Advances in Atomic, Molecular, and Optical Physics, Volume 59 (Advances in Atomic, Molecular and Optical Physics)

Editors ENNIO ARIMONDO University of Pisa Pisa, Italy PAUL R. BERMAN University of Michigan Ann Arbor, Michigan CHUN C. LIN University of Wisconsin Madison, Wisconsin

EDITORIAL BOARD P.H. BUCKSBAUM SLAC Menlo Park, California M.R. FLANNERY Georgia Tech Atlanta, Georgia C. JOACHAIN Universit�e Libre de Bruxelles Brussels, Belgium J.T.M. WALRAVEN University of Amsterdam Amsterdam, The Netherlands

ADVANCES IN

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by

E. Arimondo PHYSICS DEPARTMENT UNIVERSITY OF PISA PISA, ITALY

P. R. Berman PHYSICS DEPARTMENT, UNIVERSITY OF MICHIGAN, ANN ARBOR, MI, USA

C. C. Lin DEPARTMENT OF PHYSICS, UNIVERSITY OF WISCONSIN, MADISON, WI, USA

Volume 59

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CONTRIBUTORS Numbers in parentheses indicate the pages on which the author’s contributions begin.

JAMES F. BABB (1), ITAMP, Harvard-Smithsonian Center for Astrophysics, MS 14, 60 Garden St., Cambridge, MA 02138, USA VISHAL SHAH (21), Symmetricom Technology Realization Center, 34 Tozer Road, Beverly, MA 01915, USA JOHN KITCHING (21), Time and Frequency Division, NIST, 325 Broadway, Boulder, CO 80305, USA RAINER JOHNSEN (75), Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 16260, USA STEVEN L. GUBERMAN (75), Institute for Scientific Research, 22 Bonad Road, Winchester, MA 01890, USA TIM CHUPP (129), FOCUS and MCTP, Physics Department, University of Michigan, Ann Arbor, MI 48109, USA PAUL R. BERMAN (175), Michigan Center for Theoretical Physics, Physics Department, University of Michigan, 450 Church Street, Ann Arbor, MI 48109-1040, USA GEORGE W. FORD (175), Michigan Center for Theoretical Physics, Physics Department, University of Michigan, 450 Church Street, Ann Arbor, MI 48109-1040, USA SHAUL MUKAMEL (223), Department of Chemistry, University of California, Irvine, CA 92697, USA SAAR RAHAV (223), Department of Chemistry, University of California, Irvine, CA 92697, USA

ix

PREFACE Volume 59 of the Advances Series contains six contributions, covering a diversity of subject areas in atomic, molecular, and optical physics. James Babb presents an interesting discussion of the Casimir effect in atomic, molecular, and optical physics. Casimir effects are quantum in origin and are related to vacuum fluctuations that give rise to forces between atoms, molecules, and surfaces. The Casimir effect has received a great deal of attention over the last several years. Babb reviews both neutral atom— neutral atom and ion—neutral atom interactions and looks at how these effects vary with distance. He compares results obtained via traditional methods and those involve “dressing” of the atoms by the vacuum field. In doing so, he provides new insight into the numerical factors that arise in these theories. The chapter by Rainer Johnsen and Steven Guberman focuses on the dissociative recombination of H+3 ions with electrons. For several decades, this seemingly simple process had been a great puzzle to researchers in this field, with a strong disparity between the results of theoretical and experimental studies. Johnsen and Guberman discuss recent progress, which has reduced many of the “contradictions” and reconciled the remaining discrepancies. In particular, they discuss and compare disso­ ciative combination that is produced in beam experiments with those employing plasma afterglow techniques. In their contribution, Vishal Shah and John Kitching review recent advances in the field of coherent population trapping as applied to atomic frequency standards and atomic clocks. The very narrow absorp­ tion lines associated with atomic coherence quantum interference has stimulated a large interest within the atomic clock community, leading to the development of new atomic clocks employing both standard and “chip-scale” atomic vapor cells. The authors review the progress that has been made in improving the resonance contrast, decreasing the clock line width, and the reducing light shifts that affect the long-term stability of these devices. The expected impact of these new approaches on future generations of laboratory and commercial instruments is examined. Timothy Chupp provides an overview of the search for permanent electric dipole moments of atoms, molecules, and elementary particles. Attempts to measure an electric dipole moment of the electron or the neutron have been underway for decades. In recent years, these and other xi

xii

Preface

searches have been connected with predictions of theories that go beyond the Standard Model of particle physics. Chupp reviews several experi­ mental techniques that have been used to date, along with the current experimental limits on electric dipole moments of atoms, molecules, and elementary particles. He then gives a critical discussion of proposed experimental techniques that may lead to improved precision and impor­ tant tests of physics beyond the Standard Model. Spontaneous emission from an isolated atom is the subject of the contribution of Paul Berman and George W. Ford. Although this is an old subject, it is one that has been plagued by mathematical anomalies. Berman and Ford present a detailed calculation of both the excited state decay dynamics and the spectrum of the emitted radiation. Using differ­ ent models for the atom—vacuum field interaction, they show that, while exponential decay and the Lorentzian spectrum originally predicted by Weisskopf and Wigner are good approximations to the actual decay and spectral density associated with spontaneous emission, the actual decay and spectrum must differ from the Weisskopf—Wigner result. An integral expression is obtained for the excited state probability amplitude and an analytic expression for the spectrum. Shaul Mukamel and Saar Rahav present a diagrammatic approach to calculating the response of molecules to a number of applied optical fields. Their approach provides a consistent treatment of multi-wave mixing in which both the fields and the atoms are quantized. In effect, they are able to use an amplitude approach to keep track of the various multi-photon processes that contribute to the observed signals. A closedtime-path-loop diagrammatic method plays a critical role in their analy­ sis. They apply the formalism to pump—probe and coherent anti-Stokes Raman spectroscopy to elucidate the role played by two-photon absorp­ tion and stimulated Raman scattering. The editors would like to thank all the contributing authors for their contributions and for their cooperation in assembling this volume. They would also like to express their appreciation to Ms. Gayathri Venkata­ samy at Elsevier for her invaluable assistance. Ennio Arimondo Paul Berman Chun Lin

CHAPTER

1

Casimir Effects in Atomic, Molecular, and Optical Physics James F. Babb ITAMP, Harvard-Smithsonian Center for Astrophysics, MS 14, 60 Garden St., Cambridge, MA 02138, USA

Contents

Introduction What’s a Micro Effect; What’s a Macro Effect? Relativistic Terms Yet Another Repulsive Interaction Nonrelativistic Molecules and Dressed Atoms Not a Trivial Number Reconciling Multipoles 7.1 Two Atoms 7.2 An Electron and an Ion 8. Conclusion Acknowledgments References

Abstract

The long-range interaction between two atoms and the longrange interaction between an ion and an electron are compared at small and large intersystem separations. The vacuum dressed atom formalism is applied and found to provide a framework for interpretation of the similarities between the two cases. The van der Waals forces or Casimir–Polder potentials are used to obtain insight into relativistic and higher multipolar terms.

1. 2. 3. 4. 5. 6. 7.

1 2 4 5 9 12 13 13 15 16 16 17

1. INTRODUCTION Distance changes everything. The same is the case for electromagnetic inter­ action potential energies between polarizable systems. In atomic, molecular, and optical physics, the small retarded van der Waals (or Casimir—Polder) Advances in Atomic, Molecular, and Optical Physics, Volume 59
2010 Elsevier Inc. ISSN 1049-250X, DOI: 10.1016/S1049-250X(10)59001-3 All rights reserved.

1

2

James F. Babb

potentials between pairs of polarizable systems (either of which is an atom, molecule, surface, electron, or ion) for separations at long ranges where exchange forces are negligible have been well studied theoretically. There are also three- and higher-body potentials (cf. Salam, 2010), antimatter appli­ cations (Voronin et al., 2005), and more. Much of the current interest in the Casimir interactions between atoms and walls is due to interests related to nanotechnologies (cf. Capasso et al., 2007) and related attempts to engineer repulsive forces at nanoscales (Marcus, 2009). Numerous topical surveys and reviews, monographs, book chapters, conference proceedings, and popular texts touching on particular pairwise potentials are in print–literally a “mountain of available information” (Bonin & Kresin, 1997, p. 185)–and many sources contain extensive bibliographies. It is not uncommon to run across statements indicating that there has been a rapid increase in the number of available papers related to the Casimir effect. Even a recent book, Advances in the Casimir Effect (Bordag et al., 2009), focusing mainly on recent results, comes to over 700 pages. This chapter is concerned with bringing to light some connections between theoretical results from various formulations for the zerotemperature limits of interactions between ground state atoms, ions, or molecules. The case has been advanced that the practical relevance of zero-temperature results is questionable (see Wennerstrom et al., 1999), and although the case is reasonable, you have to start some­ where. This chapter is therefore more selective than comprehensive and it is organized as follows. In Section 2 the microscopic and macroscopic natures of Casimir effects are very briefly surveyed, and the interaction between two atoms is reviewed in Section 3 including discussion of the terms of relativistic origin arising for small atomic separations. In Sec­ tion 4 the change in the form of the interaction, when one of the two polarizable systems is charged, is studied. The vacuum dressed atom approach is introduced and applied to the case of an electron and an ion in Section 5, and it is used in Section 6 to gain insight into the origin of the numerical factor “23” in expressions for potentials related to the Casimir effect. Finally in Section 7 the treatment of multipoles beyond the electric dipole is discussed for two atoms and for an electron and an ion.

2. WHAT’S A MICRO EFFECT; WHAT’S A MACRO EFFECT? The picture of two well-spaced systems interacting through fluctuating electromagnetic fields can describe many phenomena. The usual definitions are that the Casimir–Polder potential (Casimir & Polder, 1948) is the retarded interaction between two atoms or an atom and a wall and a Casimir effect (Casimir, 1948) is the “observable non-classical electromagnetic force of

Casimir Effects in Atomic, Molecular, and Optical Physics

3

attraction between two parallel conducting plates” (Schwinger, 1975). Milton (2001, p. 3) traced the change in Casimir’s perspective from action at a distance (Casimir & Polder, 1948) to the local action of fields (Casimir, 1948) or an equivalence in physical pictures of fluctuating electric dipoles or fluctuating electric fields. The conceptual realizations of the Casimir effect and of the Casimir—Polder potential have been extended well beyond their original theoretical models; an extensive tabulation can be found in Buhmann and Welsch (2007). The term Casimir effect will be used rather more loosely in the present work recognizing in advance the connection already established in the literature with the more general pictures of “dispersion forces” (Mahanty & Ninham, 1976) or “van der Waals forces” (Barash & Ginzburg, 1984; Parsegian, 2006). Also as noted by Barton (1999), “By tradition, ‘Casimir effects’ denote macroscopic forces and energy shifts; yet for connected bodies the macroscopic must be matched to microscopic physics, and no purely macroscopic model can be guaranteed in advance to reproduce the results of this matching adequately for whatever purpose is in hand.” And as Barash and Ginzburg (1984) write, “The fluctuation nature of van der Waals forces for macroscopic objects is largely the same as for individual atoms and molecules. The macroscopic and microscopic aspects of the theory of van der Waals forces are therefore intimately related.” Moreover, there are macroscopic formulations that can yield results for microscopic systems by taking various limits (Buhmann & Welsch, 2007; Milonni & Lerner, 1992; Spagnolo et al., 2007), but local field corrections require close study (Henkel et al., 2008). For the present purposes, the concern is largely with pair-wise poten­ tials and their comparison with results from various approaches. Atomic units with h
= e = me = 1 are used throughout, wherein the fine structure constant is
= 1/c, though for some formulae h
and c are restored. It is useful to define the reduced Compton wavelength of the electron h/mec. The notational convention of Spruch and Tikochinsky l
C

(1993) is followed where the subscripts At, Ion, and El denote, respec­ tively, an atom, ion, and electron. The Casimir—Polder potential for the interaction between two identical atoms is written as (Casimir & Polder, 1948) 1 VAtAt ðrÞ ¼  6 r

1 ð

d! expð2
!rÞ½
e ði!Þ 2 Pð
!RÞ;

ð1Þ

0

with PðxÞ ¼ x4 þ 2x3 þ 5x2 þ 6x þ 3;

ð2Þ

and where the dynamic (frequency dependent) electric dipole polariz­ ability is

4

James F. Babb


e ð!Þ ¼

X u

fu ; E2u0  !2

ð3Þ

fu is the electric dipole oscillator strength from the ground state 0 to the excited state u, Eu0 = Eu—E0 is the energy difference, the summation includes an integration over continuum states, and ! is the frequency. An alternative form of Equation (1) (Boyer, 1969; Spruch & Kelsey, 1978) is hc
VAtAt ðrÞ ¼  lim  !0

1 ð

dk k6 e  k ½
e ð! Þ 2 IðkrÞ;

ð4Þ

0

with ! = kc and where IðxÞ ¼ sinð2xÞðx  2  5x  4 þ 3x  6 Þ þ cosð2xÞð2x  3  6x  5 Þ:

ð5Þ

The interaction potentials given in Equations (1) and (4) are valid for all separations larger than some tens of a0, and do not take into account electron charge cloud overlap, spin, and magnetic susceptibilities, for example, though these have all been studied. The interaction potential VAtAt(r), given by either Equation (1) or (4), does contain the van der Waals interaction, certain relativistic effects, and higher order effects, as well as the asymptotic form first obtained by Casimir and Polder (1948),
hc ½
e ð0Þ2 ; 4r7 For the hydrogen atom,
e(0) = 9/2. VAtAt ðrÞ ! 23

r ! 1:

ð6Þ

3. RELATIVISTIC TERMS Before studying the long-range Casimir—Polder interaction potential in detail, it is useful to look at the “small r” expansion1 of the full potential Equation (1) for distances, say, of the order 20 a0. Expanding Equation (1) for small r, the potential is VAtAt ðrÞ  

C6 W4
3 þ
2 4 þ O 3 ; 6 r r r

r  20 a0 :

ð7Þ

1 The term “short-range” is avoided and reserved for exchange, overlap, and forces that are, for example, exponentially decaying (Barash & Ginzburg, 1984). Thus, the term “long-range” interaction potential here will indicate the form valid for intersystem separations, typically from several to tens of a0 to infinity, such as those in Equations (1) and (4), which have a “small r” expansion [Equation (7)] and a “large r” expansion [Equation (6)].

Casimir Effects in Atomic, Molecular, and Optical Physics

5

The first term in the expansion is the van der Waals potential with van der Waals constant, 1 ð 3 C6 ¼ d! ½
e ði! Þ  2 ; ð8Þ  0

and for two H atoms, C6 = 6.499 026 705 405 84 (Watson, 1991). The term of order
2 relative to the van der Waals potential is a relativistic correction 1 ð 1 W4 ¼ d! !2 ½
e ði!Þ2 ; ð9Þ  0

which can be traced back (Power & Zienau, 1957) to the “orbit—orbit” effective potential appearing in the Breit—Pauli reduction of the Dirac equation (Meath & Hirschfelder, 1966). The numerical value of W4 for two H atoms is 0.462 806 538 843 273 according to Watson (1991), who used a momentum space approach and expansion in Pollaczek polynomials; he also obtained the highly accurate value of C6 quoted above. Certain exact representations of the dynamic polarizability function of H also facilitate evaluations of W4 (Deal & Young, 1971) and of C6 (O’Carroll & Sucher, 1968). Small relativistic terms were applied in a few cases to potential energy functions of light diatomic molecules, see for example Przybytek et al. (2010), and where improved accuracies were sought for precision calcu­ lations, for example, such as those of low-energy ultra-cold atomic colli­ sions (Zygelman et al., 2003) or of the ionization potential of the hydrogen molecule (Piszczatowski et al., 2009). Recently, Pachucki (2005) reanalyzed the Casimir—Polder potential complete to terms of Oð
2 Þ, but expressed it in such a way that its form is valid over all distances sufficiently large that the atomic wave functions do not overlap, not just in the large r limit.

4. YET ANOTHER REPULSIVE INTERACTION In the previous section, the original Casimir—Polder potential was intro­ duced and seen to be attractive, but there are several known cases where repulsive potentials have been predicted theoretically.2 Thus, Feinberg and Sucher (1968, 1970) used a general dispersion-theoretic scattering approach to show that the potential given in Equation (6) can be general­ ized for two systems A and B to an expression bilinear in the electric and the magnetic polarizabilities of each system, 2 V. Hushwater, Survey of Repulsive Casimir Forces, unpublished talk, ITAMP Casimir workshop, Cambridge, MA, November 16, 2002.

6

James F. Babb


c   A B h A B 23
e
e þ
m
m 7 4r     B A 7
eA
m þ
eB
m  þ O r9 ;

VAtAt ðrÞ  

r  1;

ð10Þ

B where
A e and
e are the static polarizabilities
e(0), respectively, of A and A B , with
m

m(0), are, respectively, the static magnetic B, and
m and
m polarizabilities of A and B. Note that the cross term in the potential contain­ ing the product of
e and
m leads to a repulsive force.3 More detailed discussions concerning the treatment of the magnetic terms for the interac­ tion between two atoms can be found in Salam (2000, 2010). There is another repulsive Casimir—Polder potential, perhaps not as well known. For the scattering interaction between a charged, structure­ less particle B and a neutral polarizable particle A (Bernabeu & Tarrach, 1976; Spruch & Kelsey, 1978) or for the interaction between a charged, structureless particle B and an ion A (Kelsey & Spruch, 1978b; Spruch & Kelsey, 1978), there is an interaction (for B an electron) given by  l–C e2  A r  1: ð11Þ 11
A VAtEl ðrÞ  VIonEl ðrÞ  e þ 5
m ; 5 4r

For either of the two cases (the target is neutral or it is charged), the asymptotic result [i.e., Equation (11)] applies, but the complete poten­ tials including other corrections are not identical, due to the remnant 1/r Coulomb interaction in the charged particle and ion case higher order corrections at large r differ, as emphasized by Au (1986, 1989). The long-range Casimir potential VIonEl(r) is the present object of interest, but it is useful sometimes to write the full potential with “instantaneous” Coulomb interactions as well. Therefore, the full poten­ tial U(r), including the charged particle and ion electric interactions (but neglecting the dominant 1/r Coulomb potential), is at large r 1 11 ð12Þ UðrÞ ¼  e2
e r  4 þ l–C e2
e r  5 . . . r  1; 2 4 where
e is the polarizability of the ion. The first term in Equation (12) is the polarization potential. The second term is the asymptotic Casi­ mir—Polder-type interaction, which was confirmed theoretically (Au, 1986; Feinberg & Sucher, 1983). The Casimir—Polder potential for the interaction between an electron and an ion was expressed in a fashion similar to the result for two atoms given in Equation (1) by Au et al. (1984) using a dispersion relation analysis, and later using Coulomb

B 3 The
A e
m term supports the result that the interaction between two plates is repulsive, if one plate (A) has only infinite permittivity and one plate (B) is only infinitely permeable (Boyer, 1974); an alternative argument not making use of Equation (10) is given by Schaden and Spruch (1998).

Casimir Effects in Atomic, Molecular, and Optical Physics

7

gauge, old-fashioned perturbation theory by Babb and Spruch (1987) and Au (1988). It can be expressed (Babb & Spruch, 1987) in the compact form similar to Equation (4), e2 VIonEl ðrÞ ¼ l–C lim !0  where FðkÞ ¼

1 ð

dk e  k k4 FðkÞI ðkrÞ;

ð13Þ

0

X

fu

u

½Eu0 ðEu0 þ Ek Þ

:

ð14Þ

Taking account of the Coulomb interactions, the potential has the expansion for small r,  2 1
UðrÞ þ VIonEl ðrÞ   e2
e r  4 þ 3r  6 þ r  4 . . . ; r  a few a0 ; 2 Z2 ð15Þ where Z is the charge of the ion, and  = 43/(8Z6) (Dalgarno et al., 1968; Kleinman et al., 1968). Note the disappearance for small r of the r5 term that was present in the large r potential, Equation (12). It was emphasized by Kelsey and Spruch (1978a) and Feinberg et al. (1989) that the 3r—6 term disappears at large r. Thus, the potential can be written in the form (Babb & Spruch, 1987) ð1 hc
dk k6 e  k IðkrÞ VIonEl ðrÞ þ 3r  6 ¼  lim  !0 0  hc

e ðkÞ
out ðkÞ  2 2
out ðkÞ½ðkÞ   ; ð16Þ e using e2 ; m!2 k FðkÞ ¼
e ðkÞ  2 ðkÞ;

out ð!Þ ¼ 

and

ðkÞ ¼

1X f   u  : 2 u Eu0 E2u0 þ E2k

ð17Þ ð18Þ

ð19Þ

The effective potential Equation (13) was evaluated numerically and used to study theoretically the energy shift arising from the interaction between an electron bound in a high Rydberg state |lni with principal quantum number n, and with angular momentum l  n. Experiments on highly excited n = 10 Rydberg states of He were carried out and several

8

James F. Babb

theoretical formulations were developed. The details about experiments and theories, and their comparisons, are very completely presented in the book edited by Levin and Micha (1992) (see also Hessels, 1992; Stevens & Lundeen, 2000; Drake, 1993; Lundeen, 2005). Many other terms must be considered carefully in the theoretical calculations; the details do not directly relate further to this chapter. Recently two groups have reproduced the electric (
e) part of Equation (11) using different arguments, but both approaching the interaction between a charged particle and a neutral particle as oneloop quantum field theoretic calculations. Panella et al. (1990) obtained Equation (11) and traced the r5 result back to a change in the mass induced by “condensed-matter renormalization” of the elec­ tromagnetic fluctuations (Panella & Widom, 1994). The repulsive potential is attributed to soft-photon infrared renormalization. In contrast, Holstein and Donoghue (2004) were seeking to identify cases where classical effects are found within one-loop diagrams. Using an effective field theory approach, they find quantum correc­ tions to the classical polarization potential; their result is identical to the
e part of Equation (11). They identify the r5 term as a quantum correction to the polarization potential, which arises from the infrared behavior of the Feynman diagrams, when at least two massless pro­ pagators occur in a loop contribution. In a subsequent study, Holstein (2008) obtained Equation (11) and observed that the r5 term in Equation (11) might be associated with zitterbe­ wegung and he noted that, under the influence of this effect, “in the quantum mechanical case the distance between two objects is uncertain by an amount of order the Compton wavelength due to zero point motion,” r  l
C ; hence VðrÞ 

1 1 1 1 ! » +4l–C 5 : r4 ðr – r Þ 4 r4 r

ð20Þ

It is an intriguing argument, though the ambiguity of the sign is unresolved. Zitterbewegung would normally be attributed to virtual electron­ positron transitions (Milonni, 1994, p. 322) at the length scale less than l
C, which would seemingly place the effect outside of the realm of the lowenergy fluctuation arguments used by, for example, Spruch and Kelsey (1978) in deriving Equation (11). Nevertheless, the calculations of Holstein and Donoghue (2004) are concerned with large r, so there must be a connec­ tion to the scale of l
C and this will be addressed in the next section. The effective field theory of Holstein (2008) allows the longest-range parts of electromagnetic scattering processes to be isolated, and he extended the asymptotic (large r) results for the interactions between two systems, with and without spin, to the case where one or both systems are electrically neutral; see also Sucher and Feinberg (1992).

