strongfield limit with E = 104 V/cm. Each row corresponds to a particular intermolecular separation, which is compared to the characteristic radius R0 . However, as noted above, there are two possible ways for the molecule to have its lowest energy, as illustrated by parts (a) and (b) of Figure 2.1. Interestingly, it turns out that these give rise to rather different adiabatic surfaces. To illustrate this, we show in the left column of Figure 2.2 the surface fora pair of type | a molecules, which corresponds at infinitely large R to the channel 21 +, 21 +; s = 1 ; and in the right column we show combinations of one type | a and one type | b molecule. In the latter case, there are two possible symmetries corresponding to s = ±1, both of which are shown. Finally, for comparison, the unperturbed “pure dipole” result is shown in all panels as a dotted line. Consider first two molecules of type | a (left column of Figure 2.2). For large distances R > R0 (Figure 2.2a), the adiabatic potential deviates only slightly from the pure-state result, reducing the repulsion at θ = π/2. When R approaches the © 2009 by Taylor and Francis Group, LLC

64

Cold Molecules: Theory, Experiment, Applications |aaÒ

(a)

|ab, sÒ

(d)

Energy (K)

0.02 0.00 1.5R0 –0.02

Energy (K)

(b) 0.08

(e)

0.00

R0 s = +1 s = –1

–0.08

(f )

Energy (K)

(c) 0.0

0.5R0 –0.5 –1.0 0

p/2 q

p

0

p/2 q

p

FIGURE 2.2 Angular dependence of adiabatic potential energies for various combinations of molecules at different interparticle spacings R, which are indicated on the right side. Dotted lines: diagonal matrix element of the interaction, assuming both molecules remain strongly aligned with the electric field. Solid and dashed lines: adiabatic surfaces. These surfaces are based on the j = 1/2 model discussed in the text, using μ = 1.68 D, Δ = 0.056 cm−1 , mr = 8.5 amu, and E = 104 V/cm, yielding R0 ∼ 70a0 . The left-hand column presents results for two molecules of type | a, as labeled in Figure 2.1; in the right column are results for one molecule of type | a and one of type | b, which necessitates specifying an exchange symmetry s.

characteristic radius R0 (Figure 2.2b), the effect of mixing in the higher-energy channels becomes apparent. Finally, when R < R0 (Figure 2.2c), the mixing is even more significant. In this case the dipole–dipole interaction is the dominant energy, with the threshold energies serving as a small perturbation. As a consequence, the quan tum numbers 21 +, 21 +; s = 1 can no longer identify the channel. It is beyond the scope of this chapter to discuss the corresponding eigenstates in detail. Nevertheless, we find that for R < R0 the channel | aa (repulsive at θ = π/2) is strongly mixed with the channel | ac (attractive at θ = π/2). The combination is just sufficient that the two channels nearly cancel out one another’s θ-dependence. At the ends of the range, however, the adiabatic curve is contaminated by a small amount of channels containing centrifugal energy ∝ 1/ sin2 θ. The right column of Figure 2.2 shows adiabatic curves for the mixed channels, one molecule of type | a and one of type | b. In this case there are two possible signs of s; © 2009 by Taylor and Francis Group, LLC

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these are distinguished by using solid lines for channel 21 +, − 21 −; s = 1 and dashed lines for channel 21 +, − 21 −; s = −1 . Strikingly, these surfaces are different both from one another, and from the surfaces in the left column of the figure. Ultimately this arises from different kinds of channel couplings in the potentials (Equation 2.29). Note that, while type | a and type | b molecules are identical in their interaction energy with the electric field, they still represent different angular momentum states. Nevertheless, molecules in these channels still closely reflect the pure dipolar potential at large R, and become nearly θ-independent for small R. A further important point is that the potentials described here represent large energies as compared to the mK or μK translational kinetic energies of cold molecules, and will therefore significantly influence their dynamics. Further, the potentials depend strongly on the value of the electric field of the environment, both through the direct effect of polarization on the magnitude of the dipole moments, and through the influence of the field on the characteristic radius R0 . It is this sensitivity to field that opens the possibility of control over interactions in an ultracold dipolar gas (see Chapter 12). Although we have limited the discussion here to the lowest adiabatic state, interesting phenomena are also expected to arise due to avoided crossings in excited states. Notable is a collection of long-range quasibound states, whose intermolecular spacing is roughly centered around R0 [7]. Such states could conceivably be used to associate pairs of molecules into well-characterized transient states, furthering the possibilities of control of molecular interactions.

2.5.4 ADIABATIC POTENTIAL ENERGY CURVES IN ONE DIMENSION: PARTIAL WAVES For many scattering applications, it is not necessarily convenient to express the dipole– dipole interaction as a surface (more properly, a set of surfaces) in the variables (R, θ, φ) describing the relative position and orientation of the molecules. Rather, it is useful to expand the relative angular coordinates (θ, φ) in a basis as well. To do this, the basis set 2.28 is augmented by spherical harmonics describing the relative orientation of the molecules, to become α1 β1 γ1 | jm1 ω ¯ 1 α2 β2 γ2 | jm2 ω ¯ 2 θφ | lml , with

θφ | lml = Ylml (θφ) =

2l + 1 Clml (θφ). 4π

The total wavefunction is therefore described by the superposition ΨMtot (R, θ, φ) =

1

F Y (θφ) | i , R i ,l ,ml l ml i ,l ,ml

which represents a conventional expansion into partial waves. The wavefunction is, as above, restricted by conservation of angular momentum to require Mtot = m1 + © 2009 by Taylor and Francis Group, LLC

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m1 + ml to have a constant value. Moreover, the effect of the symmetry operation 2.31 on | lml is to introduce a phase factor (−1)l . Therefore, the wavefunction is restricted to s(−1)l = 1 for bosons, and s(−1)l = −1 for fermions. The effect of this extra basis function is to replace the C2−q factor in Equation 2.29 by its matrix element C2−q → lml | C2−q | l ml =

l (2l + 1)(2l + 1)(−1)ml 0

2 0

l 0

l −ml

2 −q

l . ml

(2.34)

From this expression it is seen that the angular momentum q, lost to the internal degrees of freedom of the molecules, appears as the change in their relative orbital angular momentum, that is, ml = ml + q. From here, the effects of field dressing are exactly as treated above. The quantum number l, as is usual in quantum mechanics when treated in spherical coordinates, represents the orbital angular momentum of the pair of molecules about their center of mass. Following the usual treatment, this leads to a set of coupled radial Schrödinger Equations for the relative motion of the molecules: −

2 d2 Filml 2 l(l + 1) + Filml + i | Vd1D | i Fi l m + i | HS | iFilml = EFilml . 2 2 l 2mr dR 2mr R i l ml

(2.35) The second term in this expression represents a centrifugal potential ∝1/R2 , which is present for all partial waves l > 0.

2.5.5 ASYMPTOTIC FORM OF THE INTERACTION Casting the Schrödinger equation as an expansion in partial waves, and the interaction as a set of curves in R, rather than as a surface in (R, θ, φ), allows us to explore more readily the long-range behavior of the dipole–dipole interaction. The first 3-j symbol in Equation 2.34 vanishes unless l + 2 + l is an odd number, meaning that even (odd) partial waves are coupled only to even (odd) partial waves. Moreover, the values of l and l can differ by at most two. Thus the dipolar interaction can change the orbital angular momentum state from l = 2 to l = 4, for instance, but not to l = 6. Finally, the interaction vanishes altogether for l = l = 0, meaning that the dipole–dipole interaction nominally vanishes in the s-wave channel. Because all other channels have higher energy, due to their centrifugal potentials, it appears that the lowest adiabatic curve is trivially equal to zero. This is not the case, however, because the s-wave channel is coupled to a nearby d-wave channel with l = 2. Ignoring higher partial waves, the Hamiltonian corresponding to a particular channel i at long range has the form

0 A20 /R3


A02 /R3 . 32 /mr R2 + A22 /R3


67

Here A02 = A20 and A22 are coupling coefficients that follow from the expressions derived above. Note that all these coefficients are functions of the electric field E. Now, the comparison of dipolar and centrifugal energies defines another typical length scale for the interaction, namely, the one where μ2 /R3 = 2 /mr R2 , defining a “dipole radius” RD = μ2 mr /2 . (More properly, one could define an electric-field-dependent radius by substituting A02 for μ2 .) For OH, this length is ∼6800a0 , while for NiH it is ∼9000a0 . When the molecules are far apart, R > RD , the dipolar interaction is a perturbation. The size of this perturbation on the s-wave interaction is found through second-order perturbation theory to be A202 mr 1 (A02 /R3 )2 − 2 ∼− . 3 /mr R2 32 R4 Therefore, at very large intermolecular distances, the effective potential, as described by the lowest adiabatic curve, carries an attractive 1/R4 dependence on R, rather than the nominal 1/R3 dependence. At closer range, when R < RD , the 1/R3 terms dominate the 1/R2 centrifugal interaction, and the potential reduces again to the expected 1/R3 dependence on R.

ACKNOWLEDGMENTS Many fruitful discussions of molecular dipoles over the years are acknowledged, notably those with Aleksandr Avdeenkov, Doerte Blume, Daniele Bortolotti, Jeremy Hutson, and Chris Ticknor. This work was supported by the National Science Foundation.

REFERENCES 1. Brink, D.M. and Satchler, G.R., Angular Momentum, 3rd ed., Oxford University Press, 1993. 2. Brown, J. and Carrington, A., Rotational Spectroscopy of Diatomic Molecules, Cambridge University Press, 2003. 3. Jackson, J.D., Classical Electrodynamics, 2nd ed., Wiley, New York, 1975. 4. Bethe, H.A. and Salpeter, E.E., Quantum Mechanics of One- and Two-Electron Atoms, Plenum Press, 1977. 5. Hougen, J.T., The Calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic Molecules, NBS Monograph 115, available at http://physics.nist.gov/Pubs/Mono115/ 6. Allen, L. and Eberly, J.H., Optical Resonance and Two-Level Atoms, Wiley, New York, 1975. 7. Avdeenkov, A.V., Bortolotti, D.C.E., and Bohn, J.L., Field-linked states of ultracold polar molecules, Phys. Rev. A, 69, 012710, 2004.


Inelastic Collisions and 3 Chemical Reactions of Molecules at Ultracold Temperatures Goulven Quéméner, Naduvalath Balakrishnan, and Alexander Dalgarno CONTENTS 3.1 3.2

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inelastic Atom–Molecule Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Vibrational and Rotational Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.1 Collisions at Cold and Ultracold Temperatures . . . . . . . . . . . 3.2.1.2 Shape Resonances in Molecular Collisions . . . . . . . . . . . . . . . 3.2.1.3 Feshbach Resonances in Molecular Collisions . . . . . . . . . . . 3.2.2 Quasiresonant Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Atom–Molecular Ion Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Chemical Reactions at Ultracold Temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Tunneling-Dominated Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.1 Reactions at Zero Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.2 Feshbach Resonances in Reactive Scattering . . . . . . . . . . . . . 3.3.2 Barrierless Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.1 Collision Systems of Three Alkali Metal Atoms . . . . . . . . . . 3.3.2.2 Role of PES in Determining Ultracold Reactions . . . . . . . . 3.3.2.3 Relaxation of Vibrationally Excited Alkali Metal Dimers 3.3.2.4 Reactions of Heteronuclear and Isotopically Substituted Alkali-Metal Dimer Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Inelastic Molecule–Molecule Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Molecules in the Ground Vibrational State . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Vibrationally Inelastic Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70 71 71 71 79 80 82 83 84 86 87 92 95 95 101 104 106 108 108 112 115 116 117

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3.1


INTRODUCTION

The development of techniques for cooling and trapping of a wide variety of atomic and molecular species in recent years has created exciting opportunities for probing and controlling atomic and molecular encounters with unprecedented precision [1]. While many of the initial studies of cold atoms and molecules were centered on the creation of dense samples of cold and ultracold matter, more recent work has focused on the manipulation and control of intermolecular interactions, with the ultimate aim of achieving quantum control of atomic and molecular collisions [2]. Although the ideas of quantum control of chemical reactions were proposed many years ago, the ability to create ultracold molecules in specific quantum states has given further stimulus to this field. Its development requires that molecular properties and collisional behavior be well understood at cold and ultracold temperatures where the dynamics of molecules are dramatically different compared to collisions at elevated temperatures. Over the last ten years significant progress has been achieved both in theoretical and experimental works. The experimental methods such as photoassociation spectroscopy, magnetic tuning of Feshbach resonances, buffer-gas cooling, and Stark deceleration [3–6] have been developed and applied to a variety of molecular systems. Novel methods to study ultracold chemical reactions involving ion–molecule systems in a linear Paul trap have been proposed [7]. External control of chemical reactions using electric and magnetic fields is another area of active interest [8]. The aim of this chapter is to provide an overview of recent progress in characterizing molecular processes and chemical reactions at cold and ultracold temperatures, with particular emphasis on theoretical developments in quantum-dynamics simulations of atom–molecule collision systems over the last ten years. In contrast to scattering at thermal energies, ultracold collisions offer fascinating and unique opportunities to study molecular encounters in the extreme quantum regime where the entire collision can be dominated by a single partial wave. One of the main motivations of current experimental efforts to create dense samples of ultracold molecules is to study the possibility of chemical reactions at temperatures close to absolute zero. While Wigner’s law [9,10] predicts that rate coefficients of exothermic processes are finite in the zero-energy limit, it does not say if the rate coefficient will be large enough for reactions to be observable in an experiment. Nor does it say anything about how the rate coefficients at zero temperature depend on the interaction potential. Most chemical reactions between neutral atoms and molecules involve an energy barrier and it is not clear if chemical reactions between them generally occur with measurable rates at ultracold temperatures. Calculations for the F + H2 reaction, which proceeds by tunneling at low energies, have shown that the reaction may occur with a significant rate at ultracold temperatures [11]. There is also experimental and theoretical evidence that chemical reactions involving heavy-atom tunneling of carbon [12] and fluorine [13] atoms can occur with significant rate coefficients at low temperatures. Photoassociation experiments involving alkali-metal systems have stimulated considerable interest in chemical reactivity in alkali-metal dimer–alkali-metal atom collisions at ultracold temperatures [14]. Rearrangement collisions in identical particle alkali-metal trimer systems occur without energy barrier, and recent studies have


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indicated that chemical reactions in alkali-metal atom–alkali-metal dimer collisions may be very fast at ultracold temperatures. Unlike tunneling-dominated reactions, the limiting values of the rate coefficients for alkali-metal systems are less sensitive to vibrational excitation of the dimer. In this chapter, we give an overview of recent studies of ultracold atom–molecule collisions, focusing on nonreactive and reactive systems and the effect of vibrational excitation of the molecule on the collisional outcome. We will discuss both tunneling-dominated and barrierless reactions and examine recent efforts in extending these studies to ionic systems as well as molecule–molecule systems. We consider mostly the novel aspects of collisional dynamics of atom–diatom systems at cold and ultracold temperatures with illustrative results for specific systems. For more comprehensive discussion of cold and ultracold collisions including reactive and nonreactive processes and the effect of external fields we refer the reader to several review articles [6,8,13–15] that have appeared in the last few years. For details of the theoretical formalisms we refer to the chapters by Hutson and by Tscherbul and Krems.

3.2

INELASTIC ATOM–MOLECULE COLLISIONS

Theoretical studies of ultracold molecules received intense interest after the success of ultracold atom photoassociation and buffer-gas-cooling experiments, which demonstrated that a wide array of molecular systems with thermal and nonthermal vibrational energy distributions can be created in ultracold traps. The collisional loss of trapped molecules is an important issue in these experiments. Photoassociation produces molecules in highly excited vibrational levels. Whether the excited molecules decay by vibrational quenching or through chemical reaction is an intriguing question. Although an extensive literature exists on the collisional relaxation of vibrationally excited molecules at elevated temperatures not much was known about the magnitude of the relaxation rate coefficients at temperatures lower than 1 K. A few earlier reports [16–18] on atom–diatom collisions in the Wigner threshold regime had been published, but a detailed investigation of the dependence of the relaxation rate coefficients on the internal energy of molecules and their sensitivity to details of the interaction potential has not been carried out. Here, we give a brief account of recent quantum-dynamics calculations of vibrational and rotational energy transfer in atom–diatom collisions at cold and ultracold temperatures. We focus on a few representative systems to illustrate the main features of energy transfer in nonreactive atom–molecule collisions at ultracold temperatures, and we show how the corresponding rate coefficients are influenced by rotational or vibrational excitation of the molecule.

3.2.1 VIBRATIONAL AND ROTATIONAL RELAXATION 3.2.1.1

Collisions at Cold and Ultracold Temperatures

As discussed in the chapter by Hutson, at very low energies, scattering is dominated by s-waves, and the scattering cross-section can be expressed in terms of a single parameter called the scattering length. For single-channel scattering where only elastic © 2009 by Taylor and Francis Group, LLC

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scattering is possible, the scattering length is a real quantity and the magnitude of the cross-section in the s-wave limit is given by σ = 4πa2 , where a is the scattering length. For multichannel scattering, as in vibrationally or rotationally inelastic collisions of molecules, the scattering length is a complex number and is denoted as avj = αvj − iβvj where v and j are, respectively, the initial vibrational and rotational quantum numbers of the molecule [10,19]. The limiting value of the elastic cross-section in el = 4π|a |2 = 4π(α2 + β2 ). The the presence of inelastic scattering is given by σvj vj vj vj total inelastic quenching cross-section from a given initial rovibrational level of the molecule is related to the imaginary part of the scattering length through the relation in = 4πβ /k where k is the wavevector in the incident channel. The quenching σvj vj vj vj in = rate coefficient becomes constant at ultralow temperatures and is given by kvj 4πβvj /μ at zero temperature, where μ is the reduced mass of the collisional system. Thus, the rovibrational relaxation rate coefficients attain finite values for different initial vibrational and rotational levels of the molecule. The dependence of the rate coefficients on v and j has been an important issue in cold molecule research because exothermic vibrational and rotational relaxation collisions are a major pathway for trap loss in cooling and trapping experiments. Initial studies of rotational and vibrational relaxation of atom–molecule systems at cold and ultracold temperatures have mostly focused on van der Waals systems such as He–H2 [19–22], He–CO [23,24], and He–O2 [25]. Owing to the importance of some of these systems in astrophysical environments, extensive calculations of low-temperature behavior of rate coefficients have been performed for collisions of H2 [19] and CO [23,24] with both 3 He and 4 He. For both systems reasonably accurate intermolecular potentials have been reported. The initial calculations on the He–H2 system employed the potential energy surface (PES) of Muchnick and Russek (MR) [26]. For He–H2 , vibrational excitation of the H2 molecule has a dramatic effect on the zero-temperature quenching rate coefficients. As illustrated in Figure 3.1, the vibrational quenching rate coefficients increase by about three orders of magnitude between v = 1 and v = 10 of the H2 molecule [19]. The quenching rate coefficients exhibit a minimum at around 10 K, which roughly corresponds to the depth of the van der Waals interaction potential. This behavior appears to be a characteristic of vibrational quenching rate coefficients. For incident energies lower than the well depth the rate coefficient exhibits a minimum and with subsequent decrease in temperature the rate coefficient begins to increase before attaining the Wigner limit. For systems with deeper van der Waals wells the minimum is shifted to higher temperatures. Measurements of vibrational relaxation rate coefficients for the H2 −CO system have confirmed this behavior [27]. Balakrishnan, Forrey, and Dalgarno [28] investigated vibrational relaxation of H2 in collisions with H atoms for vibrational quantum numbers v = 1 to 12 of the H2 molecule. They adopted a nonreactive scattering formalism and neglected the rotational motion of the H2 molecule. The calculations showed that vibrational relaxation rate coefficients are strongly dependent on the initial vibrational level of the H2 molecule. The relaxation rate coefficients were found to increase by about seven orders of magnitude between vibrational levels v = 1 and v = 12. The dramatic variation in the rate coefficients with increase in vibrational excitation was explained in terms of the matrix elements of the interaction potential between © 2009 by Taylor and Francis Group, LLC

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Rate coefficient (cm3/molecule/sec)

10−12

10−14

10−16

u = 10 u=9 u=8 u=7 u=6 u=5 u=4 u=3 u=2 u=1

10−18

10−20 −6 10

10−4

10−2 100 Temperature (K)

102

104

FIGURE 3.1 Rate coefficients for the quenching of H2 (v, j = 0) by collisions with He atoms as functions of the temperature for v = 1 to 10 of the H2 molecule. (From Balakrishnan, N. et al., Phys. Rev. Lett., 80, 3224, 1998. With permission.)

the vibrational wavefunctions as functions of the atom–molecule center-of-mass separation. Vibrational relaxation rate coefficients in atom–molecule systems are often influenced by van der Waals complexes formed during the collision process. Decay of these complexes leads to resonances in the energy dependence of the relaxation crosssections (see Figure 3.2 for Ar–D2 collisions). Applying effective range theory to describe ultracold collisions of the He–H2 system, Balakrishnan and colleagues [19] and Forrey and colleagues [20] demonstrated that vibrational predissociation lifetimes of resonances that lie close to the energy threshold can be derived accurately from the value of the zero-temperature quenching rate coefficient. This formalism was extended to describe vibrational relaxation of trapped molecules and it was shown that the vibrational relaxation rate is controlled by the most weakly bound state of the van der Waals complex [22]. In a related work Dashevskaya and colleagues [29] have shown that vibrational quenching of H2 (v = 1, j = 0) at low temperatures can be described using a two-channel approximation within the quasiclassical method provided appropriate parameters are employed in the calculations. Subsequently, Côté and colleagues [30] generalized this method to predict vibrational relaxation lifetimes of atom–diatom van der Waals complexes with energies near the dissociation threshold. © 2009 by Taylor and Francis Group, LLC

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Cold Molecules: Theory, Experiment, Applications 10–4

j¢ = 2 j¢ = 4 j¢ = 6 j¢ = 8

Cross-section (10–16 cm2)

10–6

10–8

10–10

10–12 –8 10

10–6

10–4

10–2

Kinetic energy

100

102

104

(cm–1)

FIGURE 3.2 Cross-sections for quenching of the v = 1, j = 0 level of D2 in collisions with Ar atoms resolved into the different rotational levels j in v = 0 as functions of the incident kinetic energy. (From Uudus, N. et al., J. Chem. Phys., 122, 024304, 2005. With permission.)

Unlike in thermal energy collisions, the presence of a weakly bound state in the vicinity of a channel threshold can dramatically influence the cross-sections in the ultracold regime. This is illustrated in Figure 3.2 where the cross-sections for vibrational relaxation of D2 (v = 1, j = 0) in collisions with Ar atoms [31] are presented over an energy range of 10−8 to 103 cm−1 . The cross-sections exhibit a curvature characteristic of a resonant enhancement in the energy range 10−5 to 10−3 cm−1 . Such enhancement of the cross-section can occur when the interaction potential supports a virtual state or a very weakly bound state near the channel threshold leading to a zero-energy resonance. The virtual state is characterized by a large negative scattering length while the bound state is characterized by a large positive scattering length. For the present case the real part of the scattering length is large and positive (α10 = 97.0 Å) and the resonance occurs from the decay of a loosely bound van der Waals complex supported by the entrance channel potential. For energies below 10−2 cm−1 the cross-section is dominated by s-wave scattering in the incident channel, and the zero-energy resonance arises from s-wave scattering in the entrance channel. The resonant enhancement is more clearly seen in the plot of the reaction probability as a function of the kinetic energy, shown in Figure 3.3. The probability peaks at an energy of 6.0 × 10−4 cm−1 , which roughly corresponds to the binding © 2009 by Taylor and Francis Group, LLC

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Inelastic Collisions and Chemical Reactions of Molecules 1e–11

8e–12

Probability

6e–12

4e–12

2e–12

0e+00 10–6

10–5

10–3

10–4

Incident kinetic energy

10–2

(cm–1)

FIGURE 3.3 Total probability of quenching of the v = 1, j = 0 level of D2 in collisions with Ar atoms as a function of the incident kinetic energy. The peak value of the probability corresponds to a zero energy resonance. (From Uudus, N. et al., J. Chem. Phys., 122, 024304, 2005. With permission.)

energy of the quasibound state. The resonance appears in the scattering calculations at energies above the threshold due to its close proximity to the channel threshold. The binding energy of the quasibound state can be estimated using the scattering length approximation [10,31]. The magnitude of the binding energy is given by |Eb | = 2 cos 2γ10 /(2μ|a10 |2 ) where μ is the reduced mass of the Ar–D2 system, a10 = α10 − iβ10 is the scattering length for the v = 1, j = 0 level, and γ10 = tan−1 (β10 /α10 ). This yields a value of |Eb | = 4.9 × 10−4 cm−1 , in reasonable agreement with the exact value derived from scattering calculations. A more accurate value of the binding energy can be obtained using the effective range formula given by Forrey and colleagues [20]: α10 r0 2 2α10 r0 |Eb | = 1− − 1− |a10 |2 |a10 |2 μr02 where r0 is the effective range of the potential, which may be evaluated by fitting the low-energy behavior of the phase shift for the elastic channel to the standard 2 /2. For Ar–D (v = 1, j = 0) effective range formula, k10 cot δ10 = −1/α10 + r0 k10 2 collisions, the effective range formula yields r0 = 16.32 Å. The resonance position © 2009 by Taylor and Francis Group, LLC

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calculated using the effective range approximation is |Eb | = 5.95 × 10−4 cm−1 , in excellent agreement with the value of 6.0 × 10−4 cm−1 obtained from the scattering calculations. It is generally very difficult to accurately evaluate energies of such weakly bound states using standard bound state codes and the effective range formula provides a convenient and reliable method to calculate binding energies of weakly bound states that lead to zero-energy resonances. One of the challenging aspects of cold and ultracold collisions is the sensitivity to details of the interaction potential. Even the best available methods for the electronic structure calculations of PESs result in errors much larger than the collision energies in the cold and ultracold regime, and the dynamics calculations are often sensitive to small changes in the interaction potential. To explore the sensitivity of cold and ultracold collisions to details of the interaction potential, Lee and colleagues [32] performed a comparative study of the ultracold collision dynamics of the He–H2 system using the MR potential and a more recent ab initio potential developed by Boothroyd, Martin, and Peterson (BMP) [33]. The BMP potential was considered to be an improvement, approaching chemical accuracy, over all conformations compared to the MR potential. However, significant differences were observed for vibrational relaxation of the v = 1, j = 0 state of the H2 molecule in collisions with He computed using the two surfaces. The limiting value of the quenching rate coefficient on the BMP surface was found to be about three orders of magnitude larger than that of the MR surface. The difference was attributed to the more anisotropic nature of the BMP surface leading to larger values of the off-diagonal elements responsible for driving vibrational transitions. Indeed, it was found that the vibrational quenching of the v = 1, j = 0 level was dominated by the transition to the v = 0, j = 8 level, which is driven by the high-order anisotropic terms of the interaction potential. To explore the behavior of inelastic collisions involving polar molecules at ultracold temperatures, Balakrishnan, Forrey and Dalgarno [23] investigated vibrational and rotational relaxation of CO in 4 He–CO collisions. The dynamics of the He–CO system was found to exhibit significantly different features at low temperatures compared to the He–H2 system. Quantum scattering calculations of 4 He and 3 He collisions with the CO molecule revealed that the larger reduced mass and the deeper van der Waals interaction potential of the He–CO system give rise to a number of shape resonances in the energy dependence of vibrational relaxation cross-sections [23,24]. The effect of shape resonances on low-temperature vibrational relaxation rate coefficients will be discussed in the next subsection. The computed values of vibrational relaxation rate coefficients for both 3 He and 4 He collisions with CO(v = 1) have been found to be in good agreement with the experimental data of Reid and colleagues [34] in the temperature range 35 to 100 K. Calculations of vibrational relaxation rate coefficients for the He–CO system in the temperature range 35 to 1500 K have also been reported by Krems [35]. He has shown that inclusion of the centrifugal distortion of the vibrational wavefunction enhances the relaxation process, and that the quenching rate coefficients are sensitive to high-order anisotropic terms in the angular expansion of the interaction potential [36]. Bodo, Gianturco, and Dalgarno [37] have extended the work of Balakrishnan and colleagues [23] to study the vibrational relaxation of excited CO(v = 2, j = 0, 1) molecules in collision with 4 He atoms at ultralow energies. They found that © 2009 by Taylor and Francis Group, LLC


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vibrational quenching of CO(v = 2, j = 0, 1) in collisions with 4 He is dominated by the v = 2 → v = 1 transition. The cross-sections for the v = 2 → v = 0 transition were found to be about four orders of magnitude smaller than the single quantum transitions for both j = 0 and j = 1 initial rotational levels. Ultracold vibrational relaxation of a number of other molecules in collisions with He atoms has been reported by a number of other investigators in recent years. Stoecklin, Voronin, and Rayez [38] reported the vibrational relaxation of F2 in collisions with 3 He atoms. In this study, they constructed the PES of He–F2 using ab initio points obtained by high-level molecular electronic-structure calculations, and reported the cross-sections for elastic scattering and inelastic relaxation of F2 (v = 0, 1, j = 0) for collision energies in the range 10−6 to 2000 cm−1 . A similar study has been reported by the same authors for the 3 He + HF(v = 0, 1, j = 0, 1) system [39]. The vibrational quenching cross-sections were found to be very small compared to pure rotational quenching, in agreement with the results for the He–CO system. This is due to the weak dependence of the He–HF PES on the HF internuclear distance and the strong anisotropy of the interaction potential. Bodo and Gianturco [40] presented a comparative study of vibrational relaxation of CO(v = 1, 2, j = 0), HF(v = 1, 2, j = 0), and LiH(v = 1, 2, j = 0) in collisions with 3 He and 4 He atoms in the Wigner regime. The quenching rate coefficients were found to depend strongly on the collision partner. They reported rate coefficients in the range 10−21 to 10−19 cm3 /sec for CO, 10−16 to 10−15 cm3 /sec for HF, and 10−14 to 10−11 cm3 /sec for LiH. The differences were attributed to the features of the intermolecular forces between the diatomic molecules and the He atoms. The interaction potential of the He–CO system is almost isotropic and is characterized by small vibrational couplings elements. The He–HF system is more anisotropic and the couplings between vibrational states are more significant. The interaction potential of the He–LiH system is very anisotropic and it exhibits strong vibrational couplings. There is considerable ongoing experimental interest in cooling and trapping NH [41,42] and OH [43,44] molecules using the buffer gas cooling and Stark deceleration methods described in the chapters by Doyle and by Meijer. Krems and colleagues [45] and Cybulski and colleagues [46] reported cross-sections and rate coefficients for elastic scattering and Zeeman relaxation in 3 He–NH collisions from ultralow energies to 10 cm−1 . The calculations were performed using the rigid rotor approximation and an accurate He–NH PES. It was demonstrated that the elastic scattering of NH molecules with He atoms in weak magnetic fields is at least five orders of magnitude faster than the Zeeman relaxation, which suggests that the NH molecule is a good candidate for buffer-gas cooling. In a related study González–Sánchez and colleagues [47] examined rotational relaxation and spin-flipping in collisions of OH with He atoms at ultralow energies. They found that the rotational relaxation processes dominate the elastic process as the collision energy is decreased to zero. While theoretical prediction of rate coefficients for vibrational and rotational relaxation in a number of atom–diatom systems has been made, comparable experimental results are not available for the majority of these systems. The first measurements of vibrational relaxation of molecules at temperatures below 1 K were reported by Weinstein and colleagues [48]. In their study, CaH molecules slowed down by elastic collisions with 3 He buffer gas atoms were trapped in an inhomogeneous magnetic © 2009 by Taylor and Francis Group, LLC

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field. An upper bound of the rate coefficients for spin-flipping transitions in CaH as well as vibrational relaxation of CaH molecules in the v = 1 vibrational level in collisions with 3 He atoms were estimated at a temperature of about 500 mK. Balakrishnan and colleagues [49] presented a theoretical analysis of the vibrational relaxation of CaH in collisions with 3 He atoms based on quantum close-coupling calculations and an ab initio PES for the He−CaH system developed by Groenenboom and Balakrishnan [50]. In a related study, Krems and colleagues [51] reported cross-sections for spin-flipping transitions in CaH induced by collisions with 3 He and obtained results in close agreement with the experimentally derived values of Weinstein and colleagues [48]. Krems and colleagues demonstrated that at low energies, spin-flipping transitions in the N = 0 rotational level of 2 Σ molecules induced by structureless atoms occur through coupling to the rotationally excited N > 0 levels and that the corresponding rate coefficients are determined by the spin–rotation interaction with the transiently rotationally excited molecule. Table 3.1 provides a compilation of zero-temperature quenching rate coefficients for vibrational and rotational relaxation in a number of atom–diatom systems.

TABLE 3.1 Zero-Temperature Inelastic Rate Coefficients for Different Atom–Molecule Systems System

Initial (v, j )

kT =0 (cm3 /sec)

Reference

H + H2

(v = 1, j = 0)

1.0 × 10−17

[28]

(v = 1, j = 0) (v = 10, j = 0)

3 × 10−17 3.6 × 10−14

[19] [19]

4 He + CO

(v = 1, j = 0) (v = 1, j = 1)

6.5 × 10−21 9.0 × 10−19

[23] [23]

3 He + CO

(v = 1, j = 0) (v = 2, j = 0) (v = 1, j = 0) (v = 2, j = 0)

1.3 × 10−19 2.1 × 10−19 5.3 × 10−21 1.3 × 10−20

[40] [40] [40] [40]

3 He + CaH

(v = 0, j = 1) (v = 1, j = 0)

3.5 × 10−12 2.6 × 10−17

[49] [49]

3 He + HF

(v = 1, j = 0) (v = 2, j = 0) (v = 1, j = 0) (v = 2, j = 0)

3.1 × 10−16 2.6 × 10−15 8.1 × 10−16 6.5 × 10−15

[40] [40] [40] [40]

(v = 1, j = 0) (v = 2, j = 0) (v = 1, j = 0) (v = 2, j = 0)

9.0 × 10−14 3.6 × 10−12 3.8 × 10−13 1.5 × 10−11

[40] [40] [40] [40]

3 He + H

2

4 He + CO

4 He + HF

3 He + LiH 4 He + LiH


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3.2.1.2

Shape Resonances in Molecular Collisions

At energies above the onset of the s-wave regime, cross-sections will be dominated by contributions from nonzero angular momentum partial waves. If the interaction potential includes an attractive part, the effective potentials for nonzero angular momentum partial waves may possess centrifugal barriers that introduce shape resonances in the collision energy dependence of the cross-section. This is illustrated in Figure 3.4 for the vibrational relaxation of CO(v = 1, j = 0) in collisions with 4 He atoms. The sharp features in the energy dependence of the cross-section for energies between 0.1 and 10.0 cm−1 arise from shape resonances supported by the van der Waals interaction potential between He and the CO molecule. As shown in Figure 3.5, when integrated over the velocity distribution of the colliding species, the shape resonances lead to significant enhancement of the vibrational relaxation rate coefficient for temperatures between 0.1 and 10.0 K. Similar results have been found for vibrational relaxation of CO [24], O2 [25], and CaH [49] in collisions with 3 He atoms. The sharp features in the energy dependence of the vibrational relaxation cross-sections for the Ar + D2 sytem shown in Figure 3.2 also arises from shape resonances supported by the Ar–D2 van der Waals potential. The effect is generally more pronounced for systems composed of heavier diatomic molecules and

10−5

Cross-section (10−16 cm2)

10−6

10−7

10−8

10−9 10−5

10−3

10−1 Energy (cm−1)

101

103

FIGURE 3.4 Cross-section for the quenching of the v = 1, j = 0 level of CO in collisions with 4 He as a function of the incident kinetic energy. (From Balakrishnan, N. et al., J. Chem. Phys., 113, 621, 2000. With permission.)


80

Cold Molecules: Theory, Experiment, Applications 10−18

Rate coefficient (cm3/sec)

u = 1, j = 1

10−19

10−20 u = 1, j = 0

10−21 10−5

10−3

10−1

101

103

T (K)

FIGURE 3.5 Rate coefficients for the quenching of CO(v = 1, j = 0, 1) by collisions with 4 He as functions of the temperature. (From Balakrishnan, N. et al., J. Chem. Phys., 113, 621, 2000. With permission.)

interaction potentials with deep van der Waals wells for which the density of states will be much higher, leading to rich resonance structures in the cross-sections. 3.2.1.3

Feshbach Resonances in Molecular Collisions

Feshbach resonances occur in multichannel scattering in which an unbound (continuum) channel is coupled to a bound state of another channel. If the energy of the interacting system in the unbound channel lies close to that of the bound state and the coupling between the two channels is strong, the cross-section may change dramatically in the vicinity of the resonance. In the Feshbach resonance method for producing ultracold molecules, an external magnetic field is used to tune the energy of the bound pair to that of the separated atoms. In atom–diatom systems, the bound state may correspond to a quasibound state of the atom–diatom van der Waals complex. Channel potentials corresponding to different initial vibrational and rotational levels of the diatom may induce Feshbach resonances. For the He–CO system Feshbach resonances were found to occur near channel thresholds corresponding to the j = 1 rotational level in the v = 0 and v = 1 vibrational levels. Figure 3.6 shows the Feshbach resonance in the elastic scattering cross-sections in the v = 1, j = 0 channel in the vicinity of the v = 1, j = 1 level. The presence of the Feshbach resonance close © 2009 by Taylor and Francis Group, LLC

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260

240

220

200 3.8080

3.8090

3.8100

3.8110

Energy (cm−1)

FIGURE 3.6 Feshbach resonance in the elastic scattering cross-section of CO(v = 1, j = 0) by 4 He atoms. The resonance occurs just below the opening of the v = 1, j = 1 level shown by the vertical line. The energy is relative to the v = 1, j = 0 level of the CO molecule. (From Balakrishnan, N. et al., J. Chem. Phys., 113, 621, 2000. With permission.)

to the opening of the j = 1 level has a dramatic effect on the vibrational quenching cross-sections from the v = 1, j = 1 level of the CO molecule. Since the Feshbach resonance occurs so close to the threshold of the v = 1, j = 1 channel, its effect on scattering in the v = 1, j = 1 level is similar to that of the zero-energy resonance discussed previously for the Ar + D2 system. This is illustrated in Figure 3.5 (see also Table 3.1) where we compare the rate coefficients for vibrational relaxation from the v = 1, j = 0 and v = 1, j = 1 levels of the CO molecule. The zero-temperature limiting value of the quenching rate coefficient of the v = 1, j = 1 level is about two orders of magnitude larger than for the v = 1, j = 0 level. Similar Feshbach resonances have also been shown to occur in the vibrational and rotational predissociation of He–H2 van der Waals complexes [20]. Forrey and colleagues [20] has successfully used the effective range theory to predict the predissociation lifetimes of these resonances. The Feshbach resonances can be used as a very sensitive probe for the interaction potential and also to selectively break or make bonds in chemical reactions. The coupling between the bound and unbound states can be modified by applying an external electric or magnetic field, and this provides an important mechanism for creating or eliminating Feshbach resonances and thereby controlling the collisional © 2009 by Taylor and Francis Group, LLC

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outcome. Krems have shown that weakly bound van der Waals complexes can be dissociated by tuning a Feshbach resonance using an external magnetic field [52]. In this case the dissociation occurs through coupling between Zeeman levels of the bound and unbound channels and the magnitude of the coupling is varied by changing the external magnetic field.

3.2.2

QUASIRESONANT TRANSITIONS

Although the properties of cold and ultracold collisions are quite different from scattering at thermal energies and quantum effects dominate at low temperatures, a remarkable correlation between classical and quantum-dynamics has been discovered in the relaxation of rovibrationally excited diatomic molecules. Experiments performed nearly two decades ago [53,54] showed that collisions of rotationally excited diatomic molecules with atoms may result in very efficient internal energy transfer between specific rotational and vibrational degrees of freedom. The energy transfer becomes highly efficient when the collision time is longer than the rotational period of the molecule. This effect has since been termed “quasiresonant rotation–vibration energy transfer.” The experimental results revealed that the quasiresonant (QR) transitions satisfy the propensity rule Δj = −4Δv or Δj = −2Δv where Δv = vf − vi and Δj = jf − ji [53,54]. This inelastic channel dominates over all other rovibrational transitions. The QR transfer is generally insensitive to details of the interaction potential. Rather, the QR process involves conservation of the action I = nv v + nj j, where nv and nj are small integers. Forrey and colleagues [21] found that the QR transitions also occur in cold and ultracold collisions of rotationally excited diatomic molecules with atoms and that the process is largely insensitive to the details of the interaction potential even in the ultracold regime. The Δj = −2Δv QR transition in He + H2 collisions [55] is illustrated in Figure 3.7 where the zerotemperature vibrational and rotational transition rate coefficients for different initial vibrational levels of the H2 molecule are plotted as functions of the initial rotational level. For initial rotational levels greater than 12 the QR transition becomes the dominant energy transfer mechanism compared to pure rotational quenching. The gap at j = 22 occurs because the Δj = −2Δv transition is energetically not accessible for this initial state at zero temperature. Forrey and colleagues [21] found that the QR process is even more dominant at low temperatures than at thermal energies. Remarkably, classical trajectory calculations [21] were successful in correctly predicting the correlation between Δj and Δv at ultracold temperatures even though the changes in v and j were fractional. Extensive studies of QR energy transfer in cold and ultracold temperatures have been reported by Forrey and colleagues [56–59]. Ruiz and Heller [60] have recently published a review paper providing a detailed analysis of QR phenomenon using semiclassical techniques. McCaffery and colleagues [61–63] have also reported a number of quasiclassical trajectory calculations of the QR process in thermal energy collisions and successfully interpreted a large body of experimental data based on the QR phenomenon and simple parametric models. © 2009 by Taylor and Francis Group, LLC

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Inelastic Collisions and Chemical Reactions of Molecules 10−10 Dv = 0, Dj = −2 10−12


Dv = +1, Dj = −2

10−14 u=0 u=1

10−16

u=2 u=3 u=4

10−18

10−20

Dv = −1, Dj = +2

0

4

8

12

16

20

24

28

j

FIGURE 3.7 Zero-temperature rate coefficients for 4 He + H2 (v, j) collisions as functions of the initial vibrational and rotational quantum numbers. (From Forrey, R.C. et al., Phys. Rev. A, 64, 022706, 2001. With permission.)

3.2.3 ATOM–MOLECULAR ION COLLISIONS Dynamics of ionic systems are different from collisions of neutral species. The shortrange part of the interaction potential for ionic systems is usually more anisotropic and the long-range part has an attractive component that is determined by the polarizability of the atom and it vanishes as 1/R4 where R is the atom–molecule center-of-mass distance [64]. Due to the strong polarizability term, the interaction potential for ionic systems extends to longer range compared to neutral atom–molecule systems. Therefore, it is important to understand the effect of both the short- and long-range part of the interaction potential on the scattering dynamics. For this purpose, several studies have focused on ultracold collisions between molecular ions and neutral atoms as well as neutral molecules and atomic ions. Bodo and colleagues [64] investigated rotational quenching in Ne+ 2 + Ne and + He2 + He collisions at ultralow energies. They found that the Wigner regime begins at a collision energy of 10−4 cm−1 for the He system and 10−6 cm−1 for the Ne system. In general, the s-wave Wigner regime was found to occur at lower energies for ionic systems compared to neutral species. For example, in He + H2 [19] and He + O2 [25] collisions the Wigner regime begins at collision energies of about 10−2 cm−1 . The differences are attributed to the long range of the ion–neutral interaction potential, © 2009 by Taylor and Francis Group, LLC

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which enhances contributions from higher partial waves. The differences between the He and Ne systems can be attributed to the mass difference and to the strength of the long-range interaction potentials. For the heavier Ne system the long-range part is more attractive, which increases the contribution of higher-order partial waves at ultralow energies. The magnitude of the zero-energy rate coefficients for rotational quenching in these molecular ions is on the order of 10−9 cm3 /sec, which is considerably larger than for collisions involving neutral species. Stoecklin and colleagues [65] have performed a comparative study of collisions of 3 4 N+ 2 (v = 1, j = 0) molecular ions with neutral He or He atoms, and the scattering of neutral N2 (v = 1, j = 0) molecules in collisions with 3 He or 4 He atoms. The vibrational quenching cross-sections for these systems are presented in Figure 3.8. While the behavior of the quenching cross-sections of neutral N2 molecules with 3 He and 4 He atoms was found to be similar, they observed a striking difference 3 4 between the quenching cross-sections of N+ 2 in collisions with He and He atoms. In particular, the resonance positions in the quenching cross-sections were significantly 3 shifted and the number of resonances was different. The N+ 2 + He system has a −2 −1 shape resonance at a collision energy of 10 cm , which is absent for the N+ 2 + 4 He system. Furthermore, the zero-energy quenching rate coefficient is an order of 3 4 magnitude larger for collisions of N+ 2 (v = 1, j = 0) with He than with He, whereas 3 it is comparable for collisions of N2 (v = 1, j = 0) with both He and 4 He. The 3 differences may be due to a virtual state in the N+ 2 (v = 1, j = 0) + He collision system. More recently, Guillon and colleagues [66] studied the effect of spin-rotation interaction on vibrational and rotational quenching for the He–N+ 2 system. They found that the vibrational quenching is not modified by the spin–rotation coupling, while rotational transitions are sensitive to the fine-structure interactions. Table 3.2 provides a compilation of zero-temperature quenching rate coefficients for vibrational and rotational relaxation in several ion atom–molecule systems.

3.3

CHEMICAL REACTIONS AT ULTRACOLD TEMPERATURES

The last three decades have seen impressive progress in the experimental and theoretical descriptions of chemical reactions between atoms and small molecular systems. While fully state-resolved experiments have been performed for a large number of collision systems, accurate quantum dynamics calculations have been restricted to systems involving light atoms such as H + H2 , F + H2 , Cl + H2 , C + H2 , N + H2 , O + H2 and some of their isotopic analogs [67,68]. Most experimental studies of small molecular systems focused on reactions at thermal or elevated collision energies, although some recent measurements have been extended to temperatures as low as 10 K for some astrophysically relevant systems [69]. Due to the possibility of achieving coherent chemistry there has been substantial interest in understanding the behavior of chemical reactions at cold and ultracold temperatures. At these temperatures perturbations introduced by external electric and magnetic fields are significant compared to the collision energies involved and external fields can be employed to control and manipulate the reaction outcome. Over the last seven years a number of studies of ultracold atom–diatom chemical reactions have been reported both for © 2009 by Taylor and Francis Group, LLC

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(a)

103 Isotope 3 Isotope 4

102

s (10–16 cm2 )

101

100

10–1

10–2

10–3

10–4 –8 10

10–6

10–4

10–2 E

(b)

100

102

(cm–1)

10–2

Isotope 3 Isotope 4

s (10–16 cm2)

10–3

10–4

10–5

10–6

10–7

10–6

10–4

10–2

100

102

–1

E (cm ) + 3 FIGURE 3.8 Vibrational quenching cross-sections for N+ 2 (v = 1, j = 0) + He and N2 (v = 1, j = 0) + 4 He (a) and N2 (v = 1, j = 0) + 3 He and N2 (v = 1, j = 0) + 4 He (b) systems. (From Stoecklin, T. and Voronin, A., Phys. Rev. A, 72, 042714, 2005. With permission.)


86


TABLE 3.2 Zero-Temperature Inelastic Rate Coefficients for Different Ion Atom–Molecule Systems Initial (v, j )

kT =0 (cm3 /sec)

Reference

Ne+ 2 + Ne

(v = 0, j = 2) (v = 0, j = 4) (v = 0, j = 6)

3.3 × 10−10 7.3 × 10−10 7.0 × 10−10

[64] [64] [64]

He+ 2 + He

(v = 0, j = 2) (v = 0, j = 4) (v = 0, j = 6)

6.7 × 10−10 8.4 × 10−10 1.2 × 10−9

(64] [64] [64]

3 N+ 2 + He

(v = 1, j = 0)

4 × 10−14

[65]

4 N+ 2 + He

(v = 1, j = 0)

3 × 10−15

[65]

System

reactions with and without an energy barrier. Chemical reactions at low temperatures often behave quite differently from reactions at elevated temperatures. In particular, the weak van der Waals interaction potential, which does not play any significant role at high temperatures, may have a dramatic effect on the outcome of reactions at low temperatures.

3.3.1 TUNNELING-DOMINATED REACTIONS In the course of the last 10 to 15 years there has been much interest in understanding the role of resonances in chemical reactions that involve an energy barrier for which the reactivity is primarily driven by tunneling at low temperatures. Recent studies on F + H2 /HD/D2 [11,70–80], Cl + HD [81,82], H + HCl/DCl [83], Li + HF/LiF + H [84,85], O + H2 [86,87], and F + HCl/DCl [88] reactions have demonstrated that decay of quasibound states of the van der Waals interaction potential in the entrance and exit channels may give rise to narrow Feshbach resonances in the cross-sections. The reactions of F with H2 and HD have been the subject of numerous quantum scattering calculations over the last two decades. The two reactions have emerged as benchmark systems for experimental and theoretical studies of resonances in chemical reactions. A detailed analysis of the low-energy resonance features in the F + H2 reaction was presented by Castillo and colleagues in 1996 [70]. They demonstrated that several resonances that appear in the energy dependence of the cumulative reaction probability for the F + H2 reaction arise due to the van der Waals interaction potential in the product HF + H channel. In particular, the origin of the resonances has been attributed to the van der Waals potential associated with the HF(v = 3, j = 0 − 3) channels. A large number of experimental and theoretical papers has appeared that examined various aspects of these resonance features [11,70–72,75–80]. Among these studies, the work of Takayanagi and Kurosaki [71] deserves special attention. They showed that for F + H2 , F + HD, and F + D2 reactions, reactive scattering © 2009 by Taylor and Francis Group, LLC

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resonances occur due to the decay of quasibound states of the van der Waals potential in the entrance channel of the reaction. These Feshbach resonances are associated with the decay of quasibound states of adiabatic potentials corresponding to F· · · H2 (v = 0, j = 0, 1), F· · · HD(v = 0, j = 0 − 2), and F· · · D2 (v = 0, j = 0 − 2) complexes, obtained by diagonalizing the J = 0 Hamiltonian in a basis set of the asymptotic rovibrational states of the reactant molecules. 3.3.1.1

Reactions at Zero Temperature

Balakrishnan and Dalgarno [11] showed that quasibound states of the F + H2 van der Waals complex have a dramatic effect on the reactivity in the Wigner threshold regime. They found that the F + H2 (v = 0, j = 0) reaction has a rate coefficient of 1.25 × 10−12 cm3 /sec in the zero-temperature limit. Their calculation was based on the widely used PES of Stark and Werner [89] for the F + H2 system. The relatively large value of the zero-temperature rate coefficient is due to the presence of a small narrow energy barrier for the reaction so that tunneling of the H atom is efficient. The vibrational level resolved cross-sections for the F + H2 (v = 0, j = 0) reaction are shown in Figure 3.9 for the total angular momentum quantum number J = 0.

102


100

u¢ = 2

10−2

u¢ = 1

u¢ = 0

10−4

10−6 10−7

10−6

10−5

10−4 10−3 Kinetic energy (eV)

10−2

10−1

FIGURE 3.9 J = 0 cross-sections for F + H2 (v = 0, j = 0) → HF(v ) + H reaction for v = 0 − 2 as functions of the incident kinetic energy. (From Balakrishnan, N. and Dalgarno,A., Chem. Phys. Lett., 341, 652, 2001. With permission.)


88


The v = 2 level is the dominant channel at low energies in agreement with the behavior at higher energies. Subsequent calculations showed that at low energies the F + HD reaction is dominated by the formation of the HF product with an HF/DF branching ratio of about 5.5 [74]. The formation of the DF product is suppressed because tunneling of the heavier D atom is less efficient. In earlier quantum calculations, Baer and colleagues [90, 91] reported HF/DF branching ratios ranging from 1.5 at 450 K to 6.0 at 100 K. Their low-temperature value is in close agreement with the zero-temperature limit obtained by Balakrishnan and Dalgarno [74]. Figure 3.10 compares the J = 0 cross-sections for the F + H2 (v = 0, j = 0) and F + HD(v = 0, j = 0) reactions over a kinetic energy range of 10−7 to 1.0 eV. In the Wigner threshold regime, the reactivity of the F + H2 system is an order of magnitude greater than that of the F + HD reaction. A rigorous analysis of the scattering resonances in the F + HD reaction was recently presented by De Fazio and colleagues [80], who provided a detailed characterization of the resonances supported by the entrance and exit channels of the van der Waals potential and discussed the effect of higher total angular momenta on the position and lifetime of the resonances. Stereodynamical aspects of the F + H2 collision and the effect of polarization of the H2 molecule on the outcome of the reaction at low energies were recently 102


101

100

10−1

10−2

10−3

10−4 10−7

10−6

10−5

10−4 10−3 Kinetic energy (eV)

10−2

10−1

FIGURE 3.10 Comparison of J = 0 cross-sections for F + HD(v = 0, j = 0) → HF + D (solid curve), F + HD(v = 0, j = 0) → DF + H (short-dashed curve) and F + H2 (v = 0, j = 0) → HF + H (long-dashed curve) reactions as functions of the incident kinetic energy. (From Balakrishnan, N. and Dalgarno, A., J. Chem. Phys. A, 107, 7101, 2003. With permission.)


89


explored by Aldegunde and colleagues [77]. They argued that a reactant polarization scheme can be exploited to control state-to-state dynamics of the reaction. To explore the role of tunneling in chemical reactions at cold and ultracold temperatures Bodo, Gianturco, and Dalgarno [73] investigated the dynamics of the F + D2 system at low and ultralow energies. They found that compared to the F + H2 reaction, the reactivity of F + D2 is significantly suppressed in the Wigner regime with an HF/DF ratio of about 100. This is illustrated in Figure 3.11 where the J = 0 cumulative

Cumulative reaction probabilities

(a) 0.02

0.015

H2

0.01

D2 (×10)

0.005

0

0

0.0002

0.0004 0.0006 Collision energy (eV)

0.0008

0.001

10–3

10–2

(b) 103

Reaction cross-section (Å2)

102 101 100 10–1 10–2 10–3 10–4 –8 10

10–7

10–6 10–5 10–4 Collision energy (eV)

FIGURE 3.11 Comparison of the J = 0 cumulative reaction probabilities (a) and crosssections (b) of F + H2 and F + D2 reactions as functions of the incident kinetic energy. (From Bodo, E. et al., J. Phys. B: At. Mol. Opt. Phys., 37, 3641, 2004. With permission.)


90


reaction probability and cross-sections for the F + H2 and F + D2 reactions are plotted as functions of the incident kinetic energy from the Wigner limit to 0.01 eV. The dramatic suppression of the F + D2 reaction in the Wigner regime cannot be explained based on tunneling alone. A closer examination of the reaction probabilities for the F + H2 and F + D2 reaction revealed that an unusual enhancement of the reactivity occurs for the F + H2 reaction at about 3 × 10−5 eV [75] (see Figure 3.11a). This feature was attributed to the presence of a virtual state. The enhancement of the limiting value of the rate coefficient for the F + H2 reaction was also ascribed to the virtual state. Evidence for the virtual state is the presence of a Ramsauer–Townsend minimum in the elastic cross-section at an energy of about 3 × 10−5 eV and a negative value of the real part of the scattering length for the F + H2 reaction [75]. To explore how isotope substitution modifies reactivity at low temperatures Bodo and colleagues [75] artificially varied the mass of the hydrogen atoms in the calculation for the F + H2 reaction from 0.5 to 1.5 amu. As illustrated in Figure 3.12, for an H atom mass of 1.12 amu the virtual state induces a zero energy resonance at which the real part of the scattering length diverged to infinity. For the same value of the H atom mass, the zero-temperature rate coefficient of the reaction attains a value of 1.0 × 10−9 cm3 /sec, which is about three orders of magnitude larger than that of the F + H2 reaction in the Wigner limit. The variation of the scattering length of the F + H2 system as a function of the mass of the pseudo-hydrogen atom is similar to the variation of scattering length as a function of the magnetic field in the vicinity of a Feshbach resonance. Another example of a tunneling-dominated reaction is the Li + HF → LiF + H reaction. At cold and ultracold temperatures the reaction occurs by tunneling of the relatively heavy fluorine atom. The LiH + F channel is energetically not accessible at low energies and tunneling of the H atom is not involved in this reaction at low temperatures. Due to strong electric dipole forces exerted by the HF molecule, the van der Waals interaction potential of the LiHF system is deeper compared to the F + H2 system. The Li· · · HF van der Waals potential well is about 0.24 eV (1936.0 cm−1 ) and the H· · · LiF potential well is about 0.07 eV (565.0 cm−1 ). Quantum scattering calculations for Li + HF and LiF + H reactions by Weck and Balakrishnan [84,85] with the PES of Aguado and colleagues [92] have shown that for incident energies below 10−3 eV the reaction cross-section exhibits a large number of resonances. The energy dependence of the J = 0 cross-section for Li + HF(v = 0, j = 0) collisions is shown in Figure 3.13. Detailed bound-state calculations of the LiHF van der Waals complexes revealed that for the Li + HF(v = 0, j = 0) reaction the resonances correspond to the decay of Li· · · HF(v = 0, j = 1 − 4) van der Waals complexes. Calculations with vibrationally excited HF molecules showed that the reaction becomes about 600 times more efficient in the Wigner regime when HF is excited to the v = 1 vibrational level. As seen in Figure 3.13, a unique feature of the Li + HF(v = 0, j = 0) reaction is the presence of a strong peak at 5.0 × 10−4 eV at which the reaction cross-section is about six orders of magnitude larger than the background cross-section. The results for both Li + HF [84] and LiF + H [85] reactions with thermal and nonthermal vibrational excitation suggest that heavy-atom tunneling may play an important role in chemical reactions at cold and ultracold temperatures. An experimental study [12] of an organic ring expansion reaction at 8 K has © 2009 by Taylor and Francis Group, LLC

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Inelastic Collisions and Chemical Reactions of Molecules (a)

150 100

a (Å)

50 0 –50 –100 –150 (b) 10–8

R(T = 0) (cm3/sec)

10–9 10–10 10–11 10–12 10–13 10–14

|Re(Ep)| (meV)

(c)

100 10–1 10–2 10–3 10–4 10–5

0.6

0.8

1

1.2 1.4 1.6 Mass of both H atoms

1.8

2

FIGURE 3.12 Real part of the scattering length (a), zero-temperature limiting values of the rate coefficient (b), and positions of quasibound states (c) of the F· · · H2 complex as functions of the mass of a pseudo hydrogen atom. (From Bodo, E. et al., J. Phys. B: At. Mol. Opt. Phys., 37, 3641, 2004. With permission.)

shown that the reaction occurs almost exclusively by carbon tunneling. The tunneling contribution was found to be orders of magnitude greater than over the barrier contribution. In a recent work, Tscherbul and Krems [93] have explored the Li + HF and LiF + H reactions in the presence of an external electric field. They have shown that, for temperatures below 1 K, the reaction probability can be significantly influenced by electric fields. © 2009 by Taylor and Francis Group, LLC

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1 0.1

u=1

0.01 0.001 0.0001 1e–05 1e–06 1 u=0

0.1 0.01 0.001 0.0001 1e–05 1e–06 1e–07 1e–07

1e–06

1e–05

0.0001 0.001 Kinetic energy (eV)

0.01

0.1

FIGURE 3.13 Cross-sections for LiF formation (solid curve) and nonreactive scattering (dashed curve) in Li + HF(v, j = 0) collisions as functions of the incident kinetic energy: results for v = 0 (a); results for v = 1 (b). (From Weck, P.F. and Balakrishnan, N., J. Chem. Phys., 122, 154309, 2005. With permission.)

3.3.1.2

Feshbach Resonances in Reactive Scattering

Van der Waals complexes formed during collisions can either undergo vibrational predissociation or vibrational prereaction leading to sharp features in the energy dependence of the cross-sections. The term “prereaction” refers to the process in which a rotationally or vibrationally excited van der Waals complex decays through chemical reaction rather than by rotational or vibrational predissociation. For reactions with energy barriers the chemical reaction pathway may involve tunneling. The reactions of Cl with H2 and HD are dominated by tunneling at low temperatures [81–83]. Compared to the F + H2 and F + HD reactions, the energy barrier for the Cl + H2 reaction is much larger and the reactivity at low energies is significantly suppressed. Nearly a decade ago, Skouteris and colleagues [81] showed that the van der Waals interaction potential between Cl and HD determines the reaction outcome, despite the fact that the depth of the van der Waals interaction potential for the Cl· · · HD system is less than one-tenth of the height of the reaction barrier. Quantum scattering calculations of the Cl + HD reaction on PESs without the van der Waals interaction potential predict nearly equal probabilities for HCl and DCl products [81]. However, if the potential surface includes the van der Waals interaction a strong preference for the DCl product occurs at thermal energies, in agreement with experimental results. The effect of these weakly bound states on the reactivity at cold and ultracold temperatures has recently been explored by Balakrishnan [82]. © 2009 by Taylor and Francis Group, LLC

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The cross-sections for HCl and DCl formation and nonreactive rovibrational transitions in Cl + HD(v = 1, j = 0) collisions for total angular momentum quantum number J = 0 are shown in Figure 3.14 as functions of the total energy [82]. The sharp features in the cross-sections correspond to Feshbach resonances arising from the decay of quasibound van der Waals complexes formed in the entrance channel of the reaction. The quasibound states can be identified by examining bound states of adiabatic potentials correlating with the v = 1, j = 0 and v = 1, j = 1 levels of the HD molecule. They are displayed in Figure 3.15 as functions of the atom–molecule separation. The adiabatic potential curves are computed by diagonalizing the diabatic potential energy matrix obtained in a basis set of rovibrational levels of the HD molecule at each value of the atom–molecule separation. The Feshbach resonances labeled B, C, D, and E in Figure 3.14 result from the decay of the corresponding quasibound states shown in Figure 3.15. The cross-sections in Figure 3.14 do not show a peak corresponding to the metastable state A. This is because the state A is too deeply bound and it is not accessible through scattering in the v = 1, j = 0 channel. Figure 3.14 also shows that the quasibound states preferentially undergo prereaction than predissociation. This

102

101


100

10–1

Cl + HD – nonreactive B

C

D

10–2

E

HCl + D – reactive 10–3

10–4

DCl + H – reactive 10–5

10–6 0.685

0.69

0.695

0.7

0.705

Total energy (eV)

FIGURE 3.14 Reactive and nonreactive scattering cross-sections for Cl + HD(v = 1, j = 0) reaction as functions of the incident kinetic energy. (From Balakrishnan, N. J. Chem. Phys., 121, 5563, 2004. With permission.)


94


0.7 E

u = 1, j = 1

D C 0.69

B

Energy (eV)

u = 1, j = 0

A 0.68

0.67

5

10

15

20

R (au)

FIGURE 3.15 Adiabatic potential energy curves of the Cl + HD system correlating with the HD(v = 1, j = 0) and HD(v = 1, j = 1) levels as functions of the atom–molecule separation. Quasibound levels responsible for the resonances in Figure 3.14 are labeled by B, C, D, and E. (From Balakrishnan, N., J. Chem. Phys., 121, 5563, 2004. With permission.)

is due to the enhanced coupling of the resonance states with the reactive channels compared to those with the nonreactive channels. The wavefunctions of quasibound states B and E are shown in Figure 3.16 as functions of the atom–molecule separation. Although the wavefunction of the weakly bound state E extends far beyond the transition state region of the reaction, it preferentially undergoes prereaction rather than predissociation. Thus, regions of the interaction potential far away from the transition state region may have a significant effect on reactivity, especially when part of the wavefunction in that region is sampled by a resonance state that is coupled to the reactive channel. Figure 3.14 also demonstrates that because the reaction is dominated by tunneling at low temperatures, the formation of the DCl + H product, which involves the tunneling of the D atom, is severely suppressed in the threshold regime. This is more clearly illustrated in Figure 4 of Ref. [82] where the reaction cross-section is plotted as a function of the incident kinetic energy. Table 3.3 provides a compilation of zero-temperature quenching rate coefficients for a number of atom–diatom chemical reactions, which are dominated by tunneling at low energies. © 2009 by Taylor and Francis Group, LLC

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Inelastic Collisions and Chemical Reactions of Molecules 0.02

0.02

B E Adiabatic potential

0

0

–0.01

–0.01

–0.02

0.01 Potential energy (eV)

Wavefunction (arb. units)

0.01

0

5

10

15

20

25

30

–0.02 35

R (au)

FIGURE 3.16 Adiabatic potential and the wavefunctions of the quasibound levels B and E shown in Figure 3.15 as functions of the atom–molecule separation. Amplitudes of the wavefunctions have been reduced by a factor of 10 for the convenience of plotting. (From Balakrishnan, N., J. Chem. Phys., 121, 5563, 2004. With permission.)

3.3.2 3.3.2.1

BARRIERLESS REACTIONS Collision Systems of Three Alkali Metal Atoms

As discussed by Hutson and Soldán in two recent reviews [6,14], progress on the production of ultracold molecules and the creation of molecular Bose–Einstein condensates of alkali-metal systems have motivated theoretical studies on ultracold atom–dimer alkali-metal collisions. Here, we give an overview of recent calculations for spin-polarized triatomic alkali-metal systems, Li + Li2 [94–97], Na + Na2 [98,99], and K + K2 [100]. These results have all been obtained using the reactive scattering code written by Launay and Le Dourneuf [101], based on a time-independent quantum formalism. Refs. [94,95,97] and [100] describe the details of the PESs and dynamics calculations. The quantum dynamics studies of Refs. [98] and [99] have been performed using the Na3 PES reported by Higgins and colleagues [102], while the results of Ref. [96] have been obtained using the Li3 PES calculated by Colavecchia and colleagues [103]. Another PES for Li3 has been constructed by Brue and colleagues [104]. © 2009 by Taylor and Francis Group, LLC

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TABLE 3.3 Zero-Temperature Quenching Rate Coefficients for Tunneling Dominated Reactions (the Final Arrangement has also been Specified) Initial (v, j )

kT =0 (cm3 /sec)

Reference

F + D2

(v = 0, j = 0) (v = 0, j = 0) (v = 0, j = 0) (v = 0, j = 0)

1.3 × 10−12 (H + HF) 2.8 × 10−14 (D + HF) 0.5 × 10−14 (H + DF) 2.1 × 10−14 (D + DF)

[11] [74] [74] [73]

F + HCl F + HCl

(v = 0, j = 0) (v = 1, j = 0)

F + HCl

(v = 2, j = 0)

F + DCl

(v = 1, j = 0)

1.2 × 10−17 4.0 × 10−15 5.3 × 10−15 5.2 × 10−13 3.3 × 10−13 4.4 × 10−21

(Cl + HF) (Cl + HF) (F + HCl) (Cl + HF) (F + HCl) (Cl + DF)

[88] [88] [88] [88] [88] [88]

H + HCl

(v = 0, j = 0) (v = 1, j = 0) (v = 2, j = 0) (v = 1, j = 0) (v = 2, j = 0)

2.4 × 10−19 (Cl + H2 ) 7.2 × 10−14 (Cl + H2 ) 1.9 × 10−11 (Cl + H2 ) 7.8 × 10−15 (Cl + HD) 1.7 × 10−12 (Cl + HD)

[83] [83] [83] [83] [83]

Cl + HD

(v = 1, j = 0)

1.7 × 10−13 (D + HCl) 7.1 × 10−16 (H + DCl) 7.8 × 10−14 (Cl + HD)

[82] [82] [82]

Li + HF

(v = 0, j = 0) (v = 1, j = 0) (v = 1, j = 0)

4.5 × 10−20 2.8 × 10−17 3.8 × 10−15 1.6 × 10−14 1.7 × 10−14 2.7 × 10−13

[84] [84] [85] [85] [85] [85]

System F + H2 F + HD

H + DCl

H + LiF

(v = 2, j = 0)

(H + LiF) (H + LiF) (Li + HF) (H + LiF) (Li + HF) (H + LiF)

Quantum scattering calculations show that ultracold reactions of alkali metal atoms with alkali metal dimers are much more efficient than tunneling-driven processes. This can be explained based on the following considerations. First, the indistinguishability of the atoms may play a role. For a homonuclear triatomic system, the three possible arrangement channels are the same. Also, the presence of identical two-body potentials in all three arrangement channels may enhance the depth of the three-body interaction potential. Consider the two-body term (additive term) of a homonuclear triatomic system at equilateral geometries. Because the distances between the three identical atoms are the same, the same diatomic potential term is added three times to yield the two-body terms of the triatomic system. If the diatomic potential is deep or repulsive, the two-body term will be three times deeper or repulsive at equilateral geometries. In contrast, for triatomic systems with distinguishable atoms, the diatomic © 2009 by Taylor and Francis Group, LLC


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pairs are not the same. For example, for the Li + HF system, the three diatomic fragments are distinctly different. They have completely different electronic structures and properties such as minima, equilibrium distances, turning points, and the nature and range of the interaction. Thus, at an equilateral configuration for which the three diatomic distances are the same but the diatomic potential energies are different (one may be attractive while the other two may be repulsive), the overall two-body term can be weaker than the individual two-body interaction potential. This could lead to a triatomic interaction region that is less strong than that of a homonuclear triatomic system. Second, the topology of the PES plays a significant role. For all alkali-metal trimer systems, the minimum energy configurations arise at equilateral and collinear geometries as shown by Soldán and colleagues [105]. Furthermore, the surface is barrierless so that all atom–dimer alkali-metal collisional approaches are energetically possible. In contrast, most of the nonalkali systems discussed above are characterized by a collinear (or bend) atom–diatom approach, and dominated by a repulsive barrier in the triatomic transition-state region. The particular topology of the PES arises from the three-body interaction potential. Also, the couplings between different electronic surfaces of the triatomic system give rise to conical intersections, which create a repulsive barrier in the transition-state region. If the energy barrier is high and its width large, tunneling will be very inefficient, leading to very small values for the reaction rate coefficients in the Wigner regime. Although reactivity can be enhanced when resonances are present, the background scattering in reactive cross-sections is generally quite small. In contrast, for almost all alkali-metal trimer systems, the conical intersections arise at small interatomic distances where the PES is sufficiently repulsive that it plays no significant role at ultralow energy and the reactivity is not influenced by such repulsive barriers. Third, the density of states of the system plays an important role. If the density of states is small as for light systems, the typical energy spacing is rather large. For heavy systems such as alkali-metal systems, the density of states is very large and the narrow energy spacing can lead to very strong couplings, especially in the vicinity of avoided crossings. The strong couplings will lead to efficient energy transfer between different quantum states. This may also explain the relatively weak dependence of the vibrational quenching rate coefficients on the initial vibrational state of the molecule in atom–dimer alkali-metal collisions. Figure 3.17 provides a comparison between elastic and quenching rate coefficients for 39 K + 39 K2 (v = 1, j = 0) [100] collisions at energies ranging from 10−9 K to 10−2 K. The results include contributions from the total angular momentum quantum numbers J = 0 to 5. For low vibrational states of the K2 dimer, quenching processes are more efficient than elastic scattering at ultracold energies. For example, at a temperature of 10−9 K, the quenching rate coefficient is about 10−10 cm3 /sec compared to 10−13 cm3 /sec for elastic scattering. The quenching processes lead to collisional relaxation of the molecules to lower rovibrational states resulting in trap loss. Similar results have been obtained for Na + Na2 [98,99] and Li + Li2 [94,96,97] collisions. The zero-energy quenching rate coefficients for these systems are found to be on the order of 10−11 to 10−10 cm3 /sec. The quenching processes are more efficient than the elastic collisions at ultracold temperatures. Thus, quenching of © 2009 by Taylor and Francis Group, LLC

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Cold Molecules: Theory, Experiment, Applications (a)

Elastic rate coefficient (cm3/sec)

Total 10–10

10–11

J = (0–5) 10–12

10–13 –9 10

10–8

10–7

10–6 10–5 Collision energy (K)

10–4

10–3

10–2

10–4

10–3

10–2

Quenching rate coefficient (cm3/sec)

(b) Total Langevin

10–10

J = (0–5)

10–11 –9 10

10–8

10–7

10–6 10–5 Collision energy (K)

FIGURE 3.17 Elastic (a) and quenching (b) rate coefficients vs. the collision energy for 39 K + 39 K (v = 1, j = 0) scattering. The rate coefficient from a capture model is also shown 2 for the quenching processes. (From Quéméner, G. et al., Phys. Rev. A, 71, 032722, 2005. With permission.)

vibrationally excited alkali-metal dimers in collisions with alkali-metal atoms occurs with significant rates at ultracold temperatures. The typical magnitude of these rate coefficients from the scattering calculations is in reasonable agreement with experimental results. In a recent experiment, Staanum and colleagues [106] reported a value of 9.8 × 10−11 cm3 /sec for the relaxation of low-lying vibrational levels of Cs2 © 2009 by Taylor and Francis Group, LLC


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in collisions with Cs atoms at a temperature of 60 × 10−6 K. In a separate experiment, Zahzam and colleagues [107] determined a quenching rate coefficient of 2.6 × 10−11 cm3 /sec for the same system at a temperature of 40 × 10−6 K. Wynar and colleagues [108] estimated inelastic rate coefficients of 8 × 10−11 cm3 /sec for Rb + Rb2 collisions. Mukaiyama and colleagues [109] reported an inelastic rate coefficient of 5.5 × 10−11 cm3 /sec for collisions of Na atoms with Na2 molecules created by the Feshbach resonance method, while Syassen and colleagues [110] obtained a value of 2 × 10−10 cm3 /sec for collisions of Rb atoms with Rb2 molecules. At large atom–diatom separations R, the interaction potential can be approximated by an effective potential, composed of a repulsive centrifugal term and the long-range interaction potential. For alkali-metal trimer systems, the atom–diatom long-range potential is a van der Waals interaction potential and behaves as −C6 /R6 . The classical capture model (also known as the Langevin model) has been shown to work quite well for these systems [100] at certain energy regimes. The Langevin model is based on the assumption that if the total energy of the system can overcome the effective barrier, then the elastic probability is zero and the quenching probability is one. Otherwise, the elastic probability is one and the quenching probability is zero. Classically, the barrier prevents the atom and the molecule from accessing the region of strong coupling with the other channels. Results obtained using the Langevin model for the K + K2 system are shown in Figure 3.17 along with the quantum results. As Figure 3.17 illustrates, three different regions can be distinguished for the K + K2 system. The first region, for collision energies below 10−6 K, corresponds to the Wigner regime where the threshold laws apply. In this regime, the quenching rate coefficient tends to a constant while the elastic component approaches zero as the square root of the collision energy. A quantum description is required and classical models cannot describe the dynamics in this regime. The second region, above a collision energy of 10−3 K, corresponds to the Langevin regime. Here, the difference between the full quantum calculation and the classical model is within 10%. Similar results have been found for 7 Li + 7 Li2 (v = 1, 2, j = 0) and 6 Li + 6 Li2 (v = 1, 2, j = 1) collisions by Cvitaš and colleagues [94], who showed that in the Langevin regime the rate coefficients become independent of the vibrational level of the molecule. The third region is intermediate between the Wigner and Langevin regimes, which typically corresponds to a quantum calculation with two or three partial waves. As shown in Figure 3.17b the height of each J-resolved effective barrier depicted by vertical lines for J = 1 to 5 corresponds approximately to the maximum of each J-resolved quenching rate coefficient. This indicates that the quenching process becomes significant when the barrier height is energetically overcome. Whenever the elastic and quenching rate coefficients have comparable values, the Langevin regime is reached, as shown in Figure 3.17. Since the quenching probability is almost equal to one and the elastic probability is almost equal to zero, elastic and quenching transition matrix elements are both equal to one and the cross-sections and rate coefficients for the two processes become similar. The above analysis shows that at energies above the onset of the Wigner regime the Langevin model can correctly describe the dynamics for barrierless atom–dimer alkali-metal collisions because of the strong inelastic couplings. However, the © 2009 by Taylor and Francis Group, LLC

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classical model is not suitable for tunneling-dominated reactions. Accurate quantum dynamics calculations are computationally demanding for heavier systems such as Cs + Cs2 , Rb + Rb2 , Rb + RbCs, or Cs + RbCs and the classical model may be used to qualitatively describe the dynamics of these systems at energies above the s-wave regime. The temperature dependence of the rate coefficients predicted by the Langevin model is given by the simple formula in atomic units: kLang (T ) = π

8kB T πμ

1/2

2C6 kB T

1/3 Γ

2 3

kLang (10–10 cm3/sec)

where μ is the reduced mass of the atom–diatom collisional system and C6 the dominant atom–diatom long-range coefficient. In Figure 3.18 we show the rate coefficients predicted by the Langevin model as functions of the temperature for different atom–diatom and diatom–diatom collisions involving Rb or Cs atoms. For Rb–RbCs, Cs–RbCs, and RbCs–RbCs collisions we used the corresponding C6 coefficients calculated by Hudson and colleagues [111]. In the absence of similar data for Cs + Cs2 and Rb + Rb2 collisions, we approximated the atom–dimer C6 coefficient by multiplying the corresponding atom–atom value by a factor of two. For Cs + Cs2 collisions we used the C6 coefficient for Cs–Cs interaction calculated by Amiot and colleagues [112] and for Rb + Rb2 collisions we adopted the C6 coefficient for Rb–Rb interaction reported by Derevianko and colleagues [113]. As shown in Figure 3.18 the Langevin model predicts rate coefficients within an order of

1 Rb + Rb2 Rb + Rb2, Wynar et al. (exp, 2000) Cs + Cs2 Cs + Cs2, Staanum et al. (exp, 2006) Cs + Cs2, Zahzam et al. (exp, 2006) Rb + RbCs Rb + RbCs, Hudson et al. (exp, 2008) Cs + RbCs Cs + RbCs, Hudson et al. (exp, 2008) RbCs + RbCs

0.1 0

100

200

300

400

500

600

T (mK)

FIGURE 3.18 Rate coefficients as functions of the temperature predicted by the Langevin model compared with experimental data for different collisional processes involving Rb or Cs atoms. The curves correspond to the Langevin results and the symbols denote the experimental results.



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magnitude of the experimental values for the different systems. The predicted results for the Cs + Cs2 , Rb + RbCs, and Cs + RbCs collisions agree with the corresponding experimental data of Refs. [106,107], and [111] within the reported error bars. For Rb + Rb2 collisions the Langevin model predicts results in very close agreement with the experimental result of Wynar and colleagues [108]. Thus, the Langevin model appears to be valid for describing collisional properties of these systems at μK temperatures. The model also confirms that cold and ultracold collisions of alkali-metal atoms and dimers are essentially characterized by the leading term in the long-range part of the interaction potential. 3.3.2.2

Role of PES in Determining Ultracold Reactions

Potential energy surfaces are the key ingredients that enter in quantum dynamics calculations. While the two-body terms are generally well known and accurate, the three-body terms are more difficult to compute with high precision because they are nonadditive and involve correlations between the three atoms. As a consequence, quantum dynamics calculations may suffer from the quality and degree of accuracy of the three-body terms. The sensitivity of the ultracold collision cross-sections to the details of the PES has been investigated for atom–dimer alkali-metal systems by Quéméner and colleagues [99] for Na + Na2 (v = 1 − 3, j = 0) and by Cvitaš and colleagues [97] for Li + Li2 (v = 0 − 3, j = 0). In these studies, a linear scaling factor λ has been included to tune the three-body interaction term in the PES and cross-sections were calculated as a function of λ. Other studies have compared the dynamics with and without the three-body terms for Na + Na2 (v = 1, j = 0) [98] and for Li + Li2 (v = 0 − 10, j = 0) [96] collisions. The dependence of the total quenching cross-sections on the three-body term for Na + Na2 (v = 1 − 3, j = 0) scattering [99] at a collision energy of 10−9 K is shown in Figure 3.19. The contribution of each final vibrational level is also plotted. For the v = 1 vibrational level, the cross-sections are very sensitive to the details of the three-body potential. A change of 1% in λ leads to a significant change of 75% in the cross-sections. At the minimum of the Na3 potential, a change of 1% corresponds approximately to 10 K. At present, ab initio calculations of PESs cannot be done with such accuracy. Thus it appears that it is difficult to get an accuracy better than two orders of magnitude in the cross-sections for Na + Na2 (v = 1, j = 0) collisions. However, for v = 2 and 3, the total cross-sections show a weaker dependence on the three-body term. The state-resolved cross-sections do not show strong dependence on the three-body term, compared to collisions of molecules in v = 1. Figure 3.20 shows the dependence of the state-to-state cross-sections on the three body term for the reaction Na + Na2 (v = 3, j = 0) → Na + Na2 (vf = 2, 1, 0, jf ) at a collision energy of 10−9 K. The oscillations in the cross-sections, when λ is modified, are due to Feshbach resonances which arise when a triatomic quasibound state (or a virtual state) crosses the energy threshold as the strength of the interaction potential is decreased (increased). When such a Feshbach resonance occurs, Cvitaš and colleagues [97] have argued, strong resonant peaks appear in the cross-sections if inelastic couplings are weak. In contrast, weak oscillations of one order of magnitude at best appear when inelastic couplings are strong, as in alkali-metal trimer © 2009 by Taylor and Francis Group, LLC

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Cold Molecules: Theory, Experiment, Applications 10–8 Na + Na2 (u = 1, j = 0)

10–9

10–10

Quenching cross-section (cm2)

10–8 Na + Na2 (u = 2, j = 0)

Total uf = 1 uf = 0

10–9

10–10 10–8 Total uf = 0

Na + Na2 (u = 3, j = 0)

uf = 1 uf = 2

10–9

10–10 0.98

0.99

1

1.01

1.02

l

FIGURE 3.19 Dependence of the quenching cross-sections on the three-body term at a collision energy of 10−9 K for Na + Na2 (v = 1 − 3, j = 0). (From Quéméner, G. et al., Eur. Phys. J. D, 30, 201, 2004. With permission.)

systems. A generalization of this effect has been discussed by Hutson [114]. Larger modifications of the three-body term affect the cross-sections significantly. This can also be seen in Figure 3.21 for Li + Li2 collisions [96] in which the three-body term is excluded from the dynamics calculation. The state-to-state cross-sections exhibit a stronger dependence on the three-body term. © 2009 by Taylor and Francis Group, LLC

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(a) u f = 2

8e–11 6e–11 1.02

4e–11

1.01

2e–11 0

1.00

l

2 on (cm ) Cross-secti

1e–10

16

12

0.99

8

4

jf

0

0.98

(b) u f = 1

2 on (cm )

1e–10

Cross-secti

8e–11 6e–11 1.02

4e–11

1.01

2e–11

l

1.00 0 24 20

16 12 jf

0.99 8

4

0

0.98

(c) u f = 0

Cross-secti

2 on (cm )

1e–10 8e–11 6e–11 4e–11

1.02 1.01

2e–11

l

1.00 0

28 24 20 16 12 8 jf

0.99 4

0

0.98

FIGURE 3.20 Variation of state-to-state cross-sections at a collision energy of 10−9 K for Na + Na2 (v = 3, j = 0) → Na + Na2 (vf = 2, 1, 0, jf ) as a function of the parameter λ. See text for details. (From Quéméner, G. et al., Eur. Phys. J. D, 30, 201, 2004.) © 2009 by Taylor and Francis Group, LLC

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Cold Molecules: Theory, Experiment, Applications 7Li

+ 7Li2 (u, j = 0)

10–9 Full calculation V3 = 0


10–10 Quenching 10–11

10–12 Elastic 10–13

10–14

0

1

2

3

4

5

6

7

8

9

10

Initial vibrational number, u

FIGURE 3.21 Dependence of the elastic and quenching rate coefficients on the vibrational excitation of the molecule for 7 Li + 7 Li2 (v = 0 − 10, j = 0) scattering at a collision energy of 10−9 K. The bold line corresponds to the full calculation and the thin line corresponds to the calculation without the three-body term of the PES. (From Quéméner, G. et al., Phys. Rev. A, 75, 050701(R), 2007. With permission.)

3.3.2.3

Relaxation of Vibrationally Excited Alkali Metal Dimers

Ultracold diatomic molecules produced in photoassociation or Feshbach resonance methods are usually created in excited vibrational states. Theoretical studies involving highly vibrationally excited molecules are challenging due to the large number of energetically open reaction channels present in the quantum calculation. This puts severe restriction on the calculations for vibrationally excited molecules at low temperatures. Ultracold quantum dynamics calculations for collisions of highly vibrationally excited molecules have been reported in 2007 by Quéméner and colleagues [96] for the Li + Li2 system. A three-atom problem is generally described by two different kinds of asymptotic channels. The first kind are the single-continuum states (SCSs). They correspond to configurations where the triatomic system dissociates asymptotically into an atom and a diatomic molecule. The second kind are the double-continuum states (DCSs). They correspond to configurations where the triatomic system dissociates asymptotically into three separated atoms. Because highly vibrationally excited molecular states lie close to and below the triatomic dissociation limit, they are also coupled with the DCSs, which lie above the dissociation limit. As a consequence, DCSs have to be included in quantum simulations of atom–diatom systems involving © 2009 by Taylor and Francis Group, LLC


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highly vibrationally excited diatomic molecules [96]. This dramatically increases the size and complexity of the quantum dynamics problem. Figure 3.21 shows the dependence of the rate coefficients for 7 Li + 7 Li2 (v = 0 − 10, j = 0) collisions on the vibrational quantum number at a collision energy of 10−9 K [96]. Quenching processes are more efficient than the elastic scattering for both high and low vibrational levels. Similar results have been found for 6 Li + 6 Li (v = 0 − 9, j = 1) system composed of fermionic atoms [96]. Quenching 2 rate coefficients show a slight decrease when the molecule is in its highest vibrational state. This is because the overlap of the wavefunctions of highly excited diatomic molecules with low-lying vibrational levels is very small, leading to small values for the interaction potential coupling matrix elements between the initial and final states [115]. These results do not directly apply to ultracold molecules composed of fermionic atoms created near a Feshbach resonance. In these experiments, the quenching processes are suppressed because the atom–atom scattering length is tuned to large and positive values, leading to an efficient Pauli blocking mechanism, as explained by Petrov and colleagues [116]. In the theoretical study of Li + Li2 collisions, the atom–atom scattering length is small and negative, and no suppression of the quenching processes is found for the molecule in the last vibrational state. Thus, the sign and magnitude of the Li–Li scattering length play a crucial role in the mechanism that suppresses quenching collisions. The quenching rates displayed in Figure 3.21 show an irregular dependence on the vibrational state of the molecule. This has already been seen for H + H2 collisions [28,117]. In contrast, experimental measurements of quenching rate coefficients for Cs + Cs2 collisions [106,107] do not show any dependence on the vibrational state of the molecule. The differences between these systems can be explained based on the following considerations. First, the theoretical study applies to spin-polarized atom–dimer alkali-metal systems, whereas it is not the case for the experiment. A full theoretical treatment should involve the electronic and nuclear spins of the alkali metal atoms as well as couplings between electronic surfaces of different spins. This is beyond the scope of quantumdynamics calculations at present and will involve significant new code development and massive computational efforts. Second, the dynamics of the two systems are different. The lithium system is lighter and has a more attractive three-body term than the cesium system [105]. The triatomic adiabatic potential energy curves are well separated for a light system such as Li3 and very dense for a heavy system such as Cs3 . The density of states strongly influences the nature of vibrational relaxation. The effect of the density of states has been illustrated in calculations for the Li + Li2 system, which exclude the three-body term. Removing the three-body term makes the energy levels more sparse. This results in a more regular and monotonic dependence of the rate coefficients on the vibrational levels v = 3 to 9 as illustrated in Figure 3.21. This conclusion is in agreement with a previous work of Bodo and colleagues [117] for the H + H2 system. When they increased the density of states of the triatomic system, they found no significant dependence on the vibrational states of the molecule. However, for v = 10, they obtained the same results with and without the three-body term. The three-body term thus appears to be less significant for collisions of molecules in high vibrational levels. Since the © 2009 by Taylor and Francis Group, LLC

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three-body term vanishes at large separations it is only important in the short-range interaction region while high vibrational states involve spatially extended molecules and sample the long-range part of the interaction potential. Therefore, three-body terms, which are the most difficult interaction energy terms to compute numerically for triatomic systems, may be neglected to a first approximation in dynamics of highly vibrationally excited molecules. In the experiments, rate coefficients have been measured for temperatures of 40 × 10−6 K [107] and 60 × 10−6 K [106] for Cs + Cs2 , while the theoretical rate coefficients for Li + Li2 were reported for a temperature of 10−9 K, which corresponds to the Wigner threshold regime. Thus, it is likely that the experimental measurements did not probe the limiting values of the rate coefficients in the Wigner regime. 3.3.2.4

Reactions of Heteronuclear and Isotopically Substituted Alkali-Metal Dimer Systems

Reactive collisions involving heteronuclear molecules are currently of great interest. A major goal of recent experiments has been to produce heteronuclear alkali-metal dimers in their electronic ground state. Examples include RbCs [111,118,119], NaCs [120,121], KRb [122,123], LiCs [124], and mixed isotopes 6 Li7 Li [125]. Quantum dynamics calculations for heteronuclear systems are more difficult. They are more interesting from a chemistry perspective because the reactive channels in collisions of heteronuclear molecules can be distinguished from inelastic channels. Cvitaš and colleagues [95,97] have explored the quantum dynamics of 7 Li + 6 Li7 Li(v = 0, j = 0), 7 Li + 6 Li (v = 0, j = 1), 6 Li + 7 Li (v = 0, j = 0), 2 2 and 6 Li + 6 Li7 Li(v = 0, j = 0) collisions. The J = 0 cross-sections for elastic scattering and chemical reactions in collisions of 7 Li with 6 Li7 Li(v = 0, j = 0) are presented in Figure 3.22a for a wide range of collision energies. The cross-sections are believed to be converged for energies up to 10−4 K. The reactive process, which leads to 6 Li + 7 Li2 (v = 0, j = 0) dominates over the elastic scattering at ultralow energies. However, the reactive process is less efficient than the vibrational relaxation process in homonuclear alkali systems discussed above. For instance, at a collision energy of 10−9 K, the ratio of the cross-sections for reactive and elastic collisions presented in Figure 3.22 is larger for the homonuclear system compared to the heteronuclear system. Cvitaš and colleagues attributed the smaller ratio to the presence of only one open channel in the heteronuclear reaction. The J = 1 elastic and reactive cross-sections for 7 Li + 6 Li2 (v = 0, j = 1) collisions are presented in Figure 3.22b. The reactive process leading to the formation of 6 Li7 Li molecules is slightly more efficient than elastic scattering in the Wigner regime. The other possible collision processes are 6 Li + 6 Li7 Li(v = 0, j = 0) and 6 Li + 7 Li (v = 0, j = 0). However, only elastic scattering occurs in these systems at 2 ultralow collisions energies. These results provide important implications for the experiments on the production of 6 Li7 Li in an ultracold mixture of 6 Li and 7 Li atoms. Cvitaš and colleagues proposed removing quickly the 7 Li atomic gas after the formation of the 6 Li7 Li molecules © 2009 by Taylor and Francis Group, LLC

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Inelastic Collisions and Chemical Reactions of Molecules (a) 10−10 Elastic: uf = 0, jf = 0 10−11

Inelastic: uf = 0, jf = 1

10−12

Reactive: uf = 0, jf = 2

Cross-section (cm2)

Reactive: uf = 0, jf = 0

10−13 10−14 10−12

10−15

10−13 10−14

10−16

10−15 10−16

10−17 10−18 −9 10

10−17

0

10−8

0.2

10−7

0.4

10−6

0.6

10−5

0.8

10−4

1

10−3

10−2

10−1

100

Collision energy (K)

(b) 10−8 Elastic: uf = 0, jf = 0 Reactive: uf = 0, jf = 0 Reactive: uf = 0, jf = 1 Reactive: uf = 0, jf = 2 Reactive: uf = 0 (total)

10−9 10−10

s (cm2)

10−11 10−12 10−13 10−14 10−15 10−16 10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

E(K)

FIGURE 3.22 Elastic and reactive s-wave cross-sections for 7 Li + 6 Li7 Li(v = 0, j = 0) (a) and 7 Li + 6 Li2 (v = 0, j = 1) (b). (From Cvitaš, M.T. et al., Phys. Rev. Lett., 94, 200402, 2005. With permission.)


108


in their ground state to prevent the destructive reactive processes. But keeping the 6 Li atomic gas is recommended for sympathetic cooling of the 6 Li7 Li molecules because only elastic collisions are possible. Removing the 6 Li atoms will leave the fermionic heteronuclear 6 Li7 Li molecules in the trap. In the absence of 6 Li atoms, evaporative cooling by collisions between fermionic 6 Li7 Li molecules will not be effective because s-wave collisions of identical fermionic dimers will be suppressed due to the Pauli exclusion principle. Table 3.4 provides a compilation of zero-temperature quenching rate coefficients for different alkali-metal trimer systems.

3.4

INELASTIC MOLECULE–MOLECULE COLLISIONS

Many of the studies of cold and ultracold molecules have focused on reactive and nonreactive scattering in atom–molecule collisions. At high densities of trapped molecules, molecule–molecule collisions need to be considered. The presence of rotational and vibrational degrees of freedom in both collision partners make molecule– molecule systems especially interesting. However, quantum dynamics calculations of molecule–molecule collisions are significantly more challenging. Most dynamical calculations of molecule–molecule scattering have relied on the rigid rotor approximation, and some recent studies have adopted the coupled-states approximation. Calculations performed at high collision energies have employed more approximate methods based on semiclassical techniques. Here, we present a brief account of recent dynamical calculations for the H2 + H2 system and discuss some ongoing work on full-dimensional quantum calculations of rovibrational transitions in H2 –H2 collisions. A brief discussion of hyperfine transitions in molecule–molecule systems is also provided for the illustrative examples of the O2 + O2 and OH + OH/OD + OD systems.