Casimir Effects in Atomic, Molecular, and Optical Physics

9

5. NONRELATIVISTIC MOLECULES AND DRESSED ATOMS In the theoretical “vacuum dressed atom” approach, a ground state “bare” atom interacts with the vacuum electromagnetic field. The combi­ nation system of the atom and the field is taken to be in the lowest possible energy state of the noninteracting atom—field system, and the zero-point fluctuations of the field are seen as inducing virtual absorption and re-emission of photons in the atom–the “vacuum dressed atom” is then the system comprised of the atom and the associated cloud of virtual photons (Compagno et al., 1995a,b). A good account of the development of the concept of atoms dressed by the vacuum electromagnetic field is given by Compagno et al. (2006). The energy density can be calculated and used to obtain expressions for long-range potentials and other phy­ sical quantities, as was shown in quantum optics (Cirone & Passante, 1996; Compagno et al., 1995b; Salam, 2010) for the nonrelativistic free electron interacting with the vacuum electromagnetic field and the non­ relativistic hydrogen atom. Compagno and Salamone (1991) (see also Compagno et al., 1995b) considered a slow electron interacting with the vacuum field. The key observation is that the cloud around the electron is due to emission and reabsorption of virtual photons in the course of recoil events. They point out that the zitterbewegung due to relativistic fluctuations would enter and contribute a cloud of size of order l
C. This effectively limits their nonrelativistic model to distances  l
C, so that the positron cloud can be neglected and the electron has the physical charge e; accordingly, only low-frequency photons enter. In this picture the virtual cloud affects the field surrounding the charge and changes the average values of the squares of the electric and magnetic fields. They calculate the classical and quantum contributions to the energy density around the electron both moving and at rest, and for the electron at rest, they find (for r  l
C)
X hc e2 5 e2 l–C þ þ k ð21Þ hEe ðrÞi ¼ 4 2 4 16 r r 4V k 8r and hEm ðrÞi ¼ 


X hc 5 e2 l–C þ k; 2 4 4V k 16 r r

ð22Þ

where Ee(r) and Em(r) are, respectively, the electric and magnetic energy densities at a distance r from the electron, V is the quantization volume, and the sum is over the vacuum field modes. For the present purpose, a key observation is the appearance of the r5 contribution which Compagno and Salamone (1991) attribute to the vir­ tual photon cloud surrounding the electron fluctuations that arise due to

10

James F. Babb

interference between the virtual photons emitted and absorbed by the electron and zero-point field fluctuations. The r5 term is deemed purely quantum in nature (Compagno et al., 1995b). The energy densities can be directly related to the Casimir—Polder potential, as noted by Passante and Power (1987). What is striking is the similarity in form between Equations (21) and (22) and the large r potential for the interaction of an electron and an ion, Equation (12). Evidently, both the classical polarization potential and the retarded asymptotic Casimir—Polder potential are present. As discussed above, Holstein and Donoghue (2004) showed that, within a diagrammatic, effective field theory approach, classical effects can arise. In particular, the energy density of a particle in a plane wave calculated by Holstein and Donoghue (2004) agrees in form with the dressed electron result containing both a r4 polariza­ tion potential and the “purely quantum” r5 asymptotic retarded potential. For the nonrelativistic hydrogen atom, the analysis was carried out again within the vacuum dressed atom formalism; the extensive calcula­ tions can be found in Passante and Power (1987) and Compagno et al. (1995b, 2006). The analysis is complicated, but it is similar to that carried out by Babb and Spruch (1987) and Au (1989). For example, Equation (7.148) of Compagno et al. (1995b) describes the longitudinal electric field and transverse electric field contributions to the energy density around a hydrogen atom, 1 0 1 hjE jj ðxÞ  E\ ðxÞji0  3 4 r

ð

k2 j2 ðkrÞ dk ! N þ !k

ð23Þ

and it is almost identical to VIT, Equation (4.16) found by Babb and Spruch (1987) for the contribution of one instantaneous Coulomb photon and one transverse photon to the effective potential in the case of an electron and an ion. In an earlier study using the virtual photon cloud picture, Passante and Power (1987) note that the r6 term in the description of the energy density around a ground state hydrogen atom disappears at large r similarly to the way the van der Waals form r6 form is replaced at asymptotic distances by the r7 form. They find that nonretarded effects of order r6 in the energy density are absent in the far zone of the hydrogen atom, and they obtain the simple form for the energy density with an Oðr  7 Þ term related to the virtual charge cloud, 1 23
hc
e ; hjF2 jiEzp ¼ 8 162 r7 where F is the electric field.

ð24Þ

Casimir Effects in Atomic, Molecular, and Optical Physics

11

_ Radozycki (1990) carried out a relativistic calculation of the electro­ magnetic virtual cloud of the ground-state hydrogen atom using a Dirac formalism. According to Compagno et al. (2006), his work was supposed to be an independent calculation of the energy density of the vacuum dressed hydrogen atom. For the energy density due to the electric field, his result in the large r limit is 1 1 2 13
hce2 X 1 h1jxjnihnjxj1i 7 hF ðrÞi¼ 2 r 2 16 n En1 þ

^ri^rj 7
hce2 X 1 h1jxi jnihnjxj j1i 7 : 2 16 n En1 r

ð25Þ

Identifying the tensor electric dipole polarizability in Equation (25)

2

X 1 h1jxi jnihnjxj j1i¼
e; ij ; En1 n

ð26Þ

Equation (25) agrees with the two-level “Craig—Power” model (Compagno et al., 1995b) result for the energy density in the large r limit, hEe ðrÞi¼

1  1

ij 13ij þ 7^ri^rj 7 : hc 2 r 32

ð27Þ

In an unrelated study of the Casimir—Polder potential for an electron interacting with a hydrogen molecular ion core, Babb and Spruch (1994) obtained an expression almost identical to Equations (25) and (27). The tensor polarizability arises from the anisotropic interaction arising from the cylindrical symmetry of the diatomic molecule core. Compagno et al. (1995b) interpret the large r Casimir—Polder potential as the interaction between the vacuum dressed “source” atom with polarizability
S and the “test” atom with polarizability
T VðrÞ ¼

23 1
S
T 7 ¼ 4
T hESe ðrÞi; hc
4 r

ð28Þ

where the energy density is generated by the source at point r in the absence of the test atom. Another interesting point emphasized by Compagno et al. (1995b) is that VðrÞ ¼ 4
T hESe ðrÞi;

ð29Þ

12

James F. Babb

“thus the van der Waals force provides a means of measuring directly the electric energy density of a source both in the near and in the far regions.”

6. NOT A TRIVIAL NUMBER In his contribution to the proceedings of a conference held in Maratea, Italy,4 Casimir (1987) wrote, “In the theory of the so-called Casimir effect two lines of approach are coming together. The first one is concerned with Van der Waals forces, the second one with zero-point energy.” Today, that connection is well established, though the “reality” of zeropoint energy is still debatable; see the very accessible article by Rugh et al. (1999) and also Jaffe (2005). In Equation (6), it was shown that the asymptotic potential for the interaction between two electrically polarizable particles contains the factor 23, as does the asymptotic potential for two magnetically polariz­ able particles, see Equation (10). The factor 23 has reappeared in other situations. In the asymptotic interaction between an electron and an ion, expanding Equation (16) for large r and keeping one more term past that given in Equation (12) (Feinberg & Sucher, 1983), the potential is 1 UðrÞ  e2
e r  4 þ VIonEl ðrÞ 2 1 11
e 23 e2 l–C ð0Þ þ . . .; r  1: ð30Þ   e2
e r  4 þ l–C e2 5 þ 2 4 r 4 a20 r7 According to Feinberg et al. (1989), when told of this result [i.e., Equation (30)], at the Maratea conference, Casimir replied, “23 is not a trivial number. I am happy to see that.” However, the appearance of the 23 in a way nearly identical to the result for the asymptotic atom—atom poten­ tial was not explained completely (Feinberg & Sucher, 1983). In addition, the sign of the term containing 23 is opposite to that for the case of two atoms. Evidently, while the complete potential for the electron—ion case can be expressed as Equation (13), expansion for large r yields the two terms of Equation (11). I conjecture that the Oðr  5 Þ term can be interpreted as the effective potential arising from the energy density of the weakly bound, vacuum dressed, electron “source” interacting with the ion core “test”
e,   1 1 ð31Þ
e hE2out i
e 4 þ l–C 5 ; r r

4

June 1—14, 1986.

Casimir Effects in Atomic, Molecular, and Optical Physics

13

in accord with the ideas of Compagno and Salamone (1991), supported by the large r one-loop calculations of Holstein and Donoghue (2004) and Holstein (2008). The other term r7 can be interpreted as the source term of the fluctuating vacuum dressed polarizable ion core “source” acting on the electron “test” particle, 23
 7; hcr ð32Þ 4 where E Se indicates that the electric field energy density is modified due to the Coulomb binding and ! is a characteristic energy. This is to be expected based on arguments given by Au (1986, 1989) for the Rydberg helium case and by Compagno and Salamone (1991) for the vacuum dressed slow electron and vacuum dressed hydrogen atom. Using Equa­ tion (17) for
out(k) = l
Cak2, where kc = !, evaluated at ! ¼
c=a0 (Feinberg & Sucher, 1983). Equation (32) yields a term in general agree­ ment with Equation (30), 4
out ð!ÞhE Se ðrÞi
out ð!Þ

23 4
out ð! ÞhE Se ðrÞiþl–C r  7 : 4

ð33Þ

The approach of adding the two interactions is consistent with the inter­ pretation of the fluctuating field approach to Casimir—Polder interactions proposed by Power and Thirunamachandran (1993). Namely, that the dipole in each particle is induced by the vacuum fluctuations of the electromagnetic field.

7. RECONCILING MULTIPOLES 7.1 Two Atoms The extension beyond electric and magnetic dipoles for the retarded van der Waals (or Casimir—Polder) potential between two neutral spinless systems was achieved by Au and Feinberg (1972). Using scattering ana­ lysis, they were able to obtain integral forms for the complete potential for each multipole, valid for all separations greater than some tens of a0, and they gave the first several terms in each of the small r and large r expansions of the potentials. Their result, as noted by Feinberg (1974), and as emphasized by Power and Thirunamachandran (1996), included the property “that the expansions of the electric (magnetic) form factors include high-order magnetic (electric) susceptibilities in addition to elec­ tric (magnetic) polarizabilities.” The first several electric multipole results of Au and Feinberg (1972) were used for applications to calculations of the binding energy of the helium dimer by Luo et al. (1993) and by Chen and Chung (1996) and for applications to ultra-cold atom scattering by Marinescu et al. (1994), who

14

James F. Babb

evaluated the expressions for a pair of hydrogen atoms and for a pair of like alkali-metal atoms. A few years later, in a thorough analysis, Salam and Thirunamachandran (1996) and Power and Thirunamachandran (1996) pointed out that the results of the Au and Feinberg (1972) analysis did not concur with other results obtained evidently independently by Jenkins et al. (1994), who used a different approach. Power and Thirunamachandran (1996) argued that the correct form for the next order Casimir—Polder interaction, arising from the interaction of an electric dipole and an electric quadrupole, is 1 V12 ðrÞ ¼  3

1 ð

d!expð2
!rÞ
e ði!Þ
e ; 2 ði!ÞP12 ð!
rÞ;

ð34Þ

0

where
e;2 is the electric quadrupole polarizability and P12 ðxÞ ¼

1 6 27 x þ 3x5 þ x4 þ 42x3 þ 81x2 þ 90x þ 45: 2 2

ð35Þ

C8 W6 þ
2 6 þ . . .; r8 r

ð36Þ

d!!2
e ði!Þ
e ; 2 ði!Þ;

ð37Þ

For small r, V12 ðrÞ   where 15 C8 ¼ 

1 ð

0

and the coefficient of the relativistic correction is 3 W6 ¼ 

1 ð

d!!2
e ði!Þ
e ; 2 ði!Þ:

ð38Þ

0

Meath and Hirschfelder (1966) obtained the relativistic corrections of relative order
2 for two hydrogen atoms. For the orbit—orbit interaction HLL;0, the corresponding effective potential contributions arise as powers of r4 and r6. Their approach is not valid in the large r limit, but it should agree with the small r form of VAtAt(r) and V12(r). In turn Marinescu et al. (1994) noticed a discrepancy between the result of Meath and Hirschfelder and Johnson et al. (1967) for the r6 relativistic term for small distances and the result obtained by expanding the potential of Au and Feinberg for small r. In contrast, expansion of the revised dipole—quadrupole potential V12(r) of Jenkins et al. (1994) for small r provides a value for W6, see Equation (38), in agreement with

Casimir Effects in Atomic, Molecular, and Optical Physics

15

the expression of Meath and Hirschfelder (1966) and Johnson et al. (1967) for the r6 term in the expansion of the Breit—Pauli equation. Accordingly, earlier results, such as the results of Chen and Chung (1996), for W6 should be multiplied by 3/2. Later, Marinescu and You (1999) rederived the atom—atom potential accounting for magnetic and other terms to higher order (see also Salam, 2000). Marinescu and You note that numerically, at least, for their evalua­ tions of the like alkali-metal atom pairs, the relative error between results from the two approaches is smaller than 105. In any case, other terms, such as mass polarization, Darwin interaction terms, and Lamb shifts, would have to be included at the correct order for a complete description. Asymptotically, for large r (Jenkins et al., 1994; Marinescu & You, 1999; Thirunamachandran, 1988; Yan et al., 1997) V12 ðrÞ  

531
hc
e ð0Þ
e ; 2 ð0Þ: 16r9

ð39Þ

Feinberg (1974) expected that for atoms–with the exception of possibly the magnetic—magnetic case for two hydrogen atoms (see Feinberg and Sucher, 1968)–magnetic and higher order multipole effects would be negligible for domains where retardation was important. Thus, Feinberg was motivated to use the scattering approach to study the case of two superconducting spheres, and he obtained a series in powers of the sphere—sphere separation distance. Using a new formalism based on a scattering approach, Emig (2008) and Emig and Jaffe (2008) investigated the Casimir energy between two spheres. For large separations, they obtain an expansion in the separation distance r. The lead term is of order r7 and is given by Equation (10), where the polarizabilities correspond to those of the spheres. Moreover, the next term of order r9 is given by Equation (39). In another calculation using the scattering approach, Emig (2010) obtains the large r interaction potential for two anisotropic objects; his result is in agreement with the earlier results for two anisotropic particles given by Craig and Power (1969). 7.2 An Electron and an Ion Expanding Equation (13) for small r, it was shown above in Equation (15) that there is a term (a2/Z2)r4. This relativistic term is identical to the atom—atom case, which is known to result from perturbation treatment of the Breit interaction with the Coulomb interaction (Au, 1989; Power & Zienau, 1957). Some time after Equation (4) was obtained, the complete long-range potential including multipoles for an ion and a neutral spinless system

16

James F. Babb

was obtained by Feinberg and Sucher (1983) and by Au (1985). According to Hessels (1992), the result of Au (1985) for the next term is VE ; 1 ðrÞ ¼

9 2 6
r : 16

ð40Þ

Hessels (1992) carried out a perturbation theoretic calculation of the relativistic corrections for the ion—electron system, analogously to the calculation of Meath and Hirschfelder (1966) for the atom—atom interac­ tion. His analysis is in disagreement with Au (1985), but in agreement with comprehensive calculations by Drake (1992), indicating an unresolved discrepancy between the dispersion theoretic result and perturbation theoretic results at Oð
2 r  6 Þ for small r limit of the ion—electron system.

8. CONCLUSION The Casimir effects for the interaction between two atoms and for the inter­ action between an ion and an electron were investigated and, respectively, their expansions lead to asymptotic terms of order r7 and r5. The second correction at large r for the ion and electron case is similar to the leading term at large r for the case of two atoms. It was shown that the vacuum dressed atom picture provides a framework for interpretation of this similarity. Reconciliation of interaction potentials for electric dipole and electric quadrupole multipoles between atom—atom and ion—electron cases led to insight concerning a discrepancy between a scattering dispersion theore­ tic calculation and a perturbation theoretic calculation of the ion—electron interaction for the electric quadrupole relativistic term. As interest in the potential applications of Casimir effects in atomic, molecular, and optical physics increases, limiting results for interaction potentials at zero temperature–such as those presented here–may pro­ vide useful insights and checks on calculations for more complicated geometries. Hopefully, it will be a long time until it is true that nothing can be added to vacuum studies.

ACKNOWLEDGMENTS I am indebted to several colleagues who have shared their knowledge with me over the years on topics related to this chapter. In particular, Larry Spruch Alex Dalgarno, Joe Sucher, Akbar Salam, and Peter Milonni provided helpful insights. ITAMP is partially supported by a grant from the NSF to Harvard University and the Smithsonian Astrophysical Observatory.

Casimir Effects in Atomic, Molecular, and Optical Physics

17

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Compagno, G., & Salamone, G. M. (1991). Structure of the electromagnetic field around the free electron in nonrelativistic QED. Physical Review A, 44, 5390—5400. Craig, D. P., & Power, E. A. (1969). The asymptotic Casimir-Polder potential from secondorder perturbation theory and its generalization for anisotropic polarizabilities. Interna­ tional Journal of Quantum Chemistry, 3, 903—911. Dalgarno, A., Drake, G. W., & Victor, G. A. (1968). Nonadiabatic long-range forces. Physical Review, 176, 194—197. Deal, W., & Young, R. (1971). The long-range retarded interaction between two hydrogen atoms. Chemical Physics Letters, 11, 385—386. Drake, G. W. F. (1992). High-precision calculations for the Rydberg state of helium. In F. S. Levin & D. Micha (Eds.) Long range Casimir forces: Theory and recent experiments in atomic systems (pp. 107—217). New York: Plenum Press. Drake, G. W. F. (1993). Energies and asymptotic analysis for helium Rydberg states. In Advances in atomic, molecular, and optical physics (Vol. 31, pp. 1—62). San Diego: Academic Press. Emig, T. (2008). Fluctuation-induced quantum interactions between compact objects and a plane mirror. Journal of Statistical Mechanics, 2008, P04007. Emig, T. (2010). Casimir physics: Geometry, shape and material. arxiv.org/abs/1003.0192. Emig, T., & Jaffe, R. L. (2008). Casimir forces between arbitrary compact objects. Journal of Physics A, 41, 164001. Feinberg, G. (1974). Retarded dispersion forces between conducting spheres. Physical Review B, 9, 2490—2496. Feinberg, G., & Sucher, J. (1968). General form of the retarded van der Waals potential. Journal of Chemical Physics, 48, 3333—3334. Feinberg, G., & Sucher, J. (1970). General theory of the van der Waals interaction: A modelindependent approach. Physical Review A, 2, 2395—2415. Feinberg, G., & Sucher, J. (1983). Long-range forces between a charged and neutral system. Physical Review A, 27, 1958—1967. Feinberg, G., Sucher, J., & Au, C. K. (1989). The dispersion theory of dispersion forces. Physics Reports, 180, 83—157. Henkel, C., Boedecker, G., & Wilkens, M. (2008). Local fields in a soft matter bubble. Applied Physics B, 93, 217—221. Hessels, E. A. (1992). Higher-order relativistic corrections to the polarization energies of helium Rydberg states. Physical Review A, 46, 5389—5396. Holstein, B. R. (2008). Long range electromagnetic effects involving neutral systems and effective field theory. Physical Review D, 78, 013001. Holstein, B. R., & Donoghue, J. F. (2004). Classical physics and quantum loops. Physical Review Letters, 93, 201602. Jaffe, R. L. (2005). Casimir effect and the quantum vacuum. Physical Review D, 72, 021301. Jenkins, J. K., Salam, A., & Thirunamachandran, T. (1994). Retarded dispersion interaction energies between chiral molecules. Physical Review A, 50, 4767—4777. Johnson, R. E., Epstein, S. T., & Meath, W. J. (1967). Evaluation of long-range retarded interaction energies. Journal of Chemical Physics, 47, 1271—1274. Kelsey, E. J., & Spruch, L. (1978a). Retardation effects and the vanishing as r  1 of the nonadiabatic r6 interaction of the core and a high-Rydberg electron. Physical Review A, 18, 1055—1056. Kelsey, E. J., & Spruch, L. (1978b). Retardation effects on high Rydberg states–retarded r5 polarization potential. Physical Review A, 18, 15—25. Kleinman, C. J., Hahn, Y., & Spruch, L. (1968). Dominant nonadiabatic contribution to the long-range electron-atom interaction. Physical Review, 165(1), 53—62. Levin, F. S., & Micha, D. (Eds.) (1992). Long range Casimir forces: Theory and recent experiments in atomic systems. New York: Plenum Press. Lundeen, S. R. (2005). Fine structure in high-L Rydberg states: A path to properties of positive ions. In Advances in atomic, molecular, and optical physics (Vol. 52, pp. 161—208). San Diego: Elsevier Academic. Luo, F., Kim, G., Giese, C. F., & Gentry, W. R. (1993). Influence of retardation on the higherorder multipole dispersion contributions to the helium dimer potential. Journal of Chemical Physics, 99, 10084—10085.

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Spruch, L., & Tikochinsky, Y. (1993). Elementary approximate derivations of some retarded Casimir interactions involving one or two dielectric walls. Physical Review A, 48, 4213—4222. Stevens, G. D., & Lundeen, S. R. (2000). Experimental studies of helium Rydberg fine structure. Comments on Atomic and Molecular Physics, Comments on Modern Physics, Part D, 1, 207—219. Sucher, J., & Feinberg, G. (1992). Long-range electromagnetic forces in quantum theory: Theortical formulations. In F. S. Levin & D. Micha (Eds.) Long range Casimir forces: Theory and recent experiments in atomic systems (pp. 273—348). New York: Plenum Press. Thirunamachandran, T. (1988). Vacuum fluctuations and intermolecular interactions. Phy­ sica Scripta, T21, 123—128. Voronin, A. Y., Froelich, P., & Zygelman, B. (2005). Interaction of ultracold antihydrogen with a conducting wall. Physical Review A, 72, 062903. Watson, G. I. (1991). Two-electron perturbation problems and Pollaczek polynomials. Journal of Physics A, 24, 4989—4998. Wennerstrom, H., Daicic, J., & Ninham, B. W. (1999). Temperature dependence of atom— atom interactions. Physical Review A, 60, 2581—2584. Yan, Z.-C., Dalgarno, A., Babb, J. F. (1997). Long-range interactions of lithium atoms. Physical Review A, 55, 2882—7. Zygelman, B., Dalgarno, A., Jamieson, M. J., & Stancil, P. C. (2003). Multichannel study of spin-exchange and hyperfine-induced frequency shift and line broadening in cold colli­ sions of hydrogen atoms. Physical Review A, 67, 042715.

CHAPTER

2

Advances in Coherent Population Trapping for Atomic Clocks Vishal Shaha and John Kitchingb a

Symmetricom Technology Realization Center, 34 Tozer Road,

Beverly, MA 01915, USA

b Time and Frequency Division, NIST, 325 Broadway, Boulder,

CO 80305, USA

Contents

1. 2.