3.4.1

MOLECULES IN THE GROUND VIBRATIONAL STATE

The H2 + H2 system is the simplest neutral tetra-atomic system and it serves as a prototype for describing collisions between diatomic molecules. Although there have been a number of experimental and theoretical studies over the past several years on the H2 + H2 system (see [126] and references therein), only a few studies have explored the collision dynamics in the cold and ultracold regime. Forrey [127] presented a study of rotational transitions in H2 (v = 0, j = 2) + H2 (v = 0, j = 2) collisions at cold and ultracold collision energies using a rigid rotor model. He calculated the real and imaginary components of the complex scattering length for H2 (v = 0, j = 2, 4, 6, 8) + H2 (v = 0, j = j) collisions and showed that the imaginary parts decrease with increasing j, j . The imaginary parts of the scattering lengths are found to be small compared to the real parts, leading to small inelastic cross-sections. Maté and colleagues [128] reported an experimental study of the rate coefficient for the H2 (v = 0, j = 0) + H2 (v = 0, j = 0) → H2 (vf = 0, jf = 0) + H2 (vf = 0, jf = 2) transition at temperatures between 2 and 110 K. They found good agreement between the experimental results and quantum-dynamics calculations based on the rigid rotor model using a PES reported by Diep and Johnson © 2009 by Taylor and Francis Group, LLC

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TABLE 3.4 Zero-Temperature Quenching Rate Coefficients for Different Atom–Dimer Alkali Metal Systems Initial (v, j )

kT =0 (cm3 /sec)

Reference

39 K + 39 K

(v = 1, j = 0) (v = 1, j = 1) (v = 1, j = 0)

1.1 × 10−10 8.0 × 10−11 9.8 × 10−11

[100] [100] [100]

7 Li + 7 Li

(v = 1, j = 0) (v = 2, j = 0) (v = 3, j = 0) (v = 4, j = 0) (v = 5, j = 0) (v = 6, j = 0) (v = 7, j = 0) (v = 8, j = 0) (v = 9, j = 0) (v = 10, j = 0) (v = 1, j = 1) (v = 2, j = 1) (v = 3, j = 1)

2.1 × 10−11 1.5 × 10−11 4.4 × 10−11 3.0 × 10−11 1.2 × 10−10 8.9 × 10−11 3.3 × 10−10 1.6 × 10−10 2.9 × 10−10 2.4 × 10−11 3.3 × 10−11 2.0 × 10−11 5.1 × 10−11

[96] [96] [96] [96] [96] [96] [96] [96] [96] [96] [96] [96] [96]

(v = 1, (v = 2, (v = 1, (v = 2,

j = 0) j = 0) j = 1) j = 1)

5.6 × 10−10 9 × 10−11 2.8 × 10−10 4 × 10−10

[94] [94] [94] [94]

(v = 1, j = 0) (v = 2, j = 0) (v = 3, j = 0)

2.9 × 10−10 1.1 × 10−10 6.1 × 10−11

[99] [99] [99]

(v = 0, (v = 1, (v = 2, (v = 3, (v = 0, (v = 1, (v = 2, (v = 3, (v = 1, (v = 2, (v = 3, (v = 1, (v = 2, (v = 3,

4.1 × 10−12 2.1 × 10−10 4.4 × 10−10 4.0 × 10−10 4.4 × 10−11 5.2 × 10−10 2.6 × 10−10 3.0 × 10−10 2.6 × 10−10 3.5 × 10−10 4.4 × 10−10 2.8 × 10−10 5.3 × 10−10 4.6 × 10−10

[97] [97] [97] [97] [97] [97] [97] [97] [97] [97] [97] [97] [97] [97]

System 2 40 K + 40 K 2 41 K + 41 K 2 2

6 Li + 6 Li 2

7 Li + 7 Li

2

6 Li + 6 Li

2

23 Na + 23 Na

2

7 Li + 6 Li7 Li

7 Li + 6 Li

2

6 Li + 6 Li7 Li

6 Li + 7 Li

2

j = 0) j = 0) j = 0) j = 0) j = 1) j = 1) j = 1) j = 1) j = 0) j = 0) j = 0) j = 0) j = 0) j = 0)

(DJ) [129]. Montero and colleagues [130] have investigated cold inelastic collisions of n-H2 molecules in the ground vibrational state, using a 3:1 gas mixture of orthoand para-hydrogen. They obtained good agreement between the experimental data and the theoretical calculations based on the DJ PES. © 2009 by Taylor and Francis Group, LLC

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Using a quantum formalism based on the rigid rotor model, Lee and colleagues [126] have recently presented a comparative analysis of cross-sections for rotationally inelastic collisions between H2 molecules at low and ultralow energies. The elastic cross-sections for H2 (v = 0, j = 0) + H2 (v = 0, j = 0) collisions obtained in this study are presented in Figure 3.23 for two different PESs. The limiting value of the elastic scattering cross-section in the ultralow energy regime is 1.91 × 10−13 cm2 with the PES of Boothroyd, Martin, Keogh, and Peterson (BMKP) [131] and 1.74 × 10−13 cm2 with the DJ PES [129]. For low collision energies the dynamics is sensitive to higher-order anisotropic terms in the angular expansion of the interaction potential. A diatom–diatom scattering length of 5.88 Å was obtained for the DJ PES and 6.16 Å for the BMKP PES. Cross-sections for the quenching of rotationally excited H2 molecules in H2 (v = 0, j = 2) + H2 (v = 0, j = 0) and H2 (v = 0, j = 2) + H2 (v = 0, j = 2) collisions were presented at low and ultralow energies. Quantum calculations of rotational relaxation of CO in cold and ultracold collisions with H2 have recently been performed by Yang and colleagues [132,133]. They reported quenching rate coefficients for j = 1 to 3 of the CO molecule in collisions with both ortho- and para-H2 [132]. Due to the relatively deep van der Waals interaction potential for the H2 –CO system the cross-sections exhibit a number of narrow resonances for collision energies between 1.0 and 40.0 cm−1 . The signatures of these resonances are present in the temperature dependence of the rate coefficient, which shows broad oscillatory features in the temperature range of 10−2 to 50 K [132].

5 ¥ 103 This work: DJ PES

Elastic cross-section (10–16 cm2)

This work: BMKP PES 4 ¥ 103

3 ¥ 103

2 ¥ 103

1 ¥ 103

0 10–9

10–8

10–7

10–6

10–5 10–4 10–3 Collision energy (eV)

10–2

10–1

100

101

FIGURE 3.23 Elastic cross-section of H2 (v = 0, j = 0) + H2 (v = 0, j = 0) as a function of the collision energy. (From Lee, T.G. et al., J. Chem. Phys., 125, 114302, 2006. With permission.) © 2009 by Taylor and Francis Group, LLC

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Bohn and coworkers have reported extensive calculations of hyperfine transitions in ultracold molecule–molecule collisions. Avdeenkov and Bohn studied ultracold collisions between O2 molecules [134]. They used a quantum-mechanical formalism based on the rigid rotor model and included the electronic spin structure of the O2 molecules. Couplings between the rotational angular momentum and the electronic spin of the molecules lead to rotational fine structure. Avdeenkov and Bohn discussed the elastic and inelastic loss processes in 17 O2 + 17 O2 and 16 O2 + 16 O2 collisions. They found that for collision energies below 10−2 K, elastic collision cross-sections are larger than inelastic spin-flipping transitions. Based on relative magnitudes of elastic and inelastic spin-flipping cross-sections they concluded that 17 O molecules would be good candidates for evaporative cooling. In contrast, for 2 16 O + 16 O collisions, inelastic processes are more efficient than elastic collisions 2 2 so that 16 O2 molecules are prone to collisional trap loss. Avdeenkov and Bohn also studied ultracold collisions between OH [135,136] and OD radicals [137] in the presence of an applied electric field. They showed that elastic scattering is more efficient than inelastic processes for ultracold collisions between fermionic OD molecules [137], inhibiting state-changing collisions. The energy dependence of elastic and inelastic cross-sections for OH + OH and OD + OD collisions is illustrated in Figure 3.24 for an applied electric field of ε = 100 V/cm. While the elastic cross-sections approach finite values in the Wigner regime for the bosonic system of OH molecules and the fermionic system of OD molecules, the

10–11

s (cm2)

10–12

10–13

10–14

10–8

10–6

10–4 Energy (K)

10–2

100

FIGURE 3.24 Elastic and inelastic cross-sections for OD + OD (thick gray line) and OH + OH (thin black line) collisions for an applied electric field of ε = 100 V/cm. Solid and dashed lines refer to elastic and inelastic cross-sections, respectively. (From Avdeenkov, A.V. and Bohn, J.L., Phys. Rev. A, 71, 022706, 2005. With permission.)


112


inelastic cross-sections exhibit a totally different behavior. At ultralow energies when molecules interact with an electric field, s-wave scattering yields an inelastic cross−1/2 section diverging as Ecoll for bosonic systems, while p-wave scattering yields an 1/2 inelastic cross-section vanishing as Ecoll for fermionic systems. Thus, the inelastic processes are suppressed for the fermionic system. The differences are attributed to the bosonic/fermionic character of the molecules and to the applied electric field. In the absence of an applied electric field, the elastic cross-section of the fermionic 2 , faster than the inelastic cross-section, which decreases system would decrease as Ecoll 1/2 as Ecoll . Ticknor and Bohn [138] subsequently studied OH–OH collisions in the presence of a magnetic field. They showed that magnetic fields of several thousand gauss reduce inelastic collisions by about two orders of magnitude. Based on these results, they concluded that magnetic trapping may be favorable for OH molecules.

3.4.2 VIBRATIONALLY INELASTIC TRANSITIONS The theoretical studies presented above for molecule–molecule collisions refer to rigid rotor molecules. Pogrebnya and Clary [139,140] have investigated vibrational relaxation in collisions of hydrogen molecules using a full-dimensional quantumdynamics formalism with an angular momentum decoupling approximation in the body fixed frame. They studied H2 (v = 1, j) + H2 (v = 0, j ) collisions involving both para–para and ortho–ortho combinations for collision energies of 1 meV (11.6 K) to 1 eV (11,604 K). Time-dependent quantum-mechanical calculations based on the multiconfiguration time-dependent Hartree approach have been recently applied to study full-dimensional quantum dynamics of the H2 –H2 system [141,142]. These methods are suitable at high collision energies and have not been applied to cold and ultracold collisions. Very recently, a vibrational energy transfer mechanism for ultracold collisions between para-hydrogen molecules has been explored by Quéméner and colleagues [143] using a full-dimensional quantum theoretical formalism implemented in a new code written by Krems [144]. The quantum scattering calculations are based on the theory described by Arthurs and Dalgarno [145], Takayanagi [146], Green [147], and Alexander and DePristo [148]. The H2 molecules were initially in different quantum states characterized by the vibrational quantum number v and the rotational angular momentum j. A combination of two rovibrational states of H2 was referred to as a combined molecular state (CMS). A CMS, denoted as (vjv j ), represents a unique quantum state of the diatom–diatom system before or after a collision. The cross-sections for H2 (v = 1, j = 0) + H2 (v = 0, j = 0), H2 (v = 1, j = 2) + H2 (v = 0, j = 0), and H2 (v = 1, j = 0) + H2 (v = 0, j = 2) collisions are presented in Figure 3.25a. Using the CMS notation, this corresponds, respectively, to initial CMSs (1000), (1200), and (1002). The elastic cross-sections are almost independent of the different initial rovibrational states of the molecules, but the inelastic cross-sections are strongly dependent on the initial rotational and vibrational levels of the H2 molecules. At 10−6 K, the inelastic relaxation of H2 (v = 1, j = 0) is almost six orders of magnitude more efficient in collisions of H2 (v = 0, j = 2) than in collisions with H2 (v = 0, j = 0) and two orders of magnitude more efficient than in collisions of H2 (v = 1, j = 2) with H2 (v = 0, j = 0). At 25.45 K, which © 2009 by Taylor and Francis Group, LLC

113


(a)

H2(u = 1, j) + H2(u = 0, j ¢)

104 103

Elastic


102

Inelastic

101 100

j = 0, j¢ = 0 j = 2, j¢ = 0 j = 0, j¢ = 2

10–1

Inelastic

10–2 10–3 Inelastic

10–4 10–5 10–6

10–4

10–3 10–2 10–1 Collision energy (K)

100

101

102

02 10

H2(u = 1, j = 0) + H2(u = 0, j = 0)

12

103

00

H2(u = 1, j = 0) + H2(u = 0, j = 2) H2(u = 1, j = 2) + H2(u = 0, j = 0)

10

104

00

Ecoll = 10–6 K

105

102 101

00

10–3

10–5

02 08

04 06

00 06

00

00

10–4

04 00

02

04

02

10–2

02

04

06 02

04

10–1

08

00

100

02


(b)

10–5

10–6 Final combined molecular state

FIGURE 3.25 Elastic and inelastic cross-sections for the collisions H2 (v = 1, j = 0) + H2 (v = 0, j = 0), H2 (v = 1, j = 2) + H2 (v = 0, j = 0), and H2 (v = 1, j = 0) + H2 (v = 0, j = 2): total cross-sections (a); state-to-state cross-sections at 10−6 K (b). (From Quéméner, G. et al., Phys. Rev. A, 77, 030704 (R), 2008. With permission.)


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corresponds to the energy difference between the CMSs (1002) and (1200), the inelastic cross-sections become comparable for H2 (v = 1, j = 2) + H2 (v = 0, j = 0) and H2 (v = 1, j = 0) + H2 (v = 0, j = 2) collisions. The inelastic scattering depends on the type and the combination of rovibrational levels involved in the collision, whether it involves ground-state molecules, H2 (v = 0, j = 0), vibrationally excited molecules, H2 (v = 1, j = 0), rotationally excited molecules H2 (v = 0, j = 2), or rovibrationally excited molecules, H2 (v = 1, j = 2). H2 molecules are weakly interacting and characterized by shallow van der Waals interaction at large separations and the computed rate coefficients are not representative of strongly interacting alkali-metal dimer systems. For example, Mukaiyama and colleagues [109] reported an inelastic rate coefficient of 5.1 × 10−11 cm3 /sec for collisions between two weakly bound Na2 molecules created by the Feshbach resonance method. In a similar study Syassen and colleagues [110] reported a rate coefficient of 3 × 10−10 cm3 /sec for collisions between two Rb2 molecules. Zahzam and colleagues [107] estimated a rate coefficient of about 10−11 cm3 /sec for collisions between Cs2 molecules. Ferlaino and colleagues [149] measured a rate coefficient of 9 × 10−11 cm3 /sec for collisions between Cs2 molecules in the nonhalo regime. These last authors also measured the variation of the inelastic rate coefficient as a function of the atom–atom scattering length for collisions between tunable halo dimers of Cs2 . The large values of the inelastic rate coefficients for alkali-metal dimer systems are attributed to strong inelastic couplings and deeper potential energy wells. The state-to-state cross-sections for three rovibrational combinations of the H2 –H2 system are presented in Figure 3.25b. The magnitude of the inelastic cross-sections depends on the propensity of the diatom–diatom system to conserve the internal energy and the total rotational angular momentum of the colliding molecules [143]. The final state-to-state distribution in H2 (v = 1, j = 0) + H2 (v = 0, j = 0) collisions shows that there is no preferential population of rotational quantum number of either of the colliding molecules. The conservation of the total rotational angular momentum would entail a large change in the internal energy of the molecules. Thus, the purely vibrational transition (1000) → (0000) is not efficient because the energy gap is large. On the other hand, (near) conservation of the internal energy requires a large change in the total rotational angular momentum of the colliding molecules and the transition (1000) → (0800) is not dominant either. However, the state-to-state cross-sections for H2 (v = 1, j = 2) + H2 (v = 0, j = 0) collisions indicate that the transition (1200) → (1000) is more efficient than all the other transitions combined. For this transition, the total rotational angular momentum change and the internal energy transfer are both minimized, leading to a more efficient energy transfer process. The state-to-state cross-sections for H2 (v = 1, j = 0) + H2 (v = 0, j = 2) collisions present an interesting scenario in which the transition (1002) → (1200) is highly efficient and selective. In this case, the total rotational angular momentum is conserved and the internal energy is almost unchanged [the energy gap between (1200) and (1002) is only 25.45 K]. This creates a favorable situation leading to a near-resonant energy transfer process. This particular mechanism cannot occur in atom–diatom systems because simultaneous conservation of rotational angular momentum and internal energy of the molecule cannot occur. The mechanism is reminiscent of quasiresonant energy transfer in collisions of rotationally excited diatomic molecules © 2009 by Taylor and Francis Group, LLC


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TABLE 3.5 Zero-Temperature Inelastic Rate Coefficients for Different Molecule–Molecule Systems System

Initial (v, j , v , j )

kT =0 (cm3 /sec)

Reference

H2 + H2

(v = 1, j = 0, v = 0, j = 0) (v = 2, j = 0, v = 0, j = 0) (v = 3, j = 0, v = 0, j = 0) (v = 4, j = 0, v = 0, j = 0) (v = 5, j = 0, v = 0, j = 0) (v = 6, j = 0, v = 0, j = 0) (v = 1, j = 0, v = 1, j = 0) (v = 2, j = 0, v = 1, j = 0) (v = 3, j = 0, v = 1, j = 0) (v = 4, j = 0, v = 1, j = 0) (v = 2, j = 0, v = 2, j = 0)

8.9 × 10−18 3.9 × 10−17 1.2 × 10−16 1.8 × 10−16 6.9 × 10−16 2.9 × 10−15 2.7 × 10−16 1.3 × 10−14 1.6 × 10−14 9.0 × 10−15 6.1 × 10−16

[143] [143] [143] [143] [143] [143] [143] [143] [143] [143] [143]

H2 + H2

(v = 0, j = 2, v = 0, j = 0) (v = 1, j = 2, v = 0, j = 0) (v = 2, j = 2, v = 0, j = 0) (v = 3, j = 2, v = 0, j = 0) (v = 4, j = 2, v = 0, j = 0) (v = 1, j = 2, v = 1, j = 0) (v = 2, j = 2, v = 1, j = 0)

3.9 × 10−14 3.7 × 10−14 8.7 × 10−14 1.5 × 10−13 2.5 × 10−13 8.8 × 10−14 6.1 × 10−14

[143] [143] [143] [143] [143] [143] [143]

H2 + H2

(v (v (v (v (v

= 2) = 2) = 2) = 2) = 2)

5.6 × 10−12 3.4 × 10−12 2.7 × 10−12 2.1 × 10−12 5.2 × 10−12

[143] [143] [143] [143] [143]

H2 + CO

(v = 0, j = 0, v = 0, j = 1) (v = 0, j = 0, v = 0, j = 2) (v = 0, j = 0, v = 0, j = 3)

2.0 × 10−12 3.0 × 10−11 1.2 × 10−10

[132] [132] [132]

H2 + CO

(v = 0, j = 1, v = 0, j = 1) (v = 0, j = 1, v = 0, j = 2) (v = 0, j = 1, v = 0, j = 3)

1.2 × 10−11 4.0 × 10−11 8.5 × 10−11

[132] [132] [132]

= 1, = 2, = 3, = 4, = 2,

j j j j j

= 0, v = 0, v = 0, v = 0, v = 0, v

= 0, j = 0, j = 0, j = 0, j = 1, j

with atoms discussed previously [21,53,54,60], but with a purely quantum origin. The near-resonant process may be an important mechanism for collisional energy transfer in ultracold molecules formed by photoassociation of ultracold atoms and for chemical reactions producing identical molecules. Table 3.5 provides a compilation of zero-temperature rate coefficients for rotational and vibrational quenching in molecule–molecule systems.

3.5

SUMMARY AND OUTLOOK

In this chapter we have given an overview of recent theoretical studies of atom– molecule and molecule–molecule collisions at cold and ultracold temperatures. © 2009 by Taylor and Francis Group, LLC

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Although such systems have been extensively studied at higher collision energies over the last few decades, the new experimental breakthroughs in creating dense samples of cold and ultracold molecules have provided unprecedented opportunities to explore elastic, inelastic, and reactive collisions at temperatures close to absolute zero. These studies have revealed unique aspects of molecular collisions and energy transfer mechanisms that are otherwise not evident in thermal energy collisions. The long duration of collisions combined with large de Broglie wavelengths at cold and ultracold temperatures leads to interesting quantum effects. Calculations have shown that reactions with insurmountable energy barriers may still occur at temperatures close to absolute zero, and in certain cases, with appreciable rate coefficients. Such tunneling-dominated reactions have been the topic of many recent investigations and may soon be amenable to experimental investigation in the cold and ultracold regime. That the rates of these reactions can be enhanced by vibrational excitation of the molecule is an interesting scenario for experimental studies of ultracold chemical reactions. The studies of alkali-metal homonuclear and heteronuclear trimer systems have stimulated considerable experimental interest in investigating chemical reactivity at ultracold temperatures. The challenge for theory is to describe collisions involving highly vibrationally excited molecules. Recent theoretical calculations have indicated that the three-body interaction potential can be neglected for highly vibrationally excited molecules, offering significant savings in computational effort. Even so, heavier alkali-metal trimer systems pose a daunting computational challenge. The quantitative description of ultracold molecule–molecule collisions is another challenging topic. The recent progress on the H2 –H2 system will be difficult to implement for heavier systems due to the large number of rovibrational levels of the molecules. The study of H2 –H2 collisions has shown that, for certain combinations of rovibrational levels, the energy transfer may occur to specific final rovibrational states. In such cases, the calculations can use a much smaller basis set without compromising the accuracy. Currently there is substantial interest in controlling the collisional outcome using external electric and magnetic fields. While the idea of coherent control of molecular collisions and chemical reactivity has existed for a long time and some important progress has been achieved, the possibility of creating coherent and dense samples of molecules in specific quantum states has given further impetus to the field of controlled chemistry. We expect that the coming years will see a far greater activity in this direction driven by cold and ultracold molecules and also by the possibility of controlling chemical reactivity using external electric and magnetic fields. Electronically nonadiabatic effects in ultracold collisions is a largely unexplored area, which can be expected to attract much attention.

ACKNOWLEDGMENTS This work was supported by NSF grants PHY-0555565 (N.B.), AST-0607524 (N.B.), and by the Chemical Science, Geoscience and Bioscience Division of the Office of Basic Energy Science, Office of Science, U.S. Department of Energy (A.D.). © 2009 by Taylor and Francis Group, LLC


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Effects of External 4 Electromagnetic Fields on Collisions of Molecules at Low Temperatures Timur V. Tscherbul and Roman V. Krems CONTENTS 4.1 4.2

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collisions in Magnetic Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Zeeman Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Tunable Shape Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Tunable Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Collisions in Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Stark Relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Scattering of Molecular Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Electric-Field-Induced Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Collisions in Superimposed Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Effects of Electric Fields on Magnetic Feshbach Resonances . . . . . 4.4.2 Collisions near Tunable Avoided Crossings . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Effects of Field Orientations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Differential Scattering in Electromagnetic Fields . . . . . . . . . . . . . . . . . . 4.5 Collisions in Restricted Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Cold Controlled Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1

125 126 127 131 134 134 134 136 137 138 140 141 144 150 156 162 164 164

INTRODUCTION

As discussed in Chapters 9 through 17 of this book, diatomic molecules containing unpaired electrons have recently become of increased interest to researchers of lowtemperature gases, condensed-matter physics, precision spectroscopy, and quantum 125 © 2009 by Taylor and Francis Group, LLC

126


computation. The interaction of open-shell molecules with magnetic fields allows for magnetic trapping and thermal isolation of molecular ensembles at cold and ultracold temperatures (Chapter 13), novel methods to study fundamental symmetries of nature (Chapters 15 and 16) and mechanisms to control molecular collisions externally. These and other applications of open-shell molecules stimulated the design of laboratory superconducting magnets [1], which can now generate magnetic fields of up to 6 T. The experimental work on Stark deceleration of molecular beams described in Chapter 14 stimulated the development of techniques for the generation of tunable dc electric fields up to 200 kV/cm. As described in Chapter 12, dc electric fields can be used in combination with microwave laser radiation to engineer long-range interaction potentials between ultracold polar molecules confined by an optical lattice. Electric fields can also be used to confine ultracold molecules in electrostatic traps [2] and to generate cold molecules with tunable velocities [3]. Interactions with external electromagnetic fields thus play an important role in most studies of cold and ultracold molecules, both theoretical and experimental. The perturbations exerted by external electromagnetic fields on molecular energy levels are often larger than the kinetic energy of molecules at temperatures below 1 K. Collisions of molecules in a cold gas may therefore be significantly affected by the presence of external fields. The purpose of this chapter is to discuss the effects of external fields on dynamics of molecular collisions at cold and ultracold temperatures and outline the prospects for new discoveries in the research of molecule–field interactions at low temperatures. The experimental work on collision dynamics of low-temperature molecules in external fields may lead to the development of the research field of cold controlled chemistry [4] and we will particularly focus the discussion on mechanisms for external field control of intermolecular interactions. Most of the results presented are based on rigorous quantum-mechanical calculations. The quantum theory of molecular collisions in the presence of external fields is described in Chapter 1.

4.2

COLLISIONS IN MAGNETIC TRAPS

The development of the experimental techniques for confining cold molecules in magnetic [5], electrostatic [2], or laser-field traps [6] has opened up exciting possibilities for new research in molecular physics. For example, confining molecules in external field traps allows for spectroscopy measurements with enhanced interrogation time [7]. This has been used for measuring radiative lifetimes of molecular energy levels with unprecedented precision [8]. External field traps provide thermal isolation, which is necessary for cooling molecules to ultracold temperatures. Trapping fields modify the symmetry of intermolecular interactions and molecules in traps may exhibit interesting dynamics, not observable with thermal gases. Chapter 13 provides a detailed description of magnetic trapping experiments. Magnetic traps have been designed to confine and thermally isolate large ensembles of paramagnetic molecules (∼1013 particles) in a wide range of temperatures (∼1 μK to 700 mK). This makes magnetic traps a particularly useful instrument for experimental studies of molecular collisions at cold and ultracold temperatures. The versatility of magnetic traps is, however, limited by collision-induced Zeeman relaxation. © 2009 by Taylor and Francis Group, LLC

Effects of External Electromagnetic Fields on Collisions of Molecules

4.2.1

127

ZEEMAN RELAXATION

The rotational energy levels of open-shell molecules placed in a magnetic field split into manifolds of Zeeman sublevels. Figure 4.1 shows the Zeeman energy sublevels of a CaD molecule in the ground electronic state 2 Σ as functions of an applied magnetic field. CaH was the first molecule thermally isolated in a magnetic trap [5]. Magnetic trapping selects molecules in the state with the lowest-energy rotational angular momentum N = 0 and the electron spin projection aligned along the magnetic field axis. This state is shown by the dashed line in Figure 4.1. Zeeman energy levels with the positive derivative with respect to the magnetic field are often called “lowfield-seeking (LFS) states,” and Zeeman levels with the negative derivative are referred to as “high-field-seeking (HFS) states.” LFS states are collisionally unstable and may decay to the lowest-energy HFS state. This process leads to trap loss and heating of molecular samples in the buffer-gas-cooling experiments described in Chapter 13. Collision-induced Zeeman relaxation of magnetically trapped molecules has therefore been studied by several authors, both theoretically [9–16] and experimentally [18,19]. Translational energy thermalization of molecules in buffer-gas-cooling experiments is mediated by elastic collisions with helium atoms. In order to estimate the efficiency of buffer-gas-cooling experiments, it is necessary to understand the mechanism of Zeeman relaxation and evaluate the cross-sections for elastic scattering and 20

15

Energy (cm–1)

N=2

10

5

N=1

0 N=0 0

1

2

3 4 5 Magnetic field (T)

6

7

FIGURE 4.1 Zeeman energy levels of the CaD(2 Σ) molecule. The dashed line indicates the energy of the molecule confined in a magnetic trap.


128


Zeeman relaxation in collisions of molecules with He atoms in the cold temperature regime (T ∼ 0.5 to 1 K). Figure 4.2 presents the ratio of cross-sections for elastic scattering and Zeeman relaxation in collisions of NH molecules in the ground rotational state with 3 He atoms as a function of the collision energy and the magnetic field strength. NH molecules are currently being studied in the buffer-gas-cooling experiments [19]. The ground electronic state of NH is 3 Σ and the structure of molecular energy levels for NH in a magnetic field is slightly more complicated than the illustration in Figure 4.1. The ground rotational state of the NH molecule is characterized by the total angular momentum value J = 1. The total angular momentum includes the electron spin and the rotational angular momentum of the molecule. The J = 1 state gives rise to three Zeeman sublevels corresponding to the projections MJ = 0, −1 and +1 of J on the magnetic field axis. The MJ = +1 sublevel is the LFS state and the buffer-gas-cooling experiments confine molecules in this state in a magnetic trap. Zeeman relaxation involves transitions from the MJ = +1 state to the MJ = 0 and MJ = −1 states. The results presented in Figure 4.2 represent the sum of the two transitions. Figure 4.2 illustrates that the probability of Zeman relaxation in NH–He collisions is very small and extremely sensitive to the magnitude of an external magnetic field at low collision energies. The efficiency of Zeeman relaxation in collisions of molecules in the Σ electronic states is determined by the magnitude of the spin–rotation and spin–spin interactions giving rise to the molecular fine-structure [13]. The results of Figure 4.2 indicate that the fine-structure interactions in the NH molecule are weak.

1010 B=0 Elastic-to-inelastic ratio

B = 0.01 T 108

B=1T

106 B=2T

B=3T 104 10–4

10–3

10–2

10–1

100

101

Collision energy (cm–1)

FIGURE 4.2 Ratio of cross-sections for elastic scattering and Zeeman relaxation in collisions of rotationally ground-state NH(3 Σ) molecules with 3 He atoms (1 K = 0.695 cm−1 ). The curves are labeled by the magnetic field magnitude. (Adapted from Cybulski, H. et al., J. Chem. Phys., 122, 094307, 2005. With permission.)


Effects of External Electromagnetic Fields on Collisions of Molecules

129

It should be noted that the effects of the spin–rotation and spin–spin interactions in most other stable radicals in the Σ electronic states are much more significant and Zeeman relaxation in collisions of other diatomic molecules with He atoms should be more efficient. The He atom is a weak perturber and the interaction of NH molecules with He atoms is weak. Angular momentum transfer in molecular collisions is induced by the anisotropy of the interaction potential between the collision partners. Therefore, Zeeman relaxation in collisions of molecules with other atoms and in molecule– molecule collisions should be expected to be more efficient. Figure 4.2 also demonstrates that the relative probability of Zeeman relaxation is significantly enhanced in a narrow interval of collision energies around 1 K, which indicates the presence of a scattering resonance. The influence of scattering resonances on inelastic collisions has been discussed in Chapter 3 and will be further examined in the following sections. Volpi and Bohn were the first to discover that Zeeman relaxation in collisions of ultracold molecules is extremely sensitive to the magnitude of an external magnetic field [12]. When molecules react, the rotational motion of the collision complex gives rise to a centrifugal force that may suppress collisions at low temperatures. As explained in Chapter 1, the total scattering wavefunction of the collision complex can be decomposed into contributions from different angular momenta of the rotational motion called partial waves. Collisions at ultracold temperatures ( 100. This suggests that the collision cross-sections of molecular dipoles in the presence of high electric fields can be evaluated for any collision system based on the results of a quantum calculation for a single model system.

4.3.3

ELECTRIC-FIELD-INDUCED RESONANCES

Electric fields can be used to tune shape and Feshbach scattering resonances through a variety of mechanisms. The Stark effect may lead to tunable shape resonances due to trapping behind the centrifugal barriers in the outgoing Stark states. Such resonances would be analogous to the shape resonance in the outgoing Zeeman state illustrated by Figure 4.5. As mentioned above, however, the couplings between the Stark states are usually much larger than the matrix elements coupling different Zeeman states in Σ-state molecules. The resonances in the lower-energy Stark states may therefore © 2009 by Taylor and Francis Group, LLC

138


NaCs

102

T

101

x1/2

CsRb

4

100

1010

Rydberg

RbK 10–1

109 1019

10–2 –2 10

10–1

100

101

102 x

103

1020

104

1021

105

106

FIGURE 4.9 Collision T -matrix elements for scattering of molecules in the absolute ground state as functions of ξ. (Adapted from Ticknor, C., Phys. Rev. Lett., 100, 133202, 2008. With permission.)

have a significantly different effect on the dynamics of molecular collisions than the resonances in the lower-energy Zeeman states. For example, the field-dependent resonance of Figure 4.5 does not modify the cross-section for elastic scattering of molecules in the LFS Zeeman state. Similar resonances in the HFS Stark states may influence the scattering wavefunction in the elastic channel of LFS molecules. Electric fields shift bound energy levels of collision complexes involving polar molecules with respect to the collision threshold and may lead to diagrams of molecular energy levels similar to those discussed in Chapter 1. The Stark effect may therefore give rise to zeroenergy Feshbach resonances, similar to magnetic-field-induced Feshbach resonances in collisions of paramagnetic molecules. Unlike magnetic fields, electric fields couple different partial waves of the collision complex and states of different parity. These couplings may have a dramatic effect on differential scattering of polar molecules (see Section 4.4 below) and modify shape scattering resonances. Figure 4.10 demonstrates the effect of electric fields on the probabilities of the chemical reaction of LiF molecules with H atoms and the crosssections for vibrational relaxation of LiF molecules in the first vibrationally excited state in collisions with H atoms at low collision energies. The presence of electric fields apparently results in couplings that shift the position of the shape resonance, which suppresses the cross-sections for the chemical reaction and vibrationally inelastic processes at collision energies near 1 K [28].

4.4

COLLISIONS IN SUPERIMPOSED ELECTRIC AND MAGNETIC FIELDS

Superimposed electric and magnetic fields can be used to control microscopic interactions between atoms and molecules in a wider range of field parameters than © 2009 by Taylor and Francis Group, LLC

Effects of External Electromagnetic Fields on Collisions of Molecules Chemical reaction

(a)

Cross-section (Å2)

139

10–2

10–4

10–6 10–3 (b)

10–2

10–1 Vibrational relaxation

102

No electric field E = 200 kV/cm E = 50 kV/cm E = 150 kV/cm

101 Cross-section (Å2)

100

100 10–1 10–2 10–3 10–3

10–2

10–1 Collision energy (cm–1)

100

FIGURE 4.10 (a) Total cross-sections for the chemical reaction of LiF(v = 1, N = 0) molecules with H atoms as functions of the collision energy at zero electric field (circles) and electric fields of 100 kV/cm (triangles), 150 kV/cm (squares), and 200 kV/cm (diamonds). (b) Cross-sections for nonreactive vibrational relaxation of LiF(v = 1, N = 0) molecules in collisions with H atoms. (Adapted from Tscherbul, T.V. and Krems, R.V., J. Chem. Phys., 129, 034112, 2008. With permission.)

may be possible with a single field. For example, electric fields may shift the positions of magnetic Feshbach resonances or broaden the resonances. The following sections discuss the effects of combined electric and magnetic fields on collision dynamics of atoms and molecules at cold and ultracold temperatures. In particular, Section 4.4.1 describes how electric fields may modify magnetic Feshbach resonances or induce new resonances in ultracold collisions by coupling scattering states with different orbital angular momenta. Experimental studies of cold molecules in external field traps may be particularly interesting as a novel approach to explore the effects of combined electric and magnetic fields on molecular collisions. When molecules are trapped, their magnetic or electric dipole moments are aligned by the confining field © 2009 by Taylor and Francis Group, LLC

140


(cf. Chapter 13). The alignment of molecular dipole moments restricts the symmetry of the interaction potential between the collision partners. Superimposed external fields can then be used to induce reorientation of trapped molecules, leading to nonadiabatic transitions between different electronic states. This may modify the mechanisms of inelastic scattering and chemical reactions at low temperatures and result in enhancement or suppression of the collision rates. Section 4.4.2 demonstrates that a combination of electric and magnetic fields can be used to create magnetically aligned molecules with the orientation of the magnetic dipole moments that is extremely sensitive to collisional perturbations. Section 4.4.3 presents a few examples of molecular collision processes that can be sensitive not only to the magnitude but also to the relative orientation of superimposed electric and magnetic fields. Section 4.4.4 describes the effects of electromagnetic fields on angle-resolved differential scattering of cold polar molecules.

4.4.1

EFFECTS OF ELECTRIC FIELDS ON MAGNETIC FESHBACH RESONANCES

The duration of an ultracold collision is very long. The scattering dynamics of ultracold atoms and molecules is therefore often sensitive to weak interactions with external fields. The results of Refs. [29] and [30], for example, show that collisions of ultracold atoms can be controlled by dc electric fields. When two different atoms collide, they form a heteronuclear collision complex with an instantaneous dipole moment that can interact with external electric fields. The dipole moment function of the collision complex is typically peaked around the equilibrium distance of the diatomic molecule in the vibrationally ground state and quickly decreases as the atoms separate. Only a small part of the scattering wavefunction samples the interatomic distances, where the dipole moment function is significant and the oscillatory structure of the scattering wavefunction diminishes the interaction of the collision complex with external electric fields. Collisions of atoms are therefore usually insensitive to dc electric fields of moderate strength ( 0 is summarized in the scattering wavefunction for R → ∞ by a unitary S-matrix. Only the lowest few partial waves can contribute to cold collisions, and in the limit E → 0, only s-wave channels with = 0 have non-negligible collision cross-sections. Using the complex scattering length a − ib to represent the s-wave S-matrix element Sαα = exp [−2ik(a − ib)] in the limit E → 0, the contribution to the elastic scattering cross-section from s-wave collisions in channel α is σcl = lim g E→0

π 2 2 2 |1 | = 4gπ a + b , − S αα k2

(6.1)

√ where k = 2μE is the relative collision momentum in the center-of-mass frame for the atom pair with reduced mass μ. The rate coefficient Kloss = σloss v for E → 0 s-wave inelastic collisions that remove atoms from channel α is Kloss = lim g E→0

h π 1 − |Sαα |2 = 2g b, μk μ

(6.2)

where v = k/μ is the relative collision velocity. The symmetry factor g is equal to 1 when the atoms are bosons or fermions that are not in identical states, g = 2 or g = 1 respectively for two bosons in identical states in a normal thermal gas or a Bose– Einstein condensate, and g = 0 for two fermions in identical states. If there are no exoergic inelastic channels present, then b = 0 and only elastic collisions are possible. The Schrödinger equation also determines the bound states with discrete energies Ei < 0. While the conventional picture of molecules counts the bound states by vibrational quantum number v = 0, 1, . . . from the lowest energy ground-state up, it is more helpful for the present discussion to count the near-threshold levels from the E = 0 dissociation limit down by quantum numbers i = −1, −2, . . . . In the special © 2009 by Taylor and Francis Group, LLC

223

Molecular States Near a Collision Threshold

case where a → +∞, the energy of the last bound s-wave state of the system with i = −1 depends only on a and μ and takes on the following “universal” form: E−1 = −

2 as a → +∞. 2μa2

(6.3)

Section 6.2 describes the bound and scattering properties of a single potential with a van der Waals long range form. Section 6.3 extends the treatment to multiple states and scattering resonances. Sections 6.4 and 6.5 respectively discuss the properties of magnetically and optically tunable molecular resonance states.

6.2

PROPERTIES FOR A SINGLE POTENTIAL

In this section let us ignore any complex internal atomic structure and first consider two atoms A and B that interact by a single adiabatic Born–Oppenheimer interaction potential V (R), illustrated schematically in Figure 6.1. The wavefunction for the system is |α|ψ /R, where |α represents the electronic and rotational degrees of freedom, and the wavefunction for relative motion is found from the radial Schrödinger equation 2 ( + 1) 2 d2 ψ ψ = Eψ . + V (R) + (6.4) − 2μ dR2 2μR2 Solving Equation 6.4 gives the spectrum of bound molecular states ψi with energy 2 /(2μ) < 0 and the scattering states ψ (E) with collision kinetic energy Ei = −2 ki Short-range

V(R) E=0

Long-range

AB

Separated

A+B V ~ –Cn / Rn

1 – 10 eV

kB(1 mK) = h(21 kHz) = 0.86 neV

a

Rbond R

FIGURE 6.1 Schematic figure of the potential energy curve V (R) as a function of the separation R between two atoms A and B. The horizontal lines labeled AB indicate a spectrum of molecular bound states leading up to the molecular dissociation limit at E = 0, indicated by the dashed line. The long-range potential varies as −Cn /Rn . See text for a definition of Rbond and a¯ . © 2009 by Taylor and Francis Group, LLC

224


E = 2 k 2 /(2μ) > 0, where ki and k have units of (length)−1 . As R → ∞, the bound states decay as e−ki R and the scattering states approach ψ (E) → c sin(kR − π/2 + η )/k 1/2 .

(6.5)

Bound states are normalized to unity, |ψi |ψj |2 = δij δ . We choose the normal ization constant c = 2μ/2 π so that scattering states are normalized per unit energy, ψ (E)|ψ (E ) = δ(E − E )δ . Thus, the energy density of states is included in the wavefunction when taking matrix elements involving scattering states. The long-range potential between the two atoms varies as −Cn /Rn . We are especially interested in the case of n = 6 for the van der Waals interaction between two neutral atoms. This is the lead term in the long-range expansion of the potential in inverse powers of R that applies to many atoms that are used in ultracold exper iments. This potential has a characteristic length scale of Rvdw = 4 2μC6 /2 /2, which depends only on the values of μ and C6 [2]. Values of C6 are tabulated by Derevianko [12] for alkali–metal species and by Porsev and Derevianko [13] for alkaline–earth species. We prefer to use a closely related van der Waals length introduced by Gribakin and Flambaum [14]. a¯ = 4π/Γ(1/4)2 Rvdw = 0.955978 . . . Rvdw ,

(6.6)

where Γ(x) is the Gamma function. This length defines a corresponding energy scale E¯ = 2 /(2μ¯a2 ). The parameters a¯ and E¯ occur frequently in formulas based on the van der Waals potential. The wavefunction approaches its asymptotic form when R a¯ and is strongly influenced by the potential when R a¯ . Table 6.1 gives the values of a¯ and E¯ for several species used in ultracold experiments. Samples of cold atoms can be prepared with kinetic temperatures on the order of nK to mK. The energy associated with temperature T is kB T where kB is the Boltzmann constant. For example, at T = 1 μK, kB T = 0.86 neV and kB T /h = 21 kHz. This ultracold energy scale is 9 to 10 orders of magnitude smaller than the energy scale of 1 to 10 eV associated with ground- or excited-state interaction energies when a

TABLE 6.1 Characteristic van der Waals Scales a¯ and E¯ for Several Atomic Species Species

Mass (amu)

C6 (au)

a¯ (a0 )

E¯ /h (MHz)

E¯ /kB (mK)

6 Li

6.015122 22.989768 39.963999 86.909187 87.905616 132.905429 173.938862

1393 1556 3897 4691 3170 6860 1932

29.88 42.95 62.04 78.92 71.76 96.51 75.20

671.9 85.10 23.46 6.668 7.974 2.916 3.670

32.25 4.084 1.126 0.3200 0.3827 0.1399 0.1761

23 Na 40 K 87 Rb 88 Sr 133 Cs 174Yb

Note: 1 amu = 1/12 mass of a 12 C atom, 1 au = 1 Eh a06 where Eh is a hartree and 1 a0 = 0.0529177 nm.


225


molecule is formed at small interatomic separation Rbond on the order of a chemical bond length. In a cold collision, the initially separated atoms have very low collision energy E = 2 k 2 /(2μ) ≈ 0 and very long de Broglie wavelength 2π/k. The atoms come together from large distance R and are accelerated by the interatomic potential V (R), so that when they reach distances on the order of Rbond they have very high kinetic energy on the order of |V (Rbond )|. The local de Broglie wavelength 2π/k(R, E) √ in the short-range classical part of the potential, where k(R, E) = 2μ(E − V (R))/, is orders of magnitude smaller than the separated atom de Broglie wavelength and is nearly independent of the value of E, which is close to 0. This separation of scales is illustrated in Figure 6.2, which shows examples of s-wavefunctions at a collision energy E/kB = 1 μK, where dividing E by kB allows us to express energy in temperature units. This example uses three isotopic combinations of pairs of Yb atoms, which have a spinless 1 S0 electronic configuration and a single ground-state electronic Born–Oppenheimer potential V (R). The species Yb makes a good example case to illustrate the principles in this section, because it has seven stable isotopes and 28 different atom pairs of different isotopic composition for which the threshold properties have been worked out [15]. All combinations have the same V (R) but different reduced masses. This mass-scaling approximation, which ignores very small mass-dependent corrections to the potential, is normally quite good except for very light species such as Li. Figure 6.2 shows that the three examples have similar phase-shifted sine waves with a common long de Broglie wavelength of 2π/k √ = 6300a . For small R where kR 1 the sine function vanishes as c sin k(R − a)/ k→ 0 √ c k(R − a). The actual wavefunction oscillates rapidly at small R due to the influence of the potential. Because the asymptotic form for kR 1 varies as k 1/2 as k → 0,

y (R)

30 20

174–174 171–171 170–173

10

E/kB = 1 mK

0 –10 4

a

–20 –30 0 –40

0 0

100 2000

200 4000

6000

8000

R [a0]

FIGURE 6.2 Radial wavefunction ψ0 (R) for = 0 at E/kb = 1 μK for the pairs 174Yb– 174Yb (solid), 171Yb–171Yb (dashed), and 170Yb–173Yb (dot-dashed), which have respective scattering lengths of 105a0 , −3a0 , and −81a0 [15]. The inset shows an expanded view of the wavefunction on a smaller length scale on the order of a¯ , the characteristic length of the van der Waals potential. The 174Yb–174Yb case shows the oscillations that develop when R < a¯ .


226


the short-range oscillating part also has an amplitude proportional to k 1/2 in order to connect smoothly to the asymptotic form as k → 0. This property ensures that the threshold matrix elements that characterize Feshbach resonances and s-wave inelastic scattering are proportional to k 1/2 . Figure 6.3 illustrates more clearly the nature of threshold short-range scattering and bound-state wavefunctions. When given an appropriate short-range normalization, near-threshold scattering and bound-state wavefunctions have a common amplitude and phase in the region of R small compared to the range a¯ of the long-range potential. Although this can be put on a rigorous quantitative ground within the framework of quantum defect theory [8], it is easy to show using the familiar JWKB approximation [2,3,7]. We can always write the wavefunction in phase-amplitude form ψ (R, E) = α (R, E) sin β (R, E) and transform the Schrödinger equation 6.4 into a set of equations for α and β . The asymptotic ψ (R, E) in Equation 6.5 clearly corresponds to this form with α → c/k 1/2 as R → ∞. Another familiar form is the JWKB semiclassical wavefunction ψJWKB (R, E), for which (R, E) = c/k (R, E)1/2 αJWKB R π (R, E) = k (R , E)dR + , βJWKB 4 Rt

(6.7) (6.8)

where Rt is the inner classical turning point of the potential.

2

1

y (R)

0

–1 i=–1 bound state E/kB=1 mK scattering

–2

–3

a

10

100 R (a0)

FIGURE 6.3 wavefunctions for the last s-wave i = −1 bound state (solid line) with E−1,0 /h = −10.6 MHz and for the s-wave scattering state (dashed line) for E/h = 0.02 MHz (E/kB = 1 μK) for two 174Yb atoms. Both wavefunctions are given a common JWKB normalization at small R a¯ and are nearly indistinguishable for R < a¯ . The potential supports N = 72 bound states, and the wavefunction for this i = −1 and v = 71 level has N − 1 = 71 nodes.



227

When the collision energy E is sufficiently large, so there are no threshold effects, the JWKB approximation is an excellent approximation at all R, and the form of αJWKB (R, E) in Equation 6.7 applies at all R, transforming into the correct quantum limit as R → ∞. On the other hand, the JWKB approximation fails for s-waves with very low collision energy. This failure occurs in a region of R near a¯ and for collision energies E on the order of E¯ or less. The consequence is that the JWKB wavefunction, with the normalization in Equation 6.7, is related to the actual wavefunction, with the asymptotic form in Equation 6.5, by a multiplicative factor C (E), so that as E → 0 (R, 0). ψ (R, E) = C (E)−1 ψJWKB

(6.9)

As k → 0 for a van der Waals potential varying as 1/R6 , the s-wave threshold form is C0 (E)−2 = k a¯ [1 + (r − 1)2 ], where r = a/¯a is the dimensionless scattering length in units of a¯ [8]. Equation 6.9 gives an excellent approximation for the threshold ¯ C0 (E)−1 approaches ψ0 (R, E) for R < a¯ and k < 1/a. At high energy, when E E, unity and the JWKB approximation for ψ0 (R, E) applies at all R. The unit normalized bound-state wavefunction ψi (R) can be converted to an “energy normalized” form by multiplying by |∂i/∂Ei |1/2 , where −∂i/∂Ei > 0 is the energy density of states. Away from threshold, this is just the inverse of the mean spacing between levels, whereas for s-wave levels near threshold for a van der Waals ¯ −1 as k−1,0 = 1/a → 0 [8]. The relation of ψi to the potential, ∂i/∂Ei0 → r/(2πE) energy-normalized JWKB form in the classically allowed region of the potential is ∂i ψi (R, Ei ) = ∂E

−1/2 ψJWKB (R, Ei ).

(6.10)

i Ei

Figure 6.3 plots C0 (E)ψ0 (R, E) ≈ ψJWKB (R, 0) for the scattering state and 0 JWKB 1/2 (R, 0) for the i = −1 bound-state. Thus the near|∂i/∂Ei0 | ψi0 (R, Ei0 ) ≈ ψ0 threshold bound and scattering wavefunctions, when given a common short-range normalization, are nearly identical and are well approximated by ψJWKB (R, 0) in the 0 region R < a¯ . For R > a¯ the wavefunctions begin to take on their asymptotic form as R → ∞. The shape of the wavefunction at very small R on the order of Rbond is usually independent of E for ranges of E/kB on the order of many K. The short-range shape is even independent of for small , because the rotational energy is very small compared to typical values of V (Rbond ). However, the amplitudes of the wavefunctions depend strongly on the whole potential, which determines a, and are analytically related to the form of the long range potential. The separation of scales for R > a¯ and R < a¯ is a key feature of ultracold physics that enables much physical insight as well as practical approximations to be developed about molecular bound and quasibound states and collisions. Given that C6 , μ, and the s-wave scattering length a are known, the Schrödinger equation 6.4 can be integrated inward using the form of Equation 6.5 as k → 0 as a boundary condition, thus giving the wavefunction and nodal pattern for R < a¯ as E → 0. Assume that it is possible to pick some R = Rm such that Rbond Rm a¯ and V (Rm ) is well-represented by its van der Waals form. Then the log of the derivative of the wavefunction at Rm , which also can be calculated, provides an inner boundary condition, independent of E over © 2009 by Taylor and Francis Group, LLC

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a wide range of E, for matching the wavefunction at E propagated from large R. All that is needed to do this is to know a, μ, and the long-range potential. Thus, it is readily seen that all of the near-threshold bound and scattering states, even those for > 0, can be calculated to a very good approximation for R > Rm once C6 , μ, and a are known. ¯ for up to 5 Figure 6.4 shows the spectrum of bound states Ei , in units of E, for two cases of scattering length, based on the van der Waals quantum defect theory of Gao [9,10]. Panel (a) shows the case of a = ±∞, where there is a bound state at E = 0. The locations of the bound states for a = ±∞ define the boundaries of the “bins” in which, for any a, there will be one and only one s-wave bound state, ¯ The panel also for example, −36.1E¯ < E−1,0 < 0 and −249E¯ < E−2,0 < −36.1E. shows the rotational progressions for each level as increases. The a = ±∞ van der Waals case also follows a “rule of 4,” where partial waves = 4, 8, . . . also have a bound state at E = 0. Panel (b) shows how the spectrum changes when a = a¯ , for which the there is a d-wave level at E = 0. Similar spectra can be calculated for any a. Gribakin and Flambaum [14] showed that the near-threshold s-wave bound state for a van der Waals potential in the limit a a¯ is modified from the universal form in Equation 6.3 as E−1 = −

2 . 2μ(a − a¯ )2

(6.11)

This approaches the universality limit when a √ a¯ , in which case the s-wave wavefunction takes on the universal form ψ0 (R, E) = 2/ae−R/a . Such an exotic bound state, known as a “halo molecule,” exists primarily in the nonclassical domain beyond

a = +•

a=a

(a)

(b) 0

0

–50

–50

–100

s

p

d

f

g

h

–100

E/E –150

–150

–200

–200

–250

–250

–300

–300

s

p

d

f

g

h

FIGURE 6.4 Dimensionless bound-state energies Ei /E¯ for partial waves = 0 . . . 5 (s, p, d, f , g, h). (a) The case where a = ±∞; (b) The case for a = a¯ .



229

the outer classical turning point of the long-range potential with an expectation value of R of a/2, which grows without bound as a → +∞ [1]. Bound-state and scattering properties are closely related. It is instructive to imagine that there is some control parameter λ that can be varied to make the scattering length vary over its whole range from +∞ and −∞, changing the corresponding boundstate spectrum. One way to do this would be to vary the reduced mass. Of course, this is not physically possible. However, there are elements with many isotopes, so that a wide range of discrete reduced masses are possible. An excellent physical system to illustrate this is the Ytterbium atom, used in the examples of Figures 6.2 and 6.3. The stable isotopes with masses 168, 170, 172, 174, and 176 are all spinless bosons and the 171 and 173 isotopes are fermions with spin 1/2 and 5/2, respectively. Yb atoms can be cooled into the μK domain and all isotopes, including the fermionic ones in different spin states, have s-wave interactions. The locations of several = 0 and 2 threshold bound states of different isotopic combinations of Yb atoms in Yb2 dimer molecules have been measured, and the long-range potential parameters and scattering lengths determined [15]. Figure 6.5 shows the s-wave scattering length and bound state binding energies versus the continuous control parameter λ = 2μ. Physically, there are 28 discrete values between λ =168 and 176. The scattering length has a singularity, and a new bound state occurs with increasing λ, at λ = 167.3, 172.0, and 177.0. The range between 167.3 and 172 corresponds to exactly N = 71 bound states in the model potential used. Near λ = 167.3 the last s-wave bound state energy E−1,0 → 0 as −2 /(2μa2 ) as a → +∞. The binding energy |E−1,0 | gets larger as λ increases and a decreases, so that for a van der Waals potential E−1,0 approaches the lower edge of its “bin” at −36.1E¯ as a → −∞. As λ increases beyond 172.0, the i = −1 level becomes the i = −2 level as a new i = −1 “last” bound state appears in the spectrum. The variation of scattering length with 2μ is given by a remarkably simple formula. While semiclassical theory breaks down at threshold, Gribakin and Flambaum [14] showed that the correct quantum-mechanical relation between a and the potential is π a = a¯ 1 − tan Φ − , 8 where

Φ=

∞

Rt

−2μV (R)/2 = βJWKB (∞, 0) − π/4. 0

(6.12)

(6.13)

The number of bound states in the potential is N = [Φ/π − 5/8] + 1, where [. . .] means the integer part of the expression. These expressions work remarkably well in practice. Although the results in Figure 6.5 are obtained by solving the Schrödinger equation for a realistic potential, virtually identical results are obtained for a from Equation 6.12. In fact, a and Ei0 are nearly the same on the scale of Figure 6.5 if the simple hard-core van der Waals model of Ref. 14 is used for the potential, namely V (R) = −C6 /R6 if R ≥ R0 and V (R) = +∞ if R < R0 , where the cutoff R0 is chosen to fit a or E−1,0 data from two different isotopes. With the mass scaling √ ∝ μ in Equation 6.13, knowing C6 and E−1,0 for two isotopic pairs determines a © 2009 by Taylor and Francis Group, LLC

230

Cold Molecules: Theory, Experiment, Applications (a) 400

a(l) (a0)

200 0 –200

(b)

1

–Ei0(l)/h (MHz)

–400

10

100

1000 166

i = –1

i = –1

i = –2

i = –2

168

170

172 174 l = 2μ (amu)

176

178

FIGURE 6.5 (a) s-Wave scattering length and (b) bound-state binding energies −Ei0 (λ) for Yb2 molecular dimers vs. the control parameter λ = 2μ. The vertical dashed lines show the points of singularity of a(λ). The horizontal dashed lines show the boundaries of the bins in which the i = −1 and i = −2 levels must lie.

and E−1,0 for all isotopic pairs. The approximation is fairly good even for levels with larger |i| or > 0, although it will become worse as |i| or increase. In summary, it is very useful to take advantage of the enormous difference in energy and length scales associated with the cold separated atoms and deeply bound molecular potentials. This allows us to introduce a generalized “quantum defect” approach for understanding threshold physics [3,6–8,10]. Threshold bound state and scattering properties are determined mainly by the long-range potential, once the overall effect of the whole potential is known through the s-wave scattering length. A similar analysis can be developed for other long-range potential forms, for example, 1/R4 ion-induced dipole or 1/R3 dipole–dipole interactions.

6.3

INTERACTIONS FOR MULTIPLE POTENTIALS

Generally, the cold atoms used in experiments have additional angular momenta (electron orbital and/or electron spin and/or nuclear spin), so that more than one scattering channel α can be involved in a collision. Each channel has a separated atom © 2009 by Taylor and Francis Group, LLC

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channel energy Eα . Figure 6.1 could be modified to illustrate such channels by adding additional potentials and their corresponding spectra dissociating to the Eα limits. If Etot is the total energy of the colliding system, the designation open or closed is used for channels with Etot > Eα or Etot < Eα , respectively. Inelastic collisions from entrance channel α are possible to open exit channels β when Eα > Eβ , whereas closed channels β can support quasibound states as scattering resonances when Eα < Etot < Eβ . The ability to tune resonance states to control scattering properties or to convert them into true molecular bound states is an important aspect of ultracold physics that has been exploited in a wide variety of experiments with bosonic or fermionic atoms [1]. Let us first examine the basic magnitude of the s-wave inelastic collision rates that are possible when open channels are present. The rate constant is determined by the magnitude of b in Equation 6.2, for which a typical order of magnitude is b ≈ a¯ for an allowed transition, that is, one with relatively large short-range interactions in the system Hamiltonian. The rate constant can be written Kloss = 0.84 × 10−10 g

b[au] cm3 /sec, μ[amu]

(6.14)

where b is expressed in atomic units (1 au = 0.0529177 nm) and μ in atomic mass units (μ = 12 for 12 C). Allowed processes will typically have the order of magnitude of 10−10 cm3 /sec for Kloss . The s-wave Kloss can be even larger, with an upper bound of bu = 1/(4k) being imposed by the unitarity property of the S-matrix, that is, 0 ≤ 1 − |Sαα |2 ≤ 1. Because the lifetime relative to collision loss is τ = 1/(Kloss n), where n is the density of the collision partner, allowed processes result in fast loss with τ 1 msec at typical quantum degenerate gas densities. This applies to atom– molecule and molecule–molecule collisions as well as atom–atom collisions. Such losses need to be avoided by working with atomic or molecular states that do not experience fast loss collisions, such as the lowest energy ground-state level, which does not have exoergic 2-body exit channels. Alternatively, placing the species in a lattice cell that confines a single atom or molecule can offer protection against collisional loss. An alternative formulation of the collision loss rate is possible by rewriting Equation 6.2, not taking the E → 0 limit but introducing a thermal average over a Maxwellian distribution of collision energies E, Kloss = g

+ 1 kB T * 1 − |Sαα |2 , T QT h α

(6.15)

where QT is the translational partition function, 1/QT = (2πμkB T /h2 )3/2 = Λ3T where ΛT is the molecular thermal de Broglie wavelength. The . . .T expression implies a thermal average over the velocity distribution. The sum represents a dynamical factor fD that varies as T 1/2 as T → 0 and has an upper bound of unity for s-waves and ≈ 2max if max partial waves contribute at the unitarity limit. Although Equation 6.15 reduces to Equation 6.14 in the T → 0 s-wave limit, it lets us see that the collision rate is given by an expression having the form τ−1 = Kloss n = g(nΛ3T )


kB T fD . h

(6.16)

232


This form embodies some general principles for any collisions of atoms and molecules. The dimensionless nΛ3T factor shows that the collision rate is proportional to phase space density of the collision partner (scale by mass ratios to convert to an atomic phase space density). The kB T /h factor sets an intrinsic rate scale (dimension of inverse time) associated with T . The dimensionless factor fD embodies all of the detailed collision dynamics. Even using fast time-dependent manipulations to control fD does not change the fundamental thermodynamic limits imposed by the phase space density and kB T /h factors. Given Equations 6.14 and 6.15 and plausible assumptions about b or fD , it is possible to estimate the time scales for a wide variety of atomic and molecular collision processes under various kinds of conditions. Now we will examine the important case of tunable resonant scattering when a closed channel is present. Assume that open entrance channel α, with Eα chosen as Eα = 0, is coupled through terms in the system Hamiltonian to a closed channel β with 0 < E < Eβ . Then a molecular bound state in channel β becomes a quasibound state that acts as a scattering resonance in channel α. Using Fano’s form of resonant scattering theory [16], let us assume a “bare” or uncoupled approximate bound state |C = ψc (R)|c with energy Ec in the closed channel β = c and a bare or background scattering state |E = ψbg (R, E)|bg at energy E in the entrance channel α = bg. The scattering phase shift η(E) = ηbg (E) + ηres (E) of the coupled system picks up a resonant part due to the Hamiltonian coupling W (R) between the bare channels. Here ηbg is the phase shift due to the uncoupled single background channel, as described in the last section, and 1 −1 2 Γ(E) , (6.17) ηres (E) = − tan E − Ec − δE(E) has the standard Breit–Wigner resonance scattering form. The two key features of the resonance are its width Γ(E) = 2π|C|W (R)|E|2 , and its shift

δE(E) = P

∞ −∞

|C|W (R)|E |2 dE . E − E

(6.18)

(6.19)

The primary difference between an “ordinary” resonance and a threshold one as E → 0 is that for the former we normally make the assumption that Γ(E) and δE(E) are evaluated at E = Ec and are independent of E across the resonance. By contrast, the explicit energy dependence of Γ(E) and δE(E) are key features of threshold resonances [11,17,18]. In the special case of the E → 0 limit for s-waves, 1 Γ(E) → (kabg )Γ0 2 Ec + δE(E) → E0 ,

(6.20) (6.21)

where Γ0 and E0 are E-independent constants. Note that Γ(E) is positive definite, so that Γ0 has the same sign as abg . Assuming an entrance channel without inelastic loss, © 2009 by Taylor and Francis Group, LLC

233


so that ηbg (E) → −kabg , and for the sake of generality, adding a decay rate γc / for the decay of the bound state |C by irreversible loss processes, gives in the limit of E → 0, abg Γ0 a˜ = a − ib = abg − . (6.22) E0 − i(γc /2) This formalism accounts for both kinds of tunable resonances that are used for making cold molecules from cold atoms, namely, magnetically or optically tuned resonances. We now give our attention to each of these in turn.

6.4

MAGNETICALLY TUNABLE RESONANCES

Cold alkali metal atoms have a variety of magnetically tunable resonances that have been exploited in a number of experiments to control the properties of ultracold quantum gases or to make cold molecules. For the most part, experiments have succeeded with species that either do not have inelastic loss channels, or, if they do, the loss rates are very small. Thus, for practical purposes, we can set the resonance decay rate γc = 0 in examining a wide class of magnetically tunable resonances. While general coupled channel methods can be set up to solve the multichannel Schrödinger equation [1], we will use simpler models to explain the basic features of tunable Feshbach resonance states. Many resonances occur for alkali metal species in their 2 S electronic ground state because of their complex hyperfine and Zeeman substructure with energy splittings very large compared to kB T . Thus, closed spin channels that have bound states near Eα of an entrance channel α can serve as tunable scattering resonances for threshold collisions in that channel. The key to magnetic tuning of a resonance is that the resonance state |C has a different magnetic moment μc than the moment μatoms of the pair of separated atoms in the entrance channel. The bare bound-state energy can be tuned by varying the magnetic field B Ec (B) = δμ(B − Bc ),

(6.23)

where δμ = μatoms − μc is the magnetic moment difference and Bc is the field where Ec (Bc ) = 0 at threshold. The scattering length is real with b = 0 and takes on the following resonant form Δ , B − B0

(6.24)

B0 = Bc + δB.

(6.25)

a(B) = abg − abg where Δ=

Γ0 δμ

and

Note that the interaction between the entrance and closed channels shifts the point of singularity of a(B) from Bc to B0 . Such magnetically tunable Feshbach resonances are characterized by four parameters, namely, the background scattering length abg , the magnetic moment difference δμ, the resonance width Δ, and position B0 . © 2009 by Taylor and Francis Group, LLC

234


Figure 6.6 shows an example of the scattering length and bound-state energies for the 40 K87 Rb molecule near the lowest energy spin channel of the separated atoms. The spin quantum numbers and hyperfine splitting in their respective electronic ground states are 1, 2, and 6.835 GHz for 87 Rb and 9/2, 7/2, and −1.286 GHz (inverted) for 40 K. There are 11 other closed spin channels in this system with E > E that have α 1 the same total projection quantum number as the lowest energy α = 1 spin channel. Because of their different magnetic moments the energy of a bound state of one of these closed channels can be tuned relative to the energy of the two separated atoms in the α = 1 s-wave channel, as shown in the figure. Due to coupling terms in the Hamiltonian among the various channels, bound states that cross threshold couple to the entrance channel and give rise to resonance structure in its a(B). The resonance with B0 near 54.6 mT (546 G) has been used to associate a cold 40 K atom and a cold 87 Rb atom to make a 40 K87 Rb molecule in a near-threshold state with a small binding energy on the order of 1 MHz or less [19]. It is extremely useful to introduce the properties of the long-range van der Waals potential and take advantage of the separation of short- and long-range physics discussed in the previous section. Assuming that the interaction W (R) is confined

–100

a (a0)

–150 –200 0 E/h (GHz)

2(–1) –0.2 –0.4

5(–2)

4(–2)

12(–3)

1(–1) 11(–3)

40

50

60

70

B (mT)

FIGURE 6.6 Molecular bound state energies (lower panel) and scattering length (upper panel) vs. magnetic field B in mT (1 mT = 10 G) for the lowest energy α = 1 s-wave spin channel of the 40 K87 Rb fermionic molecule. The bound-state energies are shown relative to the channel energy E1 of the two separated atoms taken to be zero. This α = 1 spin channel has respective 40 K and 87 Rb spin projection quantum numbers of −9/2 and +1, giving a total projection of −7/2. In this species there are 11 additional closed s-wave channels with Eα > E1 and with the same projection of −7/2. The bound-state quantum numbers are α(i), where i is the vibrational quantum number relative to the dissociation limit of closed channel α = 2, . . . , 12. Four bound states cross threshold in this range of B, giving rise to singularities in the scattering length.



235

to distances R a¯ , the matrix element in Equation 6.18 defining Γ(E) can be factored as ¯ Γ(E) = Cbg, (E)−2 Γ, (6.26) where Γ¯ is a measure of resonance strength that depends only on the energyindependent short-range physics near E = 0, and is completely independent of the asymptotic boundary conditions. It thus can be used in characterizing the properties of both scattering and bound states when E = 0. The extrapolation of resonance properties away from E = 0 depends on two additional parameters associated with the long-range potential, μ and C6 , which determine ¯ Let us define a dimensionless resonance strength parameter a¯ and E. sres =

abg δμΔ Γ0 = rbg a¯ E¯ E¯

(6.27)

where rbg = abg /¯a. Using the threshold van der Waals form of Cbg,0 (E)−1 given in the previous section, we can write Γ¯ 1 ¯ = (sres E) . 2 1 + (1 − rbg )2

(6.28)

The above-threshold scattering properties are found from the scattering phase shift η(E) = ηbg (E) + ηres (E), where ηres (E) is found from Equation 6.17 once Ec , Γ(E), and δE(E) are known. The first two are given by Equations 6.23 and 6.26, and δE(E) =

Γ¯ tan λbg (E), 2

(6.29)

where tan λbg (E) is a function determined by the van der Waals potential, given abg . It has the limiting form tan λbg (E) = 1 − rbg as E → 0, and tan λbg (E) = 0 for E E¯ [3,8]. Thus the position of the scattering length singularity is shifted by δB = B0 − Bc = Δ

rbg (1 − rbg ) 1 + (1 − rbg )2

(6.30)

from the crossing point Bc of the bare bound state. Scattering phase shifts calculated from the van der Waals potential with the “quantum defect” forms in Equations 6.26 and 6.29 are generally in excellent agreement with complete coupled channels methods for energy ranges on the order of E¯ and even larger [11]. The properties of bound molecular states near threshold can also be calculated from the general coupled-channels quantum defect method using the properties of the longrange potential. When the energy Eb (B) = −2 kb (B)2 /(2μ) of the threshold s-wave bound state is small, that is, |Eb (B)| E¯ or kb (B)¯a 1, then the expression for Eb (B) from the quantum defect method is

1 Γ¯ − kb (B)¯a = . (Ec (B) − Eb (B)) rbg − 1 2 © 2009 by Taylor and Francis Group, LLC

(6.31)

236


If Γ¯ = 0, we recover the uncoupled, or bare, bound states of the system, whereas when Γ¯ > 0, this equation gives the coupled, or “dressed,” bound states. The threshold bound state “disappears” into the continuum at B = B0 , where a(B) has a singularity. The shift in Equation 6.30 follows immediately upon solving for Ec (B0 ) where Eb (B0 ) = 0. Threshold bound-state properties are strongly affected by the magnitudes of sres and rbg . When the coupled bound-state wavefunction is expanded as a mixture of closed and background channel components, |c and |bg respectively, an important property is the norm Z(B) of the closed channel component; the norm of the entrance channel component is 1 − Z(B). The value of Z can be calculated from a knowledge of Eb (B), because Z = |δμ−1 ∂Eb /∂B| [1]. There are two basic classes of resonances. One, for which sres 1, are called entrance channel dominated resonances. These have Z(B) 1 as B − B0 varies over a range that is a significant fraction of |Δ|. In addition, the bound-state energy is given by Equation 6.11 over a large part of this range. On the other hand, closedchannel-dominated resonances are those with sres 1. They have Z(B) large, on the order of unity, as |B − B0 | varies over a large fraction of |Δ|, and only have a “universal” bound state (Equation 6.3) over a quite small range |Δ| near B0 . ¯ so that Entrance-channel-dominated resonances have Γ(E, B) > E when 0 < E < E, no sharp resonance feature persists above threshold, where a(B) < 0 and the last bound state has disappeared. By contrast, closed-channel-dominated resonances with ¯ so that a sharp resonant |rbg | not too large will have Γ(E, B) < E when 0 < E < E, feature emerges just above threshold, continuing as a quasibound state with E > 0 into the region where a(B) < 0. Figure 6.7 shows an expanded view of the 4(−2) resonance of 40 K87 Rb near 54.6 mT. The figure shows the character of the bound state as it merges into threshold at B0 . It tends to be a universal “halo” bound state over a range of |B − B0 | that is less than about 1/3 of Δ. As |B − B0 | increases, the bound state increasingly takes on the character of the closed channel 4(−2) level as Z increases toward unity. Figure 6.8 shows an example of the very broad 6 Li resonance in the lowest energy α = 1 s-wave channel, which requires two 6 Li fermions in different spin states. This is a strongly entrance-channel-dominated resonance, where Z 1 over a range of |B − B0 | nearly as large as Δ. The last bound state is a universal halo molecule over a range larger than 100 G. The corrected Equation 6.11 is a good approximation over an even larger range. The scattering length graph shows that the size ≈ a(B)/2 of the halo state is very large compared to a¯ = 30a0 (see Table 6.1) over this range. Magnetically tunable scattering resonances have proven very useful in associating two cold atoms to make a molecule in the weakly bound states near threshold. This work is reviewed in detail in Ref. 1. The magnetoassociation (MA) process works by first preparing a gas with a mixture of both atomic species at B > B0 (assuming δμ > 0), where there is no threshold bound state. By ramping the B field down in time so that B < B0 , colliding pairs of atoms with E > 0 can be converted to diatomic molecules in a bound state with energy E < 0. The conversion efficiency will depend on both the ramp rate and the phase space density of the initial gas. If the initial atom pair is held in a single cell of an optical lattice instead of a gas, the conversion efficiency can approach 100%. A simple Landau–Zener picture has been found to be © 2009 by Taylor and Francis Group, LLC

237

Molecular States Near a Collision Threshold 1.0

Z

0.5

Eb/h (MHz)

0.0

B0 –5

–10 54.2

54.4

54.6

54.8

B (mT)

FIGURE 6.7 The lower panel shows an expanded view of Eb (B) near B0 for the 40 K87 Rb resonance with B0 = 54.693 mT (546.93 G) in Figure 6.6. The solid line comes from a coupled-channels calculation that includes all 12 channels with the same −7/2 projection quantum number. The dashed and dotted lines respectively show the universal energy of Equation 6.3 and the van der Waals corrected energy of Equation 6.11. The upper panel shows the closed channel norm Z(B). The width Δ = 0.310 mT (3.10 G), abg = −191a0 , and δμ/h = ¯ = 13.9 MHz, this is a marginal 33.6 MHz/mT (3.36 MHz/G). With a¯ = 68.8a0 and E/h entrance-channel-dominated resonance with sres = 2.08.

quite accurate for such lattice cells, where the conversion probability of the atom pair in the trap ground state i = 0 is 1 − e−A , where A=

2π Wci2 . E˙c

(6.32)

Here i ≥ 0 represents the above-threshold levels of the atom pair confined by the trap, continuing the below threshold series of dimer levels with i ≤ −1. For a three-dimensional harmonic trap with frequency ωx = ω√ y = ωz = ω, √the matrix element Wci = C|W (R)|i is well-approximated as Wci = Γ(Ei )/2π ∂Ei /∂i, where ∂Ei /∂i = 2ω and√Γ(Ei ) = 2ki abg δμΔ for the i = 0 trap ground state of relative motion with ki = 3μω/ (see Equations 6.18, 6.20, and 6.25). The trick used here in getting a matrix element Wci between two bound states from the matrix element C|W (R)|E involving an energy-normalized scattering state is to introduce the density of states as in Equation 6.10. In a similar manner, the matrix element can be obtained between the bare closed channel state and the bound states i < 0 of the entrance channel. Such matrix elements characterize avoided crossings like the one in Figure 6.6 for E/h near −0.4 MHz and B near 43 mT. Finally, it should be noted that a Landau–Zener model can also be used for molecular dissociation by a fast magnetic field ramp. An alternative phenomenological model has been developed to describe © 2009 by Taylor and Francis Group, LLC

238

Cold Molecules: Theory, Experiment, Applications (a) a(B) (a0)

8000 4000 0

Eb/h (MHz)

(b)

0 –1

B0

–2 –3 60

70

80

90

B (mT)

FIGURE 6.8 (a) Scattering length and (b) molecular bound state energy vs. magnetic field B for the lowest energy α = 1 s-wave spin channel of the 6 Li2 molecule. This channel has one 6 Li atom in the lowest +1/2 projection state and the other in the lowest −1/2 projection state for a total projection of 0. There are four additional closed channels with projection 0. In this range of B there is only one bound state that crosses threshold at B0 = 83.4 mT (834 G). The lower panel shows Eb (B) from a coupled channels calculation (solid circles), the universal limit of Equation 6.3 (dashed line) and the corrected limit of Equation 6.11 (solid line). The width Δ = 30.0 mT (300 G), abg = −1405a0 , and δμ/h = 28 MHz/mT (2.8 MHz/G). This is a strongly entrance-channel-dominated resonance with sres = 59, and Z < 0.06 over the range of B shown.

molecular association in cold gases, which are more complex than two atoms in a lattice cell [20].

6.5

PHOTOASSOCIATION

Cold atoms can also be coupled to molecular bound states through photoassociation (PA), as discussed in Chapters 5, 7, 8, and 9. Figure 6.9 gives a schematic description of PA, a process by which the colliding atoms can be coupled to such bound-state resonances through one or two photons. Reference 2 reviews theoretical and experimental work on PA spectroscopy and molecule formation. Molecules made using the magnetically tunable resonances described in the last section are necessarily very weakly bound, with binding energies limited by the small range of magnetic tuning. Photoassociation has the advantage that laser frequencies are widely tunable, so that a range of many bound states becomes accessible to optical methods, even the lowest v = 0 vibrational level of the ground state. In addition, the light can be turned off and on or varied in intensity for time-dependent manipulations. Photoassociation naturally lends itself to the resonant scattering treatment of a decaying resonance in Equation 6.22, which applies to the one-color case with position Ec = Ev∗ − hν1 , strength abg Γ0 (I) = Γ(E, I)/(2k), and shift δE(I). The latter two are linear in laser intensity I when I is low enough. Photoassociation is usually detected © 2009 by Taylor and Francis Group, LLC

239


Ev*

A* + B

n2

n1

Decay gc

V(R)

E RC

A+B

R

FIGURE 6.9 Schematic representation of one- and two-color photoassociation (PA). The two colliding ground-state atoms at energy E can absorb a laser photon of frequency ν1 and be excited to an excited molecular bound state at energy Ev∗ . The bound state decays via spontaneous emission at rate γc /. If a second laser is present with frequency ν2 , the excited level can also be coupled to a ground-state vibrational level v at energy Ev , if h(ν2 − ν1 ) = E − Ev . The PA process depends on the ground-state wavefunction at the Condon point RC of the transition, where hν1 equals the difference between the excited- and ground-state potentials.

by the inelastic collisional loss of cold atoms it causes, due to the spontaneous decay of the excited state to make hot atoms or deeply bound molecules. In the limit E → 0 the complex scattering length is a(ν1 , I) = abg − Lopt b(ν1 , I) =

γc E 0 E02 + (γc /2)2

γc2 1 Lopt 2 , 2 E0 + (γc /2)2

(6.33)

(6.34)

where E0 = Ev∗ − hν1 + δE(I) is the detuning from resonance, including the intensity-dependent shift, and the optical length is defined by Lopt = abg Γ0 (I)/γc . Photoassociation spectra, line shapes, and shifts have been widely studied for a variety of like and mixed alkali–metal species. At the higher temperatures often encountered in magneto-optical traps, contributions to PA spectra from higher partial waves, such as, p- or d-waves, have been observed in a number of cases. The theory can be readily extended to higher partial waves. By introducing an energy-dependent complex scattering length the theory for s-waves can be extended to finite E away from threshold and to account for effects due to reduced dimensional confinement in optical lattices [21]. The optical length formulation of resonance strength is very useful for a decaying resonance. It also applies to decaying magnetically tunable resonances, if Γ0 from © 2009 by Taylor and Francis Group, LLC

240


Equation 6.25 is used to define a resonance length abg δμΔ/γc equivalent to Lopt [22]. The scattering length has its maximum variation of abg ± Lopt when the laser is tuned to E0 = ±γc /2, and losses are maximum at E0 = 0 where b = Lopt . When detuning is small, on the order of γc , significant changes to the scattering length on the order of a¯ are thus normally accompanied by large loss rates (see Equation 6.14). Losses can be avoided by going to large detuning, because when (γc /E0 ) 1, b = (Lopt /2)(γc /E0 )2 , whereas the change in a only varies as a − abg = −Lopt (γc /E0 ). To make the change a − abg large enough while requiring (γc /E0 ) 1 means that Lopt has to be very large compared to a¯ . The magnitude of Lopt depends on the matrix element C|Ω1 (R)|E where Ω1 (R) represents the optical coupling between the ground and excited state. Using Equations 6.18 and 6.20 and the above definition of Lopt , and factoring out the relatively constant Ω1 value, Lopt = π

|Ω1 |2 F(E) . γc k

(6.35)

The Franck–Condon overlap factor is F(E) =

0

≈

∞

2 ∗ ψv (R)ψ0 (R, E) dR

∂Ev∗ 1 |ψ0 (RC , E)|2 , ∂v DC

(6.36) (6.37)

where DC is the derivative of the difference between the excited- and ground-state potentials evaluated at the Condon point RC , and ∂Ev∗ /∂v is the excited-state vibrational spacing. Equation 6.37 is known as the reflection approximation, generally an excellent approximation where F(E) is proportional to the square of the groundstate wavefunction at RC , the Condon point where the molecular potential difference matches hν1 (see Figure 6.9). Thus, F(E) can be evaluated using expressions like Equations 6.5 or 6.9 for RC a¯ or RC a¯ , respectively. The reflection approximation is quite good over a wide range of E and for higher partial waves than the s-wave. By selecting a range of excited levels v by changing laser frequency ν1 , thus changing RC , the shape and nodal structure of the ground-state wavefunction can be mapped out over a range of R. The optical length has several important properties evident from Equation 6.35. First, because both Ω1 and γc are proportional to the same squared transition dipole moment, Lopt does not depend on whether the transition is strong or weak, but can be large for both kinds of transitions. Second, Lopt ∝ |Ω1 |2 so it can be increased by increasing laser intensity. Third, because F(E) ∝ k as E → 0 for entrance channel s-waves, Lopt ∝ F(E)/k is independent of E or k at low energy. However, it does depend strongly on the molecular structure through the Franck–Condon factor. In practice, using strong transitions with large decay rates such as those in alkali–metal species leads to the requirement to use excited molecular levels far from threshold with large binding energies. This is necessary to achieve large detuning from atomic and molecular resonance. This requirement means such levels have small F(E) factors, © 2009 by Taylor and Francis Group, LLC


241

due to the very large value of DC in Equation 6.37. On the other hand, weak transitions with small decay rates, such as those associated with the 1 S0 → 3 P1 intercombination line transition for alkaline earth species such as Sr, can lead to quite large values of Lopt . This is because large detuning in γc units can be achieved for levels that are still quite close to the excited-state threshold. Such levels typically have large Franck– Condon factors. In fact, PA transitions near the weak intercombination line of Sr have been observed to have Lopt several orders of magnitude larger than was observed for strongly allowed molecular transitions involving Rb [23]. Thus, there are good prospects for some degree of optical resonant control of collisions in ultracold gases of species like Ca, Sr, or Yb. Two-color PA is also possible when a second laser with frequency ν2 is added, as shown in Figure 6.9. When the the frequency difference is chosen so that h(ν2 − ν1 ) = E − Ei , the ground i molecular level is in resonance with collisions at energy E. By keeping ν1 fixed and tuning ν2 , two-color PA spectroscopy can be used to probe the level energies. This is how the data on binding energies of the Yb2 molecule were obtained [15] so as to be able to construct Figure 6.5. In this case, 12 different levels from different isotopic species were measured, among which were levels with i = −1 and −2 and = 0 and 2. Two-color spectroscopy has also been carried out for several alkali–metal homonuclear species. Two-color processes are also an excellent way to assemble two cold atoms into a translationally cold molecule. Early work along these lines was done using the spontaneous decay of the excited level to populate a wide range of levels in the ground state. The disadvantage of spontaneous decay is that it is not selective. However, by using a laser with a precise frequency, a specific level can be chosen as the target level. One early experiment did this to associate two 87 Rb atoms in a Bose–Einstein condensate to make a molecular level at a specific energy of h(−636) MHz [24]. It is highly desirable to be able to make translationally cold molecules in their vibrational ground state v = 0. This is especially true of polar molecules, which have large dipole moments in v = 0. On the other hand, threshold levels have negligible dipole moments, because there is no charge transfer because of the large average atomic separation ≈ a¯ . A promising technique is to use MA using a tunable Feshbach resonance to associate the atoms into a threshold molecular level, then use a twocolor Raman process to move the population in that state to a much more deeply bound level. Although molecules in a gas are subject to fast destructive collisions with cold atoms or other molecules in the gas (see Equation 6.14), the molecules can be protected against such collisions by forming them in individual optical lattice trapping cells. Then the two-color Raman process could be used to produce much more deeply bound molecules that are stable against destructive collisions. This has been done successfully with 87 Rb2 [25] molecules. In the future, such methods are likely to produce v = 0 polar molecules, with which a range of interesting physics can be explored [26,27].

REFERENCES 1. Köhler, T., Góral, K., and Julienne, P.S., Production of cold molecules via magnetically tunable Feshbach resonances, Rev. Mod. Phys., 78, 1311, 2006.


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2. Jones, K.M., Tiesinga, E., Lett, P.D., and Julienne, P.S., Photoassociation spectroscopy of ultracold atoms: long-range molecules and atomic scattering, Rev. Mod. Phys., 78, 483, 2006. 3. Julienne, P.S. and Mies, F.H., Collisions of ultracold trapped atoms, J. Opt. Soc. Am. B, 6, 2257, 1989. 4. Moerdijk, A.J., Verhaar, B.J., and Axelsson, A., Resonances in ultracold collisions of 6 Li, 7 Li, and 23 Na, Phys. Rev. A, 51, 4852, 1995. 5. Vogels, J.M., Verhaar, B.J., and Blok, R.H., Diabatic models for weakly bound states and cold collisions of ground-state alkali-metal atoms, Phys. Rev. A, 57, 4049, 1998. 6. Burke, J.J.P., Greene, C.H., and Bohn, J.L., Multichannel cold collisions: simple dependences on energy and magnetic field, Phys. Rev. Lett., 81, 3355, 1998. 7. Vogels, J.M., Freeland, R.S., Tsai, C.C., Verhaar, B.J., and Heinzen, D.J., Coupled singlet-triplet analysis of two-color cold-atom photoassociation spectra, Phys. Rev. A, 61, 043407, 2000. 8. Mies, F.H. and Raoult, M., Analysis of threshold effects in ultracold atomic collisions, Phys. Rev. A, 62, 012708, 2000. 9. Gao, B., Zero-energy bound or quasibound states and their implications for diatomic systems with an asymptotic van der Waals interaction, Phys. Rev. A, 62, 050702, 2000. 10. Gao, B., Angular-momentum-insensitive quantum-defect theory for diatomic systems, Phys. Rev. A, 64, 010701, 2001. 11. Julienne, P.S. and Gao, B., Simple theoretical models for resonant cold atom interactions, in Atomic Physics 20, Roos, C., Häffner, H., and Blatt, R., Eds., AIP, Melville, New York, 2006, p. 261–268, physics/0609013. 12. Derevianko, A., Johnson, W.R., Safronova, M.S., and Babb, J.F., High-precision calculations of dispersion coefficients, static dipole polarizabilities, and atom–wall interaction constants for alkali–metal atoms, Phys. Rev. Lett., 82, 3589, 1999. 13. Porsev, S.G. and Derevianko, A., High-accuracy calculations of dipole, quadrupole, and octupole electric dynamic polarizabilities and van der Waals coefficients C6 , C8 , and C10 for alkaline-earth dimers, JETP 102, 195, 2006, [Pis’ma Zh. Eksp. Teor. Fiz., 129, 227–238 (2006)]. 14. Gribakin, G.F. and Flambaum, V.V., Calculation of the scattering length in atomic collisions using the semiclassical approximation, Phys. Rev. A, 48, 546, 1993. 15. Kitagawa, M., Enomoto, K., Kasa, K., Takahashi, Y., Ciurylo, R., Naidon, P., and Julienne, P.S., Two-color photoassociation spectroscopy of ytterbium atoms and the precise determinations of s-wave scattering lengths, Phys. Rev. A, 77, 012719, 2008. 16. Fano, U., Effects of configuration interaction on intensities and phase shifts, Phys. Rev. A, 124, 1866, 1961. 17. Bohn, J.L. and Julienne, P.S., Semianalytic theory of laser-assisted resonant cold collisions, Phys. Rev. A, 60, 414, 1999. 18. Marcelis, B., van Kempen, E.G.M., Verhaar, B.J., and Kokkelmans, S.J.J.M.F., Feshbach resonances with large background scattering length: Interplay with open-channel resonances, Phys. Rev. A, 70, 012701, 2004. 19. Ospelkaus, C., Ospelkaus, S., Humbert, L., Ernst, P., Sengstock, K., and Bongs, K., Ultracold heteronuclear molecules in a 3D optical lattice, Phys. Rev. Lett., 97, 120402, 2006. 20. Hodby, E., Thompson, S.T., Regal, C.A., Greiner, M., Wilson, A.C., Jin, D.S., Cornell, E.A., and Wieman, C.E., Production efficiency of ultracold Feshbach molecules in Bosonic and Fermionic systems, Phys. Rev. Lett., 94, 120402, 2005. 21. Naidon, P. and Julienne, P.S., Optical Feshbach resonances of alkaline–earth atoms in a 1D or 2D optical lattice, Phys. Rev. A, 74, 022710, 2006.



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22. Hutson, J.M., Feshbach resonances in ultracold atomic and molecular collisions: threshold behaviour and suppression of poles in scattering lengths, New J. Phys., 9, 152, 2007. 23. Zelevinsky, T., Boyd, M.M., Ludlow, A.D., Ido, T., Ye, J., Ciurylo, R., Naidon, P., and Julienne, P.S., Narrow line photoassociation in an optical lattice, Phys. Rev. Lett., 96, 203201, 2006. 24. Wynar, R., Freeland, R.S., Han, D.J., Ryu, C., and Heinzen, D.J., Molecules in a BoseEinstein condensate, Science, 287, 1016, 2000. 25. Winkler, K., Lang, F., Thalhammer, G., van der Straten, P., Grimm, R., and Hecker Denschlag, J., Coherent optical transfer of Feshbach molecules to a lower vibrational state, Phys. Rev. Lett., 98, 043201, 2007. 26. Lewenstein, M., Polar molecules in topological order, Nature Phys., 2, 309, 2006. 27. Büchler, H.P., Micheli, A., and Zoller, P., Three-body interactions with cold polar molecules, Nature Phys., 3, 726, 2007.


Prospects for Control of 7 Ultracold Molecule Formation via Photoassociation with Chirped Laser Pulses Eliane Luc-Koenig and Françoise Masnou-Seeuws CONTENTS 7.1

7.2

Introduction: Can Ultrafast Meet Ultracold?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Making Ultracold Molecules by Photoassociation of Ultracold Atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Link with the Coherent Control Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Outline of the Present Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling of Photoassociation with Chirped Laser Pulses . . . . . . . . . . . . . . . . 7.2.1 The Physical Problem: Example of Photoassociation into a Long-Range Well of Cs2 and Choice of a Pulse in the 100 picosecond Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1.1 Choice of Cs2 as a Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1.2 A Qualitative Interpretation of Photoassociation . . . . . . . . 7.2.1.3 Timescales and Characteristic Distances for the Vibrational Motion in the Excited State. . . . . . . . . . . . . . . . . . 7.2.1.4 Description of the Initial Collision State . . . . . . . . . . . . . . . . . 7.2.1.5 Position of the “Last” Node RN . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Choice of a Linearly Chirped Pulse in the Picosecond Domain . . 7.2.2.1 The Chirped Pulse, Central Frequency, Energy, Spectral and Temporal Widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2.2 The Photoassociation Process: Time Window, Resonance Window, Concept of a Photoassociation Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 The Two-Channel Coupled Equations and the Choice for a Rotating Wave Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247 247 248 248 249

249 249 251 251 252 254 254 254

256 256


246

7.3

7.4

7.5


7.2.3.1 Rotating Wave Approximation with the Instantaneous Frequency: Definition of the Photoassociation Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3.2 Rotating Wave Approximation with the Central Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of Numerical Simulations: Interpretation as Adiabatic Transfer Within a Photoassociation Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Results of Calculations for One Collision Energy. . . . . . . . . . . . . . . . . 7.3.2.1 Photoassociation within the Resonance Window . . . . . . . . 7.3.2.2 Formation of Halo Molecules via Optically Induced Feshbach Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2.3 Selectivity of the Resonance Window: Dependence of the Final Distribution of Population on the Pulse Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Analysis within a Two-State Model: The Concept of Adiabatic Transfer within a Photoassociation Window . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Averaging Over Initial Velocity Distribution: Use of Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4.1 Boltzmann Average on a Finite-Size Grid . . . . . . . . . . . . . . . 7.3.4.2 Average Introducing Box-Independent Energy-Normalized States: Use of a Scaling Law Near Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 What is the Absolute Number of Photoassociated Molecules? . . . 7.3.6 Transient Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shaping Vibrational Wavepackets in the Excited State to Optimize Stabilization into Deeply Bound Levels of the Lower State . . . . . . . . . . . . . . 7.4.1 Shaping the Vibrational Wavepacket in the Excited State . . . . . . . . 7.4.2 Proposal for a Two-Color Pump–Dump Experiment . . . . . . . . . . . . . 7.4.2.1 The Time-Dependent Franck–Condon Overlap . . . . . . . . . 7.4.2.2 A Two-Color Experiment for Creating Stable Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Dynamical Hole in the Initial State Wavefunction: Compression Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Phenomenological Observation of a Depletion Hole, a Momentum Kick, and a Compression Effect . . . . . . . . . . . . . . . . . . . . 7.5.2 Analysis of the Momentum Transfer with Partially Integrated Mass Current and Population. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Advantage of the Compression Effect for Photoassociation with a Second Pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4 Redistribution of Population in the Lower State after a Photoassociation Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4.1 Redistribution in the a3 Σ+ u State in the Case of Cesium Photoassociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4.2 Correlated Pairs of Hot Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . .


258 259 260 260 261 261 261

262 264 266 267

267 268 270 271 271 272 273 274 276 276 278 279 281 281 281

Control of Ultracold Molecule Formation via Photoassociation

Beyond the Impulsive or Adiabatic Approximations: New Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Controlling the Compression Effect with a Nonimpulsive Pulse Inducing Many Rabi Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Nonadiabatic Broadband Pulse: Excitation at Large Distances. . . 7.6.2.1 Large Transfer of Population Outside the PA Window . . 7.6.2.2 Thermal Average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Conclusion and Prospects for the Near Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247

7.6

7.1 7.1.1

283 283 284 284 284 286 287 287

INTRODUCTION: CAN ULTRAFAST MEET ULTRACOLD? MAKING ULTRACOLD MOLECULES BY PHOTOASSOCIATION OF ULTRACOLD ATOMS

The formation of ultracold molecules in the lowest rovibrational level of the ground electronic state is presently a subject of intense investigation. By ultracold, we mean molecules at a temperature well below 1 mK. As discussed in Chapters 5 and 6 of the present book by W. Stwalley and colleagues, and by Paul Julienne, the photoassociation (PA) route [1,2] is a very efficient one. Whereas laser cooling techniques cannot directly be applied to molecules, it is possible to start from an assembly of laser-cooled atoms, and then make cold molecules in an excited electronic state by photoassociating atom pairs. For that purpose, most experiments use a cw laser red-detuned, relative to the atomic resonance line, to a frequency coinciding with the transition to a loosely bound vibrational level of the dimer. Being in an excited electronic state, the photoassociated molecule is short-lived, and a second step is necessary to form a stable molecule, hereafter named the stabilization step. When it decays by spontaneous emission, the molecule may either break apart again into a pair of cold atoms, or reach various vibrational levels of the ground electronic state (or of the lowest triplet state in the case of alkali dimers) [3–5]. In the latter case stable molecules are indeed formed, and schemes to increase their formation rate depend very much upon details in their spectroscopy (such as the existence of long-range wells in the potential curves of the excited state, or of resonant coupling between two excited channels [6,7]). Therefore, the molecular spectroscopy foundation of the field reveals itself as essential. Compared to nonoptical routes toward cold molecules discussed in this book, the main advantage of the PA route is the ability to produce a large number of stable ultracold molecules, at the same translational temperature (a few μK, or even less) than the precursor atoms. In contrast with the “halo” molecules produced by sweeping magnetic Feshbach resonances [8], the stable molecules are formed in vibrational levels, which can be relatively deeply bound. The drawback is that stabilization through the spontaneous emission process spreads the population into a variety of excited vibrational levels, so that the product molecules are not in a pure vibrational state. Schemes to cool down the vibrational and rotational degrees of freedom have to be implemented. The present chapter explores one scheme, making use of chirped laser pulses, or more generally shaped pulses, rather than cw lasers, for PA, and implementing © 2009 by Taylor and Francis Group, LLC

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stabilization through induced emission. The idea is to fully exploit the possibilities of the optical techniques in order to gain control of the PA and stabilization processes. Besides improving molecule formation rates, the use of ultrafast lasers (picosecond and femtosecond range) and of laser shaping techniques is expected to make dynamical phenomena observable and controllable in real time.

7.1.2

LINK WITH THE COHERENT CONTROL FIELD

The formation of a bound molecule out of two colliding ultracold atoms is a simple example of a laser-induced chemical reaction at very low temperature. At room or even higher temperatures, as discussed in Chapter 8 by Evgueny Shapiro and Moshe Shapiro, the well-established field of coherent control relies upon the possibility of shaping laser pulses to control the output of chemical reactions. Whether similar schemes can be applied to the PA and stabilization reactions at low temperatures has been an open question. To give an answer, one should carefully explore the feasibility and the efficiency of pump–dump experiments to selectively create ultracold molecules in the v = 0 level of their ground electronic state. On the experimental side, the implementation of short pulse techniques looks controversial: indeed, the ultracold field being linked to precision spectroscopy, the cw-laser technology seems at first sight to be the best adapted. Changing to short and shaped pulses, one might lose the sensitivity to spectroscopic details that has been a key to success. Moreover, the spectroscopic analysis of the photoassociated or stabilized molecules, identifying which rovibrational levels of which electronic states are populated, is not well established in time-dependent experiments. Nevertheless, several groups [9–11] have started experiments to create molecules in a rubidium magneto-optical trap by photoassociating with femtosecond lasers. Although such experiments are not yet fully convincing, because the first pulse mainly destroys the molecules already present in the trap, rapid progress is occurring as the field develops. On the theoretical side, the novelty in the ultracold field lies in the description of the initial state of two colliding cold atoms. At room temperature, the Hamiltonian is dominated by kinetic energy, and time-dependent treatments using Gaussian wavepackets are well established. In the microKelvin range, a Gaussian wavepacket spreads more rapidly than it moves, so that the initial state must be a distribution of stationary collision states. The adequacy of such states to describe the collision dynamics has been revealed in early PA spectroscopy experiments [12,13]: when sweeping the frequency of the cw laser, the minima in the experimental signal are a signature of the nodes in the radial wavefunction associated with s-wave scattering. In this chapter, we shall therefore present numerical treatments calculating the time evolution, once a laser pulse is turned on, of a stationary collision wavefunction, delocalized over a wide range of internuclear distances.

7.1.3

OUTLINE OF THE PRESENT CHAPTER

An important theoretical effort has been devoted to the subject by our theoretical group in Orsay, in collaboration with Ronnie Kosloff and the Jerusalem group. It started as early as 1996 with the thesis work of Mihaela Vatasescu [14–16] and later © 2009 by Taylor and Francis Group, LLC


249

on with the contribution of Jiri Vala [17]. Owing to the creation at the end of 2002 of a European Research Training Network, funded by the European Commission under contract HPRNCT 2002 00290, and named “Cold Molecules: Formation, Trapping and Dynamics” this activity was able to develop further. The first calculations using a stationary collision wavefunction for the initial state of time-dependent calculations were published in 2004 [18,19]. The active participation of network post-docs Christiane Koch [20–24], Jordi Mur-Petit [25] and Shimshon Kallush [26–28], as well as the thesis works of Pascal Naidon [29,30] and Kai Willner [31,32], led to many developments. This chapter presents a synthesis of most of the latter work, and of the ongoing debate about the possibility of overlap between the two fields of coherent control and ultracold molecules. It is organized as follows: • Section 7.2 presents modeling of PA with a chirped laser pulse, choosing cesium as a case study. • Section 7.3 presents results of the numerical calculations. It is shown that, for well-chosen pulse parameters, total adiabatic population transfer can be achieved, so that all the pairs of atoms with interatomic distances within a range defining a “photoassociation window” are transformed into bound molecules. An estimation of the total number of such photoassociated molecules is given. • Section 7.4 presents the possibility of shaping vibrational wavepackets in the excited state, in view of a focusing effect that optimizes the formation of stable ground-state molecules in a pump–dump experiment. • Section 7.5 analyzes the dynamical hole formed in the initial-state wavepacket after PA, and the compression effect due to a momentum kick. It is suggested that one should design a second pulse red-detuned relative to the first one in order to populate efficiently more deeply bound levels of the excited state. • Section 7.6 is devoted to another possible PA mechanism involving nonadiabatic transfer at large distances. • Section 7.7 gives a conclusion and prospects for future theoretical as well as experimental work.

7.2

MODELING OF PHOTOASSOCIATION WITH CHIRPED LASER PULSES

7.2.1 THE PHYSICAL PROBLEM: EXAMPLE OF PHOTOASSOCIATION INTO A LONG-RANGE WELL OF Cs2 AND CHOICE OF A PULSE IN THE 100 PICOSECOND RANGE 7.2.1.1

Choice of Cs2 as a Case Study

The Cs2 molecule is a good candidate because the formation of a large number of stable ultracold molecules has been observed in experiments [3,13]. The PA reaction 2Cs(6S, F = 4) + (ω(t)) → Cs2 (0g− (6S + 6P3/2 ); v, J) © 2009 by Taylor and Francis Group, LLC

(7.1)

250


into the external well of the 0− g (6S + 6P3/2 ) electronic state is considered here as a case study. In this reaction two ultracold cesium atoms, in their ground 6S state, colliding at a temperature in the 50 μK range, absorb a photon red-detuned from the atomic resonance line to form a bound level (v, J) in the outer potential well of the excited state. The frequency of the laser is time-dependent. As for many PA experiments, the spectroscopy is very well known [33]. The potential curves have been described elsewhere [18] and are displayed in Figure 7.1. We use vibrational numbering vtot for the double-well potential and v for the external well; the latter supports at least 226 levels, from v = 0 (vtot = 25) to v = 225 (vtot = 256). The initial electronic state is the a3 Σ+ u (6S + 6S); only s-wave scattering is considered, and rotational excitation of the molecule is not allowed. Therefore, the calculations are performed with realistic potentials, fitted to the experimental spectra, but it is still a simplified model, because we neglect the hyperfine structure and also the possibility of two- (or more) photon absorption with population of Rydberg excited electronic states. tP+tc

tP–tc

tP

c 42 in case of the cesium dimer. An estimation for other alkali dimers [15], using the scaling laws for a potential with −C3 /R3 asymptotic behavior, has shown that for a binding energy of 1 cm−1 Tvib varies typically from ∼120 psec for Li2 to ∼550 psec for Cs2 . Since the dependence of the period on the binding energy is |Ev |−5/6 , the ∼100 psec order of magnitude is typical of many PA experiments. The potentials being highly anharmonic, we note the significant variation of the level spacing and vibrational period with the vibrational quantum number. For the discussion on optimization of the chirped pulse, it is relevant to consider the revival period, defined [35] as 4π Trev (v) ≈ , (7.3) |Ev+1 − 2Ev + Ev−1 | and characterizing the coincidences in the vibrational motion of neighboring levels with different periods. Indeed, because the vibrational periods Tvib (v) and Tvib (v − 1) © 2009 by Taylor and Francis Group, LLC

252


TABLE 7.1 Characteristic Constants for Several Levels in the Long-Range Potential Well of the Cs2 0− g (6S + 6P3/2 ) Electronic State, Located within Two Resonance Windows vtot

v

Rout (a0 )

Ev (cm−1 )

159 153 149 137 129 122 29

128 122 118 106 98 92 4

135 RL122 = 148.5 176 107.5 RL98 = 93.7 85.5 30

−0.456 −0.675 −0.869 −1.74 −2.65 −3.57 −70.3

Tvib (psec) 1095 784 635 350 250 196 20.2

Trev (nsec) 37.0 28.5 24.3 15.7 15.3 10 24.5

vtot is the vibrational number, while the numbering v is restricted to the levels in the outer well. Rout is the outer classical turning point, showing the extension of the vibrational motion up to distances of ∼100 a0 . For the two levels v = 98 and v = 122, the outer turning points are labeled RL98 and RL122 , respectively, and the radial wavefunction is drawn in Figure 7.2. Ev is minus the binding energy, Tvib is the classical vibrational period, defined in Equation 7.2, and Trev is the revival period, defined in Equation 7.3. Note the marked variation of the vibrational spacing and of Tvib as a function of the vibrational index, due to the strong anharmonicity of the potential. The set of levels [92–106] defines the resonance window for the 98 defined below, while the set [118–128] defines the resonance window for P 122 . The last line pulses P± ± shows the characteristic constants for a much deeper level.

differ, the motion of the wavepackets for two neighboring levels are out of phase until revival occurs. Also indicated in the table and the figure are the distances Rout , outer classical turning point of the levels within the resonance window. Note the large value of Rout , typical of a long-range molecule. For two levels considered in the discussion below, v = 98 (Rout = RL98 = 93.7a0 ), and v = 122 (Rout = RL122 = 150.4a0 ), the radial vibrational wavefunctions ϕe,v (R) are drawn in Figure 7.2a and d. The strong probability maximum in the vicinity of Rout justifies the picture of the “photoassociation distance.” 7.2.1.4

Description of the Initial Collision State

We have chosen for the initial state at t = tinit a stationary collision wavefunction Ψg (R, tinit ) = ϕg,E (R). Because the kinetic energy E is very small, there is a wide range of distances where the relative motion is completely governed by the potential energy. This is illustrated in Figures 7.2b and c, where the radial wavefunctions ϕg,E for various collision energies are drawn together with the radial wavefunction of the last bound level, v = 53, in the a3 Σ+ u potential. (Note that, as in Refs. 18 and 19, the data for this potential are such that the scattering length is 539 a0 , to be compared with the experimental value of 2440 a0 [36].) © 2009 by Taylor and Francis Group, LLC

253

Control of Ultracold Molecule Formation via Photoassociation R (a0) 0

200

400

(a)

600

800

1000 0.2

ϕe,u

u = 122

0

RL122

–0.2 (b)

u″ = 53 0.32 mK

ϕg,E, l=0(R)

0.33 μK

0.005 0 –0.005 –0.01

37 nK RN

Ra(54 μK)

0.11mK

54 μK

a

0.01

(c) ϕg,E, l=0(R)

0.005 0 RN

–0.005

RL98

u = 98

ϕe,u

(d)

0

20

40

60

80 R (a0)

100

120

140

–0.01 0.2 0 –0.2 –0.4

FIGURE 7.2 Wavefunctions for the initial collisional state and the resonantly photoassociated level and Franck–Condon overlap. (a) Radial wavefunction ϕe,v (R) for the v = 122 vibrational level in the external potential well of the Cs2 0− g (6S + 6P3/2 ) electronic 122 , is indicated by a dashed box, and state. The PA window, centered at RL122 , for the pulses P± coincides with a maximum of ϕg,E (R), drawn in the figure below. (b) Radial wavefunctions ϕg,E (R) for s-wave scattering at various energies E, and for the last vibrational level (v = 53) in the a3 Σ+ u potential. (c) Short-range behavior of the initial states: for R ≤ RN = 82.3a0 , the nodal structure is independent of the energy. For R > RN , the position of the nodes depends upon the energy. Because a > RN , the first node after R > RN is located at R = a for zero energy, and at Ra (54 μK) for E = 54 μK. (d) Same as (a) for the v = 98 level. The 98 , for the pulses P 98 , and indicated by a dashed box, now is close PA window, centered at RL ± to a node of ϕg,E (R). (b and c from Koch, C.P. et al., J. Phys. B: Atom. Mol. Opt. Phys., 39, S1017–S1041, 2006.)

It is clear that for R ≤ RN = 82.3a0 , the nodal structure of ϕg,E is independent of the energy. The energy dependence is concentrated in the normalization factor: this property will be used below in Section 7.3.4 to perform the Boltzmann averaging from calculations at a single energy. It is also clear that because a node of ϕg,E (R) is located at RN , not far from RL98 = 93.7a0 , the Franck–Condon overlap between the level v = 98 and the continuum levels of the initial state will be small (see Figure 7.2d). In contrast, for the level v = 122, the outer turning point coincides with a maximum of the ϕg,E functions, and a larger overlap is expected (see Figure 7.2a). © 2009 by Taylor and Francis Group, LLC

254

7.2.1.5


Position of the “Last” Node RN

The region where the nodal structure of ϕg,E is energy independent is bounded by RN , which is the “last” node of the threshold wavefunction E = 0, except for systems with a scattering length a > RN . For a low-energy s-wave radial scattering wavefunction, in a potential with asymptotic −C6 /R6 behavior, RN can be estimated [37] as a function of the scattering length a through 1 2

b RN

2 = arctan

ηb − a 3π + + pπ. ηb 8

(7.4)

2π 6 1/4 In Equation 7.4, b = ( 2mC ) is a characteristic length, η = Γ(1/4) 2 = 0.477989 is 2 a constant, and m the reduced mass. The integer number p is the smallest integer ensuring a positive value for the right-hand side of Equation 7.4.

7.2.2

CHOICE OF A LINEARLY CHIRPED PULSE IN THE PICOSECOND DOMAIN

In order to deal with a reasonable number of photoassociated levels, to be able to follow the behavior of the various components of the vibrational wavepacket during and after the pulse, and in some cases to shape this wavepacket (see Section 7.4), we have chosen to perform calculations with pulses in the 10 to 100 psec range. The dynamics is studied on a timescale that is still much smaller than the lifetime of the excited state, and therefore spontaneous emission will be neglected. It is also shorter than Tvib , so that the relative motion of the atoms is negligible during the pulse. 7.2.2.1 The Chirped Pulse, Central Frequency, Energy, Spectral, and Temporal Widths Let us consider a Gaussian chirped pulse of energy Epulse , centered at time tP , with a frequency ωL (Figure 7.3b) ω(t) = ωL + χ · (t − tP ) =

d (ωL t + φ(t)), dt

(7.5)

that varies linearly around the carrier frequency ωL . χ is the linear chirp rate in the time domain, so that the phase of the electric field is φ(t) =

1 χ(t − tP )2 + φ(tP ). 2

(7.6)

The laser is red-detuned from the atomic D2 line at ωat by δL = (ωat − ωL ),

(7.7)

and at resonance with bound levels v in the excited state of energy |Ev | ∼ δL . The spectral bandwidth δω, defined as the full-width at half-maximum (FWHM) of the intensity profile, is related to the duration τL of the transform limited pulse with the same bandwidth by δω = 4 ln 2/τL ≈ 14.7 cm−1 /τL (in psec). The instantaneous © 2009 by Taylor and Francis Group, LLC

255

Control of Ultracold Molecule Formation via Photoassociation Amplitude of the electric field

(a) 1

χ=0

0.8

f (t)

0.6 |χ| ≠ 0

0.4 2τL 2τC tP

0.2 0 50

100

(b)

150 t (psec)

0.25

200

250

Instantaneous frequency sweeping 4 χ>0

ћ χ (t–tP) cm–1

2

0

2ћ |χ|τC 2τC

–2

2τL

χ 100% should be fulfilled. The two last lines correspond to pulses for which the impulsive approximation or the opt nad provides an example of nonadiabatic adiabatic approximation are no longer valid, as discussed in Section 7.6. P− optimizes the compression effect, while P− population transfer at large distances.


TABLE 7.2

257


258


Schrödinger equation. In such a unitary treatment, the population is conserved, so that the sum of populations in the lower and in the excited state remains constant. Complete sets of vibrational (bound + continuum) levels in the two electronic states are introduced. Therefore, quantum threshold effects (see Section 7.3.4.2) are automatically accounted for. In this chapter, only s-wave scattering is considered, and one introduces a two-component radial wavefunction Ψ(R, t) describing the relative motion of the nuclei both in the lower electronic state (Ψground (R, t)) and in the excited state (Ψexc (R, t)). ∂ ˆ ˆ mol + W(t))Ψ(t) ˆ HΨ(t) = (H = i Ψ(t). ∂t

(7.13)

ˆ mol = Tˆ + Vˆ el is the sum of the kinetic energy operThe molecular Hamiltonian H ator Tˆ and electronic potential energy operator Vˆ el , with components Vground and Vexc on the lower and excited surfaces, respectively. The coupling term is written in the dipole approximation: ˆ = −D ge (R ) · eL E(t) W (7.14) ge (R ) between the lower and excited elecinvolving the transition dipole moment D tronic state of the molecule and the electric field. The latter is defined by a polarization vector eL assumed to be constant and by an amplitude E(t) (see Equation 7.11). The ˆ due to the oscillations in E(t) is rapid temporal dependence of the Hamiltonian H eliminated in the framework of the rotating wave approximation. Two choices for the reference frequency have been considered. 7.2.3.1

Rotating Wave Approximation with the Instantaneous Frequency: Definition of the Photoassociation Window

One possibility is to use the instantaneous frequency defined in Equation 7.5 to introduce new radial wavefunctions in the two channels Ψg (R, t) = Ψground (R, t) exp(−i[ωL t + φ(t)]/2)

(7.15)

Ψe (R, t) = Ψexc (R, t) exp(+i[ωL t + φ(t)]/2),

(7.16)

where the phase φ(t) was defined in Equation 7.6. When the high-frequency component in the coupling term can be neglected (rotating wave approximation), the coupled radial equations now read ⎞ ⎛ dφ ˆ W (R) + T + V g ∂ Ψg (R, t) ⎟ Ψg (R, t) ⎜ 2 dt i =⎝ dφ ⎠ Ψe (R, t) , ∂t Ψe (R, t) W Tˆ + Ve (R) − 2 dt (7.17) where the potentials Vg (R) and Ve (R), dressed with the carrier frequency, Vg (R) = Vground + ωL /2, Ve (R) = Vexc − ωL /2,


(7.18)


259

cross at the distance RL . The coupling term is real and its time-dependence is restricted to the Gaussian envelope f (t) 1 R ) · eL E0 f (t). W (R, t) = − D( 2

(7.19)

Considering the diagonal terms in Equation 7.17, such that dφ = χ(t − tP ), it is also dt convenient to define time-dependent dressed potentials, χ(t − tP ) = Vground + ω(t)/2, 2 V¯ e (R, t) = Ve − χ(t − tP ) = Vexc − ω(t)/2. 2

V¯ g (R, t) = Vg +

(7.20)

During the time window, their crossing point at a distance RC (t) spans the range of distances χ > 0 → Rmin = RC (tP − τC ),

Rmax = RC (tP + τC ),

(7.21)

χ < 0 → Rmax = RC (tP − τC ),

Rmin = RC (tP + τC ),

(7.22)

defining the PA window [Rmin , Rmax ]. 7.2.3.2

Rotating Wave Approximation with the Central Frequency

An alternative possibility is to consider the central laser frequency ωL , so that the new radial wavefunctions in the two channels are ˜ g (R, t) = Ψground (R, t) exp(−iωL t/2) Ψ

(7.23)

˜ e (R, t) = Ψexc (R, t) exp(+iωL t/2). Ψ

(7.24)

When the high-frequency component in the coupling term can be neglected (rotating wave approximation), the radial coupled equations become ∂ i ∂t

˜ g (R, t) Ψ Tˆ + Vg (R) ˜ e (R, t) = W exp(−iφ) Ψ

W exp(iφ) Tˆ + Ve (R)

˜ g (R, t) Ψ ˜ e (R, t) , Ψ

(7.25)

where the potentials dressed with the carrier frequency have been defined in Equation 7.18, and where the coupling term is no longer real, 1 i R ) · eL E0 f (t) exp − χ(t − tP )2 exp(−iφ(tP )), W (R, t) exp(−iφ(t)) = − D( 2 2 (7.26) but includes a time-dependent phase varying as the phase of the electric field (see Equation 7.6). This formulation of the dynamics has been used in the calculations, while Equation 7.17 was used for the physical interpretation. In fact, for a linearly chirped pulse, © 2009 by Taylor and Francis Group, LLC

260


when the rotating wave approximation is valid, the results of the calculations do not depend upon the choice of the reference frequency, because the probability density ˜ e |2 ) is the same, and because the slowly in each channel (e.g., |Ψexc |2 = |Ψe |2 = |Ψ ∗ ˆ Ψexc exp (−iωL t) is oscillating contribution to the nondiagonal quantity Ψground W ∗ equal to Ψg W Ψe exp(iφ) independent of the model. It is the phase of this nondiagonal quantity that is responsible for the momentum control, described in Section 7.5. The advantage of the rotating wave approximation at the instantaneous frequency is that it leads to the more familiar picture of a two-level system, as described in Section 7.3.3. The coupling term then has a slow time dependence determined by the envelope of the electric field.

7.3

7.3.1

RESULTS OF NUMERICAL SIMULATIONS: INTERPRETATION AS ADIABATIC TRANSFER WITHIN A PHOTOASSOCIATION WINDOW NUMERICAL METHOD

The main ingredients in the numerical solution of the time-dependent Schrödinger equation are given as follows: 1. The numerical representation of the wavefunctions is based on a collocation scheme [39], involving the value of the function at grid points and a set of interpolation functions. For applications to long-range molecules, Kokoouline and colleagues [40] have implemented a Mapped Fourier Grid method, where the grid step is rescaled to the value of the local de Broglie wavelength. In the present work, the interpolation functions are a set of sine functions [31,32]. Using a grid of finite extension L discretizes the continuum, with levels such that ϕg,E (R) has a node at both ends of the grid, and a level density dn L mL 2 = , where m is the reduced mass. A very large extension L is dE En n2 π2 thus necessary to represent an initial continuum state with an energy in the μK range, as well as the neighboring continuum levels and the last bound level of the a3 Σ+ u potential populated during the PA process. For the Cs2 problem considered, a typical grid has an extension L ≈= 20,000a0 and 1024 grid points: energies as low as 36 nK can be represented. 2. The propagation is performed in discrete steps, choosing Δt much shorter than the characteristic times of the problem (pulse durations, vibrational periods, Rabi periods) all larger than 10 psec: in typical calculations, we have chosen Δt ∼0.05 psec. 3. The time-dependent Schrödinger equation is then solved between t and ˆ t + Δt by expanding the evolution operator exp[−iHΔt/] in Chebychev p ˆ polynomials [39,41]. This requires the evaluation of H Ψ for all values of the index p between 1 and the number N of Chebychev polynomials, typically of the order of 100. © 2009 by Taylor and Francis Group, LLC


7.3.2

261

RESULTS OF CALCULATIONS FOR ONE COLLISION ENERGY

After the pulse, bound molecules are formed both in the excited state and in the ground state. As an example, we now present results of numerical calculations for excitation of an ensemble of ground-state cesium atoms, at a temperature T = 54 μK, by the 98 pulse, as described in Table 7.2. P− 7.3.2.1

Photoassociation within the Resonance Window

In Figure 7.4a, we have displayed the probability of population transfer to vibrational levels in the Cs2 0− g (6S + 6P3/2 ) excited state. During the pulse, many levels are populated, including highly nonresonant levels, due to time-energy uncertainty at short times. Rabi oscillations are visible during the pulse. Pulse “transients” are also observed, due to interferences between amplitudes of population coherently transferred to one level at different times: they are discussed in Section 7.3.6. After the pulse, only ≈15 vibrational levels in the excited state remain populated. Such levels lie in the energy range swept by the instantaneous laser frequency, defined previously in Section 7.2.2 as the resonance window. In contrast, the population does not remain in the levels outside the resonance window. The very small value (in the range of 10−4 ) of the population transferred is linked to our choice of a unity normalized initial continuum state in a very large box; most of the pairs of atoms are at very large distances, and are not affected by excitation in the range of the PA window. To be more realistic, on the right vertical scale of Figure 7.4, the number of molecules for a typical trap of volume V = 10−3 cm3 , containing 108 atoms, is estimated including the thermal average procedure described below in Section 7.3.4. The order of magnitude of one molecule per pulse will be discussed below. 7.3.2.2

Formation of Halo Molecules via Optically Induced Feshbach Resonance

After the pulse, there is also population of the two upper vibrational levels of the lower state, v = 52 and v = 53 (see Figure 7.4b); stable molecules are thus formed in a one-color scheme, because the time-dependent frequency of the pulse is sweeping an optical Feshbach resonance. In the dressed potential picture, the initial continuum level of the ground potential V¯ g (R, t) is at resonance with a bound level of the excited potential V¯ e (R, t). Note the efficiency of the process: the number of molecules formed in these two levels of the lower state is equivalent to the number of photoassociated molecules in 15 levels in the excited state. The efficiency of various PA pulses for this population transfer has been discussed in Ref. [19]. The levels v = 53 and v = 52 are respectively bound by 5 × 10−6 and 0.042 cm−1 , to be compared with the resonance window of ∼1.74 cm−1 in the excited state. Due to the very small value of the binding energy, these molecules are “halo molecules,” as defined by Koehler and colleagues [8]: their creation as a byproduct of the PA process should be further investigated. Recently, Kallush and Kosloff [28] have discussed the nonperturbative character of the PA process, where the conservation of the total population requires a © 2009 by Taylor and Francis Group, LLC

262

Cold Molecules: Theory, Experiment, Applications Excited-state population

(a)

Peb,oux(t)

(b) Nue(t) 0.8

Pulse

Lower-state population

Pbg,oux″(t)

u≤ = 52

u = 95 – 107 3 × 10–4

0.6 1.5 × 10–4

2 × 10–4

0.4

1 × 10–4

u≤ = 53

1 × 10–4

0.2 0.5 × 10–4

100

150 200 t (psec)

250

0.0 50

0.3

0.2

0.1 tP

tP u = 122 – 135

0.0 50

Nug″ (t) 0.4

Pulse

100

150 200 t (psec)

250

FIGURE 7.4 Molecules formed by PA within the resonance window and by an optical Feshbach resonance. (a) Probability of population transfer Pbox e,v (t), during and after the pump 98 pulse P− , into several vibrational levels v of the excited state. The levels v = 95–107 (solid line) lie in the resonance window and remain populated after the pulse. In contrast, for the levels outside the resonance window (dash–dotted line), no population remains after the pulse. (b) Population transferred Pbox g,v (t) in the two last vibrational levels v = 52 (solid line) and

v = 53 (dash–dotted line) of the a3 Σ+ u state by sweeping an optical Feshbach resonance. The initial state is a stationary collision state, at energy E = kB T , where T = 54 μK, unity normalized in a box with L = 19,250a0 . The scale on the left vertical axis of each figure is the population transfer probability, for an initial state unity normalized in the box, while the scale on the right vertical axis is the number of photoassociated molecules for a trap of volume V = 10−3 cm3 , containing 108 atoms, after a thermal average for T = 50 μK either in g the excited state Nve (t) or in the ground state Nv (t)(see Section 7.3.4). (From Luc-Koenig, E. et al., Phys. Rev. A, 70, 033414, 2004. With permission.)

significant modification of the initial state, with population of bound levels (discussed here) and of continuum levels (as discussed further in Section 7.5.4). 7.3.2.3

Selectivity of the Resonance Window: Dependence of the Final Distribution of Population on the Pulse Parameters

We have shown an example of results where the photoassociated molecules are selectively formed in vibrational levels within the resonance window. It is interesting to discuss the robustness of this concept by varying the pulse parameters, as illustrated in Figure 7.5. 122 show that the levels populated For negative chirp, calculations with the pulse P− − in 0g (6S + 6P3/2 ) (see Figure 7.5a) stay within the resonance window when the intensity IL is increased to 9 IL . Moreover, the population depends weakly upon IL . © 2009 by Taylor and Francis Group, LLC

263

Control of Ultracold Molecule Formation via Photoassociation (a)

(b) 5 × 10–3 PA window

1 × 10–4

4 × 10–3

Ng(u") = | 0, this recycling effect is much weaker: once a level is populated, the instantaneous frequency is no longer resonant with its energy, and the population remains in this photoassociated level. In contrast, the population of the last bound level in the lower state is insensitive both to the sign of the chirp and to the intensity (see Figure 7.5b). Finally, we must stress that the concept of the resonance window is linked to an adiabatic transfer, as will be explained in the following section (Section 7.3.3), and that a minimum intensity is required. © 2009 by Taylor and Francis Group, LLC

264


7.3.3 ANALYSIS WITHIN A TWO-STATE MODEL: THE CONCEPT OF ADIABATIC TRANSFER WITHIN A PHOTOASSOCIATION WINDOW When the duration of the pulse is short compared to the vibrational period, that is, when τC Tvib , which is fulfilled in the present calculations, the impulsive approximation [43] is valid. One can then ignore the motion of the nuclei during the pulse, so that the kinetic energy term may safely be neglected in Equation 7.17, allowing interpretation of the numerical results in the framework of an R- and t-dependent two-level system. Following the notation of Ref. [44], we define for each internuclear distance R and each time t a splitting between the lower and the excited state 2Δ(R, t) = Ve (R) − Vg (R) − χ(t − tP ) = Vexc (R) − Vground (R) − ωL − χ(t − tP ),

(7.27)

so that 2Δ(∞, tP ) = δL , and a coupling term, due to the laser, written W (t) = WL × f (t),

(7.28)

provided we neglect the R-dependence of the transition dipole moment. It is then straightforward to diagonalize the matrix

Δ(R, t) W (t)

W (t) , −Δ(R, t)

(7.29)

by introducing a mixing angle for the adiabatic basis through tan θ(R, t) =

|W (t)| . Δ(R, t)

(7.30)

If the adiabatic picture is valid, at distances where Δ(R, t) changes sign, that is within the PA window, the two dressed states should change their character between the beginning and the end of the pulse, allowing population inversion. The dressed potential curves are drawn in Figure 7.6a, and their crossing point RC (t) spans the PA window [Rmin , Rmax ], as discussed above. The mixing angle θ(R, t), computed from Equation 7.30, is displayed in Figure 7.6b, for three values of the time (t = tP − τC , tP , tP + τC ); within the PA window, θ(R, t) varies by π, thus verifying the condition for population inversion. Outside this range, the population transferred to the excited state during the pulse goes back to the lower electronic state. This simple model, combined with the reflection approximation, where a vibrational level in the excited state is assumed to be populated through a vertical transition at © 2009 by Taylor and Francis Group, LLC

265

Control of Ultracold Molecule Formation via Photoassociation Δ(R,t) < 0

(a)

Resonance window 2ћ|χ|τC

Δ(R,t) > 0 − V g(t)

χ 50 μK, and the Wigner threshold law (C −2 (E) ∼ E) is observed at small energy. For sufficiently large scattering length |a|, a strong enhancement of the scaling factor is observed in the intermediate energy region. The values of the scattering length are equal to, respectively, 3650a0 for 85 Rb2 [47], 90.6a0 for 87 Rb [48] and 2440a for 133 Cs (long broken medium line) [36]. The dot-dashed medium 2 0 2 line corresponds to the value a = 538a0 for the 133 Cs2 scattering length, chosen in the present chapter and markedly reducing the quantum enhancement effect. A systematic description of collisions in ground singlet and lowest triplet potentials of Rb2 and Cs2 dimers is presented in Ref. [37]. (From Koch, C.P. et al., J. Phys. B: Atom. Mol. Opt. Phys., 39, S1017–S1041, 2006. With permission.) 98 to P 122 is 1. Effect of the detuning. The increase of the PA efficiency from P− − mainly due to a larger density of atom pairs in the region of the PA window; the ratio 0.0026/0.0003 in the probability density is comparable to the ratio 20/3.2 obtained for the PA probability Pebox in Table 7.2. As illustrated in Figure 7.2, in the first case the PA window is located in the vicinity of a node of the initial stationary wavefunction ϕg,E , in the second case in the vicinity of a maximum, leading to a better Franck–Condon overlap. 2. Effect of the chirp sign. Changing the sign of the chirp does not significantly 98 , since total adiabatic population transfer change the results obtained for P− within the PA window is effective and since the number of Rabi oscillations during the pulse (1.5) is small; the final population transferred by the pulse


270

Cold Molecules: Theory, Experiment, Applications 98 is slightly larger (7% increase). In contrast, for P 122 , the final population P+ + 122 in agreement with the transferred is much larger (45% increase) than for P− increase in Pebox (see Table 7.2). Indeed, although the adiabaticity condition is better satisfied for this pulse (parameter A in the table), the large number of Rabi oscillations (5.3) results in an important recycling of the population for the negative chirp. The resonant condition of the laser excitation RC (t) is “following” the wavepacket motion, toward short distances, in the excited potential. In contrast, for a positive chirp, the population resonantly excited to Vexc at time t is out of resonance with the optical frequencies for t > t, preventing recycling.

When all partial waves are considered the number of molecules per pulse increases to 1.4 and 10, respectively. In Ref. [24] results for rubidium PA are also presented, revealing a slightly less favorable situation for the typical number of molecules formed by one pulse. In the latter case the PA occurs at shorter distances, with a different mechanism (resonant coupling), so that direct comparison is not possible. The previous discussion is restricted to one pulse. In an experiment, the total number of molecules will depend upon the repetition rate of the laser and the efficiency with which the dynamical hole in the initial scattering wavefunction, carved out due to full population transfer within the PA window, and discussed below in Section 7.5, will be refilled. Nevertheless, for the conditions of a typical MOT, the number of molecules formed by PA with a cw laser is about 106 per second. Using the chirped pulses we have considered, with a repetition rate of 108 Hz (10 nsec between two pulses), and assuming that one pulse is not destroying the molecules formed by the previous one, the number of molecules should be one to two orders of magnitude larger than for cw PA.

7.3.6 TRANSIENT EFFECTS During the pulse there is a coherent transfer of population from the initial state to the electronically excited state. The partial amplitude of population transferred during the time-interval [t, t + dt] coherently interferes with the amplitude of population already transferred at t < t. For a level v, the partial transfers are maxima at tP , where v , when the level is resonantly excited, the instanthe intensity is maximum and at tres taneous crossing point RC (t) being located at its outer turning point. As illustrated in Figure 7.8, oscillations are observed for the levels that come to resonance with the instantaneous frequency of the laser ω(t) before the maximum of the pulse. The amplitudes of the oscillations are strong for χ < 0, where the motion of the wavepacket in the excited state follows the laser excitation. For χ > 0 the vibrational motion in the excited state prevents deexcitation at t > t of the population transferred at time t. Such oscillations are a signature of significant nonadiabatic effects in the population transfer, which explains that they are observed during a time larger than the time window. The adiabaticity criterion is not fulfilled at the instantaneous crossing point in © 2009 by Taylor and Francis Group, LLC

271

Control of Ultracold Molecule Formation via Photoassociation box(t) Pe,u

box(t) Pe,u

tP

c 98 (v < 98) are resonant before tP , and levels v < 98 (v > 98) are resonant after tP . Strong oscillations in the population are observed for levels resonantly excited before tP , especially for a negative chirp rate χ, when the crossing point RC (t) of the laser excitation follows the vibrational motion of the excited wavepacket toward shorter distances R. (From Luc-Koenig, E. et al., Phys. Rev. A, 70, 033414, 2004. With permission.)

the wings of the pulse. “Coherent transients” have been previously studied in the perturbative limit in the coherent excitation of a two-level system with a linearly chirped pulse [49], and have been recently observed in time-dependent PA of rubidium [50].

7.4

SHAPING VIBRATIONAL WAVEPACKETS IN THE EXCITED STATE TO OPTIMIZE STABILIZATION INTO DEEPLY BOUND LEVELS OF THE LOWER STATE

When a chirped laser pulse is used for the PA process, a coherent wavepacket is formed in the excited state, and has components in all the vibrational levels within the resonance window. After the pulse, this wavepacket propagates toward short distances. Because population transfer back to the initial state with a second (dump) pulse is a coherent process, it is convenient to use this property to optimize the formation of ultracold stable molecules. For instance, in the chosen example of Cs2 0− g (6S + 6P3/2 ), the time-dependent Franck–Condon overlap with the bound levels in the lower electronic state can be optimized by achieving a focused wavepacket. © 2009 by Taylor and Francis Group, LLC

272

7.4.1


SHAPING THE VIBRATIONAL WAVEPACKET IN THE EXCITED STATE

Indeed, for a given vibrational level v of a photoassociated molecule, the transfer of population to bound levels of the lower electronic state is optimal at the inner turning point, half a vibrational period after the excitation. Therefore the dump pulse should be delayed by Tvib /2. However, because the 0− g (6S + 6P3/2 ) potential is strongly anharmonic, the vibrational period Tvib varies markedly from one level to the next one (see Table 7.1). This can be compensated by choosing a negative chirp parameter for the PA pulse, in order to populate first the levels with a larger vibrational period (see Figure 7.9). Due to a scaling law on Tvib , and on the revival time Trev , a linear chirp χfoc can be chosen [18,23] so that after the pulse all the components reach the inner barrier at the same time td = tP + 21 Tvib (vL ), where vL is the level resonant at the maximum of the pulse. One thus obtains a focused wavepacket, as displayed in Figure 7.10a. Previous work has suggested ways of designing laser fields for generation of spatially squeezed molecular wavepackets [51,52]. Because they considered low vibrational levels, a numerical optimization procedure appeared to be necessary. Here, we are dealing with long-range molecules, where the motion is controlled by the asymptotic potential, and the availability of scaling laws makes the optimization easier, as for atomic Rydberg wavepackets.

7.4.2

PROPOSAL FOR A TWO-COLOR PUMP–DUMP EXPERIMENT

In Ref. [22], the possibility of creating stable molecules via a two-color pump–dump experiment has been discussed in detail. The vibrational wavepacket Ψe (R, t = tP +

Vibration of the wavepacket

Rd(u1) Rd(u2)

u1 u2

RC (u1)

E1 E2

RC (u2)

t1

t2 Photoassociation td Stabilization

Chirped pulse

FIGURE 7.9 How to create a focused vibrational wavepacket: the PA pulse coherently excites vibrational levels with slightly different periods Tvib . With a negative chirp, higher levels (v1 ) with a longer period Tvib (v1 ) are populated first (t1 ), lower levels with a shorter period Tvib (v2 ) being populated later (t2 ). After half a vibrational period, provided the time delay t2 − t1 compensates for the difference (Tvib (v1 ) − Tvib (v2 ))/2, the partial wavepackets corresponding to various vibrational levels simultaneously reach the inner turning point at the same time td . (From Koch, C.P. et al., Phys. Rev. A, 73, 033408, 2006. With permission.)



273

(a) 0.005 | Ψe (R, td = tP + Tvib/2) | 0.004 0.003

χ = χfoc

0.002 χ=0

0.001 0.000

0

50

100

150

200

R (a0) (b)

Maximum probability to form bound stable molecules

χ = χfoc

0.2

0.1 tP

td

χ=0

0 –100

0

100

200 t (psec)

300

FIGURE 7.10 (a) Example of a focused wavepacket |Ψe (R, td )| (thick solid line) obtained in the excited state half a vibrational period, td = tP + 21 Tvib (vL ), after PA with the negatively 98 , which corresponds to the chirp χ chirped pulse P− foc and has a maximum intensity at t = tP . For comparison, the wavepacket obtained by photoassociating with a transform-limited pulse P098 with χ = 0 is drawn in a thin solid line. Note the strong increase of population (typically ∼8) due to the chirp. (From Luc-Koenig, E. et al., Phys. Rev. A, 70, 033414, 2004. With permission.) (b) Variation of the time-dependent Franck–Condon factor F (tdyn ), defined in Equation 7.36. (From Koch, C.P. et al., Phys. Rev. A, 73, 033408, 2006. With permission.) 98 , hereafter referred to as the pump pulse, moves freely τC ), created by the pulse P− in the excited state potential for a while; then a conveniently delayed second (dump) pulse transfers the population to the lowest triplet state. The choice for the optimal delay is discussed below.

7.4.2.1 The Time-Dependent Franck–Condon Overlap According to the value chosen for the time tdyn = t − tP , one finds very different values for the Franck–Condon overlap matrix elements between the wavepacket Ψe (R, t = tP + tdyn ) and the stationary vibrational wavefunctions ϕg,v (R) of the bound levels in the initial state. We define time-dependent Franck–Condon overlap © 2009 by Taylor and Francis Group, LLC

274


factors Fv (tdyn ) = | < Ψe (R, t = tP + tdyn )|ϕg,v (R) > |2 F(tdyn ) =

=53 v

Fv (tdyn ),

(7.36)

v =0

where F(tdyn ) is the sum of the overlap factors with all the bound levels in the lower state. Its time dependence is displayed in Figure 7.10b. Just after the PA pulse, at time tP + 40 psec ∼ tP + τC , the factors Fv (tdyn ) are negligible for most bound levels of the lower state, except for the last level v = 53, which is very loosely bound, so that F(tdyn ) is negligible. During the motion of the wavepacket Ψe (R, t = tP + tdyn ) toward short distances, the time-dependent Franck–Condon factors increase, especially when the wavepacket is localized in the vicinity of the inner turning point. Due to the focusing effect, F(tdyn = td ) has indeed a strong maximum in a short time period around td = Tvib /2. However, this maximum is far from 1; at most 20% of the population of the excited state can be transferred to bound levels of the a3 Σ+ u state. The rest is transferred to continuum levels, creating pairs of ground-state atoms. This conclusion is due to the marked difference between the excited and the a 3 Σ+ u state potentials. A situation where the two potentials are more alike (for instance in the case of heteronuclear alkali dimers) would allow a larger percentage of transfer to bound levels. The importance of the spectroscopy is again manifest. 7.4.2.2 A Two-Color Experiment for Creating Stable Molecules The proposed experiment, as described in Ref. [22], is given schematically in Figure 7.11. A first (pump) pulse creates a shaped wavepacket in the excited state. After half a vibrational period, when the focusing effect is maximum, a second (shorter) pulse transfers the population into bound levels v of the a3 Σ+ u state. Two strategies, described in Ref. [22] are possible: 1. Choosing short dump pulses, it is possible to transfer a maximum of population toward several bound levels in the a3 Σ+ u electronic state. For instance, 98 , a short π pulse in the 15 fsec range can dump all when the pump pulse is P− the population to the lower state. Then, at most 20% of this population stays in bound levels, while the remaining 80% goes into the continuum, mainly as pairs of hot ground-state atoms. Chirping the dump pulse is efficient in suppressing Rabi oscillations in the intensity dependence. 2. By narrowing the spectral width of the dump pulse, it is possible to populate only bound levels, and even to select a single level. In the chosen example, a dump pulse in the 5 psec range selectively transfers population to the rather deeply bound level v = 14 (binding energy = 113 cm−1 ). It is not possible to go directly to the v = 0 level, because the outer well in the


275


tP + 40 psec

11,900

tP + 250 psec

tP + 120 psec = td 11,800

Energy (cm–1)

11,700

t=0

100

tP + 40 psec

0 –100 –200 20

40

60

80

100

120

Internuclear distance (a0)

FIGURE 7.11 A two-color pump–dump experiment. The two potentials Vground (R) and Vexc (R) are drawn as thin lines. The square of the stationary wavefunction in the a3 Σ+ u triplet state before the pulse is drawn as a full line. The PA pulse (on the right side with an up arrow) transfers population to the 0− g (6S + 6P3/2 ) electronic state. It has a maximum intensity at t = tP . After this pump pulse, at t = tP + 40 psec, a vibrational wavepacket has been created in the excited state, the modulus |Ψe (R, td )|2 is drawn as a long dashed line, while a hole has been carved in |Ψg (R, t)|2 (long dashed line). Half a vibrational period later, at t = td , a focused wavepacket (thin line) has reached the inner turning point and is optimal for population transfer to the a3 Σ+ u state by the dump pulse (left side down arrow). After one vibrational period at t = tP + 250 psec, the wavepacket in the excited state is back to the outer turning point (dash–dotted line). (From Koch, C.P. et al., Phys. Rev. A, 73, 033408, 2006. With permission.)

excited curve is located at large distances from the minimum in Vground (R). In such a situation, the dump pulse at td = tP + Tvib /2 transfers only 12% of the population to the a3 Σ+ u state; the remaining part stays in the excited state, can decay by spontaneous emission, or could be transferred, one vibrational period after td , at t = tP + 3Tvib /2, when the vibrational wavepacket is focused for the second time, by a another dump pulse toward bound levels in the lower state. Designing a series of pump and dump pulses then seems an interesting perspective. The previous discussion shows again the crucial importance of the spectroscopy of the systems considered.


276


7.5 THE DYNAMICAL HOLE IN THE INITIAL STATE WAVEFUNCTION: COMPRESSION EFFECT If a second pump pulse identical to the first one is illuminating the system after a short delay, its efficiency should be markedly reduced, because all the pairs of atoms at distances R within the PA window [Rmin , Rmax ] have been transformed into excited molecules. This void in the pair wavefunction of the initial state, hereafter identified as the “dynamical hole,” is well known from previous work on coherent control [43,53] as one of the main outcomes of excitation of a molecular system with a short laser pulse. It is associated to a “momentum kick.” The effect is also present in the ultracold regime [27] and has been analyzed in detail in Ref. [38]. A look at Figure 7.1 shows that in the region of the PA window, the potential in the initial state is negligible, so that the classical acceleration is zero, whereas this is not the case for the excited state where the Vexc (R) potential has an R−3 asymptotic behavior.

7.5.1

PHENOMENOLOGICAL OBSERVATION OF A DEPLETION HOLE, A MOMENTUM K ICK, AND A COMPRESSION EFFECT

The generation of the depletion hole on the initial a3 Σ+ u electronic state due to PA with a short pulse can be understood by employing a phase-amplitude representation for both the lower- and the excited-state wavepackets,

iSg/e (R, t) Ψg/e (R, t) = Ag/e (R, t) exp R pg/e (R, t)dR i , = Ag/e (R, t) exp

(7.37)

where, within a semiclassical interpretation, pg (R, t) (pe (R, t)) is the local momentum in the lower (excited) state [54,55]. Let us emphasize that a nonvanishing local momentum exists in the spatial range where there is a R-variation in the phase Sg/e (R, t) of the wavepacket. From the numerical calculations reported above, the evolution of the amplitude 122 is displayed in Figure 7.12. It is clear Ag (R, t) after illuminating with the pulse P− that after the pulse a hole has been carved out in the region of the PA window, defined in Table 7.2 as 148a0 < R < 176a0 . In this region, the potential and the classical acceleration are negligible in the lower state. After the pulse, the hole starts moving to smaller internuclear distances at a velocity ∼4.2 m/sec, two orders of magnitude larger than the classical velocity of the atoms colliding at T = 54 μK in the asymptotically R−6 a3 Σ+ u potential. The motion is restricted to a localized part of the wavepacket, the wavefunction at distances larger than Ra = 314a0 not being modified, and the node at Ra remaining fixed. (As indicated in Figure 7.2, Ra (54 μK), denoted as Ra below, is the first node in the “outer” region for the scattering state at 54 μK, corresponding to the node at R ∼ a in the threshold wavefunctions at T ∼ 0 K. In the outer region the position of the nodes is energy-dependent). © 2009 by Taylor and Francis Group, LLC

Control of Ultracold Molecule Formation via Photoassociation |Ψg(R,t)|

RN

PA window

tinit

1 × 10–2

277

t = tP

5 × 10–3 0 1×

10–2

t = tP + τC = tP + 110 psec

5 × 10–3 0 1×

10–2

t = tP + 200 psec

5 × 10–3 0 1×

10–2 t = tP + 350 psec

5 × 10–3 0 1×

10–2

5 × 10–3

t = topt = tP + 850 psec

0 1×

10–2 t = tP + 1300 psec

5 × 10–3 0

0

100 Rmin

200

300

Rmax

400

Ra R (a0)

FIGURE 7.12 Dynamical hole and compression effect. Top panel: amplitude |Ψg (R, t = tinit )| (dotted grey line) of the initial stationary scattering wavefunction for a pair of ground-state cesium atoms colliding with energy E = kB T , with T = 54 μK, and zero angular momentum, in the absence of an external field. Note the position of the “first outer node” at Ra and the last inner node at RN . Top panel: amplitude |Ψg (R, t = tP )|(thick black line), when illuminating 122 , at the maximum of the pulse t = t ; the pulse has carved out a hole in the with the pulse P− P initial wavefunction. This hole is indeed located in the region [Rmin , Rmax ] of the PA window, delimited by the two vertical lines. Next panels: after the pulse, the Ψg (R, t) “wavepacket” moves inwards: note the increase of the maximum value of its amplitude in the inner region. At t − tP ∼ 850 psec, the wavepacket has been partly reflected by the a3 Σ+ u inner wall located at R ∼ 10a0 . Note that this reflection takes place on a timescale close to the classical vibrational half-period (∼[300,600] psec) for the photoassociated levels in the excited state. (From Mur-Petit, J. et al., Phys. Rev. A, 75, 061404(R), 2007. With permission.)

In contrast, we observe an increase of the probability density at distances shorter than the PA window, manifesting a “compression” of the wavepacket. A secondary maximum is created, which moves toward the repulsive wall of the lower state potential on a timescale of a few hundred picoseconds, typical of half the vibrational period © 2009 by Taylor and Francis Group, LLC

278


in the excited state. Due to the laser coupling between the two states, a “momentum kick” has indeed been given to the initial state wavepacket. The “compression” effect is maximum at t = topt , and later the wavepacket is reflected on the inner wall of the potential.

7.5.2 ANALYSIS OF THE MOMENTUM TRANSFER WITH PARTIALLY INTEGRATED MASS CURRENT AND POPULATION An analysis related to the classical limit of the Bohmian dynamics [56] has been proposed in Ref. [38]. It considers the time variation of the local probability current in each channel defined as ∗ ∂Ψg/e (R, t) ∂Ψg/e (R, t) ∗ mjg/e (R, t) = Ψg/e (R, t) − Ψg/e (R, t) 2i ∂R ∂R = pg,e (R, t)|Ψg,e (R, t)|2 ,

(7.38)

and the local probability density |Ψg/e (R, t)|2 = [Ag/e (R, t)]2 . Both quantities are partially integrated in the region R < Ra (54 μK) which includes the PA window. The position Ra (54 μK) of the first node in the outer region, hereafter referred to Ra , has been discussed in the previous section. We define a partially integrated mass current Ra part mIg/e (t) 0

=

Ra

pg/e (R, t)|Ψg/e (R, t)|2 dR,

(7.39)

0

and a partial population, between 0 and Ra , in each channel: Ra part Ng/e (t) = 0

Ra

|Ψg/e (R, t)|2 dR.

(7.40)

0

The average value of the momentum gained during the pulse by the inner part of the lower- or excited-state wavepacket, that is, the momentum kick, can be evaluated from the mass current and the partial population by part

part

< pg/e (t) >=

mIg/e (t) part

.

(7.41)

Ng/e (t)

As illustrated in Figure 7.13, such analysis has been performed in the case of PA 122 with negative chirp. If one compares the partially integrated mass by the pulse P− current and population to the same quantities integrated over the complete extension L of the box, mIg/e (t) and Ng/e (t), one observes that during the time window there is a large exchange of population between the two channels mainly located at © 2009 by Taylor and Francis Group, LLC

279


large distances R > Ra . In contrast, the momentum exchange is localized to a short part distance because mIg/e (t) ∼ mIg/e (t). Indeed for R > Ra , the amplitude Ag/e (R, t) and the phase Sg/e (R, t) in the wavepackets become R-independent. This formalism is consistent with the concept of the PA window, most convenient once the delocalized character of the initial state is introduced in the model. It is more efficient than the procedure described in Refs [9,27], which consider physical quantities integrated over the whole R-domain and use the Heisenberg equation to implement local control theory. part

part

A comparison of the time variation of mIg/e (t), Ng/e (t), in Figure 7.13.

part

dNe dNe dt , and dt

is reported

1. At the beginning of the time window, for tP − τC < t < tP , the mass currents mIg and mIe in the two channels are oscillating below and above their mean value with a π phase difference with opposite phase, while their sum is not oscillating. Due to the large negative acceleration in the excited potential this negative current is smoothly increasing in absolute value. On average, the two channels are equally sharing the strong classical acceleration −3 asymptotic behavior. due to the 0− g (6S + 6P3/2 ) potential with R 2. Then, we observe that for tP < t < tP + τC the mass current gain in the lower state becomes larger than in the excited state: the acceleration is no longer equally shared between the two channels. part 3. After the pulse, for t > tP + τC , Ig remains constant during a long time interval of ∼600 psec. This can be explained by the negligible value of the part classical force in the lower state. Because the population Ng no longer varies, this mass current corresponds to a constant mean velocity of the order of −2.7 m/sec. By comparing the results obtained with various pulses, Ref. [38] concludes that the creation of an important flux of population toward short distances is favored both by an important population cycling during the pulse and by a significant transfer of this population back to the initial state. In other words, the optimal pulse should be as close as possible to a (2n)π-like pulse, with a negative chirp, and a large number n of cycles. This can be achieved with a pulse that is “following” the excited wavepacket over a long time and causes many complete cycles of Rabi oscillations. The choice of an “optimal” pulse is described in Section 7.6.1.

7.5.3 ADVANTAGE OF THE COMPRESSION EFFECT FOR PHOTOASSOCIATION WITH A SECOND PULSE To explore the possibilities offered by the compression of the wavepacket, we have represented in Figure 7.14 the variation of the Franck–Condon overlap between Ψg (R, t = topt ), when the compression effect is maximum, and various stationary wavefunctions ϕe,v of the bound vibrational levels in the external well of the excited state. Important overlap is obtained with almost all levels in the outer well


280

Cold Molecules: Theory, Experiment, Applications Time window 2 × 10–3 1 × 10–3 mI part(t)

0 –1 × 10–3 –2 × 10–3 –3 × 10–3

N part(t)

8 × 10–3 6 × 10–3 4 × 10–3 2 × 10–3

dNe/dt(t)

0 5 × 10–3

0 –5 × 10–3 0

200

400

600

800

1000

t (psec) 122 maximum at t = t = 350 psec. The time FIGURE 7.13 Analysis of PA with the pulse P− P window [tP − τC , tP + τC ] is indicated by the horizontal arrow: outside this time window, part there is no radiative coupling. Upper panel: the variation of the mass currents mIg (R, t) in part

the lower state (long dashed line), mIe (R, t) in the excited state (solid line), and of their sum (dashed–dotted line) is displayed as a function of time. Note the Rabi oscillations during the pulse. In the region of the PA window, the classical acceleration is zero in the ground state. part After the pulse, Ig remains constant but negative over a long time delay. The inner part of the wavepacket is moving to short distances with a constant average velocity of ∼2.7 m/sec. Later (not visible in this figure) it is accelerated by the short-range potential and reflected. In the excited state, the wavepacket is moving to short distances with a large velocity, and reflected after 600 psec. Middle panel: time variation of the partial populations in the lower part part (Ng (t), long dashed line) and excited (Ne (t), full line) state during the pulse. Note the part

part

Rabi oscillations with opposite phase, while the sum Ng (t) + Ne (t) (dash–dotted line) remains constant. Lower panel: comparison of the time variation of the total population in part

dNe e the excited state dN dt (solid line) and of the partial population dt (broken line). During the pulse there is a large temporary transfer of population, mainly at large distances, outside the PA window. (From Mur-Petit, J. et al., Phys. Rev. A, 75, 061404(R), 2007. With permission.)



281

of Vexc . The deepest level populated is v = 29, vext = 4, with a binding energy of 70 cm−1 and an inner turning point at R = 30a0 . Such a level has a good Franck– Condon overlap with the v = 42 and 33 levels of the a3 Σ+ u potential, respectively bound by 3.7 and 20.5 cm−1 , which could improve the efficiency of the stabilization step. In contrast, for the initial wavepacket Ψg (R, t = tinit ), non-negligible overlap is obtained only with vibrational levels close to the dissociation limit in the Vexc potential. As discussed in Ref. [38], the compression effect suggests PA with a second PA pulse red-detuned relative to the first one, and populating deeply bound levels in the excited state. In favorable cases, the latter levels could contribute efficiently to the stabilization step toward deeply bound levels of the lower electronic state, and hence to the formation of vibrationally cold stable molecules. Indeed, the bottleneck in the PA reaction being the small probability density for pairs of cold atoms at short relative distance, the existence of a compression effect seems an interesting procedure to improve PA.

7.5.4

REDISTRIBUTION OF POPULATION IN THE LOWER STATE PHOTOASSOCIATION PULSE

AFTER A

After the pulse, the presence of a dynamical hole means that the wavepacket in the a 3 Σ+ u state differs from the initial stationary collision state, with well defined collision energy Einit ; it has now significant projection on bound levels ϕg,v and on continuum levels with different energy.

7.5.4.1

Redistribution in the a3 Σ+ u State in the Case of Cesium Photoassociation

Population of the two last bound levels has already been mentioned in Section 7.3.2.2 and illustrated in Figure 7.4. Redistribution of the population in various continuum levels of the Vground potential is analyzed in Figure 7.15 for different intensities and sign of the chirp. The density of population dNg (E )/dE for energy-normalized continuum levels is independent of the sign of the chirp; in contrast, when the maximum intensity of the pulse is increased by augmenting IL (see Equation 7.8), the width of the redistribution of collision energies is clearly reduced, the area below the curve being roughly conserved. As for the creation of halo molecules (see Section 7.3.2.2), it is independent of the pulse parameters.

7.5.4.2

Correlated Pairs of Hot Atoms

Due to Rabi cycling during the pulse, the redistribution of population in the lower state creates correlated pairs of hot atoms, as illustrated in Figure 7.15. Possible applications to condensates are being discussed. © 2009 by Taylor and Francis Group, LLC

282

Cold Molecules: Theory, Experiment, Applications 1 × 10–2 8 × 10–3

Ngpart|0R(t)

PA window

6 × 10–3 4 × 10–3 2 × 10–3

|Ψg(R,t)|

0 2 × 10–2

1 × 10–2

0

25

50

75

0

100 125 150 175 200 225 250 275

||2

R (a0)

2 × 10–4 1 × 10–4 0

–80

–70

–60

–50

–40

–30

–20

–10

Eu (cm–1)

FIGURE 7.14 The compression effect and its optimization; various quantities are compared related to the initial wavepacket for two colliding atoms (dashed lines), and to the compressed wavepackets when the compression effect is maximum, at topt = tP + 950 psec, after illu122 (solid lines). Also displayed are the compressed wavepackets, minating with the pulse P−

= t + 350 psec, after illuminating with the optimized pulse P now at topt P − (shaded curves). opt

part

Upper panel: integrated population Ng |R 0 (t) = Int for the initial collision state (dashed line); due to a quantum reflection effect ([57] and Chapter 6 by Paul Julienne), the probability of finding two cesium atoms at distances shorter than ∼ 100a0 is negligible; after a PA pulse, at (shaded curve)), this the maximum of the compression effect (t = topt (solid line) and t = topt probability is no longer negligible. Middle panel: wavepacket amplitude |Ψg (R, t = tinit )| of the initial collision state (dashed line); lower state wavepacket |Ψg (R, topt )|, after illuminat122 (solid line), and |Ψ (R, t )| after illuminating with the pulse P opt ing with the pulse P− g − opt (shaded curve). Lower panel: variation of the overlap | < ϕe,v |Ψg (R, t) > |2 with the eigenfunctions for all the bound vibrational levels v in excited potential curve, as a function of their binding energy Ev , for the initial state at t = tinit (dashed line) and for the wavepackets created (shaded curve). In contrast with the initial wavepacket, the at t = topt (solid line), and t = topt compressed wavepackets display large overlaps with the lower vibrational levels of the excited potential. The shaded areas in the three panels demonstrate how the compression effect can be opt markedly improved with use of the nonimpulsive pulse P− , by a duration τC comparable to half the vibrational period in the excited state, described in Section 7.6.1. (From Mur-Petit, J. et al., Phys. Rev. A, 75, 061404(R), 2007. With permission.) © 2009 by Taylor and Francis Group, LLC

283


7.6

BEYOND THE IMPULSIVE OR ADIABATIC APPROXIMATIONS: NEW MECHANISMS

The present chapter has focused on an excitation mechanism where population transfer from the lower to the excited state takes place in the impulsive approximation within a PA window. As discussed in Refs [18,19,23,38], other mechanisms do exist, and might have interesting applications.

7.6.1

CONTROLLING THE COMPRESSION EFFECT WITH A NONIMPULSIVE PULSE INDUCING MANY RABI CYCLES

From the conclusion of Section 7.5, a pulse has been designed to optimize the comopt pression effect in the ground-state wavepacket. The new pulse, described as P− 122 by increasing the chirp rate χ and therefore the in Table 7.2, is obtained from P− duration. For this pulse, the duration τC = 376.13 psec is comparable to half the vibration period Tvib in the excited state. The intensity was increased to obtain 24 Rabi oscillations, close to a (2n)π-pulse. The probability of excitation Pebox is markedly decreased (3 × 10−4 instead of 2 × 10−3 ). As displayed in Figure 7.14, a spectacular = t + 350 psec, is observed. The compressed compression effect, maximum at topt P wavepacket is associated with large values for the time-dependent Franck–Condon factor with low-lying levels in the external well of the excited state, suited for PA with opt a second pulse red-detuned from P− and delayed by 350 psec.

dNg/dE' (E' )= |

(c)

>

|1

>

>

|0

>

|E2

>

|0

>

|1

|1 |E

>

|2

|2

> > |3> |0>

|E1

FIGURE 8.1 The PA scenarios discussed in the text. (a) Single-channel PA. (b) ARPA with two incoming channels and two intermediate bound states. (c) Realistic arrangement in the two-channel PA of KRb. The dotted lines show undesired transitions driven by the strong pump pulses.


Adiabatic Raman Photoassociation with Shaped Laser Pulses

293

the population of state |1 is (partially) transferred to the ground molecular electronic state by either a spontaneous-emission [38] process, or a stimulated-emission [39] process, induced by an additional (“dump”) laser pulse. Although quite successful as a spectroscopic tool [40–48] (see Chapter 5 by W. Stwalley, P. Gould, and E. Eyler), as a preparatory tool, two-step PA suffers from losses due to spontaneous emission. Even in the presence of the stimulated emission process induced by a “dump” pulse [39], sponaneous emission populates in an incoherent fashion a large number of vibrational and rotational levels of the ground electronic state (or the low metastable electronic state), resulting in a translationally cold but internally hot ensemble of molecules. In contrast, the single step adiabatic Raman photoassociation technique [1] is capable of overcoming these difficulties. In this scheme PA is executed coherently using the “dump-before-pump” pulse ordering of STIRAP [2–6]. In its simplest form, STIRAP involves the “Λ-type” configuration of states, consisting of an initial state |i, a final state |f, and an intermediate, energetically higher, state |1. The task of transferring, as completely as possible, the population from state |i to state |f is achieved by coupling the three states with two laser fields: a “pump” field, tuned to be in near resonance with the |i ↔ |1 transition, and a “dump” field, preceding the pump field, tuned to be in near resonance with the |1 ↔ |f transition. When a two-photon resonance with the |i ↔ |f transition is maintained, and the pulses are long enough, this so-called “counter-intuitive” pulse sequencing [3] brings about a complete population transfer from state |i to state |f: as the dump pulse is turned off and the pump pulse is turned on, the field-dressed state correlating initially with state |i (the so-called “dark state”) is observed to execute a smooth passage to state |f while leaving the intermediate state |1 unpopulated at all times, thus eliminating the unwanted spontaneous emissions from state |1. The ARPA process in a thermal ensemble of atoms as well as in a Bose–Einstein condensate of atoms, has been studied theoretically in great detail [1,7,8,11–18]. Its first experimental demonstrations [21–23] have verified the existence of the fielddressed “dark state,” predicted in Ref. [1] to arise from a strongly coupled Λ-type system made up of an initial-continuum and two (intermediate and final) bound states. In addition, the closely related process of adiabatic passage from a loosely bound excited molecular state, produced by the “Feshbach resonance” switching process, to a deeply bound low-lying state, was observed [49–51]. Initially, the lack of a clear understanding of the nature of the adiabatic transfer from the continuum during ARPA led to conflicting opinions [7,11,12] about its viability. The answer lies, as pointed out in Ref. [11], in going beyond the three-level STIRAP-like picture. In ARPA distribution of population between the (continuum) energy eigenstates is of great importance. The conflicting opinions [1,7,11,12] have resulted from the different procedures employed for taking into account this effect. It turns out, as shown below, that the nature of the initial population spread affects the nature of the (pump and dump) pulses that maximize the efficiency of the ARPA process. The theory presented below describes ARPA in two situations: (1) an initial continuum of states and (2) an initial (dense) set of bound states. In either case the PA yield is determined by the projection of the incoming multichannel wavefunction onto a © 2009 by Taylor and Francis Group, LLC

294


wavepacket correlating adiabatically with the (deeply bound) target state. The shape of such wavepackets is defined by the shapes of the pump and dump laser pulses. Thus, ARPA may be viewed as a process in which one measures a continuum wavefunction by projecting it onto a set of (experimentally controllable) wavepackets of known form. Our approach builds on the work presented in Refs. [1] and [17]. It also extends the recent work [52], which explored coherently controlled adiabatic passage in molecular dissociation. In the present study we describe the influence of temperature on the PA process and demonstrate how ARPA can be used to read the information encoded in the incoming wavefunction. Based on the calculations of Ref. [17], we discuss the role of incoherent initial conditions. This chapter is organized as follows. Section 8.3 presents the general theory of the adiabatic PA of a multichannel wavepacket. Section 8.4 illustrates our approach by simulating the measurement of the shape of the incoming single-channel wavepacket in the PA of a Rb2 molecule. We also study the possibility of reading a two-channel superposition in the PA of KRb, and discuss the role of temperature. The modeling of the input molecular data for laser-driven formation of 85 Rb2 and 40 K-87 Rb molecules is given in Section 8.5. Section 8.6 presents the conclusions and discussion.

8.3 THEORY OF MULTICHANNEL PHOTOASSOCIATION 8.3.1 TWO-STATE DESCRIPTION OF MULTICHANNEL PHOTOASSOCIATION Figure 8.2 shows the molecular potentials describing a PA for two alkali atoms, using the Rb2 potentials as an example. We consider the ARPA scenarios illustrated in Figure 8.1a and 8.1b, starting with two atoms colliding on either the X 1 Σ+ g potential 1 + alone (for PA of a homonuclear molecule), or on both the X Σ and the a3 Σ+ diatomic potentials (for PA of a heteronuclear molecule). To realize the process shown in Figure 8.1a, one could use a pair of laser pulses: a pump pulse, coupling the 3 initial wavepacket |Ψi to state |1, chosen to be one of the LS-coupled A1 Σ+ u -b Πu bound states, and an anti-Stokes dump pulse, coupling state |1 to the target state |0, chosen to be one of the deeply bound vibrational states of the ground electronic state X 1 Σ+ g. In the two-channel case shown in Figure 8.1b, there are two incoming continuum channels, two intermediate states (|1 and |2) and two pairs of pump and dump pulses, each pair acting simultaneously, with the pump pulses delayed relative to the dump pulses, in accordance with the counterintuitive scheme. In general, a collision between two atoms takes place on a superposition of several potential surfaces, distinguished by the total angular momentum F and/or its projection mF . In the derivation below we assume that there are N incoming open channels (for which Eth , the threshold energy, is less than E, the total energy). Alternatively, for PA of atoms in a trap, we assume that there are N manifolds of nondegenerate bound trap states. As the intermediate state we choose N nondegenerate bound levels of the excited electronic potentials. We use N pump and N (anti-Stokes) dump laser pulses. © 2009 by Taylor and Francis Group, LLC

295


Potential energy (au)

0.06

A1Σu+

0.04

b3Πu

0.02

+

a3Σu

0

+

X1Σg –0.02 7.5

10

12.5 15 17.5 20 Internuclear distance (a0)

22.5

25

FIGURE 8.2 The Rb2 Born–Oppenheimer potentials of Ref. [53] involved in our calculations. The dotted line shows the energy region of the intermediate ARPA states used in our calculations.

The total Hamiltonian of the system can be written as ˆ Hˆ = Hˆ M − 2μ

N

{P,n (t) cos(ωP,n t + φP,n ) + D,n (t) cos(ωD,n t + φD,n )},

(8.1)

n=1

where Hˆ M is the field-free Hamiltonian of the atoms; P,n (t), ωP,n , φP,n , D,n (t), ωD,n , and φP,n are slowly varying dump-and-pump amplitudes, frequencies, and phases, and −2μ ˆ is the electronic dipole moment for the transition between the ground and excited electronic states. As in the theory of single-channel photodissociation and PA [1,24,54], the wavefunction of the atoms can be represented (in atomic units, au) by |Ψ(t) = b0 (t)e−iE0 t |0 +

N

n=1

bn (t)e−iEn t |n +

N

∞

dE bE (t)e−iEt |E, k +

k=1 Eth,k

(8.2) for case 1 (an initial continuum). Here |0 denotes the target state, |n the intermediate bound state(s), and |E, k + a scattering state correlating as t → −∞ with the product of a free-translational state of energy E, and the kth internal (e.g., electronic) state [24,55]. En and E are the field-free energies of the states: (E0 − Hˆ M )|0 = (En − Hˆ M )|n = (E − Hˆ M )|E, k + = 0. © 2009 by Taylor and Francis Group, LLC

(8.3)

296


For case 2 (atoms in a trap), |Ψ(t) = b0 (t)e−iE0 t |0 +

N

bn (t)e−iEn t |n +

N

bmk (t)e−iEmk t |mk ,

(8.4)

k=1 mk

n=1

where Emk and |mk are, respectively, the eigenvalues and eigenstates of an atom in a trap, satisfying (Emk − Hˆ M )|mk = 0. We assume that all the dump laser fields are in exact resonance with the corresponding transitions, ω0,n ≡ En − E0 . We also assume that all the pump laser fields are in near resonance with the transitions from a given energy E (or Emk ) of the incoming states. The detunings are defined as Δn,E ≡ ωP,n − (En − E),

Δn,mk ≡ ωP,n − (En − Emk ),

for case 1,

for case 2. (8.5)

Given the above choice of pump pulse, it follows from the rotating wave approximation (RWA) [56] that the pump laser couples each kth component of the incoming wavepacket to a single intermediate bound state. The expansion coefficients of Equation 8.2 are obtained by solving a matrix Schrödinger equation of the form b˙ 0 = i

N

Ω∗n,0 bn

(8.6)

n=1 f b˙ n = i Ωn,0 b0 − Γn bn + i

N

∞

bE,k Ωn,E,k ei Δn,E t dE

(8.7)

k=1 Eth,k

b˙ E,k = i

N

bn Ω∗n,E,k e−i Δn,E t ,

(8.8)

n=1

For case 2 (Equation 8.4), the last two equations are replaced by f b˙ n = i Ωn,0 b0 − Γn bn + i

N

bmk Ωn,mk ei Δn,mk t

(8.9)

k=1 mk

b˙ mk = i

N

bn Ω∗n,mk e−i Δn,mk t .

(8.10)

n=1

The complex Rabi frequencies of Equations 8.6 to 8.10 are defined as Ωn,0 = D,n μn,0 e−iφD,n ,

Ωn,mk = P,n μn,mk e−iφP,n ,

Ωn,E,k = P,n μn,E,k e−iφP,n . (8.11)

The lower limit Eth,k of the integration in Equation 8.7 is the threshold continuum f energy of the kth incoming channel. The empirical term Γn bn in Equations 8.7 and 8.9 © 2009 by Taylor and Francis Group, LLC

297


describes nonradiative decay of the excited bound state |n. The initial |E, k + or |mk states are assumed to be unaffected by the analogous nonradiative decay process. The field-induced decay of the input and target states observed in the experiments [49] is neglected for notational simplicity. Solving formally for either bmk or bE,k , bE,k (t) = bE,k (0) + i

N

t

n=1 0

bmk (t) = bmk (0) + i

N

t

n=1 0

dt Ω∗n,E,k bn (t )e−i Δn,E t

dt Ω∗n,mk bn (t )e−i Δn,mk t

(8.12)

we obtain upon substitution in Equations 8.7 and 8.9 the equations of motion, which can be written in a matrix form as b˙0 = i Ω†D · bexc t dt Γ(full) (t − t ) · bexc (t ) + if (full) (t). b˙ exc = i b0 ΩD −

(8.13)

0

Here bexc ≡ (b1 . . . bN )T is the column vector (hence the T transpose sign) of ampliT tudes of the excited bound states, ΩD ≡ Ω1,0 . . . Ωn,0 is the vector of Rabi frequencies between the |0 state and the excited bound states, and Ω†D is its conjugate transpose. The matrix Γ(full) is given by f

(full)

Γn n (τ) = Γn δn,n +

N

k=1 Eth,k

Ωn,E,k Ω∗n ,E,k exp(iΔn,E τ)dE,

(8.14)

for case 1, and by f

(full)

Γn n (τ) = Γn δn,n +

N

Ωn,mk Ω∗n ,mk exp(iΔn,mk τ)

(8.15)

k=1 mk

for case 2. The N-component vector f (full) (t) describing the pumping from the initial multichannel wavepacket into the manifold of excited bound levels is given as fn(full) (t) =

fn(full) (t) =

k


Ωn,E,k bE,k (0) eiΔn,E t dE,

for case1,

Eth,k

k

or

∞

mk

Ωn,mk bmk (0) eiΔn,mk t ,

for case 2.

(8.16)

298


Equations 8.13 can be further simplified using the “flat continuum” or “slowly varying continuum” approximation (SVCA). This approximation assumes that the variation of the continuum-bound dipole matrix elements with energy over the pump laser bandwidth is negligible. With this assumption, and after changing the lower limit of integration over each continuum channel to −∞, we can rewrite the second term in Equation 8.14 as N

∞

k=1 Eth,k

Ωn,E,k Ω∗n ,E,k exp(iΔn,E τ)dE ≈

N

Ωn,E0 ,k Ω∗n ,E0 ,k

k=1

≈ 2πδ(τ)

N

∞ −∞

exp(iΔn,E τ)dE

Ωn,E0 ,k Ω∗n ,E0 ,k

(8.17)

k=1

where E0 is some energy within the bandwidth of the pump pulse. Equation 8.13 now becomes b˙ exc = i b0 ΩD − Γ(SVCA) · bexc (t) + if (full) (t)

(8.18)

where (SVCA) Γn,n

=

f Γn δn,n

+π

N

Ωn,E0 ,k Ω∗n ,E0 ,k .

(8.19)

k=1

(Note ∞ that when substituting Equation 8.18 into Equation 8.13, we use the relation 0 δ(τ)dτ = 1/2.) The applicability of the SVCA in the energetic region of interest here (which for cold collisions is near the threshold) depends on variation of the continuum spectrum in this region over the pump pulse bandwidth. In most cases studied, the SVCA provides a reasonably good description of the PA process. In case 2 an approximation analogous to SVCA implies that the summation over each mk goes from −∞ to ∞, and that, for each k, the Rabi frequencies depend only weakly on mk : Ωn,mk ≡ Ωn,k . Equation 8.15 now becomes N

Ωn,mk Ω∗n ,mk exp(iΔn,mk τ) ≈

k=1 mk

N

Ωn,k Ω∗n ,k

k=1

≈ 2π

N

k=1

∞

exp(iΔn,mk τ)

mk =−∞

Ωn,k Ω∗n ,k

∞

(vib) . δ t − jTk

j=−∞

(8.20) −1 (vib) Here Tk = 2π × dEmk /dmk is the vibrational period of the wavepacket in the (vib) exceeds the duration of our kth manifold of states in the trap. Assuming that Tk © 2009 by Taylor and Francis Group, LLC


299

laser pulse sequence (i.e., the trap energy levels are densely spaced compared to the bandwidths of the laser pulses), we arrive at Equation 8.18 with f

(SVCA)

= Γn δn,n + π

Γn,n

N

Ωn,k Ω∗n ,k .

(8.21)

k=1 (full)

The source vector fk

(t) factorizes within SVCA as

fn(full) (t) =

(0)

Ωn,E0 ,k fk (t) = ΩP f (0) (t),

n

(0)

fk (t) =

∞

−∞

bE,k (0) eiΔn,E t dE,

for case 1

(8.22)

and fn(full) (t) =

(0)

Ωn,k fk (t) = ΩP f (0) (t),

n (0)

fk (t) =

∞

bmk (0) eiΔn,k t ,

for case 2.

(8.23)

mk =−∞

T (0) (0) Unlike f (full) (t), the N-component vector f (0) (t) ≡ f1 (t), . . . , fk (t) . . . describes the initial state of the system without any reference to a particular pump transition. The matrix ΩP is composed of the Rabi frequencies for the transitions between each input channel and each intermediate state. Thus one can write the inhomogeneous source term as a product of the pump line shape and the Fourier transform of the initial wavepacket components. In the semiclassical regions, f (0) (t) describes the envelope of the incoming wavepacket as it moves along the classical phase space trajectory [58–61]. We now introduce the effective average Rabi frequency of the bound–bound transitions, Ωbb ≡ Ω†D · ΩD , (8.24) and the effective average amplitude of the excited bound states: beff ≡

Ω†D · bexc . Ωbb

(8.25)

With these notations, the Schrödinger equation 8.18 assumes the form b˙0 = i Ωbb beff b˙ eff = i Ωbb b1 −


(8.26) Ω†D · Γ(SVCA) · bexc (t) Ω† · ΩP · f (0) (t) +i D . Ωbb Ωbb

(8.27)

300


In the beginning of the next section we consider the problem with only one incoming channel and only one intermediate state |1. In that case, beff = b1 Ω∗1,0 /|Ω1,0 |, and the decay term in Equation 8.27 has a simple form: −Ω†D · Γ · bexc /Ωbb = −Γ1,1 b1 . In the general case of multichannel PA, this decay term may have a rather complicated time dependence as it is determined simultaneously by all the amplitudes of the excited bound states. Nevertheless, because in the adiabatic process studied here the excited bound states remain almost unpopulated, we omit the rapid dynamics of the decay term in our qualitative analysis. The validity of this approximation is verified by numerical simulations in the next section. For now, we replace the term −Ω†D · Γ · bexc /Ωbb in Equation 8.27 by the effective decay of the form −Γeff (t) beff . With the above approximation the Schrödinger equation assumes a form resembling the single-channel wavepacket photodissociation and PA case [1,24,54] d b = i{H · b(t) + f} dt where

b(t) =

and

b0 (t) , beff (t)

H=

0 Ωbb

(8.28) Ω∗bb , iΓeff

0 . f(t) = Ω†D · ΩP · f (0) /Ωbb

(8.29)

(8.30)

8.3.2 ARPA AS A PROJECTIVE MEASUREMENT We now proceed to solve Equation 8.28 in the adiabatic approximation, following the procedure described in Refs. [1] and [54]. In order to do that, we first obtain E, the matrix of eigenvalues of H, satisfying, U · H = E · U.

(8.31)

E has two nonzero elements, E1,1 = E+ , E2,2 = E− where E± =

3 2 1 iΓeff ± 4|Ωbb |2 − Γ2eff , 2

(8.32)

The diagonalizing matrix U is a complex orthogonal transformation which can be parameterized as cos θ sin θ U= . (8.33) −sin θ cos θ where θ is a complex angle. Using Equation 8.31 we see that tan θ = E+ /Ωbb . © 2009 by Taylor and Francis Group, LLC

(8.34)

301


The adiabatic approximation (which was thoroughly investigated in Ref. [57]) is obtained by neglecting the nonadiabatic coupling matrix U−1 · dU/dt. This allows us to write Equation 8.28 in the form d a = iE(t) · a + ig, dt

(8.35)

sin θ Ω†D · ΩP · f (0) /Ωbb . g(t) = cos θ Ω†D · ΩP · f (0) /Ωbb

(8.36)

where a(t) = U(t) · b(t), and

We can solve Equation 8.35 by making the following substitution,

t a(t) = exp i dt E(t ) · α(t),

(8.37)

0

to obtain that b(t) = U−1 · a(t) is given as

t dt exp i E+ (t ) dt sin θ(t ) Ω†D · ΩP · f (0) /Ωbb

t

b0 (t) = icos θ(t)

t

0

t

− isin θ(t)

t

0

beff (t) = isin θ(t) 0

t

t dt exp i E− (t ) dt cos θ(t ) Ω†D · ΩP · f (0) /Ωbb

t dt exp i E+ (t ) dt sin θ(t ) Ω†D · ΩP · f (0) /Ωbb

(8.38)

t

t

+ icos θ(t) 0

t dt exp i E− (t ) dt cos θ(t ) Ω†D · ΩP · f (0) /Ωbb . t

(8.39) The final yield of the ARPA process is defined as the probability P0 = |b0 (t → ∞)|2 . Using Equations 8.32 and 8.34, we see that cos θ(t → ∞) = 0, and all the excited bound-state amplitudes, as well as beff , vanish. Substituting cos θ(t → ∞) = 0 and sin θ(t → ∞) = 1 in Equation 8.38 we obtain b0 (t → ∞) = 0

where

∞

∗ fARPA (t)f (0) (t) dt ≡ fARPA |f (0) t ,

t E− (t ) dt cos θ(t) Ω†D · ΩP /Ωbb . fARPA (t) = −i exp i

(8.40)

(8.41)

0

Thus, the PA amplitude b0 (t → ∞) is given by a projection of f (0) onto the specific wavepacket that correlates adiabatically with the target molecular state |0. © 2009 by Taylor and Francis Group, LLC

302


∗ The shape of this wavepacket, given by fARPA , is controlled by the amplitudes and the ∗ phases of ΩD (t) and ΩP (t). Wavepackets orthogonal to fARPA do not photoassociate ∗ in the ARPA process, while the ones that project well onto fARPA do. We conclude that one must be able to measure the shape of the initial wavepacket by choosing the proper fARPA via pulse shaping, and using the ARPA to find the projection of f (0) onto it. To illustrate this, we explore in the next section the measurement of both the temporal and multichannel structure of the incoming continuum wavepacket in PA with one and two incoming continua.

8.4 8.4.1

NUMERICAL EXAMPLES SINGLE-CHANNEL ARPA OF A CHOSEN WAVE FORM

We first consider the ARPA scenario illustrated in Figure 8.1a in which we address PA of an initial wavepacket composed of near-threshold s-wave scattering states of two 85 Rb atoms colliding on the X 1 Σ+ potential. (The numerical parameters are given in g Section 8.5 and Ref. [17].) The pump pulse couples this initial state to an excited inter3 mediate bound state belonging to the A1 Σ+ u and b Πu spin–orbit-coupled states, |1 = 1 + 3 A Σu -b Πu (v = 133, J = 1) (E1 = 0.042848 au). The anti-Stokes dump pulse couples this intermediate state to the target state |0 = X 1 Σ+ g (v = 4, J = 0) (E = −0.01823 au). Figure 8.3 shows the envelopes of two choices for the initial continuum wavepacket. Panel (a) shows the envelope f (0+) (t) of a Ψ+ state whose expansion coefficients are chosen as explained below, to be (+) bE (t = 0) = (δ2E π)−1/4 exp −(E − E0 )2 /2δ2E + i(E − E0 )t0 ,

(8.42)

with E0 = 100 μK, δE = 70 μK, and t0 = 1220 nsec. Panel (b) of the same figure shows the envelope f (0−) (t) of the state Ψ− . It is obtained by multiplying f (0+) (t)

Envelope (au)

(a) 2×10–5 1×10–5 0 Envelope (au)

(b) 2×10–5

–2×10–5 0

0.5

1 Time (μsec)

1.5

2

FIGURE 8.3 The envelopes of the two continuum wavepackets, f (0+) (a) and f (0−) (b), taken as the initial conditions in the simulations.



303

by sin[2δE (t − t0 )] and scaling the wavefunction up to satisfy the normalization condition f (0) | f (0) t = 2π. The symmetries of wavepackets viewed on a progressively finer scale offer a temperature-robust way of encoding several qubits of information [62–64]. The encoding and often the full control over the quantum evolution of the wavepacket [65] can be implemented by alternating periods of free motion with phase kicks imposed by coordinate-dependent Stark shifts. Distinguishing odd from even wave forms is the essence of the decoding of the qubits of information encoded in the wavefunction by the dynamics of atoms in a trap. The calculations below demonstrate the possibility to distinguish between the even wave form f (0+) and the odd one f (0−) . In the first calculation we examine PA with a pair of simple sin2 αt-shaped laser pulses. The dump pulse has σ− type polarization, an intensity of 1.12 × 105 W/cm2 , and a central wavelength 733 nm, in resonance with the transition between the target state |0 and the intermediate state |1. The pump pulse has σ+ type polarization, an intensity of 1.6 × 105 W/cm2 , and a central wavelength of 1063.4 nm, in resonance with the transition between a scattering state on the X 1 Σ+ g potential with a mean energy 100 μK and state |1. Both pulses are of 750 nsec duration (full-width at halfmaximum in the amplitude profile), with the peak of the pump pulse delayed by 600 nsec relative to the peak of the dump pulse. Setting 1/Γf = 30 nsec, we numerically solve the Schrödinger equation in the SVCA. Figure 8.4a displays the envelopes of the two laser pulses. Figure 8.4b shows Re( fARPA (t)) for this sequence, with Im( fARPA (t)) remaining negligible at all times except for a short period at t = 0.963 μsec, where 2ΩD = Γ, and cos θ diverges. It follows from Equation 8.40 that the ARPA process must be able to distinguish between f (0+) , the even-shaped initial continuum state, and f (0−) , the odd-shaped ∗ initial continuum state. This is due to the different projections onto fARPA : ∗ ∗ (0+) 2 (0−) | fARPA | f t | = 0.9 whereas fARPA | f t = 0.05. Figure 8.4c,d shows the calculated evolution of the target population P0 (t) for the initial state given by, respectively, Ψ+ and Ψ− . The final PA probability is equal to 0.89 for the Ψ+ initial state, and 0.001 for the Ψ− initial state. The PA probability of the even incoming state is thus 900 times larger than that of the odd state. The shapes of the pump and dump laser pulses define the shape of fARPA . Therefore by shaping the pulses it may be possible to choose the waveform transferred from the continuum during PA. According to Equation 8.41 fARPA is a function of E− (t), cos θ(t), ΩP (t), and the ratio Ω∗D /Ωbb . Both E− (t) and cos θ(t) depend only on the absolute values of the laser intensities, and their phases are difficult to control. On the other hand, both the amplitude of the pump Rabi frequency ΩP and the phase between the pump and the dump pulses (i.e., the phase of the product Ω∗D ΩP ) can be easily controlled by standard experimental methods. Figure 8.5a shows the envelopes of the pump and dump laser pulses tailored to photoassociate an incoming odd wavepacket. The preferential PA of the odd state is achieved by flipping the phase of P,1 (t) at t = t0 . Panel (b) of Figure 8.5 shows Re( fARPA (t)) for this sequence. Panels (c) and (d) respectively show the dynamics of the target state population for the Ψ+ and Ψ− initial states. ∗ ∗ | f (0+) t |2 = 0.04, and | fARPA | f (0−) t |2 = 0.7. The squared projections are | fARPA © 2009 by Taylor and Francis Group, LLC

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Field envelope (au)

(a) 2×10–6 1×10–6 0

Re ( f ARPA) (au)

(b)

0 –2×10–6 –4×10–6

(c) P0

0.8 0.4

0.4

P0

(d)

0.2

0

0

0.5

1 Time (μsec)

1.5

2

FIGURE 8.4 ARPA with Gaussian laser pulses. (a) Amplitudes of the pump (solid line) and the dump (dashed line) laser pulses. (b) Re(fARPA (t)). (c) Population of the target state |0 during PA of the continuum wavepacket Ψ+ . (d) Population of the target state during the PA of Ψ− .

The PA probability P0 (τARPA ) in the simulations is 0.002 for the even initial continuum envelope, and 0.68 for the odd one. The PA probability of the even incoming wave form is thus 340 times lower than that of the odd one. We see that by simply flipping the phase of the pump laser pulse we have chosen to photoassociate an odd incoming waveform instead of an even one. It is interesting to note that the numerical ratios of the PA probabilities in the above examples are higher than the analytical ones. As seen in Figure 8.3, the numerical optimization of the PA probabilities requires a slight temporal shift of the input wavepackets relative to the maximum of the theoretical fARPA (Figures 8.4b and 8.5b).

8.4.2 THERMAL AVERAGING We now discuss the effect of thermal averaging on the efficiency of ARPA when the initial state is a thermal ensemble. We show below that the energetic spread due to © 2009 by Taylor and Francis Group, LLC

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Field envelope (au)

(a)

4×10–6

Re ( f ARPA) (au)

(b)

10–6 0 –6 –10

0

–4×10–6 0.4

P0

(c)

0.2

0 (d) P0

0.6 0.3 0 0

0.5

1 Time (μsec)

1.5

2

FIGURE 8.5 ARPA with a shaped pump pulse. (a) Amplitudes of the two laser pulses. The sudden change of the sign of the pump amplitude at t0 = 1150 nsec is due to a flip in the phase of the pulse. (b) Re( fARPA (t)). (c) Population of the target state during the PA of Ψ+ . (d) Population of the target state during the PA of Ψ− .

the existence of a nonzero temperature defines the optimal parameters of the pump and dump laser pulse shapes. These in turn define fARPA of Equation 8.41 and set the upper limit for the PA yield. The laser pulse parameters used in our ARPA simulations are optimized [17] for a 100 μK thermal ensemble of Rb atoms. The pump and dump overlap time is chosen to match the average coherence time of a colliding atom pair at 100 μK, and the spectral widths of the pulses are chosen as the energy spread of the thermal ensemble. The laser pulse durations define their intensities because the area under the pulse envelope must be sufficiently large to ensure adiabaticity and efficient population transfer. (Note, the pulse intensities in this work are higher than those in Ref. [17], our main interest being the study of interference effects rather than searching for the lowest acceptable values of laser power.) The initial ensemble can be represented as a mixed state composed of many energy eigenstates, or as a mixed-state Gaussian wavepacket in phase space. While the first representation allows for an accurate numerical averaging over the ensemble [17], the second one is more convenient for simple estimates of the ARPA yield. © 2009 by Taylor and Francis Group, LLC

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In order to estimate the fraction of atoms photoassociated per pulse pair [1,17], we need to multiply P(E), the PA probability per collision at energy E, by the number of collisions experienced by a given atom during the pulses (this is equivalent to averaging over all possible values of t0 ). The number of collisions during the pulses is calculated as follows: at a given energy E, the velocity of an atom is 1 v = (2E/m) 2 and the distance traversed by the atom during a pulse of τlaser duration is vτlaser . The collision cross-section is πb2 where b is the impact parameter, related 1 to the partial wave angular momentum J as b = (J + 1/2)/p = (J + 1/2)/(2mE) 2 . Hence, the number of collisions experienced by the atom during the two pulses is N = ρπb2 vτlaser where ρ is the density of atoms. We conclude that for J = 0 the fraction of atoms photoassociated per pulse pair is 1

f (E) =

P(E)πρτlaser P(E)πρ(2E/m) 2 τlaser = . 1 8mE 4m3/2 (2E) 2

(8.43)

Using the estimate P(E) = 1, τlaser = 750 nsec, ρ = 1011 cm−3 , E = 100 μK, and the reduced atomic mass m = 1823 × 85/2 au, we obtain f ≈ 6 × 10−7 per pulse-pair. This estimate needs to be adjusted to take into account that approximately threequarters of near-threshold collisions between Rb atoms occur on the a3 Σ+ u potential surface, and these are not affected in our arrangement due to the selection rules for the transitions at the chosen frequencies. It is possible to increase the PA yield by increasing the phase space density of the initial sample by forming a Bose–Einstein condensate, or pairing two atoms in a single potential well of an optical lattice [66,67]. Some ways of using laser pulses to drive the phase space density of colliding atoms toward shorter internuclear distance are discussed in Refs. [68] to [70], and in Chapter 7 (by E. Luc-Koenig and F. MasnouSeeuws) of this book. Alternatively, one could repeat the PA pulse sequence many times, employing an irreversible process to prevent the molecules created by one pulse pair from being destroyed by another. This might possibly be implemented either by physically removing the molecules from the laser focus, or by exciting the molecules created via ARPA by an additional laser pulse, followed by a decay into low vibrational states of the X 1 Σ+ g potential [1,17].