3.

4.

Coherent Population Trapping 1.1 Introduction 1.2 Basic Principles Atomic Clocks 2.1 Introduction 2.2 Vapor Cell Atomic Clocks 2.3 Coherent Population Trapping in Atomic

Clocks 2.4 Stability of Vapor Cell Atomic Clocks 2.5 Light Shifts Advanced CPT Techniques 3.1 Contrast Limitations due to Excited-State

Hyperfine Structure 3.2 Contrast Limitations due to Zeeman

Substructure 3.3 High-Contrast Resonances Using

Four-Wave Mixing 3.4 Push–Pull Laser Atomic Oscillator 3.5 The CPT Maser 3.6 N-Resonance 3.7 Raman–Ramsey Pulsed CPT 3.8 CPT in Optical Clocks Additional Considerations 4.1 Light-Shift Suppression 4.2 Laser Noise Cancellation 4.3 Light Sources for Coherent Population

Trapping

22

22

23

27

27

29

30

35

38

39

40

42

51

52

54

55

56

59

60

60

61

63

Advances in Atomic, Molecular, and Optical Physics, Volume 59
2010 Elsevier Inc. ISSN 1049-250X, DOI: 10.1016/S1049-250X(10)59002-5 All rights reserved.

21

22

Vishal Shah and John Kitching

4.4 Dark Resonances in Thin Cells 4.5 The Lineshape of CPT Resonances: Narrowing Effects 5. Conclusions and Outlook Acknowledgments References

Abstract

65 65 66 67 67

We review advances in the field of coherent population trapping (CPT) over the last decade with respect to the application of this physical phenomenon to atomic frequency references. We provide an overview of both the basic phenomenon of CPT and how it has traditionally been used in atomic clocks. We then describe a number of advances made with the goal of improving the resonance contrast, decreasing its line width, and reducing light shifts that affect the long-term stability. We conclude with a discussion of how these new approaches can impact future generations of laboratory and commercial instruments.

1. COHERENT POPULATION TRAPPING 1.1 Introduction Coherent population trapping (CPT) (Arimondo, 1996) refers to the pre­ paration of atoms in coherent superposition states by use of multimode optical fields. This phenomenon, as investigated using hyperfine (Alzetta et al., 1976; Arimondo & Orriols, 1976) and optical (Whitley & Stroud, 1976) transitions in 1976, has led to significant advances in a variety of areas of optical and atomic physics including laser cooling (Aspect et al., 1988), nonlinear optics (Hemmer et al., 1995), precision spectroscopy (Wynands & Nagel, 1999), slow light (Schmidt et al., 1996), atomic clocks (Kitching et al., 2000; Thomas et al., 1981, 1982; Vanier et al., 1998; Zanon et al., 2005; Zanon-Willette et al., 2006), and other precision spectroscopic instrumentation (Nagel et al., 1998; Schwindt et al., 2004). The central principle that underlies the value of CPT in this diverse set of applica­ tions is the idea that certain coherent superposition states do not absorb light from the excitation field. This reduced absorption leads both to a spectroscopic signal on the light field and to a modified atom—light interaction. The use of CPT in atomic clocks is a particularly important application that has sustained interest over three decades. Early work to use micro­ wave CPT (Arimondo & Orriols, 1976; Orriols, 1979) in atomic beam clocks (Thomas et al., 1981, 1982) has been adapted for application to

Advances in CPT for Atomic Clocks

23

vapor cell clocks (Cyr et al., 1993; Vanier et al., 1998) and led most recently to microfabricated atomic clocks (Knappe et al., 2004). A previous review of atomic microwave clocks based on CPT is given in the work of Vanier (2005). Possible future application to optical clocks has also been consid­ ered (Hong et al., 2005; Santra et al., 2005). In these clock designs, the CPT resonance is used to directly measure the atomic transition frequency. The performance of the clock therefore depends intimately on the quality of the CPT resonance, and most specifically on its line width and contrast. This chapter reviews research over the last decade to understand and extend the phenomenon of CPT with respect to its application to atomic frequency standards. Special emphasis is placed on novel excitation and detection schemes, and other new phenomena that improve the reso­ nance contrast, reduce its line width, or minimize the effect of the light fields on the resonance frequency. In Section 1.2, we review the basic phenomenon of CPT and describe how CPT resonances have traditionally been excited and detected. In Section 2, we provide an introduction to atomic clocks and discuss the differences between conventional atomic clocks and those based on CPT. In Section 3, we consider the limitations of the simplest CPT excitation schemes and describe several schemes that have been recently developed to enhance the quality of the CPT reso­ nance for use in atomic clocks. In Section 4, we address a number of outstanding issues including light shifts, light sources for CPT, and unique experimental environments in which CPT is observed. Finally in Section 5, we offer some conclusions and discuss how the new ideas connected with CPT may ultimately impact the development of future atomic frequency references. 1.2 Basic Principles A two-level atom illuminated by a monochromatic electromagnetic field is perhaps the simplest spectroscopic system. When the frequency of the illumination field is tuned into resonance with the transition between the atomic levels, radiation is scattered by the atom via spontaneous emis­ sion, causing fluorescence, and a corresponding change in the intensity and phase of the transmitted radiation (see Figure 1a). The fluorescence signal, combined with a measurement of the radiation wavelength or frequency, often gives highly precise information about the internal structure of the atom. In atoms with more than two energy levels, a variety of more complex phenomena can occur. One example of this is optical pumping, in which light resonant with one optical transition causes atomic population to accumulate in a state not excited by the light field (see Figure 1b). Once pumped into this third state, the atoms stop scattering light to the extent that the third state is stable and not resonant with the light field. Such a state is referred to as a dark state.

24

Vishal Shah and John Kitching

(a)

|3>

|1>

(b)

|3>

(c)

|3>

|1>

|1>

|2>

|2>

Figure 1 Optically excited transitions in atoms. (a) Simple excitation of a two-level atom. (b) Optical pumping in a three-level atom. Level |2> is an incoherent dark state. (c) Bichromatic excitation of a three-level atom. A superposition of levels |1> and |2> is a coherent dark state

CPT is a phenomenon that occurs in atoms with more than two energy levels excited by coherent, multimode optical fields. Under the right conditions, atoms are optically pumped into a superposition of two of the levels that does not scatter light from the multimode field (see Figure 1c). This coherent dark state has an electromagnetic moment that oscillates at one of the beat frequencies of the multimode field. Its excita­ tion can be thought of as a nonlinear process: a nonlinear resonator (the atom) is driven with a force with two spectral components (the light), and through the nonlinearity, an oscillation at the sum or difference of the two driving frequencies is established. The phase of this oscillation, with respect to the phase of the driving fields, is such that no energy is absorbed. CPT between hyperfine atomic levels was first observed experimen­ tally in a seminal paper by Alzetta et al. (1976), in which a light field from a multi-longitudinal-mode dye laser was sent into a vapor cell containing saturated Na and a buffer gas. The laser mode spacing had a harmonic near the frequency of the ground-state hyperfine splitting of Na. A long­ itudinal magnetic field gradient was applied to the cell, and the fluores­ cence from the cell was measured as a function of longitudinal position. Dark lines were observed in the fluorescence at locations where the magnetic field had shifted magnetically sensitive hyperfine levels into resonance with a mode spacing harmonic. Data from Alzetta et al. (1976) are shown in Figure 2. The observations were explained theoretically by use of a density matrix analysis, in which the excited-state population, and hence the fluorescence rate, in the three-level system was calculated as a function of the relative detuning of a pair of resonant optical fields (Arimondo & Orriols, 1976; Gray et al., 1978; Whitley & Stroud, 1976;). The dark line in the fluorescence was identified as resulting from (destructive) interfer­ ence of absorption pathways in the coherently excited atomic system.

Advances in CPT for Atomic Clocks

25

Figure 2 Data showing the fluorescence from a Na vapor cell under illumination by a multimode optical field. A magnetic field is applied with a gradient along the axis of the light propagation. Three dark lines are observed in the fluorescence spectrum.
Reprinted figure with permission from Alzetta et al. (1976);
1976 of the Societa Italiana de Fisica

A similar phenomenon on the Zeeman rather than hyperfine coherences was reported earlier by Bell and Bloom (1961), and a review of CPT is given by Arimondo (1996). A key aspect of this phenomenon is that coherences in atoms at microwave frequencies can be excited even with no microwave fields being present at the location of the atoms. The hyperfine coherences are excited entirely by two-photon processes involving only optical fields and have a line width determined by the hyperfine relaxation rate. In addition, the presence of the coherence can be detected easily by mon­ itoring the fluorescence (or absorption) by the atomic sample. CPT can also be understood as a combination of quantum interference and optical pumping. In the three-level model, a bichromatic optical field couples two long-lived states (denoted |1> and |2> in the Figure 1c) to a single upper level (denoted |3>). The energy of the ith level is denoted Ei and the optical field is denoted EðtÞ ¼ "1 e  ið!1 tþ1 Þ þ "2 e  ið!2 tþ2 Þ ;

ð1Þ

where "i, !i, and i are the (complex) amplitude, frequency, and phase of the ith field component, respectively. We may write orthogonal super­ positions of states |1> and |2> that interact with the optical field in distinct ways: jNCi ¼ c2 j1i  c1 j2i jCi ¼ c1 j1i þ c2 j2i:

ð 2Þ

i "i ffiffiffiffiffiffiffi ei½ðEi =hÞtþi  ; ci ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 j1 "1 j þ j2 "2 j2

ð3Þ

Here

26

Vishal Shah and John Kitching

and i is the electric dipole moment between state |i> and state |3>. It can be shown that when the two-photon resonance condition, !1–!2 = (E1–E2)/h, is fulfilled, the transition amplitude from the state |NC> to the excited state |3> is zero. The |NC> state is therefore a dark (or “non­ coupled”) state, because no light is scattered when the resonance condi­ tion is fulfilled. The suppression of the transition amplitude can be interpreted as a result of quantum interference between transitions from states |1> and |2> to state |3> under the influence of the exciting optical field (Alzetta et al., 1976; Arimondo, 1996; Lounis & Cohen-Tan­ noudji, 1992). An atomic sample initially in a thermal state will develop coherences when illuminated with a bichromatic field satisfying the resonance con­ dition. This can be thought of as an optical pumping effect: atoms in the “bright” (or “coupled”) state |C> will be excited to level |3> and will eventually fall into the dark state, where they no longer interact with the optical field. Population therefore builds up in the dark state and the absorption of the optical field (and fluorescence from the atomic sample) is reduced. As a function of two-photon detuning, we therefore observe a resonance in the absorption/fluorescence signal. In the data shown in Figure 2, CPT resonances are observed in the fluorescence spectrum of a cell subjected to a magnetic field gradient. CPT resonances are also frequently observed in the transmission spec­ trum of light passing through an atomic sample in the presence of a uniform magnetic field, as shown in Figure 3. In this case the frequency difference between the two excitation fields is scanned over the hyperfine transition frequency and a spectrum containing a number of absorption resonances can be observed as a function of difference frequency, corre­ sponding to transitions between different pairs of Zeeman-split hyperfine levels. In the case of circularly polarized light and a longitudinal mag­ netic field, only transitions between Zeeman levels with DmF = 0 are observed, resulting in a spectrum consisting of 2I lines for an atom with nuclear spin I. For atoms with half-integer nuclear spin, the central line (corresponding to mF = 0 ! mF = 0) occurs at a frequency close to the f1

f2

λ /4 Bichromatic light source

Full transmission

Magnetic field 87Rb

vapor cell

Photo detector f1-f2

Figure 3 Coherent population trapping resonance spectrum observed in the transmitted light through a vapor cell subject to a uniform magnetic field

Advances in CPT for Atomic Clocks

27

zero-field hyperfine splitting of the atom. The other lines are separated from the central line by even multiples of the Larmor frequency fL = B0, where  is the gyromagnetic ratio of the atoms. Atoms with higher nuclear spin have more Zeeman levels, and the CPT spectrum consists of correspondingly more lines. If the magnetic field is rotated so it has a transverse component, the CPT spectrum displays additional lines half­ way between the DmF = 0 lines corresponding to DmF = 1 transitions. Transitions with DmF = 2 can also be excited. For a more complete description of the CPT spectrum, see Wynands and Nagel (1999) and Knappe (2001). The bichromatic light needed to excite the CPT resonance can be generated in a number of ways. The two most common methods are direct modulation of the injection current of a laser diode (Cyr et al., 1993) and phase-locking of two lasers (Nagel et al., 1998). Direct modula­ tion is simpler to implement but results in an optical spectrum typically consisting of more than two frequencies with spectral amplitudes that are difficult to control. The use of injection-locked lasers requires more com­ plicated locking electronics but results in only two optical frequencies and allows a high degree of freedom in controlling the relative polariza­ tion and amplitude of these components.

2. ATOMIC CLOCKS 2.1 Introduction Most atomic clocks are based on alkali atoms (in particular H, Rb, Cs), which have a single valence electron. In these atoms, the energy spectrum is relatively simple, and long-lived ground states result in slow relaxa­ tion, narrow transition line widths, and correspondingly high precision. In addition to charge, both the atomic nucleus and the valence electron of all alkali atoms have spin angular momentum, and therefore a magnetic moment. Microwave frequency references are based on hyperfine transi­ tions between atomic states that differ in the relative orientation of the nuclear and electron magnetic moments. This difference in energies is on the order of 1—10 GHz, when translated into frequency units by dividing by Planck’s constant h. With some notable exceptions (Post et al., 2003; Taichenachev et al., 2005b), microwave atomic clocks are based on transitions between the magnetically insensitive mF = 0 substates of different hyperfine mani­ 6 0 (but DmF = 0) require a folds. Clocks based on states for which mF ¼ careful simultaneous measurement of the local magnetic field in order to prevent variations in this parameter from resulting in variations of the clock frequency. While this latter approach is not altogether prohibitive, it

28

Vishal Shah and John Kitching

adds significant complications to the operation of the device and has not found widespread popularity. On the other hand, optical clocks based on fermionic alkaline earth atoms have a small linear Zeeman shift that is effectively and routinely removed with appropriate interrogation techni­ ques (Akatsuka et al., 2008). The measurement of transitions between these atomic states can be accomplished in several ways. Perhaps the simplest measurement method is the passive excitation method, where an oscillating magnetic field is applied to the atoms at a frequency corresponding to the energy difference. When the frequency of the applied field is close to the fre­ quency of the atomic transition, an oscillating moment (coherence) is excited in the atom. Most frequently, magnetic dipole moments are excited. This coherence allows energy to be transferred from the field (atom) to the atom (field), and changes the internal state of the atom. Because this energy transfer is a resonant effect, the internal state of the atom can be monitored to determine when the frequency of the applied field corresponds to the energy splitting of the atomic states being coupled. A typical atomic frequency reference can be thought of as a series of steps. The atoms are first prepared in one specific atomic state (by magnetic state selection, optical pumping, or some other means). The oscillating magnetic field, generated by a “local oscillator” (LO), is then applied, causing some fraction of the atoms to change their state; this fraction depends on whether the frequency of the oscillating field is onresonance with the atoms. Finally the number of atoms in the final state (or the initial state) is detected, again by optical or magnetic means. Because of the resonant nature of the interaction, the number of atoms in the final state depends on the difference between the frequencies of the oscillating field and the atomic transition, and a measurement of this quantity can therefore be used in a feedback loop to lock the frequency of the oscillating field to the atomic transition frequency. The output of the clock is simply the frequency of the locked LO. The operation of a basic passive atomic frequency reference is shown in Figure 4. Atomic clocks based on alkali atoms can be divided into four main categories. Fountain clocks (Clairon et al., 1991; Kasevich et al., 1989; Zacharias, 1953), the most accurate atomic clocks at present, are large devices that often take up the better part of an entire room and require several hundred watts of power. There exist perhaps 10 such instruments worldwide and each typically takes several person-years to construct and evaluate. Hydrogen masers (Gordon et al., 1954), highly stable over long time periods, are about the size of a large filing cabinet. Cs beam clocks (Essen & Parry, 1955; Ramsey, 1950), based on beams of alkali atoms in a vacuum, are also highly accurate and are manufactured in rack-mounted enclosures. Vapor cell atomic clocks are based on atoms confined in a cell with a buffer gas (Arditi, 1958; Carver, 1957; Dicke, 1953) or wall coating

Advances in CPT for Atomic Clocks

(a)

29

(b)

LO

Number of atoms in final state

E2

E2 E1 State preparation

E1

State

detection

Transition excitation

LO frequency

Figure 4 (a) The operation of a passive atomic frequency reference typically proceeds in three steps. First the atom is prepared in some energy state, E1. The frequency from the local oscillator is then applied, causing transitions to another state with energy E2. The number of atoms in the final state is detected. (b) With this method, the frequency of the local oscillator can be determined with respect to the atomic transition

(Goldenberg et al., 1961; Robinson et al., 1958). The highest-performance vapor cell clocks are manufactured for installation on GPS satellites. These clocks are typically stable to 1013 or better over long periods but are intrinsically accurate (without calibration) only to about 109. Com­ pact vapor cell atomic clocks, developed for the telecommunications industry, can be held in the palm of one hand and are stable to about 1011. 2.2 Vapor Cell Atomic Clocks A schematic of a traditional Rb vapor cell frequency reference (Vanier & Audoin, 1992) is shown in Figure 5. The heart of the frequency reference is a vapor cell that contains the Rb vapor, along with an appropriate density of buffer gas. The vapor cell is contained within a microwave cavity into which a microwave field is injected. The microwave field is Physics package 87Rb lamp

Microwave cavity

85Rb

filter cell

Local oscillator

87Rb

vapor cell

Photo detector

Control system

Figure 5 Schematic of the major components of a traditional vapor cell frequency reference

30

Vishal Shah and John Kitching

generated by a LO, which is usually based on an electromechanical resonator (such as a quartz crystal resonator). Because the LO is an electromechanical device, it is typically rather unstable over long periods; the atoms therefore provide a stable reference frequency to which the LO can be locked. The atoms in the cell are illuminated by light generated by a Rb lamp. The Rb lamp is a second glass cell, containing Rb, through which an RF discharge is excited. Light from the lamp, with appropriate filtering, serves to prepare the atoms in the reference cell in one of the two hyperfine ground states via optical pumping. Because of this optical pumping, the atoms in the reference cell become less absorbing than they would in a thermal dis­ tribution. As the frequency of the RF field is tuned to near the atomic resonance, the population distribution in the reference cell changes again as the hyperfine populations are returned closer to their thermal distribu­ tion. This in turn increases the atomic absorption, and the increased absorption can be detected by monitoring the optical power transmitted through the cell with a photodiode. The transmitted optical power, as a function of microwave frequency, therefore becomes the “signal” to which the LO is locked. 2.3 Coherent Population Trapping in Atomic Clocks CPT was first used in atomic clocks in the early 1980s (Ezekiel et al., 1983; Hemmer et al., 1983a, 1984; Thomas et al., 1981, 1982). In these experi­ ments, modulated dye lasers were used to excite microwave transitions in a Na beam atomic clock. In effect, CPT zones replaced the microwave cavities employed in a conventional beam clock based on Ramsey’s method of separated oscillatory fields: in the first CPT zone the atomic coherence was created, and in the second, its phase was compared to that of the drive signal. Although initial investigations focused on the hyperfine transition in Na at 1.77 GHz, the use of optical fields to excite the coherence opened the door to the possibility of exciting atomic coherences in fre­ quency bands far beyond the gigahertz range. A Ramsey zone separation of 15 cm led to a resonance line width of 2.6 kHz (see Figure 6), and a corresponding frequency instability of 8  1010 at 1 second was measured, as shown in Figure 7. A subsequent experiment using a Cs atomic beam, excited by a modulated diode laser, demonstrated resonance widths of 1 kHz and a projected instability of 6  1011 at 1 second (Hemmer et al., 1993). The signal-to-noise ratio was about 10 times worse than that pre­ dicted by photon shot noise and was limited by frequency noise on the diode laser being translated into intensity noise on the measured atomic fluorescence. The conversion of FM to AM noise continues to be an important source of instability in the current generation of laser-pumped atomic clocks (Camparo & Coffer, 1999; Kitching et al., 2000).