8.4.3

PA OF A SUPERPOSITION STATE: DETERMINING THE MULTICHANNEL STRUCTURE

Equations 8.40 and 8.41 show that the time-dependent amplitudes and phases of the laser pulses define the temporal structure of the photoassociated waveform. This is true in the multichannel as well as in the single-channel ARPA. We now assume that all the laser pulses have similar simple time profiles, and concentrate on using PA for determining the multichannel structure of the input wavepacket. The measurement is based on controlling the interference of quantum pathways during ARPA, each pathway corresponding to the adiabatic passage via one of the intermediate |1 . . . |n states. © 2009 by Taylor and Francis Group, LLC

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As shown in Equations 8.25 and 8.27, the effect of the pump is equivalent to the production of a single coefficient beff that determines the multichannel dynamics. The source term for the production of beff is the (scaled) projection of the incoming vector f (0) on the multichannel (Ω†D · ΩP )† vector. Alternatively, the source term can be viewed as the projection of the vector ΩP · f (0) on the vector of the dump Rabi frequencies Ω∗D . Both the matrix ΩP and the vector Ω∗D are controlled by the pump and dump laser amplitudes. Therefore there is a large choice of possibilities to control the projective measurement of the incoming vector f (0) performed in the PA process. As an example, we assume that at a certain moment in time the incoming wave(0) function is composed of only the jth channel: fj = (0, . . . , 1j , . . . , 0)T . In that case (0) the source vector, ΩP · f , coincides with the jth column of the ΩP matrix ΩP, j , and the maximal PA probability for a fixed dump pulse area is obtained when the Ω∗D vector is parallel to ΩP · f (0) , which is realized when Ω∗D ∝ ΩP, j .

(8.44)

If, on the other hand, the vector Ω∗D is orthogonal to ΩP,j , then the PA rate would be zero. These conclusions parallel those made for adiabatic Raman dissociation in Ref. [52]. There, Equation 8.44 was a result of the definition of the prevailing dark state, which correlates at t → −∞ with the initial bound state and at t → ∞ with the final jth continuum channel. The above assessment (8.44) does not include the effect of the uncontrolled decay of the photoassociated waveform. In its absence the above analysis was verified numerically for PA of a superposition of two KRb continuum channels belonging to the singlet X 1 Σ+ and the triplet a3 Σ+ electronic states (see Figure 8.1b and 8.1c). The PA was conducted in the near-threshold region via a pair of intermediate levels belonging to the LS coupled A1 Σ+ and b3 Πu potentials. A detailed description of the KRb molecular data used in our simulation is given in Section 8.5. According to our observations, if the rates of decay from the two intermediate levels are artificially set to Γf , then Equation 8.44 correctly predicts the ratio of the two dump fields needed to provide the maximal and the minimal PA probabilities. If, however, the field-induced decay is taken into account, then the maximal and minimal PA rates are achieved at the intensity ratios, which are different from those predicted by Equation 8.44. Moreover, there are always undesired satellite transitions. For example, as shown in Figure 8.1c (the black dotted arrow), the pump frequency component that couples the lower-energy intermediate level |1 to the incoming continuum also couples the higher-energy |2 state to a more energetic continuum level, which does not exist in the initial wavepacket. In a similar fashion, as depicted in Figure 8.1c, the pump frequency component that couples the incoming continuum with the higher level |2 can also couple the lower level |1 with one of the highly excited bound states of the X 1 Σ+ g potential (denoted as state |3 in Figure 8.1c). (0)

In order to decompose the incoming wavepacket coefficient vector into its f1 (t) (0) 3 + (belonging to X 1 Σ+ g ) and f2 (t) (belonging to a Σu ) components, we need to find ∗ the complex dump-pulse component ΩD(1) which projects well and exclusively onto

ΩP f1 , and Ω∗D(2) , which projects well and exclusively onto ΩP f2 . Comparing the (0)


(0)

308


results of PA with the dump Ω∗D = Ω∗D(1) with those with Ω∗D = Ω∗D(2) one can then determine the multichannel composition of the incoming wavepacket. The above numerical analysis can be significantly simplified if we choose as the intermediate state |1 a state that couples mainly to the singlet continuum and as intermediate state |2 a state that couples mainly to the triplet continuum. The problem is further simplified if the states |1 and |2 have the same order of magnitude Franck– Condon overlaps with the target state |0. In KRb, we choose the intermediate state |1 3 as the A1 Σ+ u − b Πu (v = 108, J = 1) state (E1 = 0.04292 au), and the intermediate 3 state |2 as the A1 Σ+ u − b Πu (v = 141, J = 1) (E2 = 0.04659 au) state. State |1 couples almost exclusively to the a3 Σ+ u low-energy continuum, and state |2 couples mostly to the X 1 Σ+ continuum. Transitions from the incoming threshold-energy g continua into these levels are driven by, respectively, the 1062 nm and 977.7 nm pump laser components. The target state |0 is chosen to be the X 1 Σ+ g (v = 13, J = 0) (E = −0.01434 au) state. The dump transitions from the two intermediate states |1 and |2 to |0 occur at wavelengths of 796 nm and 748 nm, respectively. As in the above, the presence of satellite transitions leads to complications. Thus the strong 1062-nm pump component also couples state |2 to E ≈ 0.00367 au (1160 K) scattering states, and the strong 976-nm pump component couples state |1 to the level v = 55 of the X 1 Σ+ potential surface with energy E3 = −3.61 × 10−3 au. These additional transitions and states (shown in Figure 8.1c) were included in our simulation. In the first set of calculations, both pump pulses had the peak intensity IP = 5 × 107 W/cm2 . In the first simulation, I1D = 3 × 105 W/cm2 , I2D = 0. The final PA proba(0) bilities are P0(1) = 7 × 10−6 when the initial wavepacket is given by f1 , and P0(2) = (0) 0.14 when the initial state is f2 . This huge difference (a factor of 2 × 104 ) is (0) due to the fact that the X 1 Σ+ g continuum, described by f1 is practically decoupled from the intermediate state |1. In the second simulation, IP = 2 × 108 W/cm2 , I2D = 3 × 105 W/cm2 , I1D = 0. Now P0(1) = 0.04, and P0(2) = 7 × 10−5 , with P0(1) /P0(2) = 570. In the above calculations, we have neglected the off-diagonal elements of the decay matrix (Equation 8.19) which couple the intermediate states to one another via the continua. Inclusion of these terms may lead to superpositions of intermediate states that appear to be either “dark” or “bright” with respect to the decay [24,71–73], reminiscent of the interference stabilization of atoms and molecules in strong laser fields [74–76]. The possibility of utilizing such states for optimization and control of PA is an interesting open question. Adding temporal shaping to the laser pulses in order to process information encoded both in the multichannel structure and in the temporal structure of the incoming wavepacket is another unexplored possibility.

8.5

MOLECULAR DATA

The set of Rb2 Born–Oppenheimer (BO) potentials used in the above calculations are shown in Figure 8.2. Our choice of the intermediate bound states in Rb2 PA was motivated by the possibility of generating large-area microsecond-long pulses at the wavelengths needed for the continuum-bound transitions, and by the requirement that © 2009 by Taylor and Francis Group, LLC


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the chosen intermediate states be well separated from the neighboring bound states. Thus we choose the intermediate states to belong to the spin–orbit-coupled A1 Σ+ u and b3 Πu electronic potential curves of the diatomic molecules under consideration. 3 We choose those A1 Σ+ u -b Πu vibrational levels that have reasonably large Franck– Condon (FC) overlap with the incoming continuum states and are separated in energy from the incoming continua by approximately the energy of 1064-nm photons. The target vibrational levels of the X 1 Σ+ g potential are chosen such that the dump transitions can also be driven by readily available laser sources. We used the analogous set of BO potentials for the proof-of-principle simulations of multichannel PA of KRb molecules. An alternative choice of the intermediate levels [40,68–70,77,78] would be the 3 set of the highly extended “subcontinuum” bound states just below the A1 Σ+ u –b Πu continuum threshold. The continuum-bound FC factors associated with these states are much larger, allowing for the use of weaker pump lasers. This scenario necessitates either using the incoherent decay populating a wide range of excited levels and 3 + continua of X 1 Σ+ g and a Σu potentials, or controlling the long-range behavior of the subcontinuum states, which in turn depends very strongly on the (poorly known) couplings between the many potentials that determine the dynamics. Another potential difficulty in coherent control of the excited subcontinuum evolution is the necessity to use strong broadband laser pulses with frequencies close to those used to cool and trap the atoms. Recent attempts to optimize PA by controlling behavior of these states have yet not brought a clear answer as to whether a significant optimization is possible [77–80]. Photoassociation scenarios of the nonpolar Rb2 and polar KRb molecules differ due to the presence of the gerade–ungerade symmetry: the dipole transition from the 1 + 3 a 3 Σ+ u electronic state onto either A Σu or b Πu states in Rb2 is forbidden. Therefore, 1 + no ARPA to the X Σg electronic state is possible for collisions occurring on the a3 Σ+ u potential surface of Rb2 . The analysis of ARPA in Rb2 takes into account only the part of the continuum 1 + 3 + wavefunction that belongs to the X 1 Σ+ g channel. In KRb, both the X Σ and a Σ 1 + 3 continua are dipole coupled to the vibrational states of the A Σ -b Πu potentials. In general, the incoming wavefunction is given by a superposition of the X 1 Σ+ and a3 Σ+ components. The single-channel coefficients in such a superposition depend on the internuclear separation due to the hyperfine coupling and the effects of external fields. The modeling of the KRb molecular data follows the same route described in detail in Ref. [17] for Rb2 . The short-range potentials for Rb2 and KRb molecules were taken from Refs. [81] and [82], respectively. The long-range X 1 Σ+ and a3 Σ+ potentials were modeled as a −C6 /r 6 dispersion term, with the value C6 = 4426 au for Rb2 [83], and C6 = 4106.5 au for KRb [84]. The long-range and short-range potentials were smoothly joined in the region 38–52 au to ensure the correct scattering lengths for 85 Rb [85,86], and for 40 K-87 Rb [87]. The joining procedure is described in more 2 detail in Ref. [17] The coordinate-dependent spin–orbit coupling between the A1 Σ+ u and b3 Πu electronic states of Rb2 was taken from Ref. [81]. The average value of this coupling is of the order of 50 cm−1 . In KRb calculations, we set this value to 335 cm−1 in order to correctly reproduce the avoided crossing between these potential surfaces [82]. © 2009 by Taylor and Francis Group, LLC

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(a)

(b) 0

–2 –2 –4 Rb2, continuum–bound

0

KRb, continuum–bound

0

(d)

(c)

–4

–2

–2

–4

–4 Rb2, bound–bound

–6

0.045 0.04 0.03 0.035 Energy of the excited bound state (au)

Log10 of the Franck–Condon factor

Log10 of the Franck–Condon factor

0

KRb, bound–bound

0.035 0.03 0.045 0.04 Energy of the excited bound state (au)

FIGURE 8.6 Franck–Condon (FC) factors for transitions in Rb2 and KRb. (a) FC factors 1 + 3 for the X 1 Σ+ g − A Σu − b Πu continuum-bound transitions in Rb2 vs. the energy of the + 1 3 A Σu − b Πu bound state. (b) FC factors for the transitions between the continuum states of the X 1 Σ+ (open squares) and a3 Σ+ potentials (filled squares), vs. the energy of the A1 Σ+ − b3 Πu bound state of KRb. (c) FC factors for transitions between the X 1 Σ+ g (v = 4) bound 3 Π bound states vs. the energy of the A1 Σ+ − b3 Π bound state − b state and various A1 Σ+ u u u u in Rb2 . (d) FC factors for transitions between the X 1 Σ+ (v = 13) bound state and various A1 Σ+ − b3 Πu bound states vs. the energy of the A1 Σ+ − b3 Πu bound state in KRb.

The bound-state energies, wavefunctions, and the bound–bound Franck–Condon factors were found using the finite-difference algorithm FDEXTR [88]. The obtained energies of the A1 Σ+ -b3 Πu vibrational states were used as an input in the artificial channel calculation [89] of the complex FC factors for transitions from the X 1 Σ+ and a3 Σ+ continua to the bound states of the coupled A1 Σ+ and b3 Πu potentials. The results of these calculations are shown in Figure 8.6. The continuum-bound FC factors are taken for the scattering at 100 μK energy. Note that the continuumbound FC2 factor T (E) can indeed exceed one, as long as the normalization condition, |T (E)| dE = 1, is fulfilled. The s-wave scattering of two 85 Rb atoms on the X 1 Σ+ g potential at the energies of interest is influenced by a resonance enhancing the scattering length to above 2400 au [85,86]. The resonance arises from the last (quasi-) bound state lying very close to the continuum threshold [90]. Hence, the amplitude of the continuum wavefunction in the inner region is enhanced relative to the nonresonant case. This enhances the Franck– 1 + 3 Condon factors for the X 1 Σ+ g − A Σu − b Πu continuum-bound transitions in Rb2 . The required laser intensities are therefore lower than those for the nonresonant case. While the continuum-bound FC factors in KRb depend on the collision energy as © 2009 by Taylor and Francis Group, LLC


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T (E) ∼ E 1/4 [91], the behavior of the continuum-bound FC factor in Rb2 is more complicated due to the presence of the resonance [17]. 1 + In the calculations of the Rb2 PA, we estimate the X 1 Σ+ g − A Σu electronicdipole moment to be μ = 3 au. This value corresponds to the 5S1/2 (m = 1/2) − 5P3/2 (m = 3/2) transition in Rb in a circularly polarized field, and is consistent with all the other 5S − 5P matrix elements for the Rb atom in a polarized laser field. The exact values of the atomic reduced dipole matrix elements can be found in Ref. [92]. The electronic-dipole transition matrix elements in KRb are taken as 3.5 ea0 for the X 1 Σ+ ↔ A1 Σ+ transition, and 0.5 ea0 for the a3 Σ+ ↔ 13 Π transition, the order-of-magnitude estimate based on the data from Refs. [93,94].

8.6

CONCLUSIONS

In this chapter, we have addressed a question of some controversy in the past [1,7, 11,12]. Our work elucidates what, exactly, is transferred from the incoming channels during an ARPA process. We have shown that ARPA projects the incoming wavefunction onto a certain multichannel waveform, whose shape is determined by the amplitudes and the phases of the pump and dump laser pulses (Equations 8.40 and 8.41). In this way one can choose the basis of the projective measurement performed by the PA process. We have also discussed the role of temperature of the initial sample in the ARPA process and the extent to which it can limit the process. The above idea has been applied to measuring the temporal structure of a singlechannel incoming wavepacket in the PA of Rb atoms to form Rb2 , and the multichannel structure in the two-channel PA of K + Rb to form KRb. We have demonstrated the possibility of measuring a qubit of information encoded in the wavepacket’s shape, and a qubit of information encoded in the two-channel superposition. In our numerical simulations, the temporal shapes of the driving fields needed to perform PA agreed well with the predictions of our analytical theory. On the other hand, the simple rule (Equation 8.44) for the ratio of the intensities and the phases needed to control two-channel PA often disagrees with the simulations. We attribute this discrepancy to the effects of complex decay, not taken in account in the simple analytical theory leading to Equation 8.44. We have shown that the the multichannel structure of the wavefunction can be measured with the help of a numerical parameter search based on the knowledge of the level couplings and decay rates. Alternatively, one can use intermediate levels coupled to only one (say, a3 Σ+ u ) incoming channel and not to the other(s). The presence of additional bound states not included in the simplified theory does not destroy the effectiveness of the detection scheme. Such robustness was quite unexpected in the KRb simulations because the central wavelength of one of the pump pulses is just 1.2 nm away from an exact resonance with an undesired transition between the states |1 and |3 (see Figure 8.1c). We attribute the high robustness of the scheme to the counterintuitive structure of the pulse sequence: by the time the pump pulse couples the states |3 and |1, state |1 is already mixed with the target state |0 in such a way that the population of |1 remains negligible at all times. © 2009 by Taylor and Francis Group, LLC

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The most significant approximation used in our theory is the the factorization fARPA = ΩP f (0) (t) of Equation 8.41 which is justified by, but is not a necessary part of, the slowly varying continuum approximation (SVCA). If this factorization cannot be assumed, ARPA performs a measurement of the initial state filtered through the coordinate-dependent Franck–Condon window.

ACKNOWLEDGMENTS The authors thank S. Lunell and D. Edvardsson for sharing the output data of their calculation [81], and S. Kotochigova for consultations regarding molecular data. We are also pleased to thank I. Thannopulos, J. Ye and A. Pe’er for many discussions.

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Part III Few- and Many-Body Physics


Ultracold Feshbach 9 Molecules Francesca Ferlaino, Steven Knoop, and Rudolf Grimm CONTENTS 9.1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Ultracold Atoms and Quantum Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Basic Physics of a Feshbach Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Binding Energy Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Making and Detecting Feshbach Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Bosons and Fermions: Role of Quantum Statistics . . . . . . . . . . . . . . . . . 9.2.2 Overview of Association Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Purification Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Detection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Internal-State Manipulation Near Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Avoided Level Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Cruising Through the Molecular Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Halo Dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Halo Dimers and Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Collisional Properties and Few-Body Physics . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Efimov Three-Body States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Molecular BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Toward Ground-State Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Stimulated Raman Adiabatic Passage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 STIRAP Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Further Developments and Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.1

319 320 321 323 325 325 327 329 329 331 331 333 335 336 337 339 341 342 343 344 347 348 349

INTRODUCTION

The coldest molecules available for experiments are diatomic molecules produced by association techniques in ultracold atomic gases. The basic idea is to bind atoms together when they collide at extremely low kinetic energies. If any release of internal energy is avoided in this process, the molecular gas just inherits the ultralow 319 © 2009 by Taylor and Francis Group, LLC

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temperature of the atomic gas, which can in practice be as low as a few nanokelvin. If moreover a single molecular quantum state is selectively populated, there will be no increase of the system’s entropy. A very efficient method to implement such a scenario in quantum-degenerate gases, Bose–Einstein condensates, and Fermi gases relies on the magnetically controlled association via Feshbach resonances. Such resonances serve as the entrance gate into the molecular world and allow for the conversion of atomic into molecular quantum gases. In this chapter, we give an introduction into experiments with Feshbach molecules and their various applications. Our illustrative examples are mainly based on work performed at Innsbruck University, but the reader will also find references to related work of other laboratories.

9.1.1

ULTRACOLD ATOMS AND QUANTUM GASES

The development of techniques for cooling and trapping of atoms has led to great advances in physics, which have already been recognized by two Nobel prizes. In 1997 the prize was jointly awarded to Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips “for their developments of methods to cool and trap atoms with laser light” [1–3]. In 2001, Eric A. Cornell, Wolfgang Ketterle, and Carl E. Wieman jointly received the Nobel prize “for the achievement of Bose–Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates” [4,5]. Laser cooling techniques allow researchers to prepare atoms in the microkelvin range. The key techniques have been developed since the early 1980s and a detailed account can be found in Refs. [6,7]. In brief, resonance radiation pressure is used to decelerate atoms from a thermal beam to velocities low enough to be captured into a socalled magneto-optical trap or, alternatively, low-velocity atoms can be captured into such a trap directly from a vapor. In the trap the atoms are further cooled by Doppler cooling to temperatures of the order of one millikelvin. Then, for some species, subDoppler cooling methods can be applied, reducing the temperatures deep into the microkelvin range. Typical laser-cooled clouds contain up to about 1010 atoms, and the atomic number densities are of the order of 1011 cm−3 . Such laser-cooled atoms have found numerous applications; a particularly important one is the realization of ultraprecise atomic clocks [8,9]. The quantum-degenerate regime requires atomic de Broglie wavelengths to be similar to or larger than the interparticle spacing. As in laser-cooled clouds the phasespace densities are several orders of magnitude too low to reach this condition, and further cooling needs to be applied. The method of choice is evaporative cooling [10], which can be applied in conservative traps like magnetic traps. Evaporative cooling relies on the selective removal of the most energetic atoms in combination with thermal equilibration of the sample by elastic collisions. The process can be forced by continuously lowering the trap depth. By trading one order of magnitude in the particle number, one can gain typically up to three orders of magnitude in phasespace densities, and eventually reach the quantum-degenerate regime. The typical conditions are then a trapped gas of roughly a million atoms with densities of the order of 1014 cm−3 and temperatures in the nanokelvin range. © 2009 by Taylor and Francis Group, LLC

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Quantum degeneracy has been achieved with bosonic and fermionic atoms. The attainment of Bose–Einstein condensation (BEC) in dilute ultracold gases marked the starting point of a new era in physics [11–13], and degenerate Fermi gases entered the stage a few years later [14–16]. The reader may be referred to the proceedings of the Varenna summer schools in 1998 and 2006 [17,18], which describe these exciting developments. For reviews on the theory of degenerate quantum gases of bosons and fermions see Refs. [19,20], respectively, and the textbooks of Refs. [21,22]. The trapping environment plays an important role for the applications of ultracold matter, including the present one to create molecules. Most of the experiments on molecule formation are performed with optical dipole traps [23] for two main reasons. Such optical traps can confine atoms in any Zeeman or hyperfine substate of the electronic ground state; this includes the particularly interesting lowest internal state, which cannot be trapped magnetically. Moreover, optical dipole traps leave full freedom to apply any external magnetic fields, whereas in magnetic traps the application of additional magnetic fields would substantially affect the trapping geometry. A special optical trapping environment can be generated in optical lattices [24,25]. Such lattices are created in optical standing-wave patterns, and in the three-dimensional case they provide an array of wavelength-sized microtraps, each of which may contain a single molecule. The new field of ultracold molecular quantum gases rapidly emerged in 2002/2003, when several groups reported on the formation of Feshbach molecules in BECs of 85 Rb [26], 133 Cs [27], 87 Rb [28], 23 Na [29], as well as in degenerate or near-degenerate Fermi gases of 40 K [30] and 6 Li [31–33]. Signatures of molecules in a BEC had been seen before using a photoassociative technique [34]. In the fall of 2003, the fast progress culminated in the creation of molecular Bose–Einstein condensates (mBEC) in atomic Fermi gases [35–37]. Heteronuclear Feshbach molecules entered the stage a few years later. So far, such molecules have been produced in Bose–Fermi mixtures of 87 Rb–40 K [38], and in Bose–Bose mixtures of 85 Rb–87 Rb [39] and 41 K–87 Rb [40]. The field of ultracold Feshbach molecules is developing rapidly, and there is great interest in homo- and heteronuclear molecules both in weakly bound and deeply bound regimes.

9.1.2

BASIC PHYSICS OF A FESHBACH RESONANCE

Feshbach resonances represent the essential tool to control the interaction between the atoms in ultracold gases, which has been key to many breakthroughs. Here we briefly outline the basic physics of a Feshbach resonance and its connection to the underlying near-threshold molecular structure. For more detailed discussions of the theoretical background see Chapters 6 and 11. The reader is also referred to two recent review articles [41,42]. In a simple picture, we consider two molecular potential curves Vbg (R) and Vc (R), as illustrated in Figure 9.1. For large internuclear distances R, the background potential Vbg (R) asymptotically correlates with two free atoms in an ultracold gas. For a collision process with a very small energy E, this potential represents the energetically open channel, usually referred to as the entrance channel. The other potential, Vc (R), representing the closed channel, is important as it can support bound molecular states near the threshold of the open channel. © 2009 by Taylor and Francis Group, LLC

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Closed channel EC

E

Energy

0 Entrance channel or open channel

Vc(R)

Vbg(R)

0

Atomic separation R

FIGURE 9.1 Basic two-channel model for a Feshbach resonance. The phenomenon occurs when two atoms colliding at energy E in the entrance channel resonantly couple to a molecular bound state with energy Ec supported by the closed channel potential. In the ultracold domain, collisions take place near zero energy, E → 0. Resonant coupling is then conveniently realized by magnetically tuning Ec near 0 if the magnetic moments of the closed and open channel differ.

A Feshbach resonance occurs when the bound molecular state in the closed channel is energetically close to the scattering state in the open channel, and some coupling leads to a mixing between the two channels. The energy difference can be controlled via a magnetic field when the corresponding magnetic moments are different. This leads to a magnetically tunable Feshbach resonance, which is the common way to achieve resonant coupling in the experiments with ultracold gases where the collision energy is practically zero. A magnetically tuned Feshbach resonance can be described by a simple expression for the s-wave scattering length a as a function of the magnetic field B, a(B) = abg 1 −

Δ B − B0

.

(9.1)

Figure 9.2a illustrates this resonance expression. The background scattering length abg , which is the scattering length of Vbg (R), represents the off-resonant value. It is directly related to the energy of the last bound vibrational level of Vbg (R). The parameter B0 denotes the resonance position, where the scattering length diverges (a → ±∞), and the parameter Δ is the resonance width. The energy of the weakly bound molecular state near the resonance position B0 is shown in Figure 9.2b, relative to the threshold of two free atoms with zero kinetic energy. The energy approaches the threshold on the side of the resonance where a is large and positive. Away from the resonance, the energy varies linearly with B with a slope given by δμ, the difference in magnetic moments of the open and © 2009 by Taylor and Francis Group, LLC

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4

(a)

a/abg

2 D

0 –2 –4 0.0 (b)

E/(dm D)

Eb

0.00

–0.01

–0.5 –2

–1

0 (B–B0)/D

–0.1

1

0.0

2

FIGURE 9.2 Scattering length a (a) and molecular state energy E (b) near a magnetically tuned Feshbach resonance. The inset shows the universal regime near the point of resonance where a is very large and positive.

closed channels. Near the resonance the coupling between the two channels mixes in entrance-channel contributions and strongly bends the molecular state. In the vicinity of the resonance position at B0 , where the two channels are strongly coupled, the scattering length is very large. For large positive a, a “dressed” molecular state exists with a binding energy given by Eb =

2 , 2mr a2

(9.2)

where mr is the reduced mass of the atom pair. In this limit, Eb depends quadratically on the magnetic detuning B − B0 and results in the bend depicted in the inset of Figure 9.2. This region is of particular interest because of its universal properties; see discussions in Chapters 6 and 11. Molecules formed in this regime are extremely weakly bound, and they are described by a wavefunction that extends far out of the classically allowed range. Such exotic molecules are therefore commonly referred to as halo dimers; we shall discuss their intriguing physics in Section 9.4. These considerations show how a Feshbach resonance is inherently connected with a weakly bound molecular state. The key question for experimental applications is how to prepare the molecular state in a controlled way; we shall address this in Section 9.2.

9.1.3

BINDING ENERGY REGIMES

The name Feshbach molecule emphasizes the production method, as it commonly refers to diatomic molecules made in ultracold atomic gases via Feshbach resonances. © 2009 by Taylor and Francis Group, LLC

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But what are the physical properties associated with a Feshbach molecule, and what is the exact definition of such a molecule? It is quite obvious that a Feshbach molecule is a highly excited molecule, existing near the dissociation threshold and having an extremely small binding energy as compared to the one of the vibrational ground state. To give an exact definition, however, is hardly possible as the properties of molecules gradually change with increasing binding energy, and there is no distinct physical property associated with a Feshbach molecule. Molecules created via Feshbach resonances can be transferred to many other states near threshold (Section 9.3) or to much more deeply bound states (Section 9.5), thus being converted to more conventional molecules. There is no really meaningful definition of when a molecule can still be called a Feshbach molecule, or when it loses this character. We therefore use a loose definition and call a Feshbach molecule any molecule that exists near the threshold in the energy range set by roughly the quantum of vibrational energy. Any other molecule we consider a deeply bound molecule, although views may differ on this. Figure 9.3 illustrates the vast range of molecular binding energies for the example of Cs2 dimers. According to our definition Feshbach molecules may have binding energies of up to roughly h × 100 MHz for the heavy Cs2 dimer and up to h × 2.5 GHz for the light Li2 dimer. These values are very small compared to the binding energies of ground-state molecules, which are of the order of h × 10 THz or h × 100 THz for the triplet and singlet potential, respectively. Nevertheless, ultracold Feshbach molecules can serve as the starting point for state transfer to produce very deeply bound, and even ground-state molecules, as we shall discuss in Section 9.5. 1 kHz Halo dimers

Binding energy/h

1 MHz

Feshbach molecules Evdw Vibrational quantum

1 GHz

Deeply bound molecules 1 THz Triplet Singlet

Ground-state molecules

1 PHz

FIGURE 9.3 Binding energy regimes for the example of Cs2 dimers, illustrated on a logarithmic scale. For the long-range van der Waals attraction between the atoms, a characteristic energy Evdw is introduced; see Chapter 6 and Section 9.4. The vibrational quantum, here defined as the energy range below threshold in which one finds at least one bound state, corresponds to about 40 Evdw ; see Figure 6.4 in Chapter 6. The binding energies of the vibrational ground state of the triplet and the single potential are typically five or six orders of magnitudes larger than this vibrational quantum.


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Two particular regimes of Feshbach molecules can be realized close to the dissociation threshold. The halo regime requires binding energies well below the so-called van der Waals energy Evdw (Section 6.2 in Chapter 6), which corresponds to about 3 MHz for Cs2 and about 600 MHz for Li2 . Molecules in high partial-wave states (Section 9.3) can exist above the dissociation threshold, as the large centrifugal barrier prevents them from decaying; such metastable Feshbach molecules have a negative binding energy.

9.2

MAKING AND DETECTING FESHBACH MOLECULES

This section discusses how to create ultracold molecules, starting from a degenerate or near-degenerate atomic gas. The three main experimental steps are illustrated in Figure 9.4. In a first step (a) atoms are converted into molecules, which can be done by sweeping the magnetic field across a Feshbach resonance. In a second step (b) a purification scheme can be applied that removes all the remaining atoms. In a third step (c) the molecules are forced to dissociate in a reverse sweep across the resonance in order to detect the resulting atoms. As an example, Figure 9.5 shows an image taken after the Stern–Gerlach separation of an atom–dimer mixture released from a trap; the smaller cloud represents a pure sample of Feshbach molecules. Similar techniques can also be applied to trapped atoms. In the following we describe the three steps of production, purification, and dissociation after a discussion of the important role of quantum statistics.

9.2.1

BOSONS AND FERMIONS: ROLE OF QUANTUM STATISTICS

Ultracold diatomic molecules can be composed either of two bosons, two fermions, or a boson and a fermion. In the first two cases, the molecules have bosonic character, while in the third case the molecules are fermions. Bosons are described by (b) Purification

(c) Dissociation

Energy

(a) Production

Magnetic field

FIGURE 9.4 Illustration of a typical experimental sequence to create, purify, and detect a sample of Feshbach molecules: (a) association via a magnetic-field sweep across the resonance, (b) selective removal of the remaining atoms, and (c) forced dissociation through a reverse magnetic field sweep. The dissociation is usually followed by imaging of the resulting cloud of atoms. The solid line corresponds to the bound molecular state, which intersects the threshold (gray horizontal line) and causes the Feshbach resonance; see also Figure 9.2. The dashed line indicates the molecular state above threshold, where it has the character of a quasi-bound state coupling to the scattering continuum.


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FIGURE 9.5 Example for the preparation of a pure molecular cloud. Here an optically trapped BEC of 87 Rb atoms was subjected to a magnetic-field sweep across a Feshbach resonance. The Stern–Gerlach technique was then applied after release from the trap to spatially separate the atoms (left) from the molecules (right). Finally the molecules were dissociated by a reverse Feshbach ramp, and an absorption image was taken. The field of view is 1.7 × 0.7 mm. (Courtesy of S. Dürr and G. Rempe.)

wavefunctions, which are symmetric with respect to exchange of identical particles, while, in the fermionic case, the wavefunctions are antisymmetric. This fundamental difference has important consequences. Identical bosons (fermions) can collide in even (odd) partial waves. The role of partial waves is discussed in Chapter 1. Here we use as the corresponding angular momentum quantum number. The first partial-wave contribution for bosons (fermions) is then the s-wave with = 0 (p-wave with = 1) [43]. At ultralow temperatures, only s-wave collisions are dominant with the consequence that collisions between identical fermions are suppressed. The absence of s-wave collisions is also the reason why, for instance, direct evaporative cooling cannot be applied to a sample of identical fermions. The situation changes when fermionic atoms are in different internal states, like hyperfine or Zeeman sublevels. The distinguishable particles can then interact in any partial wave. Fermion-composed molecules in s-wave states are therefore generally associated from ultracold two-component spin mixtures. Interactions in all partial waves are obviously allowed if two different atomic species are involved, regardless of their fermionic or bosonic character. The quantum-statistical character of a molecule can crucially influence its collisional stability. Feshbach molecules in highly excited vibrational states are in principle very sensitive to vibrational relaxation induced by atom–molecule and molecule– molecule collisions. If such an inelastic relaxation process occurs, the energy release is very large as compared to the trap depth and will result in an immediate loss of all collision partners from the trap. A remarkable exception to this general behavior are Feshbach molecules composed of fermionic atoms in a halo state, as they exhibit an extremely high stability against inelastic decay. The reason of this different behavior is a Pauli suppression effect. In a two-component Fermi mixture, relaxation processes are unlikely because they necessarily involve at least two identical fermions; for more details see Chapter 10 and Ref. [44]. An intermediate case can be found for molecules composed in a Bose–Fermi mixture of two atomic species. Here, the collisional stability depends on the atomic partner involved in a collision, either a boson © 2009 by Taylor and Francis Group, LLC


327

or a fermion. Only in collisions with fermionic atoms can a similar Pauli suppression effect be expected. The different degree of stability of bosons and fermions leads to a surprising result. One might think that the best way to achieve mBEC is the direct conversion of an atomic BEC. However, the stability of fermions turned out to be a key ingredient to experimentally achieve the condensation of molecules; see Section 9.4.4. For attaining a collisionally stable mBEC of boson-composed molecules, their transfer into the rovibrational ground state (Section 9.5) may be the only feasible way.

9.2.2

OVERVIEW OF ASSOCIATION METHODS

The optimum association strategy depends on several factors connected to the specific system under investigation. The main factors to consider are the quantum-statistical nature of the atoms that form the molecule and the particular Feshbach resonance employed. Ultralow temperatures and high phase-space densities are essential requirements for an efficient molecule creation. Therefore all association techniques start with an ultracold trapped atomic sample. The most widespread technique for molecule association uses time-varying magnetic fields. The magnetic field modifies the energy difference between the scattering state and the molecular state. When the two states have the same energy, the scattering length diverges and a molecular state can be accessed in the region of positive scattering length (a > 0); see Section 9.1.2. Magnetic field time sequences include linear ramps, fast jumps, or oscillating magnetic fields. Figure 9.4a illustrates a magnetic-field ramp, commonly referred to as “Feshbach ramp” [45–47]. A homogeneous magnetic field is swept across a Feshbach resonance from the side where a < 0 to the side where a > 0. The coupling between atom pairs and molecules removes the degeneracy at the crossing of the two corresponding energy levels. The resulting avoided level crossing can be used to adiabatically convert atom pairs into molecules by sweeping the magnetic field across the resonance, as indicated by the arrow in Figure 9.4a. The atomic sample is partially converted into a molecular gas. In the fermionic case, the conversion efficiency does solely depend on the ramp speed and on the atomic phase-space density [48], and can reach values close to one. In the case of bosons, the conversion efficiency is affected by inelastic collision processes. Near the center of a Feshbach resonance, the atoms experience a huge increase of the three-body recombination rate, while, after the association, the atom–dimer mixture can undergo fast collisional relaxation. The best parameters for molecule association require a compromise for the optimum ramp speed, which should be slow enough for an efficient conversion, but fast enough to avoid detrimental losses. Inelastic collisional loss can be reduced by using more elaborate time sequences for the magnetic field or, alternatively, resonant radio-frequency techniques. The key idea is to introduce an efficient atom–molecule coupling while minimizing the time spent near the resonance, where strong loss and heating results from inelastic interactions. An efficient technique is to apply a small sinusoidal modulation to the homogeneous field. The oscillating field then induces a corresponding modulation of the energy difference between the scattering state and the molecular state. The molecules © 2009 by Taylor and Francis Group, LLC

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are associated when the modulation frequencies match the molecular binding energies [49]. Typical modulation frequencies range from a few kHz to a few 100 kHz. The signature of molecule production is a resonant decrease of the atom number, as observed in gases of 85 Rb [50] and 40 K [51], and in a mixture of the two isotopes 85 Rb and 87 Rb [39]. A resonant coupling between atom pairs and molecules can also be induced by a radio-frequency excitation, which stimulates a transition between the atom pair state and the molecular state without modulating the energy difference. Heteronuclear 40 K87 Rb molecules were produced with this technique, both in an optical lattice [38] and in an optical dipole trap [52]. Collisional losses can also be suppressed by using a suitable trapping environment. This is the case when atoms are prepared in an optical lattice with double site occupancy. A stable system with a single molecule in each lattice site can then be created using, for instance, a Feshbach ramp [53,54]. A completely different association strategy can be applied to create molecules in an atomic Fermi gas of 6 Li atoms. This special situation is particularly important as it has opened up a simple and efficient route toward mBEC [36,37]; see also Section 9.4.4. Near a Feshbach resonance, in the regime of very large positive scattering lengths, atomic three-body recombination efficiently produces halo dimers. In a three-body recombination event in a two-component atomic Fermi gas, two nonidentical fermions bind into a molecule while the third atom carries away the leftover energy and momentum. Usually, the energy released in three-body collisions is very large so that all collision partners immediately leave the trap. However, for the special case of halo dimers, the very small binding energies can be on the order of the

250

Atom number (104)

200

150 Nat 100 50 2Nmol 0

0

1

2 Time (sec)

3

FIGURE 9.6 Formation of 6 Li2 halo dimers via three-body recombination. The molecules are created in an optically trapped spin mixture of atomic 6 Li at a temperature of 3 μK. The experiment is performed near a broad Feshbach resonance, where a = +1420a0 and Eb = kB × 15 μK = h × 310 kHz. Nat and Nmol denote the number of unbound atoms and the number of molecules, respectively. (Adapted from Jochim, S. et al., Phys. Rev. Lett., 91, 240402, 2003.)



329

typical thermal energies of the trapped particles and thus less than the trap depth [18]. The formation of halo dimers via three-body recombination then proceeds with small heating and negligible trap losses. This way of forming molecules can also be interpreted in terms of a thermal atom–molecule equilibrium [55,56]. Typical timescales for the atom–molecule thermalization are on the order of 100 msec to 1 sec, requiring the high collisional stability of fermion-composed molecules in the halo regime. As an example, Figure 9.6 shows how an initially pure sample of 6 Li atoms changes into an atom–molecule mixture [32].

9.2.3

PURIFICATION SCHEMES

Purification is an important step in many experiments with Feshbach molecules. The removal of atoms can be required so as to avoid fast collisional losses of molecules. In other cases, pure molecular samples are needed for particular applications. Most of the purification schemes act selectively on atoms and they are applied on timescales short compared to the typical molecular lifetimes. A possible purification strategy exploits the difference in magnetic moments of molecules and atoms. The two components of the gas can be spatially separated using the Stern–Gerlach technique, which uses magnetic field gradients [27,28]; for an example see Figure 9.5. This method is usually applied to gases in free space after release from the trap. A fast and efficient purification method, well suited for application to a trapped cloud, uses resonant light to push the atoms out of the sample by resonant radiation pressure [29,53,57]. The “blast” light needs to be resonant with a cycling atomic transition, which allows for the repeated scattering of photons. In many cases, the atoms are not in a ground-state sublevel connected to a cycling optical transition, so an intermediate step is necessary. In the most simple implementation, this is an optical pumping step that selectively acts on the atoms without affecting the molecules. Alternatively a microwave pulse can be used to transfer the atoms into the appropriate atomic level. The high selectivity of this double-resonance method minimizes residual heating and losses of molecules from the trap.

9.2.4

DETECTION METHODS

A very efficient way to detect Feshbach molecules is to convert them back into atom pairs and to take an image of the reconverted atoms with standard absorption imaging techniques [27,28]. In principle, each of the above-described association methods can be reverted and turned into a corresponding dissociation scheme. A robust and commonly used dissociation scheme is to simply reverse the Feshbach ramp, as illustrated in Figure 9.4c. The reverted ramp is usually chosen to be much faster than the one employed for association. The molecules are then brought into a quasi-bound state above the atomic threshold. Here they quickly dissociate into atom pairs, converting the dissociation energy into kinetic energy. The back-ramp is usually done in free space and can be performed either immediately after release from the trap or after some expansion time (≈10–30 msec). The choice of the delay time between the dissociation and the detection sets the type of information that can be © 2009 by Taylor and Francis Group, LLC

330


extracted from the absorption image. For a long delay time, the sample expands thus mapping the information on the momentum distribution onto the spatial distribution. The image then contains quantitative information on the dissociation energy [59]. For a short delay time, the absorption image simply reflects the spatial distribution of the molecules before the dissociation. Fast and efficient dissociation requires a sufficiently strong coupling of the molecular states to the scattering continuum. Molecular states with high rotational quantum number typically have weak couplings and high centrifugal barriers. A striking example is provided by Cs2 Feshbach molecules with = 8. (Partial waves associated with even angular momentum quantum numbers = 0, 2, 4, 6, 8 are by convention denoted as s, d, g, i, and l-waves, respectively. In the case of odd angular momentum quantum numbers = 1, 3, 5, 7 the partial waves are called p, f , h, and k-waves, respectively [43].) These molecules do not sufficiently couple to the atomic continuum and dissociation is prevented. There are two ways to detect them. The first is based on a time reversal of the association path (Section 9.3.2) [60]. The second exploits the crossing and the mixing with a quasi-bound state above threshold with ≤ 4 to force the dissociation [61]. Dissociation patterns can also provide additional spectroscopic information. An example is shown in Figure 9.7. Here the dissociation patterns of 87 Rb2 show strong modifications by the presence of a d-wave shape resonance [58]. In the halo regime, molecules can be detected by direct imaging [37,62]. The imaging frequency for halo molecules is very close to the atomic frequency. Presumably, the first photon dissociates the molecule into two atoms, which then absorb the subsequent photons. In the case of heteronuclear molecules, directed imaging is possible in a wider range of binding energies. This happens because the excited and ground molecular potentials both vary as 1/R6 ; see Chapter 5. In the homonuclear case, the excited potential varies as 1/R3 [63].

(a) 0.1 G

(b) 0.7 G

(c) 1.4 G

FIGURE 9.7 Dissociation patterns of 87 Rb2 molecules near a d-wave shape resonance. The molecules can dissociate either directly into the continuum or indirectly by passing through a d-wave shape-resonance state located behind the centrifugal barrier. Initially, the direct dissociation process dominates and preferentially populates isotropic s-wave states (a). Approaching the shape resonance, the pattern reveals first an interference between s- and d-partial waves (b), and then shows a pure outgoing d-wave (c). The given magnetic field values are relative to the field where dissociation is observed to set in. (Adapted from Volz, T. et al., Phys. Rev. A, 72, 010704(R), 2005.)


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9.3

INTERNAL-STATE MANIPULATION NEAR THRESHOLD

Below the atomic threshold a manifold of molecular states exists. They correspond to different vibrational, rotational and magnetic quantum numbers, and belong to potentials correlated with different hyperfine states. Having different magnetic moments, many levels cross when the magnetic field is varied. When some coupling between two molecular states that cross is present an avoided crossing occurs. The concept of avoided level crossings is very general and has found important applications in many fields of physics. We have already discussed the application to associate molecules in Section 9.2. In the field of ultracold molecules, the information on the avoided crossings is fundamentally important to fully describe the molecular spectrum and to understand the coupling mechanisms between different states. At the same time, controlled state transfer near avoided crossings opens up the intriguing possibility of populating many different molecular states that are not directly accessible by Feshbach association.

9.3.1 AVOIDED LEVEL CROSSINGS If an avoided crossing is well separated from any other avoided crossing, including the one resulting near threshold from the coupling to the scattering continuum, a simple two-state model can be applied. Consider two molecular states (i = 1, 2), for which the magnetic field dependencies of the binding energies Ei of the diabatic states ϕi are given by Ei (B) = μi (B − Bc ) + Ec ,

(9.3)

where μi are the magnetic moments, and Bc and Ec , the magnetic field and energy at which the two states cross, respectively (Figure 9.8). The crossing is conveniently

E2

Energy

E+

Ec V E1

E–

Bc Magnetic field

FIGURE 9.8 Schematic representation of an avoided crossing between two molecular states. The dashed lines represent the diabatic states with energies E1 and E2 , with a crossing at Bc . The solid lines represent the adiabatic states with energies E+ and E− , caused by a coupling V between the two diabatic states.


332


described in terms of adiabatic states if the two states are coupled by the interaction Hamiltonian H, ϕ1 |H|ϕ2 = 0. To obtain the corresponding adiabatic energies one has to solve the following eigenvalue problem, E1 V ϕ1 ϕ1 =E , ϕ2 ϕ2 V E2 where V = ϕ1 |H|ϕ2 is the coupling strength between the diabatic states. The adiabatic energies are given by 2E± = (E1 + E2 ) ±

(E1 − E2 )2 + 4V 2 ,

(9.4)

where the energies E+ and E− correspond to the upper and lower adiabatic levels of the avoided crossing. The energy difference between the adiabatic levels is ΔE = |E+ − E− | =

(μ1 − μ2 )2 (B − Bc )2 + 4V 2 ,

(9.5)

which at the crossing is twice the coupling strength, that is, ΔE(Bc ) = 2V . Let us illustrate the occurrence of avoided level crossings by the example of Cs2 , for which the near-threshold spectrum is shown in Figure 9.9. The molecular spectrum of Cs2 provides many avoided crossings, basically for two reasons. First, there exists a high density of molecular levels because its large mass gives rise to a small vibrational and rotation spacing, and its large nuclear spin gives rise to many hyperfine substates. Second, relatively strong relativistic spin–spin dipole and second-order spin–orbit interactions couple many of these states. Most crossings in Figure 9.9 are indeed avoided crossings and the inset shows one in an expanded view as an example. As they are well separated from each other, the simple two-state model applies very well. The crossing position Bc and the coupling strength V can be measured by different methods. Magnetic moment spectroscopy relies on measuring the effect of a magnetic field gradient after the molecular cloud is released from the trap. The difference in the position of a molecular cloud after a fixed time of flight with and without a magnetic field gradient is directly proportional to the magnetic moment. In the avoided crossing region the magnetic moment of the adiabatic molecular states shows a rapid change with magnetic field. This technique has been applied to map out several avoided crossings of the Cs2 spectrum, from which one is shown in Figure 9.10a. Another method relies on direct probing of ΔE by radio-frequency excitation between the adiabatic states, as demonstrated for 87 Rb2 [64]. More sophisticated methods rely on Ramsey-type or Stückelberg interferometry, which have been applied to avoided crossings in 87 Rb2 [64] and Cs2 [60], respectively. In the Ramsey scheme a radio-frequency pulse creates a coherent superposition. After a hold time and a second pulse, the population in one of the molecular branches is measured. This population shows an oscillation as a function of the hold time with a frequency of ΔE/h. In the Stückelberg scheme a coherent superposition is obtained by a magnetic field sweep over the avoided crossing; see Section 9.3.2. After a hold time, a reverse magnetic field sweep is applied and the population in both molecular branches is measured. Here the oscillation frequency in the population is also equal © 2009 by Taylor and Francis Group, LLC

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) 4g(4

4 d(4)

6s(6)

2g (2)

6 g (6) 5

5)

5

6g(6)

6l(

6l(4 )

4

6l(3)

Binding energy Eb/h (MHz)

4g(3)

0

4g(4)

6 –12

10 10

20

30 Magnetic field (G)

40

–14

–16 50

FIGURE 9.9 Molecular energy structure of Cs2 below the threshold of two free Cs atoms in the absolute hyperfine ground-state sublevel. The molecular states near threshold are well characterized by quantum numbers | f , mf ; , m [41], where f represents the sum of the total atomic spins F1,2 of the individual atoms, and is the rotational quantum number. The respective projection quantum numbers are given by mf and m . The interaction Hamiltonian conserves f + at zero magnetic field, and always conserves mf + m . In this example, Feshbach association is performed with pair of atoms that has mf + m = 6, and only molecular states with mf + m = 6 have to be considered. Therefore the labeling f (mf ) is sufficient to characterize the molecular states. Due to relatively strong relativistic spin-spin dipole and second-order spinorbit interactions, molecular states with rotational quantum number up to l-wave ( = 8) have to be taken into account in the case of Cs. The inset shows one of the narrow avoided crossing between two molecular states. (Adapted from Mark, M. et al., Phys. Rev. A, 76, 042514, 2007.)

to ΔE/h. In Figure 9.10b the mapping of a very narrow avoided crossing with the Stückelberg scheme is shown.

9.3.2

CRUISING THROUGH THE MOLECULAR SPECTRUM

A magnetic field ramp over the avoided crossing can be used for state transfer. When the magnetic field is ramped slowly, the population follows the adiabatic states. This is called an adiabatic ramp. If the change in magnetic field is very fast, the molecules do not experience the coupling between the diabatic states, and therefore remain in their initial state. This is a nonadiabatic ramp. The well-known Landau–Zener model describes the final population in both molecular states after the ramp. The probability of adiabatic transfer is given by ˙ , p = 1 − exp −B˙c /|B| © 2009 by Taylor and Francis Group, LLC

(9.6)

334 (a)

Mol. fraction of 6g(6) state

Cold Molecules: Theory, Experiment, Applications (b) 300

6g(6)

1.5

m(mB)

DE/h (kHz)

250

1 0.5 0

50 75 100 125 150 175 Hold time (msec)

200 150 100

1.0 50

4g(4) 12

13 14 Magnetic field (G)

15

0

11.1

11.2 11.3 11.4 11.5 Magnetic field (G)

11.6

FIGURE 9.10 Examples of two different techniques to map avoided crossings, both applied to Cs2 . (a) Magnetic moment spectroscopy on the two molecular states around the 6g(6)/4g(4) avoided crossing, measuring the magnetic moment as function of the magnetic field. A fit to the data gives Bc = 13.29(4) G and V = h × 164(30) kHz. (From Mark, M. et al., Phys. Rev. A, 76, 042514, 2007. With permission.) (b) Stückelberg interferometry on the two molecular states around the 6g(6)/6l(3) avoided crossing. The solid curve is a fit according to Equation 9.5, resulting in Bc = 11.339(1) G and V = h × 14(1) kHz. Inset: raw data showing an oscillation of the population in state 6g(6) right in the center of the crossing. (From Mark, M. et al., Phys. Rev. Lett., 99, 113201, 2007. With permission.)

where B˙ is the linear ramp speed and B˙c = 2πV 2 /(|μ1 − μ2 |). An adiabatic ramp ˙ B˙c , while a nonadiabatic ramp requires corresponds to a ramp speed of |B| ˙ B˙c . |B| Manipulation of the population over avoided crossings is fully coherent as there is no dissipation. Intermediate ramp speeds result in a coherent splitting of the population over the two molecular states. This property was demonstrated by a Stückelberg interferometer, in which two subsequent passages through an avoided crossing in combination with a variable hold time led to high-contrast population oscillations, as shown in the inset of Figure 9.10b. The required ramp speed for equal splitting is on the order of 1 G/μsec for an avoided crossing with V ∼ h × 100 kHz. The magnetic field ramps over avoided crossings allow to populate states that are not directly accessible by Feshbach association. This includes states that do not cross the atomic threshold, for example the 6g(6) state in Figure 9.9. There are also states that do cross the atomic threshold, but do not induce Feshbach resonances as the coupling with the atomic continuum is too weak. This occurs for molecular states with high rotational quantum number. For Cs2 molecular states with > 4, for example, the l-wave states shown in Figure 9.9, no Feshbach resonances exist. In Refs. [57] and [61] the population of the l-wave states using avoided crossings was demonstrated. As an example, the population scheme for one of the l-wave states is shown in Figure 9.11. The ability to populate molecular states that do not induce Feshbach resonances raises the question on the lifetime of these states above the atomic threshold, as the vanishing coupling with the atomic continuum implies that dissociation is suppressed. By studying the lifetime of the molecular sample above threshold it was found that l-wave molecules are indeed stable against dissociation on a timescale of 1 sec [61]. The © 2009 by Taylor and Francis Group, LLC

335

Ultracold Feshbach Molecules –2

4) 6l(

(4)

2

4g

Eb/h (MHz)

0

4 6g(6) 6

15

20 25 Magnetic field (G)

30

FIGURE 9.11 Population scheme for the Cs2 6l(4) state. Starting with Feshbach association on the 19.8 G g-wave Feshbach resonance, the resulting 4g(4) Feshbach molecules are brought to larger binding energies by ramping down the magnetic field. At a binding energy of about h × 5 MHz, the avoided crossing with the 6g(6) state is followed adiabatically. Then a fast ramp in the opposite direction is made, jumping the avoided crossing, after which a slow magnetic field ramp toward higher magnetic fields leads to adiabatic transfer from the 6g(6) to the 6l(4) state. The molecular sample in the 6l(4) state can be brought above the atomic threshold, as the large centrifugal barrier prohibits dissociation. (From Knoop, S. et al., Phys. Rev. Lett., 100, 083002, 2008. With permission.)

preparation of long-lived Feshbach molecules above the atomic threshold opens up the possibility to create novel metastable quantum states with strong pair correlations. The method of jumping avoided crossings is in practice limited to crossings with energy splittings up to typically h × 200 kHz. For stronger crossings, fast ramps are technically impracticable and the crossings are followed adiabatically when the magnetic field is ramped, leaving no choice for the state transfer. This problem can be overcome with the help of radio-frequency excitation [64]. For 87 Rb2 the transfer of molecules over nine level crossings was demonstrated when the magnetic field was ramped down from a Feshbach resonance near 1 kG to zero. This efficiently produced molecules with a binding energy Eb = h × 3.6 GHz at zero magnetic field. The combination of these elaborate ramp and radio-frequency techniques allows for cruising through the complex molecular energy structure. The level spectrum serves as a molecular “street map” and by passing straight through crossings or performing left or right turns one can reach any desired destination.

9.4

HALO DIMERS

Very close to resonance the Feshbach molecules are extremely weakly bound and become halo dimers. This halo regime is interesting because of its universal properties, © 2009 by Taylor and Francis Group, LLC

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regarding its intrinsic behavior (Section 9.4.1) and its few-body interaction properties (Section 9.4.2), in particular in the context of Efimov physics (Section 9.4.3). The properties of fermion-composed halo dimers, extensively discussed in Chapter 10, allow the realization of molecular Bose–Einstein condensation (mBEC), which is the topic of Section 9.4.4.

9.4.1

HALO DIMERS AND UNIVERSALITY

For large scattering lengths a, the Feshbach molecule enters the halo regime, where its binding energy is given by Equation 9.2. Its magnetic field dependence is illustrated in the inset of Figure 9.2. The halo state can be described in terms of a single effective molecular potential having scattering length a. In Figure 9.12 a typical halo wavefunction is shown as function of the interatomic distance R. This highlights its remarkable property to extend far into the classically forbidden range, which is at the heart of its universal properties. The asymptotic form of the halo wavefunction is proportional to e−R/a and the average distance between the two atoms is a/2; see also Section 11.5 of Chapter 11. The halo regime requires a to be much larger than the range of the two-body potential. For ground-state alkali atoms a characteristic range can be described by the van der Waals length Rvdw = 21 (2mr C6 /2 )1/4 , where C6 is the van der Waals dispersion coefficient; see Chapters 6 and 10. The corresponding van der Waals 2 ). Thus a halo dimer can be defined through the criteria energy is Evdw = 2 /(2mr Rvdw a Rvdw and Eb Evdw . For some examples the van der Waals scales are given in Table 9.1, where Rvdw is expressed in units of the Bohr radius a0 = 0.529 × 10−10 m. The halo regime corresponds to the universal regime for a > 0. The concept of universality is that the properties of any physical system in which |a| is much larger than the range of the interaction is determined by a, and therefore all these systems exhibit the same universal behavior [66]. In the universal regime the details of the shortrange interaction become irrelevant because of the dominant long-range nature of the wavefunction. The existence of the halo dimer at large positive a is a manifestation

E/h (MHz)

200 Rf(R) ~ e–R/a

100 0 –100 0

50

100

R (a0)

150

200

250

FIGURE 9.12 Radial wavefunction of a halo dimer, where R is the interatomic distance, extending far into the classically forbidden range where the wavefunction is proportional to e−R/a . The wavefunction is calculated for a Lennard–Jones potential and mr = 3 amu [65]. The potential depth is tuned such that the highest vibrational level, in this case v = 5, is in the halo regime.


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TABLE 9.1 Characteristic Van Der Waals Scales Rvdw and Evdw for Some Selected Examples 6 Li

Rvdw (a0 ) Evdw /h (MHz)

2

31 614

40 K 2

40 K87 Rb

65 21

72 13

85 Rb

2

82 6.2

133 Cs

2

101 2.7

of universality in two-body physics [67]. Halo states are known in nuclear physics, with the deuteron being a prominent example. In molecular physics, the He dimer has for many years served as the prime example of a halo state. Feshbach molecules with halo character are special to other halo states in the sense that their properties are magnetically tunable. In particular, one can scan from the universal regime of halo dimers to the nonuniversal regime of nonhalo Feshbach molecules. In principle, every Feshbach resonance features a halo region. In practice, however, only a few resonances are suited to experimentally realize this interesting regime. Those are broad resonances with large widths |Δ| (typically 1 G) and large background scattering lengths |abg | that exceed Rvdw [41,42]. Prominent examples are found in the two fermionic systems 6 Li and 40 K, and the bosonic systems 39 K, 85 Rb and 133 Cs, some of which are discussed in Section 6.4 of Chapter 6 and Section 11.5 of Chapter 11. Another universal feature of a halo dimer regards spontaneous dissociation, which is possible when its atomic constituents are not in the lowest internal states and open decay channels are present. In the halo regime the dissociation rate scales as a−3 , as has been observed for 85 Rb2 [68], and can be understood as a direct consequence of the large halo wavefunction [69].

9.4.2

COLLISIONAL PROPERTIES AND FEW-BODY PHYSICS

The concept of universality extends beyond two-body physics to few-body phenomena, including the low-energy scattering properties of halo dimers. The inelastic collisional properties are important as inelastic collisions usually lead to trap loss, determining the lifetime of a sample of trapped halo dimers. At the same they provide a convenient experimental observable to study few-body physics. In general, the loss of dimers can be described by the following rate equation: 2 n˙ D = −αnD − βnD nA ,

(9.7)

where α and β are the loss rate coefficients for dimer–dimer and atom–dimer collisions, respectively, and nA (nD ) is the atomic (molecular) density. Fast trap loss has been observed for Feshbach molecules in the nonuniversal regime, with α and β on the order of 10−11 to 10−10 cm3 /sec; see Section 3.3 of Chapter 3. The loss coefficient α is most conveniently measured using a pure molecular sample. To determine β an atom–dimer mixture is required, the atom number greatly exceeding the molecule number being beneficial. © 2009 by Taylor and Francis Group, LLC

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In the universal regime, simple scaling laws for α and β can be derived. For halo dimers composed of two identical bosons, β scales linearly with a [70,71], resulting in an enhancement of dimer loss near a Feshbach resonance. In contrast, for halo dimers made of two fermions in different spin states an a−2.55 and an a−3.33 scaling are derived for α and β [44], respectively, connected to the collisional stability of the molecular sample near a Feshbach resonance, as discussed in Section 9.2.1. Universal scaling laws are also derived for other atom–dimer systems [52,72], as well as for the related problem of three-body recombination in atomic samples [71,72]. Inelastic dimer–dimer scattering represents an elementary four-body process. For four identical bosons a universal prediction has so far not been derived, as the four-body problem is substantially more challenging than the three-body problem. Experimentally one can investigate this process by measuring α in a pure molecular sample of halo dimers made of identical bosons, which has been done in the case of Cs2 [73]. The results are given in Figure 9.13, showing a strong scattering length dependence of α. A pronounced loss minimum around 500a0 is observed, followed by a linear increase toward larger a. The interpretation of this striking behavior is still an open issue and might stimulate progress in the theoretical description of the four-body problem. For the elastic scattering properties, universal scaling laws have been derived for fermion-composed halo dimers. Both the atom–dimer and dimer–dimer scattering lengths simply scale with the atom–atom scattering length a, namely as 1.2 a [74] and 0.6 a [44], respectively. Therefore the elastic cross-sections, being proportional to the square of these scattering lengths, are very large in the halo region. This leads to

10

α (10–11cm3/sec)

8 6 4 2 Halo

Nonhalo 0

–1.0

0.0

a (1000a0)

0.5

1.0

FIGURE 9.13 Experimental study of the interaction between halo dimers. The relaxation rate coefficient α for inelastic dimer–dimer scattering is measured as function of the scattering length a for a pure sample of optically trapped Cs Feshbach molecules. The experiment makes use of the wide tunability of the scattering length for Cs, in particular into the halo regime. The solid line is a linear fit to the data in the region a ≥ 500 a0 . The shaded region indicates the a < Rvdw regime, where the Feshbach molecules are not in a halo state. (From Ferlaino, F. et al., Phys. Rev. Lett., 101, 023201, 2008. With permission.)


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fast thermalization in an atom–dimer mixture and a pure dimer sample, opening the possibility of evaporative cooling. For boson-composed halo dimers the few-body problem is more complicated and universal predictions for their elastic scattering properties are so far lacking.

9.4.3

EFIMOV THREE-BODY STATES

Efimov quantum states in a system of three identical bosons [75,76] are a paradigm for universal few-body physics. These states have attracted considerable interest, fueled by their bizarre and counter-intuitive properties, and by the fact that they had been elusive to experimentalists for more than 35 years. In 2006, experimental evidence for Efimov states in an ultracold gas of Cs atoms was reported [77]. In the context of ultracold quantum gases, Efimov physics manifests itself in three-body decay properties, such as resonances in the three-body recombination and atom–dimer relaxation loss rates. These resonances appear on top of the nonresonant “background” scattering behavior, given by the universal scaling laws discussed in Section 9.4.2. Efimov’s scenario is illustrated in Figure 9.14, showing the energy spectrum of the three-body system as a function of the inverse scattering length 1/a. For a < 0, the natural three-body dissociation threshold is at zero energy. States below are trimer states and states above are continuum states of three free atoms. For a > 0, the dissociation threshold is given by the binding energy of the halo dimer; at this threshold a trimer dissociates into a halo dimer and an atom. Efimov predicted an infinite series of weakly bound trimer states with universal scaling behavior. When the scattering 0

1/a < 0

1/a > 0

A+A+A

A+A+A

×eπ/s0 T

Energy

×e2π/s0

A+D

T

Inverse scattering length 1/a

FIGURE 9.14 Efimov’s scenario. Appearance of an infinite series of weakly bound Efimov trimer states (T) for resonant two-body interaction, showing a logarithmic-periodic pattern with universal scaling factors eπ/s0 and e2π/s0 for the scattering length and the binding energy, respectively. The binding energy is plotted as a function of the inverse two-body scattering length 1/a. The gray regions indicate the scattering continuum for three atoms (A + A + A) and for an atom and a halo dimer (A + D). To illustrate the series of Efimov states, the universal scaling factor is artificially set to 2.


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length is increased by a universal scaling factor eπ/s0 , a new Efimov state appears, which is just larger by this factor and has a weaker binding energy by a factor e2π/s0 . For three identical bosons s0 ≈ 1.00624, so that the scaling factors for the scattering length and the binding energy are 22.7 and 22.72 ≈ 515, respectively. For a < 0, Efimov states are “Borromean” states [67], which means that a weakly bound three-body state exists in the absence of a weakly bound two-body state. This property that three quantum objects stay together without pairwise binding is part of the bizarre nature of Efimov states. The Efimov trimers influence the three-body scattering properties. When an Efimov state intersects the continuum threshold for a < 0 three-body recombination loss is enhanced [79,80], as the resonant coupling of three atoms to an Efimov state opens up fast decay channels into deeply bound dimer states plus a free atom. Such an Efimov resonance has been observed in an ultracold, thermal gas of Cs atoms [77]. For a > 0 a similar phenomenon is predicted, namely an atom–dimer scattering resonance at the location at which an Efimov state intersects the atom–dimer threshold [81,82]. Resonance enhancement of β has been observed in a mixture of Cs atoms and Cs2 halo dimers [78]; see Figure 9.15. The asymmetric shape of the resonance can be explained by the background scattering behavior, which here is a linear increase as a function of a.

40 nK 170 nK

b (cm3/sec)

10–10

10–11 –1.5

–1.0

0.0 –0.5 a (1000a0)

0.5

1.0

FIGURE 9.15 Observation of a scattering resonance in an ultracold, optically trapped mixture of Cs atoms and Cs2 halo dimers. As for Figure 9.13, one benefits from the wide tunability of the scattering length provided by Cs. The data show the loss rate coefficient for atom– dimer relaxation β for temperatures of 40(10) nK and 170(20) nK. A pronounced resonance is observed at about +400 a0 , corresponding to a magnetic field of 25 G. The solid curve is a fit of an analytic model from effective field theory [66] to the data for a > Rvdw . (Adapted from Knoop, S. et al., Nature Phys. (in press, 2009), http://arxiv.org/abs/0807.3306. With permission.) © 2009 by Taylor and Francis Group, LLC


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Efimov physics impacts not only the scattering properties of three identical bosons, but any three-body system in which at least two of the three pairwise two-body interactions are large. Interestingly, depending on the mass ratio of the different components and the particle statistics, the scaling factor eπ/s0 can be much smaller than 22.7, facilitating the observation of successive Efimov resonances in future experiments [83].

9.4.4

MOLECULAR BEC

The achievement of Bose–Einstein condensation of molecules out of degenerate atomic Fermi gases is arguably one of the most remarkable results in the field of ultracold quantum gases. The experimental realization was demonstrated in ultracold Fermi gases of 6 Li and 40 K and turned out to be an excellent starting point to investigate the so-called BEC–BCS crossover and the properties of strongly interacting Fermi gases [18,20]. The realization of an mBEC of Feshbach molecules is only possible in the halo regime, where the collisional properties are favorable; see Section 9.4.2. In a spin mixture of 6 Li, containing the lowest two hyperfine states, the route to mBEC is particularly simple [36]. Evaporative cooling toward BEC can be performed in an optical dipole trap at a constant magnetic field in the halo region near the Feshbach resonance. In the initial stage of evaporative cooling the gas is purely atomic and molecules are produced via three-body recombination; see Section 9.2.2. With decreasing temperature the atom–molecule equilibrium favors the formation of molecules and a purely molecular sample is cooled down to BEC. The large atom– dimer and dimer–dimer scattering lengths along with strongly suppressed relaxation loss facilitate an efficient evaporation process. In this way, mBECs are achieved with a condensate fraction exceeding 90%. The experiments in 40 K followed a different approach to achieve mBEC [35]. For 40 K the halo dimers are less stable because of less favorable short-range three-body interaction properties. Therefore the sample is first cooled above the Feshbach resonance, where a is large and negative, to achieve a deeply degenerate atomic Fermi gas. A sweep across the Feshbach resonance then converts the sample into a partially condensed cloud of molecules. The emergence of the mBEC in 40 K is shown in Figure 9.16. For very large a the size of the halo dimers becomes comparable to the interparticle spacing, and the properties of the fermion pairs start to be determined by manybody physics. For a < 0 two-body physics no longer supports the weakly bound molecular state and pairing is entirely a many-body effect. In particular, in the limit of weak interactions on the a < 0 side of the resonance, pairing can be understood in the framework of the well-established BCS theory, developed in the 1950s to describe superconductivity. Here the fermionic pairs are called Cooper pairs. The BEC and BCS limits are smoothly connected by a crossover regime where the gas is strongly interacting. This BEC–BCS crossover has attracted considerable attention in many-body quantum physics [18,20,84]. A theoretical description of this challenging problem is very difficult and various approaches have been developed. With tunable Fermi gases, a unique testing ground has become available to quantitatively investigate the crossover problem. © 2009 by Taylor and Francis Group, LLC

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(a)

Optical density

(b)

2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5 0.0

0.0 –200 –100 0 100 200 Position (μm)

–200 –100 0 100 200 Position (μm)

FIGURE 9.16 Emergence of a molecular BEC in an ultracold Fermi gas of 40 K atoms, observed in time-of-flight absorption images. The density distribution on the left-hand side ((a) 2D surface plot; (b) 1D cross-section) was taken for a Fermi gas that was cooled down to 19% of the Fermi temperature. After ramping across the Feshbach resonance no mBEC was observed as the sample was too hot. The density distribution on the right-hand side was observed for a colder sample at 6% of the Fermi temperature. Here the ramp across the Feshbach resonance resulted in a bimodal distribution, revealing the presence of an mBEC with a condensate fraction of 12%. (From Greiner, M. et al., Nature, 426, 537, 2003. With permission.)

9.5 TOWARD GROUND-STATE MOLECULES Ultracold gases of ground-state molecules hold great prospects for novel quantum systems with more complex interactions than existing in atomic quantum gases. For the experimental preparation of such molecular systems, the collisional stability is an important prerequisite. This makes rovibrational ground states the prime candidates, as vibrational relaxation is energetically suppressed. Stable BECs of dimers may for instance serve as a starting point to synthesize mBECs of more complex constituents. A particular motivation arises from the large electric dipole moments of heteronuclear dimers; see Chapter 2. This leads to the long-range dipole–dipole interaction, which is anisotropic and can be modified by an electric field. As a result, a rich variety of phenomena can be expected, with conceptual challenges and intriguing experimental opportunities. Proposals range from the study of new quantum phases and quantum simulations of condensed-matter system (Chapter 12), to fundamental physics tests (Chapters 15 and 16) and schemes for quantum-information processing (Chapter 17). The STIRAP method has attracted considerable interest as a highly efficient way to coherently transfer atom pairs or Feshbach molecules into a more deeply bound state [85]. The method offers a high transfer efficiency without heating of the molecular © 2009 by Taylor and Francis Group, LLC

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sample, thus allowing the preservation of the high phase-space density of an ultracold gas. Several groups are actively pursuing such experiments with the motivation to create quantum gases of collisionally stable ground-state molecules.

9.5.1

STIMULATED RAMAN ADIABATIC PASSAGE

The basic idea of STIRAP to transfer population between two quantum states relies on a particularly clever implementation of a coherent two-photon Raman transition, involving a dark state during the transfer. For a detailed description of its principles and an overview of early applications, the reader may refer to Ref. [86]. Let us consider a three-level system with |a and |b representing two different ground-state levels, and |e being an electronically excited state, as shown in Figure 9.17. Laser 1 (2) couples the state |a (|b) to the state |e and the Rabi frequency Ω1 (Ω2 ) describes the corresponding coupling strength [87]. The Rabi frequency is defined as d · E/, with E being the electric-field amplitude of the laser field and d the dipole matrix element. The states |a and |b are long lived, whereas the excited state |e can undergo spontaneous decay into lower states. An essential property of such a coherently coupled three-level system is the existence of a dark state |D as an eigenstate of the system. This state generally occurs if both laser fields have the same resonance detuning with respect to the corresponding transition, that is, if the two-photon detuning is zero. The state is dark in the sense that it is decoupled from the excited state |e and thus not influenced by its radiative decay. The dark state can be understood as a coherent superposition of state |a and state |b, |D =

1 Ω1 + Ω 2 2 2

(Ω2 |a − Ω1 |b) ,

(9.8)

with state |e not being involved. The key idea of STIRAP is to slowly change Ω1 and Ω2 in a way that the system is always kept in a dark state. This avoids losses caused by spontaneous light scattering in the excited state |e [88]. One may intuitively expect that the transfer occurs by first applying Laser 1 and then Laser 2, as in a conventional two-photon Raman transition. STIRAP instead uses a counterintuitive laser pulse scheme, in which Laser 2 is applied first. Initially, only state |a is populated Figure 9.17a; this corresponds to the dark |e>

|e>

|e>

|e>

|e>

Laser 1

Laser 2

Laser 1

Laser 2 |a>

|a>

(a)

|a> |b>

|b>

|b>

(b)

|a> |b>

(c)

|a> |b>

(d)

(e)

FIGURE 9.17 Illustration of the basic idea of STIRAP. Transfer from state |a to state |b, keeping the population always in a dark state (see Equation 9.8). Initially Ω2 = 1 and Ω1 = 0, and finally Ω2 = 0 and Ω1 = 1.


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state |D if only Laser 2 is active. If one now slowly introduces Laser 1, the dark state |D begins to evolve into a superposition of |a and |b (Figure 9.17b). By adiabatically ramping down the power of Laser 2 and ramping up the intensity of Laser 1, the character of the superposition state changes and it acquires an increasing |b admixture (Figure 9.17c, d). When finally Laser 1 is on and Laser 2 is turned off, the population is completely transfered to the state |b (Figure 9.17e). This adiabatic passage through the dark state |D does not involve any population in the radiatively decaying state |e. Note that the relative phase coherence of the two laser fields is an essential requirement for STIRAP. Moreover, it is important to note that the coherent sequence can be time-reversed to achieve transfer from |b to |a.

9.5.2

STIRAP EXPERIMENTS

In the experiments to create quantum gases of collisionally stable ground-state molecules, the initially populated state |a corresponds to a Feshbach molecule and |b is a deeply bound molecular state. The state |e is a carefully chosen electronically excited level, which has to fulfil two important conditions. To obtain large enough optical Rabi frequencies Ω1 and Ω2 the state must provide sufficiently strong coupling to both states |a and |b, demanding sufficiently strong dipole matrix elements besides the technical requirement of sufficiently intense laser fields at the particular wavelengths. Moreover, state |e needs to be well separated from other excited states, as a significant coupling to these would deteriorate the dark character of the superposition state. Proof-of-principle experiments have been performed on homonuclear 87 Rb2 molecules [89] and heteronuclear 40 K87 Rb molecules [90], demonstrating the transfer by one or a few vibrational quanta. The STIRAP transfer of 87 Rb2 started with trapped Feshbach molecules. In order to characterize and optimize the experimental parameters, the study of a “dark resonance” was an essential step (Figure 9.18). STIRAP could then be efficiently implemented to transfer the molecules from the last bound vibrational level (Eb = h × 24 MHz) to the second-to-last state (h × 637 MHz) with an efficiency close to 90%. In the experiment with the heteronuclear 40 K87 Rb molecules, STIRAP was demonstrated with final molecular binding energies of up to h × 10 GHz. Subsequent experiments then moved on to more deeply bound molecules and in particular to the rovibrational ground states of the triplet and the singlet potential. To bridge a large energy range with STIRAP the experiments become more challenging both for technical and physical reasons. In particular, the optical transition matrix elements become an important issue as they enter the optical Rabi frequencies. In molecular physics, the Franck–Condon principle states that an optical transition does not change the internuclear separation. As a consequence, the overlap of the wavefunctions between ground state and excited state, quantified by the so-called Franck–Condon factors, crucially enters into the matrix elements. A general rule of thumb can be given for finding a large Franck–Condon overlap based on a simple classical argument. The dominant part of the wavefunction occurs near the turning points of the classical oscillatory motion in a specific molecular potential. A large © 2009 by Taylor and Francis Group, LLC

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Normalized molecule number

1 0.8 0.6 0.4 0.2 0

−40

−20

0 δ/2π (MHz)

20

40

FIGURE 9.18 Dark resonance observed as a signature of a dark state between two different vibrational levels in a trapped sample of 87 Rb2 Feshbach molecules. Here both laser fields are turned on simultaneously, and the frequency of Laser 1 is varied while Laser 2 is kept at a fixed frequency, thus varying the two-photon detuning δ. The population of the optically excited state |e leads to a strong background loss caused by spontaneous decay into other levels in a region corresponding to the single-photon transition linewidth. If the two-photon resonance condition is met (δ = 0), a suppression of loss occurs in a very narrow frequency range, which indicates the decoupling from the excited state. The behavior can be modeled on the basis of a three-state system, which provides full information on the relevant experimental parameters. In practice, the study of such a dark resonance is an important step to implement STIRAP. (Adapted from Winkler, K. et al., Phys. Rev. Lett., 98, 043201, 2007.)

Franck–Condon factor for an optical transition can be expected if the classical turning points approximately coincide for the ground and excited state. Molecules with large binding energies of about h × 32 THz were demonstrated in an experiment on 133 Cs2 [91]. The experiment starts with the creation of a pure sample of Feshbach molecules in a weakly bound state |a with a binding energy of h × 5 MHz. Here the standard methods for association and purification are applied as described in Section 9.2.A suitable Franck–Condon overlap was experimentally found for the particular choice of |e that is depicted in Figure 9.19a. The vertical arrows indicate the optical transitions at internuclear separations where maximum wavefunction overlap occurs. The applied timing sequence is shown in Figure 9.19b. In a first STIRAP pulse sequence, the molecules are transferred from |a to |b within 15 μsec. After a variable holding time in state |b, the deeply bound molecules are reconverted into Feshbach molecules by a time-reversed STIRAP pulse sequence. The Feshbach molecules are then detected by the usual dissociative imaging. The experimental results in Figure 9.19c show that 65% of the initial number of molecules reappear after the full double-STIRAP sequence, pointing to a single STIRAP efficiency of about 80%. In the final stage of preparation of the present manuscript, breakthroughs have been achieved on the creation of rovibrational ground-state molecules in three different experiments. While Ref. [92] reports on polar 40 K87 Rb molecules in both the © 2009 by Taylor and Francis Group, LLC

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Cold Molecules: Theory, Experiment, Applications (b) 6S + 6P

2300 cm–1 u¢ = 225

|e>

8000

0+ u

Transfer

Rabi frequency

(a)

Holding time

Reverse transfer

Laser 1

4000

STIRAP time

Laser 1 ~1126 nm

(c)

a3∑+ u

0

u = 155

6S + 6S

|a>

–1061

u = 73

|b>

X1∑+ g u=0

–3650 6

8 10 12 14 16 18 20 22 24 Internuclear distance (a0)

Molecule number (103)

Energy (cm–1)

Laser 2 Laser 2 ~1006 nm

8 7 6 5 4 3 2 1 0 0

5

10 15 20 25 30 STIRAP time (μsec)

35

40

FIGURE 9.19 STIRAP experiment starting with ultracold 133 Cs2 Feshbach molecules. (a) The relevant molecular potential curves and energy levels are shown schematically. The vertical arrows show the optical transitions and their positions indicate the internuclear distances where maximum Franck–Condon overlap is expected near the classical turning points. (b) Variation of the Rabi frequencies during the pulse sequence. (c) Experimental results showing the measured molecule population in |a as a function of time through a double-STIRAP sequence. (Adapted from Danzl, J.G. et al., Science, 321, 1062, 2008. With permission.)

ground state of the triplet potential (Eb = h × 7.2 THz) and that of the singlet potential (h × 125 THz), Ref. [93] demonstrates homonuclear 87 Rb2 molecules in the rovibrational ground state of the triplet potential (h × 7.0 THz). A further experiment [94] has succeeded in the production of 133 Cs2 molecules in the rovibrational ground state of the singlet potential (h × 109 THz). The successful 40 K87 Rb experiment also provides a further remarkable demonstration of the importance of the Franck–Condon overlap arguments. At first glance, it may be surprising that the huge binding energy difference to the singlet ground state could be bridged in a single STIRAP step. However, an excited molecular state |e could be found for which the outer classical turning point provides good overlap with the Feshbach molecular state, while the internuclear distance at the inner turning point matches the rovibrational ground state. In the absence of such a fortunate coincidence, transfer may not be feasible in a single STIRAP sequence. Two consecutive STIRAP steps involving different excited states [85,94] promise a general way to achieve efficient ground-state transfer for any homo- or heteronuclear Feshbach molecule. © 2009 by Taylor and Francis Group, LLC


9.6

347

FURTHER DEVELOPMENTS AND CONCLUDING REMARKS

In the rapidly developing field of cold molecules, Feshbach molecules play a particular role as the link to research on ultracold quantum matter. In this chapter, we have presented the basic experimental techniques and discussed some major developments in the field. In this final Section, we briefly point to some other recent developments not discussed so far, adding further interesting aspects to our overview of the research field. Intriguing connections to condensed-matter physics can be found when Feshbach molecules are trapped in optical lattices. In this spatially periodic environment, a Feshbach molecule can be used both as a well-controllable source of correlated atom pairs, and as an efficient tool to detect such pairs. A recent experiment [95] shows how Feshbach molecules, prepared in a three-dimensional lattice, are converted into “repulsively bound pairs” of atoms when forcing their dissociation through a Feshbach ramp. Such atom pairs stay together and jointly hop between different sites of the lattice, because the atoms repel each other. This counterintuitive behavior is due to the fact that the bandgap of the lattice does not provide any available states for taking up the interaction energy. Another fascinating example along the lines of condensed-matter physics is a quantum state with exactly one Feshbach molecule per lattice site, as illustrated in Figure 9.20. Such a state has vanishing entropy and closely resembles a Mott-insulator state [96]. It offers an excellent starting point for experiments on strongly correlated many-body states (Chapter 12) and on quantum information processing (Chapter 17), and for the production of a BEC of molecules in the internal ground state [85]. In the

FIGURE 9.20 Illustration of a quantum state with one molecule at each site of an optical lattice. In the central region of the three-dimensional lattice one finds exactly one molecule per site, trapped in the vibrational ground state. Such a state was prepared with 87 Rb2 Feshbach molecules. (From Volz, T. et al., Nature Phys., 2, 692, 2006. With permission.)


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latter case, the basic idea is to convert the collisionally unstable Feshbach molecules into collisionally stable molecules in the rovibrational ground state. Then the lattice is no longer needed for isolating the molecules from each other and dynamical melting of the ordered state by ramping down the lattice may eventually lead to mBEC. Novel phenomena can also be observed with Feshbach molecules in lowdimensional trapping schemes, such as one-dimensional tubes realized in twodimensional optical lattices. Here a two-body bound state can exist also for negative scattering lengths, contrary to the situation in free space [97]. Low-dimensional traps also offer very interesting environments to create strongly correlated many-body regimes. Counterintuitively, the sensitivity of Feshbach molecules to inelastic decay can inhibit loss and lead to a stable correlated state, as recently demonstrated with 87 Rb in a one-dimensional trap [98]. 2 Various mixtures of ultracold atomic species offer a wide playground for ultracold heteronuclear molecules, and the basic interaction properties of many different combinations involving bosonic and fermionic atoms are currently being explored. A new twist to the field is given by ultracold Fermi–Fermi systems, which have recently been realized in mixtures of 6 Li and 40 K [99,100]. Molecule formation in such systems will lead to bosonic molecules, which in the many-body context may lead to new types of pair-correlated states and novel types of strongly interacting fermionic superfluids. Another interesting question is whether more complex ultracold molecules can be created than simple dimers. A first step into this direction is the observation of scattering resonances between ultracold dimers [101] which have been found in the collisional decay of an optically trapped sample of 133 Cs2 molecules and interpreted as a result of a resonant coupling to tetramer states. All the examples presented in this chapter highlight how great progress is currently being made to fully control the internal and external degrees of freedom of various types of ultracold molecules under conditions near quantum degeneracy. This will soon enable many new applications, ranging from high-precision measurements and quantum computation to the exploration of few-body physics and novel correlated many-body quantum states. Several major research themes are obvious and will undoubtedly lead to great success, but we are also confident that the field also holds the potential for many surprises and developments that we can hardly imagine now.

ACKNOWLEDGMENTS We would like to thank all members of the Innsbruck group “Ultracold Atoms and Quantum Gases” (www.ultracold.at) for contributing to our research program on Feshbach molecules. In particular, we would like to thank Johannes Hecker Denschlag and Hanns-Christoph Nägerl for long-standing collaboration on this topic. We are also indebted to Cheng Chin, Paul Julienne, and Eite Tiesinga for the many insights gained when jointly preparing a review article on Feshbach resonances together with one of us (R.G.). We are grateful to the Austrian Science Fund (FWF) for supporting our various projects related to ultracold molecules. F.F. is a Lise-Meitner fellow of the FWF, and S.K. is supported by the European Commission with a Marie Curie Intra-European Fellowship. © 2009 by Taylor and Francis Group, LLC


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Molecular Regimes in 10 Ultracold Fermi Gases Dmitry S. Petrov, Christophe Salomon, and Georgy V. Shlyapnikov CONTENTS 10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Feshbach Resonances and Diatomic Molecules . . . . . . . . . . . . . . . . . . . 10.2 Homonuclear Diatomic Molecules in Fermi Gases. . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Weakly Interacting Gas of Bosonic Molecules: Molecule–Molecule Elastic Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Suppression of Collisional Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Collisional Stability and Molecular BEC . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Heteronuclear Molecules in Fermi–Fermi Mixtures. . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Effect of Mass Ratio on Elastic Intermolecular Interaction . . . . . . . 10.3.2 Collisional Relaxation for Moderate Mass Ratios. . . . . . . . . . . . . . . . . 10.3.3 Born–Oppenheimer Picture of Collisional Relaxation . . . . . . . . . . . . 10.3.4 Molecules of Heavy and Light Fermionic Atoms . . . . . . . . . . . . . . . . . 10.3.5 Trimer States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.6 Collisional Relaxation of Molecules of Heavy and Light Fermions and Formation of Trimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Crystalline Molecular Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Born–Oppenheimer Potential in a Many-Body System of Molecules of Heavy and Light Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Gas–Crystal Quantum Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Molecular Superlattice in an Optical Lattice . . . . . . . . . . . . . . . . . . . . . . 10.5 Concluding Remarks and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The field of quantum gases is rapidly expanding in the direction of ultracold clouds of fermionic atoms, with the goal of revealing novel macroscopic quantum states 355 © 2009 by Taylor and Francis Group, LLC

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and achieving various regimes of superfluidity. The initial idea was to achieve the Bardeen–Cooper–Schrieffer (BCS) superfluid phase transition in a two-component Fermi gas, which requires attractive interactions between the atoms of different components. Then, in the simplest version of this transition, at sufficiently low temperatures, fermions belonging to different components and with opposite momenta on the Fermi surface form correlated (Cooper) pairs in the momentum space. This leads to the appearance of a gap in the single-particle excitation spectrum and to the phenomenon of superfluidity (e.g., Ref. [1]). In a dilute ultracold two-component Fermi gas, the most efficient formation is that of Cooper pairs due to the attractive intercomponent interaction in the s-wave channel (negative s-wave scattering length a). However, for typical values of a, the superfluid transition temperature is extremely low. For this reason, the efforts of many experimental groups have been focused on modifying the intercomponent interaction using Feshbach resonances. The scattering length a near a Feshbach resonance can be tuned from −∞ to +∞. This has led to exciting developments (see Ref. [2] for review), such as the direct observation of superfluid behavior in the strongly interacting regime (n|a|3 1, where n is the gas density) through vortex formation [3], and the study of the influence of imbalance between the two components of the Fermi gas on superfluidity [4–8]. We focus here on the remarkable physics of weakly bound diatomic molecules of fermionic atoms. This initially unexpected physics connects molecular and condensed matter physics. The weakly bound molecules are formed on the positive side of the resonance (a > 0) [9–12] and they are the largest diatomic molecules obtained so far. Their size is of the order of a and it has reached thousands of angstroms in current experiments. Accordingly, their binding energy is exceedingly small (10 μK or less). Being composite bosons, these molecules obey Bose statistics, and they have been Bose-condensed in experiments with 40 K2 [13,14] at JILA and with 6 Li2 at Innsbruck [15,16], MIT [17,18], ENS [19], Rice [20], and Duke [21]. Nevertheless, some of the interaction properties of these molecules reflect Fermi statistics of the individual atoms forming the molecule. In particular, these molecules are found to be remarkably stable with respect to collisional decay. Being in the highest rovibrational state, they do not undergo collisional relaxation to deeply bound states on a timescale exceeding seconds at densities of about 1013 cm−3 . This is more than four orders of magnitude longer than the lifetime of similar molecules consisting of bosonic atoms. The key idea of our discussion of homonuclear diatomic molecules formed in a two-component Fermi gas by atoms in different internal (hyperfine) states is to show how one obtains an exact universal result for the elastic interaction between such weakly bound molecules and how the Fermi statistics for the atoms provides a strong suppression of their collisional relaxation into deep bound states. It is emphasized that the repulsive character of the elastic intermolecular interaction and remarkable collisional stability of the molecules are the main factors allowing for their Bose–Einstein condensation and for prospects related to interesting manipulations with these molecular condensates. Currently, a new generation of experiments is being developed for studying degenerate mixtures of different fermionic atoms [22,23], with the idea of revealing the influence of the mass difference on superfluid properties and finding novel types of superfluid pairings. On the positive side of the resonance one expects the formation © 2009 by Taylor and Francis Group, LLC

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of heteronuclear weakly bound molecules; this attracts a great deal of interest, in particular for creating dipolar gases. The 6 Li –40 K weakly bound molecules have already been created in experiments [24]. We present an analysis of how the mass ratio for constituent atoms influences the elastic interaction between the molecules and their collisional stability. The discussion is focused on molecules of heavy and light fermions, where one expects the formation of trimer bound states and the manifestation of the Efimov effect. We then show that a many-body system of such molecules can exhibit a gas–crystal quantum transition. Remarkably, the atomic system itself remains dilute, and the crystalline ordering is due to a relatively long-range interaction between the molecules originating from the exchange of light fermions. Realization of the crystalline phase requires a very large mass ratio for the atoms forming a molecule in order to suppress the molecular kinetic energy. This can be achieved in an optical lattice for heavy atoms, where the crystalline phase of a dilute molecular system emerges as a superlattice, and we discuss the related physics. The chapter is concluded by an overview of prospects for manipulations with the weakly bound molecules of fermionic atoms. The leading ideas include the achievement of ultralow temperatures and BCS transition for atomic fermions, the creation of dipolar quantum gases, as well as observations of peculiar trimer bound states in an optical lattice.

10.1.2

FESHBACH RESONANCES AND DIATOMIC MOLECULES

At ultralow temperatures, when the de Broglie wavelength of atoms greatly exceeds the characteristic radius of interatomic interaction forces, atomic collisions and interactions are generally determined by the s-wave scattering. Therefore, in twocomponent Fermi gases one may consider only the interaction between atoms of different components, which can be tuned by using Feshbach resonances. The description of a many-body system near a Feshbach resonance requires a detailed knowledge of the two-body problem. In the vicinity of the resonance, the energy of a colliding pair of atoms in the open channel is close to the energy of a molecular state in another hyperfine domain (closed channel). The coupling between these channels leads to a resonant dependence of the scattering amplitude on the detuning δ of the closed channel state from the threshold of the open channel, which can be controlled by an external magnetic (or laser) field. Thus, the scattering length becomes field dependent (see Figure 10.1). The Feshbach effect is a two-channel problem that can be described in terms of the Breit–Wigner scattering [25,26], and various aspects of such problems have been discussed by Feshbach [27] and Fano [28]. In cold atom physics the idea of Feshbach resonances was introduced in Ref. [29], and optically induced resonances have been analyzed in Refs. [30–33]. At resonance, the scattering length changes from +∞ to −∞, and in the vicinity of the resonance one has the inequality n|a|3 1, where n is the gas density. The gas is then said to be in a strongly interacting regime. It is still dilute in the sense that the mean interparticle separation greatly exceeds the characteristic radius of the interparticle interaction Re . However, the amplitude of binary interactions (scattering © 2009 by Taylor and Francis Group, LLC

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BCS

Weakly bound molecules BEC

II

B

I

III

FIGURE 10.1 The dependence of the scattering length on the magnetic field near a Feshbach resonance. The symbols I, II, and III label the regime of a weakly interacting degenerate atomic Fermi gas, the strongly interacting regime of BCS–BEC crossover, and the regime of weakly bound molecules. At sufficiently low temperatures region I corresponds to the BCS superfluid pairing, and region III to the Bose–Einstein condensation of molecules.

length) is larger than the mean separation between particles, and in the quantum degenerate regime the conventional mean field approach is no longer valid. For large detuning from resonance the gas is in the weakly interacting regime, that is, the inequality n|a|3 1 is satisfied. On the negative side of the resonance (a < 0), at sufficiently low temperatures of the two-species Fermi gas one expects the BCS pairing between distinguishable fermions, which is well described in the literature [1]. On the positive side (a > 0), two fermions belonging to different components form diatomic molecules. For a Re , these molecules are weakly bound and their size is of the order of a. The crossover from BCS to BEC behavior has recently attracted a great deal of interest, in particular with respect to the nature of superfluid pairing, transition temperature, and elementary excitations. This type of crossover has been earlier discussed in the literature in the context of superconductivity [34–37] and in relation to superfluidity in two-dimensional films of 3 He [38,39]. The idea of resonant coupling through a Feshbach resonance for achieving a superfluid phase transition in ultracold two-component Fermi gases has been proposed in Refs. [40] and [41], and for the two-dimensional case it has been discussed in Ref. [42]. The two-body physics of the Feshbach resonance is the most transparent if the (small) background scattering length is neglected. Then, for low collision energies ε, the scattering amplitude is given by [26]: √ γ/ 2μ F(ε) = − √ , ε + δ + iγ ε

(10.1)

√ where the quantity γ/ 2μ ≡ W characterizes the coupling between the open and closed channels and μ is the reduced mass of the two atoms. The scattering length is © 2009 by Taylor and Francis Group, LLC


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a = −F(0). In Equation 10.1 the detuning δ is positive if the bound molecular state is below the continuum of the colliding atoms. Then, for δ > 0, the scattering length is positive, and for δ < 0 it is negative. Introducing a characteristic length R∗ = 2 /2μW

(10.2)

and expressing the scattering amplitude through the relative momentum of particles √ k = 2με/, we can rewrite Equation 10.1 in the form F(k) = −

a−1

1 . + R∗ k 2 + ik

(10.3)

The validity of Equation 10.3 does not require the condition kR∗ 1. At the same time, this equation formally coincides with the amplitude of scattering of slow particles by a potential with the same scattering length a and an effective range R = −2R∗ , obtained under the condition k|R| 1. The length R∗ is an intrinsic parameter of a Feshbach resonance. It characterizes the width of the resonance. From Equations 10.1 and 10.2 we see that small W and, consequently, large R∗ correspond to narrow resonances, whereas large W and small R∗ lead to wide resonances. The term “wide” is generally used when the length R∗ drops out of the problem, which, according to Equation 10.3, requires the condition kR∗ 1. In a quantum degenerate atomic Fermi gas the characteristic momentum of particles is the Fermi momentum, kF = (3π2 n)1/3 . Thus, in the strongly interacting regime and on the negative side of the resonance (a < 0), for a given R∗ the condition of wide resonance depends on the gas density n and takes the form kF R∗ 1 [43–47]. For a > 0 one has weakly bound molecular states (it is certainly assumed that the characteristic radius of interaction Re a), and for such molecular systems the criterion of the wide resonance is different [48,50]. The binding energy of the weakly bound molecule state is determined by the pole of the scattering amplitude (Equation 10.3). One then finds [48,50] that this state exists only for a > 0, and under the condition R∗ a

(10.4)

ε0 = 2 /2μa2 .