Advances in CPT for Atomic Clocks

(a)

31

520 kHz

(b) 43 kHz (c) 1.3 kHz

ω2 Figure 6 (a) and (b) Rabi fringes from a Na atomic beam excited by a modulated dye laser. (c) Raman–Ramsey fringes with a width of 2.6 kHz. Reprinted figure with permission from Thomas et al. (1982);
1982 of the American Physical Society

10−6 10−7 10−8 σy (τ) 10−9 10−10 10−11 10−12 −3 10 10−2 0.1

1 10 102 103 104 τ (seconds)

Figure 7 Allan deviation of a frequency reference based on Raman–Ramsey excitation of CPT resonances in a Na atomic beam. Reprinted figure with permission from Hemmer et al. (1983b);
1983 of the Optical Society of America

These early experiments not only demonstrated the viability of the use of CPT in atomic frequency references but also identified the major limitations to both the short-term frequency stability and the accuracy. Sources of frequency instability common to many types of CPT frequency references include the FM—AM noise conversion mentioned above, atom and photon shot noise, and instability arising from the light shift (Hemmer et al., 1989). Sources of frequency instability associated specifically with the

32

Vishal Shah and John Kitching

Raman—Ramsey scheme include misalignment of the beams from a copro­ pagating configuration, birefringence, and polarization differences between the beams and changes in the optical path length (Hemmer et al., 1986). As mentioned above, semiconductor lasers have been used for CPT excitation in atomic beam clocks (Hemmer et al., 1993). The advantages of semiconductor lasers over dye lasers in this type of experiment are clear: smaller size and simpler operation. In addition, it is possible to modulate the optical field output of the laser by directly modulating the injection current. Resonances of width 1 kHz were obtained in a 133Cs beam by use of an edge-emitting AlGaAs diode laser modulated at 4.6 GHz. The two coherent first-order sidebands created the 9.2-GHz frequency difference needed to excite the Raman—Ramsey fringes. In 1993, Cyr, Tetu, and Breton described a method for exciting and detecting CPT resonances in an alkali vapor cell by use of a single diode laser (Cyr et al., 1993). The details of the experiment are shown in Figure 8. The injection current of the laser was modulated near the sixth subharmo­ nic (1.139 GHz) of the 87Rb hyperfine frequency (6.835 GHz), creating sidebands on the optical carrier, several of which are separated by approxi­ mately the atomic resonance frequency. When one of these sideband pairs was tuned to be in optical resonance with the atomic transitions, and their (a)

(c)

νeg

νef

e

1.0 m=0

f

νo νo

g

(b)

fm

fm

νI

S

P LD

0.8 Intensity Id [a.u.]

νef νeg

Probe p Pum

θ

AN PD

m = −1

m=1

0.6 0.4 0.2 0 −60 −40 −20

0

20

40

60

Detuning of f m [kHz]

λ /4

Synthesizer

Figure 8 Excitation of CPT resonances in an alkali vapor cell with a modulated diode laser. (a) The atomic energy level spectrum and optical frequency spectrum of the modulated diode laser. The CPT resonance is excited when the frequency splitting,  0 = nfm, between two components of the diode laser optical spectrum is equal to the atomic ground-state hyperfine splitting,  eg- ef. (b) Experimental setup. LD, laser diode; P, polarizer; S, solenoid; AN, polarization analyzer; PD, photodetector; /4, quarterwave plate. (c) The photodetector signal as a function of the detuning of the laser modulation frequency from the sixth subharmonic of the atomic resonance frequency. Reprinted figure with permission from Cyr et al. (1993);
1993 IEEE

Advances in CPT for Atomic Clocks

33

difference frequency (determined by the modulation frequency) made equal to the microwave transition, a microwave coherence was excited in the atoms. This coherence was detected by monitoring the polarization rotation of a probe beam derived from the same laser (Figure 8c). This idea is particularly noteworthy in the context of miniaturized frequency references, because all components used in the experiment are small. The use of a vapor cell lends itself well to miniaturization compared to previous CPT experiments based on atomic beams because much smal­ ler cell sizes can be used to achieve a given short-term frequency stability, due to the presence of the buffer gas. Some time later it was demonstrated that a single laser beam could also be used, allowing for even further simplification of the experimental setup (Levi et al., 1997). Since then, there has been considerable study of CPT-excited vapor cell frequency references. The noise processes in these instruments have been identified (Kitching et al., 2000) and short-term instabilities as low as 1.3  1012/H have been demonstrated in large-scale systems (Zhu & Cutler, 2000). Vapor cell CPT atomic clocks have been investi­ gated in some detail theoretically (Vanier et al., 1998, 2003a,c,d) and experimentally (Godone et al., 2002d; Knappe et al., 2001, 2002; Levi et al., 2000; Merimaa et al., 2003; Stahler et al., 2002). They have also been compared both theoretically (Vanier, 2001b) and experimentally (Lutwak et al., 2002) to conventional optically pumped vapor cell refer­ ences with the conclusion that the short-term frequency stability of CPTbased instruments should be comparable to or better than conventional ones. A review of atomic clocks based on CPT is given in the work of Vanier (2005). In 2001, a compact physics package for CPT frequency reference was demonstrated by Kitching et al. (2001a,b). This device had a volume of about 14 cm3, and the short-term stability of this device was 1.31010/ H. A photograph of the device is shown in Figure 9a, and the CPT resonance and Allan deviation are shown in Figure 9b and c, respec­ tively. Similar work was being explored simultaneously (Delany et al., 2001; Vanier, 2001a), connected with the ultimate development of com­ pact, commercial CPT clocks (Deng, 2008; Vanier et al., 2005). These early efforts to use CPT to miniaturize atomic frequency standards used glass-blown vapor cells with a diameter of several millimeters or more. The cells were assembled as discrete components with a laser, optics, and photodetector to form the functioning physics package. Complete miniaturized CPT frequency references for use in a com­ mercial setting have also been demonstrated (Deng, 2008; Vanier et al., 2004, 2005). This work demonstrated integration of a compact CPT phy­ sics package with a low-power LO and compact control electronics. The opportunities for atomic clock miniaturization afforded by the use of CPT led to some significant developments related to use of

34

Vishal Shah and John Kitching

(a) Laser Lens

Attenuator

Waveplate

Magnetic shielding

Photodetector

(b)

(c) 1.003 1.002

620 Hz at 4.6 GHz

σ y (τ)

Change in absorption

Cell

1.001

10−10

10−11 1.000 −1000

−500

0

500

1000

Frequency offset from 4.6 GHz (Hz)

100 101

102

103

104

105

τ (seconds)

Figure 9 A miniaturized physics package for a CPT frequency reference. (a) Photograph of the instrument with major components identified. (b) CPT resonance and (c) fractional frequency instability (Allan deviation) as a function of integration period. Reprinted figure with permission from Kitching et al. (2001b)

micromachining processes in atomic clocks. A preliminary analysis suggested that CPT clocks based on millimeter-scale vapor cells could achieve short-term frequency instabilities of a few parts in 1011 at 1 second of integration (Kitching et al., 2002). While the stability was expected to be worse than that of their larger counterparts, it was recognized that these micromachined or “chip-scale” atomic clocks could serve an important role in providing precise timing for portable, battery-operated instruments. Because of the small size, the power required to maintain the cell at its operating temperature could be drastically reduced. When combined with similar improvements in power resulting from the use of a laser, rather than a lamp, as the light source, operation with small batteries could be envisioned. A review of chip-scale atomic frequency references can be found in the work of Knappe (2007). Because CPT played an important role in many of these new micromachined clock designs, considerable research was carried out to improve and optimize CPT techniques specifically for this new development.

Advances in CPT for Atomic Clocks

35

2.4 Stability of Vapor Cell Atomic Clocks The stability of an atomic clock is most often characterized by its Allan deviation. The Allan deviation (Allan, 1966; Barnes et al., 1971), devel­ oped to quantify the fluctuations of nonstationary random variables, is a measure of the frequency instability of the clock obtained after integrat­ ing a measurement of its frequency for a period  and is denoted y(). For passive atomic clocks, in which white frequency noise is the domi­ nant noise source, the Allan deviation is given by y ðÞ ¼

pffiffiffi ; QðS=NÞ 

ð4Þ

where Q is the Q-factor of the atomic resonance,  is the integration period, S/N is the measurement signal-to-noise ratio (in units of HHz), and is a constant of order unity related to how the resonance is mea­ sured. The frequency instability is hence proportional to the resonance line width, or relaxation rate, of the atoms and inversely proportional to the signal strength. In a conventional optical-microwave double-resonance (OMDR) fre­ quency reference, the optical transmission is high when the microwave field is tuned away from resonance and decreases on resonance as the atoms are repumped into an equilibrium state by the microwave field (see Figure 10a). The OMDR resonance is characterized by its width W and its transmission contrast A/B, according to Figure 10a. The CPT resonance, shown in Figure 10b, has low transmission when the frequency difference between the optical fields is off resonance; the transmission increases on resonance when the dark state is populated. The resonance can be characterized by its width and its absorption con­ trast, denoted by W and CA = A/B in Figure 10b in the limit that the quantity B ;

Advances in CPT for Atomic Clocks

F′=3

(a)

41

(b)

F′=2 F′=1 F′=2

νopt

F′=1 νopt F=2

F=2

δr

Δhfs

F=1

δr

Δhfs

Transmission

Transmission

F=1

Raman detuning, δr

Raman detuning, δr

Figure 11 Simplified energy level diagram of 87Rb, showing the (a) D2 and (b) D1 transitions. The gray lines show the original three-level CPT model, while the dark lines show the additional transitions due to the presence of the excited-state hyperfine structure. The solid lines indicate CPT transitions, while the dashed line indicates a single-photon transition

!

jF ¼ 1; mF ¼ 0 > ! jF 0 ¼ 2; mF ¼ 1 >

jF ¼ 2; mF ¼ 0 > :

The transition amplitudes are such that the individual dark states are in phase and therefore add constructively (Stahler et al., 2002). On the D2 transition, the same two Lambda systems are again excited. However, due to the different Clebsch—Gordon coefficients, the phases of the two dark states are quite misaligned, which results in a partial destruction of the CPT resonance. For example, apart from a constant phase factor on the D2 87Rb line on the mF = 0 ground states, the dark state for F = 1 ! F’ = 2 F = 2 is jNCi ¼ p1ffiffi2 ðj1i þ j2iÞ, and for the F = 1 ffi ð5j1i þ j2iÞ. On the other ! F’ = 1 F = 2 Lambda system, jNCi ¼ p1ffiffiffi 26 hand, on the D1 line, the dark state is given by jNCi ¼ p1ffiffi2 ðj1i þ j2iÞ for both Lambda systems. In addition, on the D2 line there is one single-photon transition that depopulates the dark state: !

!

jF ¼ 2; mF ¼ 0 > !jF 0 ¼ 3; mF ¼ 1 > : The combined influence of these two effects is that the strength of the CPT resonance on the D2 transition is significantly smaller than that on the D1 transition.

42

Vishal Shah and John Kitching

CPT resonance contrast

0.10 A

0.08 0.06 0.04

B

0.02 0.00 −8

−6

−4

−2

0

2

4

6

8

10

Detuning (kHz) Figure 12 Comparison of CPT resonances excited using light resonant with the D1 (Trace A) and D2 (Trace B) optical transitions in 85Rb. An improvement in the absorption contrast by a factor of 10 is obtained by use of the D1 resonance. The line width of the D1-excited resonance is also narrower. Reprinted figure with permission from Stahler et al. (2002);
2002 of the Optical Society of America

The influence of additional energy levels on CPT resonances becomes clearly evident when CPT resonances using D1 and D2 transitions are compared by use of the atomic vapor cell (Zhu, 2002). It is found experimentally that CPT resonances excited by use of the D1 transition are almost an order of magnitude stronger than those seen by use of the D2 transition (Lutwak et al., 2003; Stahler et al., 2002). A comparison of CPT resonances obtained using D1 and D2 excitation, but otherwise under similar conditions, is shown in Figure 12. The discrepancy between the strengths of the CPT resonances on the D1 and D2 transi­ tions is in contrast with microwave resonances seen in conventional optically pumped clocks, in which efficient optical pumping can be achieved by use of both D1 and D2 transitions. The reason CPT resonances are so sensitive to excitation pathways is that CPT reso­ nances rely equally on both optical pumping and quantum interference. This is contrasted with conventional microwave resonances that rely only on optical pumping. 3.2 Contrast Limitations due to Zeeman Substructure The multiplicity of Zeeman levels present in the ground states of alkali atoms is another primary limitation to the signal strength. In the pre­ sence of a small magnetic field, the ground-state hyperfine levels are Zeeman split into (2F þ 1) magnetic ground states. In atomic clocks, a small magnetic field is typically applied to separate the various ground

43

Advances in CPT for Atomic Clocks

states so that the magnetically insensitive mF = 0 ground states can be uniquely interrogated without interference from the magnetically sen­ sitive transitions. However, the presence of the unwanted but closely spaced mF 6¼ 0 states also influences the strength of the CPT resonance. In thermal equilibrium, atoms populate all of the ground-state mag­ netic sublevels with roughly equal probability, as shown in Figure 13a. In atomic clocks, the optimized light intensity used for CPT excitation is generally weak enough that it causes some, but not significant, redistribution of atomic population between the various magnetic sub­ levels. The useable CPT resonance signal (quantity A in Figure 10b) is generated by atoms that are in the mF = 0 states, but many atoms in states with mF 6¼ 0 contribute to absorption of the incident light (quan­ tity B in Figure 10b). The absorption contrast is therefore reduced by roughly the ratio of the number of mF = 0 ground states to the total number of ground states, under the assumption that the light intensity is weak enough that it does not significantly redistribute the level populations. Another issue that is frequently discussed is optical pumping loss to the “trap state” (Renzoni & Arimondo, 1998; Vanier et al., 2003b). When

(a)

(b)

νopt

νopt

F=2

δr

F=2

Δhfs

δr

F=1 Transmission

Transmission

F=1

Raman detuning, δr

Δhfs

or

Raman detuning, δr

Figure 13 (a) Reduction of the CPT resonance contrast due to the presence of groundstate Zeeman structure. An even distribution of atoms among the Zeeman groundstate sublevels implies that only a small fraction of atoms contribute to the magnetically-insensitive mF = 0 ! mF = 0 transition. (b) Effects of the “trap” state (population indicated by the dark bar), which does not form a CPT resonance and into which atoms are pumped by the optical fields

44

Vishal Shah and John Kitching

circularly polarized light resonant with the D1 transition is used to excite CPT resonance, for example, a significant fraction of the population can be pumped to the magnetic end states (F = I þ 1/2, mF = + F). As seen in Figure 13b, the magnetic end state is an incoherent state, which is dark due to selection rules which prohibit excitation by circularly polarized light fields (þ transitions in this case). Atoms can therefore be “trapped” in this end state (or even in other states, depending on the laser polariza­ tion) and stop interacting with the light fields. Because the optical con­ figuration described above is used in current generation of microwave CPT clocks, the problem of the trap state has received significant attention from the research community. This configuration in which a Lambda system is excited in the presence of an incoherent dark state is often referred to as an open Lambda system. The atoms that are trapped or lost to the end state contribute neither to the CPT resonance signal nor to the optical absorption. They are simply invisible to the incident light fields. To compensate for this loss of atomic population, alkali density can be increased, but this also increases relaxa­ tion through alkali density dependent decoherence mechanisms such as spin exchange (Happer, 1972), which increases the line width of the CPT resonance. There have been a number of solutions that have been pro­ posed to depopulate the end state. Some of the approaches are outlined below. 3.2.1 Depopulation Pumping Using -Polarized Light This idea involves use of an additional linearly polarized laser light resonant with the F = I þ 1/2 ground state and the F’ = I þ 1/2 excited state (Kazakov et al., 2005b). The linearly-polarized light travels in a direction perpendicular to the applied magnetic field such that it excites -transitions (DmF = 0). This secondary light field is used to depopulate the end state but does not depopulate the atomic population in the F = I þ 1/2, mF = 0 ground because of selection rules that prohibit such a transition. While this technique works well in theory, there are several difficulties associated with using it in a practical device. Besides the technical diffi­ culties arising from the use of separate laser beams traveling in perpen­ dicular directions, this technique requires selective excitation of the F = I þ 1/2 ground state by use of the F’ = I þ 1/2 excited state. This limits the amount of buffer gas that can be used in the vapor cell to approximately below 1 kPa in Rb and Cs such that the levels in the excited state can be clearly resolved. This technique therefore presents difficulties for use with smaller cells, which typically use higher buffer gas pressures to avoid broadening of the hyperfine transition due to wall

Advances in CPT for Atomic Clocks

45

collisions. In addition, the narrow optical line width resulting from the low buffer gas pressure requires tighter restrictions on the amount of laser drift and laser frequency noise that can be tolerated. 3.2.2 Excitation with Orthogonal s-Polarized Light Fields Another approach that has been proposed to reduce the atomic population in the unused Zeeman levels is simultaneous excitation of the CPT resonance by use of a combination of þ and — light fields. This technique addresses both the thermal population in all Zeeman levels and the population build-up in the trap state due to optical pumping. A linear light field traveling along the magnetic field is the simplest example of þ and — light fields. The transitions that are excited by a combination of þ and — light fields are shown in Figure 14. The þ and — light fields independently excite Lambda systems on the mF = 0 states. As can be seen from the figure, there are no end states that are dark in this optical configuration. Unfortunately, simply using linearly polarized light to excite a closed Lambda system between mF = 0 levels does not work. The reason for this is that the individual dark states excited by the þ and the — light are out of

νopt

F=2

δr

Δhfs

Transmission

F=1

Raman detuning,δr Figure 14 Excitation of CPT resonances by use of a combination of sþ and s– light fields. Destructive interference can be avoided by independently adjusting the phase of the modulation in each polarization component. Solid lines indicate optical fields with sþ polarization that excite CPT transitions, while dashed lines indicate optical fields with s– polarization. Dotted lines indicate depopulation pumping of “trap” states

46

Vishal Shah and John Kitching

phase. In other words, the atoms that appear dark to the þ polariza­ tion appear bright to the — polarization, and vice versa. The result of this destructive interference is that the overall CPT resonance is not observed. One solution to this problem is to introduce time delay between þ and the — light fields such that the individual dark states constructively add together in phase. This principle has been successfully demonstrated in two ways. Jau et al. (2004a), Taichenachev et al. (2004b), and Karga­ poltsev et al. (2004) have proposed splitting the light fields into two parts with orthogonal polarization and recombining the fields after introdu­ cing a relative path delay between the þ and — components. The relative path difference of half a microwave wavelength (ground-state hyperfine splitting) shifts the phase of the dark state such that the reso­ nances constructively interfere. This path length difference can be intro­ duced by use of polarization filtering in a copropagating geometry or by reflecting the light back through the cell. A significant gain in the CPT contrast was reported by Jau et al. (2004a). The difficulty in using additional optics in splitting and recombining the light field after introducing the path delay led to development of another similar approach (Shah et al., 2006b). In this approach two separate lasers were used to avoid the difficulty in splitting and recom­ bining the light fields. Each of the two independent lasers had opposite circular polarizations, and they independently excited CPT resonance on the mF = 0 ground states. The lasers were modulated by use of the same microwave source, and an electronic microwave phase shifter was inserted into the RF path to one laser to shift the relative phase of the microwave modulation on the two lasers by . This technique works well and can also be implemented in a miniature package; however, the use of two separate lasers adds some complexity to the overall implementation and to the control system in particular. Another strategy is to generate a coherence on the mF = 0 ! mF = 0 transition using one polarization and measure the resonant change in birefringence with a weak optical field with an orthogonal polarization (Zhu, 2003). 3.2.3 CPT Excitation on DmF = 2 Transitions A third approach that has been proposed involves the direct use of linearly polarized light and excitation of CPT resonance between mF = þ1 and mF = 1 (Taichenachev et al., 2005b). This scheme, often referred to as "lin // lin" since two optical fields with parallel linear polarization are used, is by far the simplest way of exciting a closed Lambda transition by use of a combination of þ and — light fields. The transitions that are excited are shown in Figure 15. The relative phase between the dark states excited between mF = þ1 ! mF = –1 and mF = –1 ! mF = þ1 is no

Advances in CPT for Atomic Clocks

47

νopt

F=2 Δhfs Δ hfs

δr

Δhfs

Transmission

F=1

Raman detuning, δr Figure 15 levels

Excitation of CPT resonance on DmF = 2 transitions between hyperfine

longer of concern, because the dark states are excited between two independent sets of ground states. In this scheme, the transitions have an interesting dependence on magnetic field. Because of the slightly different g-factors for states in each of the two hyperfine levels, there is a small first-order magnetic field sensitivity for each of the mF = þ1 ! mF = –1 and mF = –1 ! mF = þ1 transitions. In 87Rb for example, individual mF = +1 states are shifted by over 300 kHz in 50 mT magnetic field. However, the difference frequency between the states jF ¼ 2; mF ¼ þ1Þ and jF ¼ 2; mF ¼ 1Þ shifts by only a few hundred hertz because of their much smaller linear magnetic field dependence. In addition, the sign of this residual linear sensitivity is negative for one transition (jF ¼ 2; mF ¼ þ1i $ jF ¼ 2; mF ¼ 1i) and positive for the other (jF ¼ 2; mF ¼ 1i $ jF ¼ 2; mF ¼ þ1i). As a result, when the resonance is excited using both pairs of ground states simulta­ neously, the linear dependence on magnetic field of the center point of the resonance vanishes. The presence of a small magnetic field therefore produces only a broadening of the overall resonance. Kazakov et al. (2005a) have proposed using this mechanism to produce a pseudoreso­ nance by applying a magnetic field large enough that the resonances on the individual pairs of ground states are shifted such that there is a dip in the center to which an LO can be stabilized. The main perceived drawback of this technique is that it was predicted to be effective only in 87Rb (or other atoms that have a nuclear spin of 3/2) atoms but not in 133Cs (which has nuclear spin 7/2). Atoms with

48

Vishal Shah and John Kitching

larger nuclear spin have a greater number of levels in the excited state, which cause additional Lambda systems to be excited that destroy the CPT resonance through destructive interference. This is due to the fact that it requires selective excitation of the Lambda system by use of the F’ = 1 excited state. This requires the use of very low buffer gas pressures, possibly in combination with a wall coating to reduce the wall-induced relaxation. Interestingly, high-contrast resonances have been observed using a similar scheme in a cell containing 133Cs and a low buffer gas pressure (Watabe et al., 2009). This experimental result suggests that the excitation of multiple Lambda systems affects the resonance contrast only modestly. 3.2.4 The Use of End Resonances The diluted atomic population participating in the clock transition can be improved by use of optical pumping, for example, with -polarized light, as mentioned in Section 3.2.1 above. Ideally, the population accumulates in the mF = 0 states, which are first-order magnetically insensitive. How­ ever, population can also be pumped into the “end” mF = F states, which can then be used to measure the hyperfine frequency. Because transitions between end states are first-order sensitive to magnetic fields, the local magnetic field must be measured simultaneously in a precise manner in order to reduce the field dependence of the clock output frequency. This can be done by measuring the Zeeman resonance frequency simulta­ neously with the hyperfine end-resonance frequency. An additional advantage gained by the use of end resonances is the suppression of spin-exchange broadening. At high alkali atom densities, spin-exchange collisions can be the dominant source of hyperfine relaxa­ tion (Walter & Happer, 2002). An atomic sample perfectly polarized in the end state does not undergo spin-exchange relaxation because all atoms are oriented in the same direction, and therefore no angular momentum can be exchanged between any two colliding atoms. However, a small population in other states creates some relaxation, and hence only a partial suppression of the spin-exchange relaxation can be achieved in real atomic systems. A final advantage of this scheme is that atoms can be optically pumped even at very high buffer gas pressures where the optical transi­ tions from the ground-state hyperfine levels are broadened far beyond the state frequency splitting. The traditional OMDR configuration has considerably degraded performance in the presence of high buffer gas pressure, because the hyperfine optical pumping is very inefficient. This proposal and accompanying experiments in 87Rb are described by Jau et al. (2004b) and shown schematically in Figure 16. Microwave excitation of the hyperfine transitions was used in the experiment, as opposed to CPT transitions, although the enhanced contrast and narrow

Advances in CPT for Atomic Clocks

49

(a) νopt

F=2

ω RF ω mw

Δhfs

ωL ωL

F=1 (c)

3ω L

Δhfs Microwave frequency, ω mw

Transmission

Transmission

(b)

ωL

RF frequency, ω RF

Figure 16 “End-resonance” method for increasing the absorption contrast and decreasing the line width of CPT resonances. (a) Atoms are optically pumped into the “end” state with maximal angular momentum, resulting in high transmission of the pumping light through the cell. A microwave field w mw and RF field w RF are applied simultaneously and the pump light transmission monitored as a function of (b) w mw and (c) w RF, providing a simultaneous measurement of w L and Dhfs –3 w L (for 87Rb)

line width should be equally present in CPT resonances. Line width suppression by a factor of about three was measured, as was considerably enhanced signal contrast (Post et al., 2003). Simultaneous measurement of a magnetically sensitive hyperfine transition frequency and Larmor pre­ cession frequency by use of a “tilted 0-0 state” was demonstrated by Jau and Happer (2005). When the system was operated as an atomic clock, a short-term instability of 6  1011/H was obtained in a compact physics package (Braun et al., 2007). 3.2.5 Amplitude-Modulated Versus Frequency-Modulated CPT Excitation Sources CPT resonances are often excited in alkali vapor cells with dimensions on the order of 1 cm. In order to prevent rapid relaxation of the hyperfine coherence due to wall collisions, a buffer gas is usually added to the cell. The buffer gas species can be chosen so that its effect on the hyperfine relaxation rate is rather small. The optimal buffer gas pressure varies roughly as the inverse of the smallest cell dimension and balances colli­ sion-induced relaxation with relaxation caused by residual diffusion