(10.5)

the binding energy is given by

The wavefunction of such a weakly bound molecular state has only a small admixture of the closed channel, and the size of the molecule is ∼a. The characteristic momenta of the atoms in the molecule are of the order of a−1 , and in this respect the inequality 10.4 represents the criterion of a wide resonance for the molecular system. Under these conditions atom–molecule and molecule–molecule interactions are determined by a single parameter—the atom–atom scattering length a. In this sense, the problem becomes universal. It is equivalent to the interaction problem for the twobody potential, which is characterized by a large positive scattering length a and has © 2009 by Taylor and Francis Group, LLC

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a potential well with a weakly bound molecular state. The picture remains the same when the background scattering length cannot be neglected, although the condition of a wide resonance can be somewhat modified [51]. Most ongoing experiments with Fermi gases of atoms in two different internal (hyperfine) states use wide Feshbach resonances [52]. For example, weakly bound molecules 6 Li2 and 40 K2 have been produced in experiments [10–20] by using Feshbach resonances with a length R∗ 20 Å, and for the achieved values of the scattering length a (from 500 to 2000 Å) the ratio R∗ /a was smaller than 0.1. In this review we will consider the case of a wide Feshbach resonance.

10.2

HOMONUCLEAR DIATOMIC MOLECULES IN FERMI GASES

10.2.1 WEAKLY INTERACTING GAS OF BOSONIC MOLECULES: MOLECULE–MOLECULE ELASTIC INTERACTION As we have shown in the previous section, the size of weakly bound bosonic molecules formed at a positive atom–atom scattering length a in a two-species Fermi gas (region III in Figure 10.1) is of the order of a. Therefore, at densities such that na3 1, the atoms form a weakly interacting gas of these molecules. Moreover, under this condition, at temperatures sufficiently lower than the molecular binding energy ε0 and for equal concentrations of the two atomic components, practically all atoms are converted into molecules [53]. This is definitely the case at temperatures below the temperature of quantum degeneracy, Td = 2π2 n2/3 /M (the lowest one in the case of fermionic atoms with different masses, with M being the mass of the heaviest atom). One can clearly see this by comparing Td with ε0 , given by Equation 10.5. Thus, one has a weakly interacting molecular Bose gas and the first question is related to the elastic interaction between the molecules. For a weakly interacting gas the interaction energy in the system is equal to the sum of pair interactions and the energy per particle is ng (2ng for a noncondensed Bose gas), with g being the coupling constant. In our case this coupling constant is given by g = 4π2 add /(M + m), where add is the scattering length for the molecule– molecule (dimer–dimer) elastic s-wave scattering, and M, m are the masses of heavy and light atoms, respectively. The value of add is important for evaporative cooling of the molecular gas to the regime of Bose–Einstein condensation and for the stability of the condensate. The Bose–Einstein condensate is stable for repulsive intermolecular interactions (add > 0), and for add < 0 it collapses. We thus see that for analyzing macroscopic properties of the molecular Bose gas, one should first solve the problem of elastic interaction (scattering) between two molecules. In this section we present the exact solution of this problem for homonuclear molecules formed by fermionic atoms of different components (different internal states) in a two-component Fermi gas. The case of M = m will be discussed in Section 10.3. The solution for M = m was obtained in Refs. [49] and [50] assuming that the atom–atom scattering length a greatly exceeds the characteristic radius of interatomic potential: a Re . © 2009 by Taylor and Francis Group, LLC

(10.6)

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Then, as in the case of the three-body problem with fermions [54–57], the amplitude of elastic interaction is determined only by a and can be found in the zero-range approximation for the interatomic potential. This approach was introduced in the two-body physics by Bethe and Peierls [58]. The leading idea is to solve the equation for the free relative motion of two particles by placing a boundary condition on the wavefunction ψ at a vanishing interparticle distance r: (rψ) 1 =− , rψ a

r → 0,

(10.7)

which can also be rewritten as ψ ∝ (1/r − 1/a),

r → 0.

(10.8)

One then gets the correct expression for the wavefunction at distances r Re . When a Re , Equation 10.8 correctly describes the wavefunction of weakly bound and continuum states even at distances much smaller than a. We now use the Bethe–Peierls approach for the problem of elastic molecule– molecule (dimer–dimer) scattering, which is a four-body problem described by the Schrödinger equation: 2

2 2 (∇ − ∇r22 − ∇R2 ) + U(r1 ) + U(r2 ) m r1 3

√ U[(r1 + r2 ± 2R)/2] − E Ψ = 0, +

−

(10.9)

±

where m is the atom mass. Labeling fermionic atoms in different internal states by the symbols ↑ and ↓, the distance between two given ↑ and ↓ fermions is r1 , and r2 is the distance √between the other two. √ The distance between the centers of mass of these pairs is R/ 2, and (r1 + r2 ± 2R)/2 are the separations between ↑ and ↓ fermions in the other two possible ↑↓ pairs (see Figure 10.2). The total energy is E = −2ε0 + ε, with ε being the collision energy, and ε0 = 2 /ma2 the binding energy of a dimer. The wavefunction Ψ is symmetric with respect to the permutation of bosonic ↑↓ pairs and antisymmetric with respect to permutations of identical fermions: Ψ(r1 , r2 , R) = Ψ(r2 , r1 , −R) √ √ r1 + r2 ± 2R r1 + r2 ∓ 2R r1 − r2 = −Ψ , ,± √ . 2 2 2

(10.10)

For the weak binding of atoms in a molecule, assuming that the two-body scattering length satisfies inequality 10.6, at all interatomic distances (even much smaller than a) except for very short separations of the order of or smaller than Re , the © 2009 by Taylor and Francis Group, LLC

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r1 1 R

1

2

r2

2

FIGURE 10.2 Set of coordinates for the four-body problem.

motion of atoms in the four-body system is described by the free-particle Schrödinger equation,

mE 2 2 2 − ∇r1 + ∇r2 + ∇R + 2 Ψ = 0. (10.11) The correct description of this motion requires the four-body wavefunction Ψ to satisfy the Bethe–Peierls boundary condition for the vanishing distance in any pair √ of ↑ and ↓ fermions, that is, for r1 → 0, r2 → 0, and r1 + r2 ± 2R → 0. Due to the symmetry condition 10.10 it is necessary to require a proper behavior of Ψ only at one of these boundaries. For r1 → 0 the boundary condition reads Ψ(r1 , r2 , R) → f (r2 , R)(1/4πr1 − 1/4πa).

(10.12)

The function f (r2 , R) contains the information about the second pair of particles when the first two are on top of each other. In the ultracold limit, where ka 1, (10.13) the molecule–molecule scattering is dominated by the contribution of the s-wave channel. Inequality 10.13 is equivalent to ε ε0 and, hence, the s-wave scattering can be analyzed from the solution of Equation 10.11 with E = −2ε0 < 0. For large R the corresponding wavefunction takes the form √ Ψ ≈ φ0 (r1 )φ0 (r2 )(1 − 2add /R), R a, (10.14) where the wavefunction of a weakly bound molecule is given by φ0 (r) = √


1 2πa r

exp(−r/a).

(10.15)


363

Combining Equations 10.12 and 10.14, we obtain the asymptotic expression for f at large distances R: √ R a. (10.16) f (r2 , R) ≈ (2/r2 a) exp (−r2 /a)(1 − 2add /R), In the case of s-wave scattering, the function f depends only on three variables: the absolute values of r2 and R, and the angle between them. We now derive and solve the equation for f . The value of the molecule–molecule scattering length add is then deduced from the behavior of f at large R determined by Equation 10.16. We first establish a general form of the wavefunction Ψ satisfying Equation 10.11, with the boundary condition 10.12 and symmetry relations 10.10. In our case the total energy E = −22 /ma2 < 0, and the Green function of Equation 10.11 reads √ √ (10.17) G(X) = (2π)−9/2 (Xa/ 2)−7/2 K7/2 ( 2 X/a), √ where X = |S − S |, K7/2 ( 2X/a) is the decaying Bessel function, and S= {r1 , r2 , R} is a nine-component vector. Accordingly, |S − S | = (r1 − r 1 )2 + (r2 − r 2 )2 + (R − R )2 . The four-body wavefunction Ψ is regular everywhere except for vanishing distances between ↑ and ↓ fermions. Therefore, it can be expressed through G(|S − S |) with coordinates S corresponding to a vanishing distance the ↑ and ↓ fermions, that is, for r 1 → 0, r 2 → 0, and √ between (r 1 + r 2 ± 2R )/2 → 0. Thus, for the wavefunction Ψ satisfying the symmetry relations 10.10 we have Ψ(S) = Ψ0 + d3 r d3 R G(|S − S1 |) + G(|S − S2 |) − G(|S − S+ |) − G(|S − S− |) h(r , R ),

(10.18)

√ √ where S1 = {0, r , R }, S2 = {r , 0, −R }, and S± = {r /2 ± R / 2, r /2 ∓ R / 2 ∓ √ r 2}. The function Ψ0 is a properly symmetrized finite solution of Equation 10.11, regular at any distances between the atoms. For E < 0, nontrivial solutions of this type do not exist and we have to put Ψ0 = 0. The function h(r2 , R) has to be determined by comparing Ψ in Equation 10.18 at r1 → 0, with the boundary condition 10.12. Considering the limit r1 → 0, we extract the leading terms on the right-hand side of Equation 10.18. These are the terms that behave as 1/r1 or remain finite in this limit. The last three terms in the square brackets in Equation 10.18 provide a finite contribution: (10.19) d3 r d3 R h(r , R ) G(|S¯ 2 − S2 |) − G(|S¯ 2 − S+ |) − G(|S¯ 2 − S− |) , where S¯ 2 = {0, r2 , R}. To find the contribution of the first term in the square brackets, we subtract and add an auxiliary quantity: √ h(r2 , R) exp (− 2r1 /a). (10.20) h(r2 , R) G(|S − S1 |)d3 r d3 R = 4πr1 © 2009 by Taylor and Francis Group, LLC

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The result of the subtraction yields a finite contribution, which for r1 → 0 can be written as d3 r d3 R [h(r , R ) − h(r2 , R)]G(|S − S1 |) =P

d3 r d3 R [h(r , R ) − h(r2 , R)]G(|S¯ 2 − S1 |),

r1 → 0,

(10.21)

with the symbol P denoting the principal value of the integral over dr (or dR ). A detailed derivation of Equation 10.21 and the proof that the integral in the second line of this equation is convergent are given in Ref. [50]. In the limit r1 → 0, the right-hand side of Equation 10.20 is equal to √ h(r2 , R)(1/4πr1 − 2/4πa). (10.22) We thus find that for r1 → 0 the wavefunction Ψ of Equation 10.18 takes the form Ψ(r1 , r2 , R) =

h(r2 , R) + R, 4πr1

r1 → 0,

(10.23)

where R is the sum of regular r1 -independent terms given by Equations 10.19 and 10.21, and by the second term on the right-hand side of Equation 10.22. Equation 10.23 must coincide with Equation 10.12, and comparing the singular terms of these equations we find h(r2 , R) = f (r2 , R). As the quantity R must coincide with the regular term of Equation 10.12, equal to −f (r2 R)/4πa, we obtain the following equation for the function f : & d3 r d3 R G(|S¯ − S1 |)[f (r , R ) − f (r, R)] + G(|S¯ − S2 |) −

' √ G(|S¯ − S± |) f (r , R ) = ( 2 − 1)f (r, R)/4πa.

(10.24)

±

Here S¯ = {0, r, R}, and we omitted the symbol of the principal value for the integral in the first line of Equation 10.24. As we have already mentioned above, for s-wave scattering the function f (r, R) depends only on the absolute values of r and R and on the angle between them. Thus, Equation 10.24 is an integral equation for the function of three variables. In order to find the molecule–molecule scattering length, it is more convenient to transform √ the momentum–space function, Equation 10.24 into an equation for f (k, p) = d3 rd3 Rf (r, R) exp(ik · r/a + ip · R/ 2a), which yields the following expression:

f (k ± (p − p)/2, p ) d3 p 2 + p2 /2 + (k ± (p − p)/2)2 + (k ± (p + p)/2)2 ±

=

2π2 (1 + k 2 + p2 /2)f (k, p) f (k , −p) d3 k − . 2 2 2 2 + k + k + p /2 2 + k 2 + p2 /2 + 1


(10.25)


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By making the substitution f (k, p) = (δ(p) + g(k, p)/p2 )/(1 + k 2 ) we reduce Equation 10.25 to an inhomogeneous equation for the function g(k, p): 1 2π2 (1 + k 2 + p2 /2)g(k, p) + (1 + k 2 + p2 /4)2 − (kp)2 p2 (1 + k 2 )( 2 + k 2 + p2 /2 + 1)

g(k ± (p − p)/2, p )d3 p =− 2 2 p (2 + p /2 + (k ± (p − p)/2)2 + (k ± (p + p)/2)2 ) ± ×(1 + (k ± (p − p)/2)2 ) g(k , −p)d3 k + . (10.26) p2 (2 + k 2 + k 2 + p2 /2)(1 + k 2 ) In the case of s-wave scattering, the function g(k, p) depends on the absolute values of k and p and on the angle between these vectors. For p → 0 this function tends to a finite value independent of k. As one can easily establish on the basis of Equation 10.16 and the definition of g(k, p), the molecule–molecule scattering length is given by add = −2π2 a limp→0 g(k, p). Numerical calculations from Equation 10.26 give, with 2% accuracy [48], add = 0.6a > 0.

(10.27)

This result was first obtained in Refs. [49] and [50] from the direct numerical solution of Equation 10.24 by fitting the obtained f (r, R) with the asymptotic form 10.16 at large R. The calculations show the absence of four-body weakly bound states, and the behavior of f at small R suggests a soft-core repulsion between dimers, with a range ∼a. The result of Equation 10.27 is exact, and it indicates the stability of molecular BEC with respect to collapse. Compared to earlier studies, which assumed add = 2a [37,59], Equation 10.27 gives almost twice as small a sound velocity of the molecular condensate and a rate of elastic collisions smaller by an order of magnitude. The result of Equation 10.27 has been confirmed by Monte Carlo calculations [60] and by calculations within the diagrammatic approach [61,62]. An approximate diagrammatic approach leading to add = 0.75a has been developed in Ref. [59].

10.2.2

SUPPRESSION OF COLLISIONAL RELAXATION

The weakly bound dimers that we are considering are diatomic molecules in the highest rovibrational state (see Figure 10.3). They can undergo relaxation into deeply bound states in their collisions with each other: for example, one of the colliding molecules may relax to a deeply bound state while the other one dissociates (Including p-wave interactions, one can think of the formation of deeply bound states as two identical (↑ or ↓) fermions. So, the collision of two weakly bound molecules can lead to the creation of a deep bound state by two ↑ (or ↓) fermionic atoms, and two ↓ (or ↑) atoms become unbound.). The released energy is the binding energy of the final deep state, which is of the order of 2 /mRe2 . It is transformed into the kinetic energy of the particles in the outgoing collision channel and they escape from the trapped © 2009 by Taylor and Francis Group, LLC

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Cold Molecules: Theory, Experiment, Applications U

2 e0= ћ 2 ma

R

FIGURE 10.3 Interaction potential U as a function of the distance R between two distinguishable fermionic atoms. The dashed line shows the energy level of the weakly bound molecule, and the solid line the energy level of a deeply bound state.

sample. Therefore, the process of collisional relaxation of weakly bound molecules determines the lifetime of a gas of these molecules and possibilities to Bose-condense this gas. We now show that collisional relaxation is suppressed due to Fermi statistics for atoms in combination with a large size of weakly bound molecules [49,50]. The binding energy of the molecules is ε0 = 2 /ma2 and their size is ∼a Re . The size of deeply bound states is of the order of Re . Therefore, the relaxation process may occur when at least three fermionic atoms are at distances ∼Re with respect to each other. As two of them are necessarily identical, due to the Pauli exclusion principle the relaxation probability acquires a small factor proportional to a power of (qRe ), where q ∼ 1/a is a characteristic momentum of the atoms in the weakly bound molecular state. Relying on the inequality a Re we outline a method that allows us to establish the dependence of the relaxation rate on the scattering length a, without going into a detailed analysis of the short-range behavior of the system. It is assumed that the amplitude of the inelastic relaxation process is much smaller than the amplitude of elastic scattering. Then the dependence of the relaxation rate on a is related only to the a-dependence of the initial-state four-body wavefunction Ψ. We again consider the ultracold limit described by the condition 10.13, where the relaxation process is dominated by the contribution of the s-wave molecule–molecule scattering. The key point is that the relaxation process requires only three atoms to approach each other in short distances of the order of Re . The fourth particle can be far away from these three and, in this respect, does not participate in the relaxation process. This distance is of the order of the size of a molecule, which is ∼ a Re . We thus see that the configuration space contributing to the relaxation probability can be viewed as a system of three atoms at short distances ∼ Re from each other and a fourth atom separated from this system by a large distance ∼a (see Figure 10.4). In this case the four-body wavefunction decomposes into a product: Ψ = η(z)Ψ(3) (ρ, Ω), © 2009 by Taylor and Francis Group, LLC

(10.28)

367


Re 13.6 we have a well-known phenomenon of the fall of a particle to the center in an attractive 1/R2 potential [26]. In this case the shape of the wavefunction at distances of the order of Re can significantly influence the large-scale behavior, and a short-range three-body parameter is required to describe the system. The wavefunction of heavy atoms χ(R) acquires many nodes at short distances R, which indicates the appearance of three-body bound Efimov states.

10.3.4

MOLECULES OF HEAVY AND LIGHT FERMIONIC ATOMS

The discussion of the previous subsection shows that weakly bound molecules of heavy and light fermions become collisionally unstable for the mass ratio M/m close to the limiting value 13.6. The effect of the Pauli principle becomes weaker than the attraction between heavy atoms at distances R a, mediated by light fermions. However, this picture explains only the dependence of the relaxation rate on the twobody scattering length a. At the same time, for heteronuclear molecules the relaxation rate and the amplitude of the elastic molecule–molecule interaction can also depend on the mass ratio irrespective of the value of a and short-range physics. To elucidate this dependence, we will look at the interaction between the molecules of heavy and light fermions at large intermolecular separations. We consider the interaction between two such molecules in the Born–Oppenheimer approximation and calculate the wavefunctions and binding energies of two light fermions in the field of two heavy atoms fixed at their positions R1 and R2 . The sum of the corresponding binding energies gives an effective interaction potential Ueff for the heavy fermions as a function of the separation R = |R1 − R2 | between them. For R > a, there are two bound states, gerade (+) and ungerade (−), for a light atom interacting with a pair of fixed heavy atoms. Their wavefunctions are given by Equation 10.39, and the corresponding binding energies follow from Equations 10.40 and 10.41. For large R satisfying the condition exp(−R/a) 1, Equation 10.40 yields ± (R) ≈ −|ε0 | ∓ 2|ε0 |

Uex (R) a exp(−R/a) + , R 2

(10.43)

where the binding energy of a single molecule, ε0 , is given by Equation 10.5 with the reduced mass μ very close to the light atom mass m: Uex (R) = 4|ε0 | © 2009 by Taylor and Francis Group, LLC

a a 1− exp(−2R/a). R 2R

(10.44)

377


Because the light fermions are identical, their two-body wavefunction is an antisymmetrized product of gerade and ungerade wavefunctions: √ − + − ψR (r1 , r2 ) = [ψ+ (10.45) R (r1 )ψR (r2 ) − ψR (r2 )ψR (r1 )]/ 2. The Born–Oppenheimer adiabatic approach is valid at distances R > a, where the effective interaction potential between the molecules, Ueff , is the sum of + (R) and − (R) (one should add 2|ε0 | so that Ueff (R) → 0 for R → ∞). This potential is displayed in Figure 10.10. At sufficiently large interheavy separations where Equation 10.43 is valid, the effective potential can be written as Ueff (R) = + (R) + − (R) + 2|ε0 | ≈ Uex (R).

(10.46)

The potential Uex originates from the exchange of light fermions and thus can be treated as an exchange interaction. It is purely repulsive and, according to Equation 10.44, has the asymptotic shape of a Yukawa potential at large R. Direct calculations show that Uex is a very good approximation of Ueff for R 1.5a. We now demonstrate the calculation of the dimer–dimer scattering length add in the limit of M/m 1 [67]. In the Born–Oppenheimer approach the Schrödinger equation for the relative motion of two molecules reads (−(2 /m)∇R2 + Ueff (R) − ε)Ψ(R) = 0,

(10.47)

where ε is the collision energy. Note, that the repulsive effective potential is inversely proportional to the light mass m, whereas the kinetic energy operator in Equation 10.47 has a prefactor 1/M. Therefore, for a large mass ratio M/m, the heavy atoms approach each other at distances smaller than a with an exponentially small √ tunneling probability P ∝ exp(−B M/m), where B ∼ 1. This leads to the relaxation rate constant αrel ∝ exp(−B M/m), (10.48) which strongly decreases with increasing mass ratio M/m. Ueff

a

R

FIGURE 10.10 Interaction potential for two molecules of heavy and light fermions as a function of the separation R between the heavy atoms. © 2009 by Taylor and Francis Group, LLC

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The analysis shows that the elastic part of the scattering amplitude can be calculated with a very high accuracy using Equation 10.47 for M/m 20 and is practically insensitive to the way we choose the boundary condition for the wavefunction at R = a. The dominant contribution to the scattering comes from distances in the vicinity of R = add a, where the effective potential can be approximated by Equation 10.44 with a constant preexponential factor: Ueff (R) ≈ 22 (maadd )−1 exp(−2R/a).

(10.49)

Then, the zero-energy solution of Equation 10.47, which decays at smaller R, reads a Ψ(R) = K0 R

2M a −R/a , e m add

(10.50)

where K0 is the decaying Bessel function. Comparing the result of Equation 10.50 at large R with the asymptotic behavior Ψ(R) ∝ (1 − add /R), we obtain an equation for add : a e2γ M a add = ln . (10.51) 2 2 m add This gives add ≈ a ln

M/m,

(10.52)

and the scattering cross-section is 2 σdd = 8πadd .

(10.53)

From Equation 10.50 we see that the interval of distances near R = add , where the wavefunction changes, is of the order of a. This justifies the use of Equation 10.49. In fact, the corrections to Equation 10.51 can be obtained by treating the difference between Equations 10.44 and 10.49 perturbatively. In this way the first-order correction to the dimer–dimer scattering length is −(3/4)a2 /add , where add is determined from Equation 10.51. Qualitatively, Ueff (R) can be viewed as a hard-core potential with radius add , where the edge is smeared out on a length scale ∼a add . Therefore, the ultracold limit for dimer–dimer collisions, required for the validity of Equation 10.53, is realized for relative momenta of the dimers, k, satisfying the inequality kadd 1.

(10.54)

It can be useful (see Ref. [68]) to approximate the potential Ueff by a pure hard core with radius add . This approximation works under the condition ka 1, which is less strict than Equation 10.54. © 2009 by Taylor and Francis Group, LLC

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Let us now mention that numerical calculations [67] show that there are no resonances in the dimer–dimer scattering amplitude, which could appear in the presence of a weakly bound state of two dimers. Here we give a qualitative explanation of the absence of these bound states. Suppose there is such a state with energy ε → 0. Then, at distances R > a the wavefunction of the heavy atoms should exponentially decay √ on the distance scale ∼a m/M a, because Ueff represents a barrier with height ∼1/ma2 . This means that the heavy atoms in such a bound state should be localized mostly at distances smaller than a. The gerade light atom is also localized at these distances, as seen from the shape of the function ψ+ . The motion of the ungerade light atom relative to the localized trimer can be viewed as scattering with odd values of the angular momentum, and due to the centrifugal barrier, the bound states of this atom with the trimer should be localized at distances ∼a from the heavy atoms. In this case one would expect the Born–Oppenheimer approximation to work, because the ungerade light atom is moving much faster than the heavy atoms. However, this leads to a contradiction, because in the Born–Oppenheimer approach discussed above the ungerade state at interheavy separations R < a is unbounded. We thus conclude that weakly bound states of two dimers are absent. Although there are no resonances in the dimer–dimer collisions, there are branchcut singularities in the scattering amplitude. They are related to the presence of inelastic processes in molecule–molecule collisions. These represent the relaxation of one of the colliding dimers into a deeply bound state, with the other dimer being dissociated, and the formation of bound trimers consisting of two heavy and one light atom, the other light atom carrying away the released binding energy.

10.3.5 TRIMER STATES The trimer states, which in most cases can be called Efimov trimers, are interesting objects. Their existence can be seen from the Born–Oppenheimer picture for two heavy atoms and one light atom in the gerade state. Within the Born–Oppenheimer approach the three-body problem reduces to the calculation of the relative motion of the heavy atoms in the effective potential created by the light atom. For the light atom in the gerade state, this potential is + (R), found in the previous subsection. The Schrödinger equation for the wavefunction of the relative motion of the heavy atoms, χν (R), reads ˆ (10.55) Hχ(R) = −(2 /M)∇R2 + + (R) χν (R) = ν χν (R). The trimer states are nothing more than the bound states of heavy atoms in the effective potential + (R). Accordingly, they correspond to the discrete part of the spectrum ν , where the symbol ν denotes a set containing angular (l) and radial (n) quantum numbers. For R a the potential + (R) is proportional to −1/R2 (see Equation 10.42) and, if this effective attraction overcomes the centrifugal barrier, we arrive at the wellknown phenomenon of the fall of a particle to the center in an attractive 1/R2 potential. Then, for a given orbital angular momentum l, the radial part of χν can be written as χν (R) ∝ R−1/2 sin(sl ln R/r0 ), © 2009 by Taylor and Francis Group, LLC

R a,

(10.56)

380


where sl =

0.16M/m − (l + 1/2)2 .

(10.57)

The three-body parameter r0 determines the phase of the wavefunction at small distances and, in principle, depends on l. The wavefunction 10.56 has infinitely many nodes, which means that in the zero-range approximation there are infinitely many trimer states. This is one of the properties of three-body systems with resonant interactions discovered by Efimov [54]. We see that the fall to the center is possible in many angular momentum channels, provided the mass ratio is sufficiently large. However, for practical purposes and for simplicity, it is sufficient to consider the case where the Efimov effect occurs only for the angular momentum channel with the lowest possible l for a given symmetry. This implies that when the heavy atoms are fermions and one has odd l, in order to confine ourselves to l = 1 we should have the mass ratio in the range 14 M/m 76. For bosonic heavy atoms where l is even, we set l = 0 and consider M/m 39 to avoid the Efimov effect for l ≥ 2. In both cases we need a single three-body parameter r0 . The formation of Efimov trimers in ultracold dimer–dimer collisions is energetically allowed only if ν < −2|0 |. This means that the trimers that we are interested in are relatively well bound and their size is smaller than a. Therefore, the process of trimer formation is exponentially reduced for large mass ratios as the heavy atoms have to tunnel under the repulsive barrier Ueff (R). Moreover, this process requires all of the four atoms to approach each other at distances smaller than a, and its rate decreases with trimer size because it is more difficult for two identical light fermions to be in a smaller volume. From Equation 10.56 one sees that the behavior of the three-body system does not change if r0 is multiplied by λl = exp(π/sl ).

(10.58)

On the other hand, the dimensional analysis shows that the quantity ν /ε0 depends only on the ratio a/r0 . This means that, except for a straightforward scaling with a, properties of the three-body system do not change when a is multiplied or divided by λl . This discrete scaling symmetry of a three-body system, which shows itself in the log-periodic dependence of three-body observables, has yet to be observed experimentally. In the case of three identical bosons, where the Born–Oppenheimer approach does not work and it is necessary to solve the three-body problem exactly [54], the observation of the consequences of the discrete scaling requires a to be changed by a factor of λ ≈ 22.7, which is technically very difficult in ongoing experiments with cold atoms. In this respect three-body systems with a very large mass difference can be more favorable because of smaller values of λ. For example, in order to see one period of the log-periodic dependence in a Cs–Cs–Li three-body system, a has to be changed only by a factor of λ ≈ 5. At this point it is worth emphasizing that three-body effects can be observed in a gas of light–heavy dimers, where the interdimer repulsion originating from the exchange of the light fermions strongly reduces the decay rate associated with the relaxation of the dimers into deep bound states. The trimer formation in dimer–dimer collisions is © 2009 by Taylor and Francis Group, LLC


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very sensitive to the positions and sizes of the Efimov states, and the measurement of the formation rate can be used to demonstrate the discrete scaling symmetry of a three-body system. Indeed, this rate should have the log-periodic dependence on a and is detectable by measuring the lifetime of the gas of dimers. Besides the Efimov trimers, one light and two heavy atoms may form “universal” trimer states well described in the zero-range approximation without introducing the three-body parameter [69]. In particular, they exist for the orbital angular momentum l = 1 and mass ratios below the critical value, where the Efimov effect is absent and short-range physics drops out of consideration. One of such states emerges at M/m ≈ 8 and crosses the trimer formation threshold (tr = −2|ε0 |) at M/m ≈ 12.7. The existence of this state is already seen in the Born–Oppenheimer picture. It appears as a bound state of fermionic heavy atoms in the potential + (R) for l = 1. The other state exists at M/m even closer to the critical mass ratio and never becomes sufficiently deeply bound to be formed in cold dimer–dimer collisions. The universal trimer states also exist for l > 1 and M/m > 13.6 [69]. However, trimer formation in dimer–dimer collisions at such mass ratios is dominated by the contribution of Efimov trimers with smaller l. Therefore, below, we focus on the formation of Efimov trimers. The calculation of the intrinsic lifetime of a trimer requires a detailed knowledge of short-range physics and is a tedious task. Estimates of the imaginary part of the trimer energy, τ−1 , from the experimental data on Cs3 trimers [70] show that it is smaller approximately by a factor of 4 than the real part ν (in this case η∗ ≈ 0.06). From a general point of view, we do not expect that the trimers with a binding energy ν < −2|ε0 | are very long-lived. However, one can have relatively narrow resonances, and we demonstrate calculations for various values of the elasticity parameter η∗ [67].

10.3.6

COLLISIONAL RELAXATION OF MOLECULES OF HEAVY AND LIGHT FERMIONS AND FORMATION OF TRIMERS

Let us now discuss inelastic processes in dimer–dimer collisions and start with relaxation of the dimers into deeply bound states. The typical size of a deeply bound state is of the order of the characteristic radius of the corresponding interatomic potential. We first consider the relaxation channel that requires one light and two heavy atoms to approach each other at distances ∼Re a. Unlike trimer formation, this decay mechanism is a purely three-body process. The other light atom is just a spectator. A qualitative scenario of this process is the following. With the tunneling probability, which is exponentially suppressed for large M/m, two dimers approach each other at distances R ∼ a. Then the heavy atoms are accelerated toward each other in the potential + (R), and the light atom in the gerade state is always closely bound to the heavy ones, as is seen from the shape of the function ψ+ . The relaxation transition occurs when the heavy atoms (and the gerade light fermion) are at interatomic separations ∼Re . The relaxation rate constant certainly satisfies Equation 10.48, but we should also find out how to take into account the relaxation process in the description of the trimer states and their formation. The most convenient way to do so is to consider the three-body parameter r0 as a complex quantity and introduce the so-called elasticity © 2009 by Taylor and Francis Group, LLC

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parameter η∗ = −sl Arg(r0 ) [71]. As follows from the asymptotic expression for the wavefunction 10.56, a negative argument of r0 ensures that the incoming flux of heavy atoms is not smaller than the outgoing one: Φout /Φin = exp[4sl Arg(r0 )] = exp[−4η∗ ] ≤ 1.

(10.59)

This mimics the loss of atoms at small distances due to the relaxation into deeply bound states. In the analysis of Efimov states, the imaginary part of r0 leads to the appearance of an imaginary part of ν . This means that any Efimov state has a finite lifetime τ due to the relaxation. For small |Arg(r0 )| and for trimer states that are localized at distances smaller than a, we get τ−1 /|ν | = 4|Arg(r0 )| = 4η∗ /sl . Strictly speaking, this fact indicates that it is not possible to separate the relaxation process from trimer formation because the trimers that are formed in dimer–dimer collisions will eventually decay due to relaxation. Nevertheless, both the modulus and the argument of the three-body parameter can be determined by measuring the lifetime of a gas of dimers, leading to a number of quantitative predictions concerning the structure of Efimov states in the three-body subsystem of one light and two heavy atoms. Another relaxation channel is the one in which two light atoms approach a heavy atom at distances ∼Re a. This channel is, however, suppressed due to the Fermi statistics for the light atoms, which strongly reduces the probability of having them in a small volume. As a result, for realistic parameters this relaxation mechanism is much weaker than the one in the system of one light and two heavy atoms [67]. The study of the formation of trimer states in molecule–molecule collisions requires us to go beyond the conventional Born–Oppenheimer approximation, because this approximation breaks down for the ungerade light atom at separations between the heavy atoms R < a. Within the recently developed “hybrid Born– Oppenheimer” approach [67] the Born–Oppenheimer method is applied to the gerade light fermion, which is characterized by the wavefunction ψ+ R (r) and energy + (R) adiabatically ajusting themselves to the motion of the heavy atoms. The gerade light atom is then integrated out by introducing the potential + (R) for the heavy atoms. Once this is done the original four-body problem is reduced to a three-body problem described by the Schrödinger equation [Hˆ − (2 /2μ3 )∇r2 − E]Ψ(R, r) = 0,

(10.60)

where Hˆ is given by Equation 10.55, μ3 = 2mM/(2M + m), E = −2|ε0 | + ε is the total energy of the four-body system in the center-of-mass reference frame, and ε is the dimer–dimer collision energy. This problem is then treated exactly the same as in the Bethe–Peierls approach, and the interaction of the light atom with the heavy atoms is included in the form of the Bethe–Peierls boundary condition 10.8 for Ψ at vanishing light–heavy separations |r ± R/2|. The ungerade symmetry for this atom is taken into account by the condition Ψ(R, r) = −Ψ(R, −r). © 2009 by Taylor and Francis Group, LLC

(10.61)

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As the heavy atoms are identical fermions, we have Ψ(R, r) = −Ψ(−R, r). Combined with Equation 10.61, this leads to the condition Ψ(R, r) = Ψ(−R, −r). Therefore, Ψ(R, r) describes atom–dimer scattering with even angular momenta, and for ultracold collisions we have to solve an s-wave atom–dimer scattering problem. In order to solve Equation 10.60 we follow the method of Ref. [55] and introduce an auxiliary function f (R) and write down the wavefunction Ψ(R, r) in the form

Ψ(R, r) = χν (R)χ∗ν (R )Kκν (2r, R )f (R ), (10.62) R

ν

where Kκν (2r, R ) = and

e−κν |r−R /2| e−κν |r+R /2| − 4π|r − R /2| 4π|r + R /2|

(√ 2μ3 (ν − E)/, ν > E, κν = √ −i 2μ3 (E − ν )/, ν < E.

(10.63)

(10.64)

For ν < E, the trimer can be formed in the state ν. In this case κν is imaginary and the function 10.63 describes an outgoing wave of the light atom moving away from the trimer. The choice of the sign in Equation 10.64 ensures that there is no incoming flux in the atom–trimer channel. Using the Bethe–Peierls boundary condition 10.8 for the wavefunction 10.62 at |r ± R/2| → 0 we obtain an integral equation for the function f (R): √ 2 3 2μ(−ε0 − E)/ − 1/a ˆL − Lˆ + sin2 θ f (R) = 0, 4π

(10.65)

√ where μ = mM/(m + M), θ = arctan 1 + 2M/m, and

Lˆ f (R) = [χν (R)χ∗ν (R )Kκν (R, R ) − χ0ν (R)χ0∗ ν (R )Kκ0ν (R, R )] f (R ), R

ν

(10.66) & ' ˆ (R) = P Lf G(|R − R |) f (R)−f (R ) ± G( R2 +R2 −2R · R cos 2θ)f (R ) , R

√

G(X) =

sin 2θM(−ε0 − E)K2 ( M(−ε0 − E)X/ sin θ) , 82 π3 X 2

(10.67) (10.68)

with K2 (z) being the exponentially decaying Bessel (Macdonald) function. A detailed derivation is given in Ref. [67] and is omitted here. The operators Lˆ and Lˆ conserve angular momentum and, expanding the function f (R) in spherical harmonics, we arrive at a set of uncoupled one-dimensional integral equations for each of the radial functions fl (R). Below we present the results for s-wave dimer–dimer scattering. © 2009 by Taylor and Francis Group, LLC

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At large distances (R a) the reduced wavefunction Ψ(R, r) takes the form: Ψ(R, r) ≈ Ψ(R)ψ− R (r),

(10.69)

Ψ(R) ∝ f (R),

(10.70)

and it can be shown that R a.

Therefore, f (R) can serve as the wavefunction for the dimer–dimer motion at large distances. In particular, it contains the dimer–dimer scattering phase shift. The dimer–dimer s-wave scattering amplitude add is determined from the asymptotic behavior of the solution of Equation 10.65 at large distances for E = 2ε0 , which should be matched with f0 (R) ∝ (1/R − 1/add ) (10.71) √ at R a ln M/m. In Figure 10.11 we compare the resulting add /a with that following from Equation 10.51. The results agree quite well even for moderate values of M/m. These results also agree with the calculations based on the exact four-body equation for M/m < 13.6 [65] and displayed in Figure 10.8, and with the Monte Carlo results for M/m < 20 [72]. It is straightforward to extend this theory to account for inelastic processes of the trimer formation and the relaxation of dimers into deeply bound states. Let us first assume that the rate of the relaxation into deep molecular states is negligible and neglect this process. Then the three-body parameter is real, and the trimer formation rate is determined by the imaginary part of the s-wave scattering length. The rate constant is given by [26] 16π α=− (10.72) Im(add ). M

2

1.5

add/a (Equation 10.51) add/a (HBO)

1

20

40

60 M/m

80

100

FIGURE 10.11 The dimer–dimer s-wave scattering length add /a. The solid curve shows the results obtained in the hybrid Born–Oppenheimer (HBO) approximation, and the dotted line the results of Equation 10.51.


385


Alternatively, if it is necessary to know the rate of the trimer formation in the state ν, one can substitute the solution of Equation 10.65 into Equation 10.62 and calculate the flux of light atoms at r → ∞. The summation over ν gives the same result as Equation 10.72. We find that the contribution of the highest “dangerous” trimer state is by far dominant and α is very sensitive to its position. We now include the relaxation of the dimers into deeply bound states. As we have mentioned above, the light–heavy–heavy relaxation process can be taken into account by adding an imaginary part to the three-body parameter. The total inelastic decay rate is then still given by Equation 10.72. However, strictly speaking, we can no longer distinguish between the formation of the trimer in a particular state and the collisional relaxation since the trimers ultimately decay due to the relaxation process. In this sense the only decay channel is the relaxation. However, for a sufficiently long lifetime of a trimer, that is, if the trimer states are narrow resonances, we can still observe a pronounced dependence of the total inelastic decay rate on the position of the highest “dangerous” trimer state (see below). Figure 10.12 shows the results for the inelastic collisional rate in the case of bosonic molecules with the mass ratio M/m = 28.5 characteristic of 171Yb–6 Li dimers. The solid line corresponds to the case of a real three-body parameter. It is convenient to introduce a related quantity, a0 , defined as the value of a at which the energy,

0.008 0.06

(M/ha)α

0.004 0.04 0

1

1.02

1.04

0.02

0 1

10 a/a0

FIGURE 10.12 The inelastic rate constant for bosonic dimers with M/m = 28.5 as a function of the atom–atom scattering length a. The solid line corresponds to the case of a real threebody parameter. The results plotted in dashed, dotted, dash–dotted, and dash–dot–dot lines are obtained by taking into account the light–heavy–heavy relaxation processes for values of the elasticity parameter of η∗ = 0.1, 0.5, 1, and ∞, respectively (see text). The quantity a0 is the value of a at which the energy of a trimer state equals E = −2ε0 and a new inelastic channel opens. The inset shows the region a ≈ a0 in greater detail in order to see the threshold behavior (10.73).


386


ν , of a trimer state exactly equals E = −2ε0 . This new “dangerous” trimer state becomes more deeply bound for a > a0 and the rate constant rapidly increases. It is proportional to the density of states in the outgoing atom–trimer channel. The corresponding orbital angular momentum is equal to 1 and the threshold law reads (see also inset in Figure 10.12): α ∝ const + (E − ν )3/2 ∝ const + (a − a0 )3/2 .

(10.73)

The constant term in Equation 10.73 describes the contribution of more deeply bound states, which is typically very small. In fact, as trimer states become more compact, both light atoms should approach the heavy atoms and each other at small distances where the trimer formation takes place. Because they are identical fermions, there is a strong suppression of the trimer formation to these deeply bound states. The dependence of α on a/a0 is periodic on the logarithmic scale, with the multiplicative factor being equal to λ1 ≈ 7.3. The dashed, dotted, and dash–dotted curves are obtained for η∗ = 0.1, 0.5, and 1, respectively. The corresponding values of the ratio Φout /Φin are 0.67, 0.14, and 0.02. The horizontal line represents the limiting case of η∗ = ∞ or Φout = 0. This case is universal in the sense that physical observables depend only on the masses and the atomic scattering length. For a very weak light–heavy–heavy relaxation, the dimer–dimer inelastic collision can be viewed as the formation of a trimer (with the rate constant α) followed by its slow decay due to the relaxation. In this case one can think of detecting the trimers spectroscopically. We note, however, that even for the conditions corresponding to the dashed curve in Figure 10.12, that is, for η∗ as small as 0.1, the decay rate of the trimer, τ−1 ≈ 0.25|ν |/ is rather fast, which will likely make its direct detection difficult. For larger η∗ it is impossible to separate the formation of trimers from their intrinsic relaxational decays, and α is practically the relaxation rate constant. Remarkably, it remains quite sensitive to the positions of the trimer states (in this case resonances) for values of η∗ up to 0.5 and even larger. This suggests that measuring the lifetime of a gas of dimers as a function of a may provide important information on three-body observables. Moreover, for small η∗ it may be possible to create a stable molecular gas in sufficiently broad regions of a, where “dangerous” trimer states are far from the trimer formation threshold. The inelastic rate α for other mass ratios in the range 20 < M/m < 76 has also been found [67]. Its dependence on the scattering length has the same form as depicted in Figure 10.12. The maximum of the rate constant is well fitted by the formula αmax = √ 5.8(a/M) exp(−0.87 M/m) √ and the position of the horizontal line (η∗ = ∞) by α∞ = 1.6(a/M) exp(−0.82 M/m). The multiplicative factor in the log-periodic dependence is given by Equations 10.58 and 10.57 with l = 1. The same method was employed to estimate the formation of the universal trimer state with the orbital angular momentum l = 1 at mass ratios M/m > 12.7 but below the critical value for the onset of the Efimov effect [67]. The rate constant increases with M/m and reaches α = 0.2(a/M) close to the critical mass ratio. This corresponds to the imaginary part of the scattering length Imadd ≈ −4 × 10−3 a, which is smaller by a factor of 300 than the real part of add obtained from four-body calculations © 2009 by Taylor and Francis Group, LLC

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[66]. Thus, the formation of this state does not change the elastic scattering amplitude add shown in Figure 10.8. We can now estimate the collisional rates for 171Yb–6 Li dimers. On the basis of the results in Figure 10.12 we find that for a = 20 nm the upper bound of the inelastic rate constant is αmax ≈ 4 × 10−13 cm3 /sec. The elastic rate constant for a thermal gas with a ∼ 20 nm and T ∼ 100 nK equals αel ≈ 8π|add |2 2T /M ∼ 4 × 10−11 cm3 /sec.

(10.74)

Here we used the calculated s-wave dimer–dimer scattering length add ≈ 1.4a. We see that αel , is much larger than α and this inequality becomes even more pronounced for larger a due to the scaling relations αel ∝ a2 and αmax ∝ a. Thus, the gas of molecules of heavy and light fermions is well suited for evaporative cooling toward their Bose–Einstein condensation.

10.4 10.4.1

CRYSTALLINE MOLECULAR PHASE BORN–OPPENHEIMER POTENTIAL IN A MANY-BODY SYSTEM OF MOLECULES OF HEAVY AND LIGHT FERMIONS

A strong long-distance repulsive interaction between weakly bound molecules of light and heavy fermionic atoms has an important consequence not only for the relaxation process, but also for the macroscopic properties of the molecular system. In contrast to two-component Fermi gases of atoms in different internal states, heteronuclear Fermi–Fermi mixtures can form a molecular crystalline phase even when the mean interparticle separation greatly exceeds the size of the molecule: R¯ a.

(10.75)

Let us consider a mixture of heavy and light fermionic atoms with equal concentrations and a large positive scattering length for the interaction between them, satisfying the inequality a Re . At zero temperature all atoms will be converted into weakly bound molecules and under the condition 10.75 the molecular size (∼a) will be much smaller than the mean intermolecular separation. Then, using the Born–Oppenheimer approximation and integrating out the motion of light atoms we are left with a system of identical (composite) bosons that is described by the Hamiltonian: Hˆ = −

2

1

ΔRi + Ueff (Rij ), 2M 2 i

(10.76)

i=j

where the indices i and j label the bosons, their coordinates are denoted by Ri and Rj , and Rij = |Ri − Rj | is the separation between the ith and jth bosons. Assuming that the motion of light fermions is three-dimensional, the effective repulsive potential Ueff is given by Equation 10.44 and is independent of the mass of the heavy atom M. Therefore, at a large mass ratio M/m it dominates over the kinetic © 2009 by Taylor and Francis Group, LLC

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energy, which is inversly proportional to M, and which may lead to the formation of a crystalline phase. We will discuss the case where the motion of heavy atoms is confined to two dimensions, while the motion of light atoms can be either two- or three-dimensional. It will be shown that the Hamiltonian 10.76 with Ueff in 10.44 supports the first-order quantum gas-crystal transition at T = 0 [68]. This phase transition resembles the one for the flux lattice melting in superconductors, where the flux lines are mapped onto a system of bosons interacting via a two-dimensional Yukawa potential [73]. In this case Monte Carlo studies [74,75] identified the first-order liquid-crystal transition at zero and finite temperatures. Aside from the difference in the interaction potentials, a distinguished feature of our system is related to its stability. The molecules can undergo collisional relaxation into deeply bound states, or form weakly bound trimers. Another subtle question is how dilute the system should be to enable the use of the binary approximation for the molecule–molecule interaction, leading to Equations 10.76 and 10.44. Let us first consider the system of N molecules and derive the Born–Oppenheimer interaction potential for this system. Omitting the interaction between light (identical) fermions, it is sufficient to find N lowest single-particle eigenstates, and the sum of their energies will give the interaction potential for the molecules. For the interaction between light and heavy atoms we use the Bethe–Peierls approach, and the wavefunction of a single light atom then reads

Ψ({R}, r) =

N

Ci Gκ (r − Ri ),

(10.77)

i=1

where r is its coordinate and R denotes the set of coordinates of the heavy atoms. The Green function Gκ satisfies the equation (−∇r2 + κ2 )Gκ (r) = δ(r). The energy of the state 10.77 equals = −2 κ2 /2m, and here we only search for negative singleparticle energies. The dependence of the coefficients Ci and κ on {R} is obtained using the Bethe–Peierls boundary condition: Ψ({R}, r) ∝ Gκ0 (r − Ri ),

r → Ri .

(10.78)

Up to a normalization constant, Gκ0 is the wavefunction of a bound state of a single molecule with energy −ε0 = −2 κ20 /2m and molecular size κ−1 0 . The Bethe–Peierls boundary condition 10.78 written through the Green function Gκ0 can be used for both two- and three-dimensional motion of light atoms. In the latter case one has κ0 = a−1 , the Green function is Gκ0 (r − Ri ) ∝ (|r − Ri |−1 − a−1 ) for |r − Ri | → 0, and Equation 10.78 takes the form of Equation 10.8. 4 From Equations 10.77 and 10.78 we get a set of N equations: j Aij Cj = 0, where Aij = λ(κ)δij + Gκ (Rij )(1 − δij ), Rij = |Ri − Rj |, and λ(κ) =limr→0 [Gκ (r) − Gκ0 (r)]. The single-particle energy levels are determined by the equation det Aij (κ, {R}) = 0. © 2009 by Taylor and Francis Group, LLC

(10.79)

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For Rij → ∞, Equation 10.79 gives an N-fold degenerate ground state with κ = κ0 . At finite large Rij , the levels split into a narrow band. Given a small parameter, ˜ 0 |λκ (κ0 )| 1, ξ = Gκ0 (R)/κ

(10.80)

where R˜ is a characteristic distance at which heavy atoms can approach each other, and the bandwidth is Δ ≈ 4|ε0 |ξ |ε0 |. It is important for the adiabatic approximation that all lowest N eigenstates have negative energies and are separated from the continuum by a gap ∼|ε0 |. We now calculate the single-particle energies up to the second order in ξ. To this order we write κ(λ) ≈ κ0 + κλ λ + κλλ λ2 /2 and turn from Aij (κ) to Aij (λ): Aij = λδij + [Gκ0 (Rij ) + κλ λ∂Gκ0 (Rij )/∂κ](1 − δij ),

(10.81)

where all derivatives are taken at λ = 0. Using Aij (10.81) in Equation 10.79 gives a polynomial of degree N in λ. Its roots4λi give the light-atom energy spectrum i = −2 κ2 (λi )/2m. The total energy, E = N i=1 i , is then given by N N

2 λi + (κκλ )λ λ2i . E = −( /2m) Nκ0 + 2κ0 κλ 2

i=1

(10.82)

i=1

Keeping only the terms up to the second order in ξ and using the basic properties of determinants and polynomial roots 4we find that the first-order terms vanish and the energy reads E = −Nε0 + (1/2) i=j U(Rij ), where U(R) = −

∂G2κ0 (R) 2 κ0 (κλ )2 + (κκλ )λ G2κ0 (R) . m ∂κ

(10.83)

Thus, up to the second order in ξ, the interaction in the system of N molecules is the sum of the binary potentials (10.83). If the motion of light atoms is three-dimensional, the Green function is Gκ (R) = (1/4πR) exp(−κR), and λ(κ) = (κ0 − κ)/4π, with the molecular size κ−1 0 equal to the three-dimensional scattering length a. Equation 10.83 then gives a repulsive potential 3D : Uex (Equation 10.44) that we now denote as Uex 3D Uex (R) = 4|ε0 |(1 − (2κ0 R)−1 ) exp(−2κ0 R)/κ0 R,

(10.84)

and the criterion (Equation 10.80) reads (1/κ0 R) exp(−κ0 R) 1. For the twodimensional motion of light atoms we have Gκ (R) = (1/2π)K0 (κR) and λ(κ) = −(1/2π) ln(κ/κ0 ), where K0 is the decaying Bessel function and κ−1 0 follows from Ref. [76]. This leads to a repulsive intermolecular potential: 2D Uex (R) = 4|ε0 |[κ0 RK0 (κ0 R)K1 (κ0 R) − K02 (κ0 R)],

(10.85)

with the validity criterion K0 (κ0 R) 1. In both cases, which we denote 2 × 3 and 2 × 2 for brevity, the validity criteria are well satisfied already for κ0 R ≈ 2. © 2009 by Taylor and Francis Group, LLC

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10.4.2


GAS–CRYSTAL QUANTUM TRANSITION

The inequality κ0 R¯ 2 may be considered as the condition under which the system is described by the Hamiltonian 10.76, with Ueff given by Equation 10.84 for the three-dimensional motion of light atoms, or by Equation 10.85 in the case where this motion is two-dimensional. The state of the system is then determined by two parameters: the mass ratio M/m and the rescaled two-dimensional density nκ−2 0 . At a large M/m, the potential repulsion dominates over the kinetic energy, which should lead to the formation of a crystalline ground state. For separations Rij < κ−1 0 the adiabatic approximation breaks down. However, the interaction potential U(R) is strongly repulsive at larger distances. Hence, even for an average separation between −1 heavy atoms R¯ close to 2/κ0 , they approach each √ other at distances smaller than κ0 with a small tunneling probability P ∝ exp(−β M/m) 1, where β ∼ 1. It is then 3D (R) (or U 2D (R)) to R κ−1 in a way that provides a proper possible to extend Uex ex 0 molecule–molecule scattering phase shift in vacuum and verify that the phase diagram for the many-body system is not sensitive to the choice of this extension [69]. In Figure 10.13 we display the zero-temperature phase diagram obtained by the Diffusion Monte Carlo method [68]. Simulations were performed with 30 particles and showed that the solid phase is a two-dimensional triangular lattice. For the largest density it has been verified that using more particles has little effect on the results. For both 2 × 3 and 2 × 2 cases the (Lindemann) ratio γ of the rms displacement of molecules to R¯ on the transition lines ranges from 0.23 to 0.27.At low densities n the de 3D 2D Broglie wavelength of molecules is Λ ∼ γR¯ κ−1 0 , and Uex (R) (or Uex (R)) can be approximated by a hard-disk potential with the diameter equal to the two-dimensional

M m Crystal

103 2 × 3, γ = 0.24 2 × 2, γ = 0.23

Gas 102 0.00

0.05

0.10

0.15

0.20

0.25

nk0–2

FIGURE 10.13 Diffusion Monte Carlo gas–crystal transition lines for three-dimensional (triangles) and two-dimensional (circles) motion of light atoms. Solid curves show the lowdensity hard-disk limit, and dashed curves the results of the harmonic approach (see text).



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scattering length. Then, using the Diffusion Monte Carlo (DMC) results for hard-disk bosons [77], we obtain the transition lines shown by solid curves in Figure 10.13. At larger n, we have Λ < κ−1 0 and, using the harmonic expansion of U(R) around equlibrium positions in the crystal, we calculate the Lindemann ratio and select γ for the best fit to the Monte Carlo data points (dashed curves in Figure 10.13).

10.4.3

MOLECULAR SUPERLATTICE IN AN OPTICAL LATTICE

The mass ratio above 100, required for the observation of the crystalline order, can be achieved in an optical lattice with a small filling factor for heavy atoms. Their effective mass in the lattice, M∗ , can be made very large, and the discussed solid phase should appear as a superlattice. There is no interplay between the superlattice order and the shape of the underlying optical lattice, in contrast to the recently studied solid and supersolid phases in a triangular lattice with the filling factor of order one [78–80]. The superlattice discussed in our review remains compressible and supports two branches of phonons. The gaseous and solid phases of weakly bound molecules in an optical lattice are metastable. As well as in the gas of such molecules in free space, the main decay channels are the relaxation of molecules into deep bound states and the formation of trimer states by one light and two heavy atoms. The relaxation into deeply bound states turns out to be rather slow, with a relaxation time exceeding 10 sec even at two-dimensional densities of ∼109 cm−2 [68]. The most interesting is the formation of the trimer states. 4 In an optical lattice the trimers are eigenstates of the Hamiltonian H0 = −(2 /2M∗ ) i=1,2 ΔRi + + (R12 ). In a deep lattice it is possible to neglect all higher bands and regard Ri as discrete lattice coordinates and Δ as the lattice Laplacian. Then, the fermionic nature of the heavy atoms prohibits them to be on the same lattice site. For a very large mass ratio M∗ /m the kinetic energy term in H0 can be neglected, and the lowest trimer state has energy tr ≈ + (L), where L is the lattice period. It consists of a pair of heavy atoms localized at neighboring sites and a light atom in the gerade state. Higher trimer states are formed by heavy atoms localized in sites separated by distances R > L. This picture breaks down at large R, where the spacing between trimer levels is comparable with the tunneling energy 2 /M∗ L 2 and the heavy atoms are delocalized. In the many-body molecular system the scale of energies in Equation 10.76 is much smaller than |ε0 |. Thus, the formation of trimers in molecule–molecule “collisions” is energetically allowed only if the trimer binding energy is tr < −2ε0 . Because the lowest trimer energy in the optical lattice is + (L), the trimer formation requires the condition + (L) −2ε0 , which is equivalent to κ−1 0 1.6L in the 2 × 3 case and −1 κ0 1.25L in the 2 × 2 case. This means that for a sufficiently small molecular size or large lattice period L the formation of trimers is forbidden. At a larger molecular size or smaller L the trimer formation is possible. The formation rate has been calculated in Ref. [68] by using the hybrid Born–Oppenheimer approach, and here we only present the results and give their qualitative explanation. In order to form a bound trimer state two molecules have to tunnel toward each other at distances R κ−1 0 . This can be viewed as tunneling of particles with mass M∗ in the repulsive potential Ueff (R). Therefore, the probability of approaching at © 2009 by Taylor and Francis Group, LLC

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interheavy √ separations where the trimer formation occurs, acquires a small factor exp(−J M∗ /m) with J ∼ 1, and so does the formation rate. Thus, one can suppress the trimer formation by increasing the ratio M∗ /m. On the other hand, for M∗ /m 100 these peculiar bound states can be formed on the time scale τ 1 sec. Note that the trimer states in an optical lattice, at least the lowest ones, are much more long-lived than in the gas phase. An intrinsic relaxational decay is strongly suppressed as it requires the two heavy atoms of the trimer to approach each other and occupy the same lattice sites. The trimer state can also decay when one of the heavy atoms of the trimer is approached by its own light atom and another light atom at light–heavy separations ∼Re . However, this decay channel turns out to be rather slow, with a decay time exceeding tens of seconds even at two-dimensional densities of n ∼ 109 cm−2 [68].

10.5

CONCLUDING REMARKS AND PROSPECTS

The most distinguishing feature of weakly bound bosonic molecules formed of fermionic atoms is their remarkable collisional stability, despite the fact that they are in the highest rovibrational state. As we mentioned in the Introduction, the lifetime of such molecules can be of the order of seconds or even tens of seconds at densities ∼1013 cm−3 , depending on the value of the two-body scattering length. This allows for interesting manipulations with these molecules. One of the ideas relates to reaching extremely low temperatures in a gas of fermionic atoms at a < 0 and achieving the superfluid BCS regime. This regime has not been obtained so far because of difficulties with evaporative cooling of fermionic atoms due to Pauli blocking of their elastic collisions. The route to BCS may be the following. In the first stage, one arranges a deep evaporative cooling of the molecular Bose-condensed gas to temperatures of the order of the chemical potential. Then one converts the molecular BEC into fermionic atoms by adiabatically changing the scattering length to negative values. This provides an additional cooling, and the obtained atomic Fermi gas will have extremely low temperatures, 10−2 TF , where TF is the Fermi temperature. The gas can then enter the superfluid BCS regime [81]. Moreover, at such temperatures elastic collisions are suppressed by a very strong Pauli blocking and the thermal cloud is in the collisionless regime. This is promising for identifying the BCS-paired state through the observation of collective oscillations or free expansion [16,19,82–84]. It will also be interesting to transfer weakly bound molecules of fermionic atoms to their ground (or less excited) rovibrational state. For molecules of bosonic atoms this has been done using two-photon spectroscopy [85–87] and by magnetically tuned mixing of neighbouring molecular levels, which enables otherwise forbidden radiofrequency transitions [88]. Long lifetimes of weakly bound molecules of fermionic atoms at densities ∼1013 cm−3 may ensure an efficient production of ground-state molecules compared to the case of more short-lived molecules of bosonic atoms. One could then extensively study the physics of molecular Bose–Einstein condensation. Moreover, the ground-state heteronuclear molecules have a relatively large permanent dipole moment and can be polarized by an electric field. This may be used to create a gas of dipoles interacting via anisotropic long-range forces, which drastically changes the physics of Bose–Einstein condensation (e.g., Ref. [89] and references therein). In © 2009 by Taylor and Francis Group, LLC


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experiments at JILA weakly bound fermionic 40 K –87 Rb molecules have been created and transferred to low lying rovibrational states [90], and recently to the ground state [91,92]. They have been cooled to temperatures very close to the regime of quantum degeneracy, which opens possibilities for future studies of unconventional superfluid pairing in dipolar fermionic gases. In the last few years, the observation of the Efimov effect was one of the important goals in cold atom studies. As we discussed in Section 10.3, the Efimov trimers in the gas phase are short-lived and rather represent narrow resonances. The Efimov effect then manifests itself in the log-periodic dependence of collision rates on the two-body scattering length. In particular, this is the case for the rate of three-body recombination of atoms [70] and for the rate of trimer formation in molecule–molecule collisions [67]. In this sense, the trimer formation in gases of bosonic molecules consisting of heavy and light fermions (such as LiYb) attracts great interest as the observation of the Efimov oscillations requires a much smaller change of the two-body scattering length (by a factor of 7 or 5) than in the case of identical bosons. Of particular interest are the trimer states of two heavy and one light fermion in an optical lattice. For two-dimensional densities ∼108 cm−2 the rate of trimer formation can be of the order of seconds, and these states can be detected optically. As we mentioned in Section 10.4, the lattice trimers are long-lived, with a lifetime that can be of the order of tens of seconds. Thus, it is interesting to study to what extent these nonconventional states, in which the heavy atoms are localized in different sites and the light atom is delocalized between them, can exhibit the Efimov effect. The creation of a superlattice of molecules in an optical lattice also looks feasible. A promising candidate is the 6 Li–40 K mixture, as the Li atom may tunnel freely in a lattice while localizing the heavy K atoms to reach high mass ratios. A lattice with period 250 nm and K effective mass M ∗ = 20 M provide a tunneling rate ∼103 sec−1 sufficiently fast to let the crystal form. Near a Feshbach resonance, a value a = 500 nm gives a binding energy 300 nK, and lower temperatures should be reached in the gas. The parameters nκ−2 0 of Figure 10.13 are then obtained at two-dimensional densities in the range 107 –108 cm−2 easily reachable in experiments.

ACKNOWLEDGMENTS The work on this review was financially supported by the IFRAF Institute, by ANR (grants 05-BLAN-0205 and 06-Nano-014), by the EuroQUAM program of ESF (project Fermix), by Nederlandse Stichtung voor Fundamenteel Onderzoek der Materie (FOM), and by the Russian Foundation for Fundamental Research. LKB is a research unit no. 8552 of CNRS, ENS, and of the University of Pierre et Marie Curie. LPTMS is a mixed research unit no. 8626 of CNRS and the University Paris-Sud.

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62. Levinsen, J. and Gurarie, V., Properties of strongly paired fermionic condensates, Phys. Rev. A, 73, 053607, 2006. 63. Fuchs, J., Duffy, G.J., Veeravalli, G., Dyke, P., Bartenstein, M., Vale, C.J., Hannaford, P., and Rowlands, W.J., Molecular Bose–Einstein condensation in a versatile low power crossed dipole trap, J. Phys. B, 40, 4109, 2007. 64. Inada, Y., Horikoshi, M., Nakajima, S., Kuwata-Gonokami, M., Ueda, M., and Mukaiyama, T., Critical temperature and condensate fraction of a fermion pair condensate, Phys. Rev. Lett., 101, 180406, 2008. 65. Petrov, D.S., Salomon, C., and Shlyapnikov, G.V., Diatomic molecules in ultracold Fermi gases - novel composite bosons, J. Phys. B, 38, S645, 2005. 66. The Born–Oppenheimer approach for the three-body system of one light and two heavy atoms was discussed in: Fonseca, A.C., Redish, E.F., and Shanley, P.E., Efimov effect in a solvable model, Nucl. Phys. A, 320, 273, 1979. 67. Marcelis, B., Kokkelmans, S.J.J.M.F., Shlyapnikov, G.V., and Petrov, D.S., Collisional properties of weakly bound heteronuclear dimers, Phys. Rev. A, 77, 032707, 2008. 68. Petrov, D.S., Astrakharchik, G.E., Papoular, D.J., Salomon, C., and Shlyapnikov, G.V., Crystalline phase of strongly interacting Fermi mixtures, Phys. Rev. Lett., 99, 130407, 2007. 69. Kartavtsev, O.I. and Malykh, A.V., Low-energy three-body dynamics in binary quantum gases, J. Phys. B, 40, 1429, 2007; Universal description of the rotational-vibrational spectrum of three particles with zero-range interactions, Pis’ma Zh. Eksp. Teor. Fiz., 86, 713, 2007. 70. Kraemer, T., Mark, M., Waldburger, P., Danzl, J.G., Chin, C., Engeser, B., Lange, A.D. et al., Evidence for Efimov quantum states in an ultracold gas of cesium atoms, Nature, 440, 315, 2006. 71. Braaten, E. and Hammer, H.-W., Efimov physics in cold atoms, Ann. Phys., 322, 120, 2007. 72. von Stecher, J., Greene, C.H., and Blume, D., BEC–BCS crossover of a trapped twocomponent Fermi gas with unequal masses, Phys. Rev. A, 76, 053613, 2007. 73. Nelson D.R. and Seung, H.S., Theory of melted flux liquids, Phys. Rev. B, 48, 411, 1993. 74. Margo W.R. and Ceperley, D.M., Ground state of two-dimensional Yukawa bosons: Applications to vortex melting, Phys. Rev. B, 48, 411, 1993. 75. Nordborg, H. and Blatter, G., Vortices and 2D bosons: A path-integral Monte Carlo study, Phys. Rev. Lett., 79, 1925, 1997. 76. In the 2D regime achieved by confining the light-atom motion to zero point oscillations with amplitude l0 , the weakly bound molecular states exist at a negative a satisafying the inequality |a| l0 . See Petrov, D.S. and Shlyapnikov, G.V., Interatomic collisions in a tightly confined Bose gas, Phys. Rev. A, 64, 012706, 2001. 77. Xing, L., Monte Carlo simulations of a two-dimensional hard-disk boson system, Phys. Rev. B, 42, 8426, 1990. 78. Wessel, S. and Troyer, M., Supersolid hard-core bosons on the triangular lattice, Phys. Rev. Lett., 95, 127205, 2005. 79. Heidarian, D. and Damle, K., Persistent supersolid phase of hard-core bosons on the triangular lattice, Phys. Rev. Lett., 95, 127206, 2005. 80. Melko, R.G., Paramekanti, A., Burkov, A.A., Vishwanath, A., Sheng, D.N., and Balents, L., Supersolid order from disorder: hard-core bosons on the triangular lattice, Phys. Rev. Lett., 95, 127207, 2005. 81. Carr, L.D., Shlyapnikov, G.V., and Castin, Y., Achieving a BCS transition in an atomic Fermi gas, Phys. Rev. Lett., 92, 150404, 2004.


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82. Menotti, C., Pedri, P., and Stringari, S., Expansion of an interacting Fermi gas, Phys. Rev. Lett., 89, 250402, 2002. 83. O’Hara, K.M., Hemmer, S.L., Gehm, M.E., Granade, S.R., and Thomas, J.E., Observation of a strongly interacting degenerate Fermi gas of atoms, Science, 298, 2179, 2002. 84. Kinast, J., Hemmer, S.L., Gehm, M.E., Turlapov, A., and Thomas, J.E., Evidence for superfluidity in a resonantly interacting Fermi gas, Phys. Rev. Lett., 92, 150402, 2004. 85. Kerman, A.J., Sage, J.M., Sainis, S., Bergeman, T., and DeMille, D., Production and state-selective detection of ultracold RbCs molecules, Phys. Rev. Lett., 92, 153001, 2004. 86. Sage, J.M., Sainis, S., Bergeman, T., and DeMille, D., Optical production of ultracold polar molecules, Phys. Rev. Lett., 94, 203001, 2005. 87. Winkler, K., Lang, F., Thalhammer, G., Straten, P.v.d., Grimm, R., and Hecker Denschlag, J., Coherent optical transfer of Feshbach molecules to a lower vibrational state, Phys. Rev. Lett., 98, 043201, 2007. In this experiment the presence of an optical lattice suppressed inelastic collisions between molecules of bosonic 87 Rb atoms, which provided a highly efficient transfer of these molecules to a less excited ro-vibrational state and a long molecular lifetime of about 1 second. 88. Lang, F., Straten, P.v.d., Brandstatter, B., Thalhammer, G., Winkler, K., Julienne, P.S., Grimm, R., and Hecker Denshlag, J., Cruising through molecular bound-state manifolds with radiofrequency, Nature Physics, 4, 223, 2008. 89. Santos, L. and Pfau, T., Spin-3 chromium Bose–Einstein condensates, Phys. Rev. Lett., 96, 190404, 2006. 90. Ospelkaus, S., Pe’er,A., Ni, K.-K., Zirbel, J.J., Neyenhuis, B., Kotochigova, S., Julienne, P.S., Ye, J., and Jin, D.S., Ultracold dense gas of deeply bound heteronuclear molecules, Nature Physics, 4, 622, 2008. 91. Ni, K.-K., Ospelkaus, S., de Miranda, M.H.G., Pe’er, A., Neyenhuis, B., Zirbel, J.J., Kotochigova, S., Julienne, P.S., Jin, D.S., and Ye, J., A high phase-space-density gas of polar molecules, Science, 322, 231, 2008. 92. Ospelkaus, S., Ni, K.-K., de Miranda, M.H.G., Neyenhuis, B., Wang, D., Kotochigova, S., Julienne, P.S., Jin, D.S., and Ye, J., Ultracold polar molecules near quantum degeneracy, arXiv:0811.4618.


Theory of Ultracold 11 Feshbach Molecules Thomas M. Hanna, Hugo Martay, and Thorsten Köhler CONTENTS 11.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Microscopic Theory of Feshbach Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Zeeman Effect in the Hyperfine Structure of Alkali–Metal Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Interatomic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Singlet and Triplet Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Bound States and Scattering Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Two-Channel Two-Potential Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Two-Channel Single-Resonance Approach . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Magnetic Tuning of the Scattering Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Resonance Width and Background-Scattering Length . . . . . . . . . . . . 11.4.2 Relation between Bound-State Energy and Resonance Position . 11.5 Classification of Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Closed-Channel Dominated Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Entrance-Channel Dominated Resonances . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11.1

399 400 400 402 402 404 406 406 407 409 409 410 412 412 414 416 416 416

INTRODUCTION

Resonances in the collision cross-sections of particles have a long history of theoretical studies in atomic and molecular physics [1–3], as well as in nuclear physics [4,5]. Their recent applications to Feshbach-resonance phenomena in cold gases often refer to a situation in which a dilute assembly of atoms is exposed to a spatially homogeneous magnetic field. Such an experimental setup allows magnetic tuning of the interatomic interactions. Early suggestions referred to the possibility of manipulating collision cross-sections in dilute vapors of spin-polarized hydrogen 399 © 2009 by Taylor and Francis Group, LLC

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and deuterium [6], as well as lithium [7]. At temperatures in the sub-microkelvin regime, the proposed applications of Feshbach resonances involved magnetic tuning of the s-wave scattering length [8], as well as formation of molecules [9], in trapped gases of alkali–metal atoms at the threshold between scattering and molecular binding. While the possibility of manipulating scattering lengths was demonstrated in a sodium Bose–Einstein condensate [10], the production of ultracold molecules using magnetically tunable Feshbach resonances has been the subject of experimental studies in Bose [11–14] and Fermi [15–19] gases. A review on cold molecules in general can be found, for instance, in Ref. [20]. Their applications in the context of the cross-over from Bardeen–Cooper–Schrieffer (BCS) pairing to molecular Bose–Einstein condensation are detailed in Refs. [21–26]. This chapter gives an overview of theoretical methods for describing the properties of diatomic molecules produced from ultracold atomic gases using the technique of magnetically tunable Feshbach resonances. Section 11.2 summarizes the microscopic description of diatomic energy spectra in terms of the coupled-channels theory. This summary involves brief descriptions of the Zeeman effect in the hyperfine structure of alkali–metal atoms and of the basic interatomic interactions relevant to ultracold gases. Section 11.3 gives an overview of two-channel approximations modeling Feshbach resonances at the threshold between scattering and molecular binding. These methods are applied in Section 11.4 to describe the relation between singularities of the scattering length and the energy of the highest excited vibrational molecular bound state. Section 11.5 extends these applications to the Feshbach-resonance-enhanced collision physics, as well as the associated boundstate properties, at finite energies about the scattering threshold in ultracold gases. Section 11.6 presents the conclusions of this chapter.

11.2 11.2.1

MICROSCOPIC THEORY OF FESHBACH MOLECULES ZEEMAN EFFECT IN THE HYPERFINE STRUCTURE OF ALKALI–METAL ATOMS

Magnetic tuning of diatomic bound-state and scattering properties relies on the Zeeman effect in the hyperfine structure of alkali–metal atoms. The splitting into sublevels of the 2 S1/2 electronic ground state of such an atom which is exposed to a magnetic field B can be described by the following Hamiltonian comprised of hyperfine and Zeeman interactions [27]: Hint =

Chf s · i + μB (ge s + gn i) · B. 2

(11.1)

Here s denotes the spin of the single unpaired valence electron, i is the nuclear spin, Chf is the hyperfine-structure constant, ge and gn are the electronic and nuclear gyromagnetic factors, respectively, and μB is the Bohr magneton. The sign convention for gn used in Equation 11.1 follows Ref. [27]. At zero magnetic field, the electronic ground state is split into two hyperfine levels characterized by the quantum number f associated with the total atomic angular © 2009 by Taylor and Francis Group, LLC

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Theory of Ultracold Feshbach Molecules

momentum, f = s + i. Using the general relation, s·i=

1 2 f − s2 − i 2 , 2

(11.2)

the energy shifts of the individual levels as obtained from the hyperfine interaction of Equation 11.1 are determined by [28] ΔE =

Chf f ( f + 1) − s(s + 1) − i(i + 1) . 2

(11.3)

Here i is the angular-momentum quantum number associated with the nuclear spin, while the electron spin is characterized simply by s = 1/2. In the presence of an external magnetic field of strength B = |B| the hyperfine levels of total angular-momentum quantum number f are split into Zeeman sublevels. Because the rotational symmetry is still preserved about the field direction, these sublevels can be characterized by the magnetic quantum numbers mf associated with the component of f along this direction. This is illustrated in Figure 11.1 for the example of an i = 3/2 alkali–metal atom. Although the Hamiltonian of Equation 11.1 does not commute with f 2 , the Zeeman sublevels are often labeled by the pair of quantum numbers ( fmf ) associated with the degenerate hyperfine state they correlate with adiabatically in the limit of zero magnetic field. Nuclear-spin quantum numbers, i, and numerical values of ge and gn for the stable isotopes of all alkali– metal atomic species, as well as the hyperfine transition frequencies for their 2 S1/2 electronic ground states are tabulated in Ref. [27]. 6000 4000

mf = +2 mf = +1 mf = 0

f=2

mf = –1

E/h (MHz)

2000 0

mf = –2

Ehf/h

–2000 –4000

mf = –1

f=1

mf = 0

–6000 0

500

1000 B (G)

1500

mf = +1 2000

FIGURE 11.1 Zeeman splitting of the f = 1 (solid curves) and f = 2 (dashed curves) hyperfine-energy levels vs. magnetic field strength, B, referring to the 2 S1/2 electronic ground state of 87 Rb. Given the nuclear-spin quantum number, i = 3/2 [27], as well as s = 1/2, the hyperfine constant Chf and the hyperfine transition frequency between the f = 1 and f = 2 levels at zero magnetic field, Ehf /h = 6835 MHz [27], are related by Chf = Ehf /2 in accordance with Equation 11.3.


402

11.2.2


INTERATOMIC INTERACTIONS

Using the Born–Oppenheimer approximation, the Hamiltonian of the relative motion describing bound molecular states as well as collisions of two ground-state alkali– metal atoms in the presence of a magnetic field is given by [8,29] 2 2 int ∇ + Hn + Vint . 2μ 2

H=−

(11.4)

n=1

Here μ is the reduced mass, H1int and H2int are the Hamiltonians defined in Equation 11.1 for each atom, and Vint is the effective interaction potential depending on the relative position of the atoms, r. For many applications, such as the description of broad scattering resonances and their associated Feshbach molecules, it is sufficient to include in Vint only the rotationally symmetric singlet and triplet Born–Oppenheimer potentials, VS=0 and VS=1 , respectively. Their labels S = 0 and S = 1 refer to the possible values of the angular-momentum quantum number associated with the total spin of the two atomic valence electrons, S = s1 + s2 . In this approximation, the interaction part of Equation 11.4 can be represented by [8,29] Vint = V0 P0 + V1 P1 ,

(11.5)

where P0 and P1 project onto the subspaces of definite total electron-spin quantum number, S = 0, 1. Several applications, such as predictions of narrow scattering resonances for heavy alkali–metal atomic species or spin relaxation in collisions of atoms in excited Zeeman states, may require the inclusion of higher-order corrections (e.g., Refs. [29–32]) in Equation 11.5. Whereas the singlet and triplet potentials depend only on the inter-atomic distance, r = |r|, these additional interactions are generally directional, that is, they couple different partial waves in the relative motion of the atoms. All examples of bound-state and scattering phenomena throughout this chapter might be described, however, at least qualitatively, on the basis of the isotropic Born–Oppenheimer potentials only.

11.2.3

SINGLET AND TRIPLET POTENTIALS

The inset of Figure 11.2 shows the typical shapes of singlet and triplet potentials associated with interactions of alkali–metal atoms of the same chemical species. At short distances, their functional forms are governed by Pauli repulsion of the two electron clouds. Their behavior at asymptotically large interatomic distances is determined by an attractive van der Waals potential, that is, VS (r) ∼ −C6 /r 6 , r→∞

(11.6)

where the van der Waals coefficient C6 is the same for the singlet and triplet states. To provide accurate predictions on the magnetic-field locations of low-energy scattering resonances using Equation 11.4, the Born–Oppenheimer potentials are © 2009 by Taylor and Francis Group, LLC

403


Energy (au)

rf0(r)

0.02

3Su+

0.01 0 1S + g

–0.01 0

10

20

Triplet

Singlet 0 0

1000

at = –388 a0

2000 r (a0)

3000

4000

as = +2795 a0

FIGURE 11.2 Sketch of the relation between singlet (dashed curve) and triplet (solid curve) zero-energy scattering wavefunctions plotted vs. the interatomic distance r in units of the Bohr radius a0 = 0.0529 nm, and the associated scattering lengths, as and at , respectively. The values of as and at , referring here to the 85 Rb isotope [33], are determined by the zeros of the linear asymptotes of the scattering wavefunctions. Inset: Typical shapes of singlet (dashed curve) and triplet (solid curve) Born–Oppenheimer potentials for rubidium atoms in their 2 S1/2 electronic ground states [34,35].

usually calibrated in such a way that they recover the correct C6 coefficient (e.g., Refs. [36–39]) as well as the associated scattering lengths (e.g., Refs. [33,40–52]). The singlet and triplet scattering lengths, as and at , respectively, are determined by the solutions of the stationary Schrödinger equation

2 d 2 − + VS (r) rφ0 (r) = 0, 2μ dr 2

(11.7)

which exhibit the following long-range asymptotic behavior at zero energy [53]: rφ0 (r) ∝ r − a.

(11.8)

Here a refers to either as or at . Such wavefunctions and their linear asymptotes determining the singlet and triplet scattering lengths are sketched in Figure 11.2. Table 11.1 summarizes the parameters as , at , and C6 of the Born–Oppenheimer potentials for pairs of identical alkali–metal atoms. Ideally, the singlet and triplet potentials depend only on the chemical element, while the differences in as and at for different isotopes are only due to their different reduced masses in Equation 11.7. © 2009 by Taylor and Francis Group, LLC

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TABLE 11.1 Singlet and Triplet Scattering Lengths as and at , Respectively, as well as Long-Range van der Waals Dispersion Coefficients C6 Associated with the Interactions of Several Homonuclear Alkali–Metal Atom Pairs Species 6 Li 7 Li 23 Na 39 K 40 K 41 K 85 Rb 87 Rb 133 Cs

as (a0 )

at (a0 )

45.167(8) [50] 33(2) [41] 19.20(30) [46] 138.49(12) [52] 104.41(9) [52] 85.53(6) [52] 2795 +420 −290 [33] 90.4(2) [33] 280.37(6) [49]

−2140(18) [50] −27.6(5) [41] 62.51(50) [46] −33.48(18) [52] 169.67(24) [52] 60.54(6) [52] −388(3) [33] 98.98(4) [33] 2440(25) [49]

C6 (au) 1393.39(16) [37] 1393.39(16) [37] 1561 [38] 3927.171 [52] 3927.171 [52] 3927.171 [52] 4703(9) [33] 4703(9) [33] 6860(25) [49]

All quantities are given in atomic units.

11.2.4

BOUND STATES AND SCATTERING RESONANCES

The solutions of the stationary Schrödinger equation associated with the Hamiltonian of Equation 11.4 determine the molecular energy levels as well as the scattering resonances for a pair of alkali–metal atoms in their electronic ground states. In order to calculate molecular bound and continuum levels, it is convenient to perform a partial-wave analysis of the stationary wavefunction with respect to the orbital angular √ momentum of the relative motion of the atoms, . Given a definite modulus ( + 1) of and the projection m onto the magnetic field direction, an asymptotic scattering channel is defined through a channel state, 5 6 |α = f1 mf1 , f2 mf2 , m .

(11.9)

Here the pairs of indices ( f1 mf1 ) and ( f2 mf2 ) determine the Zeeman levels of the atoms at asymptotically large separations. The curly brackets on the right-hand side of Equation 11.9 indicate that the channel state is symmetric if the atoms are identical bosons and antisymmetric if they are identical fermions [29]. The minimum energy of an atom pair at asymptotically large separations determines an associated channel energy, Eα = Ef1 mf1 + Ef2 mf2 ,

(11.10)

where Ef1 mf1 and Ef2 mf2 are the Zeeman energies of the individual atoms. Given the energy E of a stationary wavefunction associated with the Hamiltonian of Equation 11.4, a scattering channel with index α is said to be open when the channel energy, Eα , is below E. Otherwise the channel is referred to as closed. Stable molecular bound states and continuum levels are separated by the scattering (or dissociation) © 2009 by Taylor and Francis Group, LLC


405

threshold determined by the lowest possible channel energy. Long-lived metastable molecular states might exist, however, also in the presence of open decay channels (e.g., Ref. [54]). The stationary wavefunction of an atom pair, Ψ(r, E), can be determined using the coupled-channels method [29,31,55]. To this end, Ψ(r, E) is expanded in terms of basis-set components ψα (r, E) associated with the channel states defined in Equation 11.9. Using the radial wavefunctions, Fα (r, E) = r ψα (r, E),

(11.11)

the molecular bound states and continuum levels are determined by the following set of coupled equations: ∂ 2 Fα (r, E) 2μ + 2 E δαβ − Vαβ (r) Fβ (r, E) = 0. ∂r 2

(11.12)

β

Here α and β are the channel indices and 2 ( + 1) int δαβ + Vαβ (r) Vαβ (r) = Eα + 2μr 2

(11.13)

includes the interatomic interaction potentials. Using the approximation of Equation 11.5, the scattering channels are coupled in Equation 11.12 simply because the product of Zeeman states in Equation 11.9 is not necessarily associated with a definite total electronic-spin quantum number S. For this reason, the off-diagonal part of Equation 11.13 involves linear combinations of the singlet and triplet potentials. Additional directional forces lead to coupling also between different partial waves in the relative motion of the atoms [29]. Due to rotational symmetry about the direction of the magnetic field, however, the quantum number, M = mf1 + mf2 + m , representing the projection of the total diatomic angular momentum Ftotal = f1 + f2 + onto this direction is strictly conserved [29]. Figure 11.3 shows energies of 87 Rb2 molecular bound states for = 0 vs. B below the ( f1 = 1, mf1 = +1; f2 = 1, mf2 = +1) scattering threshold calculated using only the singlet and triplet potentials in Equation 11.13. Due to the conservation law M = mf1 + mf2 = 2, there are five coupled s-wave scattering channels in this example, namely ( f1 , mf1 ; f2 , mf2 ) = (1, 1; 1, 1), (1, 1; 2, 1), (1, 0; 2, 2), (2, 1; 2, 1), and (2, 0; 2, 2). The bound-state labels ( f1 , f2 )v at zero magnetic field strength indicate the vibrational quantum number, v = −1, −2, −3, . . . , counted downward from the ( f1 , f2 ) threshold which gives rise to the series of molecular energy levels [56]. Intersections of such bound-state energies with the scattering threshold, as indicated by the filled circles in Figure 11.3, are expected to lead to resonant enhancement of the zero-energy collision cross-section for a pair of 87 Rb atoms prepared in their ( f = 1, mf = +1) Zeeman ground state. © 2009 by Taylor and Francis Group, LLC

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Cold Molecules: Theory, Experiment, Applications (1,1) –1 –9000

(1,1) –2 (1,2) –4 (1,2) –4

–10,000 E/h (MHz)

–11,000 (1,1) –3 (2,2) –5 –12,000

(2,2) –5

–13,000 0

200

400

600 B (G)

800

1000

FIGURE 11.3 Sketch of several coupled-channel bound-state energies (solid curves) associated with 87 Rb2 molecules below the ( f1 = 1, mf1 = +1; f2 = 1, mf2 = +1) scattering threshold (dotted curve) versus the magnetic field strength, B. The calculations shown were performed using a five s-wave channel spherical-box model including only the central interaction of Equation 11.5. Bound-state labels to the left refer to the quantum numbers ( f1 , f2 )v. Intersections of bound-state energies with the scattering threshold lead to resonant enhancement in collisions at zero kinetic energy between 87 Rb atoms in their lowest energetic Zeeman levels. Their experimentally observed positions are indicated by filled circles. Precise predictions on these and several other scattering resonances as well as their associated measurements can be found in Ref. [56].

11.3

FESHBACH RESONANCES

11.3.1 TWO-CHANNEL TWO-POTENTIAL APPROACH Resonant enhancements of scattering cross-sections in multichannel collision physics are often described in terms of the Feshbach theory of closed-channel resonance states [57]. Feshbach’s general formalism involves projecting the stationary Schrödinger equation onto complementary subspaces associated with the open and closed scattering channels. This theory has been applied in the context of the nearthreshold collision physics of ultracold gases consisting of alkali–metal atoms in a variety of different approaches (e.g., Refs. [9,30,58]). Within a limited range of energies and magnetic field strengths, the Feshbachresonance enhanced diatomic interactions in ultracold gases can often be described parametrically by simpler two-channel two-potential or two-channel single-resonance methods. Such approaches might be based on the following general form of a twochannel Hamiltonian [59]: Hbg W (r) H= . (11.14) W (r) Hcl (B) Here r is the interatomic distance and B denotes the strength of the homogeneous magnetic field. The diagonal matrix elements of Equation 11.14 involve kinetic- and © 2009 by Taylor and Francis Group, LLC


407

potential-energy contributions of the following form: Hbg = − Hcl (B) = −

2 2 ∇ + Vbg (r), 2μ

(11.15)

2 2 ∇ + Vcl (B, r). 2μ

(11.16)

Here Hbg is referred to as the bare entrance-channel Hamiltonian describing, in the hypothetical absence of interchannel coupling, the configuration of Zeeman states in which the individual atoms of s-wave interacting pairs in an ultracold gas are prepared. The potential Vbg (r) thus mimics a bare entrance-channel interaction. Two-channel approaches can be used when the entrance channel is coupled significantly only to a single Zeeman-state configuration of atom pairs that belongs to an energetically closed channel. Accordingly, the bare closed-channel Hamiltonian, Hcl (B), describes such atom pairs in the hypothetical absence of interchannel coupling. This implies that the dissociation thresholds of Vcl (B, r) and Vbg (r) are separated by the positive difference in energies associated with the closed-channel and entrance-channel Zeeman-state configurations. Throughout the remainder of this chapter, the zero of energy is the dissociation threshold of the entrance-channel potential. Due to this convention, the Hamiltonian of Equation 11.14 depends on the magnetic field strength B only through the bare closed-channel Hamiltonian. The off-diagonal matrix elements W (r) on the right-hand side of Equation 11.14 provide the interchannel coupling. Although Vbg (r) and Vcl (B, r) are often chosen to have the typical form of diatomic potentials, governed by Pauli repulsion at short distances and by the van der Waals attraction, −C6 /r 6 , in the limit r → ∞, the coupling-matrix element W (r) mimics the microscopic spin-exchange or dipolar interactions. Figure 11.4 illustrates such a two-channel two-potential model [60] determined by Equation 11.14, in combination with Equations 11.15 and 11.16. This example has been set up to describe weakly bound 23 Na2 molecular states leading to resonant enhancement in s-wave collision cross-sections of 23 Na atom pairs prepared in the ( f = 1, mf = +1) Zeeman ground states at magnetic field strengths in the vicinity of 907 G [60]. In this model, Vbg (r) and Vcl (B, r) have been chosen to be of exactly the same shape, with an energy offset determined by the Zeeman effect in the 23 Na hyperfine structure. Their potential well and long-distance behavior have been calibrated in such a way that they recover the exact scattering length at and position of the highest excited vibrational energy level, as well as the C6 coefficient of the sodium triplet potential. For simplicity, the number of vibrational levels supported by Vbg (r) and Vcl (B, r) is reduced to only five as compared to the 16 levels supported by the complete triplet potential. The off-diagonal matrix elements W (r) used in this model to describe the interchannel coupling have the arbitrary form of a decaying exponential function.

11.3.2 TWO-CHANNEL SINGLE-RESONANCE APPROACH Strong interchannel interactions can occur in ultracold diatomic collisions when the magnetically tunable energy of a closed-channel vibrational state, Eres (B), is © 2009 by Taylor and Francis Group, LLC

408


nearly degenerate with the entrance-channel dissociation threshold. Such a bare closed-channel Feshbach-resonance state, |φres , is given by the solution of the following Schrödinger equation: Hcl (B)|φres = Eres (B)|φres .

(11.17)

In the example of Figure 11.4, the resonance state is determined by the second vibrational level labeled v = −2 from the top of the sodium triplet√potential, which has been modeled by Vcl (B, r). Its associated radial wavefunction, 4π rφres (r), is indicated by the dotted curve. Given the assumption that the interaction between the channels is caused mainly by |φres , the closed-channel Hamiltonian might be replaced simply by the following one-dimensional term: Hcl (B) → |φres Eres (B)φres |.

(11.18)

100

0

u = –4

–100

–200 u = –5

(4π)1/2rφ b(r)

E/h (GHz)

u = –3

u = –2 0.2 0.1 0 –0.1 –0.2

–300

15

100 r (a0) 20

25

30

35 r (a0)

40

1000

45

50

FIGURE 11.4 Sketch of effective entrance-channel (solid curve) and closed-channel (dashed curve) potentials plotted vs. the interatomic separation r. Horizontal lines refer to the locations of several bare entrance-channel and closed-channel vibrational energy levels counted downward from the top of the potential wells (excluding v = −1 due to its proximity to the dissociation thresholds). In the situation shown, the v = −2 closed-channel vibrational energy level is degenerate with the entrance-channel zero-energy scattering threshold. The associated √ radial resonance-state wavefunction, 4π rφres (r), is indicated by the dotted curve. Inset: Entrance-channel (solid curve) and closed-channel (dashed curve) radial wavefunctions of the highest excited vibrational bound state as obtained from this two-channel two-potential model at a resonance detuning of Eres (B)/h = −10 MHz. The model potentials shown, as well as the off-diagonal couplings and resonance slope ∂Eres /∂B used in this figure, refer to the description of 23 Na2 molecules at magnetic field strengths in the vicinity of 907 G given in Ref. [60].


409


Here the resonance state |φres is assumed to be unit normalized, that is φres |φres = 1. Such a replacement in a two-channel Hamiltonian shall be referred to in the following as a single-resonance or configuration-interaction [3] model. Contrary to the two-channel two-potential method, Equation 11.18 always allows the stationary Schrödinger equation associated with the Hamiltonian of Equation 11.14 to be solved analytically in terms of the bare bound and continuum states of Hbg [3,59].

11.4 11.4.1

MAGNETIC TUNING OF THE SCATTERING LENGTH RESONANCE WIDTH AND BACKGROUND-SCATTERING LENGTH

Ultracold diatomic collisions can often be described by a single interaction parameter, the s-wave scattering length, a, which can depend sensitively on B through Eres (B) in Equation 11.18. Within the limited range of magnetic field strengths in which a two-channel single-resonance approach is applicable, the resonance-state energy can usually be approximated by a linear function, Eres (B) =

∂Eres (B − Bres ). ∂B

(11.19)

Here ∂Eres /∂B is the, by assumption, constant difference in magnetic moments between atom pairs in the closed-channel and entrance-channel Zeeman-state configurations, and Bres is the magnetic field strength at which Eres (B) crosses the entrance-channel dissociation threshold. Given that Equation 11.19 is valid, the analytic solution of the stationary Schrödinger equation at zero energy recovers the following parametric expression for the scattering length of Ref. [30]: a(B) = abg 1 −

ΔB . B − B0

(11.20)

Here abg is referred to as the background scattering length, which can be derived, in the hypothetical absence of interchannel coupling, from Equations 11.7 and 11.8 using the entrance-channel potential Vbg (r) instead of VS (r). According to Equation 11.20, the scattering length has a singularity at the magnetic field strength B0 , which is referred to as the resonance position, as well as a zero, whose distance from B0 is given by the resonance width ΔB. Figure 11.5 illustrates these parameters, as well as abg , for pairwise collisions of 87 Rb atoms prepared in the ( f = 1, mf = +1) Zeeman ground state at magnetic field strengths in the vicinity of 1007 G. According to Figure 11.3, it is the vibrational molecular energy level labelled ( f1 = 2, f2 = 2)v = −5 at B = 0 which approaches the scattering threshold at the measured resonance position of B0 = 1007.4 G [61]. The energy and magnetic moment of the near-resonant vibrational level below this threshold [13], as well as the low-energy diatomic collision physics directly above it, can often be described approximately by a properly adjusted two-channel model. © 2009 by Taylor and Francis Group, LLC

410


800 600 ΔB

400

abg

a(B) (a0)

200 0

a=0

–200 –400 –600

B0

–800 1006

1006.5

1007

1007.5 B (G)

1008

1008.5

FIGURE 11.5 Illustration of Equation 11.20 describing resonant enhancement of the s-wave scattering length a(B) (solid curve) as a function of the magnetic field strength B. Associated parameters include the resonance position B0 (vertical dashed line), resonance width ΔB, and background scattering length abg (horizontal dotted line). The resonance width determines the distance between the resonance position and the zero crossing of the scattering length as indicated by the vertical dotted line. The specific resonance displayed refers to collisions between 87 Rb atoms in the ( f = 1, m = +1) Zeeman ground state using parameters determined in the f experiments of Refs. [61,62].