50

Vishal Shah and John Kitching

through the buffer gas to the walls of the cell (Kitching et al., 2002; Knappe, 2007). The buffer gas also significantly broadens the optical transitions involved in the CPT resonance. As the buffer gas pressure is increased, the optical transitions from the ground-state hyperfine levels can go from being completely resolved to being completely unresolved. When a single modulated laser is used to excite the resonances, the number of modula­ tion sidebands that interact with the atoms can vary from two to many. In Figure 17a, the spectrum from the optical transitions is plotted in these

Transmission

Optical Spectrum

Δhfs

(a)

A B

Optical frequency

Saturation S

(b) 100 Ideal saturation AM Ideal saturation FM AM data FM data Computer model fit

10−1

10−2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Buffer gas Pressure p (atm at 21°C) Figure 17 (a) Comparison of atomic optical absorption spectrum with laser modulation spectrum for two qualitatively different buffer gas pressures. Trace A is for a low buffer gas pressure, for which the optical transitions from each hyperfine level are well resolved, while Trace B is for a higher buffer gas pressure, for which the transitions are not resolved. For the case of Trace A, only two of the frequencies in the optical spectrum interact with the atoms, while for the case of Trace B, almost all do. (b) Experimental data comparing the strength of the CPT resonance (identified with the saturation parameter S) as a function of buffer gas pressure for FM- and AMmodulated light fields. Reprinted figure with permission from Post et al. (2005);
2005 of the American Physical Society

Advances in CPT for Atomic Clocks

51

two limits and compared with the spectrum of a light field modulated at one-half the hyperfine splitting. When the transitions are resolved and only two optical field frequencies interact with the atoms (Trace A in Figure 17a), a dark state is created with a phase defined by the relative phase of the two relevant optical fields, and the absorption contrast can be quite large. However, when the transitions are not resolved, many optical frequencies interact with the atoms (Trace B in Figure 17), and the dark state tries to adjust its phase to correspond to the phase of each pair of sidebands separated by the hyperfine splitting. If the light source is amplitude modulated, the beatnote between each pair of sidebands has the same phase and the dark states created by each pair independently add constructively to form a single dark state with high contrast. If, on the other hand, the light source is frequency modulated, some pairs of sidebands are out of phase with the other pairs, and the dark states add destructively. For the high buffer gas pressures required for small vapor cells, therefore, the modulation properties of the light source are critically important. A careful study of this phenomenon is presented by Post et al. (2005), largely supporting the intuitive reasoning presented here. Data from Post et al. (2005) are shown in Figure 17b. 3.3 High-Contrast Resonances Using Four-Wave Mixing One of the issues associated with microwave CPT atomic clocks is that the transmission contrast of the CPT resonance is typically small (in the range of a few percent) when the atomic clock is fully optimized. This is due to several reasons, including the presence of modes generated by micro­ wave laser modulation that do not participate in the formation of CPT resonances. A very large fraction of the laser noise that affects the CPT clock performance can be eliminated by removing the background light. Shah et al. (2007) demonstrated a novel technique based on four-wave mixing in a double-Lambda system, shown in Figure 18a, to eliminate most of the background light falling on the photodetector. The experimental setup from Shah et al. (2007) is shown in Figure 18b. In this experiment, the þ light is used to create a dark state (coherence) in atoms by use of conventional CPT laser modulation. By use of a second single-frequency laser with opposite circular polarization, the coherence generated in the atoms is gently probed to stimulate emission of a con­ jugate light field whose frequency is separated from that of the original probe beam by the ground-state splitting. Through a combination of spectral and polarization filtering, the power in all of the incident light fields other than the conjugate field is then largely eliminated. When the pump beams satisfy the two-photon resonance condition, there is brightness observed on the photodetector from the incident conjugate field. When the two-photon condition is not satisfied, the conjugate field

52

Vishal Shah and John Kitching

(a)

⏐F ′ = 2, m f +1〉

⏐F ′ = 2, m f −1〉

52P1/2

Γ

σ−

Γ

Pump σ+ Beam-2

ΩP

D1

795 nm

Generated conjugate field ⏐F = 2, m f 〉 52S1/2

Ω

Ω

Pump Beam-1

γ

⏐F = 1, m f 〉 6.834 GHz (b) Beam-1 +

+

+

+

Beam-2 87Rb

λ/4

85

Rb

λ/4 Polarizar Photodetector Conjugate field power (nW)

Laser-2

Laser-1

20

Contrast: 100 × h /H = 90.3%

h H

A 0 −10 10 −5 5 0 Two photon detuning (KHz)

Figure 18 (a) Level diagram showing the polarizations and tunings of the optical fields for CPT resonance contrast enhancement using four-wave mixing and a filter cell. (b) Experimental setup and contrast measurement. From Shah et al. (2007); reproduced with permission from the Optical Society of America

is not generated, and therefore there is no light incident on the photo­ detector. Experimentally the transmission contrast seen in this way approaches 95% and is limited only by the efficiency of the optical filter­ ing in the setup. 3.4 Push–Pull Laser Atomic Oscillator Jau and Happer (2007) have demonstrated a novel technique in which they show a “mode-locking” type behavior in a laser cavity in the pre­ sence of alkali atoms. In this self-oscillating system, the frequency

Advances in CPT for Atomic Clocks

λ /4 (2) Vapor cell

(a)

Semitransparent mirror

Grain medium

λ/4 (1)

Alkali-metal atoms

53

Mirror

Photon spin of the laser light Time

Electron spin of alkali-metal atoms

Amplitude (a.u.)

(b) Fabry−Perot signal:

12

Without SPPP Single optical peak:

8

With SPPP An optical comb:

Generation of optical comb

4 0 –10

–5

0

5

Relative frequency (GHz) Figure 19 (a) Operation of the “push–pull” laser-atomic oscillator. (b) The comb of output frequencies spaced by the hyperfine frequency of the alkali atoms in the cavity. Reprinted figure with permission from Jau and Happer (2007);
2007 of the American Physical Society

separation of the spontaneously generated modes is given by the ground state mF = 0 ! mF = 0 transition frequency, introducing the prospects of operating the system as an atomic clock. The schematic of their experi­ mental setup is shown in Figure 19a. The basic idea is the following: imagine that all the atoms are initially prepared in a dark state between the mF = 0 hyperfine ground states. The atoms in the dark state precess at the hyperfine frequency and appear continuously transparent only to light fields that excite a Lambda system. Two or more optical modes that are separated by the hyperfine frequency and have a constant phase relation between them suitable for exciting a Lambda system are thus the “allowed” cavity modes in the system. Because of the fixed phase relationship between the optical modes, the cavity operation is analogous to that in a mode-locked laser. The result is that the light coupled out of the cavity has a frequency spectrum similar to that of a frequency comb (see Figure 19b).

54

Vishal Shah and John Kitching

Through the use of two /4 wave plates, the optical arrangement in the cavity was such that the light excites þ transitions traveling along one direction and — transitions when traveling in the opposite direction. The purpose of this was to prepare atoms in a pure superposition of mF = 0 states without the usual fraction in the end state. The length of the cavity was chosen to be an odd integral multiple of half the hyperfine wave­ length, such that the dark state due to oppositely traveling light fields remained in phase. This system has the important feature that no LO is needed to excite the resonance; the system here is an active system and oscillates at the hyperfine frequency. A related experiment was described by Akulshin and Ohtsu (1994), in which an alkali cell was placed in an external cavity providing optical feedback to a distributed feedback (DBF) laser. A second laser beam, separated in frequency by approximately the alkali ground-state hyper­ fine frequency, was sent through the cell parallel to the first laser beam. It was found that the laser with feedback optically locked to the second laser with a frequency difference exactly equal to the ground-state hyper­ fine splitting. CPT-induced polarization rotation has also been used in a similar context (Liu et al., 1996). A number of other self-oscillating systems based on CPT have been developed (Strekalov et al., 2003, 2004; Vukicevic et al., 2000), in which RF rather than optical feedback was used to sustain the oscillation. 3.5 The CPT Maser A CPT maser (Godone et al., 1999; Vukicevic et al., 2000) is an active frequency standard in which a coherent microwave signal is directly recovered from the atoms instead of an indirect signal in the form of change in optical transmission. This approach can also eliminate the need for an external microwave oscillator for laser modulation. Once the microwave oscillation is started, it can be sustained indefinitely by use of the microwaves obtained directly from the atoms in a feedback configuration. In a CPT maser, atoms are enclosed in a microwave cavity whose frequency is tuned close to the difference frequency between the mF = 0 hyperfine ground states (see Figure 20). The coherence generated in the atoms through dark-state excitation couples with one of the cavity modes to stimulate emission of microwaves by the atoms at the hyperfine frequency. The microwaves emitted by the atoms can, in turn, be used to modulate the laser to sustain the maser operation. A complete CPT maser prototype was demonstrated and evaluated and an instability of 31012/H was measured, integrating down to below 1013 at 1 hour, once the linear drift had been removed (Levi et al., 2004). A variety of noise contributions were also measured, with thermal

Advances in CPT for Atomic Clocks

55

B2 Pout Ptr

Lasers

rf Field Cesium cell & Buffer gas

P11

Microwave cavity

Figure 20 The CPT maser, in which a ground-state atomic coherence is generated using CPT and the power radiated by the corresponding magnetic moment, is captured in a microwave cavity. Experimental setup. Reprinted figure with permission from Vanier et al. (1998);
1998 of the American Physical Society

noise being the most important at short integration times and temperature-related effects dominating at long integration times. Considerable theoretical work was also carried out to understand the operation of the CPT maser in detail (Godone et al., 2000; Vanier et al., 1998), as well as a number of interesting independent phenomena related to its operation and underlying physics (Godone et al., 2002a,b,c,d). 3.6 N-Resonance A novel alternative to CPT, the N-resonance, was proposed (Zibrov et al., 2005) and studied subsequently (Novikova et al., 2006a,b). This scheme, which has its origins in earlier work on multiphoton resonances in alkali atoms (Zibrov et al., 2002), can be thought of as a modification of the conventional OMDR technique. Just as in the OMDR technique, atoms are optically pumped into one hyperfine level with an optical “probe” field resonant with a transition from the other hyperfine level. However, instead of exciting the microwave transition by use of a microwave field, a bichromatic optical field is used, close to Raman resonance with the microwave transition, but detuned from the optical transition. When the Raman resonance condition is achieved, atoms are optically pumped back into the depleted hyperfine level, leading to increased optical absorption of the pump field. This absorptive resonance is in contrast with the conventional CPT resonance, in which reduced absorption is seen when the two-photon condition is satisfied. Among the advantages of the N-resonance scheme is that unlike CPT resonance, this scheme produces signals of high contrast (as high as 30% transmission contrast) on both the D1 and the D2 transitions. Clock in stabilities of 1.5  1010/H have been obtained in 87Rb confined in a cell

56

Vishal Shah and John Kitching

of diameter 2.5 cm. It has also been shown that despite the inherently offresonant operation of N-resonance-based atomic clocks, the light shifts can be canceled to first order by appropriately choosing the intensity in the pump and the probe beams, allowing the possibility of building an atomic clock with good long-term stability based on N-resonances. This scheme still requires that the excited-state hyperfine resonances be resolved, and hence has the same limitations with respect to buffer gas pressure as some of the techniques discussed above. Although in its most general form requiring optical fields at three unique frequencies, this scheme can be implemented by use of only two optical fields by allowing a single field to do double duty, both as the probe field and as one of the legs of the Raman field. An energy-level diagram of the N-resonance excitation mechanism is shown in Figure 21a, and the experimental implementation using a single modu­ lated diode laser is shown in Figure 21b. One of the weak sidebands generated by the modulator serves as the probe and one of the Raman fields, while the strong optical carrier provides the second Raman field. An etalon is placed after the cell to attenuate the strong Raman field and therefore increase the resonance contrast. 3.7 Raman–Ramsey Pulsed CPT As described above, light shifts play a major role in determining the longterm stability of vapor cell atomic clocks. In CPT clocks, the presence of the off-resonant optical modes and an imbalance between the intensities in the two arms of the Lambda system can cause significant light shifts. A com­ monly used technique to avoid light shifts is to pulse the light fields and allow the atoms to evolve in the dark. An additional advantage of pulsing the light fields is that atoms can be prepared in a coherent superposition state with higher efficiency by use of strong light fields while avoiding power broadening to a large extent. A pulsed technique for CPTclocks has also been recently proposed (Zanon et al., 2004b, 2005) and demonstrated (Guerandel et al., 2007), and it has shown excellent both stability (Boudot et al., 2009) and a high degree of insensitivity to light shifts (Castagna et al., 2009). In this technique, the light fields are switched on and off at regular intervals. The operation of the clock can be understood as follows: during the first pulse, CPT light fields prepare atoms in a coherent dark state. After the state preparation is nearly complete, the light fields are turned off for a period roughly equal to the ground-state relaxation time. During this period, the atoms freely evolve at the ground-state hyperfine fre­ quency without being perturbed by the light fields. When the second light pulse is turned on, the hyperfine frequency of the atoms in the dark is inferred from the initial absorption signal of the light by the atoms. If the frequency of the microwave oscillator used to modulate the light fields is the same as that of the atomic hyperfine frequency (in the

57

Advances in CPT for Atomic Clocks

(a)

(b)

(a)

v0

Ωp

Ωp Ωp

Ωp

|b〉

lbg

Δv

A

Probe frequency

|b〉

v0

|c〉 Probe transmission

Probe transmission

|c〉

Ωp

Ωp

|b〉

v0

(c)

|a〉

|a〉

Δv A lbg

Ωp

v0

|c〉 Probe transmission

|a〉

Probe frequency

Δv

lbg A

Microwave frequency

(b) (a) Rb cell inside magnetic shielding Laser

EOM

Fabry−Perot etalon

PD

λ /4 Solenoid 6.835 GHz frequency synthesizer From H-maser

Slow frequency modulation (f m = 330 Hz)

Lock-In amplifier Output

Figure 21 (a) Atomic level diagram for the N-resonance. (b) Experimental implementation. Reprinted figure with permission from Zibrov et al. (2005);
2005 of the American Physical Society

dark), then the atoms appear transparent to the light fields the instant they are turned on. This is because the microwave modulation on the laser and the atomic evolution in the dark remain in phase; as a result, the atoms continue to appear transparent to the light fields. The later part of the second pulse repumps the fraction of the atoms that relaxed during the period when the light fields were turned off. The pulsed excitation scheme and atomic energy level diagram are shown in Figure 22a. Figure 22b shows experimentally observed Ramsey fringes when the CPTclock is operated in

58

Vishal Shah and John Kitching

Preparation pulse

(a)

Detection pulse

Free evaluation

τ1 |3〉

Δ0

Γ

|3〉

Ω32 Ω31

δR

Γ

Ω42

|2〉

γc

|4〉

Δ0 Ω32

δR

Ω91

τ2

T

Ω41 |2〉

γc

|1〉

|1〉

Normalized transmission

(b) 1024

1016

1008

1000

992 –30 –25 –20 –15 –10

–5

0

5

10

15

20

Raman detuning δR (kHz)

Figure 22 Raman–Ramsey excitation of hyperfine clock transitions. (a) Pulsed excitation scheme and atomic energy spectrum. (b) Raman–Ramsey pulses. Reprinted figure with permission from Zanon et al. (2005);
2005 of the American Physical Society

the pulsed mode. The individual fringe width can be narrower than the width of a zero-intensity continuously-excited CPT resonance if the delay between successive pulses is greater than the ground-state relaxation time. The pulsed CPT interrogation scheme has been operated as an atomic frequency reference and studied in some detail. It was shown that both the short-term frequency stability and the light shift could be improved considerably compared to continuous interrogation in the pulsed config­ uration (Castagna et al., 2007). A short-term instability of 71013/H was obtained, integrating down to 21014 at 1000 seconds when the linear frequency drift was removed. Dominant contributions to the

Advances in CPT for Atomic Clocks

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short-term instability were amplitude noise on the laser and electronic noise in the photodetection system. Recently, “pulsed” vapor cell atomic clocks, conventional OMDR clocks in which the optical pumping field and microwave field are applied at different times, have also been the subject of some research (Godone et al., 2004, 2006a,b). In addition, there has been some considera­ tion of pulsed CPT excitation of cold atom systems in order to eliminate the buffer gas shifts present in vapor cell clocks (Farkas et al., 2009; Zanon et al., 2003, 2004a). A novel system based on transient excitation of a hyperfine coherence and feedforward to correct the LO frequency has also been investigated (Guo et al., 2009). 3.8 CPT in Optical Clocks So far we have focused on the role of CPT resonances in microwave clocks. In recent years, CPT-based atomic clocks operating in the optical or the terahertz regime have been proposed (Hong et al., 2005; Santra et al., 2005; Yoon, 2007). Optical clocks have the great advantage over microwave clocks that the transition frequencies, and hence the Q-factors of the resonances, are orders of magnitude higher. In order to obtain narrow line widths, forbidden transitions such as the intercombination lines in alkaline earth atoms are often used. Some of these transitions, such as the 1S0 $ 3P0 line at 698 nm in bosonic 188Sr, are forbidden to any order and cannot be accessed by use of single-photon excitation. It is, however, possible to observe the transition indirectly by use of twophoton (CPT) excitation as shown in Figure 23. The line width of the (b)

(a)

Γ31

Γ32 Δ2

Δ1

γc 2

Ω2 Ω1

3P = |2〉 0

γc1

γc 1S = |1〉 0

Percentage in ground-state population transfer

1P = |3〉 1

1.00

Δ0 / 2π = 80 MHz

0.95 0.90 0.85 0.80

Ω2 = Ω Ω1 = 15.16 Ω −6 −4 −2 0 2 4 6 Raman detuning δr /2π (Hz)

Figure 23 CPT excitation of intercombination transitions in alkaline earth atoms. (a) Atomic level structure and optical fields. (b) Predicted Raman lineshape. Reprinted figure with permission from Zanon-Willette et al. (2006);
2006 of the American Physical Society

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Vishal Shah and John Kitching

transition can be conveniently tuned from megahertz to below 1 Hz by controlling the power broadening of this transition. It has been predicted that the accuracy of such a clock can be better than 1017. To eliminate light shifts, a pulsed CPT scheme similar to the scheme proposed for microwave clocks (see pulsed CPT section above) has been proposed and demonstrated (Zanon-Willette et al., 2006).

4. ADDITIONAL CONSIDERATIONS 4.1 Light-Shift Suppression Several interesting techniques have been developed to reduce the effects of light shift on the long-term instability of vapor cell atomic clocks. One such scheme involves the use of the multiplicity of sidebands created when the injection current of a diode laser is modulated (Zhu & Cutler, 2000). Current modulation of a diode laser results in both AM and FM modulation of the output optical field. As the modulation index is increased, a comb of optical frequencies is therefore produced, separated from the carrier by multiples of the modulation frequency. Each of these optical frequencies produces a light shift for each of the ground-state hyperfine frequencies, and each shift can be either positive or negative, depending on the detuning of the specific frequency from the relevant transition. By adjusting the modulation index, it is therefore possible to modify the light shift and reduce it to near zero. A calculation from Zhu and Cutler (2000) is shown in Figure 24a, indicating that for two different operating conditions, the first-order light shift is reduced to zero at a modulation index of about 2.5. Measurements shown in Figure 24b con­ firm the effect. A frequency instability of about 1013 was obtained at an integration period of 1000 seconds by use of this technique (Zhu & Cutler, 2000). It should be noted that, in principle, this same technique can be used to reduce or eliminate the light shift in OMDR clocks (Affolderbach et al., 2003). However, the required modulation of the laser injection current would have to be added to this configuration, while this is present quite naturally in the CPT configuration. The technique above allows for the modulation index to be set such that small changes in the light intensity do not (to first order) affect the clock frequency. Such changes in light intensity can occur, for example, if the laser generating the optical fields ages in some way. However, this aging can also result in a change in the electrical impedance of the laser. If the laser impedance changes, the coupling of the RF modulation field to modulation on the optical field in general changes. As shown in Figure 24b, a change in the coupled RF power by 1 dB can result in a frequency shift on the order of 1010.

61

Advances in CPT for Atomic Clocks

(b)

(a) 60

40

for CPT-generation

30 20 10

4

2 Δf/f (10−10)

Light shift (Hz)

50

Constant input intensity Constant Rabi frequencies

Microwave power 95 dBm (β-208) 105 dBm (β-208) 115 dBm (β-208)

0

−2

0 −10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency modulation index

−4 0

20

40

60

80

Laser beam intensity (μW/cm2)

Figure 24 Suppression of light shifts using optimized modulation of a diode laser. (a) Theoretical predictions of the light shift as a function of madulation index. (b) Measurements of the frequency shift as a function of laser beam intensity for several values of the modulation index. A local minimum is observed for  2.33 and I  40 m}go:goW/cm2. Operation of the clock at this setpoint should result in improved long-term frequency stability when light shifts dominate. Reprinted figure with permission from Zhu and Cutler (2000)

In order to address the effects of such changes in laser impedance, a modification of the RF modulation technique was suggested (Shah et al., 2006a). After adjusting the RF power to the zero-light-shift point, the power of the light field is modulated at a low frequency (17 Hz in this experiment) by use of a variable attenuator. This modulation in intensity will cause a corresponding change in the frequency of the CPT resonance if the zero-light-shift condition is not satisfied. Assuming that the synthesizer frequency is more stable than the CPT resonance at the modulation fre­ quency, the modulated frequency shift can be detected through the normal comparison of synthesizer frequency to CPT resonance frequency. The RF power can then be corrected to maintain the zero-light-shift condition. A schematic of the optical/electronic arrangement is shown in Figure 25. Under exaggerated conditions, an improvement in the insensitivity of the output frequency to RF impedance (as adjusted by modulating the laser temperature) by a factor of 10 is obtained (Shah et al., 2006a). 4.2 Laser Noise Cancellation One of the problems frequently encountered in laser-based atomic inter­ rogation is the conversion of laser frequency noise to current noise in the detector output by the optical resonance. The laser frequency noise, which otherwise cannot be seen by the photodetector, appears because the atoms act as a sharp discriminator of the optical frequency in the vicinity of the optical resonance. The amount of noise that appears on

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Physics package LCD attenuator

B

Photo detector

VCSEL

Polarizer λ/4 Vapor cell

17 Hz Signal generator

RF-synthesizer

Frequency error signal

Lock-in 2

Amplifier

RF power error signal

Servo Experimental setup

Figure 25 Schematic of a technique to maintain the RF power at the zero-light-shift point as the laser impedance ages. Reprinted figure with permission from Shah et al. (2006a);
2006 of the Optical Society of America

the photodetector signal depends on the amount of frequency noise present on the light (the laser line width), the width of the optical resonance, and the tuning of the laser frequency. There are several solutions with varying complexity that can be used to reduce this excess laser frequency to photodetector current noise. Among the simplest is to choose a laser with a narrow line width. However, this is not always feasible for commercial or other technical reasons. Another simple alternative is to broaden the optical resonance using higher buffer gas pressure which reduces the slope of frequency discrimination by the optical resonance. Here again, there are limitations since buffer gas pressure cannot be arbitrarily increased without affecting the clock performance. For example, in the regime in which most CPT/OMDR clocks operate, one of the immediate consequences of using higher buffer gas pressure is that the optical depth of the atomic medium at a given temperature is correspond­ ingly reduced. To compensate for the loss in the optical depth, the atomic vapor pressure can be increased by increasing the cell temperature. But this increases the width of the ground-state resonance by increasing alka­ li—alkali spin-exchange contribution to the ground-state relaxation. Another alternative to reducing the laser noise is to use external means such as real-time laser noise cancellation using differential detection. This was accomplished, for example, by Gerginov et al. (2008). Here, a split wave plate was used such that light in one half of the vapor cell was circularly polarized and light in the other half was linearly polarized (see Figure 26). The circular and the linear components of the light beam were collected on two spatially separated photodetectors. While the CPT

Advances in CPT for Atomic Clocks

VCSEL

λ/4 σ +

L P ND G

B

PD1

lin PD2 Rb cell + shield

–

RF

63

To exp.