11.4.2

RELATION BETWEEN BOUND-STATE ENERGY AND RESONANCE POSITION

Figure 11.6 shows a measured order of magnitude variation of the scattering length for pairs of 23 Na atoms prepared in the ( f = 1, mf = +1) Zeeman ground state in the vicinity of a singularity detected at B0 = 907 G [10]. According to the lower panel, the resonance position exactly coincides with the magnetic field strength at which the energy of the highest excited diatomic vibrational bound state Eb (B) becomes degenerate with the threshold for dissociation. This coupled-channel bound state, |φb , often termed the Feshbach molecule, is determined by the stationary Schrödinger equation H|φb = Eb |φb .

(11.21)

Here H might be either the microscopic Hamiltonian of Equation 11.4 or the two-channel Hamiltonian of Equation 11.14, depending on the level of approximation. As Eb (B) approaches the dissociation threshold, the scattering length is always positive and changes sign at the singularity where the Feshbach molecule vanishes into the continuum. Such a scenario in general is often referred to as a zero-energy resonance [53] or, in the context of ultracold gases, as a Feshbach or Fano–Feshbach resonance. Equation 11.20 gives a parametric description of such singularities of the scattering length within the range of magnetic field strengths in which a single-resonance © 2009 by Taylor and Francis Group, LLC

411

Theory of Ultracold Feshbach Molecules 6 (a)

a/abg

4

2

0

0

–2 (b)

E/h (MHz)

–20 –10–3 –10–2

–40 Eb(B)

–10–1 –1

–60

890

–10 895

900

906.6 905

906.8 910

907 915

B (G)

FIGURE 11.6 (a) Dependence of the scattering length a(B) on the magnetic field strength B for a pair of 23 Na atoms in the ( f = 1, mf = +1) Zeeman ground state in the vicinity of B0 = 907 G. The circles indicate measured data [10] normalized to the background scattering length, while the curve refers to predicted resonance parameters [63]. (b) Energies of the Feshbach molecular state (solid curve labeled Eb ) vs. the magnetic field strength. The inset shows an enlargement of the bound-state energies near resonance as obtained from the twochannel two-potential approach of Ref. [60] (filled circles), a two-channel single-resonance approach (solid curve), in addition to the universal estimate, Eb = −2 /(2μa2 ) (dotted curve). The position of the singularity of a(B) in (a) coincides with the magnetic field strength at which Eb (B) becomes degenerate with the zero-energy entrance-channel dissociation threshold. (From Inouye, S. et al., Nature, 392, 151, 1998. With permission.)

picture is applicable. Using a two-channel single-resonance approach, Figure 11.7 sketches the behavior of the Feshbach molecular bound-state energy and of the s-wave collision cross-section away from the scattering threshold for the narrow (ΔB = 1 G; Refs. [60,63]) B0 = 907 G zero-energy resonance of 23 Na and for the broad (ΔB = 10.71 G; Ref. [64]) B0 = 155 G zero-energy resonance of 85 Rb. Due to © 2009 by Taylor and Francis Group, LLC

412

Cold Molecules: Theory, Experiment, Applications 1

(a) 20

(b)

0.8

15

0.8 4

0.6 10

0.6

0.4

5

0.2 0

0 Eb(B) Eres(B) –5

0.4 E/h (MHz)

E/h (MHz)

1

6

2 0.2 0

0 Eb(B)

Eres(B)

–2

–10 –4 −15 23Na −20

900

905 B (G)

910

915

85Rb –6 140

150

160 B (G)

170

180

FIGURE 11.7 Sketch of the relation between the Feshbach molecular bound-state energy below the dissociation threshold and the s-wave collision cross-section above it as a function of the magnetic field strength, B. (a) The B0 = 907 G zero-energy resonance of 23 Na of Figure 11.6. (b) The B0 = 155 G zero-energy resonance associated with a pair of 85 Rb atoms prepared in the ( f = 2, mf = −2) Zeeman state. The s-wave collision cross-section, σ(k), is presented in terms of the dimensionless quantity, sin2 ϑ(k) = k 2 σ(k)/(8π), where k is the angular wavenumber associated with the relative motion of an atom pair and ϑ(k) is the s-wave scattering phase shift [53]. Positive energies E = 2 k 2 /(2μ) refer to diatomic collisions, while negative energies, E < 0, are associated with bound states.

the interchannel coupling, the energy of an unperturbed resonance state Eres (B) above the scattering threshold is generally shifted and broadened [1–5]. For the narrow resonance in Figure 11.7a, the maximum collision cross-section, as well as the bound-state energy Eb (B) follow the linear trend of Eres (B), apart from an unresolved energy range about the dissociation threshold. For the broad resonance in Figure 11.7b, the slope and width of the collision cross-section, as well as the dependence of the bound-state energy on the magnetic field strength, B, differ qualitatively from Eres (B) over the entire range of energies shown. A detailed analysis [58,65] indicates that the associated strong interchannel interactions result from the comparatively large width of the B0 = 155 G zero-energy resonance of 85 Rb, in combination with the large negative background scattering length of abg = −443 a0 [64].

11.5 11.5.1

CLASSIFICATION OF RESONANCES CLOSED-CHANNEL DOMINATED RESONANCES

Signatures of resonant enhancement in such s-wave collision cross-sections at finite kinetic energy have been observed, for instance, in experiments on Feshbach molecule © 2009 by Taylor and Francis Group, LLC

413


Ediss/kB (μK)

4

n(E/kB) (1/mK)

87Rb

6

3

23Na

5 4

Sudden 1 G/msec 0.2 G/msec 0.1 G/msec Asymptotic

3 2 1

2

0

0

50

100 150 . B (G/msec)

200

1

0

0

1

2

3

4

E/kB (μK)

FIGURE 11.8 Calculated probability densities for the dissociation of 87 Rb2 Feshbach molecules into free atom pairs with kinetic energy of the relative motion E for different ramp speeds B˙ at the end of a linear magnetic field sweep. In accordance with associated experiments [62], the sweep starts at a magnetic field strength Bi = B0 − 50 mG and terminates at Bf = B0 + 40 mG in the vicinity of the B0 = 685 G zero-energy resonance of 87 Rb atoms prepared in the ( f = 1, mf = +1) Zeeman ground state. Solid and dotted curves refer to the idealized scenarios of an infinitely fast switch of the magnetic field strength from Bi to Bf , and an asymptotically wide sweep across the zero-energy resonance, starting and ending infinitely far away from B0 , respectively. Inset: Comparison between calculated average kinetic energies of the atomic constituents and measurements on Feshbach molecular dissociation [66] using asymptotically wide linear magnetic field sweeps across the 23 Na zero-energy resonance of Figure 11.6. (Reprinted data from Figure 11.2 from Mukaiyama, T. et al., Phys. Rev. Lett. 92, 180402, 2004. With permission. Copyright (2004) by the American Physical Society.)

dissociation using sufficiently fast magnetic field sweeps across B0 , from positive to negative scattering lengths [62]. As an example, Figure 11.8 shows calculated energy spectra of Feshbach molecule fragments dissociated by linear sweeps of variable speed B˙ across a narrow (ΔB = 6.2(6) mG; Ref. [62]) zero-energy resonance for pairs of 87 Rb atoms prepared in their Zeeman ground state. As the ramp speed increases, a pronounced peak appears at an energy determined by the final magnetic field strength, given that the resonance is sufficiently narrow, such as the rubidium zero-energy resonance of this example, which is visible in Figure 11.3 at B0 = 685 G, or the sodium resonance of Figure 11.6. The position of this peak in the dissociation-energy spectra of Figure 11.8 coincides, to a good approximation, with the kinetic energy of maximum s-wave collision cross-section at the final magnetic field strength, Bf = B0 + 40 mG. Apart from a narrow energy range about the scattering threshold, the associated resonance state is closed-channel dominated, similar to the resonance in the s-wave collision cross-section shown in the left image of Figure 11.7a. © 2009 by Taylor and Francis Group, LLC

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As the ramp speed B˙ decreases for given initial and final magnetic field strengths, the dissociation-energy spectrum approaches its universal limit [66], as indicated by the dotted curve in Figure 11.8, which depends only on the product abg ΔB and on ˙ The inset of Figure 11.8 shows measured kinetic energies of the relative motion B. of Feshbach molecule fragments originating from dissociation by linear magnetic field sweeps across the sodium B0 = 907 G zero-energy resonance in the universal limit [66]. Whereas this asymptotic shape of a dissociation-energy spectrum is common to all zero-energy resonances, the appearance of a sharp peak as in Figure 11.8 depends on the width of the resonance above the scattering threshold being much smaller than its energy, as in Figure 11.7a.

11.5.2

ENTRANCE-CHANNEL DOMINATED RESONANCES

Broadness of zero-energy resonances tends to indicate strong interchannel interactions, as illustrated in Figure 11.7b. The strength of interactions between the entrance and closed scattering channels might be estimated on the basis of the resonance width ΔB, in addition to abg , ∂Eres /∂B, and a single parameter characterizing the long-distance behavior of the Born–Oppenheimer potentials, the van der Waals length [67], 1 lvdW = (2μC6 /2 )1/4 . (11.22) 2 Given the parameters ΔB, abg , and ∂Eres /∂B, Wigner’s threshold law [68] yields the following estimate for the energy width of a resonance [30,60] above the scattering threshold [66]: Γres (E) = k abg (∂Eres /∂B) ΔB. (11.23) Here k is the angular wavenumber related to the kinetic energy of the relative motion of the colliding atoms by E = 2 k 2 /(2μ). The interchannel interactions might be considered to be strong when the width of a resonance exceeds its energy within a 2 ), as displayed in range limited by the van der Waals energy, EvdW = 2 /(2μlvdW Figure 11.7. Using Equation 11.23 and k = 1/lvdW , this requirement leads to the following criterion characterizing a broad resonance [65,69]: 2 ) 2 /(2μlvdW 1. abg (∂Eres /∂B) ΔB/lvdW

(11.24)

The Feshbach molecule bound states associated with broad resonances, in the sense of Equation 11.24, tend to be entrance-channel dominated over an experimentally significant range of magnetic field strengths and energies below the dissociation threshold. Figure 11.9 shows the bound-state energy Eb as a function of the magnetic field strength B for a typical example, the B0 = 155 G zero-energy resonance of 85 Rb, which is shown in Figure 11.7b. As B decreases toward the resonance position B0 , that is, when a(B) diverges to infinity, the energy Eb (B) smoothly approaches the dissociation threshold in accordance with the universal formula [71,72], Eb = −2 /(2μa2 ). © 2009 by Taylor and Francis Group, LLC

(11.25)

415

Theory of Ultracold Feshbach Molecules 0

–0.2

E/h (MHz)

0 –0.4

–0.6

B0

–2 –4 –6

–0.8 –8 –1 156

–10

156

160

157

158

164 159 B (G)

160

161

162

FIGURE 11.9 Bound-state energies of 85 Rb2 Feshbach molecules vs. magnetic field strength B in the vicinity of B0 = 155 G displayed similarly to Figure 11.6b. The squares with error bars refer to measurements near resonance [64] using the atom–molecule Ramsey interferometry technique of Ref. [11], while filled circles indicate a coupled-channels calculation [70]. For comparison, the dotted curve represents the universal estimate for Eb (B) of Equation 11.25. Inset: Comparison between the different theoretical approaches showing their increasing deviations with increasing magnetic field strength away from the resonance position. (Data reprinted from Figure 11.2 from Claussen, N.R. et al., Phys. Rev. A, 67, 060701(R), 2003. With permission. Copyright (2003) by the American Physical Society.)

This dependence of Eb (B) only on the scattering length indicates that the associated bound-state wavefunction consists almost entirely of its entrance-channel component of asymptotic form [72], 1 e−r/a , (11.26) φb (r) = √ 2πa r in the limit a → +∞. As illustrated in the inset of Figure 11.6, the asymptotic boundstate energy of Equation 11.25 also applies to 23 Na2 Feshbach molecules associated with the narrow, in the sense of Equation 11.24, closed-channel dominated zeroenergy resonance, at B0 = 907 G. At magnetic field strengths just below B0 , the 23 Na Feshbach molecular wavefunction, as sketched in the inset of Figure 11.4, also 2 eventually acquires a universal entrance-channel component given by Equation 11.26. In the limit a → +∞, the bound-state wavefunction of Equation 11.26 can have an arbitrarily long range determined only by the scattering length, probing a region of distances where the classical attractive force between the atoms may be neglected. The bond length of such a universal Feshbach molecule, that is, its average interatomic distance, is found to be ∞ r = 4π r 2 dr r |φb (r)|2 = a/2. (11.27) 0


416


This quantum physical spatial extent might be compared to the outer turning point of a hypothetically classical motion determined by the bound-state energy of Equation 11.25 and by the long-range van der Waals potential through the relation 6 Eb = −C6 /rclassical , which yields 1/3 rclassical = a(2lvdW )2 a/2 = r.

(11.28)

In this sense, the Feshbach molecules associated with the measured bound-state energies of Figure 11.9 are in classically forbidden states. For this reason, their universal long-range entrance-channel wavefunction of Equation 11.26, in the limit a → +∞, formally refers to the interaction-free stationary Schrödinger equation with energy Eb of Equation 11.25.

11.6

CONCLUSIONS

In summary, this chapter has given an overview of several theoretical methods to describe magnetic tuning of the s-wave scattering length in ultracold gases of alkali– metal atoms and its relation to the physics of diatomic Feshbach molecules. These methods involve the microscopic coupled-channels theory, its reduction to effective two-channel approaches, as well as the universal description of Feshbach molecules. The presentation of these models accounts for their decreasing ranges of validity with respect to the energy about the scattering threshold and to the magnetic field strength in the vicinity of zero-energy resonances.

ACKNOWLEDGMENTS We are grateful to Chris Greene, Eleanor Hodby, Wolfgang Ketterle, Servaas Kokkelmans, Takashi Mukaiyama, Sarah Thompson, and Carl Wieman for allowing us to present their experimental and theoretical data. This work has been supported by the United Kingdom Engineering and Physical Sciences Research Council (grant number EP/E025935/2) and by a University Research Fellowship of the Royal Society.

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26.

27. 28. 29.

30. 31.

32.

33.

34. 35. 36. 37.

38. 39.

40.

41.

42.

Cold Molecules: Theory, Experiment, Applications in the BEC–BCS crossover, in Ultracold Fermi Gases, Inguscio, M., Ketterle, W., and Salomon, C., Eds., Proceedings of the International School of Physics “Enrico Fermi”, Course CLXIV, Varenna, 20–30 June 2006, IOS Press, Amsterdam, 2008, p. 845; e-print cond-mat/0701181. Stringari, S., Dynamics and superfluidity of an ultracold Fermi gas, in Ultracold Fermi Gases, Inguscio, M., Ketterle, W., and Salomon, C. Eds., Proceedings of the International School of Physics “Enrico Fermi”, Course CLXIV, Varenna, 20–30 June 2006, IOS Press, Amsterdam, 2008, p. 53; e-print cond-mat/0702526. Arimondo, E., Inguscio, M., and Violino, P., Experimental determinations of the hyperfine structure in the alkali atoms, Rev. Mod. Phys., 49, 31, 1977. Bransden, B.H. and Joachain, C.J., Physics of Atoms and Molecules, Pearson Education Ltd., Harlow, 2003. Stoof, H.T.C., Koelman, J.M.V.A., and Verhaar, B.J., Spin-exchange and dipole relaxation rates in atomic hydrogen: Rigorous and simplified calculations, Phys. Rev. B, 38, 4688, 1988. Moerdijk, A.J., Verhaar, B.J., and A.Axelsson. Resonances in ultracold collisions of 6 Li, 7 Li, and 23 Na, Phys. Rev. A, 51, 4852, 1995. Mies, F.H., Julienne, P.S., Williams, C.J., and Krauss, M., Estimating bounds on collisional relaxation rates of spin-polarized 87 Rb atoms at ultracold temperatures, J. Res. Nat. Inst. Stand. Technol., 101, 521, 1996. Kotochigova, S., Tiesinga, E., and Julienne, P.S., Relativistic ab initio treatment of the second-order spin–orbit splitting of the a3 Σ+ u potential of rubidium and cesium dimers, Phys. Rev. A, 63, 012517, 2001. van Kempen, E.G.M., Kokkelmans, S.J.J.M.F., Heinzen, D.J., and Verhaar, B.J., Interisotope determination of ultracold rubidium interactions from three high-precision experiments, Phys. Rev. Lett., 88, 093201, 2002. Burke, J.P., Bohn, J.L., Esry, B.D., and Greene, C.H., Prospects for Mixed-isotope Bose–Einstein condensates in rubidium, Phys. Rev. Lett., 80, 2097, 1998. Klausen, N.N., Bohn, J.L., and Greene, C.H., Nature of spinor Bose–Einstein condensates in rubidium, Phys. Rev. A, 64, 053602, 2001. Marinescu, M., Sadeghpour, H.R., and Dalgarno, A., Dispersion coefficients for alkali–metal dimers, Phys. Rev. A, 49, 982, 1994. Zong-Chao Yan, James F. Babb, Dalgarno, A., and Drake, G.W.F., Variational calculations of dispersion coefficients for interactions among H, He, and Li atoms, Phys. Rev. A, 54, 2824, 1996. Kharchenko, P., Babb, J.F., and Dalgarno, A., Long-range interactions of sodium atoms, Phys. Rev. A, 55, 3566, 1997. Derevianko, A. Johnson, W.R., Safronova, M.S., and Babb, J.F., High-precision calculations of dispersion coefficients, static dipole polarizabilities, and atom–wall interaction constants for alkali–metal atoms, Phys. Rev. Lett., 82, 3589, 1999. Vogels, J.M., Tsai, C.C., Freeland, R.S., Kokkelmans, S.J.J.M.F., Verhaar, B.J., and Heinzen, D.J., Prediction of Feshbach resonances in collisions of ultracold rubidium atoms, Phys. Rev. A, 56, R1067, 1997. Abraham, E.R.I., McAlexander, W.I., Gerton, J.M., Hulet, R.G., Côté, R., and Dalgarno, A., Triplet s-wave resonance in 6 Li collisions and scattering lengths of 6 Li and 7 Li, Phys. Rev. A, 55, R3299, 1997. Roberts, J.L., Claussen, N.R., Burke, J.P., Greene, C.H., Cornell, E.A., and Wieman, C.E., Resonant magnetic field control of elastic scattering in cold 85 Rb, Phys. Rev. Lett., 81, 5109, 1998.



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43. Bohn, J.L., Burke, J.P., Greene, C.H., Wang, H., Gould, P.L., and Stwalley, W.C., Collisional properties of ultracold potassium: Consequences for degenerate Bose and Fermi gases, Phys. Rev. A, 59, 3660, 1999. 44. Burke, J.P., Greene, C.H., Bohn, J.L., Wang, H., Gould, P.L., and Stwalley, W.C., Determination of 39 K scattering lengths using photoassociation spectroscopy of the 0− g state, Phys. Rev. A, 60, 4417, 1999. 45. Wang, H., Nikolov, A.N., Ensher, J.R., et al., Ground-state scattering lengths for potassium isotopes determined by double-resonance photoassociative spectroscopy of ultracold 39 K, Phys. Rev. A, 62, 052704, 2000. 46. Samuelis, C., Tiesinga, E., Laue, T., Elbs, M., Knöckel, H., and Tiemann, E., Cold atomic collisions studied by molecular spectroscopy, Phys. Rev. A, 63, 012710, 2000. 47. Modugno, G., Ferrari, G., Roati, G., Brecha, R.J., Simoni, A., and Inguscio, M., Bose– Einstein condensation of potassium atoms by sympathetic cooling, Science, 294, 1320, 2001. 48. Loftus, T., Regal, C., Ticknor, C., Bohn, J., and Jin, D., Resonant control of elastic collisions in an optically trapped Fermi gas of atoms, Phys. Rev. Lett., 88, 173201, 2002. 49. Chin, C., Vuletic, V., Kerman, A.J., Chu, S., Tiesinga, E., Leo, P.J., and Julienne, P.S., Precision Feshbach spectroscopy of ultracold Cs2 , Phys. Rev. A, 70, 032701, 2004. 50. Bartenstein, M., Altmeyer, A., Riedl, S., et al., Precise determination of 6 Li cold collision parameters by radio-frequency spectroscopy on weakly bound molecules, Phys. Rev. Lett., 94, 103201, 2005. 51. D’Errico, C., Zaccanti, M., Fattori, M., Roati, G., Inguscio, M., Modugno, G., and Simoni, A., Feshbach resonances in ultracold 39 K, New J. Phys., 9, 223, 2007. 52. Falke, S., Knöckel, H., Friebe, J., Riedmann, M., Tiemann, E., and Lisdat, C., Ground-state scattering lengths for potassium isotopes determined by double-resonance photoassociative spectroscopy of ultracold 39 K, Phys. Rev. A, 78, 012503, 2008. 53. Taylor, J.R., Scattering Theory, Wiley, New York, 1972. 54. Thompson, S.T., Hodby, E., and Wieman, C.E., Spontaneous dissociation of 85 Rb Feshbach molecules, Phys. Rev. Lett., 94, 020401, 2005. 55. Gao, B., Theory of slow-atom collisions, Phys. Rev. A, 54, 2022, 1996. 56. Marte, A., Volz, T., Schuster, J., Dürr, S., Rempe, G., van Kempen, E.G.M., and Verhaar, B.J., Feshbach resonances in rubidium 87: Precision measurement and analysis, Phys. Rev. Lett., 89, 283202, 2002. 57. Feshbach, H., Theoretical Nuclear Physics, Wiley, New York, 1992. 58. Marcelis, B., van Kempen, E.G.M., Verhaar, B.J., and Kokkelmans, S.J.J.M.F., Feshbach resonances with large background scattering length: Interplay with open-channel resonances, Phys. Rev. A, 70, 012701, 2004. 59. Child, M.S., Molecular Collision Theory, Academic, London, 1974. 60. Mies, F.H., Tiesinga, E., and Julienne, P.S., Manipulation of Feshbach resonances in ultracold atomic collisions using time-dependent magnetic fields, Phys. Rev. A, 61, 022721, 2000. 61. Volz, T., Dürr, S., Ernst, S., Marte, A., and Rempe, G., Characterization of elastic scattering near a Feshbach resonance in 87 Rb, Phys. Rev. A, 68, 010702(R), 2003. 62. Dürr, S., Volz, T., and Rempe, G., Dissociation of ultracold molecules with Feshbach resonances, Phys. Rev. A, 70, 031601(R), 2004. 63. van Abeelen, F.A. and Verhaar, B.J., Unpublished, quoted by Inouye, S. et al. in Ref. [10], 1998. 64. Claussen, N.R., Kokkelmans, S.J.J.M.F., Thompson, S.T., Donley, E.A., Hodby, E., and Wieman, C.E., Very-high-precision bound-state spectroscopy near a 85 Rb Feshbach resonance, Phys. Rev. A, 67, 060701(R), 2003.


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Condensed Matter 12 Physics with Cold Polar Molecules Guido Pupillo, Andrea Micheli, Hans-Peter Büchler, and Peter Zoller CONTENTS 12.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Overview: Strongly Interacting Systems of Cold Polar Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Effective Many-Body Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Self-Assembled Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Blue-Shielding and Three-Body Interactions . . . . . . . . . . . . . . . . . . . . . 12.2.4 Hubbard Lattice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.5 Lattice Spin Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.6 Hubbard Models in Self-Assembled Dipolar Lattices . . . . . . . . . . . . 12.3 Engineering of Interaction Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Molecular Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1.1 Rotational Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1.2 Coupling to External Electric Fields . . . . . . . . . . . . . . . . . . . . 12.3.2 Two Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2.1 Designing the Repulsive 1/r 3 Potential in 2D . . . . . . . . . 12.3.2.2 Designing ad hoc Potentials with ac Fields . . . . . . . . . . . . 12.4 Many-Body Physics with Cold Polar Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Two-Dimensional Self-Assembled Crystals . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Floating Lattices of Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3 Three-Body Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.4 Lattice Spin Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.1

421 423 423 425 427 430 431 433 436 436 436 437 438 439 444 447 447 450 455 459 463

INTRODUCTION

The realization of Bose–Einstein condensates (BEC) and quantum degenerate Fermi gases with cold atoms has been one of the highlights of experimental atomic physics during the last decade [1]. In view of recent progress in the experimental work on the production of cold molecules we expect a similarly spectacular 421 © 2009 by Taylor and Francis Group, LLC

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development for molecular physics [2–22]. The outstanding features of the physics of cold atomic and molecular gases are the microscopic knowledge of the manybody Hamiltonians, as realized in the experiments, combined with the possibility to control and tune system parameters via external fields. External field control can be achieved by confining ultracold gases with magnetic, electric and optical traps, allowing for the formation of quantum gases in one-, two-, and threedimensional geometries, and tuning contact interparticle interactions by varying the scattering length via Feshbach resonances [23,24]. This control is key to the experimental realization of fundamental quantum phases, as illustrated by the BEC– BCS (Bardeen–Cooper–Schrieffer) crossover in atomic Fermi gases [25–29], the Kosterlitz–Thouless transition [30–32] and the superfluid Mott insulator quantum phase transition with cold bosonic atoms in an optical lattice [33,34]. A recent highlight has been the realization of a degenerate magnetic dipolar gas of 52 Cr atoms [35–37]. In this chapter we will mainly discuss heteronuclear molecules prepared in the electronic and vibrational ground state. Polar molecules are characterized by large electric dipole moments associated with rotational excitations. This gives rise to large dipole– dipole interactions between molecules, which can be manipulated with external dc and ac microwave fields. The possibility to tune these strong, long-range and anisotropic interactions raises interesting prospects for cold ensembles of polar molecules as strongly correlated systems [38–54]. Much work has recently been devoted to the study of cold collisions in dipolar gases [55–61], which in this book is reviewed in Chapters 1, 2, 3, and 4 contributed by Hutson, Bohn, Dalgarno, and Krems. In the context of degenerate molecular gases many recent studies have focused on the regime of weak interactions, where the isotropic contact interaction potential competes with the anisotropic long-range dipole–dipole interaction. For example, the existence of rotons in weakly interacting dipolar gases has been predicted [62–70], while exciting prospects have been envisioned for rotating systems [71–76] and polar molecules in optical lattices [77–85]. In this chapter we will be interested in the many-body dynamics of polar molecules in the strongly interacting limit. In particular, we will develop a toolbox for engineering interesting many-body Hamiltonians based on the manipulation of the electric dipole moments with external dc and ac fields, and thus of the molecular interactions. This forms the basis for the realization of novel quantum phases in these systems. Our emphasis will be on condensed-matter physics, while we refer to the contribution by Yelin, DeMille, and Côté in the present book for applications in the context of quantum information processing [86–91]. This chapter is organized as follows. In Section 12.2 we give a qualitative tour through some of the key ideas of engineering Hamiltonians and of the associated quantum phases. This is followed by two slightly more technical sections, Section 12.3.1 and Section 12.3.2, where we provide details of the realization of a two-dimensional setup where particles interact via purely repulsive 1/r 3 potentials, and where we sketch how to design more complicated interactions by using a combination of ac and dc fields. Section 12.4 illustrates the possibility of inducing strongly correlated phases and realizing quantum simulations by tuning intermolecular interactions. © 2009 by Taylor and Francis Group, LLC

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Condensed Matter Physics with Cold Polar Molecules

12.2

OVERVIEW: STRONGLY INTERACTING SYSTEMS OF COLD POLAR MOLECULES

In this section we give a qualitative overview of many-body physics of cold polar molecules with emphasis on strongly interacting systems. In the following sections we will revisit some of the topics for a more indepth discussion.

12.2.1

EFFECTIVE MANY-BODY HAMILTONIANS

Hamiltonians underlying condensed-matter physics of N structureless bosonic or fermionic particles have the generic form Heff =

N

i=1

p2i 3D + Vtrap (ri ) + Veff ({ri }), 2m

(12.1)

where p2i /2m is the kinetic energy term, and Vtrap (ri ) is a confining potential for the 3D ({r }) represents an effective N-body interaction, which can particles. The term Veff i be expanded as a sum of two-body and many-body interactions 3D Veff ({ri }) =

N

N

V 3D ri −rj + W 3D ri , rj , rk + · · · ,

i<j

(12.2)

i<j r/ 3 the interaction becomes attractive, resulting in an instability in the many-body system. This instability can be suppressed by a sufficiently strong two-dimensional confinement with the potential Vtrap (zi ) along z, due to, for example, an optical force induced by an off-resonant light field [46]. The two-dimensional dynamics in this pancake configuration is described by the Hamiltonian

p2ρi

2D 2D + = Veff (ρij ), Heff (12.5) 2m i

i<j

which is obtained by integrating out the fast z-motion. Equation 12.5 is the sum of the two-dimensional kinetic energy in the (x, y)-plane and a repulsive two-dimensional dipolar interaction 2D (ρ) = D/ρ3 , Veff

(12.6)

with ρij ≡ (xj − xi , yj − yi ) a vector in the (x, y)-plane (solid line in Figure 12.1b). The distinguishing feature of the system described by the Hamiltonian (Equation 12.5) is that tuning the induced dipole moment d drives the system from a weakly interacting gas (a two-dimensional superfluid in the case of bosons), to a crystalline phase in the limit of strong repulsive dipole–dipole interactions. This transition and the crystalline phase have no analog in the atomic bose gases with short-range interactions modeled by a pseudopotential of a given scattering length. A crystalline phase corresponds to the limit of strong repulsion where particles undergo small oscillations around their equilibrium positions, which is a result of the balance between the repulsive long-range dipole–dipole forces and an additional (weak) confining potential in the (x, y)-plane. The relevant parameter is rd ≡

Epot D/a3 Dm = 2 = 2 , 2 Ekin /ma a

(12.7)

which is the ratio of the the interaction energy and the kinetic energy at the mean interparticle distance a. This parameter is tunable as a function of d from small rd to © 2009 by Taylor and Francis Group, LLC

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(a)

+kL

Eac(t)

Edc

ϑ

d2 r

d1

j ez

ez

ϑ

–kL

d1

j

ey

ex

ex

(b) 1

d2 r

ey

V2D eff (r) (arb. units)

C3 / r3

0 0.5

1

1.5

2

2.5 r (arb. units)

FIGURE 12.1 (a) System setup. Polar molecules are trapped in the (x, y)-plane by an optical lattice made of two counter propagating laser beams with wavevectors ±kL = ±kL ez . The dipoles are aligned in the z-direction by a dc electric field Edc ≡ Edc ez . An ac microwave field is indicated. Inset: Definition of polar (ϑ) and azimuthal (ϕ) angles for the relative orientation of the intermolecular collision axis r12 with respect to a space-fixed frame, with axis along 2D (ρ) for polar molecules z. (b) Qualitative sketch of effective two-dimensional potentials Veff confined in a two-dimensional (pancake) geometry. Here, ρ = r12 sin ϑ(cos ϕ, sin ϕ, 0) is the two-dimensional coordinate in the plane z = 0 and ρ = r12 sin ϑ (see inset of (a)). Solid line: 2D (ρ) = D/ρ3 induced by a dc electric field. Dash–dotted line: Repulsive dipolar potential Veff “Step-like” potential induced by a single ac microwave field and a weak dc field. Dashed line: Attractive potential induced by the combination of several ac fields and a weak dc field. 2D (ρ) and the separation ρ are given in arbitrary units. (From Micheli, A. The potentials Veff et al., Phys. Rev. A, 76, 043604, 2007. With permission.) © 2009 by Taylor and Francis Group, LLC


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large. A crystal forms for rd 1, when interactions dominate. For a dipolar crystal, this is the limit of large densities, where collisions become harmful. However, the crystalline phase will protect a cold ensemble of polar molecules from (harmful) close-encounter collisions. This density dependence is different from that in Wigner crystals with 1/r-Coulomb interactions, as realized for example with laser cooled trapped ions [93]. In the latter case rc = (e2 /a)/2 /ma2 ∼ a and the crystal forms at low densities. In addition, the charge e is a fixed quantitiy, while d can be varied as a function of the dc field. Figure 12.2a shows a schematic phase diagram for a dipolar gas of bosonic molecules in two-dimensional as a function of rd and temperature T . In the limit of weak interactions rd < 1, the ground state is a superfluid with a finite (quasi-) condensate. In the opposite limit of strong interactions rd 1 the polar molecules are in a crystalline phase for temperatures T < Tm with Tm ≈ 0.09D/a3 [94]. The configuration with minimal energy is a triangular lattice with excitations given by acoustic phonons. In Ref. [46] we investigated the intermediate strongly interacting regime with rd 1, using a recently developed path integral Monte Carlo technique (PIMC) [95]. We determined the critical interaction strength rQM = 18 ± 4 for the quantum melting transition from the crystal into the superfluid, a result that has been confirmed with a number of quantum Monte Carlo techniques [47,48]. In Section 12.4 below we return to the discussion of these quantum phases, and in particular the crystalline phase, to show that the relevant regime of parameters for these phases to occur is accessible with polar molecules. Besides the fundamental interest in dipolar quantum gases, the crystalline phase has interesting applications, such as in the context of quantum information [91]. We will return to self-assembled dipolar lattices below in a discussion of Hubbard models.

12.2.3

BLUE-SHIELDING AND THREE-BODY INTERACTIONS

By combining dc and ac fields to dress the manifold ofrotational energy levels it is possible to design effective interaction potentials V 3D ri − rj with (essentially) any shape as a function of distance. For example, the addition of a single linearlypolarized ac field to the configuration of Figure 12.1a leads to the realization of the two-dimensional “step-like” potential of Figure 12.1b, where the character of the repulsive potential varies considerably in a small region of space. The derivation of this effective two-dimensional interaction is outlined in Figure 12.3 and discussed in more detail in Section 12.3.2.2 below [46,61]. The (weak) dc field splits the first excited rotational (J = 1) manifold of each molecule by an amount δ, while a linearly polarized ac field with Rabi frequency Ω is blue-detuned from the (|g − |e) transition by Δ (Figure 12.3a). Because of δ and the choice of polarization, for distances ρ (d 2 /δ)1/3 the relevant single-particle states for the two-body interaction reduce to the states |g and |e of each molecule. Figure 12.3b shows that the dipole–dipole interaction splits the excited state manifold of the two-body rotational spectrum, making the detuning Δ position-dependent. As a consequence, the combined energies of the bare ground state of the two-particle spectrum and of a microwave photon become degenerate with the energy of a (symmetric) excited state at a characteristic resonant (Condon) point ρC = (d 2 /Δ)1/3 , which is represented © 2009 by Taylor and Francis Group, LLC

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Ta3/D

0.15

Normal

Tm

0.1

0.05 Superfluid

Crystal

0 0

rQM

10

30

rd

40

(b1) 1.25W0

0.17W0

0.02W0

(b2)

1.71W0

1.25W0

0.47W0

0.23W0

0.21W0

1.38W0

0.58W0

0.58W0

0.34W0

0.28W0

(b3)

FIGURE 12.2 (a) Tentative phase diagram in the T − rd plane: crystalline phase for interactions rd > rQM and temperatures below the classical melting temperature Tm (dashed line) [94]. The superfluid phase appears below the upper bound T < π2 n/2 m (dotted line) [31,32]. The quantum melting transition is studied at fixed temperature T = 0.014 D/a3 with interactions rd = 5 − 30 (dash–dotted line). The crossover to the unstable regime for small replusion and finite confinement ω⊥ is indicated (hatched region). (b) Strengths of the dominant three-body interactions Wijk appearing in the Hubbard model of Equation 12.9 for different lattice geometries: (b1) one-dimensional setup; (b2) two-dimensional square lattice; (b3) two-dimensional honeycomb lattice. The characteristic energy scale W0 = γ2 DR06 /a6 is discussed later in Equation 12.38. (From Micheli, A. et al., Phys. Rev. A, 76, 043604, 2007; Büchler, H.P., Nature Phys., 3, 726, 2007. With permission.)

by an arrow in Figure 12.3b. At this Condon point, an avoided crossing occurs in a field-dressed picture, and the new (dressed) ground-state potential inherits the character of the bare ground and excited potentials for distances ρ ρC and ρ ρC , respectively. Figure 12.3c shows that the dressed ground-state potential (which has © 2009 by Taylor and Francis Group, LLC

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Condensed Matter Physics with Cold Polar Molecules (a)

(b)

ħD ˙ eÒ

ħwe

~ E(r, z = 0)/ħD

(c)

E(r, z = 0)/ħwe

2

˙ e; eÒ

+2

ħd +1 ˙ e–1Ò

˙ e+1Ò

W

D/we

˙ g; eÒ+

1 ˙ g; eÒ–

W

0

˙ g; gÒ

2W/D

˙ g; eÒ+ –1 ˙ g; eÒ–

˙ gÒ

0

˙ g; gÒ

1 0.5

1

r/rc

2

–2 0.5

˙ e; eÒ 1

r/rc

2

FIGURE 12.3 Design of the step-like potential of Figure 12.1b: (a) Rotational spectrum of a molecule in a weak dc field. The dc field splits the (J = 1) manifold by an amount δ. The linearly polarized microwave transition with detuning Δ and Rabi frequency Ω is shown as an arrow. (b) Born–Oppenheimer√ potentials for the internal states for Ω = 0 (bare potentials), where |g; e± ≡ (|g; e ± |e; g)/ 2. The resonant Condon point ρC is indicated by an arrow. (c) ac-field-dressed Born–Oppenheimer potentials. The dressed ground-state potential has the largest energy. (From Büchler, H.P., et al., Phys. Rev. Lett., 98, 060404, 2007. With permission.)

the largest energy) is almost flat for ρ ρC and it is strongly repulsive as 1/ρ3 for ρ ρC , which corresponds to the realization of the step-like potential of Figure 12.1b. We remark that, due to the choice of polarization, this strong repulsion is present only in the plane z = 0, while for z = 0 the ground-state potential can become attractive. The optical confinement along z of Figure 12.1a is therefore necessary to ensure the stability of the system. The interactions in the presence of a single ac field are described in detail in Ref. [61], where it is shown that in the absence of external confinement, this case is analogous to the (three-dimensional) optical blue-shielding developed in the context of ultracold collisions of neutral atoms [96–98], however with the advantage of the long lifetime of the excited rotational states of the molecules, as opposed to the electronic states of cold atoms. The strong inelastic losses observed in three-dimensional collisions with cold atoms [96–98] can be avoided via a judicious choice of the field’s polarization, eventually combined with a tight confinement to ensure a twodimensional geometry (as, e.g., in Figure 12.3). For example, in Ref. [99] it is shown that in the presence of a dc field and of a circularly polarized ac field the attractive timeaveraged interaction due to the rotating (ac-induced) dipole moments of the molecules allows for the cancelation of the total dipole–dipole interaction. The residual interactions remaining after this cancelation are purely repulsive three-dimensional 3D (r) ∼ (d 4 /Δ)/r 6 . This interactions with a characteristic van der Waals behavior Veff three-dimensional repulsion provides for a shielding of the inner part of the interaction potential and thus it will strongly suppress inelastic collisions in experiments. This will possibly lead to the realization of quantum degenerate systems of molecules and perhaps to tightly packed crystalline structures in three dimensions [99]. © 2009 by Taylor and Francis Group, LLC

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A cancelation of the leading effective two-body interaction similar to the one described above in a dense ensemble of molecules can lead to the realization of systems where the effective three-body interaction W 3D (ri , rj , rk ) of Equation 12.2 dominates over the two-body term V 3D (ri − rj ) and determines the properties of the system in the ground state. This is interesting, because model Hamiltonians with strong three-body and many-body interactions have attracted a lot of interest in the search for microscopic Hamiltonians exhibiting exotic ground-state properties. Well-known examples are the fractional quantum Hall states described by the Pfaffian wavefunctions which appear as ground states of a Hamiltonian with three-body interactions [100–102]. These topological phases support anionic excitations with nonabelian braiding statistic. Three-body interactions are also an essential ingredient for systems with a low energy degeneracy characterized by string nets [103,104], which play an important role in models for nonabelian topological phases. The possibility of realizing a Hamiltonian where the two-body interaction can be manipulated independently of the three-body term has been studied in Ref. [85]. There, it is shown that a stable system of particles interacting via purely repulsive three-body potentials can be realized by combining the setup above with a tight optical confinement provided by an optical lattice. In fact, the latter serves the two-fold purpose of ensuring the collisional stability of the setup and of defining a characteristic length scale (the lattice spacing) where the exact cancelation of the two-body term occurs. The details of this derivation are given in Section 12.4.3, in connection with the derivation of an extended Hubbard model with three-body interactions introduced in Section 12.2.4.

12.2.4

HUBBARD LATTICE MODELS

Hubbard Hamiltonians are model Hamiltonians describing the low-energy physics of interacting fermionic and bosonic particles in a lattice [105]. They have the general tight-binding form H=−

i,j,σ

† Jijσ bi,σ bj,σ

+

Uijσσ i,j,σ,σ

2

ni,σ nj,σ .

(12.8)

† ) are the destruction (creation) operators for a particle at site i in the Here bi,σ (bi,σ internal state σ, Jijσ describes coherent hopping of a particle from site i to site j (typically

the nearest neighbor), and Uijσσ describes the on-site (i = j) or off-site (i = j) two-

† bi,σ . Hubbard models have a long body interactions between particles, with ni,σ = bi,σ history in condensed-matter physics, where they have been used as tight-binding approximations of strongly correlated systems. For example, for a system of electrons in a crystal hopping from the orbital of a given atom to that of its nearest neighbor, σ represents the electron spin. A (fermionic) Hubbard model comprising electrons in a two-dimensional lattice with interspecies on-site interactions is thought to be responsible for the high-temperature superconductivity observed in cuprates [106]. In recent years, Hubbard models have been shown to properly describe the lowenergy physics of interacting bosonic and fermionic atoms trapped at the bottom



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of an optical lattice [107,108]. The resulting low-energy Hamiltonians are of the form of Equation 12.8. Spectacular experiments with ultracold atoms have led to the realization of the superfluid/Mott insulator quantum phase transition for bosonic atoms [33,34], and further experimental work with fermions may resolve the phase diagram of the fermionic Hubbard model in two dimensions by performing an analog quantum simulation of Equation 12.8 with two-species cold fermions [109,110]. Since the interactions between cold atoms are short-ranged, in these systems σ,σ Hubbard Hamiltonian typically have on-site interactions only (Ui,i in Equation 12.8). However, it has been shown that the presence of moderately long-range interactions in Equation 12.8, such as nearest-neighbor interactions, can lead to interesting phases such as checkerboard solids and two-dimensional supersolids [83,84]. Polar molecules in optical lattices can provide for off-site interactions [78–82] that are strong, of the order of hundreds of kilohertz, and long-range; that is they decay with distance as 1/r 3 . Due to these strong interactions, two molecules cannot hop onto the same site, and thus the particles may be treated as effectively “hard-core.” An intriguing possibility is offered by the interaction engineering discussed above in the context of the realization of effective lattice models where particles interact via exotic (extended) Hubbard Hamiltonians. An example of this is given in Ref. [85], where it is shown how to engineer the following Hubbard-like Hamiltonian H = −J

bi† bj +

ij

Uij i=j

2

ni nj +

Wijk ni nj nk , 6

(12.9)

i=j=k

where Wijk ni nj nk is an off-site three-body term. The latter is tunable independently of the two-body term Uij ni nj , to the extent that it can be made to dominate the dynamics and determine the properties of the system in the ground state. In contrast to the common approach to derive effective many-body terms from Hubbard models involving two-body interactions, which are obtained in the J U perturbation limit, and are thus necessarily small [111], the derivation of the Hubbard model (Equation 12.9) is based directly on the effective many-particle potential given by (Equation 12.2). Thus, all the energy scales in Equation 12.9 can be tuned independently, which allows one to obtain comparatively large hopping rates determining the time and temperature scales to observe exotic quantum phases. This is important, because analytical calculations in one dimension suggest that the Hamiltonian (Equation 12.9) has a rich ground-state phase diagram, supporting valence-bond, charge-density-wave, and superfluid phases [85]. In Section 12.4 below we provide the microscopic derivation of the effective interaction potentials of Equation 12.9.

12.2.5

LATTICE SPIN MODELS

The Hamiltonian in Equation 12.3 can be generalized to include other internal degrees of freedom for each molecule in addition to rotation. This offers new possibilities to engineer effective interactions and novel many-body phases. For example, the addition of a spin-1/2 (qubit) degree of freedom to polar molecules trapped in an optical lattice allows us to construct a complete toolbox for the simulation of any permutation symmetric lattice spin models [51]. Lattice spin models are ubiquitous in © 2009 by Taylor and Francis Group, LLC

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condensed-matter physics where they are used to describe the characteristic behavior of complicated interacting physical systems. The basic building block is a system of two polar molecules strongly trapped at given sites of an optical lattice, where the spin-1/2 (or qubit) is represented by a single unpaired electron outside a closed shell of a 2 Σ1/2 heteronuclear molecule in its rotational ground state, as provided, for example, by alkaline-earth monohalogenides. As dicussed above, heteronuclear molecules possess large permanent electric dipole moments, which are responsible for strong, long-range and anisotropic dipole–dipole interactions, whose spatial dependence can be manipulated using microwave fields. The spin-rotation coupling in the molecules makes these dipole–dipole interactions spin-dependent. General lattice spin models can then be readily built from these binary interactions. Although in this review we will present results for spin-1/2 models only, we notice that the inclusion of hyperfine effects offers extensions to spin systems with larger spin. For example, the design of a large class of spin-1 interactions for polar molecules has been shown in Ref. [52], which allows, for example, for the realization of a generalized Haldane model in one dimension [112]. Two highly anisotropic models with spin-1/2 particles that can be simulated are illustrated in Figures 12.4a and 12.4b, respectively. The first takes place on a square two-dimensional lattice with nearest-neighbor interactions (I) Hspin

−1

−1

z z x x = J σi,j σi,j+1 + cos ζσi,j σi+1,j .

(12.10)

i=1 j=1

Introduced by Douçot and colleagues [113] in the context of Josephson junction arrays, this model (for ζ = ±π/2) admits a two-fold degenerate ground subspace that (a)

(b)

S2 Ÿ

D2

S1

y

z

r

E(t) E(t)

D1 Ÿ

z

Ÿ

x

Si

Sj

FIGURE 12.4 Example anisotropic spin models that can be simulated with polar molecules trapped in optical lattices. (a) Square lattice in two-dimensional with nearest-neighbor orientation-dependent Ising interactions along xˆ and zˆ . Effective interactions between the spins S1 and S2 of the molecules in their rovibrational ground states are generated with a microwave field E(t) inducing dipole–dipole interactions between the molecules with dipole moments D1 and D2 , respectively. (b) Two staggered triangular lattices with nearest neighbors oriented along orthogonal triads. The interactions depend on the orientation of the links with respect to the electric field. (Dashed lines are included for perspective.) (From Micheli, A. et al., Nature Phys., 2, 341, 2006. With permission.)


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is immune to local noise up to th order and hence is a good candidate for storing a protected qubit. The second occurs on a bipartite lattice constructed with two two-dimensional triangular lattices, one shifted and stacked on top of the other. The interactions are indicated by nearest-neighbor links along the xˆ , yˆ , and zˆ directions in real space:

y y

(II) Hspin = J⊥ σjx σkx + J⊥ σj σk + Jz σjz σkz . (12.11) x -links y-links z-links This model has the same spin dependence and nearest-neighbor graph as the model on a honeycomb lattice introduced by Kitaev [114]. He has shown that by tuning the ratio of interaction strengths |J⊥ |/|Jz | one can tune the system from a gapped phase carrying abelian anyonic excitations to a gapless phase, which, in the presence of a magnetic field, becomes gapped with nonabelian excitations. In the regime |J⊥ |/|Jz | 1 the Hamiltonian can be mapped to a model with four-body operators on a square lattice with ground states that encode topologically protected quantum memory [115]. One proposal [116] describes how to use trapped atoms in spin-dependent optical lattices (II) to simulate the spin model Hspin . There the induced spin couplings are obtained via spin-dependent collisions in second-order tunneling processes. Larger coupling strengths as provided by polar molecules are desirable. In both spin models (I and II) above, the signs of the interactions are irrelevant, although it is possible to tune the sign if needed.

12.2.6

HUBBARD MODELS IN SELF-ASSEMBLED DIPOLAR LATTICES

In Hubbard models with cold atoms or molecules in optical lattices there are no phonon degrees of freedom corresponding to an intrinsic dynamics of the lattice, as the back action on the optical potentials is typically negligible. Thus, atomic and molecular Hubbard models allow for the study of strong correlations in the absence of phonon effects. However, the simulation of models where the presence of (crystal) phonons strongly affects the (Hubbard) dynamics of the particles remains a challenge. These models are of fundamental interest in condensed-matter physics, where they describe polaronic and/or superconducting materials [117]. In the context of atoms, one example is to immerse atoms moving on a lattice into a BEC of a second atomic species, representing a bath of Bogoliubov excitations [118–120]. A second example is a self-assembled floating lattice of molecules as discussed in Section 12.2.2, which provides a periodic potential for extra atoms or molecules, whose dynamics can again be described in terms of a Hubbard model [53]. Phonon degrees of freedom enter as vibrations of the dipolar lattice. The Hamiltonian for extra atoms or molecules in a self-assembled dipolar lattice is H = −J

ci† cj +

0 1

† Vij ci† cj† cj ci + Hc + Mq eiq·Rj cj† cj (aq + a−q ). 2 i,j

q,j

(12.12) Here, the first and second terms define a Hubbard-like Hamiltonian for the extraparticles of the form of Equation 12.8, where the operators ci (ci† ) are destruction © 2009 by Taylor and Francis Group, LLC

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Cold Molecules: Theory, Experiment, Applications Vcp

~ J

dp dc

dp

~ Vi, j

dp Edc

a

dc

dc

ex

ez ey

FIGURE 12.5 Floating lattices of dipoles. A self-assembled crystal of polar molecules with dipole moment dc provides a two-dimensional periodic honeycomb lattice Vcp (darker shading corresponds to deeper potentials) for extra molecules with dipole dp dc giving rise to a lattice model with hopping J˜ and long-range interactions V˜ i,j . (From Pupillo, G. et al., Phys. Rev. Lett., 100, 050402, 2008. With permission.)

(creation) operators of the extra-particles. However, the third and fourth terms describe the acoustic phonons of the crystal and the coupling of the extra-particles to the crystal phonons, respectively. In particular, Hc is the phonon Hamiltonian

Hc = ωq aq† aq , (12.13) q

where aq destroys a phonon of quasimomentum q in the mode λ. Tracing over these phonon degrees of freedom in a strong coupling limit provides effective Hubbard models for the extra-particles dressed by the crystal phonons H˜ = −J˜

ci† cj +

1

V˜ ij ci† cj† cj ci . 2

(12.14)

i,j

The hopping of a dressed extra-particle between the minima of the periodic potential occurs at a rate J˜ , which is exponentially suppressed due to the copropagation of the lattice distortion, while off-site particle–particle interactions V˜ i,j are now a combination of direct particle–particle interactions and interactions mediated by the coupling to phonons. The setup we have in mind is depicted in Figure 12.5, © 2009 by Taylor and Francis Group, LLC

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Condensed Matter Physics with Cold Polar Molecules EJ,M – E0,0 ħw ħD

ħw + 2ħd/3

˙ f1,0Ò

ħw + ħd/3 ħd ħw ħw – ħd/3

q=0 ˙ f1,–1Ò

q = –1

q = +1

˙ f1,1Ò

0 ˙ f0,0Ò

FIGURE 12.6 Solid lines: energies EJ,M (left) and states |φJ,M (right) with J = 0, 1, for a molecule in a weak dc electric field Edc = Edc e0 with β ≡ dEdc /B 1. The dc-field-induced splitting δ and the average energy separation ω ¯ are δ = 3Bβ2 /20 and ω ¯ = 2B + Bβ2 /6, respectively. Dashed and dotted lines: energy levels for a molecule in combined dc and ac fields. (The ac-Stark shifts of the dressed states are not shown.) Dashed line: the ac field is monochromatic, with frequency ω, linear polarization q = 0, and detuning Δ = ω − (ω ¯ + 2δ/3) > 0. Dotted lines: schematics of energy levels for an ac field with polarization q = ±1 and frequency ω = ω. (From Micheli, A. et al., Phys. Rev. A, 76, 043604, 2007. With permission.)

where extra-particles, which are molecules with a dipole moment dp dc interact repulsively with the molecules in the crystal, and thus see a periodic (honeycomb) lattice potential. The distinguishing features of this realization of lattice models are as follows: (i) Dipolar molecular crystals constitute an array of microtraps with its own quantum dynamics represented by phonons (lattice vibrations), while the lattice spacings are tunable with external control fields, ranging from a micrometer down to the hundred nanometer regime, that is, potentially smaller than for optical lattices. (ii) The motion of the extra particles is governed by an interplay of Hubbard (correlation) dynamics in the lattice and coupling to phonons. The tunability of the lattice allows access to a wide range of Hubbard parameters and phonon couplings. Compared with optical lattices, for example, a small-scale lattice yields significantly enhanced hopping amplitudes, which set the relevant energy scale © 2009 by Taylor and Francis Group, LLC

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for the Hubbard model, and thus also the temperature requirements for realizing strongly correlated quantum phases.

12.3

ENGINEERING OF INTERACTION POTENTIALS

In this section we show in some detail how to realize a collisionally stable twodimensional setup where particles interact via a purely repulsive 1/r 3 potential, by using a combination of a dc field and of tight optical confinement in the field’s direction. We then sketch how to design more complicated interactions using a combination of ac, dc, and optical fields, by focusing on the step-like potential of Figure 12.1b. This engineering of interaction potentials is at the core of the realization of the strongly correlated phases and quantum simulations discussed in Section 12.4.

12.3.1

MOLECULAR HAMILTONIAN

We consider spinless polar molecules in the electronic and vibrational ground state X 1 Σ(0). In the following, we are interested in manipulating the rotational states of the molecules using dc and ac electric fields and in confining their motion using a (optical) far off-resonance trap (FORT). The application of these external fields will serve as a key element in engineering effective interaction potentials between the molecules. The low-energy effective Hamiltonian for the external motion and internal rotational excitations of a single molecule is H(t) =

p2 + Hrot + Hdc + Hac (t) + Hopt (r), 2m

(12.15)

where p2 /2m is the kinetic energy for the center-of-mass motion of a molecule of mass m, Hrot accounts for the rotational degrees of freedom, while the terms Hdc , Hac (t), and Hopt (r) refer to the interaction with electric dc and ac (microwave) fields and to the optical trapping potential for the molecule in the ground electronic-vibrational manifold, respectively. In the following we consider tight harmonic optical traps with the frequency ω⊥ = 2π × 150 kHz, the same for all the relevant rotational states of the molecule. That is, we neglect possible tensor-shifts induced by the optical trapping field in the energies of the excited rotational states of the molecules, which in general can be compensated for by an appropriate choice of additional laser fields [61]. Thus, for a confinement along ez , Hopt (r) reads Hopt (r) = mω2⊥ z2 /2, independent of the internal (rotational) quantum state [44,121]. 12.3.1.1

Rotational Spectrum

The term Hrot in Equation 12.15 is the Hamiltonian describing a rigid spherical rotor [122] Hrot = B J2 ,

(12.16)

which accounts for the rotation of the internuclear axis of a molecule with total angular momentum J [122–124]. Rotations are the lowest-energy internal excitations of the © 2009 by Taylor and Francis Group, LLC


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molecule. Here B is the rotational constant for the electronic-vibrational ground state, which is of the order of B ∼ h 10 GHz [125]. We denote the energy eigenstates of the Hamiltonian 12.16 by |J, M, where J is the quantum number associated with the total internal angular momentum and M is the quantum number associated with its projection onto a space-fixed quantization axis. The excitation spectrum is EJ = BJ(J + 1), which is anharmonic. Each J-level is (2J + 1)-fold degenerate. A polar molecule has an electric dipole moment d, which for Σ-state molecules is directed along the internuclear axis eab , that is, d = deab . Here, d is the “permanent” dipole moment of a molecule in its electronic-vibrational ground state. This dipole moment gives rise to the dipole–dipole interaction between two molecules. The spherical component dq = eq · d of the dipole operator in the space-fixed √ spherical basis {e−1 , e0 , e1 }, with eq=0 ≡ ez and e±1 = ∓(ex ± iey )/ 2, couples the rotational states |J, M and |J ± 1, M + q according to J ± 1, M + q|dq |J, M = d(J, M; 1, q|J ± 1, M + q) 2J + 1 × (J, 0; 1, 0|J ± 1, 0) , 2(J ± 1) + 1 where (J1 , M1 ; J2 , M2 |J, M) are the Clebsch–Gordan coefficients. This means that for a spherically symmetric system the eigenstates of the rotor have no net dipole moment, J, M|d|J, M = 0. However, the dipole coupling to an external electric field breaks this spherical symmetry by aligning each molecule along the field’s direction. This induces an interaction between the rotational energy levels of the molecule, and a corresponding finite dipole moment in each rotational state, as explained below. 12.3.1.2

Coupling to External Electric Fields

The terms Hdc and Hac (t) in Equation12.15 are the electric dipole interactions of a molecule with an external dc electric field Edc = Edc ez directed along e0 ≡ ez , and with ac microwave fields Eac (t) = Eac e−iωt eq + c.c., which are linearly (q = 0) or circularly polarized (q = ±1) relative to ez , respectively. Here we have neglected the spatial dependence of Eac because in the following we are interested in dressing the rotational states of the molecules with microwave fields, whose wavelengths are of the order of centimeters, and thus much larger than the size of our system. Then, the terms Hdc and Hac (t) in Equation 12.15 read Hdc = −d · Edc = −d0 Edc , Hac (t) = −d · Eac (t) = −dq Eac e−iωt + h.c.

(12.17a) (12.17b)

In the presence of a single dc electric field Edc (Eac , ω⊥ = 0), the internal Hamiltonian is that of a rigid spherical pendulum [122] H = Hrot + Hdc = BJ2 − d0 Edc which conserves the projection of the angular momentum J on the quantization axis, that is, M is a good quantum number. Thus, the energy eigenvalues and eigenstates are labeled EJ,M and |φJ,M , respectively, where each eigenstate |φJ,M is a superposition © 2009 by Taylor and Francis Group, LLC

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of various states |J, M mixed by the electric dipole interaction (here J, not conserved, is used as a simple label). The effects of a dc electric field on a single polar molecule are shown in Figure 12.6, and they amount to (1) a split in the (2J + 1)-fold degeneracy in the rotor spectrum, and (b) aligning the molecule along the direction of the field. The latter corresponds to inducing a finite dipole moment in each rotational state. For weak fields β ≡ dEdc /B 1, the state |φJ,M and its associated induced dipole moment approximately read √ β β J2 − M2 (J + 1)2 − M 2 |φJ,M = |J, M − |J − 1, M + |J + 1, M, 2 J 3 (2J + 1) 2 (J + 1)3 (2J + 1) (12.18) and φJ,M |d|φJ,M = dβ

3M 2 /J(J + 1) − 1 e0 , (2J − 1)(2J + 3)

respectively. Thus, the ground state acquires a finite dipole moment φ0,0 |d0 |φ0,0 = dβ/3 along the field axis, which is at the origin of ground-state dipole–dipole interactions between polar molecules. We notice that for a typical rotational constant B ∼ h 10 GHz and dipole-moment d ≈ 9 D, the condition β 1 corresponds to dc fields (much) weaker than B/d ≈ 2 kV/cm. Individual transitions of the internal Hamiltonian can be addressed by applying one (or several non-interfering) microwave field Eac (t), which can be linearly or circularly polarized. This is shown in Figure 12.6 for transitions coupling the J = 0 and J = 1 manifolds, where the Rabi frequency Ω and the detuning Δ are Ω ≡ Eac φ1,q |dq |φ0,0 / and Δ ≡ ω − (E1,q − E0,0 )/, respectively. Dressed energy levels of a molecule are obtained by diagonalizing the Hamiltonian H = Hrot + Hdc + Hac (t) in a Floquet picture. That is, first, the Hamiltonian matrix is evaluated in the 4basis |φJ,M , which diagonalizes the time-independent part of H as Hrot + Hdc = J,M |φJ,M EJ,M φJ,M |, and then the time-dependent wavefunction is expanded in a Fourier series in the ac frequency ω. After applying a rotating wave approximation, that is, keeping only the energy conserving terms, one obtains a time˜ whose eigenvalues correspond to the dressed energy independent Hamiltonian H, levels [61].

12.3.2 TWO MOLECULES We now consider the interactions of two polar molecules j = 1, 2 confined to the (x, y) plane by a tight harmonic trapping potential of frequency ω⊥ , directed along z. The interaction of the two molecules at a distance r ≡ r2 − r1 = rer is described by the Hamiltonian H(t) =

2

j=1


Hj (t) + Vdd (r),

(12.19)


439

where Hj (t) is the single-molecule Hamiltonian Equation 12.15, and Vdd (r) is the dipole–dipole interaction of Equation 12.4. In the absence of external fields Edc = Eac = 0, the interaction of the two molecules in their rotational ground state is determined by the van der Waals attraction Vvdw ∼ C6 /r 6 with C6 ≈ −d 4 /6B. This expression for the interaction potential is valid outside of the molecular core region r > rB ≡ (d 2 /B)1/3 , where rB defines the characteristic length at which the dipole–dipole interaction becomes comparable to the splittings of the rotational levels. In the following we show that it is possible to induce and design long-range interaction potentials, by dressing the interactions with appropriately chosen static and/or microwave fields. In fact, the combination of the latter with low-dimensional trapping allows us to engineer effective potentials whose strength and shape can both be tuned. The derivation of the effective interactions proceeds in two steps: 1. We derive a set of Born–Oppenheimer potentials by first separating Equation 12.19 into center-of-mass and relative coordinates, and diagonalizing the Hamiltonian Hrel for the relative motion for fixed molecular positions. Within an adiabatic approximation, the corresponding eigenvalues play the role of an effective three-dimensional interaction potential in a given state manifold dressed by the external field. 2. We eliminate the motional degrees of freedom in the tightly confined z-direction to obtain an effective two-dimensional dynamics with interaction 2D (ρ). Veff In the following we show how to design interaction potentials, presenting in some details the simplest case of a purely repulsive 1/r 3 potential in two-dimensional, obtained using a static electric field (Section 12.3.2.1). We then sketch how to design more elaborate potentials using a combination of static and microwave fields coupling the lowest rotor states of each molecule (Section 12.3.2.2). 12.3.2.1

Designing the Repulsive 1/r 3 Potential in 2D

Collisions in a dc field. We consider a weak static electric field applied in the z-direction E = Edc e0 with β = dEdc /B 1, and in the absence of an optical trapping potential (ω⊥ = 0). The effective interaction potentials for the collision of two particles can be obtained in the adiabatic approximation by neglecting the kinetic energy and by diagonalizing the following Hamiltonian Hrel for fixed particle positions Hrel =

2

BJj2 − Edc d0;j + Vdd (r) j=1

=

|Φn (r)En (r)Φn (r)|,

(12.20)

n

where En (r) and |Φn (r) are the nth adiabatic energy eigenvalues and two-particle eigenfunctions, respectively. In the limit r → ∞ the latter are symmetrized products © 2009 by Taylor and Francis Group, LLC

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of the single-particle states |φJj ,Mj j of Equation 12.18, while for finite r they are superpositions of several single-particle states, which are mixed by the dipole–dipole interaction Vdd (r). The quantity n ≡ (J; M; σ) is the collective quantum number labeling the eigenvalues En (r), with J = J1 + J2 being the total number of rotational excitations shared by two molecules, M ≡ |M1 | + |M2 | the total projection of angular momentum onto the electric field direction, and σ = ± the permutation symmetry associated with the exchange of the two particles. We note that, due to the presence of the dc field, J is not conserved here and used as a simple label for the various energy manifolds. Because we are mainly interested in collisions of molecules in the ground state, in the following we restrict our discussion to the Jj = 0 and 1 manifolds of each molecule, which include 16 rotational states of the combined two-particle system. Figure 12.7 shows the corresponding eigenvalues En (r) as a function of the interparticle distance r, for β = 1/5. The vector r is expressed in spherical coordinates r = (r, ϑ, ϕ), with ϑ and ϕ the polar and azimuthal angles, respectively, and z = r cos ϑ. Figure 12.7a shows that the energy spectrum has a markedly different behavior in the molecular core region r < rB and for r > rB , with rB ≡ (d 2 /B)1/3 . In fact, for r < rB the energy spectrum is characterized by a series of level crossings and anticrossings, so that the adiabatic approximation is generally not valid. For r > rB the energy levels group into well-defined manifolds approximately spaced by an energy 2B, corresponding to the energy of rotational excitation. In the following we focus on this region r > rB , where the adiabatic approximation can be used. Figures 12.7a through 12.7e present an expanded view of the two lowest-energy manifolds of Figure 12.7a in the region r > rB , for ϑ = π/2 and ϑ = 0, respectively. Figures 12.7b and 12.7d show that the excited-state manifold with one quantum of rotation (J1 + J2 = 1) is asymptotically split into two sub-manifolds. This separation corresponds to the electric-field-induced splitting of the Jj = 1 manifold of each molecule, and it is thus given by δ = 3Bβ2 /20 (see caption of Figure 12.6). More importantly, Figure 12.7c and 12.7e shows that the effective ground-state potential E0 (r) has a very different character for the cases ϑ = π/2 and ϑ = 0, respectively. In fact, in the case ϑ = π/2 (Figure 12.7c), corresponding to collisions in the (z = 0)plane, the potential is attractive for r < r , while for r > r it becomes repulsive and decays at large distances as 1/r 3 , where r is a characteristic length defined below. On the other hand, for ϑ = 0 (see Figure 12.7e) the potential is purely attractive, with dipolar character. This change in character of the ground-state potential as a function of ϑ is captured by the following analytic expression for E0;0;+ (r), derived using a perturbation expansion in Vdd (r)/B, 3D (r) ≡ E0;0;+ (r) ≈ Veff

C C3 6 1 − 3 cos2 ϑ + 6 . 3 r r

(12.21)

Here, the constants C3 ≈ d 2 β2 /9 and C6 ≈ −d 4 /6B are the dipolar and van der Waals coefficients for the ground-state potential, respectively, and the constant term 2E0,0 = −β2 B/3 due to single-particle dc Stark-shifts has been neglected. Equation 12.21 is 3D (r) has a local valid for r rB and Vdd (r)/B 1, and it shows that the potential Veff © 2009 by Taylor and Francis Group, LLC

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10–2

E2; M;+

2

– [EJ; M;s (r, p/2)–ħw–2E 0,0]/2B

(b)

EJ; M;s (r, 0)/2B

E1; 0;– E 1; 0;+

E2; M;– E1;1 ±;+

ħd

0 E1;1 ±;–

E1; 0;–

E1;1 ±;+

1 E1;1 ±;– 0

E1;0 ;+

–10–2

E0;0 ;+

(c)

1

rd

10

[EJ; M;s (r, p/2)–2E0,0]/2B

10–5 0.5

1

1.5

–10–5

ħd E1;1 ±;+

10–2 1

10

1 (e)

r

10

r/rB

[EJ; M;s (r, 0)–2E0,0]/2B

10–5

E0;0 ;+

0

E1;1 ±;– rd

E0;0 ;+

0

r/rB

– (d) [E J; M;s (r, 0)–ħw–2E0,0]/2B 10–2 E1; 0;– E1; 0;+ 0

r/rB

–10–5 r/rB

1

r

10

r/rB

FIGURE 12.7 Born–Oppenheimer potentials EJ;M;σ (r, ϑ) for two molecules colliding in the presence of a dc field, with β ≡ dEdc /B = 1/5. The solid and dashed curves correspond to symmetric (σ = +) and antisymmetric (σ = −) eigenstates, respectively. (a): Born–Oppenheimer potentials for the 16 lowest-energy eigenstates EJ;M;σ (r, ϑ). The molecular-core region is identified as the region r < rB = (d 2 /B)1/3 , while for r rB the eigenstates group into manifolds separated by one quantum of rotational excitation 2B. (b) and (d) Enlargements of the first excited energy manifold of panel (a) in the region r rB for ϑ = π/2 and ϑ = 0, respectively. Note the electric-field-induced splitting δ ≡ 3Bβ2 /20. The distance rδ where the dipole– dipole interaction becomes comparable to δ is rδ = (d 2 /δ)1/3 . (c) and (e) Enlargements of the ground-state potential E0,0;+ (r, ϑ) of panel (a) in the region r rB for ϑ = π/2 and ϑ = 0, respectively. The distance r of Equation 12.22, where the dipole–dipole interaction becomes comparable to the van der Waals attraction is indicated. Note the repulsive (attractive) character of the potential for ϑ = π/2 (ϑ = 0) and r > r . (From Micheli, A., et al., Phys. Rev. A, 76, 043604, 2007. With permission.)

maximum in the plane z = r cos ϑ = 0 at the position r , defined as 1/3 2|C6 | 1/3 3d 2 r ≡ ≈ , C3 Bβ2

(12.22)

where the dipole–dipole and van der Waals interactions become comparable. The height of this maximum is V =


C3 2 Bβ4 ≈ , 4|C6 | 54

(12.23)

442


and the curvature of the potential along z [∂z2 V (r = r , z = 0) = −6C3 /r5 ≡ −mω2c /2] defines a characteristic frequency ωc ≡

12C3 mr5

1/2 ,

(12.24)

to be used below. The latter has a strong dependence β8/3 = (dEdc /B)8/3 on the applied electric field. For distances r r the dipole–dipole interaction dominates 3D (r) ∼ C (1 − 3 cos2 ϑ)/r 3 (see over the van der Waals attractive potential, and Veff 3 Ref. [46]). Thus, if it were possible to confine the collisional dynamics to the (z = 0)plane with r r , rB , purely repulsive long-range interactions with a characteristic dipolar spatial dependence ∼1/r 3 could be attained. In the following we analyze the conditions for realizing this setup, using a strong confinement along z as provided, for example, by an optical trapping potential. Parabolic confinement. The presence of a finite trapping potential of frequency ω⊥ in the z-direction provides for a position-dependent energy shift of Equation 12.21. The new total potential reads V (r) =

C C3 1 6 2 ϑ + 6 + mω2⊥ z2 . 1 − 3 cos 3 4 r r

(12.25)

As noted before, for z = 0 the repulsive dipole–dipole interaction dominates over the attractive van der Waals potential at distances r r given by Equation 12.22. In addition, for ω⊥ > 0 the harmonic potential confines the particle’s motion in the z-direction. Thus, the combination of the dipole–dipole interaction and the harmonic confinement yields a repulsive potential that provides for a three-dimensional barrier separating the long-distance from the short-distance regime. If the collision energy is much smaller than this barrier, the particles’ motion is confined to the long-distance region, where the potential is purely repulsive. Figure 12.8 is a contour plot of V (r) in units of V , for β > 0 and ω⊥ = ωc /10, with r ≡ (ρ, z) = r(sin ϑ, cos ϑ) (the angle ϕ is neglected due to the cylindrical symmetry of the problem). Darker regions correspond to a stronger repulsive potential, while the white region at ρ ≈ 0 is the short-range, attractive part of the interaction. The repulsion due to the dipole–dipole and harmonic potentials is distinguishable at |z|/r ∼ 0 and 7, respectively. The less dark regions located at (ρ⊥ , ±z⊥ ) ≡ ⊥ (sin ϑ⊥ , ± cos ϑ⊥ ) correspond to the existence of two saddle points positioned in between √ the maxima of 2 1/5 3 V (r), with ⊥ = (12C3 /mω⊥ ) and cos ϑ⊥ = 1 − (r /⊥ ) / 5, (see circles in Figure 12.8a). These saddle points act as an effective potential barrier separating the attractive part of the potential present at r < l⊥ from the region r ⊥ ≥ r , rB where the effective interaction potential given by Equation 12.25 is purely repulsive. For collision energies smaller than the height of this barrier the dynamics of the particles can be reduced to quasi two-dimensional, by averaging over the fast particle motion in the z-direction. We notice that the existence of two saddle points at distances r ∼ ⊥ separating the long-distance from the short-distance regimes is a general feature of systems with a comparatively weak transverse trapping potential ω⊥ /ωc < 1, with ωc defined in Equation 12.24. In fact, for a strong transverse trapping potential ω⊥ ≥ ωc © 2009 by Taylor and Francis Group, LLC

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(b) 7

+5

5.78

6

0.01

0

SE/S0

5 0.01

0 –z^

0.01

4 3

7.01(w^/wc)1/5

2 1

–5 0

r^

0 10 r/r

15

20

10–6 10–5 10–4 10–3 10–2 10–1 100 10+1 10+2 w^/wc

FIGURE 12.8 (a) Contour plot of the effective potential V (ρ, z) of Equation 12.25, for two polar molecules interacting in the presence of a dc field β > 0, and a confining harmonic potential in the z-direction, with trapping frequency ω⊥ = ωc /10, where ωc ≡ (12C3 /mr5 )1/2 of Equation 12.24 and r = (2|C6 |/C3 )1/3 of Equation 12.22. The contour lines are shown for V (ρ, z)/V ≥ 0, with V = Bβ4 /54. Darker regions represent stronger repulsive interactions. The combination of the dipole–dipole interactions induced by the dc field and of the harmonic confinement leads to realizing a three-dimensional repulsive potential. The repulsion due to the dipole–dipole interaction and the harmonic confinement is distinguishable at z ∼ 0 and z/r ∼ ±7, respectively. Two saddle points (circles) located at (ρ⊥ , ±z⊥ ) separate the longdistance region where the potential is repulsive ∼ 1/r 3 from the attractive short-distance region. The gradients of the potential are indicated by dash–dotted lines. The thick dashed line indicates the instanton solution for the tunneling through the potential barrier. (b) The euclidian action SE as a function of ω⊥ /ωc (solid line). For ω⊥ < ωc ≈ 0.88 ωc (ω⊥ > ωc ) the “bounce” occurs for z(0) = 0 (within the plane z(0) = 0), see text. ωc is signaled by a circle. √ The point 2 For ω⊥ > ωc the action is SE ≈ 5.78S0 , with S0 = m|C6 |/r , which is ω⊥ -independent, consistent with the “bounce” occurring in the (z = 0)-plane (see text). (From Micheli, A., et al., Phys. Rev. A, 76, 043604, 2007. With permission.)

the two saddle points collapse into a single one located at z = 0, and ρ = ⊥ ∼ r . In this limit the dynamics is purely two-dimensional, with the particles strictly confined to the (z = 0)-plane. Collisional stability. Inelastic collisions and three body recombination in an ensemble of polar molecules may lead the system to a potential instability, associated with the attractive character of the dipole–dipole interaction [62–70]. In our discussion, this instability is associated with the population of the short-distance region r < ⊥ , which can be efficiently suppressed by strong dipole–dipole interactions and transverse confinement. In fact, for collision energies smaller than the potential barrier V (ρ⊥ , ±z⊥ ) the particles are mostly confined to the long-distance regime, where they scatter elastically. That is, when a cold ensemble of molecules is considered the barrier ensures the stability of the system by “shielding” the short-distance attractive part of the two-body potential. In this limit, residual losses are due to the tunneling through the potential barrier at a rate Γ, which can be effectively suppressed for reasonable values of β and ω⊥ , as shown below. The tunneling rate Γ = Γ0 e−SE / through the barrier V (ρ⊥ , ±z⊥ ) can be calculated using a semi-classical/instanton approach [126]. The euclidian action SE , © 2009 by Taylor and Francis Group, LLC

444


which is responsible for the exponential suppression √ of the tunneling, is plotted in Figure 12.8 as a function of ω⊥ /ωc , in units of S0 = m|C6 |/r2 . The figure shows that the form of SE is different for ω⊥ ωc and ω⊥ ωc . In fact, for ω⊥ ωc the action increases with increasing ω⊥ as SE ≈ 7.01S0 (ω⊥ /ωc )1/5 = 1.43(⊥ /a⊥ )2 (dotted line), which depends on the confinement along z, via a⊥ = (/mω⊥ )1/2 . On the other hand, for ω⊥ ωc it reads SE ≈ 5.78 S0 , which is ω⊥ -independent. The transition between the two different regimes mirrors the change in the nature of the underlying potential V (r). In particular, for ω⊥ ωc the dynamics is strictly confined to the plane z = 0 and thus it becomes independent of ω⊥ . The constant Γ0 is related to the quantum fluctuations around the semiclassical trajectory, and its value is strongly system-dependent. For the crystalline phase of Ref. [46], it is the collisional “attempt frequency,” proportional to the characteristic phonon frequency Γ0 ≈ C3 /ma5 , where a is the mean interparticle distance. In the limit of strong interactions and tight transverse confinement Γ rapidly tends to zero. We illustrate this for the example of SrO molecules, which have a permanent dipole moment of d ≈ 8.9 D and mass m = 104 amu. Then, for a tight transverse optical lattice with harmonic oscillator frequency ω⊥ = 2π × 150 kHz and for a dc-field β = dEdc /B = 1/3 we have (C32 m3 ω⊥ /85 )1/5 ≈ 3.39 and obtain Γ/Γ0 ≈ e−5.86×3.39 ≈ 2 × 10−9 . Even for a dc field as weak as β = 1/6 we still obtain a suppression by five orders of magnitudes, as Γ/Γ0 ≈ e−5.86×1.94 ≈ 10−5 . This calculation confirms that a collisionally stable system of polar molecules in the strongly interacting regime can be realized by combining the strong dipole–dipole interactions with a tight transverse confinement. Effective two-dimensional interaction. The effective two-dimensional interaction potential is obtained by integrating out the fast particle motion in the transverse direction z. For r > ⊥ a⊥ , the two-particle eigenfunctions in the z-direction approximately factorize into products of single-particle harmonic oscillator wavefunctions 2D /ω , the effective two-dimensional ψk1 (z1 )ψk2 (z2 ), and, to first order in Veff ⊥ 2D interaction potential Veff reads 2D Veff (ρ) ≈ √

1 2πa⊥

dze−z

2 /2a2 ⊥

3D Veff (r).

(12.26)

For large separations ρ ⊥ the two-dimensional potential reduces to 2D (ρ) = Veff

C3 , ρ3

which is a purely repulsive two-dimensional interaction potential. The derivation of 2D (ρ) is one of the central results of this section. We show below (Section 12.4) that Veff this interaction potential can be used to realize interesting many-body phases, in the context of condensed-matter applications with cold molecular quantum gases. 12.3.2.2

Designing ad hoc Potentials with ac Fields

We have shown how to design purely repulsive two-dimensional effective groundstate interactions that decay as ∼1/r 3 . The use of one or several noninterfering ac © 2009 by Taylor and Francis Group, LLC


445

fields allows us to engineer more complicated interactions by combining the spatial texture of the adiabatic ground-state potential of the two-particle spectrum with that of selected excited potentials, in a field-dressed picture. This mixing of ground- and excited-state potentials is favored by the dipole–dipole interactions that split the degeneracy of the excited-state manifolds of the two-particle spectrum and render the state-selectivity of the ac fields space-dependent, as explained below. In combination with a strong optical confinement, and due to the long lifetimes of excited rotational states [127], this allows for the realization of collisionally stable ensembles of molecules in a strongly interacting regime. We exemplify the situation above by considering the case of a single ac field Eac (t) = Eac e−iωt eq + c.c. which is added to the configuration of Figure 12.7 (interactions in the presence of a static electric field Edc = βBez ). The field’s polarization is chosen to be linear (q = 0) and the frequency ω is blue-detuned from the (|φ0,0 → |φ1,0 )-transition of the single-particle spectrum by an amount Δ = ω − 2B/ > 0. The ac field induces: (1) oscillating dipole moments in each molecule, which give rise to long-range dipole–dipole interactions (in addition to those determined by the static field Edc ), whose sign and angular dependence are given by the polarization q; (2) a coupling between the ground and excited-state manifolds of the two-particle spectrum at a resonant (Condon) point rC = (d 2 /3hΔ)1/3 , where the dipole–dipole interaction becomes comparable to the detuning Δ. This coupling is responsible for an avoided crossing of the field-dressed energy levels at rC , whose properties depend crucially on the polarization q. This fact is at the core of the engineering of interaction potentials, in that the three-dimensional effective dressed adiabatic ground-state interaction potential inherits the character of the bare ground and excited potentials for r rC and r rC , respectively. The setup described above is illustrated in Figure 12.9a and 12.9b, where the continuous and dashed lines are the bare (Eac = 0) symmetric and antisymmetric potentials EJ;M;σ (r) of Figure 12.7, respectively, and the presence of the ac field is signaled by a black arrow at the resonant Condon point rC . The presence of the weak dc field splits the (J = 1)-manifold asymptotically by an amount δ as in Figure 12.7b, which makes it possible the apply to adiabatic approximation in the excited-state manifold for distances r rδ = (d 2 /δ)1/3 . In fact, the energy of the E1;0;+ (r) potential becomes degenerate with the energy of other bare symmetric potentials only at distances r rδ . In addition, we notice that the presence of the splitting δ also shifts the level crossing with antisymmetric states to small distances r rδ . For distances r rδ = (d 2 /δ)1/3 , we are allowed to consider only the four states of Figure 12.9b, because all the other potentials of the (J = 1)- and (J = 2)-manifolds are far detuned by an amount that is (at least) of order δ Δ and they are not coupled by the ac field to the bare ground state E0,0;+ (r), due to the choice of the field’s polarization. Figure 12.9b shows that the splitting induced by the dipole–dipole interaction in the (J = 1)-manifold renders the detuning Δ position-dependent, so that at rC the energy of the bare ground state and that of the symmetric bare excited state become degenerate. The resulting dressed ground-state potential is outlined in Figure 12.9b (bold black line) and it roughly corresponds to the bare E0,0;+ (r) and E1,0;+ (r) potentials for r > rC and r < rC , respectively. Accordingly, Figure 12.9c shows that the dressed ground-state potential E˜ 0;0;+ (r), which has the highest energy, © 2009 by Taylor and Francis Group, LLC

446


(b) [EJ; M;s (r,p/2)–2E0,0]/ħ E2;0;+

– + 4d/3 2w

[EJ; M;s (r,p/2)–2E0,0]/ħ

– 2w +4d/3

E2;0;+

E2;1 ±;±

– + d/3 2w

w +2D

E2;2m;+

– + 2d/3 2w

w +D

– + 5d/3 w

w – w +2d/3

E1;0;±

– + 2d/3 w –– d/3 w w – 4d/3

d

w –2D

E1;1 ±;±

w –2D

W

E1;0;– W

w –4D E0;0;±

0 0

E0;0;+

0

10 20 30 40 50 rc rd r/rB

(c)

E1;0;+

D

0 10 20 30 40 50 60 70 80 90 100 rd rc r/rB

~ [EJ; M;s (r,p/2)–2E0,0]/ħD

2

1 ~ E0;0;+

÷8W

0

~ E1;0;–

–1

~ E1;0;+

~ E2;0;+

–2 0

10 20 30 40 50 60 70 80 90 100 rc r/rB

FIGURE 12.9 (a) Schematic representation of the effects of dc and ac microwave fields on the interaction of two molecules. The solid and dashed lines are the bare potentials En (r) ≡ EJ;M;σ (r, ϑ) of Section 12.3.2.1 with ϑ = π/2 for interactions in the presence of the dc field only, for the symmetric (σ = +) and antisymmetric (σ = −) states, respectively. The dc field induces a splitting δ of the first-excited manifold of the two-particle spectrum. A microwave field of frequency ω = ω + 2δ/3 + Δ is blue-detuned by Δ > 0 from the single-particle rotational resonance. The dipole–dipole interaction further splits the excitedstate manifold, making the detuning space-dependent. Eventually, the combined energy of the bare ground-state potential E0;0;+ (r) and of an ac photon (black arrow) becomes degenerate with the energy of the bare symmetric E1;0;+ (r, π/2). The resonant point rC = (d 2 /3Δ)1/3 occurs at r ≈ 46 rB . (b) Enlarged view of the potentials of panel (a) with M = 0. The dressed ground-state potential is indicated by a thick solid line. (c) The four potentials of panel (b) in the field-dressed picture. The dressed ground-state potential E˜ 0;0;+ (r, π/2) has the largest energy and is indicated by a thick solid line. (From Micheli, A. et al., Phys. Rev. A, 76, 043604, 2007. With permission.)

turns from weakly to strongly repulsive for r rC and r rC , respectively. This change in the character of the ground-state interaction potential corresponds to the design of a “step-like” interaction. This example shows that the three-dimensional ground-state interaction for two molecules can be strongly modified by combined ac © 2009 by Taylor and Francis Group, LLC


447

and dc fields, which is the central result of this section. More complicated potentials can be engineered using multiple ac fields and different polarizations. Analogous to the case (Eac = 0) of Section 12.3.2.1, the interaction potential of Figure 12.9 is actually repulsive along certain directions (e.g., for θ = π/2, as shown in the figure), while it becomes attractive at other configurations (e.g., for θ = 0, not shown). As for the (Eac = 0) case of Section 12.3.2.1, when more than two particles are considered this attraction can lead to many-body instabilities. Moreover, here the dressed potential E˜ 0;0;+ (r) of Figure 12.9c is not the lowest-energy potential, which in general can introduce additional loss channels. The latter correspond to diabatic couplings to symmetric states for particles approaching distances r rC , and are therefore present even in the simple two-particle collision process, and to couplings to antisymmetric states, which can be induced, for example, by three-body collisions or by non-compensated tensor shifts for two optically trapped particles. The presence of all of these loss channels may render impractical the realization of collisionally stable setups for strongly interacting molecular gases (see, however, Ref. [99] for a solution involving the use of a circularly polarized ac field). At the same time, we have shown above that for the system depicted in Figure 12.9 the presence of the static field shifts the various resonance points with the potentials that are responsible for these loss channels in the (ϑ = π/2)-plane (z = 0) to distances r rC . This suggests that by confining the motion of the particles to the plane z = 0 using a strong optical transverse confinement analogous to that of Section 12.3.2.1 it may be possible to realize collisionally stable systems in the region r > rC . This scheme has been shown to work in Ref. [61] and thus the main message here is that a judicious combination of the dipole–dipole interactions and of the optical confinement can act as an effective “shield” of the region r < rC where losses occur and the collision system may be stable. The use of the step-like potential above and of other engineered potentials can lead to the realization of interesting phases for an ensemble of polar molecules in the strongly interacting regime [46,61].

12.4

MANY-BODY PHYSICS WITH COLD POLAR MOLECULES

12.4.1 TWO-DIMENSIONAL SELF-ASSEMBLED CRYSTALS The above discussion of the intermolecular potentials and of the stability of collision systems in reduced dimensionality provides the microscopic justification for studying an ensemble of polar molecules in two-dimensions interacting via (modified) dipole– dipole potentials. At low temperatures T < ω⊥ , the general many-body Hamiltonian has the form of Equation 12.5. As an example of the possibilities offered by potential engineering to realize novel many-body quantum phases, we here focus on bosonic 2D (ρ) = D/ρ3 of Equation 12.6, particles interacting via the effective potential Veff derived in Section 12.3.2.1. The Hamiltonian of Equation 12.5 gives rise to novel quantum phenomena, which have not been accessed so far in the context of cold neutral atoms and molecules. The phase diagram for the two-dimensional system of bosonic dipoles is outlined in Figure 12.2a. In the limit of weak interactions rd < 1, the ground state is a superfluid (SF) characterized by a finite superfluid fraction ρs (T ), which depends © 2009 by Taylor and Francis Group, LLC

448


on temperature T with ρs (T = 0) = 1. In the opposite limit of strong interactions rd 1 the polar molecules are in a crystalline phase for temperatures T < Tm with Tm ≈ 0.09D/a3 ! 0.018rd ER,c , with ER,c ≡ π2 2 /2ma2 the crystal recoil energy, typically a few to tens of Kilohertz. The configuration with minimal energy is thus a triangular lattice with spacing aL = (4/3)1/4 a. Excitations of the crystal are acoustic phonons with the Hamiltonian given by Equation 12.13, and characteristic Debye √ frequency ωD ∼ 1.6 rd ER,c . At T = 0 the static structure factor S diverges at a reciprocal lattice vector K, and thus S(K)/N acts as an order parameter for the crystalline phase. In Ref. [46] we investigated the intermediate strongly interacting regime with rd 1, and we determined the critical interaction strength rQM for the quantum phase transition between the superfluid and the crystal. In our analysis we used a recently developed PIMC-code based on the Worm algorithm [95], which is an exact Monte (b) 0 50

7

100

4 3 2

3D

1

6

50

2D crystal

5

0

r qm

3 2 2D superfluid

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a/1 mm

rs

(d) 1 0.8 0.6 0.4 0.2 0

4

5 x/a

0.5 1

0 1

0

N = 36

(e) 0.8

N = 36

N = 90

0.6

N = 90

S(K)/N

0

3

1

1 0.1

0

2

(c)

5

1

1

0

4

g2

(d/÷m) / (D/÷ 200 amu)

8

y/a

(a) 9

2 r/d

0.4 0.2 0

5

10 15 20 25 rd

5

10 15 20 25 rd

FIGURE 12.10 (a) Quantum phases of two-dimensional dipoles. Contour plot of the interaction strength rd = Dm/2 a as a function of the dipole moment d (in Debye) and of the interparticle distance a (in μm), with m the mass of a molecule (in atomic units 200 × amu). The regions of stability of the two-dimensional superfluid and crystalline phases where ω⊥ > D/a3 are indicated, with ω⊥ = 2π × 150 kHz the frequency of the transverse confinement. (b) PIMC-snapshot of the mean particle positions in the crystalline phase for N = 36 at rd ≈ 26.5. (c) Density–density (angle-averaged) correlation function g2 (r), for N = 36 at rd ∼ 11.8. (d) Superfluid density ρs and (e) static structure factor S(K)/N as a function of rd , for N = 36 (circles) and N = 90 (squares). (From Büchler, H.P. et al., Phys. Rev. Lett., 98, 060404, 2007. With permission.)



449

Carlo method for the determination of thermodynamic quantities in continuous space at small finite temperature. In Figure 12.10d and 12.10e, the order parameters ρs and S(K)/N are shown at a small temperature T = 0.014D/a3 for different interaction strengths rd and particle numbers N = 36, 90. We find that ρs exhibits a sudden drop to zero for rd ≈ 15, while at the same position S(K) strongly increases. In addition, during the Monte Carlo simulations we observed that in a few instances ρs suddenly increased from 0 to 1, and then returned to 0, in the interval rd ≈ 15–20, which suggests a competition between the superfluid and crystalline phases. These results indicate a superfluid to crystal phase transition at rqm = 18 ± 4.

(12.27)

The step-like behavior of ρs and S(K)/N is consistent with a first-order phase transition, a result that has been confirmed in Refs. [47] and [48]. Notice that the superfluid with rd ∼ 1 is strongly interacting, and in particular the density-density correlation function is quenched at lengths R < a (see Figure 12.10c). This observation is consistent with the validity condition of the effective two-dimensional interaction potential given in Equation 12.6, indicating that two particles never approach each other to distances smaller than l⊥ . Having determined the low-temperature phase diagram, the remaining question is whether these phases, and in particular the crystalline phase emerging at strong dipole–dipole interactions, are in fact accessible with polar molecules. This question is addressed in Figure 12.10a, which is a contour plot √ of the interaction strength rd as a function of the induced dipole moment d = D (in units of Debye) and of √ the mean interparticle distance a (in μm). The dimensionless quantity m/200 amu depends on the mass m of the molecules (in atomic units, au), and it is of order one for characteristic molecules like SrO or RbCs. In the figure, stable two-dimensional configurations for the molecules exist in the parameter region where the transverse (optical) trapping frequency ω⊥ = 2π × 150 Hz exceeds the dipole–dipole interaction (⊥ ω > D/a3 ), such that l⊥ = (12D/mω2⊥ )1/5 < a, consistent with the stability discussion of Section 12.3.2.1 (notice that l⊥ ∼ (D/ω⊥ )1/3 for realistic parameters). The figure shows that for a given induced dipole d the ground state of an ensemble of polar molecules is a crystal for mean interparticle distances l⊥ a amax , where amax ≡ d 2 m/2 rqm corresponds to the distance at which the crystal melts into a superfluid. For SrO (RbCs) molecules with the permanent dipole moment d = 8.9 D (d = 1.25 D), amin ≈ 200 nm (100 nm), while amax can be several micrometers. Since for large enough interactions the melting temperature Tm can be of the order of several microkelvin, the self-assembled crystalline phase should be accessible for reasonable experimental parameters using cold polar molecules. The zero-temperature phases can be observed using Bragg scattering with optical light, which allows for probing the crystalline phase. The detection of vortices can be used as a definitive signature of superfluidity. We notice that the two-dimensional (quasi) condensate involves a fraction of the total density only, and therefore we expect only small coherence peaks in a time-of-flight experiment. Finally, we notice that by adding an additional in-plane optical confinement, it may be possible to realize strongly interacting one-dimensional phases © 2009 by Taylor and Francis Group, LLC

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analogous to the two-dimensional crystals discussed above [49,50,91]. For large enough interactions r 1, the phonon frequencies have the simple form ωq = 1/2 4 ER,c , with fq = j>0 4 sin(qaj/2)2 /j5 . The Debye frequency (2/π2 ) 12rd fq √ is ωD ≡ ωπ/a ∼ 1.4 rd ER,c , while the classical melting temperature can be estimated to be of the order of Tm ! 0.2rd ER,c /kB [91].