+

Figure 26 Experimental setup used to reduce the laser noise. VCSEL: L, lens; P, polarizer; ND, neutral density filter; /4, quarter-wave plate; G, glass plate sþ; PD, photodiode. Reprinted figure with permission from Gerginov et al. (2008);
2008 of IEEE

resonance is excited in only the first part of the vapor cell and thus seen only by PD1, the laser noise appears in both the channels. Subtracting the signals from the two photodetectors thus removes the laser noise without affecting the CPT resonance seen using PD1. Similar techniques have also been developed for OMDR clocks (Deng, 2001; Mileti et al., 1998; Rosen­ bluh et al., 2006). Yet another technique for laser noise cancellation was proposed and demonstrated by Rosenbluh et al. (2006). In this technique, the noise originating with the laser was reduced and the effects of optical pumping to the dark end state were simultaneously eliminated. A CPT resonance was excited using a combination of copropagating left and right circu­ larly polarized light obtained from a common laser. A relative path delay, equal to one quarter of the microwave wavelength, was introduced between the right and the left circular components of the light beam. After propagating the light through the vapor cell, the two components were separated using an arrangement of a quarter-wave plate (/4) and a polarizing beam splitter; each of the components was separately moni­ tored using photodetectors. Due to the /4 path delay between the two polarization components of the light beam that excite the CPT resonance, the phase of the coherent dark state that is excited by the two beams combined is partially shifted in phase with respect to the individual polarization components of the beam. This phase shift, which has equal but opposite signs for the two light components, introduces an asymme­ try into the CPT resonance lineshape by adding a dispersive component to an otherwise Lorenztian profile. While the CPT resonance seen by adding the signals from both photodetectors still has a purely Lorenztian lineshape (Figure 27a), the difference signal is purely dispersive (Figure 27b). Because differential detection is employed in the latter case, the common mode laser noise is removed, without any effect on the overall strength of the CPT resonance signal. It can be seen even from the oscilloscope traces of the CPT resonances that the trace in Figure 27b has lower noise compared to Figure 27a, in which differential detection is not employed.

Vishal Shah and John Kitching

0.665 (a) 0.660 0.655 0.650 0.645 0.640 0.635 0.630

–2 2 4 –4 0 Frequency detuning, δr [kHz]

0.015 Photodetector signal [V]

Photodetector signal [V]

64

(b)

0.010 0.005 0.000 –0.005 –0.010 –0.015

–2 2 4 –4 0 Frequency detuning, δr [kHz]

Figure 27 Lineshapes of atomic-coherence-induced resonances for phase-shifted, two-beam excitation. (a) One beam blocked, conventional CPT resonance, (b) both beams present, signals from balanced photodetectors subtracted. Reprinted figure with permission from Rosenbluh et al. (2006);
2006 of the Optical Society of America

4.3 Light Sources for Coherent Population Trapping A number of types of light sources have been used to generate the bichromatic optical field needed to excite hyperfine CPT resonances. Although the coherence requirements of the light source are not nearly as stringent as in optical spectroscopy, the lamps currently used in con­ ventional atomic clocks appear to be too incoherent to generate CPT resonances of any reasonable contrast. The requirement on the coherence is nominally that the line width of the light source be smaller than the buffer gas broadened optical transitions in the alkali atoms, typically ranging from a few hundred megahertz to several tens of gigahertz. Most lasers satisfy this requirement, which allows great latitude in laser choice to optimize the system with respect to other criteria. The earliest experiments on CPT were carried out using multimode dye lasers (Alzetta et al., 1976). Since then, a variety of more sophisticated light sources have been used, including acousto-optically modulated dye lasers (Thomas et al., 1982); injection-current-modulated edge-emitting diode lasers (Hemmer et al., 1993; Levi et al., 1997); phase-locked external cavity diode lasers (ECDLs) (Brandt et al., 1997; Zanon et al., 2005) or edge-emitting lasers (Zhu & Cutler, 2000); injection-current-modulated vertical-cavity surface-emitting lasers (VCSELs) (Affolderbach et al., 2000; Braun et al., 2007; DeNatale et al., 2008; Kitching et al., 2000; Lutwak et al., 2003; Serkland et al., 2007; Youngner et al., 2007); and electro­ optically modulated ECDLs (Jau et al., 2004a). Each of these light sources has relative merits and detriments. For example, VCSELs have very low threshold currents, making them ideal for low-power instruments based on CPT, but suffer from inflexibility with respect to the modulation side­ band spectrum. The spectrum of phase-locked ECDLs can be controlled

Advances in CPT for Atomic Clocks

65

very precisely, but the locking is difficult to implement experimentally. Acousto- and electro-optically modulated sources can have nearly perfect amplitude modulation with no associated phase modulation, but are large, cumbersome, and expensive. Novel lasers, developed for CPT experiments, include VCSELs in an extended cavity (Gavra et al., 2008), and very short edge emitting lasers with a low-reflectivity coating on one facet (Kargapoltsev et al., 2009). These lasers were developed to have high modulation bandwidths without the need for even a modest amount of output power (in many experiments 10 mW of optical power is sufficient to excite high-contrast resonances). 4.4 Dark Resonances in Thin Cells Considerable recent work has focused on the optical properties of alkali atoms confined in cells for which the longitudinal dimension is such that the transit time across the cell is shorter than the optical relaxation period (Briaudeau et al., 1996). In these cells, atoms in velocity classes perpendi­ cular to the cell walls do not build up appreciable optical coherence before the wall-induced relaxation and therefore do not contribute to the optical absorption. By contrast, atoms in velocity classes parallel to the cell walls do not collide with the walls as frequently and hence do build up coherence and exhibit corresponding absorption. Some recent work in this area has focused on the understanding of CPT resonances that are observed in these media (Failache et al., 2007; Fukuda et al., 2005; Petrosyan & Malakyan, 2000; Sargsyan et al., 2006). While the CPT line widths are typically very large (greater than 1 MHz), some work is proceeding to evaluate how these systems might be used in compact atomic clocks (Lenci et al., 2009). 4.5 The Lineshape of CPT Resonances: Narrowing Effects The simplest theories of the CPT resonance lineshape typically involve collisional or diffusion-induced relaxation processes or radiative relaxa­ tion. These relaxation mechanisms result in the typical Lorentzian line width found for most spectroscopic signals. However, there are a number of physical effects that occur in real experiments that can distort the CPT resonance lineshape. It has been found, for example, that if the transverse intensity distribution of the excitation light field is non-uniform, or if the beam diameter is smaller than the diffusion length of the atoms over the ground-state relaxation time scales, then a narrowing of the resonance near its center can result (Levi et al., 2000; Taichenachev et al., 2004a, 2005a) to form a pointed resonance lineshape. This lineshape can also be explained by diffusion-induced narrowing (Xiao et al., 2006, 2008) based on the Ramsey effect, in which atoms diffuse out and then re-enter the

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excitation beam before relaxing. Finally, propagation effects are known to cause modifications of the resonance line width compared to that observed in an optically thin medium (Godone et al., 2002d). The optical absorption coefficient is smaller when the CPT resonance condition is satisfied, and because the absorption in an optically dense medium is nonlinear as a function of propagation length, significantly more light may be transmitted when on resonance compared to when away from resonance, producing an artificial narrowing effect. However, it remains unclear whether these unusual lineshapes can be effectively used to improve the performance of a CPT atomic clock in some way.

5. CONCLUSIONS AND OUTLOOK After considering the many possibilities for improving CPT resonances for use in atomic clocks, it is perhaps important to ask to what extent these techniques have impacted the design and performance of actual devices. To some extent this question is premature, as it often takes considerable time for new knowledge, even if it allows clear performance improvements, to find its way into realized systems implemented in the laboratory or in a commercial setting. The cost, time, and risk associated with implementing a new technique to replace, for example, an already proven commercial instrument or an already-operating laboratory instru­ ment are often considered too high. In addition, until experiments are engineered at a level that their importance emerges, issues believed to be ultimately important, such as the effects of the light shift on the long-term stability of the clock, are often masked by other more technical effects, such as temperature-induced shifts. It is therefore often difficult to estab­ lish a clear measure of the improvements certain techniques will allow [see, for example, the work by Shah et al. (2006a)]. However, CPT as a whole has now been not just successful for labora­ tory instruments, but appears to be on the verge of commercial success (DeNatale et al., 2008; Deng, 2008; Lutwak et al., 2007; Vanier et al., 2005; Youngner et al., 2007). In particular, CPT has been shown to be the method of choice for microfabricated vapor cell frequency standards. This is interesting because comparisons of CPT techniques to conven­ tional OMDR techniques for vapor cell frequency references (Kitching et al., 2002; Lutwak et al., 2002; Vanier, 2001b, 2005) have suggested that there is no clear advantage to be gained through the use of CPT with respect to short-term stability. From a technical viewpoint, the biggest strengths of the CPT approach at present therefore appear to be that (a) the physics package design is simple to implement, and (b) that the considerable work over the last 10 years on miniaturized CPT frequency references has clearly established the technical viability of this approach.

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While it remains possible that highly miniaturized frequency references based on OMDR will ultimately be competitive with their CPT counter­ parts, considerable work is needed to develop the OMDR approach further for miniaturized devices. One additional, and perhaps over­ looked, advantage of CPT is that the resonances can be excited by use of modulation at a subharmonic of the hyperfine frequency. This allows the LO to be placed in close proximity to the physics package without having radiated RF power interfere with the atomic transition. Probably the most important improvement among the techniques described above has been the use of the D1 line rather than the D2 line to excite the resonances. The D2 line was used initially because of the availability of commercial diode lasers at the 852 nm D2 transition of Cs. The improvements gained by using the D1 line have, to some extent, motivated the development of new lasers, and it now appears that (a) the use of the D1 line is clearly superior with respect to the short-term stability and (b) that there is no significant disadvantage to this approach. Certain other design improvements focused on improving the resonance contrast, such as the end-resonance technique, push—pull optical pump­ ing, and the linear polarization techniques, continue to appear promising in principle, but work is needed to quantify the level of improvement in a real clock experiment. The additional system complexity and correspond­ ing impact on reliability will also be a factor in determining the extent to which these techniques will be used in real-world instruments. Techni­ ques to reduce the light shift, such as pulsed CPT and sideband spectrum engineering, are expected to be important for future generations of CPT clocks, for which the engineering has progressed to the point where technical limitations to the long-term instability have been suppressed. Whatever the outcome from an instrumentation perspective, it is clear that the understanding of CPT as it applies to atomic clocks has advanced considerably over the last decade.

ACKNOWLEDGMENTS We gratefully acknowledge valuable comments from S. Knappe and A. Post. This work is a partial contribution of NIST, an agency of the US Government, and is not subject to copyright.

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CHAPTER

3

Dissociative Recombination of H3þ Ions with Electrons: Theory and Experiment Rainer Johnsena and Steven L. Gubermanb a

Department of Physics and Astronomy, University of Pittsburgh,

Pittsburgh, PA 16260, USA

b Institute for Scientific Research, 22 Bonad Road, Winchester,

MA 01890, USA

Contents

1. 2. 3.

Introduction Basic Definitions Experimental Techniques 3.1 Afterglow Techniques 3.2 Single-Pass Merged-Beam and Ion-Storage

Ring Experiments 4. Theory 4.1 DR Mechanisms 4.2 H3þ Potential Curves and Surface 4.3 Vibrational and Rotational Considerations 4.4 One- and Two-Dimensional Theory 4.5 Three-Dimensional Treatments of H3þ DR 5. History of Experimental H3þ Recombination

Studies 6. Reconciling Afterglow and Storage Ring Results 6.1 Afterglow Measurements That Yielded

Very Low Recombination Coefficients 6.2 Afterglow Measurements That Yielded

High Recombination Coefficients 6.3 Third-Body Stabilized Recombination of H3þ 7. Comparison of Storage Ring Data 8. H3þ Product Branching 9. Isotope Effects 10. Conclusions Acknowledgments References

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Advances in Atomic, Molecular, and Optical Physics, Volume 59
2010 Elsevier Inc. ISSN 1049-250X, DOI: 10.1016/S1049-250X(10)59003-7 All rights reserved.

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76 Abstract

Rainer Johnsen and Steven L. Guberman

Four decades of experimental and theoretical studies of the dissociative recombination of the seemingly “simple” H3þ ions with electrons have often given strongly disagreeing results. The literature on the subject abounds in terms like enigma and puzzles, and several authors have asked if the “saga” is finally approaching a satisfactory ending. Fortunately, recent progress in theory and experiment has greatly reduced many of the apparent contradictions. In this review, we attempt to reconcile the remaining discrepancies, in particular those between beam experiments and those employing plasma afterglow techniques. We conclude that there are no true contradictions between those results if one examines the conditions under which the data were taken and includes effects arising from third-body-assisted recombination. The best available theoretical treatments of purely binary recombination now agree rather well with state­ of-the-art ion-storage ring results, but we think that further refinements in the complex theoretical calculations are required before it can be said that the mechanism of the recombination is understood in all details and that the “saga” has truly come to an end.

1. INTRODUCTION H3þ, the simplest of all polyatomic molecular ions, consists of three protons arranged in an equilateral triangle, held together by two elec­ trons. The physics and chemistry of this ion has occupied a special niche in the molecular physics community for many years, and it is a fair question to ask why it continues to be of interest today and what pro­ gress has been made in understanding its basic properties. The apparent simplicity of this ion makes it attractive as a test case for ab initio quantum-chemical calculations and that certainly has stimulated much theoretical work. A second important motivation comes from astrophy­ sics: H3þ is perhaps the second most abundant molecular species (after H2) in interstellar clouds, in the ionospheres of the outer planets, and plays a central role in determining the ionization balance and in building more complex ions that determine the physical properties in these starforming regions (Herbst, 2007; McCall, 2006; McCall et al., 2002). While the ion is quite stable, the relatively small proton affinity of H2 (4.2 eV) enables efficient proton transfer to other molecules. However, if H3þ ions recombine efficiently with electrons and dissociate into H2 and/or H atoms in the process, the same species from which they were formed by several slow steps, the reaction chain is essentially terminated, and recombination limits the rate of molecule formation. The effect of H3þ on

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the interstellar chemistry can be quite complicated and lead to bistable chemical evolutions, as has been discussed in detail by Pineau Des Foreˆ ts and Roueff (2000). In this review, we focus on the dissociative recombination (DR) of H3þ ions with electrons, a process that can be symbolically represented as  Hþ 3 þe !HþHþH ! H2 þ H

ð1Þ

Anticipating later discussions, we note that Equation (1) may be read either as representing an ion—electron binary collision or as a reaction equation that describes a more complex process in an ionized gas. We adopt the first interpretation but note that other electron—ion recombina­ tion mechanisms exist in which part of the energy released by recombi­ nation is transferred to third bodies (atoms, molecules, or other electrons) or is removed by emission of radiation. We will discuss such third-body­ assisted recombination only to the extent that it affects the interpretation of experimental data. All experimental studies of DR face the problem that two charged species, ions and electrons, must be brought together in a controlled manner with a small relative velocity. Theorists have an equally and perhaps even more difficult task. A slow electron that is captured by a molecular ion can give rise to numerous excited states of the molecule, and it requires extensive quantum mechanical calculations to decide which of those states eventually lead to dissociation. The task is further complicated by the fact that recombination is sensitive to the rotational and vibrational states of the ion and that the ion exists in two nuclear spin modifications, denoted as para-H3þ (two of the three proton spins aligned) and orho-H3þ (three proton spins aligned). As in other fields of physics and chemistry, experiment and theory sometimes have often given conflicting answers to some of the basic questions. For many years there was considerable doubt that efficient recombination of H3þ actually occurred! Many open and once difficult questions have been clarified in recent years by advances in theory and by new and powerful experimental techniques, especially ion-storage rings that supply more detailed information than the plasma-based experimental methods. Progress in theory has been commensurate with that in experiment: what Bates (1993), the “founding father” of DR, once described as an “enigma” has largely been solved, but some finer details may still need to be worked out. This review is intended to present a critical but not necessarily complete analysis of all experiments and theories. We seek to reconcile experiment and theory as far as possible given the current state of knowledge, and to see if remaining discrepancies are “real” in

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the sense that they indicate deficiencies in our understanding as opposed to incomplete or erroneous interpretations of experimental observations. The literature on DR is extremely large and H3þ is certainly not the only ion of interest. Several excellent reviews on DR in general have appeared in the last few years that include extensive lists of measured rate coefficients and other data (Florescu-Mitchell & Mitchell, 2006; Larsson & Orel, 2008). A previous review of H3þ recombination mea­ surements by Johnsen (2005) contains much additional material that we will not repeat here, and some proposed solutions of apparent contra­ dictions have now been ruled out by new experimental and theoretical work.

2. BASIC DEFINITIONS We begin by reviewing some basic definitions, most of which are com­ mon in the physics of atomic collisions, but others are specific to parti­ cular experiments and require a few words of explanation. Consider an ion that moves in a region containing uniformly distrib­ uted free electrons at density ne (cm3). The recombination coefficient
is defined by the probability dP that the ion captures an electron during time dt and dissociates before releasing it by autoionization, i.e., dP ¼
ne dt:

ð2Þ

This defines a “raw” or “effective” recombination coefficient that still depends on the distribution of the relative ion—electron speeds f(vrel). If the recombination is purely binary, one can define a recombi­ nation cross section (vrel), which is related to the rate coefficient by the average
¼ hðvrel Þvrel f ðvrel Þi;

ð3Þ

where the brackets indicate averaging over all vrel. If f(vrel) is sufficiently narrow to be reasonably approximated by a delta function centered at , the cross section is closely given by the ratio
: ð4Þ ðvrel Þ ffi hvrel i This approximation is fairly good in merged-beam experiments, but fails at very low vrel. For that reason merged-beam experimenters often report their raw results not as cross section but as a nonthermal recombi­ nation coefficient as a function of the “detuning energy.” However, they usually deconvolute the recombination coefficient to obtain the cross section, and then compute the thermal recombination coefficient by

Dissociative Recombination of H3þ Ions with Electrons

79

convolving the cross section again with a Maxwell distribution. The deconvolution may require extrapolation to very low energies. Plasma afterglow experiments directly yield the thermal recombina­ tion coefficient, although often only over a narrow range of temperatures. Those results are typically given in the form of a power-law dependence
 Te  x
ðTe Þ ¼
ð300KÞ ; ð5Þ 300 as a function of the electron temperature Te. In such experiments, the ion translational ion temperature Ti is almost always the same as the gas temperature Tg, but Te can be greater than Ti. It can hardly ever be assumed that the internal degrees of freedom of the ions, especially their vibrations, are in thermal equilibrium at the translational temperature. Theoretical calculations usually generate cross sections for a set of discrete collision energies. To facilitate comparison to experiment, theor­ ists often calculate (a) the thermal rate coefficient and (b) an “effective” rate coefficient that should be measured in beam experiments with a finite energy resolution. The procedure “washes out” some of the finer structure in the theoretical cross section but, unlike the thermal rate coefficient, retains some of its structure.

3. EXPERIMENTAL TECHNIQUES The experimental techniques used to study DR can be divided into two broad categories, plasma afterglow experiments and merged-beam experiments. In afterglow experiments, electron—ion recombination rate coefficients and product yields are derived from observations of ion and electron densities, optical emissions, and neutral products during the afterglow phase of a plasma. The analysis of afterglow plasmas can be complicated by reaction processes that occur in addition to electron—ion recombina­ tion, and it also is not always obvious that recombination in a plasma involves only simple binary recombination. However, what is regarded as a “complication” in the context of recombination may be of great interest to the physics of ionized gases in general and this should be kept in mind. Merged-beam and ion-storage ring methods, while requiring far greater experimental effort, are closer to the theorists’ ideal experiment and can provide more detailed information. The outstanding progress that has been made in refining these techniques now permits studies with very high energy resolution as well as determinations of the chemical

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identity of neutral reaction products and their kinetic energy. Such data, of course, provide far more sensitive tests of theoretical calculations of recombination than the thermally averaged rate coefficients obtained by afterglow techniques. 3.1 Afterglow Techniques The two principal subcategories, “stationary” or “flowing” afterglows, have much in common, but they differ in the way that the plasma is produced and observed. We will discuss them together while pointing out relative strengths and weaknesses. In the stationary afterglow, more appropriately called a “pulsed” after­ glow, a plasma in a pure gas or gas mixture is created by repeated pulses of microwaves, high voltages applied to discharge electrodes, ultraviolet light, or other ionizing radiation (see Figure 1). All afterglow observations are carried out in the same volume as a function of time. In the flowing afterglow method (see Figure 2), a pure gas (helium most often) is first ionized, usually in a microwave discharge, and then flows at high speed down the flow tube and is eventually discharged into a fast pump. At some point, reagent gases are added that convert the primary ions and metastable atoms to the desired molecular ion species. Recombination occurs in the region downstream from the reagent inlet, and observations are carried out as a function of distance from the gas inlet. The flow tube method has the advantage of greater chemical flex­ ibility and it avoids exposing the molecular gases directly to an intense discharge, which can lead to undesired excitation or dissociation. It also has some disadvantages: There is only an approximate correspondence between time and distance since the gas flows faster at the center of the tube than it does near the wall and the spatial distribution of particles in the plasma is not necessarily uniform. Also, the mixing of gases at the reagent inlet is not instantaneous and this can complicate the data analysis. A frequently employed method to convert the active species flowing out of the discharge to ions consists of adding argon at a point upstream from the reagent inlet. This converts metastable helium to argon ions, which are subsequently used as precursors for the ion—molecule reactions that generate the desired ion species. What is often ignored is that along with the argon ions some undesired energetic particles and ultraviolet photons also enter the region downstream from the reagent inlet, for instance metastable argon atoms (see, e.g., Skrzypkowski et al., 2004) that are produced by collisional radiative recombination of argon ions. Ultraviolet photons, in particular “trapped” helium resonance radiation, can enter the reaction zone unless one adds a sufficient amount of argon to destroy them by photoionization of argon. Fortunately, such effects do

Dissociative Recombination of H3þ Ions with Electrons

81

Pulsed microwave for plasma generation

Vacuum

Mass spectrometer Gas inlet

Langmuir probe

Figure 1 Schematic diagram of a stationary or pulsed afterglow apparatus. A Langmuir probe or a microwave frequency method is used to record the decay of the electron density subsequent to an ionizing pulse. Typical linear dimensions of the plasma chamber are 10–40 cm

Microwave discharge

Mass spectrometer

Reagent gas inlet

z

Langmuir probe

Figure 2 Schematic diagram of a flowing afterglow Langmuir probe apparatus (FALP). A movable Langmuir probe records the electron density as a function of distance from the reagent inlet

not interfere much with measurements of recombination coefficients, but they can be important in spectroscopic studies of reaction products. Different afterglow experiments employ different reaction sequences to produce H3þ ions. One frequently used scheme makes use of the fast two-step reaction sequence

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Rainer Johnsen and Steven L. Guberman

Arþ þ H2 ! ArHþ þ H þ 1:53eV

ð6Þ

ArHþ þ H2 ! Ar þ Hþ 3 þ 0:57eV

ð7Þ

followed by þ

which releases sufficient energy to produce H3 in vibrational states up to v = 5. If argon is present in sufficient concentration, subsequent proton transfer to Ar, þ Hþ 3 ðvÞ þ Ar ! ArH þ H2 ;

ð8Þ

þ

destroys all H3 ions with internal energies above 0.57 eV, leaving only those in the ground state [A1 (0, 00)], in the v2 = 1 bending-mode vibration [E (0, 11) at 0.3126 eV], and in the v1 = 1 breathing-mode vibra­ tion [A1 (1, 00) at 0.394 eV]. The radiative lifetime of the v1 = 1 state is very long (1.2 s). Radiative decay of ions in the v2 = 1 level is faster (4 ms), but does not necessarily occur at the time scale of recombination measurements. The electron density can be measured by several methods: Langmuir probes return local values of ne, while microwave methods have low spatial resolution and yield a “microwave-averaged electron density.” The flow tube has the significant practical advantage that the gas is exchanged rapidly, on a time scale of milliseconds. In stationary after­ glows, outflow of gases occurs only through the small sampling orifice used for mass spectrometric sampling of ions, but the gas exchange time is usually on the order of many minutes. For this reason, impurity problems tend to be less serious in flow tubes than in stationary afterglows. The methods to measure recombination coefficients are essentially the same in both types of afterglows. In the simplest case, when only a single ion species is present and the plasma is quasi-neutral, e.g., ne = ni, the electron continuity equation is given by @ne ðt; ! rÞ r Þ; ð9Þ ¼ 
n2e ðt; ! r Þ þ Da r2 ne ðt; ! @t where Da is the ambipolar diffusion coefficient of the ion. If diffusion is sufficiently slow that it can be ignored, the reciprocal electron density varies with time as 1 1 Þ ¼ Þ þ
t; r ne ð0; ! r ne ðt; !

ð10Þ

and hence the recombination coefficient can be obtained directly from the slope of a graph of the measured reciprocal electron densities as a func­ tion of time. This simple form of analysis yields reasonably accurate

Dissociative Recombination of H3þ Ions with Electrons

83

recombination coefficients only if the diffusion current of ions into or out of the volume in which ne is measured is very small compared to the volume loss rate of electrons due to recombination. A frequently used, but not entirely satisfactory, approximation “corrects” for the diffusion loss of electrons by fitting the observed electron density decays to an equation of the form Da ne dne ðtÞ ¼ 
n2e ðtÞ  ; dt 2

ð11Þ

in which L2 is the fundamental diffusion length of the plasma container, and the electron density is measured at the center of the container (or points on the axis of a flow tube). The equation is correct only in the limits when either of the two loss terms greatly outweighs the other since it ignores the fact that quadratic recombination loss tends to “flatten” the spatial distribution of electrons and ions. As a conse­ quence, the diffusion current away from the center is reduced, and Equation (11) overestimates the diffusion loss, but underestimates the recombination loss. The pulsed microwave afterglow measurements often employed numerical solutions of the continuity equations to ana­ lyze the data while the analysis of flow tube data is usually carried out using Equation (11). The time scale of recombination experiment is of practical interest. From Equation (10), it follows that the electron density during the after­ glow decays by a factor of 2 from its value at time t whenever the time increases by the “half-time”  1/2, given by 1 : ð12Þ 1=2 ¼
ne ðtÞ Accurate determinations of recombination coefficients require obser­ vation of ne over a significant range, a factor of 4 or preferably more. Hence, for an initial electron density of ne(t = 0) = 1010 cm3 and a typical recombination coefficient
= 107 cm3/s, one must measure ne(t) over a time of at least (1 þ 2) = 3 ms, longer if the initial electron density is only 109 cm3. Obviously, the ion—molecule reactions that form the desired ions should go essentially to completion in a time short compared to the time scale of recombination, and the ions under study must not convert to a different type during this time by reacting with any of the gases in the afterglow plasma or impurity gases. We will show later (see Section 6.1) that serious errors ensue when these requirements are not fulfilled. The gas temperature in afterglows can be adjusted fairly easily over a limited range from liquid-nitrogen temperature (77 K) to roughly 600 K by heating the entire apparatus. This is more useful as a means to control equilibrium concentrations of weakly bound ions (for instance shifting the chemical equilibrium from H3þ to H5þ ions) than as a means to measure the

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temperature variation of recombination coefficients. Much higher electron temperatures (up to 10,000 K) can be reached by microwave heating of the electron gas, a technique that was used extensively in stationary afterglows and that has provided data on many important ion species, including H3þ. The technique is subject to complications in the presence of molecular additives (Johnsen, 1987). In the afterglow measurements on H3þ, such effects are not important and we will not discuss this subject further. Higher gas temperatures (up to nearly 10,000 K) can be reached by employing shock heating of the afterglow plasma (Cunningham et al., 1981), a technique that has been applied to several recombination processes of atmospheric interest but not to H3þ. 3.2 Single-Pass Merged-Beam and Ion-Storage Ring Experiments With the development of ion-storage rings, experiments on DR were transformed from small-scale “table-top” experiments to large-scale mul­ tiuser type operations that made use of technologies from nuclear and high-energy physics. The impact of these new machines cannot be over­ stated: the considerable investment in the experimental facilities revita­ lized and revolutionized experimental studies of DR. We will only summarize the basic principles and current capabilities since extensive reviews have been written by authors who are more familiar with experi­ mental details (Larsson & Orel, 2008). The predecessor of the storage rings, the single-pass merged-beam method was developed at the University of Western Ontario (see, e.g., Auerbach et al., 1977). While it was an important step forward and resulted in many important results, the single-pass merged beam has been superseded by the more powerful ion-storage ring technique. Both have in common that recombination of ions and electrons takes place between parallel ion and electron beams of nearly the same velocity. In a single-pass merged beam, the ion beam passes through the electron target beam once and is then discarded; in a storage ring the ions circulate in the ring and pass through the interaction region (see Figure 3) many times. It is not the more “efficient” use of ions in storage rings that makes them preferable but the fact that the longer storage time (up to 10 seconds) in a ring removes all excited ions that radiate on that time scale, for instance infrared active vibrationally excited ions. In merged beams, the relative velocity between the two beams can be made very small. More importantly, the velocity spread in the electron target gas can be greatly reduced by accelerating the (initially “hot”) electrons to a high velocity that closely matches that of the ions. The narrowing of the electron velocity distribution in the direction of the beam (but not transverse to it) is a purely kinematic effect that follows from the classical equations of motion. However, at finite electron

Dissociative Recombination of H3þ Ions with Electrons

85

Electron collector

Electron gun

Electron beam

Circulating ion beam

Beam deflector Merging region

Circulating ion beam

Demerging region Neutral products

Interaction length

Detector

Figure 3 Schematic diagram of the electron cooler and interaction region of an ion-storage ring. The length of the interaction region is typically on the order of 1 m

densities Coulomb interactions between electrons occur and the actual velocity spread in the beam direction is somewhat larger than that calcu­ lated from the kinematic equations. In addition, the effective energy resolution for ion—electron collisions depends also on the velocity com­ ponents transverse to the beam. It is common practice to model the electron velocity distribution by a two-temperature Maxwellian function with temperatures Ti for the parallel velocity component and T\ for the two transverse components. Several methods are available to reduce the transverse velocity spread and thereby improve the energy resolution: In the single-pass merged-beam apparatus (Auerbach, 1977), improvements in the energy resolution were made by using trochoidal analyzers to merge electron and ion beams, while storage rings employ “electron coolers” in which the electrons are cooled by expansion in a magnetic guiding field. Cooling and recombination can be accomplished either in the same section of the ring or in the two separate sections. In addition, the coolers also cool the ion beam by a “friction” effect and reduce the diameter of the ion beam. In all merged-beam techniques, recombination events are detected by counting recombination products using an energy-sensitive barrier detec­ tor. The detector ideally registers one count of full pulse height when all products from a single event strike the detector simultaneously. In that case the number of counts received for a single traversal of a single ion through the electron target is N ¼
ne Dt:

ð13Þ

Here
is the recombination rate coefficient appropriate to the experi­ mental velocity distribution, ne the electron density, and Dt the time of traversal of an ion through the interaction region. If the ion beam is much narrower than the electron beam, which is the case in storage rings, there

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is no need to consider overlap factors. To obtain absolute values of
, one needs to know the ion beam current, which can be measured either by collecting ions or using a current transformer. While ion-storage rings come very close to the theorist’s perception of an ideal experiment, there are some, fortunately minor, imperfections that should be mentioned. Ion—electron collisions also occur in the mer­ ging and demerging regions (magnets in storage rings) where the ion and electron beams are obviously not parallel and their relative velocities are larger than those in the straight part of the interaction region; however, this “toroidal correction” is not large and can be taken into account. In addition to providing high-resolution recombination cross sections, storage rings have an outstanding ability to determine the relative abun­ dance of recombination products by placing grids in front of the detector and analyzing the pulse-height spectra (for details, see Larsson & Orel, 2008). The single-pass merged beam also has been employed for such studies but the small event rate made quantitative product determina­ tions tedious and time consuming.

4. THEORY 4.1 DR Mechanisms If all internuclear distances in a polyatomic molecule are held constant except for the dissociation coordinate, a potential curve similar to that for a diatomic molecule can be used to illustrate the fundamental features of DR. Figure 4 shows such a slice through the potential surfaces of the ion, Rydberg, and dissociative states.  denotes the electron energy at which capture takes place into a repulsive state of the neutral molecule from an ion in some vibrational level. Note that any electron energy will do, even zero, since varying the electron energy only varies the point of capture. Once in the repulsive state, the neutral molecule can emit the captured electron or dissociate. If dissociation takes the internuclear distance beyond the crossing point of the neutral and ion curves, electron emission (autoionization) is no longer possible and dissociation is completed. This is the direct mechanism for DR originally proposed by Bates (1950). Superexcited states of the neutral molecule are generally found at the same total energies as that for the ion ground state. Electron capture also occurs into these superexcited states and competes with capture into the dissociative state. Among these states are the vibrationally excited Rydberg states that have the ground state of the ion as core. The v = 0 ground core Rydberg levels all lie below the ion, but the v = 1 ion level is the energetic limit of an infinite number of Rydberg levels as are the other excited ion vibrational levels. Capture into one of these levels at electron

Dissociative Recombination of H3þ Ions with Electrons

87

Energy (Hartrees)

−112.60

−112.61

ε� ε

Ion

−112.62 Rydberg

Dissociative state

−112.63

−112.64 1.8

2.0

2.2

2.4

Internuclear distance (Bohr) Figure 4 The direct (at electron energy e) and indirect (at electron energy e 0 ) mechanisms of dissociative recombination

energy "0 is shown in Figure 4. After capture, the electron can be emitted or the Rydberg level can be predissociated by the dissociative state of the direct mechanism. This is the indirect DR mechanism, first introduced by Bardsley (1968). Both the direct and indirect mechanisms are paths to the same dissociation products and can interfere with each other. Any tech­ nique for calculating the DR cross section must account for this inter­ ference. A recent addition to the indirect DR mechanism, Rydberg states having an excited ion core (Guberman, 2007), will not play a role in H3þ DR at low electron energies since the first excited ion states lie too high above the ground state (Shaad & Hicks, 1974). A second-order mechan­ ism (Guberman & Giusti-Suzor, 1991; Hickman, 1987; O’Malley, 1981) also can take place in which the neutral repulsive state acts as an inter­ mediate between the electron—ion and a bound Rydberg state. In this manner, an electron can be captured by an electron—electron interaction into a Rydberg state. 4.2 H3þ Potential Curves and Surface Figure 6 has potential curves for H3þ and for several H3 states that are important for DR. These states have been calculated in C2v symmetry with the nuclear configuration shown in Figure 5, i.e., R1, the distance between two H atoms has been kept constant at the equilibrium separa­ tion, 1.63 a0. The remaining atom moves along R2, which is perpendicular

88

Rainer Johnsen and Steven L. Guberman

H

R1

Θ

H R2

H Figure 5 Jacobi coordinates for H3

to R1 and intersects R1 at its midpoint. The potential curves are calculated with [4s, 3p, 2d, 1f] Gaussian basis sets centered on each H atom. For the description of Rydberg surfaces, this basis set is supplemented with six diffuse s and six diffuse p basis functions placed at the center of mass. Orbitals are determined in Hartree-Fock (HF) calculations on H3þ, and the final energies are obtained from CI wave functions calculated by taking all single and double excitations to the virtual orbitals from a large reference set of configurations. The potential curves are identified by the symmetries in C2v as well as the symmetries at the equilateral triangle configuration in D3h. It is clear from Figure 6 that no neutral state potential curves cross the X1A1 ground-state ion curve (X1A0 1 at the equilateral triangle configuration), the highest potential curve in the figure. The two possible dissociative routes are the lowest curves, 12A1 and 12B2. These curves are degenerate at the equilateral triangle position where they have 12E0 symmetry, and they have asymptotes that lie below

−1.0

H3+ X1A1(1A1�)

Energy (Hartrees)

−1.1 −1.2

22B2(22E�)

22B1(B2A2′′)

−1.3 −1.4

22A1(12A1�)

−1.5

12B1(12A2′′)

−1.6

12B2(12E�)

−1.7

32A1(22E�) 12A1(12E�)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Internuclear distance (Bohr) Figure 6 Potential curves for the approach of one H in C2v geometry along R2 (as in Figure 5) to the midpoint of the other two H atoms held at R1 = 1.63 ao

Dissociative Recombination of H3þ Ions with Electrons

89

the ground vibrational state of H3þ. No other states can provide DR routes at low electron energies. These states consist of a 2px or 2py orbital bound to the ion ground state where the xy plane is the plane of the molecule. Because these states do not cross the ion, DR was initially thought to be slow for H3þ. This is discussed further below. The neutral curves shown in Figure 6 are qualitatively similar to those shown in Figure 3 of Petsalakis et al. (1988). A precise comparison is not possible due to the different geometries used in their figure. However, at the equilateral triangle geometry, the curves in Figure 6 are about 0.12 eV lower than those of Petsalakis et al. (1988). Also shown in the figure are the 32A1 and 22B2 states, which are the components of the 22E0 doubly degenerate state at the equilateral triangle configuration. These states are too high in energy to be dissociative channels at low electron energies. Figure 7 shows a two-dimensional surface for 12A1, 12B2 and the ion ground state. In the plot, both R1 and R2 are varied and , as shown in Figure 5, is fixed at 90. Both neutral surfaces intersect at the equilateral triangle configuration. As shown in the figure, the 12A1 surface leads to H2 þ H and both 12A1 and 12B2 can generate H þ H þ H.

−1.26 −1.62

−1.44

Energy (Har trees)

−1.08

H3 + X1A1

1.0

H+H+H

4.0 3.0 ) r 2.0 (Boh R2

H312B2 H2 + H

H312A1

5.0

.00 .00 5 .00 4 3 0 2.0 hr) 1.00 R 1 (Bo

Figure 7 Potential surfaces for H3 and H3þ using the coordinates of Figure 5 with  = 90

90

Rainer Johnsen and Steven L. Guberman

4.3 Vibrational and Rotational Considerations The nuclear configuration of the ground state of H3þ is an equilateral triangle and belongs to symmetry group D3h. The normal modes are shown in Figure 8, labeled using the notation of Herzberg (1945). The first normal mode, labeled  1, is the symmetric stretch or breath­ ing mode. The remaining normal modes,  2a and  2b, are degenerate, i.e., they have the same frequency. Indeed,  2a and  2b, as shown in Figure 8, are not unique. An infinite number of pairs of modes can be obtained by taking orthogonal linear combinations of  2a and  2b, and they are all equally valid. If one degenerate mode is superposed upon another with different phases for the vibrational motion, the H atoms will move in ellipses (Herzberg, 1945). If the motion in the two modes is out of phase by 90 (i.e., when the atoms in one mode are passing through the equilibrium position, the atoms in the other mode are at the maximum displacement), the H atoms will move on circles and the motion can be described with a vibrational angular momentum quantum number, ‘. Instead of describing the vibrational state of the molecule with quantum numbers, v1, v2, and v3, it is now common practice to use (v1, v2‘). For H3þ (Watson, 2000), ‘ = —v2, —v2 þ 2, . . . , v2 — 2, v2. Since H3þ is a symmetric top (i.e., two of its moments of inertia are equal), the quantum numbers specifying the rotational energy levels are Nþ, the total angular momentum, and Kþ, the projection of Nþ, upon the molecular symmetry axis. Each proton has a spin of 1/2 and the total nuclear spin, I, can be 3/2 (ortho) or 1/2 (para). For the ortho states, Kþ = 3n, where n is an integer (Pan & Oka, 1986). For the para states, Kþ = 3n + 1 (Pan & Oka, 1986). It can be shown that the state with (Nþ, Kþ) = (0, 0) does not exist. The lowest energy rota­ tional state is for (1, 1) and is para. The second level, at 23 cm1 above (1, 1), is (1, 0) and is ortho. The (1, 0) level is highly metastable since an ortho—para transition is forbidden. The lowest ortho levels are (1, 0), (3, 3), (3, 0), and (4, 3). The lowest para levels are (1, 1), (2, 2), (2, 1), and (3, 2). It is interesting to note, especially for the

H1

H1

ν1 H2

H1

ν2a H3

H2

ν2b H3

H2

Figure 8 The three normal mode vibrations of the ground state of H3þ

H3

Dissociative Recombination of H3þ Ions with Electrons

91

interpretation of DR experiments, that all of these levels have very long lifetimes (Pan & Oka, 1986). The radiative lifetimes are 1.2  106 seconds for (2, 2), 15  106 seconds for (2, 1), 3.3  104 seconds for (3, 2), 2.2  104 seconds for (3, 0), and 2.2  104 seconds for (4, 3) (Pan & Oka, 1986). Once generated, these ions will not decay by photoemission during DR experiments. 4.4 One- and Two-Dimensional Theory 4.4.1 Direct Recombination The direct recombination cross section for vibrational level v0 , v0 , is given by (Bardsley, 1968; Flannery, 1995; Giusti, 1980) v 0 ¼

2 Gv 0 r 2 k ð1 þ Sv Gv Þ2

ð14Þ

where Gv 0 ¼ 2 jðYd Xd jHjYi Xv 0 Þj2 , r is the ratio of the statistical weights of the neutral and ion states, k is the wave number of the incident electron, v runs over the open ion vibrational levels, Xd and Xv0 are dissociative and bound vibrational wave functions, respectively, Yd and Yi are electronic wave functions of the dissociative and the ion states, respectively, and H is the electronic Hamiltonian. Equation (14) does not account for the inter­ mediate Rydberg levels. In the expression for Gv0, the integration is over the electronic and nuclear coordinates. If the dissociative potential curve does not cross within the turning points of the ion vibrational level, the small vibrational overlap will lead to a small v0 . Figure 6 shows that the dissociative potential curves, 12A1 and 12B2, in a one-dimensional view, do not cross the ion. This feature alone led theorists (Kulander & Guest, 1979; Michels & Hobbs 1984) to predict that the DR rate constant for H3þ is small. At the time, the direct recombination process was thought to be much more important than the indirect process. 4.4.2 Multichannel Quantum Defect Theory Because of the large literature on Multichannel Quantum Defect Theory (MQDT), a full description of the technique is not given here. Instead we guide the reader to the most relevant literature. The primary advantages of MQDT for the study of DR is that one can account for interference between direct DR and indirect DR with both being treated equally and one can treat entire Rydberg series rather than concentrating upon individual states as would be the case with a scattering theory approach. The pioneering studies which introduced MQDT to the study of DR were those of Lee (1977) and Giusti (1980). The approach of Giusti (1980) modified by Nakashima et al. (1987) to incorporate Seaton’s (1983) closed-channel

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elimination procedure for the S matrix is the approach used today by most theorists. The theory involves a K or reaction matrix which contains the interaction matrix elements between all channels. The K matrix is calculated perturbatively from the Lippmann—Schwinger equation. The first papers used a K matrix limited to first order. The usage of a second-order K matrix was introduced by Guberman and Giusti-Suzor (1991). The original approach has been revised to include rotation (Schneider et al., 1997; Takagi, 1993; Takagi et al., 1991), derivative couplings (Guberman, 1994), Rydberg states with excited cores (Guberman, 2007), and spin—orbit coupling (Guberman, 1997). An excellent reference on MQDT is the volume by Jungen (1996) and the papers contained therein. 4.4.3 Dissociative Recombination of HeHþ and One-Dimensional H3þ A clue that the theoretical view of H3þ could be wrong came in calcula­ tions on a diatomic molecule that shares the noncrossing features of H3þ. Because we can think of HeHþ as H3þ with two of the protons super­ posed, they are expected to have similar recombination mechanisms. Figure 9 shows the ground-state potential curve for HeHþ and curves for seven HeH states (Guberman, 1994, 1995). All the HeH states in Figure 9 are Rydberg with the exception of the ground state. None of the states cross the ion curve. For this case, it was shown that electron capture could occur by breakdown of the Born—Oppenheimer principle, which also drives indirect DR. Because all the states found to be involved in DR are adiabatic Rydberg states, there are no electronic couplings between these states. Instead, derivative couplings were introduced to drive DR between the adiabatic states. The cross section was calculated for 3HeH up to 0.3 eV, using the MQDT approach (Giusti, 1980; Guber­ man & Giusti-Suzor, 1991), and over most of this region, the indirect process was much more important than direct recombination. Indeed, inclusion of the indirect mechanism increased the cross section by a factor of 49 (Guberman, 1995). For 3HeH, it was also found that He þ H(2s) are the main dissociation products at low electron energies. The total rate coefficient at 300 K was 2.6  108 cm3/s, giving a clear example of how DR, dominated by the indirect mechanism, can have a high rate coeffi­ cient. Indeed, the rate would have been higher if it had been calculated for the true analog of H3þ, the unphysical 2HeH. The potential curves shown in Figure 8 apply also to 2HeH, but the lower mass, compared to 3 HeH, raisesPthe vibrational levels in the well leading to higher overlap with the C2 þ dissociative state. Other calculations (Sarpal et al., 1994) for 4HeH using an R-matrix approach did not report a rate coefficient but also found that indirect recombination dominated the cross section. The main dissociation products were He þ H (1s) at low electron energies. This was surprising since the identity of the dissociation products found