12.4.2

FLOATING LATTICES OF DIPOLES

An interesting possibility offered by the realization of the self-assembled crystals discussed above is to utilize them as floating mesoscopic lattice potentials to trap other particles, which can be atoms or polar molecules of a different species. We show below that within an experimentally accessible regime of parameters extended Hubbard models with tunable long-range phonon-mediated interactions describe the effective dynamics of the extra-particles dressed by the lattice phonons. The systems that we have in mind are shown in Figures 12.5, 12.11a, and 12.11b, where extra particles confined to a two-dimensional 4 crystal plane or a one-dimensional tube scatter from the periodic lattice potential j Vcp (Rj − r). Here, r and Rj = Rj0 + uj are the coordinates of the particle and crystal molecule j, respectively, with Rj0 the (a)

dc

a

b

Vcp ~ J dp

~ Vi, j

(b) Vcp

~ J

dc

~ Vi, j

a

~ Vi, i

FIGURE 12.11 A dipolar crystal of polar molecules provides a periodic lattice Vcp for extra atoms or molecules giving rise to a lattice model with hopping J˜ and long-range interactions V˜ i,j (see text and Figure 12.5). (a) A one-dimensional dipolar crystal with lattice spacing a provides a periodic potential for a second molecular species moving in a parallel tube at distance b (Configuration 1). (b) One-dimensional setup with atoms scattering from the dipolar lattice (configuration 2). (From Pupillo, G. et al., Phys. Rev. Lett., 100, 050402, 2008. With permission.)



451

equilibrium positions and uj small displacements. If the extra-particles are molecules, this potential is given by the repulsive dipole–dipole interaction Vcp (Rj − r) = dp dc /| Rj − r |3 with dp dc the induced dipole moment, and for atoms we assume that the interaction is modeled by a short range pseudopotential proportional to an elastic scattering length acp . In addition, extra molecules and atoms will interact via dipolar, or short-range interactions, respectively. We are interested in a situation where the extra particles in the lattice are described by a single band Hubbard Hamiltonian coupled to the acoustic phonons of the lattice as given in Equation 12.12 [128]. In the latter equation, the first and second terms describe the nearest-neighbor hopping of the extra-particles with hopping amplitudes J, and interactions V , computed for each microscopic model by band-structure calculations for uj = 0, respectively. The third term is the phonon Hamiltonian. The fourth term is the phonon coupling obtained in lowest order in the displacement uj = i

0 † (/2mc Nωq )1/2 ξq (aq + a−q )eiq·Rj , q

with Mq = V¯ q q · ξq (/2Nmc ωq )1/2 βq . Here, ξq and N are the phonon polarization and the number of lattice molecules, ¯ the Fourier transform of the particle–crystal interaction Vcp , respectively, while Vq 2is iqr and βq = dr|w0 (r)| e , with w0 (r) the Wannier function of the lowest Bloch band [128]. The validity of the single-band Hubbard model requires that J, V < Δ, and the temperature kB T < Δ with Δ the energy gap to the first excited Bloch band. The Hubbard parameters of Equation 12.12 are of the order of magnitude of the recoil energy, J, V ≈ ER,c , which is (much) smaller than the Debye frequency √ ωD ∼ ER,c rd , for rd 1 [129]. This separation of timescales J, V ωD , combined with the fact that the coupling to phonons is dominated by high frequencies ω > J, V (see the discussion of Mq below) is reminiscent of polarons as particles dressed by (optical) phonons, where the dynamics is given by coherent and incoherent hopping on a lattice [117,128]. This physical picture is brought out in a master equation treatment within a strong coupling perturbation theory. The starting point is a Lang–Firsov transformation of the Hamiltonian H → SHS † with a density-dependent displacement ⎡

⎤

Mq iqR0 † † ⎦. S = exp⎣− e j cj cj aq − a−q ωq q,j

This eliminates the phonon coupling in the second line of Equation 12.12 in favor of a transformed kinetic energy term −J


ci† cj Xi† Xj ,

452


where the displacement operators ⎡

⎤

Mq iqR0 † ⎦ Xj = exp⎣ e j aq − a−q ω q q can be interpreted as a lattice recoil of the dressed particles in a hopping process. In addition, the bare interactions are renormalized according to (1) V˜ ij = Vij + Vij ,

4 (1) with Vij = −2 q cos(q(Ri0 − Rj0 ))Mq2 /ωq , that is, the phonon couplings induce and modify off-site interactions. The onsite interaction is given by V˜ j,j = Vj,j − 2Ep with Ep =

Mq2 q

ωq

,

the polaron self-energy or polaron shift. For J = 0 the new Hamiltonian is diagonal and describes interacting polarons and independent phonons. The latter are vibrations of the lattice molecules around new equilibrium positions with unchanged frequencies. A stable crystal requires the variance of the displacements Δu around these new equilibrium positions to be small compared to a. A Born–Markov approximation with the phonons the finite-temperature heatbath with J, V ωD (see above), and the transformed kinetic energy −J

ci† cj (Xi† Xj − Xi† Xj )

as the system-bath interaction with Xi† Xj the equilibrium bath average, provides the master equation for the reduced density operator of the dressed particles ρt in the Lindblad form [130]

Γδ,δ i j,l ˜ + ρ˙ t = [ρt , H] [bjδ , ρt blδ ] + [blδ , ρt bjδ ] , 2

(12.28)

j,l,δ,δ

† with bjδ = cj+δ cj . The effective system Hamiltonian H˜ takes the form of Equation 12.14, which is of the extended Hubbard type, valid for J˜ , V˜ ij , Ep < Δ. Coherent hopping of the dressed particles is then described by

J˜ = JXi† Xj ≡ J exp(−ST ),



where ST =

453

Mq 2 [1 − cos(qa)](2nq (T ) + 1) ωq q

characterizes the strength of the particle-phonon interactions, and nq (T ) is the thermal occupation at temperature T [128]. The dissipative term in Lindblad form in Equation 12.28 corresponds to ther mally activated incoherent hopping with rates Γδ,δ j,l , which can be made negligible for the energies of interest kB T min(Δ, Ep , kB TC ) [117,131]. Corrections to Equation 12.14 proportional to J 2 are small relative to H˜ provided J Ep [117] (also J ωD in one-dimensional [131,132]). Thus, in the parameter regime of interest the dynamics of the dressed particles is described by the extended Hubbard Hamil˜ In the following, we verify the existence of this parameter regime and tonian H. calculate the effective Hubbard parameters from the microscopic model for the onedimensional configuration of Figure 12.11a, where extra-particles are polar molecules of a different species. An analogous calculation for the configuration of Figure 12.11b is reported in Ref. [53]. In the configuration of Figure 12.11a molecules of a second species are trapped in a tube at a distance b from the crystal tube under one-dimensional trapping conditions. For crystal molecules fixed at the equilibrium positions with lattice spacing a, the extra particles feel a periodic potential Vcp (x) = dc dp

−3/2

b2 + (x − ja)2 , j

which determines the band structure. The lattice depth V0 ≡ Vcp (a/2) − Vcp (0) ∼ rd

dp mp e−3b/a ER,p dc m (b/a)3

is shown in Figure 12.12a as a function of b/a, where the thick solid lines indicate the parameter regime 4J < Δ, and ER,p = 2 π2 /2mp a2 . The potential is comb-like for b/a < 1/4, because the particles resolve the individual molecules forming the crystal, while it is sinusoidal for b/a 1/4. The strong dipole–dipole repulsion between the extra-particles acts as an effective hardcore constraint [85]. We find that for 4J < Δ and dp dc the bare off-site interactions satisfy Vij ∼ dp2 /(a|i − j|)3 < Δ, which justifies a single-band approximation for the dynamics of the extra-particles in the static potential. The particle–phonon coupling is dc dp Mq = ab


2 Nmc ωq

1/2 q2 K1 (b|q|)βq

454


V0 /ER,p

S0, E0/J

(b)

rd = 500

100 100

rd = 500

10 1

10

0.1 1

rd = 50

0.01

rd = 50

0.1 b/a

0 (1)

(c)

0

1

Vj, j+1/ER,p

b/a

1

(d)

0

rd 1

10

200

rd = 500

1

100 rd = 50

50

0.1

–0.1

0

20

–0.5 1

b/a

0.2

0.4

0

0.5

0

10

b/a

0.8

FIGURE 12.12 Configuration 1 (Figure 12.11a): Hubbard parameters for dp /dc = 0.1 and m = mp . (a) Lattice depth V0 in units of ER,p vs. b/a for rd = 50 and 500. Thick continuous lines: tight-binding region 4J < Δ. (b) Reduction factor S0 (dashed–dotted lines) and polaron shift Ep /J (solid lines), for 4J < Δ. (c) Continuous lines: phonon-mediated (1) interactions V . Horizontal (dashed) lines: Vj,j+1 . (d) Contour plot of V˜ j,j+1 /2J˜ (solid j,j+1

lines) as a function of b/a and rd . A single-band Hubbard model is valid left of the dashed region (4J˜ , V˜ ij < Δ), and right of the black region (Ep < Δ).

√ with K1 being the modified Bessel function of the second kind, and Mq ∼ q for q → 0. In the regime of interest b/a < 1 where the single-band approximation is valid (4J, Vij < Δ), we find that Mq is peaked at large q ∼ π/a, so that the main contribution to the integrals in the definition of ST and Ep is indeed dominated by large frequencies ωq > J. Together with the separation of timescales J, Vij ωD , this is consistent with the picture of the system’s dynamics as given by particles dressed by fast (optical) phonons, as discussed above. We note that this so-called antiadiabatic regime is generally hard to achieve in cold atomic setups [129]. A plot of S0 as a function of b/a is shown in Figure 12.12b. We find the scaling S0 ∝

√ rd (dp /dc )2 ,

and within the regime of validity of the single-band approximation, S0 can be tuned from S0 1 (J˜ ∼ J) to S0 1 (J˜ J) corresponding to the large and small polaron limit, respectively. The polaron shift Ep generally exceeds the bare hopping rate J, © 2009 by Taylor and Francis Group, LLC


455

and in particular, Ep J for S0 1; see Figure 12.12b. Together with the condition ωD J this ensures that the corrections to Equation12.14 which are proportional to J 2 , are indeed small, and thus Equation12.14 fully accounts for the coherent dynamics of the dressed particles. The extended Hubbard model corresponding to the configuration of Figure 12.11a is characterized by tunable off-site interactions, which are a combination of the direct dipole–dipole interactions between the extra-particles and of the phonon-mediated (1) (1) interactions Vi,j . For b/a 1/4 we find that the interactions Vi,j decay slowly with (1)

the interparticle distance as ∼1/|i − j|2 , and are thus long-ranged. The sign of Vi,j is a function of the ratio b/a. Thus, depending on b/a the phonon-mediated interactions can enhance or reduce the direct dipole–dipole repulsion of the extra particles. As an (1) example, Figure 12.12c shows that the sign of the term Vj,j+1 alternates between attractive and repulsive as a function of the ratio b/a, and that for small enough values of b/a the phonon-mediated interactions can become larger than the direct dipole–dipole interactions. The effective Hubbard parameters V˜ j,j+1 and J˜ are summarized in Figure 12.12d, which is a contour plot of V˜ j,j+1 /2J˜ as a function of rd and b/a. The ratio V˜ j,j+1 /2J˜ increases as b/a decreases or rd increases, and can be much larger than one. This appearance of strong off-site interactions in the effective dynamics is a necessary ingredient for the realization of a variety of new quantum phases [78–82]. As an example of the possible quantum phases that can be realized in this setup, at half filling, and considering nearest-neighbor interactions only, the particles in the configuration described above undergo a transition from a (Luttinger) liquid (V˜ i,i+1 < 2J˜ ) to a charge-density wave (V˜ i,i+1 > 2J˜ ) as a function of b/a and rd . Figure 12.12d shows that the parameter regime V˜ i,i+1 ≈ 2J˜ (see Ref. [133]), where this transition occurs can be satisfied for various choices of rd and b/a, for example, for rd = 100 and b/a ≈ 0.5.

12.4.3 THREE-BODY INTERACTIONS As discussed in Section 12.2, it is of interest to design systems where effective manybody interactions dominate over two-body interactions, and determine the properties of the system in the groundstate. Here we describe how an effective low-energy 3D of the form given in Equation 12.2 can be derived in the interaction potential Veff Born–Oppenheimer approximation for 1 Σ polar molecules interacting via dipole– dipole interactions by dressing low-energy rotational states of each molecule with external static and microwave fields, in analogy to the discussion of Section 12.3.2. We here focus on a system of molecules in a static electric field E = Eez directed along the z-axis (Figure 12.13), where the two states |gi ≡ |φ0,0 i and |e+ i ≡ |φ1,+1 i with energies Eg and Ee,± are coupled by a circularly polarized microwave field propagating along the z-axis. The microwave transition is characterized by the (blue) detuning Δ > 0 and the Rabi frequency Ω/. While the following discussion can be readily generalized to include the degenerate case [85], here we assume that the degeneracy of the states |e− ≡ |φ1,−1 i and |e+ is lifted, for example, by an additional microwave field coupling the state |e− near-resonantly to © 2009 by Taylor and Francis Group, LLC

456


(a)

(b)

˙ e Òi Ee

6B 2B

D

Ee , ± Wd

˙ e+ Òi ˙ e Òi

2B

D ˙ e– Ò i

W

˙ e+ Òi

0

W

0

1

2

3

dE/B

Eg

Eg ˙ g Òi

˙ g Òi

FIGURE 12.13 Spectrum of a polar molecule. (a) Level structure for Ed/B = 3: the circular polarized microwave field couples the ground state |g with the excited state |e+ with Rabi frequency Ω/ and detuning Δ. The excited state |e+ is characterized by a finite angular momentum Jz |e+ = |e+ . Applying a second microwave field with opposite polarization and Rabi frequency Ωd / allows us to lift the degeneracy in the first excited manifold by resonantly couple the state |e− to the next manifold. (b) Internal excitation energies for a single polar molecule in a static electric field E = Eez . (From Büchler, H.P. et al., Nature Phys., 3, 726, 2007. With permission.)

the next state manifold (see Figure 12.13). Then, the internal structure of a single polar molecule reduces to a two-level system and is described as a spin-1/2 particle via the identification of the state |gi (|e+ i ) as eigenstate of the spin operator Siz with positive (negative) eigenvalue. Using the rotating frame and the rotating wave approximation, the Hamiltonian describing the internal dynamics of the polar molecule can be written as (i)

H0 =

1 Δ 2 Ω

Ω = hSi −Δ

(12.29) y

with the effective magnetic field h = (Ω, 0, Δ) and the spin operator Si = (Six , Si , Siz ). The eigenstates of this Hamiltonian √ are denoted as |+i = α|gi + β|e+ i and |−i = −β|gi + α|e+ i with energies ± Δ2 + Ω2 /2. For distances |rij | (D/B)1/3 with D = |g|di |e+ |2 and di the dipole operator, the dipole–dipole interaction in Equation 12.4 between two polar molecules can be mapped onto the effective spin interaction Hamiltonian Hd = Hdint + Hdshift . The first term describes an effective spin–spin interaction Hdint = −

1

y y Dν(rij ) Six Sjx + Si Sj − η2− Siz Sjz , 2 i=j


(12.30)

457


√ where η± = ηg ± ηe is determined by the induced dipole moments ηg = ∂E Eg / D √ and ηe = ∂E Ee,+ / D. The anisotropic behavior of the dipole–dipole interaction is accounted for by ν(r) = (1 − 3 cos2 ϑ)/r 3 with ϑ the angle between r and the z-axis. In addition, the asymmetry of the induced dipole moments gives rise to a positiondependent renormalization of the effective magnetic field and an energy shift 1

η− η+ z η2+ Si + . = Dν(rij ) 2 2 4

Hdshift

(12.31)

i=j

Within the Born–Oppenheimer approximation, an analytic expression for the effective interaction Veff ({ri }) between two polar molecules each prepared in the state |+i can be derived in second-order perturbation theory in the dipole–dipole interaction Vdd (r)/h as 3D Veff ({ri }) = E (1) ({ri }) + E (2) ({ri })

(12.32)

where D/(a3 |h|) = (R0 /a)3 1 is the (small) parameter controlling the perturbative expansion, a is the characteristic length scale of the interparticle separation, and R0 = √ (D/ Δ2 + Ω2 )1/3 is a Condon point, analogous to that discussed in Section 12.3.2.2. The energy shift E (1) ({ri }) =

2 1 2 α ηg +β2 ηe −α2 β2 Dν(rij ), 2

(12.33)

i=j

gives rise to a dipole–dipole interaction between the particles, while the term 2

|N|2 Dν rij D2 ν (rik ) ν rjk + √ 2 2 Δ2 + Ω 2 i<j 2 Δ + Ω k=i,k =j (12.34) corresponds to a correction to the two-particle interaction potential and an additional three-body interaction. The matrix elements M and N take the form E (2) ({ri }) =

√

|M|2

α2 ηg + β2 ηe ηe − ηg − (α2 − β2 )/2 , 2 N = α2 β2 ηe − ηg + 1 .

M = αβ

Therefore, the effective interaction potential Veff up to second order in (R0 /a)3 reduces to the form in Equation 12.2 with the two-particle interaction potential V (r) = λ1 D ν (r) + λ2 DR03 [ν (r)]2 ,

(12.35)

and the three-body interaction W (r1 , r2 , r3 ) = γ2 R03 D [ν(r12 )ν(r13 ) + ν(r12 )ν(r23 ) + ν(r13 )ν(r23 )]. © 2009 by Taylor and Francis Group, LLC

(12.36)

458


2 The dimensionless coupling parameters are λ1 = α2 ηg + β2 ηe − α2 β2 , λ2 = 2|M|2 + |N|2 /2, and γ2 = 2|M|2 . These parameters can be tuned by varying the strength of the electric field Ed/B and the ratio of the Rabi frequency and the detuning, Ω/Δ (Figure 12.14). Of particular interest are the values of the external fields where the leading two-particle interaction vanishes, that is, λ1 = 0. Then, the interaction is dominated by the second-order contribution with λ2 and γ2 , which includes the three-body interaction (Figure 12.14d), while a small deviation away from the line λ1 = 0 allows us to change the character of the two-particle interaction. Note, that an n-body interaction term (n ≥ 4) appears as (n − 1)-th order perturbation in the small parameter (R0 /a)3 . Therefore, the contribution of these terms is suppressed and can be safely ignored.

l1 (a) 4

10l2 (b) 4

1 0.8 0.4

2

3.0

3

0.6 dE/B

dE/B

3

0.2

1.8

2

1.2

0

1

2.4

1

0.6

–0.2 0

1

2

3

0

4

1

W/D (c)

2

3

4

3

4

W/D

10g2

(d)

4

3.0

0.1 l2

2.4

3 dE/B

1.8 2

0.05

1.2

g2

0.6 1

0

1

2 W/D

3

4

0

1

2 W/D

FIGURE 12.14 Parameters of the effective interaction potential. (a)–(c): Strength of the interaction parameters λ1 , λ2 , and γ2 as a function of the external fields Ed/B and Ω/Δ. The leading dipole–dipole interaction vanishes for λ1 = 0 [dashed line in (b) and (c)], and the second order contributions dominate the interaction. (d) Strength of λ2 (dashed line) and γ2 (solid line) along the line in parameter space with λ1 = 0. (From Büchler, H.P. et al., Nature Phys., 3, 726, 2007. With permission.)



459

The perturbative expansion requires that a R0 . We notice that particles can be confined to interparticle distances larger than R0 by combining repulsive dipole–dipole interactions with a strong (optical) transverse confinement ω⊥ , in analogy to Section 12.3.2.1. Then, for a repulsive two-particle potential with λ1 −λ2 (R0 /a)3 and for ω⊥ > D/R03 , the particles approach distances |ri − rj | < R0 at an exponentially small rate Γ ∼ (/ma2 ) exp(−2SE /), with SE / ∼ Dm/R0 2 . This exponential suppression ensures the stability of the collision system for the duration of an experiment. Applying an optical lattice provides a periodic structure for the polar molecules 3D given by Equation 12.32. described by the Hamiltonian of Equation 12.1, with Veff In the limit of a deep lattice, a standard expansion of the field operators ψ† (r) = 4 † i w(r − Ri )bi in the second-quantized expression of Equation 12.1 in terms of lowest-band Wannier functions w(r) and particle creation operators bi† [107] leads to the realization of the Hubbard model of Equation 12.9, characterized by strong nearest-neighbor interactions [85]. We notice that the particles are treated as hardcore because of the constraint a R0 . The interaction parameters Uij and Vijk in Equation 12.9 derive from the effective interaction V ({ri }), and in the limit of well-localized Wannier functions reduce to a3 a6 + U1 , 3 |Ri − Rj | |Ri − Rj |6

Uij = U0 and

Wijk = W0

a6 + perm , |Ri − Rj |3 |Ri − Rk |3

(12.37)

(12.38)

respectively, with U0 = λ1 D/a3 , U1 = λ2 DR03 /a6 , and W0 = γ2 DR03 /a6 . The dominant contributions and strengths of the three-body terms in different lattice geometries are shown in Figure 12.2b. For LiCs with a permanent dipole moment d = 6.3 D trapped in an optical lattice with spacing a ≈ 500 nm, the leading dipole–dipole interaction can give rise to very strong nearest-neighbor interactions with U0 ∼ 55Ekin , and Ekin = 2 /ma2 . On the other hand, tuning the parameters via the external fields to λ1 = 0 results in the characteristic energy scale for the three-body interaction W0 ≈ (R0 /a)3 Ekin . Then, controlling the hopping energy J by the strength of the optical lattice allows for the regime with dominant three-body interactions. For bosonic particles, an analytic calculation has suggested that the ground-state phase diagram of Equation 12.9 with Uij = 0 in one-dimensional is characterized by the presence of valence bond states at specific rational fillings of the lattice, charge-density waves and superfluid phases [85].

12.4.4

LATTICE SPIN MODELS

Cold gases of polar molecules can be used to construct in a natural way a complete toolbox for any permutation symmetric two spin-1/2 (qubit) interaction, based on techniques of interaction engineering discussed in the previous sections. © 2009 by Taylor and Francis Group, LLC

460


The system we consider in this section is comprised of heteronuclear molecules in the 2 Σ1/2 electronic state, such as, for example, alkaline-earth monohalogenides with a single electron outside a closed shell. We adopt a model molecule where the rotational excitations are described by the Hamiltonian Hm = BN2 + gN · S,

(12.39)

with N the rotational angular momentum of the nuclei, and S the dimensionless electronic spin (assumed to be S = 1/2 in the following). Here B denotes the rotational constant and γ is the spin-rotation coupling constant. The typical values of B are a few tens of gigahertz, and γ is usually in the hundred megahertz regime. The coupled basis of a single molecule i corresponding to the eigenbasis of i is {|N , S , J ; M } where J = N + S with eigenvalues E(N = 0, 1/2, 1/2) = Hm i i i Ji i i i 0, E(1, 1/2, 1/2) = 2B − γ, and E(1, 1/2, 3/2) = 2B + γ/2. The Hamiltonian describing the internal and external dynamics of a pair of molecules trapped in wells of an optical lattice is denoted by H = Hin +4 Hex . The i , interaction describing the internal degrees of freedom is Hin = Hdd + 2i=1 Hm where Hdd is the dipole–dipole interaction. The Hamiltonian describing the external, 4 or motional, degrees of freedom is Hex = 2i=1 Pi2 /(2m) + Vi (xi − x¯ i ), where Pi is the momentum of molecule i with mass m, and the potential generated by the optical lattice Vi (x − x¯ i ) describes an external confinement of molecule i about a local minimum x¯ i with one-dimensional rms width z0 . We assume that the traps are approximately harmonic near the trap minimum with a vibrational spacing ωosc . Furthermore, we assume that the molecules can be prepared in the motional ground state of each local potential using dissipative electromagnetic pumping [3]. It is convenient to direct the quantization axis zˆ along the line connecting the two molecules, x¯ 2 − x¯ 1 = Δzˆz with Δz corresponding to a multiple of the lattice spacing. The ground subspace of each molecule is isomorphic to a spin-1/2 particle. Our goal is to obtain an effective spin–spin interaction between two neighboring molecules. Static spin–spin interactions due to spin-rotation and dipole–dipole couplings do exist but are very small in our model: HvdW (r) = −(d 4 /2Br 6 ) 1 + (γ/4B)2 1 + 4S1 · S2 /3 − 2S1z S2z . The first term is the familiar van der Waals 1/r 6 interaction, while the spin-dependent part is strongly suppressed as γ/4B ≈ 10−3 1. However, dipole–dipole coupled excited states can be dynamically mixed using a microwave field. The molecules are assumed to be trapped with a separation Δz ∼ rγ ≡ (2d 2 /γ)1/3 , where the dipole–dipole interaction is d 2 /rγ3 = γ/2. In this regime the rotation of the molecules is strongly coupled to the spin and the excited states are described by Hunds case (c) states in analogy to the dipole–dipole coupled excited electronic states of two atoms with fine structure. The ground states are essentially spin-independent. In the subspace of one rotational quantum (N1 + N2 = 1), there are 24 eigenstates of Hin which are linear superpositions of two electron spin states and properly symmetrized rotational states of the two molecules. There are several symmetries that reduce Hin to block diagonal form. First, Hdd , conserves the quantum number Y = MN + MS where MN = MN1 + MN2 and MS = MS1 + MS2 are the total rotational and spin projections along the intermolecular axis. Second, parity, defined as the interchange of the two © 2009 by Taylor and Francis Group, LLC

461


molecules followed by parity though the center of each molecule, is conserved. The σ = ±1 eigenvalues of parity are conventionally denoted g(u) for gerade(ungerade). Finally, there is a symmetry associated with reflection R of all electronic and rotational coordinates through a plane containing the intermolecular axis. For |Y | > 0 all eigenstates are even under R but for states with zero angular momentum projection there are ±1 eigenstates of R. The 16 distinct eigenvalues correspond to degenerate subspaces labeled |Y |± σ (J), with J indicating the quantum number in the r → ∞ asymptotic manifold (N = 0, J = 1/2; N = 1, J). Remarkably, the eigenvalues and eigenstates can be computed analytically yielding the Movre–Pichler potentials [134] plotted in Figure 12.15a. In order to induce strong dipole–dipole coupling we introduce a microwave field E(x, t)eF with a frequency ωF and Rabi-frequency Ω tuned near resonance with the N = 0 → N = 1 transition. The effective Hamiltonian acting on the lowest-energy states is obtained in second-order perturbation theory as Heff (r) =

gf |Hmf |λ(r)λ(r)|Hmf |gi i,f λ(r)

ωF − E(λ(r))

|gf gi |,

(12.40)

where {|gi , |gf } are ground states with N1 = N2 = 0 and {|λ(r)} are excited eigenstates of Hin with N1 + N2 = 1 and with excitation energies {E(λ(r))}. The reduced interaction in the subspace of the spin degrees of freedom is then obtained by tracing over the motional degrees of freedom. For molecules trapped in the ground motional states of isotropic harmonic wells with rms width z0 , the wave function is separable in center of mass and relative coordinates, and the effective spin–spin Hamiltonian is Hspin = Heff (r)rel . The Hamiltonian in Equation 12.40 is guaranteed to yield some entangling interaction for appropriate choice of field parameters, but it is desirable to have a systematic way to design a spin–spin interaction. The model presented here possesses sufficient structure to achieve this essentially analytically. The effective Hamiltonian of molecules 1 and 2 in a microwave field is Heff (r) =

3 |Ω| α β σ1 Aα,β (r)σ2 , 8

(12.41)

α,β=0

where {σα }3α=0 ≡ {1, σx , σy , σz } and A is a real symmetric tensor. Equation 12.41 describes a generic permutation symmetric two-qubit Hamiltonian. The components A0,s describe a pseudo magnetic field which acts locally on each spin and the components As,t describe two-qubit coupling. The pseudo magnetic field is zero if the microwave field is linearly polarized but a real magnetic field could be used to tune local interactions and, given a large enough gradient, could break the permutation invariance of Hspin . For a given field polarization, tuning the frequency near an excited state induces a particular spin pattern on the ground states. These patterns change as the frequency is tuned though multiple resonances at a fixed intermolecular separation. Table 12.1 presents the parameters needed to simulate the Ising and Heisenberg interactions in © 2009 by Taylor and Francis Group, LLC

462 (a)


g

0+g

2g

1+ B

1g 1u

E/2B

1

g

1– B

(b) z/D z 1u

4

0u–

1 (0, 2 ;

3 1, 2 )

3

0–g

3g 4B

1g 2u

0+g

0u+

1

E(t)

1

(0, 2 ; 1, 2 )

2

Ÿ

z 1u

0+ 0– 1g u 0u– g

1 1

Ÿ

0

0 –1

–1 0 0.5 D z/rg (g /4B)1/3

1

1.5

Ÿ

x

1

(0, 2 ; 0, 2 ) 2

x/D z

1

0

1

y –2 y/D z

r/rg

FIGURE 12.15 (a) Movre–Pichler potentials for a pair of molecules as a function of their separation r. The potentials E(gi (r)) for the four ground states (dashed lines) and the potentials E(λ(r)) for the first 24 excited states (solid lines). The symmetries |Y |± σ of the corresponding excited manifolds are indicated, as are the asymptotic manifolds (Ni , Ji ; Nj , Jj ). (b) Implemen(II)

tation of spin model Hspin . Shown is the spatial configuration of 12 polar molecules trapped by two parallel planes) with separation normal to the plane √ triangular lattices (indicated by shaded √ of Δz/ 3 and in plane relative lattice shift of Δz√ 2/3. Nearest neighbors are separated by b = Δz and next nearest neighbor couplings are at 2b. The graph vertices represent spins and the edges correspond to pairwise spin couplings. The edge width and shading indicate the nature of the dominant pairwise coupling for that edge (thick dark edge = σz σz , thin dark edge = σy σy , thin light edge = σx σx , black = “other”). For nearest neighbor couplings, the edge width indicates the relative strength of the absolute value of the coupling. For this implementation, the nearest-neighbor separation is b = rγ . Three fields all polarized along zˆ were used to generate the effective spin–spin interaction with frequencies and intensities optimized to approximate the (II) ideal model Hspin . The field detunings at the nearest-neighbor spacing are ω1 − E(1g (1/2)) =

−0.05γ/2, ω2 − E(0g− (1/2)) = 0.05γ/2, ω3 − E(2g (3/2)) = 0.10γ/2 and the amplitudes are |Ω1 | = 4|Ω2 | = |Ω3 | = 0.01γ/. For γ = 40 MHz this generates effective coupling strengths Jz = −100 kHz and J⊥ = −0.4Jz . The magnitude of residual nearest-neighbor couplings are less than 0.04|Jz | along x- and y-links and less than 0.003|Jz | along z-links. The size of longer-range couplings Jlr are indicated by edge line style (dashed: |Jlr | < 0.01|Jz |, dotted: |Jlr | < 10−3 |Jz |). Treating pairs of spins on z-links as a single effective spin in the low-energy sector, the model approximates Kitaev’s 4-local Hamiltonian [115] on a square grid (shown here are one plaquette on the square lattice and a neighbor plaquette on the dual lattice) with an effective coupling strength Jeff = −(J⊥ /Jz )4 |Jz |/16 ≈ 167 Hz. (From Micheli, A. et al., Nature Phys., 2, 341, 2006. With permission.)

this way. Using several fields that are sufficiently separated in frequency, the resulting effective interactions are additive creating a spin texture on the ground states. The anisotropic spin model HXYZ = λx σx σx + λy σy σy + λz σz σz can be simulated using three fields: one polarized along zˆ tuned to 0u+ (3/2), one polarized along yˆ tuned to 0g− (3/2), and one polarized along yˆ tuned to 0g+ (1/2). The strengths λj can be tuned by adjusting the Rabi frequencies and detunings of the three fields. Using © 2009 by Taylor and Francis Group, LLC

463


TABLE 12.1 Some Spin Patterns that Result from Equation 12.41 Polarization

Resonance

Spin Pattern

xˆ zˆ zˆ yˆ yˆ √ ( yˆ − xˆ )/ 2 cos ξˆx + sin ξˆz

2g 0u+ 0g− 0g− 0g+ 0g+ 1g

cos ξˆy + sin ξˆz

1g

σz σz σ · σ σx σx + σy σy − σz σz σx σx − σy σy + σz σz −σx σx + σy σy + σz σz −σx σy − σy σx + σz σz λ1 (σx σz + σz σx ) + λ2 σz σz +λ3 (σx σx + σy σy ) λ1 (σy σz + σz σy ) + λ2 σz σz +λ3 (σx σx + σy σy )

The field polarization is given with respect to the intermolecular axis zˆ , and the frequency ωF is chosen to be near resonant with the indicated excited state potential at the internuclear separation Δz. The sign of the interaction will depend on whether the frequency is tuned above or below resonance. Source: Micheli, A. et al., Nature Physics, 2, 341, 2006.

an external magnetic field and six microwave fields with, for example, frequencies and polarizations corresponding to the last six spin patterns in Table 12.1, arbitrary permutation symmetric two-qubit interactions are possible. The Kitaev model of Equation 12.11 (Spin model II) can be obtained in the following way. Consider a system of four molecules connected by edges of three length b forming an orthogonal triad in space. There are several different microwave field (II) configurations that can be used to realize the interaction Hspin along the links. One choice is to use two microwave fields polarized along zˆ , one tuned near resonance with a 1g potential and one near a 1u potential. A realization of model II using a different set of three microwave fields is shown in Figure 12.15b. The obtained interaction is close to ideal with small residual coupling to next nearest neighbors.

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110. Jördens, R., Strohmaier, N., Günter, K., Moritz, H., and Esslinger, T., A Mott insulator of Fermionic atoms in an optical lattice, arXiv:0804.4009. 111. Tewari, S., Scarola, V.W., Senthil, T.S.D., and Sarma, S. Emergence of artificial photons in an optical lattice, Phys. Rev. Lett., 97, 200401, 2006. 112. Haldane, F.D.M., Two-Dimensional Strongly Correlated Electron Systems, Gan, Z.Z. and Su, Z.B., Eds., Gordon and Breach, 1988. 113. Douçot, B., Feigel’man, M.V., Ioffe, L.B., and Ioselevich, A.S., Protected qubits and Chern-Simons theories in Josephson junction arrays, Phys. Rev. B, 71, 024505, 2005. 114. Kitaev, A.Yu., Anyons in an exactly solved model and beyond, Annals of Physics, 321, 2, 2006. 115. Dennis, E., Kitaev, A.Yu., Landahl, A., and Preskill, J., Topological quantum memory, J. Math. Phys., 43, 4452, 2002. 116. Duan, L.M., Demler, E., and Lukin, M.D., Controlling spin exchange interactions of ultracold atoms in optical lattices, Phys. Rev. Lett., 91, 090402, 2003. 117. Alexandrov, A.S., Theory of Superconductivity, IoP Publishing, Philadelphia, 2003. 118. Illuminati, F. and Albus, A., High-temperature atomic superfluidity in lattice BoseFermi mixtures, Phys. Rev. Lett., 93, 090406, 2004. 119. Wang, D.-W., Lukin, M.D., and Demler, E., Engineering superfluidity in Bose-Fermi mixtures of ultracold atoms, Phys. Rev. A, 72, R051604, 2005. 120. Bruderer, M., Klein, A., Clark, S.R., and Jaksch, D., Polaron physics in optical lattices, Phys. Rev. A, 76, 011605(R), 2007. 121. Friedrich, B. and Herschbach, D., Alignment and trapping of molecules in intense laser fields, Phys. Rev. Lett., 74, 4623, 1995. 122. Herzberg, G., Molecular Spectra and Molecular Structure I, Spectra of Diatomic Molecules, Van Nostrand Reinhold, New York, 1950. 123. Brown, J.M. and Carrington, A., Rotational Spectroscopy of Diatomic Molecules, Cambridge University Press, New York, 2003. 124. Judd, B.R., Angular Momentum Theory for Diatomic Molecules, Academic Press, New York, 1975. 125. See e.g. http://physics.nist.gov/PhysRefData/MolSpec/ 126. Coleman, S., Fate of the false vacuum: Semiclassical theory, Phys. Rev. D, 15, 2929, 1977. 127. Kotochigova, S., Tiesinga, E., and Julienne, P.S., Photoassociative formation of ultracold polar KRB molecules, Eur. Phys. J. D, 31, 189, 2004. 128. Mahan, G.D., Many Particle Physics, Kluwer Academic/Plenum Publishers, NewYork, 2000. 129. This antiadiabatic regime is hard to achieve with cold atoms, see, e.g., Refs. [118] and [119]. 130. Carmichael, H.J., Statistical Methods in Quantum Optics 1, Springer-Verlag, Berlin, 1999. 131. Ortner, M., Micheli, A., Pupillo, G., and Zoller, P., Quantum simulations of extended Hubbard models with dipolar crystals, 2009 (submitted for publication). 132. For single-frequency phonons in 1D, see: Datta, S., Das, A., and Yarlagadda, S., Manypolaron effects in the Holstein model, Phys. Rev. B, 71, 235118, 2005. 133. Hirsch, J.E. and Fradkin, E., Phase diagram of one-dimensional electron-phonon systems. II. The molecular-crystal model, Phys. Rev. B, 27, 4302, 1983; Niyaz, P., Scalettar, R.T., Fong, C.Y., and Batrouni, G.G., Phase transitions in an interacting boson model with near-neighbor repulsion, Phys. Rev. B, 50, 362, 1994. 134. Movre, M. and Pichler, G., Resonant interaction and self-broadening of alkali resonance lines I. Adiabatic potential curves, J. Phys. B: Atom. Molec. Phys., 10, 2631, 1977.


Part IV Cooling and Trapping


Cooling, Trap Loading, 13 and Beam Production Using a Cryogenic Helium Buffer Gas Wesley C. Campbell and John M. Doyle CONTENTS 13.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Buffer-Gas Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Loading of Species into the Buffer Gas. . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1.1 Laser Ablation and LIAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1.2 Beam Injection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1.3 Capillary Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1.4 Discharge Etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Rotational and Vibrational Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Buffer-Gas Loading of Magnetic Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Lifetime of Trapped Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1.1 Evaporative Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1.2 Buffer Gas Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1.3 Spin Relaxation Loss (Atoms) . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Zeeman Relaxation Collisions between Molecules and Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2.1 Inelastic Collisions of 2Σ Molecules with He . . . . . . . . . 13.3.2.2 Inelastic Collisions of 3Σ Molecules with He . . . . . . . . . 13.4 Buffer-Gas Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Thermalization and Extraction Conditions . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Boosting Condition and Slow Beam Constraints . . . . . . . . . . . . . . . . . 13.4.3 Studies with Diffusively Extracted Beams . . . . . . . . . . . . . . . . . . . . . . . 13.4.4 Studies with Hydrodynamically Extracted Beams . . . . . . . . . . . . . . . 13.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

474 474 476 476 478 480 480 483 484 485 485 487 488 491 493 494 496 497 498 499 501 504 504 504


474

13.1


INTRODUCTION

Trapping and cooling of neutral particles, starting 23 years ago with the first magnetic trapping of an atom [1], has completely transformed the toolbox of atomic, molecular, and optical physics. The breadth of scientific impact is astounding, ranging from the creation of new quantum systems, observation of new collisional processes, enhancements in precision measurement techniques, and new approaches to quantum information and simulation. However, we feel that the current situation is only a beginning. Compared to the situation with atomic and molecular beams, the number of trapped or cooled species is paltry—no more than 30 different species in comparison to hundreds for beams. With new species come new interactions (and new complications). There is much to look forward to. Examples of “unfinished business” include the creation of polar molecules in optical lattices (predicted to be useful as a tunable Hubbard model system), strongly interacting dipolar gases, dramatic improvements in the search for permanent dipole moments, in-laboratory study of the myriad of astrophysical cold collisional processes, quantum computers based on single atoms or molecule qubits, and arbitrary species cooling for precision studies such as variation of fundamental constants (see Chapter 16 by Flambaum and Kozlov). All of these are on the near horizon, thanks to the continued large efforts toward expanding trapping and cooling techniques. The development of these new techniques and their rapid application to new systems has brought a continuing harvest of scientific findings. This chapter describes the experimental techniques of buffer-gas cooling, loading, and slow beam formation, the latter two building upon the first. Buffer-gas cooling uses a helium refrigerator to cool a gas of helium, which, in turn, cools atoms or molecules. The extraordinary generality and simplicity of buffer-gas cooling results from a combination of (1) helium being chemically inert and effectively structureless, (2) the typical cold elastic cross-section of helium with any atom or molecule being around 10−14 cm2 (allowing for centimeter size cooling cells), (3) the appreciable saturated vapor pressure of helium at temperatures down to 200 mK and (4) existing techniques to create gas-phase samples of nearly any species, including radicals. There are three main sections to this chapter. First, we review buffer-gas cooling, in which hot atoms or molecules are introduced into a cold helium gas, where they thermalize. Second, we discuss buffer-gas loading of traps. Third, we describe the production of cold molecular and atomic beams made by adding a hole in the side of the cryogenic buffer-gas cell. These three processes are depicted in Figure 13.1. Although buffer-gas cooling could be used with other gases to cool to temperatures above 4 K (e.g., H2 or Ne to temperatures around 15 K), this review covers only buffergas cooling to temperatures around and below the boiling point of liquid helium, 4.2 K. Much of the buffer gas physics described in this review would apply to higher temperature systems, especially those using neon.

13.2

BUFFER-GAS COOLING

The technique of buffer-gas cooling [2] relies on collisions with cold buffer gas atoms to thermalize atoms or molecules to low temperature. The buffer gas serves to dissipate translational energy of the target species and, in the case of molecules, © 2009 by Taylor and Francis Group, LLC

475

Cooling, Trap Loading, and Beam Production (a)

(b)

Cold buffer-gas cell

Molecule

(c)

Helium atom

FIGURE 13.1 Buffer-gas (a) cooling, (b) loading, and (c) beam creation. (a) Buffer-gas cooling occurs when the hot target species A (molecules or atoms) enters cold buffer gas inside a cold cell, which are both at a low temperature T . The A particles collide with the buffer gas, cooling to a temperature close to T in approximately 100 collisions. The particles of A then diffuse to the wall, where they stick and are lost. The sizes of the A–He and He–He cross-sections determine the efficiency of indirect cooling of A by the cell wall after the initial thermalization. (b) In buffer-gas loading, thermalized A particles feel a potential due to some type of trapping field (e.g., magnetic or electric). The A particles in trapping states feel a force pulling them to the center of the trap, whereas those in untrapped (or antitrapped) states move to the walls where they stick. The time for this “fall in” is about the diffusion time of A particles across the cell with the trapping field turned off. (c) Thermalized A particles can exit a hole in the side of a cryogenic cell and form a cold beam of A. The extraction efficiency and velocities of A in the beam depend on the density of helium, size of the cell, and size of the exit hole.

their rotational energy as well. Because this dissipation scheme does not depend on any particular energy level pattern, any target species is amenable to it. As with the case of evaporative cooling of a trapped ensemble, buffer-gas loading relies on elastic collisions. At temperatures of ∼1 K, all substances except for He (and certain spin-polarized species, such as atomic hydrogen) have negligible vapor pressure, so the question arises as to how to bring the species to be cooled into the gas phase and then into the buffer gas. Five methods have been used to accomplish this: laser ablation, beam injection, capillary filling, discharge etching, and laser-induced atom desorption (LIAD). The translational thermalization process of the target species with cold helium can be modeled by assuming elastic collisions between two mass points, m (buffer gas atom) and M (target species). From energy and momentum conservation in a hardsphere model, we find, after thermal averaging, that the difference ΔT in temperature of the atom or molecule before and after a collision with the buffer gas atom is given by ΔT = (T − T )/κ, with T denoting the temperature of the buffer gas, T the initial temperature of the atom or molecule, and κ ≡ (M + m)2 /(2Mm). The equation for the temperature change can be generalized and recast in differential form: dT = −(T − T )/κ d © 2009 by Taylor and Francis Group, LLC

(13.1)

476


where T is the temperature of the atom or molecule after collisions with the buffer gas atom. Equation 13.1 has a solution T /T = (T /T − 1)e−/κ + 1.

(13.2)

Under the conditions of T ≈ 1000 K and M/m ≈ 50, of order 100 collisions are required for the atoms or molecules to fall to within 30% of the He buffer-gas temperature T = 0.25 K. This 100 collisions typically corresponds to a time of order 0.1–10 msec, depending on the buffer-gas density; this is consistent with our observations of buffer-gas cooling. Figure 13.2 shows the thermalization of laser-ablated VO in cold helium buffer gas for different delays after the ablation pulse. The spectra show complete translational thermalization in less than 10 msec, in accordance with the above simple model. In order to ensure that the target species thermalizes before impinging on the wall of the cell, it is necessary that the density of the buffer gas be large enough to allow for thermalization on a path smaller than the size of the cell. Cells are typically of order 1 cm in diameter. Assuming an elastic collision cross-section of about 10−14 cm2 between the target species and helium (an assumption accurately borne out by numerous experiments [3–6]), the minimum density required is typically 3 × 1014 cm−3 . This requirement puts a lower limit on the temperature of the buffer gas. Figure 13.3 shows the dependence of number density on temperature for 3 He [7] and 4 He [8] at about 1 K. One can see that 3 He can be used at temperatures as low as 180 mK and 4 He as low as 500 mK. In the above discussion of thermalization, it was assumed that the temperature of the buffer gas was fixed to its final temperature (typically around 1 K). For short times (less than the diffusion time of helium to the cell walls) one must consider the heat capacity of the buffer gas, that is, when there is no other mechanism present to take energy away from the hot molecules. To cool to a temperature within 30% of T , the ratio Υ of the minimum number of (precooled) helium atoms to the number of initially hot (T ) target species is given by Υ = (T /T − 1)/0.3. For T ≈ 4 K and T ≈ 1000 K this amounts to Υ ≈ 1000. This is a kind of “worst case.” Depending on the experimental setup, goals, and elastic cross-sections, it may be possible to use the cold walls of the chamber to cool the buffer gas with little loss in the number of molecules. This is especially true while, for example, molecules are being drawn through the buffer gas into the center of a magnetic trap.

13.2.1

LOADING OF SPECIES INTO THE BUFFER GAS

We know of five demonstrated species introduction methods (the first four of which have been used in our laboratory): laser ablation, beam injection, capillary filling, discharge etching and LIAD. A schematic of the first four methods is depicted in Figure 13.4. 13.2.1.1

Laser Ablation and LIAD

Numerous species have been laser-ablated and buffer-gas cooled. Laser ablation of solid materials is well established as an important tool in many scientific and technological endeavors, including surface processing, surgery, mass spectrometry and © 2009 by Taylor and Francis Group, LLC

477

Cooling, Trap Loading, and Beam Production 0.4 0–0.5 msec 0.3 0.2 0.1 0 0.4 0.5–1 msec 0.3 0.2

Absorption

0.1 0 0.4 1–2 msec 0.3 0.2 0.1 0 0.4 0.3

4–10 msec 3/2

1/2

P1 (0)

R1 (–1)

0.2

3/2

P1 (0)

3/2

P1 (0)

0.1 0

–3

–2

–1

0 Frequency (GHz)

1

2

3

FIGURE 13.2 Laser absorption spectra showing the thermalization of laser-ablated VO in helium buffer gas. The molecules thermalize to the temperature of the cell walls in less than 4 msec. (From Weinstein, J.D. et al., J. Chem. Phys., 109(7), 2656–2661, 1998. With permission.)

growth of materials. Despite prolific applications, sorting out the fundamental mechanisms for laser ablation and identifying all the physical processes involved in laser ablation has proven quite difficult. For example, the coupling mechanism between the laser light and the sample can be very complex because the optical and thermal properties may change upon laser exposure due to the formation of excited states and plasma. These processes become even more complicated and essentially intractable © 2009 by Taylor and Francis Group, LLC

478

Cold Molecules: Theory, Experiment, Applications 1022 3He

1020

4He

Density (cm–3)

1018

1016

1014

1012

1010

0.2

0.5 1 Temperature (K)

2

5

FIGURE 13.3 The saturated vapor density of helium at buffer-gas cooling temperatures. The 4 He curve is extrapolated from ITS-90 [8] and the 3 He curve is from Ref. [7] with the old ITS90 curve (thin line) shown for reference. The shaded region represents the range of saturated vapor densities used for buffer-gas loading, which sets a minimum temperature of 180 and 500 mK for 3 He and 4 He buffer gas, respectively.

when chemical reactions are required to produce the species of interest. For example, CaF, CaH, and VO radicals were produced by the same 532 nm (4 nsec pulse length, approximately 10 mJ energy) laser ablation of stable solids, CaF2 , CaH2 , and V2 O5 , respectively. The yields, however, varied by five orders of magnitude. In the case of CaH, only 10−8 of the pulse energy went into breaking the single chemical bond to produce CaH from CaH2 . Because of the heating limitations of buffer-gas cooling, the pulse energies used are typically below 20 mJ. LIAD is a laser-induced desorption process that has been demonstrated exclusively with atomic Rb and K. We refer the reader to Ref. [9] for more details. In short, laser light excites a plasmon resonance in the solid metal, driving atoms off the surface and into the buffer gas. 13.2.1.2

Beam Injection

Beam injection is perhaps the most general way of introducing target species into a cold buffer gas. However, it is also the most difficult. The idea is to have an orifice in the side of the buffer-gas cell, typically millimeter to centimeter size, to allow a beam of molecules or atoms to pass into the cell. Making beams of essentially any atom or molecule is possible. The density of helium in the cell is not high enough to form clusters (at least for atoms and simple molecules). However, with the addition of the molecular beam input orifice, a new host of problems arises. First, the escaping © 2009 by Taylor and Francis Group, LLC

479

Cooling, Trap Loading, and Beam Production (a)

(b) Cryostat

Cryostat Room temp. enclosure

Solid precursor

Helium buffer gas

High temp. beam source

Molecules of interest

Ablation laser beam (d)

(c)

Cryostat

Cryostat

Room temp. gas reservoir

Capillary tube

RF discharge coil

FIGURE 13.4 Introduction of molecules into the buffer gas through (a) laser ablation of a solid precursor, (b) molecular beam injection, (c) capillary injection, and (d) discharge etching.

helium can build up density just outside the cell, knocking incoming molecules out of the molecular beam, preventing them from entering the cell. Second, even if the molecules enter the cell, they may be hydrodynamically pulled back out of the cell as helium is continually moving through the orifice into the vacuum region. Third, helium must continually be replenished, adding a potential heat load to the cryogenic system. Despite these difficulties, we have been able to show efficient beam injection loading and buffer-gas cooling of Rb, N, NH, and ND3 , including trapping of N and © 2009 by Taylor and Francis Group, LLC

480


NH and electric guiding of ND3 (Tables 13.1 and 13.2). The key to success was rapid pumping of the buffer gas in the beam region (outside the orifice) via a cryogenic sorption pump for helium made of charcoal. 13.2.1.3

Capillary Filling

Capillary filling is, perhaps, the first idea that comes to mind when thinking about how to introduce atoms or molecules into a buffer gas. Indeed, it was the first method used to buffer-gas cool a molecule [10]. In essence, it is guiding using solid walls. Gas-phase species are transported from high temperature (typically at or above 300 K) to 4 K via a fill capillary. This arrangement has its own peculiar technical problems. First, because one end of the fill capillary is exposed to 4 K buffer gas, not only is there a heat load from the tube into the buffer gas, but the tube near the buffer-gas cell could become cold enough to freeze the incoming species. Second, because species require the fill tube to be maintained above their freezing point, most species require temperatures at or above room temperature, which can become a problem due to the cell capillary depositing heat into the cold buffer-gas cell. Third, depending on the flow conditions the species coming down the tube can suffer intraspecies collisions and collisions with the walls of the tube. As such, this method may be of limited utility for chemically reactive species. However, it is possible to partially overcome the wall issues by using high enough flow of gas through the capillary so that the target species does not hit the tube wall before entering the cold cell. One variation on this approach is coflow of an inert gas in the capillary. If the flow is rapid enough, the diffusion time of a molecule through the inert gas to the wall of the capillary can be less than the time the molecule spends in the capillary (the “residence time”). One can show that the condition for the diffusion time to be equal to the residence time in a capillary of length L (cm) and with a flow rate N˙ (sccm) is approximately given by L ≈ 0.1N˙ at T ≈ 4 K. Without resorting to coflow the capillary method is generally limited to a small subset of atomic and molecular species. However, that small subset contains some very important molecules, including O2 , NH3 , HCN, and N2 . Although in principle atomic samples could be introduced into a buffer-gas cell through a heated capillary (particularly for low boiling point metals like Hg and Cs), to our knowledge no atom has ever been buffer-gas cooled using capillary filling. 13.2.1.4

Discharge Etching

A final method for introducing species into the cold buffer gas is discharge etching, in which an electrical discharge plasma in the buffer gas “etches” off species that are frozen to the cell wall, bringing them into the gas phase. Once liberated, the discharge plasma can even excite/dissociate a fraction of them (e.g., into a metastable state). To date we have only used this for one species, He*, but based on the success with He* and previous work with low-temperature production of atomic hydrogen, it is likely that this method will find further application. The type of discharge used for this purpose in our lab is a λ/4 radio-frequency helical resonator coil discharge. The coil is wound around or inside the buffer-gas cell [11]. Our He∗ experiments begin with an entirely empty cell with an inner surface precoated with a few monolayers of atomic helium and with one end open to high © 2009 by Taylor and Francis Group, LLC

481

Cooling, Trap Loading, and Beam Production

TABLE 13.1 A List of Molecules That have been Helium Buffer-Gas Cooled to Less than 10 K Species

Na

CaF CaHf CH3 Fg CrH CO DClg H2 Sg H2 COg HCN MnH NDh NH NHh,i (v = 1)j ND3 ND3 ND3 NOg O2 PbO PbO SrO SrO SrO ThOk ThOl VO

5 × 1013

a b

c d e f g h i j k l

1 × 108

1 × 105

1 × 105 1 × 108 5 × 1011 2 × 108 2 × 107

na (cm−3 ) 8 × 107 ≤2 × 1013 1 × 106 ≤5 × 1012 ≤5 × 1012 ≤5 × 1012 ≤1 × 1013 ≤7 × 1013 1 × 106

1 × 109 ≤2 × 1012 1 × 1012 2 × 1012 1 × 1013

Fa (sec−1 )

1 × 1015 1 × 1012 7 × 1010 3 × 1012 1 × 109

1 × 1013 5 × 1010

1 × 1012

Tb (K) 2 0.40 1.2 0.65 1.3 1.85 1 1.7 1.3 0.65 0.55 4 0.55 0.615 7 7 5 1.8 1.6–25 4 4 4 Beam Guide 4

1 × 1015 1.5

Result

τc (sec)

γd [×104 ]

Loadinge

Cold 0.08 1.3 Ablation Trap 0.50 103 Ablation Cold – – Capillary Trap 0.12 0.9 Ablation Cold – – Capillary Cold – – Capillary Cold – – Capillary Cold – – Capillary Cold – – Capillary Trap 0.18 0.05 Ablation Trap 0.50 7 From beam Cold – – From beam Trap 0.95 20 From beam Trap 0.033 >5 From beam Beam – – Capillary Guide – – Capillary Guide Capillary Cold – – Capillary Guide – – Capillary Cold 0.020 – Ablation Cold – – Ablation Cold >0.02 – Ablation – – Ablation [57] – – Ablation [57] Cold 0.03 – Ablation Beam – – Ablation Cold 0.06 – Ablation

Ref. [33,42] [4,18] [43,44] [45] [46,47] [48] [13,49] [15] [50] [45] [30,51] [6] [52] [16,51] [53] [53] [54] [14,55] [39] [42] [37] [57]

[58] [58] [18,59]

N, n, F, maximum number, density, and flux. T , lowest temperature reached with this species in a helium buffer-gas-based experiment. This temperature does not necessarily correspond to the maximum number or density given in the N, n, F columns. In the case of trapping this temperature indicates the lowest trapped temperature achieved. In the case of a beam, it generally indicates the transverse temperature of the beam. Results indicate whether the molecules were just cooled or cooled and trapped, made into a beam, or made into a guided beam. τ is the maximum lifetime observed. In the case of trapping, this is the approximate maximum lifetime observed in the trap. γ is the ratio of diffusion to spin relaxation cross-section. Loading refers to the method used to introduce species into the buffer gas—either laser ablation, capillary injection, LIAD, or loading from a beam. Data from Refs. [4,18] were reanalyzed for γ. Steady-state flow of molecules into the cell, we estimate helium densities used equivalent to about 100 msec diffusion lifetime. 15 NH and 15 ND also trapped. γ = 7 × 104 for 3 He. Lifetime limited by spontaneous emission. Simultaneous metastable and ground-state production and observation. Instantaneous flux over a 3 msec wide pulse.


482


TABLE 13.2 Atoms That have been Helium Buffer-Gas Cooled Species Ag Au Bi Ceh Cr Cr Cs Cu Dy Er Eu Fe Gd He∗i Hf Ho Ho K Lij Mn Mo N Nak Na Nd Nil Pr Rbm Rbn Rb Re Sc Tb Ti Tm Y Yb Yb Zr a b

Na

na (cm−3 )

Fa (sec−1 )

4 × 1013 1 × 1013 5 × 1011 1 × 1012 1 × 1012 1 × 1011 2 × 109 3 × 1012 2 × 1012 2 × 1011 1 × 1012 5 × 1011 1 × 1010 5 × 1011 1 × 1012 1 × 1010 9 × 1011 4 × 109 2 × 1013 2 × 1012 2 × 1010 1 × 1011

1 × 1013 1 × 1012

5 × 1012

5 × 1011

2 × 1012

5 × 1012 1 × 1015

5 × 1012 1 × 1012 1 × 1013 3 × 1011 1.2 × 1012

1.5 × 1011 8 × 109 1 × 109

1 × 1012 1 × 1011 2 × 1011 4 × 1010 2 × 1011 1 × 1011 2 × 1013

Tb (K) 0.42 0.40 0.50 0.60 20 0.05 >100 20 0.008 100 >100 >100 12 – 0.3 1 0.02 0.12 0.018 – 10 0.45 0.14 0.12 0.15 0.03 0.18 0.10 – 0.10

300 >10 1, Equation 13.14 can be expressed as Re > D/d,

(13.15)

where we note that D/d is always greater than 1. Nonetheless, the huge flux enhancement attained by running hydrodynamically (as opposed to diffusively) makes this alternative attractive for certain experiments. The choice between running a cold beam source with diffusive vs. hydrodynamic cell conditions depends on the specific experimental goals. A hydrodynamically extracted beam yields about a factor of 1000 higher flux than a beam made from a diffusive-mode cell, but the forward velocity is higher, around 100 m/sec for a 4 K cell. On the other hand, an effusive cold beam (which must be extracted under diffusive cell conditions) can have forward velocities as low as around 10 m/sec, corresponding to forward kinetic energies 100 times lower than a hydrodynamically extracted beam.

13.4.3

STUDIES WITH DIFFUSIVELY EXTRACTED BEAMS

The diffusive regime was studied in detail using PbO molecules and Na atoms. A full description is given in Ref. [37]. The basic layout of the beam apparatus is shown in Figure 13.11a. The buffer-gas cell is a brass box of ∼10 cm on edge. The exit hole has d = 3 mm and is centered on one side face. Several ablation targets are mounted on the inside top face of the cell about 6 cm from the exit aperture. Na atoms are ablated from Na metal or NaCl targets, and PbO molecules are ablated from a vacuum hotpressed PbO target. Buffer gas continuously flows into the cell. A good vacuum is maintained in the beam region by means of a charcoal sorption pump with a pumping speed for helium of ∼1000 l/sec. For this particular cell, using an assumed cold diffusion cross-section σc ≈ 3 × 1015 cm−3 , the crossover between effusive and boosted flow (the condition d = λ) occurs for nHe ≈ 1015 cm−3 . We characterize the beam source for both species within a range of densities around the anticipated optimal condition for Na, namely nHe ≈ 0.2 − 5 × 1015 cm−3 . For Na, in-cell laser absorption spectroscopy is used to determine the number of thermalized Na atoms, NNa , and the cold collision diffusion cross-section, σc,Na . We find NNa ≈ 1014 /pulse for both the metallic Na and NaCl ablation targets. We measure diffusion lifetimes of τ (msec) ≈ 4 × 10−15 × nHe (cm−3 ). From this we infer σc,Na ≈ 3 × 10−15 cm2 . Previous work with PbO has measured an ablation yield of ≈ 1012 /pulse [38], and our in-cell LIF measurements indicate a comparable yield. In Figure 13.13 we plot the number NA,beam of thermalized particles of species A exiting the hole as a function of nHe . The ablation plume is not directed at the aperture in order to ensure that particles only exit the cell and form a beam by first © 2009 by Taylor and Francis Group, LLC

500


Number in beam per pulse, NA,beam

1011 Na PbO Na simulation

Plateau

1010 3 nHe

109

108

107

0

1

2 3 Buffer-gas density, nHe (1015 cm–3)

4

5

FIGURE 13.13 The number of cold A particles, NA,beam , that emerge in the beam as a function of the buffer-gas density (nHe ) in the diffusive cell regime. Curves with specific functional forms have been inserted to show the different scaling regimes. (From Maxwell, S.E. et al., Phys. Rev. Lett., 95, 173201, 2005. Copyright 2005 by the American Physical Society. With permission.)

colliding with the cold buffer gas. We can therefore monitor the thermalization of the ablated particles by looking for their appearance in the beam. In the Na data 3 ) up and simulations, we find that NA,beam increases rapidly (approximately ∝ nHe to a critical value of nHe , above which NA,beam is roughly constant at its maximum 3 scaling is consistent with a simple picture in which A value. The low-density nHe 3 ≈ (N λ)3 after a near particles are uniformly distributed over a volume of Rtherm thermalization to T . The intersection of the solid curves corresponds to the condition where the thermalization length matches the distance between the ablation target and the aperture. This saturation corresponds to a maximum fraction in the beam ( fmax ) given by the fractional solid angle of the aperture compared to the rest of the cell, indicating that the motion of particles of A in the cell is fully diffusive and randomized. For this cell geometry we have fmax ≈ 3 × 10−4 . The condition for full thermalization (i.e., all of the hot Na produced is thermalized to a temperature close to T ) is apparent in both the experimental and simulated data for Na. We find that the highest effusive-regime helium density is almost exactly equal to the density necessary for thermalization of an ablated sample of Na. Thus it should be anticipated that this cell is near the optimal geometry for producing a maximal flux of slow (effusive), cold Na. By contrast, the different initial conditions produced by the ablation of PbO makes the helium density necessary for thermalization larger than for Na, implying that our cell geometry is not optimal for PbO. Nonetheless, we find the translational and rotational temperature of the PbO in the beam is close to T , because beam particles are produced only through collisions with buffer-gas atoms. © 2009 by Taylor and Francis Group, LLC

501

Cooling, Trap Loading, and Beam Production 160

Na PbO

Forward velocity, vF (m/sec)

140

Mean velocity of an effusive He beam

120 100 80 Mean velocity of an effusive Na beam

60 40

Mean velocity of an effusive PbO beam

20 0

0

1

2

3

4

5

Buffer-gas density, nHe (1015 cm–3)

FIGURE 13.14 PbO and Na beam mean forward velocities vF as a function of buffer-gas density nHe . Extrapolation of the data to zero buffer-gas density is illustrated by best-fit lines (dashed). (From Maxwell, S.E. et al., Phys. Rev. Lett., 95, 173201, 2005. Copyright 2005 by the American Physical Society. With permission.)

In order to verify that the cell is being operated in the diffusive regime, we can calculate ξ. The emptying time of the cell can be calculated using the known cell and exit hole sizes. For this cell, this results in a pump-out time of τpumpout = 100 msec, which is longer than the 1 to 10 msec diffusion times for the buffer-gas densities accessed in these experiments. This verifies that this work is strongly in the diffusive cell regime, that is, ξ 1. Figure 13.14 shows the average forward velocity vF of the beams of A particles as the buffer-gas density nHe is varied. For both Na and PbO, the data show a linear increase of vF with nHe . Here we can see that for this diffusive-regime source, the output beam can be tuned all the way from effusive to strongly boosted (see Figure 13.12). The linear increase of vF with buffer-gas density is consistent with the following simple picture: a slowly moving particle of A takes a time te to exit the hole, where te ∼ d/vA ; during this time, it undergoes Ne collisions with fast, primarily forward-moving He atoms, where Ne ∼ nHe σc vHe te , resulting in a net velocity boost given by ΔvA ∼ vA d/λ ∝ nHe . This picture should be roughly valid for densities below the fully supersonic regime. The velocities we measure for Na are approximately reproduced by modeling the beam formation process with our measured value of σc .

13.4.4

STUDIES WITH HYDRODYNAMICALLY EXTRACTED BEAMS

The cell used for our studies of hydrodynamically extracted beams is smaller than that used in the diffusive regime beam studies, making it possible to achieve ξ = 1 © 2009 by Taylor and Francis Group, LLC

502


while still operating in the convenient nHe ≈ 1015 cm−3 region. The heart of the experimental apparatus is a ≈2.5 cm size cell anchored to the cold plate of a cryostat. This cell is a copper box with two fill lines (one for helium and one for molecular oxygen) on one side, the exit aperture on the opposite side, a Yb ablation target inside, and windows for laser access. With this cell, we make beams of either atomic Yb or O2 . In order to produce beams of O2 , we flow He and O2 continuously into the cell where they mix and thermalize to the temperature of the cell. To produce beams of Yb, we flow He continuously into the cell and ablate the Yb target with a pulsed YAG laser. Helium typically flows into the cell at 1 × 1017 to 8 × 1018 atom/sec. Despite the large He gas flow, the vacuum in the beam region is maintained at a low 3 × 10−8 torr by two stages of differential pumping with high speed cryopumps made of activated charcoal. We run two different types of experiments with this cell, either magnetically guided (with O2 ) or unguided (with Yb). In the guided work, we couple O2 into a magnetic guide and measure the flux of O2 exiting the guide with a residual gas analyzer (RGA). This work is described in detail in Ref. [39] and we direct the reader there. In the unguided work, we use Yb to characterize the beam source. The flux and velocity profile of the Yb beam and the density and temperature of the Yb gas in the cell are all measured using laser absorption spectroscopy. Two exit hole configurations were demonstrated to produceYb beams. Figure 13.15 shows the output efficiency of a simple slit aperture (1 × 4 mm), where this efficiency is defined as the ratio of the number of Yb in the beam to the number of cold Yb produced by ablation in the cell. At high buffer-gas flows, ξ > 1 (the hydrodynamic cell regime) and up to 40% of the cold Yb atoms in the cell are detected in the resultant beam. The divergence of the beam is measured by comparing the Doppler shifts in the absorption spectra parallel and transverse to the atomic beam. The beam is more collimated than a pure effusive source, with a divergence of ≈0.1 steradian. This observed effect of angular peaking of the beam has been seen in room-temperature beam experiments using two species of different mass and is often called “Machnumber focusing” [40,41]. The higher mass species has its angular beam distribution narrowed due to the same boosting effect described earlier—the light species knocks the heavy species from behind, randomizing its transverse motion while always kicking it forward. This effect peaks in the middle of the intermediate regime, Re ≈ 30, becoming much less pronounced as one approaches either the effusive or fully supersonic regime. A peak on-axis flux per unit solid angle of 6 × 1015 atom/sec/sr (5 × 1012 atoms/pulse) has been measured. For trapping, where total kinetic energy of the trappable species must be less than the trap depth (typically a few Kelvin) the high forward velocity of a boosted beam (vF = 130 m/sec > 100 K effective temperature forYb) makes it unsuitable. Trapping work would ideally use a beam with both high flux (available for ξ > 1, implying Re > 1) and vF as close as possible to vA,thermal . In an attempt to achieve the best of the two worlds, we developed a two-stage aperture called the suppressor nozzle (by analogy to a suppressor or silencer for a gun, which works much the same way) [39]. With this two-stage aperture, the He-Yb mixture still passes through the 1 × 4 mm slit out of the buffer-gas cell, resulting in a large hydrodynamic flux enhancement but correspondingly boosted forward velocity (vF > vA,thermal ). To suppress this boost © 2009 by Taylor and Francis Group, LLC

503

Cooling, Trap Loading, and Beam Production 0.6

Fraction into beam

0.5

0.4

0.3

0.2

0.1

0

0

0.5

1

1.5

2

2.5

x = tdiffusion /tpumpout

FIGURE 13.15 The ratio of the number of ablated and cooled ytterbium atoms to the number emitted into the beam for a simple slit aperture (no suppressor nozzle). At high buffer-gas densities, which correspond to long diffusion times and therefore high ξ (see main text), up to 40% of the cold ablated atoms are detected in the peak instantaneous flux represented here of 5 × 1015 atom/sec sr−1 . (From Patterson, D., and Doyle, J.M., J. Chem. Phys., 126, 154307, 2007. With permission.)

of the forward beam velocity while maintaining the high brightness provided by the hydrodynamic regime, the cell is fitted with the suppressor nozzle just outside the slit aperture. The suppressor nozzle consists of a small chamber with a large exit hole opposite the cell slit. This second, larger exit hole is covered by a stainless steel mesh with pore size 140 μm (28% transparency) and creates a near-effusive beam (actually a large number of near-effusive beams in parallel) in the vacuum region, where there are no more collisions. The He density in the suppressor nozzle volume would be far too low to thermalize hot atoms entering the cell. However, this density is high enough to collisionally slow the internally cold (but forward boosted) Yb that comes through the first orifice and low enough so that ξ 1; that is, diffusion dominates the dynamics. In this sense, the suppressor nozzle itself is a small buffer-gas cooling cell operated in the diffusive cell regime. The Yb beam from the suppressor nozzle has a mean velocity of 35 m/sec, with a spread of 20 m/sec. The measured peak on-axis beam flux of 5 × 1012 atom/sec/sr represents about 1% of the cold atomic Yb produced in the ablation, or about 3% of the output of the one-stage aperture (compare to 0.03% for the diffusive cell regime). One of the future directions for producing cold beams is to inject hot molecules along the same direction and into the stream of cold helium gas flowing inside a long tube L > D. The tube length would be set so that particles of A would diffuse a distance D/2 in a time L/Vflow for a helium density set so that nHe = ΥnA , where © 2009 by Taylor and Francis Group, LLC

504


nA is the density of particles of A and Υ is defined in Section 13.2. The cooling of particles of A in such a system could be enhanced with suppressed loss of A particles in a case where σHe−He < σHe−A .

13.5

SUMMARY

Buffer-gas cooling of molecules promises to continue to provide researchers with a general method for obtaining gas-phase samples of cold molecules. The applicability of the technique to a wide variety of atoms and molecules has already led to extensions of the fields of magnetic trapping, quantitative spectroscopy, and slow beam creation. Furthermore, due to the fact that buffer-gas cooling accepts the full Boltzmann distribution of the initial molecular sample, the high numbers of molecules that can be cooled with this method (often also translating to high densities or high beam fluxes) exceed other methods, often by orders of magnitude. Because diatomic molecules have rotational splittings that are often of the order of 1 K, the use of cryogenic helium buffer gas for collisional cooling is a conceptually straightforward process that seems to fit naturally into the world of diatomic molecules. For many types of experiments, the rotational state population enhancement alone is sufficient to make buffer-gas cooling an attractive option. Buffer-gas cooling is also useful for experiments requiring certain degrees of freedom to remain out of equilibrium with the buffer-gas temperature, such as magnetostatic trapping and metastable lifetime measurements. It is important to continue to study the collisional thermalization processes in detail to gain a thorough understanding of the relevant mechanisms. We expect the knowledge base that will be developed for cold collisional processes can be applied in the future to sympathetically or evaporatively cool trapped molecules down to the ultracold regime.

ACKNOWLEDGMENTS We would like to acknowledge the many members of the experimental and theoretical groups at Harvard who contributed to the development of buffer-gas cooling. Others who are deserving of acknowledgment for advancing the field include F. DeLucia for his pioneering work on collisional cooling, D. DeMille, G. Meijer, A. Peters, M. Stoll, D. Kleppner, T. Greytak, and W. Ketterle. This work was supported by the U.S. NSF, DOE, and ARO.

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45. Stoll, M., Buffer-gas cooling and magnetic trapping of CrH and MnH molecules, Ph.D. thesis, Humboldt-Universität zu Berlin, 2008. 46. Willey, D.R., Crownover, R.L., Bittner, D.N., and De Lucia, F.C., Very low temperature spectroscopy: The pressure broadening coefficients for CO–He between 4.3 and 1.7 k. J. Chem. Phys., 89, 1923, 1988. 47. Beaky, M.M., Goyette, T.M., and De Lucia, F.C., Pressure broadening and line shift measurement of carbon monoxide in collision with helium from 1 to 600 K, J. Chem. Phys., 105, 3994, 1996. 48. Willey, D.R., Choong, V.E., and De Lucia, F.C., Very low temperature helium pressure broadening of DCl in a collisionally cooled cell, J. Chem. Phys., 96, 898–902, 1992. 49. Willey, D.R., Bittner, D.N., and De Lucia, F.C., Pressure broadening cross-sections for the H2 S–He system in the temperature region between 4.3 and 1.8 K, J. Molec. Spec., 134, 240, 1989. 50. Ronningen, T.J. and De Lucia, F.C., Helium induced pressure broadening and shifting of HCN hyperfine transitions between 1.3 and 20 K, J. Chem. Phys., 122, 184319, 2005. 51. Campbell, W.C., Magnetic trapping of imidogen molecules, Ph.D. thesis, Harvard University, 2008. 52. See, for example, Euro. Phys. J. D, 2004, Special Issue on Cold Molecules. 53. Patterson, D., Rasmussen, J., and Doyle, J.M., Intense atomic and molecular beams via neon buffer gas cooling, New Journal of Physics, 2009 (to be published). 54. van Buuren, L.D., Sommer, C., Motsch, M., Pohle, S., Schenk, M., Bayerl, J., Pinske, P.W.H., and Rempe, G., Electrostatic extraction of cold molecules from a cryogenic reservoir. Phys. Rev. Lett., 102, 033001, 2009. 55. Willey, D.R., Bittner, D.N., and De Lucia, F.C., Collisional cooling of the NO–He system: The pressure broadening cross-sections between 4.3 and 1.8 K. Mol. Phys., 66, 1, 1988. 56. Ball, C.D. and De Lucia, F.C., Direct observation of Λ-doublet and hyperfine branching ratios for rotationally inelastic collisions of NO–He at 4.2 K, Chem. Phys. Lett., 300, 227–235, 1999. 57. DeMille, D., et al. Cold beam of SrO for trapping studies. Unpublished. 58. Vutha, A.C., Baker, O.K., Campbell, W.C., DeMille, D., Doyle, J.M., Gabrielse, G., Gurevich, Y.V., and Jansen, M.A.H.M., Cold beam of ThO for EDM studies. Unpublished. 59. Weinstein, J.D., deCarvalho, R., Amar, K., Boca, A., Odom, B.C., Friedrich, B., and Doyle, J.M., Spectroscopy of buffer-gas cooled vanadium monoxide in a magnetic trapping field, J. Chem. Phys., 109(7), 2656–2661, 1998. 60. Brahms, N., Newman, B., Johnson, C., Kleppner, D., Greytak, T., and Doyle, J.M., Magnetic trapping of silver and copper, and anomolous spin–relaxation in the Ag–He system, submitted to PRL. 61. Newman, B., Brahms, N., Johnson, C., Kleppner, D., Greytak, T., and Doyle, J., Buffergas cooled cerium, 2008. Unpublished. 62. Bakker, J.M., Stoll, M., Weise, D.R., Vogelsang, O., Meijer, G., and Peters, A., Magnetic trapping of buffer-gas-cooled chromium atoms and prospects for the extension to paramagnetic molecules, J. Phys. B, 39, S1111, 2006. 63. Parsons, M., Chakraborty, R., Campbell, W., and Doyle, J.M., Ablation studies, Unpublished, 2008. 64. Hancox, C.I., Doret, S.C., Hummon, M.T., Luo, L., and Doyle, J.M., Buffer-gas cooling of gadolinium. Unpublished.


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65. Nguyen, S., Doret, S.C., Connolly, C., Michniak, R., Ketterle, W., and Doyle, J.M., Evaporation of metastable helium in the multi-partial-wave regime, Phys. Rev. A, 92, 060703(R), 2005. 66. Nguyen, S.V., Doret, S.C., Helton, J., Maussang, K., and Doyle, J.M., Buffer gas cooled hafnium, 2008. Unpublished. 67. deCarvalho, R., Brahms, N., Newman, B., Doyle, J.M., Kleppner, D., and Greytak, T., A new path to ultracold hydrogen, Can. J. Phys., 83, 293–300, 2005. 68. Nguyen, S.V., Helton, J.S., Maussang, K., Ketterle, W., and Doyle, J.M., Magnetic trapping of an atomic 55 Mn–52 Cr mixture, Phys. Rev. A, 71, 025602, 2005. 69. Nguyen, S.V., Harris, J.G.E., Doret, S.C., Helton, J., Michniak, R.A., Ketterle, W., and Doyle, J.M., Spin-exchange and dipolar relaxation of magnetically trapped Mn, Phys. Rev. Lett., 99, 2007. 70. Hancox, C.I., Hummon, M.T., Nguyen, S.V., and Doyle, J.M., Magnetic trapping of atomic molybdenum, Phys. Rev. A, 71, 031402, 2004. 71. Hummon, M.T., Campbell, W.C., Lu, H., Tsikata, E., Wang, Y., and Doyle, J.M., Magnetic trapping of atomic nitrogen (14 N) and cotrapping of NH (X 3 Σ− ), Phys. Rev. A, 78, 050702(R), 2008. 72. Nguyen, S.V., Michniak, R., and Doyle, J., Trapping of Na in the presence of buffer gas, 2004. 73. Hong, T., Gorshkov, A.V., Patterson, D., Zibrov, A.S., Doyle, J.M., Lukin, M.D., and Prentiss, M.G., Realization of coherent optically dense media via buffer-gas cooling. Phys. Rev. A, 79, 013806, 2009. 74. Hancox, C.I., Magnetic trapping of transition-metal and rare-earth atoms using buffer gas loading, Ph.D. thesis, Harvard University, 2005. 75. Patterson, D. and Doyle, J.M., Buffer gas cooled Yb in cell, 2008. Unpublished. 76. mJ = ±J. 77. We will focus here on the case where the magnetic moment comes entirely from electron spin, although it is possible for the magnetic moment to have contributions from electron orbital angular momentum. 78. A more recent analysis of the data gives this value, which is larger than that quoted in Refs. [4], [18], and [32].


Slowing, Trapping, 14 and Storing of Polar Molecules by Means of Electric Fields Sebastiaan Y.T. van de Meerakker, Hendrick L. Bethlem, and Gerard Meijer CONTENTS 14.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Stark Deceleration of Neutral Polar Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 The Stark Decelerator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Phase Stability in the Stark Decelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.3 Transverse Focusing in a Stark Decelerator . . . . . . . . . . . . . . . . . . . . . . . 14.2.4 Stark Deceleration of OH Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.5 Longitudinal Focusing of a Stark Decelerated Molecular Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.6 Deceleration of Molecules in High-Field Seeking States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 The Zeeman, Rydberg, and Optical Decelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Trapping Neutral Polar Molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 DC Trapping of Molecules in Low-Field Seeking States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2 Storage Ring and Molecular Synchrotron . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.3 AC Trapping of Molecules in High-Field Seeking States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.4 Trap Lifetime Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Applications of Decelerated Beams and Trapped Molecules . . . . . . . . . . . . . 14.5.1 High-Resolution Spectroscopy and Metrology . . . . . . . . . . . . . . . . . . . . 14.5.2 Collision Studies at a Tunable Collision Energy . . . . . . . . . . . . . . . . . . 14.5.3 Direct Lifetime Measurements of Metastable States . . . . . . . . . . . . . . 14.6 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

510 516 516 518 521 522 525 527 528 530 530 532 533 537 537 538 540 542 543 547 548


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14.1


INTRODUCTION

The importance of the role that atomic and molecular beams have played in physics and chemistry cannot be overstated [1]. Nowadays, sophisticated laser-based techniques exist to sensitively and quantum-state-selectively detect atoms and molecules in a beam. In the early days, such detection techniques were lacking and the particles in a beam were detected, for instance, by a “hot wire” (Langmuir–Taylor) detector, by electron-impact ionization, or by deposition and ex situ inspection of the deposit on a substrate placed at the end of a beam-machine [2]. In order to achieve quantum-state selectivity of the detection process, these detection techniques were combined with inhomogeneous magnetic and/or electric fields to characteristically alter the trajectories of the particles on their way to the detector. This was the approach pioneered by Otto Stern and Walther Gerlach in 1922 [3]. The key concept of their experiment, that is, the sorting of quantum states via space quantization, has been extensively used ever since. The experimental geometries were devised to create strong magnetic or electric field gradients on the beam axis to efficiently deflect particles. In 1939, Isidor Rabi introduced the molecular beam magnetic resonance method by using two magnets in succession to produce inhomogeneous magnetic fields of oppositely directed gradients. In Rabi’s setup, the deflection of particles caused by the first magnet was compensated for by the second magnet such that the particles were directed on a sigmoidal path to the detector.A transition to “other states of space quantization,” induced between the two magnetic sections, could be detected via the resulting reduction of the detector signal [4]. Later, both magnetic [5,6] and electric [7] field geometries were designed to focus particles in selected quantum states onto a detector. An electrostatic quadrupole focuser, that is, an arrangement of four cylindrical electrodes alternately energized by positive and negative voltages, was used to couple a beam of ammonia molecules into a microwave cavity. Such an electrostatic quadrupole lens focuses ammonia molecules that are in the upper level of the inversion doublet while simultaneously defocusing those that are in the lower level. The inverted population distribution of the ammonia molecules in the microwave cavity that was thus produced led to the invention of the maser by James Gordon, Herbert Zeiger, and Charles Townes in 1954–1955 [8,9]. Apart from making it possible to observe a spectacular amplification of the microwaves by stimulated emission, these focusing elements made it possible to record, with high resolution and good sensitivity, microwave spectra in a molecular beam. By using several multipole focusers in succession, interlaced with interaction regions with electromagnetic radiation, versatile setups to unravel the quantum structure of atoms and molecules were developed. In scattering experiments, multipole focusers were exploited to study the steric effect, that is, how the orientation of an attacking molecule affects its reactivity [10]. Variants of the molecular beam resonance methods as well as scattering machines that employed state selectors were implemented in many laboratories, and have yielded a wealth of detailed information on stable molecules, radicals, and molecular complexes alike. Manipulation of beams of atoms and molecules by electric and magnetic fields is about as old as atomic and molecular beams themselves; atomic and molecular beam techniques could not have been developed without the ability to manipulate the atoms or molecules. In his autobiography, Norman Ramsey recalls that, when he arrived © 2009 by Taylor and Francis Group, LLC

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in Rabi’s lab in 1937, Rabi was rather discouraged about the future of molecular beam research; his discouragement only vanishing when he invented the molecular beam magnetic resonance method [11]. However, even though the manipulation of beams of molecules by external fields has been used extensively and with great success in the past, it concerned exclusively the transverse motion of the molecules. Only in 1999 was it experimentally demonstrated that appropriately designed arrays of timedependent electric fields can be used to also influence and control the longitudinal (forward) velocity of the molecules in a beam. This so-called “Stark decelerator” was used to slow down a beam of neutral polar molecules [12]. Since then, the ability to produce focused packets of state-selected accelerated or decelerated molecules has made possible a host of new experiments. How to achieve a full control over the three-dimensional motion of neutral polar molecules using an electric field, and how exerting this control can be made use of in a variety of novel experiments is the subject of this chapter. For the sake of simplicity, we will restrict our discussion to the interaction of polar molecules (i.e., of molecules that carry a permanent electric dipole moment) with electric fields. However, we note that the same arguments and principles hold for the interaction of particles with a magnetic dipole moment with magnetic fields. In a quadrupole or hexapole focuser, the magnitude of the electric field is zero on the symmetry axis, which is normally made to coincide with the molecular beam axis. Close to this axis, the electric field strength is—to a good approximation— cylindrically symmetrical, and it increases with distance r from the axis, proportionately to r or r 2 for the quadrupole or hexapole field, respectively (Figure 14.1). For polar molecules in a so-called low-field seeking (LFS) quantum state, that is, a state in which their space-fixed dipole moment is antiparallel to the external electric field, the Stark energy increases with increasing electric field strength. The force that a molecule is subjected to in an electric field is given by the negative gradient of the Stark energy, which implies that in a multipole field there is always a restoring force toward the molecular beam axis acting on a molecule in a LFS (b)

(a) +

–

–

–

– –

(c) +

+

+ +

+ –

|E|

FIGURE 14.1 The electrode geometries for generating (a) dipole, (b) quadrupole, and (c) hexapole fields. The lower part shows how the magnitude |E| of the electric field varies along a line passing perpendicularly through the center for each geometry. Whereas the dipole geometry generates a constant electric field, the quadrupole (hexapole) geometry generates an electric field that increases linearly (quadratically) with distance from the center.


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state. A multipole focuser operates like a perfect lens when this restoring force is linearly proportional to r. For a molecule whose Stark energy scales linearly with the strength of the electric field, it is the hexapole focuser that acts as a perfect lens; on the other hand, a quadrupole focuser is needed to perfectly focus a molecule whose Stark energy scales quadratically with the electric field strength. The depth of the confining potential depends on the properties of the molecule as well as on the magnitude of the electric field that can be experimentally realized. In Figure 14.2, the Stark shifts are shown for the relevant low rotational levels of OH, CO(a3 Π1 ), ND3 , and H2 CO molecules, which have been used in the studies highlighted in this chapter. It can be seen from these plots that at feasible electric field strengths of up to 100 kV/cm, potential energy wells with a depth of typically 1 cm−1 (≈1.44 K) can be created. This is enough to transversally confine such molecules about the beam axis because, in a typical molecular beam experiment, the transverse velocity distribution

3 2 1 0 –1 –2 –3

MJ W 2

–9/4 –3/4

J = 3/2

3/4 OH (X 2P3/2) 0

50

Energy (cm–1)

Energy (cm–1)

MJ W

9/4 100

150

J = 1, K = 1

0

0

–1 –2

ND3 0

1 50

100

150

Electric field strength (kV/cm)

200

0

–1 –2

CO (a3P1)

1

MJ K

MJ K –1 Energy (cm–1)

Energy (cm–1)

1

J=1 0

0 50 100 150 200 Electric field strength (kV/cm)


2

1

–3

200

–1

2 Jt 1 11 0 10 –1 –2 –3 –4 H2CO –5 0

–1

0

1 50

100

150

200


FIGURE 14.2 Stark energy curves of OH(X 2 Π3/2 ), CO(a 3 Π1 ), ND3 , and H2 CO in low rotational states. In zero field, there are two closely spaced levels with opposite parity. This zero-field energy splitting is caused by Λ-doubling (OH, CO), inversion doubling (ND3 ), or K-doubling (H2 CO). In an electric field, the opposite parity levels are coupled, leading to a large linear Stark splitting. Levels that undergo a positive (negative) Stark shift in an increasing electric field are referred to as “low-field seeking, LFS” (“high-field seeking, HFS”). If the Stark shift becomes comparable with the spacing of the rotational levels, the interaction with higher rotational states needs to be taken into account. Since this interaction pushes the rotational levels down, ultimately, all states become HFS at sufficiently high electric fields.


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is centered around zero, with a full-width at half-maximum (FWHM) spread of several tens of m/sec, corresponding to a sub-cm−1 transverse kinetic energy spread. Normally, the multipole focusers are made just long enough to focus a beam of molecules onto the interaction or detection point somewhere downstream. Inside the multipole, the molecules follow a sinusoidal path. This is schematically shown in Figure 14.3. The position of the focus can be controlled by varying the voltages on the multipole electrodes, or, for fixed voltages, by varying the time during which the voltages are switched on; with the second approach, one can eliminate the chromatic aberration. Figure 14.3a pertains to the case when the focusing lens is perfect, whereas Figure 14.3b shows what happens when the focusing lens is imperfect, due to a nonlinear behavior of the force. If the molecules spent a longer time inside the focusing lens or if the trapping potential were steeper, the molecules would undergo multiple transverse oscillations. The latter is the case, for instance, in a guide [13] or when a long multipole focuser is bent into a torus, confining the particles on a circular orbit, such as in an electrostatic storage ring [14]. In multipole focusers as well as in the deflection elements that had been used in atomic and molecular beam experiments in the past, the field gradients are perpendicular to the beam axis, and thus the velocity component of the particles along the beam axis is not affected. In a typical molecular beam experiment, the forward velocity distribution is centered at a high velocity (300 to 2000 m/sec) with a FWHM spread of about 10% of this central velocity. Even at the low end of this velocity range, the kinetic energy of the molecules, whose Stark energies are shown in Figure 14.2, is on the order of 100 cm−1 . This is much larger than the depth of any single potential well that can be realized for these molecules with an electrostatic field. Therefore, a direct (a)

(b)

4

r (mm)

3 2 1 0 A –1

0

2

B 4

A 6 z (cm)

8

10

0

2

B 4

6 z (cm)

8

10

FIGURE 14.3 Trajectories in the (r, z)-plane of molecules in a LFS state, traversing a hexapole at a fixed velocity. Here z is the hexapole’s axis of cylindrical symmetry. The hexapole is switched on (off) when the molecules are at position A (B). The molecules are assumed to originate from a single point. (a) The case when the force in the hexapole is perfectly linear. As a result, the molecular beam is focused onto a single point. (b) The case when a zero-field energy splitting due to Λ-doubling, for instance, is included. In this case, the force is nonlinear at low electric field strengths, resulting in a blurring of the focus.


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longitudinal confinement of the molecules in a static potential well is impossible. On the other hand, the FWHM longitudinal velocity spread of the beam molecules corresponds to a kinetic energy spread on the order of only 1 cm−1 , which is comparable to the FWHM of the transverse kinetic energy spread. As potentials with a depth of 1 cm−1 can be produced, such a potential—with a gradient along the molecular beam axis—can longitudinally confine the molecules, provided this potential moves along with the molecular beam. A multipole focuser would be able to produce a strong-enough gradient of the electric field along the molecular beam axis if it were mounted with its symmetry axis perpendicular to the molecular beam axis. If such a perpendicularly mounted multipole focuser could be moved along the molecular beam axis at the most probable velocity of the molecules, it would provide for the requisite longitudinal confinement; the molecules on the beam axis would oscillate, both in position and velocity, around the center of the multipole. Although the most probable velocity of the molecules in the beam would thereby not be changed, the pack of molecules would remain flocked together while moving in the forward direction. Such a device would thus have the same advantage for the longitudinal motion of the beam molecule as a multipole focuser normally has for their transverse motion. Moreover, if the velocity of this moving potential well could be gradually changed, (a fraction of ) the beam molecules could be brought to any desired final velocity. In order to, for instance, decelerate the beam molecules, the potential well would have to be gradually slowed down, such that the molecules in the beam would spend more time on the leading slope of the longitudinal potential well, thereby feeling a force opposing their motion. In order to accelerate the molecules, the potential well would have to be gradually sped up, thus pushing the molecules forward on the trailing slope of the potential well. The hypothetical situation just described is actually almost exactly what happens in a real Stark decelerator [12]. However, rather than mechanically moving an electrode geometry that generates the confinement potential along the molecular beam, a static array of electrode pairs that create field gradients along the molecular beam axis is used. By switching the fields on adjacent pairs of electrodes on and off at the appropriate times, a traveling potential well is created [15]. In a Stark decelerator, the molecules can either be transported along the beam axis at a constant velocity or gradually decelerated or accelerated to any desired final velocity. In the first experimental demonstration of Stark deceleration, a beam of metastable CO(a3 Π1 , J = 1) molecules was slowed down from 225 to 98 m/sec [12]. Experiments of this kind have been considered and tried before. Electric field deceleration of neutral molecules was first attempted by John King at MIT in 1958. King intended to produce a slow ammonia beam, to obtain a maser with an ultranarrow linewidth. Much better known, especially in the physical chemistry community, is the experimental effort of Lennard Wharton to demonstrate electric field acceleration of a molecular beam. In the 1960s, at the University of Chicago, he constructed an 11 m long molecular beam machine for the acceleration of LiF molecules in HFS states from 0.2 to 2.0 eV, with the goal of using these high-energy beams for reactive scattering studies [16]. Both of the above experiments were unsuccessful, and were not continued after the graduation of the PhD students involved [17,18]. Whereas the interest in slow molecules as a maser medium declined, owing to the invention of the © 2009 by Taylor and Francis Group, LLC


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laser, the molecular beam accelerator was superseded by gas-dynamic acceleration of heavy species in seeded supersonic beams of He or H2 [19]. The state-selected molecular beam, which exits a Stark decelerator, with a tunable velocity and a tunable velocity spread, is ideally suited for many applications. Decelerated beams can be used, for instance, for high-resolution spectroscopic studies [20,21], taking advantage of the increased interaction times. We also anticipate that these beams are advantageous for future molecular interferometry and molecule optics experiments. Decelerated beams also enable the study of (in)elastic collisions and reactive scattering as a function of collision energy, down to zero collision energy [22]. Last but not least, a Stark decelerator enables three-dimensional trapping of neutral polar molecules. “If one extends the rules of two-dimensional focusing to three dimensions, one possesses all ingredients for particle trapping.” This is how Wolfgang Paul put it in his Nobel lecture [23], and as far as the underlying physics principles of particle traps are concerned, it is indeed as simple as that. In order to experimentally realize the trapping of neutral particles, however, one has to face the challenge of producing sufficiently slow particles that can be trapped in a relatively shallow trap. When the particles are confined along a line, rather than around a point, the requirements on the kinetic energy of the particles are more relaxed, and storage of neutrons in a 1 m diameter magnetic hexapole torus could thus be demonstrated [24]. Trapping of atoms in a three dimensional trap only became feasible when Na atoms were laser cooled to sufficiently low temperatures to enable confinement in a quadrupole magnetic trap [25]. The Stark decelerator made possible the first demonstration of three-dimensional trapping of neutral ammonia molecules in a quadrupole electrostatic trap [26], even prior to the first demonstration of an electrostatic storage ring for neutral molecules [14]. The quadrupole trap permits the observation of packets of molecules, isolated from their environment, for times of up to several seconds. Among other things, this enables the direct measurement of lifetimes of metastable states [27]. The electrostatic trap, loaded from a Stark decelerator, also holds great promise for studies of cold collisions. More generally, electrostatic traps are key to a further development of the field of cold molecules, with the production and study of quantum degenerate gases of polar molecules as a prominent goal. In the remainder of this chapter, Stark deceleration of a molecular beam will be presented in more detail, followed by a description of the process of trapping of neutral polar molecules. An overview of the applications of slow beams and trapped samples of molecules will also be given. The experiments described in this chapter, which exemplify the operational characteristics of the various components, have been performed in different molecular beam machines and using different molecules. The deceleration and three-dimensional trapping of molecules in LFS states is explained using the OH radical as a model molecular system; a photograph of the Stark decelerator that was used in these experiments is shown in Figure 14.4. The buncher, used for longitudinal focusing of a molecular beam, and the storage ring and synchrotron were demonstrated using the ND3 molecule. The ND3 molecule, decelerated to a near standstill while in a LFS state, and then transferred, by microwave radiation, to a HFS state, was also used in the demonstration of an ac trap for neutral polar molecules. In order to explain the operation of a decelerator for molecules in HFS states, the © 2009 by Taylor and Francis Group, LLC

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FIGURE 14.4 Photograph of a Stark decelerator. The molecular beam passes through the 4 × 4 mm2 opening between the electrodes, shown enlarged in the inset from the perspective of the molecules. The forward velocity of the molecules is affected by switching, at appropriate times, a voltage difference of 40 kV between opposing electrodes. (From van de Meerakker, S.Y.T. et al., Annu. Rev. Phys. Chem., 57, 159–190, 2006. Copyright (2006) by Annual Reviews www.annualreviews.org. With permission.)

so-called alternating gradient (AG) decelerator, the experiments on metastable CO in the appropriate quantum state will be highlighted. Throughout this chapter we restrict ourselves to the discussion and demonstration of the basic concepts of the various elements; for more detail, we refer the reader to the original literature.

14.2

STARK DECELERATION OF NEUTRAL POLAR MOLECULES

14.2.1 THE STARK DECELERATOR The Stark decelerator (or accelerator) for neutral polar molecules is the equivalent of a linear decelerator (or accelerator) for charged particles. The Stark decelerator exploits the quantum-state specific force that a polar molecule is subjected to in an electric field. This force is rather weak, typically some eight to ten orders of magnitude weaker than the force that the molecule, when singly ionized, would experience in an equivalent electric field. Nevertheless, this force suffices to exert a complete control over the motion of polar molecules using principles akin to those developed to manipulate charged particles. In a Stark decelerator, the longitudinal velocity of a beam of polar molecules is manipulated using a longitudinally inhomogeneous electric field array. Let us consider a single electric field stage of such an array, itself composed of two opposing electrodes that are connected to power supplies of opposite polarity, as shown in Figure 14.5. A polar molecule in a LFS state, as it approaches the plane of the electrodes, experiences the increasing electric field as a potential hill, and thus loses kinetic energy as it climbs the upward slope of the hill. However, on leaving the high-field region along the longitudinal coordinate, the molecule regains the same amount of © 2009 by Taylor and Francis Group, LLC


517

20 kV

W(z)

–20 kV

FIGURE 14.5 The potential energy W (z) of a polar molecule as a function of position z along the molecular beam axis. The molecule is in a LFS state and the electric field is created by two electrodes energized by high voltages of opposite polarity.

kinetic energy it lost during its ascent. The acceleration on the downward slope of the hill can be avoided if a time-varying electric field is used: if the electric field is switched off abruptly, before the molecule has left the region of high electric field, the kinetic energy or velocity of the molecule will not return to its original value. As noted in the previous section, the effect of a single electric field stage on the forward velocity of a beam molecule is rather small. In order to obtain a significant change of the velocity, the process of hill climbing needs to be repeated many times. Therefore, a Stark decelerator consists of an array of electric field stages, as shown in Figure 14.6. Each field stage consists of two parallel cylindrical metal rods of radius r, held apart at a distance 2r + d. One of the rods is connected to a positive, and the other to a negative switchable power supply. Alternating rods are connected to each other. The adjacent field stages are separated by a distance L from one another. At a given time, the even stages are switched to a high voltage while the odd stages are grounded. The potential energy W (z) of a molecule as a function of the position z along the beam axis is shown in Figure 14.6 as well. The deceleration procedure is now straightforward: when the molecule has reached a position that is close to the top of the first potential hill, the even stages are grounded and the odd stages are switched to a high voltage. As a result, the molecule will find itself in front of a potential hill again and will lose kinetic energy while climbing it. When the molecule has reached the high electric field region, corresponding to the hilltop, the voltages are switched back to the original configuration. By repeating this process many times, the velocity of the molecule can be arbitrarily reduced in a stepwise fashion. The amount of kinetic energy that is lost per stage, and thus the velocity with which the molecule exits the decelerator, depends on the exact position of the molecule © 2009 by Taylor and Francis Group, LLC

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Cold Molecules: Theory, Experiment, Applications 20 kV

20 kV

Time t1

–20 kV W(z)

–20 kV

20 kV

20 kV

Time t2

–20 kV

W(z)

–20 kV

FIGURE 14.6 The potential energy W (z) of a polar molecule as a function of position z along the axis of an array of field stages (electrode pairs) with voltages applied as shown. Repeated switching between the upper and the lower field configuration is performed after a time interval t2 − t1 .

at the time when the fields are switched. A key property of the Stark decelerator is that it not only decelerates a single molecule in the beam, but all molecules whose positions and velocities are from within a certain range, called the acceptance of the decelerator. As a result, one can decelerate (or accelerate) a part of the beam to any desired velocity while keeping the selected part of the beam flocked together as a compact packet. Note that it is essential that the molecules remain in the same quantum state throughout the deceleration process. In order to achieve this, the orientation of a molecule needs to adiabatically follow the field, which requires that the electric field varies slowly enough.

14.2.2

PHASE STABILITY IN THE STARK DECELERATOR

Central to the understanding of the operation principle of a Stark decelerator are the concepts of a synchronous molecule and of phase stability. Returning to Figure 14.6, we call the position z of a molecule at the time when the fields are switched the © 2009 by Taylor and Francis Group, LLC


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“phase angle,” φ = zπ/L (while the spatial periodicity of the field stages is 2L, the periodicity of the corresponding phase angle is 2π). We define the position corresponding to φ = 0◦ as a position between two adjacent field stages such that the electrodes at φ = 90◦ become grounded just after the fields were switched.A molecule with velocity v0 is called synchronous, if its phase φ0 is always the same at the time when the fields are switched; that is, φ0 remains constant. As a consequence, a synchronous molecule loses a constant amount of kinetic energy, ΔK(φ0 ), per stage. This happens because the synchronous molecule travels exactly a distance L during the time interval between two successive switchings. Hence the synchronous molecule is always “in phase” with the switching of the fields in the decelerator. A molecule that has a slightly different phase φ and/or velocity v than the synchronous molecule will experience an automatic correction toward the equilibrium values φ0 and v0 . For instance, a molecule whose phase is slightly higher than φ0 at a particular switching time will lose more kinetic energy than the synchronous molecule, and so will slow down with respect to the synchronous molecule. This, in turn, will decrease its phase, until it begins to lag behind the synchronous molecule, at which point the process reverses. Molecules from within a certain region in phase-space, bounded by the so-called separatrix, undergo stable phase-space oscillations around the synchronous molecule. This feature of the process is referred to as phase stability; it ensures that some nonsynchronous molecules will also be decelerated, and that these molecules will remain bunched together in a packet throughout the deceleration process. In order to better understand phase stability, it is helpful to consider the trajectories of the molecules along the molecular beam axis and to derive the corresponding longitudinal equation of motion. The most important elements of the derivation are reproduced here; more details can be found elsewhere [29,30]. A mathematically more rigorous derivation, based on a spatial and temporal Fourier representation of the potential, has also been given [31]. The Stark energy of a molecule W (zπ/L) is symmetric about the position of a field stage (a pair of electrodes) and can be conveniently written as a Fourier series: W

zπ L

∞ zπ π a0

+ + an cos n 2 L 2 n=1 zπ zπ zπ a0 − a1 sin − a2 cos 2 + a3 sin 3 + ··· = 2 L L L

=

(14.1)

By definition, the synchronous molecule travels a distance L during the time interval between two successive switching times. The change in the kinetic energy per stage, ΔK(φ0 ) = −ΔW (φ0 ), for a synchronous molecule with phase φ0 and velocity v0 at a certain switching time is then given by the difference in the potential energy at the positions φ0 and φ0 + π: ΔW (φ0 ) = W (φ0 + π) − W (φ0 ) = 2a1 sin φ0 .

(14.2)

As a result, the average force, F, acting on the synchronous molecule is given by F(φ0 ) = −


2a1 ΔW (φ0 ) =− sin φ0 , L L

(14.3)

520


provided terms with n > 2 in Equation 14.1 can be neglected. The average force acting on a nonsynchronous molecule with a phase φ = φ0 + Δφ, but with a velocity v0 , is given by − 2aL1 sin(φ0 + Δφ). Hence, to a good approximation, the equation of motion of a nonsynchronous molecule with respect to the synchronous one is mL d2 Δφ 2a1 + [sin(φ0 + Δφ) − sin(φ0 )] = 0 π dt 2 L

(14.4)

where m is the mass of the molecule. In the phase-stability diagrams of Figure 14.7, contours of constant energy are shown that result from a numerical integration of Equation 14.4 for OH radicals in the J = 3/2, MΩ = −9/4 state. The computations were carried out with the parameters of a Stark decelerator operated in our laboratory, at the values of the synchronous 60

f0 = 0°

vz (m/sec)

40 20 0

–20 –40 –60 –p

0

p

2p f (rad)

3p

4p

5p

0

p

2p f (rad)

3p

4p

5p

60 f = 70° 0

vz (m/sec)

40 20 0

–20 –40 –60 –p

FIGURE 14.7 Phase-stability diagram for OH(J = 3/2, MΩ = −9/4) radicals when the decelerator is operated at a synchronous phase angle φ0 = 0 ◦ (upper panel) or φ0 = 70 ◦ (lower panel), with vz the velocity of the nonsynchronous molecule along the longitudinal coordinate z and φ its phase. The value vz = 0 corresponds to the velocity of the synchronous molecule. The positions of the electrodes of the decelerator are indicated by dashed lines. (From van de Meerakker, S.Y.T. et al., Annu. Rev. Phys. Chem., 57, 159–190, 2006. Copyright (2006) by Annual Reviews www.annualreviews.org. With permission.)


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Slowing, Trapping, and Storing of Polar Molecules 40 vz (m/sec)

20 f0 = 0°

20°

40° 60° 0

80°

f (rad) p

–p –20

–40

FIGURE 14.8 Longitudinal acceptance of a Stark decelerator for OH radicals for different values of the synchronous phase angle φ0 . Here vz is the velocity of the nonsynchronous molecule along the longitudinal coordinate z and φ is its phase. The value vz = 0 corresponds to the velocity of the synchronous molecule.

phase angle φ0 = 0◦ and φ0 = 70◦ . The solid curves show the phase-space trajectories of the molecules. The dashed lines indicate the positions of the electrodes of the decelerator. The closed curves in the phase-space diagram correspond to bound orbits; molecules within such a “bucket,” bound by the thick contour, oscillate in the phase space about the phase and velocity of the synchronous molecule. Note that the operation of the decelerator at φ0 = 0◦ corresponds to a transport of (a part) of the molecular beam through the decelerator without a change of velocity. Acceleration or deceleration of the beam occurs for −90◦ < φ0 < 0 and 0 < φ0 < 90◦ , respectively. The separatrix defines the longitudinal acceptance of the Stark decelerator, and is shown for different phase angles φ0 in Figure 14.8. One can see that the acceptance is larger for smaller values of φ0 , while the deceleration per stage increases for higher values of φ0 . Because either is desirable, there is a tradeoff between the two. A more extensive description of phase stability in a Stark decelerator revealed that additional phase-stable regions exist; these have indeed been observed in an experiment [30]. The higher-order phase-stable regions can be understood as resulting from higher partial waves in the Fourier expansion of the time-dependent inhomogeneous electric field and from their interferences [31].

14.2.3 TRANSVERSE FOCUSING IN A STARK DECELERATOR Phase stability only ensures that the molecules remain in a “bucket” of the longitudinal phase space. However, what is indispensable is that the molecules also stay together, throughout the deceleration process, in the transverse direction. We have opted for a compact design of the decelerator, whose electric field stages serve simultaneously for deceleration and transverse focusing. Indeed, in the electrode geometry © 2009 by Taylor and Francis Group, LLC

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shown in Figure 14.6, molecules in LFS states remain transversely confined to the molecular beam axis. This is because the electric field is always weaker on the axis than at the electrodes. In order to focus the molecules in both transverse directions (x and y), the rod-electrode pairs that make up a deceleration stage are alternately positioned horizontally (along the x-axis) and vertically (along the y-axis); see inset of Figure 14.4. The three-dimensional trajectories of molecules passing through a Stark decelerator are rather complex. In the longitudinal direction, a molecule oscillates in position and velocity about the synchronous molecule, while in the transverse direction it oscillates about the longitudinal (molecular beam) axis z. The oscillation frequencies involved are generally similar in magnitude, but depend strongly on the phase angle φ. We studied the influence of the transverse motion of the molecules on the longitudinal phase stability, and found that, for high values of the synchronous phase φ0 , the transverse motion actually enhances the longitudinal phase-space region corresponding to phase-stable deceleration. For low values of φ0 , however, the transverse motion reduces the acceptance of the Stark decelerator and unstable regions in the longitudinal phase space appear. These effects can be quantitatively explained in terms of a coupling between the longitudinal and transverse motion, and coupled equations of motion have been derived that reproduce the observations. This coupling does not significantly impair the overall performance of the Stark decelerator, provided the number of the deceleration stages is limited [32]. At low longitudinal velocities, the assumptions used in deriving the analytical models of the longitudinal and transverse motion [29,30,32] are no longer valid. Therefore, in order to avoid losses, special care must be taken when designing the last few electrode pairs of the decelerator.

14.2.4

STARK DECELERATION OF OH RADICALS

A schematic of the Stark deceleration and trapping machine that has been used to decelerate and trap OH radicals in our laboratory is shown in Figure 14.9. A pulsed beam of OH radicals is produced by photodissociation of HNO3 molecules, which are coexpanded with a rare gas through a room-temperature pulsed solenoid valve. In most experiments, either Kr or Xe is used as the carrier gas, producing a beam with the most probable velocity of 450 m/sec or 360 m/sec, respectively. In the supersonic expansion, the beam is rotationally and vibrationally cooled, as a result of which, after the expansion, most of the OH radicals reside in the lowest rotational (J = 3/2) and vibrational level of the electronic ground state X 2 Π3/2 . This level has a Λ-doublet splitting of 0.055 cm−1 and each Λ-doublet component is split (neglecting hyperfine structure) into a MJ Ω = −3/4 and a MJ Ω = −9/4 component when an electric field is applied. This is shown by the Stark-energy diagram in Figure 14.2. The MJ Ω = −9/4 component offers a Stark shift three times larger than the MJ Ω = −3/4 component. Only molecules that are in the LFS MJ Ω = −3/4 or MJ Ω = −9/4 components of the upper Λ-doublet level participate in the electric field manipulation process. The molecular beam passes through a skimmer, and enters a second vacuum chamber, containing the decelerator. In the decelerator chamber, the beam of OH radicals enters a short hexapole that focuses the beam onto the Stark decelerator. © 2009 by Taylor and Francis Group, LLC


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FIGURE 14.9 Schematic of the experimental setup. A pulsed beam of OH radicals is produced via ArF-laser photodissociation of HNO3 seeded in a heavy carrier gas. The molecular beam passes through a skimmer, hexapole, and Stark decelerator into the detection region. State-selective laser-induced fluorescence (LIF) detection is used to measure the arrival time distribution of the OH(J = 3/2) radicals in the detection zone. (From van de Meerakker, S.Y.T. et al., Annu. Rev. Chem., 57, 159–190, 2006. Copyright (2006) Annual Reviews www.annualreviews.org. With permission.)

The 1188-mm-long Stark decelerator consists of an array of 108 equidistant electric field stages, with a center-to-center distance L of adjacent stages of 11 mm. Each stage consists of two parallel 6-mm-diameter polished hardened-steel rods that are centered 10-mm apart, and located symmetrically around the beam axis. Alternating stages are rotated by 90◦ with respect to one another, providing a 4 × 4 mm2 spatial transverse acceptance area. A photograph of the decelerator, with a close-up of the 4 × 4 mm2 opening between the electrodes, is shown in Figure 14.4. The decelerator is operated using a voltage of ±20 kV applied to the opposing electrodes of a field stage, thus creating a maximum electric field strength near the electrodes of 115 kV/cm. The high voltage pulses are applied to the electrodes using fast semiconductor high-voltage switches. The OH radicals are state-selectively detected 21 mm downstream from the last electric field stage (1.307 m from the nozzle) using an off-resonant laser-induced fluorescence (LIF) detection scheme. The performance of the Stark decelerator can be studied by recording the time-offlight (TOF) profile of the OH radicals exiting the decelerator, that is, by scanning the timing of the detection laser relative to the dissociation laser. Alternatively, the phase-space distribution of the molecules can be studied inside the decelerator by propagating a laser beam along the molecular beam axis, and by performing spatially resolved LIF detection from there. The latter strategy has been implemented by Jun Ye and coworkers at JILA, also using the OH radical [33]. Typical TOF profiles of OH radicals at the exit of the decelerator, obtained in our laboratory, are shown in Figure 14.10. © 2009 by Taylor and Francis Group, LLC

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237 m/sec (a)

LIF signal (arb. units)

194 m/sec (b)

142 m/sec (c)

(d)

95 m/sec

2.0

3.0

4.0

5.0

6.0

Time of flight (msec)

FIGURE 14.10 Observed and simulated TOF profiles of a molecular beam of OH radicals exiting the Stark decelerator when the decelerator is operated at a phase angle of 70◦ for a synchronous molecule with an initial velocity of (a) 470 m/sec, (b) 450 m/sec, (c) 430 m/sec, and (d) 417 m/sec. The molecules that are accepted by the decelerator are split off from the molecular beam and arrive in the detection region at later times, and with the indicated final velocities. (From van de Meerakker, S.Y.T. et al., Annu. Rev. Phys. Chem., 57, 159–190, 2006. Copyright (2006) by Annual Reviews www.annualreviews.org. With permission.)

These TOF profiles were obtained using Kr as a carrier gas, producing a molecular beam with the most probable velocity of 460 m/sec. The decelerator was operated at a phase angle φ0 = 70◦ and an initial velocity of the synchronous molecule of 470 m/sec (curve a), 450 m/sec (b), 430 m/sec (c), and 417 m/sec (d). With these settings, the decelerated bunch of molecules exits the decelerator with final velocities of 237, 194, 142, and 95 m/sec, respectively. The gaps in the profiles of the nondecelerated © 2009 by Taylor and Francis Group, LLC


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beams, which result from the removal of the decelerated bunch of the OH radicals, are indicated by the vertical arrows. The width of the arrival-time distribution of the decelerated packet becomes larger for lower values of the final velocity. This is due to the spreading of the molecular beam while in flight from the exit of the decelerator to the LIF detection zone, and due to the spatial extent of the detection laser beam. Both effects are incorporated in the simulated TOF profiles, shown below the observed ones. In curves (c) and (d), a rich oscillatory structure of the TOF profile of the fast beam is observed (its enlargement is shown in the inset). This structure results from the modulation of the phase-space distribution of the nondecelerated beam in the decelerator [30].

14.2.5

LONGITUDINAL FOCUSING OF A STARK DECELERATED MOLECULAR BEAM

In a Stark decelerator, the force acting on a polar molecule is conservative, because it depends, at any given time, solely on the molecule’s position. Consequently, the product of the velocity and position spreads remains constant throughout the deceleration process, in accordance with Liouville’s theorem. Cooling of a packet of molecules while maintaining the packet’s density is therefore fundamentally impossible in a Stark decelerator. However, what is possible is to reduce the velocity spread at the expense of the position spread, and vice versa—as long as the product of the two spreads remains constant. The “swap” can be done using a so-called buncher. A buncher is an additional array of field stages, mounted some distance behind the decelerator. In the buncher, a beam of polar molecules is exposed to a harmonic potential in the forward, longitudinal direction. This results in a uniform rotation of the longitudinal phase-space distribution of the packet of molecules. By switching the buncher on and off at the appropriate times, it can be used to either produce a narrow spatial distribution at a certain position downstream from the buncher, or a narrow velocity distribution. Both possibilities have been experimentally demonstrated; in particular, a beam of Stark-decelerated ND3 molecules with a longitudinal temperature of 250 μK was produced [34–36]. Figure 14.11 shows schematically the principle of bunching. When the Stark decelerator is operated at a phase angle of 70◦ , the longitudinal position spread of the decelerated ammonia molecules is about 1 mm and the longitudinal velocity spread is about 6.5 m/sec. The phase-space distribution of the decelerated molecules is shown in Figure 14.11 underneath the last stage of the decelerator. While in flight from the exit of the decelerator to the buncher, the packet of ammonia molecules spreads out along the molecular beam axis. This results in the elongated and tilted distribution in the longitudinal phase-space, shown in Figure 14.11. The geometry of the buncher is identical to that of the decelerator, except for an overall scaling factor. The fields are switched such that the synchronous molecule spends an equal time on the downward and upward slope of the potential well, and therefore keeps its original velocity. Molecules that are originally slightly ahead of the synchronous molecule (faster molecules), will spend more time on the upward slope than on the downward slope of the potential well, and will, therefore, be decelerated relative to the synchronous molecule. Molecules that are originally slightly behind the synchronous molecule (slower molecules), will spend less time on the upward slope © 2009 by Taylor and Francis Group, LLC

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Decelerator

Buncher

Laser

20 10 5 –2.5 –5

2.5

–10

–5

5

10

–10 20

–20

10

vz (m/sec)

10

–10

–5

5 –10

z (mm)

10

–5

5 –10

–20

FIGURE 14.11 Schematic of the end part of the decelerator, buncher, and detection region. The calculated longitudinal phase-space distributions of the ammonia molecules pertaining to the exit of the Stark decelerator, the entrance and exit of the buncher, and the detection region are shown. The position and velocity are plotted with respect to the synchronous molecule. The solid curves show lines of equal energy in the buncher potential (compare to Figure 14.7). (From Crompvoets, F.M.H. et al., Phys. Rev. Lett., 89, 093004, 2002. Copyright the American Physical Society. With permission.)

than on the downward slope of the potential well, and will, therefore, be accelerated relative to the synchronous molecule. The calculated distribution at the time when the buncher is switched off is also displayed in Figure 14.11, where the contours of equal energy of the ammonia molecules in the potential well are shown relative to the phase-space position of the synchronous molecule. As the potential well is approximately harmonic the phase-space distribution of the molecules undergoes a clockwise uniform rotation. When the buncher is switched off, the slow molecules are speeding ahead while the fast ones are lagging behind, giving rise to a longitudinal spatial focus at some position downstream. The angle by which the packet is rotated, and thus the exact position where the packet comes to a spatial focus, can be controlled by varying the voltage on the buncher electrodes or by varying the time during which the buncher fields are on [35]. In a Stark deceleration beamline, hexapoles and bunchers are used © 2009 by Taylor and Francis Group, LLC


527

to map the phase-space distribution of the molecules at the exit of one element onto the entrance of the next in, respectively, the transverse and longitudinal direction.

14.2.6

DECELERATION OF MOLECULES IN HIGH-FIELD SEEKING STATES

It may appear straightforward to apply the above deceleration and bunching techniques to molecules in HFS states. One could simply let the HFS molecules fly out, instead of into, a high-field region. For the longitudinal motion of the molecules, this is indeed all that is required. However, achieving a simultaneous transverse stability is significantly more difficult for high-field seekers than for low-field seekers. This has to do with a prohibition, imposed by Maxwell’s equations, to create a maximum of field strength of a static electric field in free space. As a result, simple means to transversally focus high-field seekers are lacking [37]. Molecules in HFS states have a tendency to crash into the electrodes where the electric fields are the highest. However, this fundamental problem can be overcome by making use of alternating gradient (AG) focusers [38]. Figure 14.12a captures the general features of an AG decelerator. The AG lenses are formed from pairs of cylindrical electrodes to which a voltage difference is applied. Molecules in HFS states are defocused in the plane containing the electrode center lines, and focused in the orthogonal plane. As the molecules move down the beamline, the orientation of the lenses, and thus the focusing and defocusing directions, alternate. In any transverse direction, the defocusing lenses have less effect than the focusing ones, not because they are weaker (which they are not), but because the molecules are closer to the axis while acted upon by a defocusing lens than by a focusing lens. Molecules in HFS states are accelerated when entering the field of an AG lens and decelerated when leaving the field. By simply switching the lenses on and off at the appropriate times, AG focusing and deceleration of polar molecules can be achieved simultaneously. Figure 14.12b shows the potential energy along the z-axis of a single lens for a metastable CO(a3 Π1 , v = 0, J = 1) molecule in the MΩ = +1 HFS state. The molecules enter each lens with the electric fields turned off, so that their speed is not affected. Subsequently, the fields are rapidly turned on, and the HFS molecules are decelerated as they leave the lens and move from a region of high field to a region of low field. This process is repeated until the molecules reach the desired speed. A prototype machine of this type, consisting of 12 AG lenses, has been used to decelerate HFS metastable CO molecules [41]. In the meantime, the technique has been implemented by the group of Ed Hinds at Imperial College, London, and used to decelerate ground-state YbF molecules [42]. See also the chapter of Tarbutt and colleagues (Chapter 15). The transverse acceptance of the AG decelerator was estimated to be about a factor of 100 smaller than that of a decelerator for molecules in LFS states (of the same aperture) [39]. The acceptance can be increased by using a more sophisticated lens design, consisting of four electrodes rather than two. At Imperial College, a new AG decelerator was constructed to bring YbF molecules to a standstill. At the Fritz-Haber-Institut, AG focusing and deceleration has been applied to larger molecules such as benzonitrile [40]. We note that larger molecules may even lack any LFS states, in which case AG deceleration is the only option to slow them down. © 2009 by Taylor and Francis Group, LLC

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Cold Molecules: Theory, Experiment, Applications Lens 4

(a)

Lens 3 Lens 2

y z x

Lens 1

Molecular beam HV on

(b)

HV off

Stark shift, W (cm–1)

Leff 0 –0.5 –1 –1.5 –30

Wmax W –20

–10 Position, z (mm)

0

10

FIGURE 14.12 (a) Layout of an AG decelerator for polar molecules showing the first four deceleration stages. Each electrode pair both focuses and decelerates the molecules. (b) Crosssection of a single lens formed from two 20-mm-long rods with hemispherical end-caps, 6 mm in diameter and spaced 2 mm apart. The potential energy is shown along the longitudinal axis z for metastable CO molecules in the a3 Π1 , J = 1, MΩ = +1 state, when the potential difference between the electrodes is 20 kV. The procedure of switching the high voltages is indicated: the voltages are turned on when the bunch of molecules reaches the “HV on” position, and are turned off once they reach the “HV off” position. (From Bethlem, H.L. et al., J. Phys. B, 39, R263–R291, 2006. Copyright IOP Publishing Ltd. With permission.)

14.3 THE ZEEMAN, RYDBERG, AND OPTICAL DECELERATOR Inspired in part by the manipulation of polar molecules by electric fields, a magnetic analog of the Stark decelerator has recently been developed. Deceleration based on the magnetic interaction allows the manipulation of a wide range of atoms and molecules to which the Stark deceleration technique cannot be applied. The requisite rapid switching of the magnetic fields posed a considerable experimental challenge. © 2009 by Taylor and Francis Group, LLC


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The Zeeman deceleration technique was first experimentally demonstrated by slowing down ground-state H and D atoms, initially using six [43,44] and later twelve [45] pulsed magnetic field stages. The deceleration stages consist of 7.8-mm-long copperwire solenoids in which magnetic fields of up to 1.5 T are achieved. The coil design provides a cylindrically symmetric transverse restoring force that keeps the molecular beam focused on the longitudinal axis. When the current through the coils is switched, rise and fall times of the magnetic pulses as short as 5 μsec are achieved. These experiments have demonstrated that magnetic fields can be switched fast enough to allow for Zeeman deceleration of paramagnetic atoms and that the deceleration proceeds under the conditions of phase stability. The Zeeman deceleration technique has also been applied to decelerate metastable Ne atoms [46] in an 18-stage decelerator. Using electromagnetic coils that are encased in magnetic-steel shells with Permendur discs, even higher magnetic field strengths, of 3.6 T were achieved. Recently, the deceleration of metastable Ne atoms [47] and oxygen molecules [48] to velocities as low as 50 m/sec using 64 field stages has been reported. Compared to polar molecules, atoms or molecules in Rydberg states offer a much larger electric dipole moment. Hence, these particles can be manipulated using only modest electric field strengths and either a single or just a few field stages. Level crossings in the dense Rydberg manifolds limit the magnitude of the electric field strength that can be applied. The electric field manipulation of atoms and molecules in high Rydberg states has been pioneered using H2 molecules [49] and Ar atoms [50]. Using a Rydberg decelerator, H atoms could be stopped and electrostatically trapped in two [51] or three [52] dimensions. The short lifetimes of Rydberg states inherently limit the time that is available to store and study such species in a trap. However, if fluorescence to the ground state is the dominant decay process, cold samples of ground-state atoms or molecules can thus be produced. As all atoms and molecules possess Rydberg states, this may provide a versatile route to cold samples of atoms and molecules. Optical fields provide another general means to manipulate the motion of neutral particles. An intense optical field polarizes and aligns molecules [53]. In a laser focus, the molecules are subject to a force that is proportional to the gradient of the laser intensity. This force can be used to focus and trap the molecules. Optical manipulation of molecules was experimentally demonstrated by focusing [54] or deflecting [55] a beam of CS2 using a high-intensity pulsed laser beam. Optical forces have also been used to reduce the translational energy of molecular beams [56]. Benzene molecules were decelerated from 320 to 295 m/sec, while at the same time the xenon carrier gas was decelerated from 320 to 310 m/sec, illustrating the scope of this method. Rather than using a single laser beam, much higher forces can be generated by using two nearly counterpropagating laser beams. The two laser beams interfere and give rise to an optical lattice, that is, a periodic array of potential wells for polarizable atoms and molecules. The lattice can be set in motion by using laser beams with slightly different frequencies. By carefully controlling the frequency difference, the lattice can be moved at the same velocity as the molecules in the molecular beam. By lowering the velocity of the lattice, the molecules can be decelerated to any given velocity [57]. In this so-called optical Stark decelerator, the chirp of the laser beams needs to be very well controlled. In combination with the high intensities required, © 2009 by Taylor and Francis Group, LLC

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this is an experimental challenge. A simpler scheme was recently implemented using a constant-frequency offset between the two laser beams, such that the lattice moved with a velocity slightly below that of a molecular beam [58]. For a suitable choice of the parameters, the molecules make exactly half an oscillation within the optical potential. In this way, NO molecules were decelerated from 400 to 270 m/sec.

14.4 TRAPPING NEUTRAL POLAR MOLECULES 14.4.1

DC TRAPPING OF MOLECULES IN LOW-FIELD SEEKING STATES

The first electrostatic trap for polar molecules was demonstrated in 2000 using Starkdecelerated ND3 molecules [26]. This trap, consisting of a ring electrode and two hyperbolic end-caps in a quadrupole geometry, was originally proposed by Wing for Rydberg atoms [59]. Such a quadrupole trap can be mounted directly behind a Stark decelerator, and, through holes in the end-caps, a slow packet of molecules can be loaded into the trap using a procedure that is shown schematically in Figure 14.13. The specific values of the parameters referred to in the figure apply to the OH trapping experiments [60]. In this case, the Stark decelerator is programmed to produce a packet of molecules with a velocity of approximately 20 m/sec. The slow beam of OH radicals is loaded into the electrostatic trap with voltages of 7, 15, and −15 kV on the first end-cap, the ring electrode, and the second end-cap, respectively. This “loading configuration” of the trap is shown on the left side in Figure 14.13. In the loading geometry, a potential hill in the trap is created that exceeds the remaining kinetic energy of the molecules. The incoming OH radicals therefore come to a standstill near the center of the trap. Subsequently, the trap is switched into the “trapping configuration,” shown on the right side in Figure 14.13; the first end-cap is switched from 7 to −15 kV to create a (nearly) symmetric 500 mK deep potential well. A typical TOF profile, obtained in an OH deceleration and trapping experiment, is shown in Figure 14.14. In this experiment, the fluorescence signal of the OH radicals in the trap is detected through the hole in the end-cap. The TOF profile was recorded with Kr as carrier gas, and is therefore complementary to the series of TOF profiles shown in Figure 14.10. The gap in the TOF profile of the fast beam, which is due to the removal of the decelerated OH radicals, is indicated by a vertical arrow. The decelerated OH radicals come to a standstill about 7.4 msec after their production. At that time, indicated by another arrow, the voltages on the trap electrodes are switched to the trapping configuration. After some initial oscillations, a steady LIF signal is observed from the OH radicals in the trap, attesting to the fact that the molecules are confined. In the inset, the signal of the trapped OH radicals is shown on a 10 sec timescale, from which a 1/e trap lifetime of 1.6 sec can be deduced. We note that when Xe is used as a carrier gas, the most intense part of the molecular beam pulse can be selected, decelerated, and trapped [28]. The efficiency of the trap loading process could be increased by 40% by feedback control optimization using evolutionary strategies [61]. Electrostatic traps with other electrode geometries have been developed and tested as well. A four-electrode trap geometry that combines a dipole, quadrupole, and hexapole fields has been tested using decelerated ND3 molecules. By applying © 2009 by Taylor and Francis Group, LLC

531

Slowing, Trapping, and Storing of Polar Molecules Trapping geometry

Loading geometry

7 kV

–15 kV

15 kV –15 kV

0.6 Energy (cm–1)

Energy (cm–1)

0.6 0.4 0.2 0 –12

15 kV –15 kV

–6 0 6 Position (mm)

12

0.4 0.2 0 –12

–6 0 6 Position (mm)

12

FIGURE 14.13 Schematic of the loading procedure of the electrostatic quadrupole trap. In the “loading configuration.” the voltages on the trap electrodes are set such that a potential hill is created in the trap that is higher than the remaining kinetic energy of the incoming molecules. At the time the molecules come to a standstill in the center of the trap, the trap is switched into the “trapping configuration.” In this geometry a (nearly) symmetric 500 mK deep potential well is created, which confines the molecules. (From van de Meerakker, S.Y.T. et al., Annu. Rev. Phys. Chem., 57, 159–190, 2006. Copyright (2006) by Annual Reviews www.annualreviews.org. With permission.)

different voltages to the electrodes, a double-well or donut trapping potential can be generated. Rapid switching between the various trap potentials offers good prospects for studying collisions as a function of collision energy at low temperatures [62]. Confinement of Stark decelerated molecules in combined magnetic and electric fields has recently been demonstrated in the JILA group with the OH radical. An electric field was superimposed on a magnetic field to create a combined magnetoelectric trapping potential [63]. The adjustable electric field might be advantageous in the study of low-energy dipolar interactions. Rather then actively manipulating fast molecules to generate slow ones, a mere velocity selection of slow molecules from an effusive source has also been successfully used; the velocity selection relied on a bent quadrupole guide with a longitudinal curvature such that only slow molecules could follow [13,64]. © 2009 by Taylor and Francis Group, LLC

532

LIF signal (arb. units)

Normalized intensity

Cold Molecules: Theory, Experiment, Applications 1.00 0.75 0.50 0.25 0.00 0 Trap on

2.0

4.0

6.0 Time of flight (msec)

2

4 6 8 Time (sec)

8.0

10

10.0

FIGURE 14.14 Measured TOF profile of OH( J = 3/2) radicals for a trapping experiment. The time at which the trap is switched on is indicated. In the inset, the signal of the trapped OH radicals is shown on a 10 sec timescale. (Figure partially reproduced from van de Meerakker, S.Y.T. et al., Phys. Rev. Lett., 94, 023004, 2005. Copyright the American Physical Society. With permission.)

14.4.2

STORAGE RING AND MOLECULAR SYNCHROTRON

In its simplest form, a storage ring is a trap in which the particles are subject to a minimum potential energy on a circle rather than at a point. The advantage of a storage ring over a trap is that it can confine packets of particles with a nonzero longitudinal velocity. While circling the ring, these particles can be made to interact repeatedly, at well-defined times and at distinct positions, with electromagnetic fields and/or other particles. Figure 14.15a shows the experimental setup used in the demonstration of an electrostatic storage ring for neutral molecules [65]. A beam of ammonia molecules was decelerated to a velocity of 92 m/sec, longitudinally cooled to a temperature of 300 μK by a buncher, and focused into a 25 cm diameter hexapole ring. The density of ammonia molecules inside the storage ring was probed via a laser-ionization detection scheme. In Figure 14.15b, the ion signal is shown as a function of the storage time in the ring; the time axis originates at the time when the high voltages on the ring are switched on. Peaks are observed when the packet of molecules passes the detection zone. Upon making successive round trips, the packet of molecules gradually spreads out as a result of the residual velocity spread, until it fills the entire ring. In order to counteract the spreading of the packet in the ring, we have constructed a storage ring consisting of two half-rings separated by a 2 mm gap, schematically shown in Figure 14.16a. By appropriately switching the voltages applied to the electrodes as the molecules pass through the gaps between the two half-rings, molecules can be accelerated, decelerated, or bunched. This device is the neutralparticle analog of a synchrotron for charged particles. Figure 14.16b shows again the density of molecules stored in the ring as a function of storage time. It is seen that, after an initial decrease during the first 25 round trips, the width of the stored packet remains constant. Bunching not only ensures a high density of the © 2009 by Taylor and Francis Group, LLC

533

Slowing, Trapping, and Storing of Polar Molecules Storage ring

(a)

Stark decelerator

Buncher Hexapole Hexapole

Ion detector Pulsed UV laser

Pulsed valve 0.2 ND3 density (arb. units)

(b)

4 0.1

Dv = 0.83 ± 0.13 m/sec T = 300 ± 100 mK

49

50

51

3 0

2

0.42 1

10

20

0.43 0.44 t (sec) 30

0 0

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100

150 t (msec)

200

250

300

FIGURE 14.15 (a) Schematic of an experimental setup with a storage ring. A pulsed beam of ND3 molecules is Stark-decelerated to 92 m/sec, cooled to 300 μK using a buncher, and focused into a hexapole torus storage ring. (b) Density of ammonia molecules at the detection zone inside the ring as a function of storage time up to the 33rd round trip. Due to the spreading of the packet, the peak density decreases as 1/t (shown as the dashed line). In the inset, a measurement of the ammonia density after 49 to 51 round trips is shown, together with a multipeak Gaussian fit. (From Crompvoets, F.M.H. et al., Phys. Rev. A, 69, 063406, 2004. Copyright the American Physical Society. With permission.)

stored molecules, but, in addition, it makes it possible to inject multiple co- or counterpropagating packets into the ring, without affecting the packet(s) already stored [66].

14.4.3 AC TRAPPING OF MOLECULES IN HIGH-FIELD SEEKING STATES Trapping molecules in HFS states is of particular interest for two reasons: (1) The ground state of a system is always lowered by an external perturbation. Therefore, the ground state of any molecule is HFS. In the ground state, trap loss due to inelastic collisions is absent, making it possible to cool such molecules further using evaporative or sympathetic cooling. This is particularly relevant as the dipole–dipole interaction is expected to lead to large cross-sections for inelastic collisions of polar molecules in excited rovibrational states [67]. (2) Molecules composed of heavy atoms or many light atoms, such as polycyclic hydrocarbons, have small rotational constants. Consequently, all states of these molecules become HFS in relatively weak magnetic or electric fields. © 2009 by Taylor and Francis Group, LLC

534

Cold Molecules: Theory, Experiment, Applications Ion detector

Laser

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(b)

ND3-density (arb. units)

55 msec 139.5

139.6 139.7 Time (msec)

139.8

35 msec

10 20

372.3 372.4 372.5 Time (msec)

30 40

0

100

200 Storage time (msec)

300

400

FIGURE 14.16 (a) Schematic of a molecular synchrotron. The synchrotron consists of two hexapole half-rings with a 12.5 cm radius, separated by a 2 mm gap. (b) Density of ammonia molecules at the detection zone inside the synchrotron as a function of storage time for up to the 40th round trip. Expanded views of two TOF profiles are shown in the insets, illustrating the narrow widths of these peaks. (From Heiner, C.E., Nature Phys., 3, 115–118, 2007. Copyright MacMillan Publishers Ltd. With permission.)

The problem of trapping molecules in HFS states is essentially the same as that of decelerating molecules in HFS states (discussed in Section 14.2.6). Ideally, one would like to have an electrode geometry that creates a maximum electric field strength at some position away from the electrodes; however, this is prohibited by Maxwell’s equations [37]. Although it is not possible to generate a maximum of the electric field in free space, it is possible to generate an electric field that has a saddle point. This is accomplished by superimposing an inhomogeneous electric field and a homogeneous electric field. In such a field, molecules are focused along one direction, while being defocused along the other. By reversing the direction of the inhomogeneous electric field, the focusing and defocusing directions can be interchanged. If this is done periodically, molecules will be further away from the saddle point along the focusing direction and closer to the saddle point along the defocusing direction, leading to a net time-averaged focusing force in any direction. Such a trap works both for molecules in HFS and LFS states. There are three possible electrode geometries that can be used © 2009 by Taylor and Francis Group, LLC

535


to create the desired electric three-dimensional trapping field [68]. By now, all these geometries have been demonstrated for atoms and/or molecules. Figure 14.17a shows schematically the electrode geometry of our cylindrical ac trap that has been used to trap both ND3 molecules [68,69] and Rb atoms [70]. The trap has a hexapole symmetry. Consequently, when positive and negative voltages are applied alternately to the four electrodes of the trap, a perfect hexapole field is obtained. In order to create a saddle point, the voltage applied to one of the end-caps is increased, while the voltage on the other end-cap is decreased. This adds a dipole term to the electric field. If the direction of the hexapole term is reversed, the focusing and defocusing directions are interchanged. Figure 14.17b,c shows the strength of the electric field along the symmetry axis z and along the radial distance r from the z-axis, when the field is either focusing along z or r. 4.6 mm

1.8 mm

(a)

10 mm 2 mm

2.9 mm 9.1 mm (c)

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Electric field (kV/cm)

(b)

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–2

0 z (mm)

2

4

30 z-focusing 20 10 0 –4

r-focusing

-2

0 2 r (mm)

4

FIGURE 14.17 (a) Schematic of a cylindrically symmetric ac trap. (b) Electric field strength along the symmetry axis z. (c) The electric field as a function of the radial distance r from the z-axis. When voltages of 5, 7.5, −7.5, and −5 kV are applied to the electrodes, molecules in HFS states are focused toward the center along the z-axis and defocused in the radial direction r. When voltages of 11, 1.6, −1.6, and −11 kV are applied to the electrodes, molecules in HFS states are radially focused along r and defocused along the z-axis. (From Bethlem, H.L. et al., Phys. Rev. A, 74, 063403, 2006. Copyright the American Physical Society. With permission.)


536


ND3 molecules were loaded into the ac trap by Stark-decelerating them to a standstill using their LFS state, and subsequently pumping about 20% of the molecules, with a microwave pulse, to a HFS state. Figure 14.18 shows the density of ND3 molecules at the center of the ac trap as a function of the switching frequency for molecules in either the HFS or LFS component of the ground state of para-ammonia [68]. For clarity, the signal for molecules in LFS states has been vertically offset. The signal for the HFS is scaled up by a factor of five, to correct for the smaller initial density of HFS. At low frequencies of the applied voltages, the trajectories of the molecules in the ac trap are unstable and no signal is observed. Above a frequency of about 900 Hz, the trap becomes abruptly stable. Maximum signal is observed at 1100 Hz. When the frequency is increased further, the molecules have less time to move between switching times, and the net time-averaged force on the molecules decreases. As a result the depth of the trap decreases. Higher-order terms in the trapping potential give rise to a (frequency-independent) potential that reduces the trap depth for molecules in HFS states and increases the trap depth for molecules in LFS states. Therefore, the signal of the HFS drops faster with increasing frequency than the signal of the LFS.

ND3 density (arb. units)

LFS

HFS ×5

500

1000

1500 Wdriven/2p (Hz)

2000

2500

FIGURE 14.18 Density of 15 ND3 molecules in LFS and HFS states of the |J, K = |1, 1 level at the center of the trap as a function of the switching frequency Ωdriven /2π. The measurements were performed 80 msec after the molecules had been loaded into the trap. The signal of the HFS is scaled up by a factor of 5 to compensate for the fact that the initial density of HFS is only about 20% of that of the LFS. (From Bethlem, H.L. et al., Phys. Rev. A, 74, 063403, 2006. Copyright the American Physical Society. With permission.)



537

It is instructive to compare the ac trap with the dc trap, discussed in Section 14.4.1. The depth of the electrostatic traps is on the order of 1 K (depending on the molecular species and details of the trap design), and their volume is typically 1 cm3 . An ac electric trap has a depth of about 1 to 10 mK and a volume of about 10−2 cm3 [68].

14.4.4 TRAP LIFETIME LIMITATIONS Returning to Figure 14.14, we see that the time during which the OH radicals were stored in our electrostatic trap was on the order of a few seconds. Molecules can leave the trap via several distinct mechanisms. As there is a zero electric field at the center of the dc quadrupole or hexapole traps, the molecules can undergo transitions to degenerate quantum states that cannot be trapped. In practice, however, such Majorana transitions have not (yet) become a limitation of the lifetime, both because the volume in which these transitions could occur is very small and because, when hyperfine structure is included, molecules are often in states that are exclusively LFS; that is, there is no degeneracy [71]. Molecules can collide with particles in the residual gas in the vacuum chamber, leading to a kinetic energy transfer that results in a direct loss of trapped molecules from the rather shallow trap. Molecules in the trap can also collide with one another. In this case, inelastic collision processes can transfer the molecules from the trapped state to a different quantum state that is either antitrapped (HFS) or that is subjected to a lower trapping potential. Finally, the molecules can absorb black-body radiation from the (room-temperature) environment, leading to a change of the internal quantum state of the molecule. For future studies of collisions between trapped molecules, a quantitative understanding of all trap-loss mechanisms is essential. The trap losses due to optical pumping by black-body radiation and due to collisions with the background gas have been studied by monitoring the population decay of OH and OD radicals in a room-temperature electrostatic trap [72]. By comparing two isotopes of the same molecular species under otherwise identical conditions, the trap-loss mechanisms could be disentangled and quantified. The optical pumping rate by room-temperature black-body radiation was determined as 0.49 sec−1 for the OH radical and 0.16 sec−1 for the OD radical. Trap loss due to black-body radiation is thus a major limitation for the room-temperature trapping of OH radicals. The trapped molecules would have to be shielded from thermal radiation if longer trapping times were required. Most polar molecules exhibit strong electric dipole-allowed rovibrational transitions within the room-temperature black-body spectral region. In Table 14.1, the calculated blackbody pumping rates out of a specified initial quantum state are given for a number of polar molecules for which trapping is being pursued using the currently available techniques.

14.5 APPLICATIONS OF DECELERATED BEAMS AND TRAPPED MOLECULES As already mentioned in the Introduction, the three-dimensional focused packets of decelerated molecules with their tunable velocity and their narrow velocity spread can, for instance, be used for high-resolution spectroscopy and metrology, © 2009 by Taylor and Francis Group, LLC

538


TABLE 14.1 Pumping Rates Due to Black-body Radiation at Two Different Temperatures, Out of a Specified Initial State, for a Number of Polar Molecules Pumping Rate (sec−1 ) System OH/OD NH/ND NH/ND NH3 /ND3 SO 6 LiH/6 LiD CaH/CaD RbCs KRb CO

Initial State X2Π

3 3/2 , J = 2

a1 Δ, J = 2 X 3 Σ− , N = 0, J = 1 X˜ 1 A1 , J = 1, |K| = 1 X 3 Σ− , N = 0, J = 1 X 1 Σ+ , J = 1 X 2 Σ+ , N = 0, J = 21 X 1 Σ+ , J = 0 X 1 Σ+ , J = 0 a3 Π1,2 , J = 1, 2

295 K 0.49/0.16 0.36/0.12 0.12/0.036 0.23/0.14 0.01 1.64/0.81 0.048/0.063 / plane. 428 Cold trapped coulomb clusters. 658-670. See also Multispecies ensembles charge-coupled device (CCD). 656-670 collisional heating of ion crystals. 664-666 ion diffusion in. 660 ion ensembles in fluid phase. 659 ion-neutral head-on collision. 665 ion numbers, determination. 664 mixed-species ion crystals, 662 molecular dynamics (MD) simulations. 658-664 multispecies ensembles, heating effects in. 666-670 properties of. 658-670 spatial separation of species. 662 Collision systems of three alkali metal atoms. 95-101 Collisional heating of ion crystals. 664-666 Collisional properties, halodimers. 337-339 Collisional relaxation suppression in fermi gases. 365-368 Collision-induced quenching. 483 Collisions

atom-molecule, at cold and ultracold temperatures, 71-78 theory. 3-33. See also Atomic and molecular interactions calculations, 5 Combined molecular state (CMS). 112 Compound state resonance. 24 Compression effect. 276-282. See also Dynamical hole controlling with a nonimpulsive pulse, 283 dynamical hole in initial slate wavefunction. 276-282 in lower state after a PA pulse. 282 for PA with second pulse dynamical hole in initial state wavefunction. 279-281 Condensed matter Physics with cold polar molecules. 421-463. See also Many-body physics Confined geometries, ultracold molecules in. 163 Control qubit. 632 Controlled not (CNOT) gate. 632 Cooling using buffer-gas, 475 Cooper pair boxes, xxv Coulomb clusters, cold trapped. See Cold trapped coulomb clusters Counter-intuitive pulse sequencing. ARPA. 293 Coupled equations, 17-23 bound states, 21 decoupling approximations, 20-21 log-derivative matrix. 20 numerical methods for bound states. 21-23 for scattering, 19-20 scattering calculations, 19 Coupled-channel bound state. 410 Coupled-channel methods, 21 Courant-Snyder matrix, 589 Cross-sections, collisions. 5-6 Cryo-bakcout. 488 Cryogenic Helium buffer gas. 473-504. See also Buffer gas; Zeeman relaxation collisions atoms. 482 beam production using. 473-504. See also Beam injection; Beam production capillary filling. 480 cooling using. 473-504. See also Buffer-gas cooling discharge etching. 480-483 rotational and vibrational relaxation. 483-484 spin relaxation loss (atoms). 488-491 trap loading using. 473-504. See also Loading Crystalline molecular phase, in ultracold Fermi gases. 387-392 Born-Oppenheimer potential

Index

in a many-body system erf molecules of heavy and light fermions. 387-389 gas-crystal quantum transition. 390-391 molecular superlallice in an optical lattice. 391-392 C$2 astrophysical experiments with, 615-618 Curve-crossing. 24

D Dc field, cold polar molecules collisions in. 439 DCSs. See Double-continuum states DC trapping of molecules in low-field seeking states. 530-532 De Broglie wavelengths. 4. 357 Debye frequency, 451 Decelerated beams, applications, 537-543 collision studies at tunable collision energy. 540-542 crossed-beam scattering machine, 542 direct lifetime measurements of metastable states. 542-543 high-resolution spectroscopy and metrology. 538-540 Decoherence. in switchable dipoles. 643-645 Decoupling approximations, coupled equations. 20-21 Deflection angle. 5 Depletion laser (DEP). 199 DEP. See Depletion laser Destructive detection technique, multispecies ensembles. 672-673 Detection methods, ultracold Feshbach molecules. 329-330 Detuning effect, in adiabatic transfer within PA window. 269 Diatomic molecules narrow close-lying levels of. 610-615 molecular ion HfF + , 613 natural widths of quasidegenerate states. estimate. 614-615 with quasidegenerate fine-structure and vibrational levels, 61 1-613 quasidegenerate hyperfine level molecules. 610-611 rotational level molecules. 610-61 I ultracold Fermi gases. 357-360 Diep and Johnson (DJ). 108 Differential crosssections (DCSs). 5. 16. 150-162 Differential scattering in electromagnetic fields. 150-162 Diffusion Monte Carlo gas-crystal transition. 390 Diffusively extracted beams, buffer-gas, 499-501 Dipolar relaxation, buffer gas. 489 Dipole blockade. 641-642 Dipole dipole interaction strength. 643-644 Dipole field, polar molecules in. 51 I

709

Dipoles, 39-67. See also Electric dipoles; Field due to a dipole; Interaction of dipoles classical dipoles. 40-42 interaction of, 57-67 Dirac equation. 557 Direct methods of cold molecules production, xiii. xx Discharge etching, buffer-gas. 479-483 Dissociation patterns, ultracold Feshbach molecules. 330 DiVincen/o criteria, 646 DJ. See Diep and Johnson Doppler cooling methods. 320 Doppler shift. 567 Double pulses, experiments with, 576-579 Double sum over partial waves, 10 Double-continuum states (DCSs). 104 Double-minimum potentials, enhancement in. 202-204 Dump-before-pump pulse ordering of STIR AP. 293 D-wave, 10 Dynamical hole in initial state wavefunction. 276-282 compression effect. 276-282 observation. 276-278 for PA with second pulse. 279-281 depletion hole, observation of, 276-278 momentum kick, observation of. 276-278 momentum transfer analysis with partially integrated mass current and population. 278-279 population redistribution in lower state after a PA pulse, 281-282 correlated pairs of hot atoms. 281-282 in a 3 £J" state in Cesium PA. 281

E EDM. See Electric dipole moment Effective interaction potential, cold polar molecules. 458 Effective many-body Hamiltonians, 423-425 Effective potential, cold polar molecules, 443 Effective range theory. 12 Efimov effect/trimers. xix. 339-341, 357. 379-382, 393 Elastic collisions. 485 Elastic intermolecular interaction, mass ratio effect on, 370-373 Elastic scattering, 6, 338 Electric dipole moment (EDM). 209. 540, 557-558 Electric dipoles at ultralow temperatures, 39-67 field due to a dipole. 51-57 fundamental fact of. 44 lambda-doubling, molecules with. 48-51

Index

710

Electric dipoles at ultralow temperatures {Continued) quantum-mechanical dipoles in fields. 42-51 Electric fields, collisions in. 134-138 electric-field-induced resonances. 137-138 low temperature molecular collisions. 125-163. See also Low temperature molecular collisions scattering of molecular dipoles, 136-137 Stark relaxation. 134-136 Electric fields, polar molecules in alternating gradient (AG) decelerator. 516 decelerated beams, applications. 537-543 dipole field, 511 full-width at half-maximum (FWHM). 513 hexapole fields. 511 high-resolution spectroscopy and metrology. 538-540 low-field seeking (LPS) quantum state, 511-512 optical decelerator. 528-530 quadrupole field, 511 Rydbcrg decelerator. 528-530 slowing. 509-547 Stark decelerator, 511-528. See also Neutral polar molecules storing of, 509-547 trapped molecules, applications. 537-543 trapping. 509-547 Zeeman decelerator. 528-530 Electromagnetic fields, phot dissociation in. 192-193 Electron orbital angular momentum, 7 Electrospray ionization (ESI), 656 Electrostatic traps, polar molecules. 530 Energy-normalized radial wavefunctions, 267 Entangled states ultracold polar molecules. quantum information and, 631-632 control qubit. 632 controlled not (CNOT) gate, 632

target qubit. 632 Entrance channel dominated resonances. 414-416 near a collision threshold. 236 Entrance channel. Feshbach resonance. 321 Errors, in switchable dipoles. 643-645 ESI. See Electrospray ionization Evaporative cooling. 320 Evaporative loss, in buffer-gas loading, 485-487 Extended Hubbard model, cold polar molecules, 455 External electric fields, cold polar molecules in. coupling. 437-438 External fields collisions. 26-30 high-field-seeking states. 28 low-field-seeking states. 28 spin relaxation, 28

total angular momentum representations. 27-28 totally uncoupled representation. 27 Extraction conditions, buffer gas beam production. 497-498 F Fano-Feshbach resonance. 410 Far off-resonance trap (FORT). 436 Fermi gases, homonuclear diatomic molecules in. 360-370 Fermi statistics, 366, 368 Fermions. 325-327 Feshbach molecules. See also Ultracold Feshbach molecules bosons. 325-327 detecting. 325-330 fermions. 325-327 Feshbach molecule bound states, 414 making, 325-330 microscopic theory of. 400-406 quantum statistics role. 325-327 Slern-Gerlach separation of an alom-dimer mixture. 325 Feshbach resonance, basic physics. 321-323 background scattering length. 322 closed channel. 321 entrance channel. 321 halodimers, 323 magnetically tunable Feshbach resonance. 322 resonance position. 322 resonance width. 322 two-channel model for, 322 Feshbach resonances. 24. 27-30. 356-357.

406-409 Feshbach resonance threshold. 192 halo molecules formation via optically induced. 261-262 magnetic field near, scattering length dependence on, 358 in molecular collisions. 80-82 non-optical. 170 in reactive scattering, 92-95

switching process, 293 tunable Feshbach resonances. 134 two-body physics of, 358 two-channel two-potential approach. 406-407 Few-body physics, halo dimers, 337-339 Field due to a dipole. 51-57 j= 1/2.53-55 j= 1.55-57 Field orientations effects. 144-150 Field-dressed energy levels, of cold polar molecules with ac fields, 445 Fine tuning, 612 Floating lattices of dipoles, cold polar molecules, 450-455

Index

phonon Hamiltonian, 451 single band Hubbard Hamiltonian. 451 Formalism. 40 FORT. See Far off-resonance trap Four-body problem with two heteronuclear molecules. 371 Four-body system, set of coordinates for, 362 Four-electrode trap geometry. 530 Franck-Condon (FC) factors. 181. 201. 240-241. 269. 271. 273-274. 279, 309. 344-346 for transitions in Rl>2 and KRb. 310 Full-width at half-maximum (FWHM). 254. 513.562 Fundamental physics tests. See also Beams of cold polar radicals alternating gradient deceleration of polar molecules. 580-592 beams of cold polar radicals. 560-568 fundamental constants nature of. 556-557 testing, 557-560 invariance principles, testing. 556-560 Lorent/ invariance. 560 parity violation. 559 preparation and manipulation of molecules for. 555-593 FWF. See Austrian Science Fund

G Gas-crystal quantum transition, in ultracold Fermi gases, 390-391 Gaussian laser pulses. ARPA with. 304 Ground vibrational state molecules. 108-112 H 2 + Hi system. 108 Ground-state molecules, ultracold gases of, 342-346 stimulated raman adiabatic passage (STIRAP), 343-344 STIRAP method. 342

H H2. astrophysical observations of. 602-604 Haldane model. 432 Halo dimers. 323. 335-342 collisional properties. 337-339 Fllmov three-body stales, 339-341 few-body physics. 337-339 inelastic dimer-dimer scattering. 338 interaction between two, experiment, 338 and universality. 336-337 Halo molecules. 228. 247 formation via optically induced Feshbach resonance, 261-262 Halo regime. 325

711

Hamiltonian for the colliding pair, 15 Hamiltonian matrix, 23 Hartree approach. 112 HBO. See Hybrid Born-Oppcnheimer Heavy fermion molecules. 376-379. 381-387 Helium atoms, metastable. scattering. 16-17 Heteronuclear alkali-metal dimer systems. reactions of, 106-108 Heteronuclear molecules in Fermi-Fermi mixtures. 370-387 collisional relaxation Born-Oppenheimer approximation. 374-376 of heavy fermion molecules. 381-387 of light fermions. 381-387 for moderate mass ratios. 373-374 (rimers formation. 381-387 four-body problem. 371 mass ratio effect on elastic intermolecular interaction. 370-373 molecules of heavy and light fermionic atoms. 376-379 trimer states. 379-381 Hexapole fields, polar molecules in. 511 HFS. See High-field seekers HIBRIDON package. 20 High-field seekers (HFS) states, 127, 484 AC trapping of molecules in, 533-537 molecules deceleration in. 527-528 High-resolution spectroscopy of molecular hydrogen ions. 695 Homonuclear diatomic molecules in fermi gases, 360-370 collisional relaxation, suppression. 365-368 collisional stability, 368-370 dimer-dimer collisions. 367 *K2 ultracold molecular gas. 369 molecular BEC. 368-370 weakly interacting gas of bosonic molecules. 360-365 Hubbard lattice models, cold polar molecules. 430-431 analog quantum simulation, 431 Hubbard models in self-assembled dipolar lattices. 433-436 floating lattices of dipoles, 434 realization of. 435 Hund'srule, 183-186 Hybrid Born-Oppenheimer (HBO) approach. 382-384.391 Hydrodynamically extracted beams, buffer-gas. 501-504 Hydrogen molecular ions Ht and HD+ astrophysical experiments. 618-619 Hyperfine interaction. Zeeman effect in alkali-metal atoms. 400-401

712

Hyperfine states/levels stark shifts of, 568-571 zeeman shifts of. 568-571 Hyperfine structure. 601 Hyperspherical coordinates, 31

I Impulsive or adiabatic approximations. 283-285 compression effect controlling with a nonimpulsive pulse. 283 nonadiabatic broadband pulse, 284-285 thermal average. 284-585 Incoherent hopping, cold polar molecules. 453 Indirect methods of cold molecules production. xiii. xx Inelastic atom-molecule collisions. 71-84 atom-molecular ion collisions. 83-84 Boothroyd. Martin, and Peterson (BMP) potential. 76 at cold and ultracold temperatures, 71-78 Feshbach resonances. 80-82 potential energy surface (PES). 72 quasiresonant transitions. 82-83 rotational relaxation. 71-82 shape resonances in molecular collisions, 79-80 vibrational relaxation. 71-82 zero-temperature rate coefficients for 4 He + U2(vJ) collisions, 83 Inelastic collisions loss, ultracold Feshbach molecules. 327 of 2 £ molecules with He. 493-494 of 3 E molecules with He. 494-496 of molecules at ultracold temperatures, 69-116 Inelastic dimer-dimer scattering. 338 Inelastic molecule-molecule collisions. 108-115 ground vibrational state molecules. 108-112 OD + OD and OH + OH collisions. I 11 rovibralional combinations of the HT — H2. 114 vibrational!)' inelastic transitions. 112-115 Inelastic scattering. 6 Infrared spectroscopy of HD + ions. 695-697 Inner classically forbidden region, single-channel scattering, 14 Inner turning point, single-channel scattering. 14 Integral cross-sections. 5-6. 16 Interaction of dipoles. 57-67 adiabatic potential energy surfaces in one dimension, 65-66 in two dimensions, 61-62 asymptotic form of interaction. 66-67 j = 1/2 molecules. 62-65 potential matrix elements, 58-61 Interaction potentials of cold polar molecules, engineering. 436-447

Index

ad hoc potentials with ac fields, designing. 444-447 dressed adiabatic ground-state interaction potential. 445 effects of dc and ac microwave fields on, 446 field-dressed energy levels, 445 oscillating dipole moments, 445 two-particle collision process, 447 two-particle spectrum. 445 effective interactions derivation. 439 external electric fields, coupling to. 437-138 molecular Hamiltonian. 436-438 repulsive Mr potential in 2D. designing, 439-444 Born-Oppenheimer potentials. 441 collisional stability. 443 dc field, collisions in. 439 effective potential, contour plot. 443 effective two-dimensional interaction. 444 parabolic confinement, 442 quasi two-dimensional. 442 three-dimensional barrier. 442 rotational spectrum. 436—437 two molecules, 438—447 Interactions, cold polar molecules attractive time averaged interaction. 429 PfatTian wave functions, 430 purely repulsive three-dimensional interactions, 429 in single ac field. 429 Interatomic interactions. 402 Interatomic separation, effective entrance-channel and closed-channel potentials vs.. 408 Internal states, coherent manipulation of. 568-579 Internal-state manipulation near threshold ultracold Feshbach molecules. 331-335 avoided level crossings, 331-333 Invariance principles, testing. 556-560 fundamental constants, nature of. 556-557 fundamental symmetries, testing. 557-560 parity violation. 559 Inversion spectrum of ammonia. 605-609 Ion trapping and cold molecules production. 654-658 atomic ion cooling. 656 linear radio-frequency trap. 655 radio-frequency ion traps. 654-656 Ion-neutral chemical reactions, sympathetically cooled molecular ions. 678-685 Ion-neutral head-on collision, 665 Isotope effect, 601 Isotopically substituted alkali-metal dimer systems, reactions of, 106-108

Index

J Jacobi-Anger expansion. 158 Jeffreys-Wentzel-Kramers-Brillouin(JWKB), 11. 226-227 JWKB. See Jeffreys-Wentzel-Kramers-Brillouin K Kaser-cooled (LC), 658 ^-dependent scattering length. I I Kosterlitz-Thouless transition. 422 L Lab frame. 46 Laboratory coordinates. 4-5 Ladder technique, 177 Lamb shift. 601 Lambda (A)-doubling. molecules with. 48-51 Lambda two-color PA approach. 179 Landau-Zener model. 236-237. 333 Langevin model for barrierless reactions. 32-33.99 Lang-Firsov transformation. 451 Larmor precession. 572 Laser ablation, xxi. 476-478 Laser cooling techniques. 320 Laser-cooled atomic ions reactions, 679-681 Laser-induced atom desorption (LIAD). 475. 476-478 Laser-induced fluorescence (LIF) detection, 523, 561 Lattice spin models, cold polar molecules, 431^33.459-463 anisotropic spin models. 432 dipole-dipole coupling. 461 Haldane model, 432 Movre-Pichler potentials. 462 rotational excitations. 460 self-assembled dipolar lattices Hubbard models in. 433-436 spatial dependence. 432 spin-dependent. 432 LC. See Kaser-cooled LFS. See Low-field seeking LIAD. See Laser-induced atom desorption LIF. See Laser-induced fluorescence Light fermion molecules, 381-387 Light fermionic atoms, molecules of, 376-379 Like atoms, pholoassocialion of, 172-182 Linear radio-frequency trap. 655 Linearly chirped pulse in picosecond domain, 254-256 central frequency, 254-256 chirped pulse. 254-256 energy, 254-256

713

spectral width, 254-256 temporalwidths. 254-256 Liouville's theorem. 525 Loading, buffer-gas. 475 of magnetic traps. 484-496 effective volume of trap. 486 evaporative loss, 485-487 field-free diffusion lifetime. 486 trapped molecule lifetime vs. buffer-gas density. 485 trapped molecules, lifetime of. 485-491 Log-derivative matrix. 20 Longitudinal focusing of stark decelerated molecular beam. 525-527 Lorentz invariance. 560 Low temperature molecular collisions cold controlled chemistry. 162-163 electric fields, collisions in. 134-138 external electromagnetic fields effects on. 125-163. See also Superimposed electric and magnetic fields magnetic traps, collisions in. 126-134 tunable feshbach resonances. 134 tunable shape resonances. 131-134 Zeeman relaxation. 127-131 Low-energy collisions, single-channel scattering. 11-13 Low-energy scattering. 25-26 Low-field seeking (LFS) quantum state. 127. 484. 511-512

M MA. See Magnetoassociation Machnumber focusing. 502 Magnetic Feshbach resonances. 192 electric fields effect on. 140-141 Magnetic moment spectroscopy. 334 Magnetic traps buffer-gas loading of. 484-496 collisions in, 126-134 Magnetic tuning of scattering length, 409-412 background-scattering length, 409-410 bound-state energy and resonance position, relation between, 410-412 resonance width, 409-410 Magnetically tunable Feshbach resonance. 322 Magnetically tunable resonances, near a collision threshold. 233-238 Magnetoassociation (MA). 170. 191,236-241 Magneto-optical trap (MOT). 170 Many-body dynamics of cold polar molecules. 423-425 Many-body physics with cold polar molecules, 447-463. See also Lattice spin models antiadiabatic regime, 454 Born-Markov approximation. 452 coherent hopping, 452

Index

714

Many-body physics with cold polar molecules {Continual) effective interaction potential. 458 extended Hubbard model. 455 floating lattices of dipoles. 45()-455 incoherent hopping. 453 particle-phonon coupling, 453 polaron shift. 452 quantum phases of two-dimensional dipoles. 448 spectrum of polar molecule. 456 three-body interactions. 455-459 two-dimensional self-assembled crystals. 447-450 Matrix elements, dipoles. 58-61 Maxwell's equations. 557 Maxwcll-Boltzmann distribution, xx mBEC. See Molecular Bose-Einstein condensates MD. See Molecular dynamics Mean scattering length. 12 Metastable Feshbach molecules. 325 Metastable helium atoms, scattering. 16-17 Microscopic theory of Feshbach molecules. 400-406 Microwave molecular spectra, astrophysical observations of, 604-605 Molecular beam injection, buffer-gas, 479 Molecular Bose-Einstein condensates (mBEC). 321.336.341-342 in fermi gases. 368-370 Molecular collisions Feshbach resonances in. 80-82 shape resonances in. 79-80 Molecular data, shaped laser pulses. 308-311 Molecular dipoles, scattering of, 136-137 Molecular dynamics (MD) simulations. 658-664 Molecular flux. 564-567 Molecular Hamiltonian. of cold polar molecules, 436-438 Molecular hydrogen ions, high-resolution spectroscopy of. 695 -E Molecules, inelastic collisions with He. 493-494 Molecular ion HfF + . 613 Molecular ion production. 656-658 Molecular ions reactions, sympathetically cooled ions. 681-685 cold H^ and room-temperalure HT. reaction between, 682 sequential chemical reactions. 684 Molecular ions, ultracold. 208-209 Molecular regimes in ultracold Fermi gases. 355-393. See also Crystalline molecular phase: Heteronuclear molecules; Homonuclear diatomic molecules diatomic molecules, 357-360

Feshbach resonances. 357-360 Molecular state decoherence. 644 Molecular states near a collision threshold. 221-241 bound-state binding energies. 230 closed channel-dominated resonances, 236 entrance channel dominated resonances, 236 interactions for multiple potentials, 230-233 magnetically tunable resonances. 233-238 photoassociation. 238-241 one-color PA. 239-241 two-color PA, 239-241 single potential, properties for, 223-230 A-wave scattering length. 229-230 Molecular superlattice in optical lattice. 391-392 Molecular synchrotron, in trapping neutral polar molecules. 532-533 Molecular thermometry, sympathetically cooled molecular ions, 693-695 MOLSCAT package, 20 Momentum transfer analysis with partially integrated mass current and population. 278-279 MOT. See Magneto-optical trap Motional resonance coupling, multispecies ensembles. 674-676 Movre-Pichler potentials, cold polar molecules, 462 MR. See Muchnick and Russek Much nick and Russek (MR). 72 Multichannel photoassociation. 294-302 two-Slate description. 294-300 Multichannel scattering. 15-17 atom-diatom scattering, 15-16 metastable helium atoms, scattering. 16-17 Multichannel structure determination, of PA superposition state. 306-308 Multiple potentials, interactions near a collision threshold. 230-233 Multispecies ensembles characterization, 670-678 crystal shapes, 670-672 ellipsoidal plasmas. 670 heating effects in, 666-670 interaction-induced coupling between radial modes. 675 manipulation of, 670-678 motional frequency spectra. 676 motional resonance coupling, 674-676 particle identification destructive, 672-673 nondestructive. 672-673 species-selective ion removal, 676-678

N Narrow close-lying levels of diatomic molecules, 610-615

Index

Neutral polar molecules, stark deceleration of. 516-528. See also OH radicals: Trapping high-field seeking states deceleration of molecules in. 527-528 longitudinal focusing, 525-527 phase stability in, 518-521 for OH radicals, 520 transverse focusing in. 521-522 Nodal structure. 267 Nonadiabatic broadband pulse. 284-285 Nonadiabatic transfer. PA mechanism. 285 Nondestructive detection technique, multispecics ensembles. 672-673 Nonimpulsive pulse, compression effect controlling with. 283 Non-.V-state spin relaxation. 490 Numerical methods for bound states. 21-23 for scattering, 19-20 single-channel scattering, 13-15 Numerov method. 14-15, 20

OH radicals, stark deceleration of. 522-525 laser-induced fluorescence (LIF) detection, 523 phase stability. 518-521 time-offlight (TOF) profile, 524-525 Oklo natural nuclear reactor. 598-599 One-color PA technique. 177 molecular states near a collision threshold. 239-241 Open channels. 18 Optical decelerator. 528-530 Optical dipole traps. 321 Optical lattices. 321 molecular superlatlice in, 391-392 photoassociation in. 193-194 Optical quantum computing in polar ensembles. 646-647 Orbital angular momentum. 8 Oscillating dipole moments, of cold polar molecules with ac fields. 445 Outer classically forbidden region. 14 Outer turning point, 14

PA. See Photoassociation Parabolic confinement, cold polar molecules interaction potentials. 442 Parity (P). xxiii Partial waves. 8. 25. 65-66 Partial width. 25 Path integral Monte Carlo technique (PIMC), 427

715

Pan I i principle, xxiii PauIi repulsion, 402 PDOODR. See Pump-dump optical-optical double resonance Penning traps, 670 Permanent dipoles, 635-638 CNOT gate implementation. 636 decoherence. 637-638 experimental parameters. 637-638 NMR-based quantum computers. 636 PES. See Potential energy surface Phase gate, 631 Phase shift, 10 Phase stability, 519 in stark decelerator. 518-521 Plionon Hamiltonian. cold polar molecules. 451 Photoassociation (PA). 70-71. 170-172. 221. 238. 247, 292 mechanisms of. 285 adiabatic transfer. 285 nonadiabatic transfer. 285 of molecules near a collision threshold. 238-241 photoassociation window concept, 256. 258-259 of superposition state multichannel structure determination, 306-308 of ultracold atoms, xvi-xvii, 172-194 in an electromagnetic field. 192-193 detection techniques, 176 Hund'srule, 183 involving molecules. 188-189 ladder technique. 177 like atoms. 172-182 nonalkali metal atoms. 177 one-color technique. 177 in an optical lattice, 193-194 STIRAP technique. 182 in a quantum degenerate gas, 189-192 unlike atoms, 182-188 ultracold molecule formation by. 172-194 Photofragmentation of polyatomic molecules, 686-688 sympathetically cooled molecular ions. 678-688 Photoionization(PI). 178 Physics tests, 555-593. See also Fundamental physics tests PIMC. See Path integral Monte Carlo technique PI. See Photoionization Platforms to implement quantum computers. 632-634 Polar molecules. See also Neutral polar molecules condensed matter Physics with. 421-463. See also Cold polar molecules properties, 634—635

Index

716

Polar molecules (Continued) choice of qubits. 634 cooling and trapping. 634 coupling strength. 634 dccoherencc, 634 robust dipole-dipole interaction. 634 slowing, 509-547. Sec also Electric fields storing of. 5()9-547. See also Electric fields trapping. 509-547. See also Electric fields Polaron shift. 452 Polyatomic molecules, photofrag mental ion of. 686-688 Population redistribution in lower state after a PA pulse. 281-282 compression effect and its optimization, 282 correlated pairs of hot atoms. 281-282 in a' £,, state in Cesium PA. 281 Potential energy surface (PES), 72 Potential near a collision threshold, properties for, 223-230 PPOODR. See Pump-probe optical-optical double resonance Preexisting molecules cooling of, xx-xxiii trapping of. xx-xxiii Prereaction. 92 Projective measurement. ARPA as. 300-302 Proof-of-principlc STIRAP experiments. 344 Propagation methods, single-channel scattering, 13 P-statc atom, 490 Pump-dump optical-optical double resonance (PDOODR). 178 Pump-probe optical-optical double resonance (PPOODR). 177-179 Purification schemes, ultracold Feshbach molecules. 329 P-wave symmetry. 10 in three-body system, 368

Q QCD. See Quantum chromodynamics QED. See Quantum electrodynamics QR. See Quasiresonant Q-switched laser, 562 Quadrupole field, polar molecules in, 511 Quantum chromodynamics (QCD), 598 Quantum collision theory, 7-30 close-coupling calculations, 16 coupled equations. 17-23 low-energy scattering, 25-26 multichannel scattering. 15-17 quasibound states. 23-25 scattering resonances. 23-25 single-channel scattering. 7-15 Quantum computing with cold molecules, xxiv-xxv

with ultracold polar molecules. 210 Quantum defect, 230, 235 Quantum degenerate gases, photoassociation in, 189-192 Quantum electrodynamics (QED), 652-690 Quantum enhancement effect. 269 Quantum formalism, 110 Quantum information processing with ultracold polar molecules. 629-647 entangled states. 631-632 optical quantum computing in polar ensembles. 646-647 permanent dipoles. 635-638 CNOTgate implementation. 636 NMR-based quantum computers. 636 polar molecules, properties. 634-635 quantum computers. 631-635 platforms to implement. 632-634 superconducting microwave resonators. 645-646 switchable dipoles, schemes with. 638-645. See also Switchable dipoles topological quantum computing. 632 Quantum number at dissociation, 13 Quantum parallelism, 631 Quantum scattering. 92 Quantum-mechanical dipoles in fields. 42-51 atoms. 42-45 rotating molecules, 45-48 Quantum-mechanical formalism, 111 Quasi two-dimensional interaction potential, cold polar molecules. 442 Quasibound states. 23-25 Quasidegenerate fine-structure molecules. 611-613 Quasidegenerate hyperfine level molecules. 610-611 Quasidegenerate states, natural widths of. 614-615 Quasi-electrostatic optical trap (QUEST). 210 Quasiresonant (QR) transitions, 82-83 QUEST. See Quasi-electrostatic optical trap Qubit.631

R Rabi cycling, 281 Rabi frequencies, 462 Rabi oscillation. 572 Radial basis set methods. 21 Radial strength functions, 19 Radio frequency (RF). 192 Radio-frequency ion traps. 654-656 Ramsauer-Townsend minimum. 90 Ramsey fringes. 573. 576-578 Ramsey-type interferometry. 332 Rate coefficients, collisions. 5-6 Reactive scattering. 30-33

Index

Feshbach resonances in, 92-95 Langevin model for barrierless reactions. 32-33 Reactive scattering. 6 Relaxation, collisional Born-Oppenheimer approximation. 374-376 for moderate mass ratios. 373-374 suppression in fermi gases, 365-368 REMPD. See Resonance enhanced multiphoton dissociation REMPI. See Resonance-enhanced multiphoton ionization Renormalized Numerov method. 14 Repulsive Mr potential in 2D. of cold polar molecules. 439-444. See also Interaction potentials Repulsive three-dimensional interactions, cold polar molecules. 429 Resonance enhanced multiphoton dissociation (R EM PD ).69()-691 black-body radiation (BBR), 690-691 (1 + 17) REMPD spectroscopy of HD+ ions, 690-691 spontaneous emission rates, 691 Resonance position, Feshbach resonance. 322 Resonance width, 409-410 Feshbach resonance, 322 Resonance window PA, 256. 261 Resonance-enhanced multiphoton ionization (REMPI). 637 Resonances, classification. 412-416 closed-channel dominated resonances, 412^14 entrance-channel dominated resonances. 414-416 85 Rb 2 Feshbach molecules. 415 Resonances, electric-field-induced, 137-138

Resonant coupling of excited states, enhancement by. 204-208 KRb. 205 s5 Rb 2 , 205 stimulated Raman transfer to deeply bound levels. 206-208 Resonant scattering length. 30 Restricted geometries, collisions in. 156-162 RF. See Radio frequency Rhodamine (R6G) ions. 686-687 Riccati-Bessel functions, 19 Rotating molecules. 45-48 Rotatingwave approximation (RWA). 256-260. 296 with central frequency. 259-260 with instantaneous frequency, 258-259 Rotational level molecules. 610-611 Rotational relaxation buffer-gas. 483-484 inelastic atom-molecule collisions. 71-82

717

Rotational scheme, switchable dipoles. 641 Rotational spectra, aslrophysical observations. 604-605 Rotational spectrum, of cold polar molecules, 436-437 Rotational temperature, 567-568 "R-transfer" technique. 201-202 Rovibrational spectroscopy of sympathetically cooled molecular ions. 688-697 in electronic ground state. 689 high-resolution spectroscopy of molecular hydrogen ions, 695 molecular thermometry. 693-695 resonance enhanced multiphoton dissociation (REMPD). 690-691 black-body radiation (BBR). 690-691 (1 + I') REMPD spectroscopy ofHD+ ions, 690-691 spontaneous emission rates. 691 sub-MHz accuracy infrared spectroscopy of HD+ ions, 695-697 RWA. See Rotatingwave approximation Rydberg atoms, xxv Rydberg decelerator. 528-530 Rydberg excitations, 642

s Saturated vapor density of helium, at buffer-gas cooling temperatures. 478 SC. See Sympathetically cooled Scaling factor, 268 Scaling laws. 266-268 near threshold, 267-268 Scattering amplitude. 8 Scattering calculations, coupled equations, 19 Scattering length, magnetic tuning, in ullracold Feshbach molecules. 409-412. See also Magnetic tuning Scattering resonances. 23-25 ultracold Feshbach molecules. 404-406 Schrodinger equation. 11. 61. 222. 367. 382. 408 SCSs. See Single-continuum states SE. See Spontaneous emission Self-assembled crystals. 425-427 Self-assembled dipolar lattices. Hubbard models in, 433-436 Separatrix. 519 SF. See Superfluid Shape resonance. 23 in molecular collisions. 79-80 Shaped laser pulses adiabatic raman photoassociation (ARPA) with. 291-312. See also Adiabatic Raman photoassociation: Multichannel photoassociation molecular data. 308-31 1

Index

718

Shaped laser pulses (Continued) Franck-Condon (FC) factors for transitions in R02 and KRb, 310 thermal averaging, 3()4-3()6 Short pulse techniques. 248 Single band Hubbard Hamiltonian, 451 Single energy. 268 Single pulses, experiments with. 574-576 Single step ARPA. 292-293 Single-channel ARPA of a chosenwave form. 302-304 Single-channel photodissociation. 295 Single-channel scattering. 7-15 bound states. 11 classically allowed region. 14 effective potential for, 9 inner classically forbidden region. 14 inner turning point. 14 low-energy collisions. 11-13 numerical methods. 13-15 outer classically forbidden region. 14 outer turning point. 14 Single-continuum states (SCSs), 104 Single-qubit rotations. 639 Singlet potentials. 402^404 Slow beam constraints, buffer gas beam production. 498-499 Slowly varying continuum approximation (SVCA). 298 SM. See Standard model \-Matrix elements. 18. 25. 31 Source noise, 568 Spatial dependence, cold polar molecules, 432 Species-selective ion removal, multispecies ensembles. 676-678 Spin depolarization. 494 Spin relaxation loss (atoms), buffer gas. 488-491 dipolar relaxation, 489 non-.S-state spin relaxation. 490 P-state atom. 490 spatial charge cloud, 489 spin-exchange. 489 5-state atoms. 489 submerged shell structure. 490 Spin relaxation, 28 Spin texture, 462 Spin-dependence, cold polar molecules. 432 Spontaneous emission (SE). 196, 198-199 Sri astrophysical experiments with. 615-618 Raman spectroscopy, 618 S-state atoms. 489 Stability, collisional cold polar molecules interaction potentials, 443 in fermi gases. 368-370 Standard model (SM ).598

Stark deceleration, xx. 511 Stark energies. 44, 512 Stark interaction. 47 Stark relaxation. 134-136 in ground-state OH molecules. 135 Stark shifts of hyperfine states. 568-571 Stark-decelerated beams. 540 Stationary Schrodinger equation. 403-404. 410 Steric effect, 540 Stern-Gerlach technique. 329 Stimulated Raman adiabatic passage (STIRAP) technique. 182, 206-207, 291 dump-before-pump pulse ordering of. 293 Stimulated Raman adiabatic passage (STIRAP). ultracold Feshbach molecules, 343-344 '• • Cs2 Feshbach molecules, 346 experiments. 344-346 proof-of-principle experiments. 344 Rb 2 Feshbach molecules. 345 Stimulated Raman transfer to deeply bound levels. 206-208 STIRAP. See Stimulated Raman adiabatic passage Storage ring, in trapping neutral polar molecules, 532-533 Stripline resonators. 645 Strongly bound .1 state levels, ultracold alkali dimers, 201-208 double-minimum potentials, enhancement in. 202-204 production by photoassociation. 201-202 resonant coupling of excited states. enhancement by, 204-208 "R-transfer" technique. 201-202 Strongly interacting systems of cold polar molecules. 423-436 Structured particles, 7 Stuckelberg interferometry. 332, 334 Sub-MHz accuracy infrared spectroscopy of HL)+ ions. 695-697 Superconducting microwave resonators. 645-646 Superfluid (SF). 447 Superimposed electric and magnetic fields, collisions in, 138-162 CaD molecule, energy levels of. 147-150 differential scattering in electromagnetic fields. 150-162 field orientations effects. 144-150 magnetic feshbach resonances. 140-141 near tunable avoided crossings, 141-144 restricted geometries, collisions in, 156-162 Supersonic expansion, 561 Suppressor nozzle. 502-503 SVCA. See Slowly varying continuum approximation .y-wave, 10, 13 87

Index

.v-wave dimer-dimer scattering, 383 .v-wave scattering length, 229-230 Switchable dipoles, 638-645 decoherence. 643-645 dipolc-dipole interaction strength. 643-644 errors. 643-645 molecular state decoherence. 644 phase gate, 639 rotational scheme. 641 single-qubit rotations, 639 trap-induced decoherence. 644-645 Sympathetically cooled (SC) molecular ions. 651 -698. See also Cold trapped coulomb clusters advantage of, 653 applications. 652 chemical reactions. 678-688 ion-neutral chemical reactions. 678-685 ion trapping and production of cold molecules, 654-658 atomic ion cooling, 656 linear radio-frequency trap. 655 molecular ion production. 656-658 radio-frequency ion traps. 654-656 laser-cooled atomic ions reactions. 679-68! molecular ions, reactions of, 681-685. See also Rovibrational spectroscopy coldffif"and room-temperature 1¾. reaction between, 682 sequential chemical reactions, 684 photofrag me nt at ion. 67 8-688 of polyatomic molecules. 686-688 Rhodamine (R6G) ions, 686-687 rovibrational spectroscopy of molecular ions. 688-697 Synchronous molecule, 519

T Target qubit. 632 Thermal averaging, 284-285, 304-306 Thermalizalion. buffer gas beam production, 497^198 Tbiee-body interactions, cold polar molecules. 427-430, 455-459 Three-body recombination. 328 Lh halo dimers formation. 328 Three-dimensional barrier, cold polar molecules interaction potentials. 442 Three-level system, two-pulse interferometry of. 571-574 Thresholds. 18 Time window. 256 Time-dependent dressed potentials. 259 Time-dependent Franck-Condon overlap. 271, 273-274 Time-dependent methods. 7

719

Time-dependent quantum-mechanical calculations. 112 Time-of-flight (ToF) profiles, 194, 523. 562-563, 672 OH radicals, 524-525 Time-reversal symmetry (T). xxiii ToF. See Time-of-flight Topological quantum computing, 632 Total angular momentum representations. 27-28 Totally uncoupled representation, 27 Transient effects, in adiabatic transfer within PA window. 270-271 Translational temperature. 562-564 Translational thermalization process, 475 Transverse focusing in stark decelerator. 521 -522 Transverse motion. 586-592 effective potential. 587 Trap-induced decoherence. 644-645 Trapped molecules, applications. 537-543 collision studies at tunable collision energy, 540-542 crossed-beam scattering machine. 542 direct lifetime measurements of metastable states. 542-543 high-resolution spectroscopy and metrology, 538-540 Trapped molecules, buffer-gas loading. 485-491 Trapping neutral polar molecules, 530-537 AC trapping in high-field seeking states, 533-537 DC trapping in low-field seeking states. 530-532 electrostatic traps, 530 four-electrode trap geometry. 530 loading configuration. 531 molecular synchrotron. 532-533 storage ring. 532-533 trap lifetime limitations. 537 trapping configuration. 530 Trimer states, heteronuclear molecules in Fermi-Fermi mixtures. 379-381 Trimers formation. 381-387 Triplet potentials, 402-404 Tunable avoided crossings, collisions near. 141-144 Tunable Feshbach resonances. 134 Tunable shape resonances. 131 -134 Tunneling. 24 Tunneling-dominated reactions. 86-95 F + H D reaction, 88 Feshbach resonances in reactive scattering. 92-95 HC1 and DC1 formation, 93 J = 0 cumulative reaction probabilities. 89 LiH + F channel. 90-91 reactions at zero temperature. 87-92

Index

720

Tunnel tag-dominated reactions {Continued) zero-temperature quenching rate coefficients tor. 96 Two-channel coupled equations, 256-260 Two-channel model for Feshbach resonance. 322 Two-channel single-resonance approach. 407-409, 411 Two-channel two-potential approach. 406-407 Two-color experiment for creating stable molecules. 274-275 Two-color PA, molecular stales near a collision threshold. 239-241 Two-color pump-dump experiment proposal. 272-275 Two-dimensional interaction, cold polar molecules. 444 Two-dimensional self-assembled crystals, 447_450 Two-particle collision process, cold polar molecules. 447 Two-pulse interferometry of three-level system, 571-574 7t/2-pulse, 572 ir-pulse. 572. 579 Two-state description of multichannel PA. 294-300 Two-state model, adiabalic transfer within a PA window, 264-266

U Ultracold alkali dimers. 194-208. See also Strongly bound x state levels characterization. 194-208 KRb experiments, 195-197 levels near dissociation. 194-201 pulsed ionization detection method, 198 state-to-state transfer of. 194-208 Ultracold and cold molecules, collisions of. xiv-xvi Ultracold atoms, 320-321 photoassociation. xvi-xvii Ultracold collisions and chemistry. 210-211 Ultracold Fermi gases, molecular regimes in, 355-393. See also Molecular regimes Ultracold Feshbach molecules theory. 319-348. 399-416. See also Feshbach resonances: Halo dimers: Stimulated Raman adiabalic passage (STIRAP) association methods. 327-329 binding energy regimes. 323-325 of Cs2 dimers. 324 bound states, 404-406 cruising through molecular spectrum. 333-335 detection methods. 329-330 dissociation patterns. 330

effective entrance-channel and closed-channel potentials vs. interatomic separation. 408 inelastic collisional loss. 327 interatomic interactions. 402 internal-state manipulation near threshold. 331-335 magnetic tuning of scattering length. 409-412 background-scattering length, 409-410 resonance width, 409-410 microscopic theory. 400-406 molecular BEC. 341-342 molecular energy structure of Cs2 below the threshold of two free Cs atoms. 333 purification schemes. 329 Scattering resonances. 404-406 singlet potentials, 402-404 Stern-Gerlach separation of an atom-dimer mixture. 325 three-body recombination. 328 toward ground-state molecules, 342-346 triplet potentials. 402-404 two-channel single-resonance approach, 407-409 Zeeman effect in hyperfine structure of alkali-metal atoms. 400-401 Ultracold molecule formation, 169-211, 245-287. See also Chirped laser pulses: Photoassociation Ultracold polar molecules quantum computing with. 210 quantum information processing with. 629-647 Ultracold quantum gases. 320-321 Ultracold temperatures chemical reactions at, 84-108. See also Chemical reactions inelastic collisions and chemical reactions of molecules at, 69-1 16 PES role in determining. 101-104 Na + Na 2 . 102-103 Ultrafast and ultracold. 247-249 Unity box-normalized functions. 267 Universality, halo dimers and. 336-337 Unlike atoms, photoassociation of. 182-188

V Vacuum expectation value (VEV). 598 van der Waals potential. 402 VEV See Vacuum expectation value Vibrational level molecules. 61 1-613 Vibrational relaxation buffer-gas. 483-484 inelastic atom-molecule collisions. 71-82

Index

rate coefficients, 73 two-channel approximation. 73 Vibrational wavepackets shaping In excited state creation. 272 to optimize stabilization into deeply bound levels of lower state. 271-275 narrowing spectral width of the dump pulse. 274 Short dump pulses, selecting. 274 two-color experiment for creating stable molecules. 274-275 two-color pump-dump experiment proposal. 272-275 Vibrationally excited alkali metal (timers, relaxation. 104-106 Vibrationally inelastic transitions, 112-115 Virtual microwave photons. 646

W Wavenumber. 8 Wavevector, 8 Wentzel-Kramers-Brillouin (WKB) approximation. 607

721

WKB. See Wentzel-Kramers-Brillouin Wigner regime. 90. 99 Wigner rotation matrices, 46 Wigner threshold laws. 25-26. 414

Z Zeeman decelerator. 528-530 Zeeman effect in hyperfine structure of alkali-metal atoms. 400-401 Zeeman relaxation, 127-131 inelastic collisions between molecules and helium, 491-496 of 2 E molecules with He. 493-494 of 3 E molecules with He. 494-496 zero-temperature rate constant for. 131 Zeeman shifts of hyperfine states, 568-571 Zeeman transition. 489 Zero temperature, reactions at. 87-92 Zero-energy Feshbach resonances. 27. 28-30. 134.414 Zero-energy scattering length. 12 Zero-range approximation. 370 Zero-temperature quenching rate coefficients, 96

Cold molecules: theory, experiment, applications

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