93

HeH+ X1Σ+

−3.0

He+H (n = 3)

A2Σ+,C2Σ+,D2Σ+,3p2Σ+

−3.3

−3.2

−3.1

He+H (n = 2)

HeH X2Σ+

−3.5

−3.4

Energy (Hartrees)

−2.9

−2.8

Dissociative Recombination of H3þ Ions with Electrons

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Internuclear distance (Bohr) Figure 9 Potential curves for HeH and HeHþ from Guberman (1995). Reprinted by permission from the AIP Press

with the MQDT approach was a qualitative and not a quantitative result. Experiments (Stro¨ mholm et al., 1996) have since verified that the main products are He þ H (2s). Takagi (2003) reported a one-dimensional MQDT treatment of DR using the potential curves of Michels and Hobbs (1984). He found that the rate coefficient of the recombining ion was highly sensitive to the initial rotational level with the N  4 levels of the vibrational ground state having large rate coefficients. 4.4.4 Derivative Couplings for H3þ In a study of the predissociation of H3 (Schneider & Orel, 1999), d/dR1 and d/dR2 (see Figure 5 in their paper) derivative couplings connecting the lowest 2A1 dissociative state with 2s2A1 and 3s2A1 were reported. For the 2s state, the d/dR1 coupling at the ion equilibrium separation (R1 = 1.65 ao and R2 = 1.43 ao) is 0.15 ao1 and that for d/dR2 is —0.20 ao1. (The phase of the coupling is arbitrary since it depends upon the phases of the orbitals and the total wave function.) The largest d/dR1 coupling is 0.75 ao1 at R1 = 1.15 ao1 and R2 = 0.92 ao1 and for |d/dR2| it is >0.95 near R1 = 1.15—1.85 and R2 = 0.93. The largest coupling in this case is for R1 near the equilibrium separation but for R2 smaller than the equilibrium separation. The couplings with the 3s Rydberg state, as

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Rainer Johnsen and Steven L. Guberman

expected, are much smaller. Tashiro and Kato (2002) have reported derivative couplings calculated in hyperspherical coordinates between the two 2pE 0 (12A1 and 12B2 states in C2v) dissociative states and the 2s2A1. They found a large coupling that peaks at 5 ao1 for a hyperradius of 1.5 ao and hyperangles of  = 1/2 and = /6 radians and for the upper 2pE 0 state (12B2). Couplings with the lower 2pE 0 state (12A1) were found to be much smaller in agreement with the results of Schneider and Orel (1999). 4.4.5 Two-Dimensional Cross Sections Using a combined wave packet MQDT approach and derivative cou­ plings, a two-dimensional calculation (varying R1 and R2 as in Figure 5) was performed for DR along the 2B2 surface (Schneider et al., 2000). For direct recombination, they found that the calculated cross section is 4—5 orders of magnitude below the experimental cross section (Larsson et al., 1997). However, the inclusion of Rydberg states coupled together by the R1 and R2 dependence of the quantum defect led to a dramatic increase in the cross section although the theory was still two orders of magni­ tude less than the experimental cross section. The authors concluded that the Rydberg channels, via the indirect mechanism, played a crucial role in the DR of H3þ. They attributed the difference between theory and experiment to the lack of a full three-dimensional treatment and to the absence of the 2A1 dissociative state in the theoretical treatment. They also tested the proposal of Bates that DR in H3þ may occur via inter­ connected Rydberg states in which the connection is mainly between states differing by Dv = 1. They found that Dv  1 connections are also very important. 4.5 Three-Dimensional Treatments of H3þ DR The first three-dimensional theoretical treatment of the DR of a polyatomic molecule (Kokoouline et al., 2001) combined several new theore­ tical methods for the study of DR with aspects of the MQDT approach. In these pioneering calculations, a new driving mechanism, not present in diatomic molecules, was introduced. The next section contains a brief description of the adiabatic hyperspherical approach. Section 4.5.2 describes the role of Jahn—Teller (JT) coupling in the DR of H3þ. Section 4.5.3 sum­ marizes the role of the nuclear spin. The approach to calculating the cross sections used in the first paper (Kokoouline et al., 2001) is given in Section 4.5.4. The revised approach used in later papers is discussed throughout and described further in Section 4.5.5. The last section contains suggestions for future theoretical research.

Dissociative Recombination of H3þ Ions with Electrons

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4.5.1 Hyperspherical Coordinates and the Adiabatic Approximation The calculations describe the nuclear motion with hyperspherical coordi­ nates consisting of a hyperspherical radius, R, and two hyperangles,  and . The coordinates can be defined in terms of the distances between the H atoms. Taking ri to be the distance between atom i and the center of mass, the hyperradius is given by R2 = H3 (r12 þ r22 þ r32) (Kokoouline et al., 2001). In later papers (Kokoouline & Greene, 2003a,b), the expres­ sion for R remains the same but ri is taken to be the distance between atoms j and k and the coordinates are given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 2 r1 ¼ 3  1 = 4 R 1 þ sin sin þ ; 3

r2 ¼ 3

1=4

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 2 R 1 þ sin sin  ; 3

and r3 ¼ 3  1 = 4 R

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ sin  sin :

ð15Þ

ð16Þ

ð17Þ

From Equations (15)—(17) one can derive expressions for  and , which become intuitively meaningful by consulting Figure 6 in the work of Kokoouline and Greene (2003a) for a valuable demonstration of the mean­ ings of these angles. [For further discussion of hyperspherical coordinates, the reader is referred to the review by Lin (1995)]. The general idea is that the hyperradius describes the overall size of the molecule, whereas the hyperangles, which are not explicitly defined in the first paper, describe the shape of the molecule. These considerations lead to the adiabatic hyperspherical approximation in which motion in R is considered to be much slower than the motion in the hyperangles, i.e., as the atoms traverse the potential surface, the shape of H3þ changes more rapidly than the overall size of the molecule. With the motion in the hyperangles separated from the motion in R, a Schro¨dinger equation at a single value of R can be written in which the eigenvalue is a point on the potential curve. The hyperradius, R, is identified as the polyatomic analog to the familiar diatomic internuclear distance. But is this analogy appropriate? The famil­ iar Born—Oppenheimer approximation is an adiabatic treatment of the nuclear motion and is justified by the great difference in the electron and nuclear masses. However, in the adiabatic hyperspherical approach for H3þ, the particles are all of equal mass. This approach is tested (Kokoouline & Greene, 2003a and 2003b) by solving for the nuclear vibrational energies within the generated potential curves. The eigenvalues for several lowlying levels differ by less than 23 cm1 from a full three-dimensional

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diagonalization (Jaquet et al., 1998), and the results support the use of this approximation. In later work (Fonseca dos Santos et al., 2007), this approx­ imation is partially dropped (see below). While the adiabatic hyperspherical approximation for H3þ appears to be successful for the vibrational energies, there have been no reported tests of the accuracy of the amplitudes of the vibrational wave functions. Inaccuracies in the amplitudes may significantly affect the values of important matrix elements between Rydberg states. It is interesting to note that the adiabatic hyperspherical approach fails for H2Dþ and D2Hþ (Kokoouline & Greene, 2005). 4.5.2 Potential Surfaces and Jahn–Teller Coupling The potential surface of H3þ used by Kokoouline et al. (2003) is from Cencek et al. (1998) and Jaquet et al. (1998), and the H3 surfaces are from the work of Siegbahn and Liu (1978), Truhlar and Horowitz (1978), and Varandas et al. (1987). These surfaces need to be interpolated to be converted to a grid in hyperspherical coordinates, but this is not covered in the published papers. The main driving force for DR, introduced for the first time in these calculations, is the JT coupling (Jahn & Teller, 1937), a coupling which does not occur in diatomic molecules. Figure 6 shows that the two lowest states of H3 intersect. The intersection point is at the equilateral triangle configuration where the molecule has D3h symmetry. The two lowest states have two electrons distributed in the three H 1s orbitals and one electron that is either in a 2px or 2py orbital where the molecule is in the xy plane. As R2 (see Figure 5) moves away from the equilateral triangle value (1.43 a0) but with  fixed at 90, the molecular symmetry is lowered to C2v and the degeneracy is split. The splitting is known as the static JT effect. The degeneracy at the D3h configuration appears as a point of conical intersection when the potential surfaces are plotted in normal coordinate space, not Cartesian coordinate space. A further splitting of the energies of the original vibrational levels (in the wells of the 2px or 2py electronic states) occurs when the levels are determined in the mixed state. This splitting arises from the dynamic JT effect. An excellent description of the JT effects can be found in the work of Herzberg (1966). For H3 (Greene et al., 2003), the JT mixing has used the K-matrix form of Staib and Domcke (1990) and the JT mixing parameter and quantum defects, , of Mistrık et al. (2000), which was obtained from a fit to ab initio surfaces. The nature of the conical intersection allows one to represent the coupling with two parameters [see Equation (4.7) of Mistrık et al. (2000)]. This is an enormous simplification compared to other situations where a non-JT coupling may need to be represented by a surface of derivative couplings. Of further importance, Staib and Domcke (1990) reported that

Dissociative Recombination of H3þ Ions with Electrons

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the fit to the ab initio results of Nager and Jungen (1982) shows that the conical shape of the potential surfaces is close to a true cone, although data to support this observation were not reported. This observation means that higher JT interaction terms beyond linear may not be needed and that the interaction of the np series with the ns or nd series is not important since if they were important a distorted conical shape would occur. Here, n is the principal quantum number. These observations provide some justification for the use of only an ‘ = 1 partial wave for the incoming electron in the calculations of Kokoouline and Greene (2003). On the other hand, Mistrık et al. (2000, 2001) found evidence for strong mixing of the ns and nd Rydberg states with 3p, 4p, and 5p states built on H3þ cores having the degenerate asymmetric vibrational motion. A point on the 5pE0 surface was found to have only 80% p character. The JT coupling parameter and the quantum defects used (Kokoouline et al., 2003a, b) are for the 4p state of H3 (Mistrık et al., 2000). Ideally, the best coupling parameters and the best quantum defects would vary with the Rydberg or continuum orbital energy. However, these are not available. In addition, for n  3 the quantum defect varies only slightly with n but that for n = 2 differs considerably from those for n  3. Because the MQDT approach requires a single coupling parameter and a single quantum defect surface, one must choose a compromise value. Usage of the n = 4 quantum defects (Kokoouline et al., 2003) should produce only very small errors in the positions of resonances, but the n = 2 states will suffer the largest shift in energy from the true positions. The JT interaction is generally thought of as that between states having the same n. Here n is the effective principal quantum number, i.e., n = n — , and is the quantum defect. In the MQDT approach used by Greene and coworkers, the JT interaction not only describes the interac­ tion between the two E0 states having outer orbitals 2px and 2py but also accounts for the mixing of Rydberg states with different n, the mixing of Rydberg states with continuum states, and the mixing of the Rydberg and continuum states with the 2pE 0 dissociative states. This assumes that these mixings are symmetry allowed. Rydberg orbitals of differing n, although orthogonal to each other, are quite similar near the nuclei except for a normalization factor of 1/n3/2. Since it is the Rydberg amplitude near the nuclei that is most important, the JT effect will occur between Rydberg states and the two dissociative states, scaled by the 1/n3 factor. The normalization constant is squared since the width [see the expression below Equation (14)] has the square of the interaction matrix element. If the incoming electron is in a px continuum orbital, it can be captured into an npy orbital also scaled by the 1/n3 factor. For two different Rydberg states having effective principal quantum numbers of n1 and n2, the connecting width would scale as 1/(n1n2)3.

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The couplings mix the Rydberg and continuum states with the lowest n = 2 dissociative states along which DR is finalized. The mixing of all the Rydberg states with each other and that of the continuum state with all the Rydberg states means that many possibilities for DR can occur. The continuum electron can be directly captured into the 2px,y states followed by dissociation, or it can be captured into a higher state which, via couplings to other intermediate levels, can even­ tually lead to the dissociative levels. The mechanism is remarkably simi­ lar to one originally proposed by Bates et al. (1993). 4.5.3 Nuclear Spin The nuclear spin has been included in prior theoretical studies of homo­ nuclear diatomic DR whenever molecular rotation is considered. The calculations reported by Greene and coworkers also include nuclear spin in the theory. Nuclear spin cannot be ignored because in H3, the nuclei are fermions and the total wave function must change sign for an interchange of any two protons. (The exchange is equivalent to a rotation by 180 around an axis perpendicular to the main symmetry axis.) This requirement places restrictions upon the allowed values for the rotational quantum numbers and requires that the total symmetry (i.e., the product of the symmetries of the vibrational, rotational, nuclear spin and electro­ nic wave functions) be that of the A02 or A2† representations of D3h. The ortho and para states have total nuclear spin of 3/2 and 1/2, respectively. 4.5.4 Calculation of the DR Cross Section and Rate Coefficient The first paper (Kokoouline et al., 2001) reported preliminary calculations which made use of the hyperspherical adiabatic approach and an expres­ sion derived by O’Malley (1966) for the direct DR cross section, , of diatomic molecules: ¼

X 2 G R     YR 2 0   Eel U  R 

ð18Þ

In Equation (18)  is an index that runs over the dissociative routes, Eel is the electron energy, R is the value of the hyperradius for the  th dissociative route at an energy, Eel, above the ion rovibrational level undergoing DR, G is the width for capture into the  th dissociative 0 route, Ub is the slope of the th dissociative route, and Y(R) is the dissociative nuclear wave function. The use of this expression follows from the observation that when the potential curves are plotted as a function of the hyperradius, all the Rydberg states cross the ion ground state.

Dissociative Recombination of H3þ Ions with Electrons

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Equation (18) omits the survival factor [the denominator within parentheses in Equation (14)] and thereby does not account for autoioni­ zation. There are several other caveats to consider. This expression was derived for diatomic molecules and its use for H3þ entails replacing the internuclear distance by the hyperradius. This replacement is not likely to lead to quantitative results. In diatomics, the nuclear configuration depends solely upon the internuclear distance, and the Franck—Condon factor in G has a rigorous dependence upon this distance. In a triatomic, in hyperspherical coordinates, the Franck—Condon factors will depend upon both the hyperradius and the hyperangles. Since the hyperradius is often viewed as a measure of the size of the molecule, taking G to depend only upon R is making the approximation that the Franck— Condon factors depend more upon molecular size than upon the details of molecular shape. This approximation is not expected to be reliable. It is probably for these reasons that the results (Kokoouline et al., 2001) are referred to as preliminary and approximate. Both upper and lower bound cross sections were reported. The lower bound cross sections included only the 2p states. Both the 2p and higher np states are included in the upper bound cross section. In a later paper (Kokoouline & Greene, 2003b), it is noted that the cross sections reported in the first paper (Kokoouline et al., 2001) need to be multiplied by a factor of 2 due to inconsistencies in the literature concerning the definition of the K matrix. Surprisingly, if one multiplies the 2001 results by 2, the upper bound cross section is in quite good agreement with the storage ring results (Jensen et al., 2001). The calculated cross sections are structureless as are the experimental results to which they were compared. Using only the 2p states, it is estimated that 70% of the DR events lead to H þ H þ H compared to the experimental result (Datz et al., 1995a, b) of 75% + 8%. For the H þ H2 channel, the peak H2 vibrational distribution occurs at v = 5—6 compared to the broad distribution found experimentally, which peaks at v = 5 (Strasser et al., 2001). The upper bound thermal rate coefficient at 300 K is 1.2  107cm3/s after correction by the p2 factor and compares well to the storage ring results of 1.0  107cm3/s (Jensen et al., 2001) and 1.15  107cm3/s (Sundstro¨m et al., 1994). The usage of Equation (18) to calculate these results would lead one to conclude that this agreement must be fortuitous. However, the agreement reported not only for the cross section and rate constant but also for the branching fraction and vibrational distribution argues otherwise. 4.5.5 Improved Cross Sections The lack of structure in the calculated cross section was corrected in a later detailed paper (Kokoouline & Greene, 2003b), which used an MQDT approach instead of Equation (18). The use of the adiabatic

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Rainer Johnsen and Steven L. Guberman

hyperspherical approximation has been described above as has the K matrix having the JT coupling. The calculated rate coefficients are reported (Kokoouline & Greene, 2003a and 2003b) to be accurate to better than 20% due to the incomplete set of states that are included in the calculations. The states are character­ ized by the quantum numbers [I, G, Nþ, N], where I represents the two values for the total nuclear spin, 3/2 (ortho) and 1/2 (para), G denotes the 0 total molecular symmetry (A2 or A2†), and Nþ and N denote the rotational quantum number for H3þ and H3, respectively. (The total molecular symmetry is determined by the need to have the total wave function change sign upon a swap of any two nuclei.) In the first detailed report of the calculations (Kokoouline & Greene, 2003b), 17 sets of these quantum numbers were used, each with 8—12 vibrational wave functions (includ­ ing the continuum) and 50—100 hyperspherical potential curves. A tabu­ lation of the levels is not included. In the most recently reported calculations (Fonseca dos Santos et al., 2007), several improvements were incorporated into the cross section and rate constant calculations. The adiabatic hyperspherical approximation was relaxed by including couplings between the adiabatic channels. The slow variable discretization approach was used to incorporate these couplings, but the details of these new calculations are not reported. A comparison of the calculated vibrational energies for 26 low-lying vibrational states with a full three-dimensional diagonalization (Jaquet et al., 1998) shows a clear improvement over the earlier full adiabatic approach (Kokoouline & Greene, 2003b). The positions of the Rydberg resonances are improved with this revision. However, the physical interpretability of these calculations is some­ what problematic. Potential curves plotted as a function of the hyperra­ dius are much more difficult to interpret than the more familiar surfaces plotted as a function of Cartesian coordinates. Furthermore, if one improves upon the adiabatic hyperspherical approach by including more couplings between the curves, the concept of a potential curve as a function of the hyperradius becomes weak. In the limit of completely dropping the adiabatic hyperspherical approximation, potential curves are no longer meaningful. These considerations must be balanced against the reasonable agreement that has been obtained to date between these calculations and experiment. This is discussed further below. An important additional improvement in the most recent calculations (Fonseca dos Santos et al., 2007) is the addition of more resonance states. Rotational states up to Nþ = 5 are included compared to the prior calcula­ tions which included levels up to Nþ = 3 (but not Kþ = 1) and (4, 3) for the ground vibrational level. A detailed accounting of the included vibra­ tional levels is not presented, which makes it difficult to assess whether or not the theoretical treatment is adequate at particular electron energies.

Dissociative Recombination of H3þ Ions with Electrons

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4.5.6 Toroidal Correction In the storage ring experiments, a beam of molecular ions circulates in a large ring (51.6-m circumference) (Stro¨ mholm et al., 1996) and merges with a beam of electrons in only a small section (0.85 m) (Stro¨ mholm et al. 1996) of the ring known as the electron cooler (the region between the merging and demerging regions in Figure 3). The electron beam is bent by a toroidal magnetic coil at the beginning and end of the overlap region. Collisions between the continuously renewed electron beam and the ions serve to reduce the random motions of the ions leading to a high energy resolution. The ion beam is generally a few mm in diameter compared to the electron beam which is a few cm in diameter. The cooler is also the location where DR takes place. For measure­ ments at “zero” center of mass energy, the electron beam is velocity matched with the ion beam. For other center of mass energies, the elec­ tron beam energy is shifted up or down from the “zero” energy measure­ ment. For most of the length of the cooler, the electron beam is very closely collinear with the ion beam and the intended center of mass energy is appropriate. However, in the merging and separating regions at both ends of the cooler, the ion and electron beams are not parallel and the center of mass energy changes with the angle between the two beams. The result is that a measurement of the DR rate constant at a single center of mass energy (appropriate in the straight section of the electron beam cooler) is actually an average of rate constants for different center of mass energies over the length of the cooler from the beginning of the merging region to the end of the separating region. The bending region comprises only about 15% (Amitay et al., 1996) of the full length of the overlap of electron and ion beams and was thought to not play a significant role in deriving the value of the rate constants. However, an important recent study by Kokoouline and Greene (2005) on H3þ indi­ cates that the experimental data deviate considerably from the theoretical values near 0.03 eV, 0.1 eV, and above 0.8 eV. In the latter region the difference between experiment and theory is over an order of magnitude. If the theoretical results are averaged over the full cooler length, account­ ing for the higher relative center of mass energies at the ends of the cooler, the theory agrees with experiment above 0.8 eV and shows improved agreement at 0.03 and 0.1 eV. The results indicate that raw storage ring data must be corrected to remove the effect of the electron bending regions. The deconvolution procedure for accomplishing the correction (Lampert et al., 1996) introduces considerable uncertainty because the rate constants needed at higher energies have often not been measured, and in the case of those that have been measured, they too must be corrected. The result is an iterative procedure which is usually carried out to first order (i.e., a single iteration) (Stro¨ mholm et al., 1996).

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4.5.7 Breit–Wigner Cross Sections If one assumes that all electron captures into Rydberg states lead to dissociation in one way or another and that there is no direct dissociative channel that would interfere with the dissociation through the Rydberg states (as is the case for H3), the Breit—Wigner expression can be used for calculating DR cross sections. An important innovative approach along these lines has been reported by Jungen and Pratt (2009). They treated the linear JT effect, restricting capture into v2 = 1 Rydberg levels (from the ion ground state). Capture into v1 = 1 Rydberg levels was not considered. Using spectroscopic data for the 3pE 0 state and previously determined JT coupling parameters, they show that after averaging over the closely spaced v2 = 1 resonances, a simple cross section expression results which is independent of n and structureless. 4.5.8 Comparison of Theory and Experiment For the four isotopomers, Jungen and Pratt (2009) show that there is a factor of two disagreement with the experimental rate coefficient at some energies for D3þ and a factor of 2—3 disagreement for D2Hþ. For H2Dþ and H3þ the agreement is even better except near 0.006 eV for H3þ. The resulting rate coefficients show remarkable agreement with experimental results for the four isotopomers considering the simplicity of the cross section expression. Figure 14 has the latest results of Greene and coworkers (Fonseca dos Santos et al., 2007), Jungen and Pratt (2009), and the CRYRING (McCall, 2004) data for H3þ. The theoretical results of Fonseca dos Santos et al. (2007) show much more structure than the CRYRING data. Although the theory and experiment are in generally good agreement, there is clearly room for improvement. 4.5.9 Suggestions for Future Theory The pioneering research of Greene, Kokoouline, and coworkers has made an enormous contribution to our understanding of the DR of H3þ. Never­ theless, many of the details remain to be uncovered. We still do not know which Rydberg states drive DR. The identities of the important states will change with electron energy as will the details of the mechanism. An important contribution in this regard has been the theoretical work of Tashiro and Kato (2002, 2003) on the predissociation lifetimes of H3 0 Rydberg states. They found that the 2s2A1 state has a large coupling with the upper 2pE 0 state (see Section 4.4.4) and may be a feeder state for DR from higher Rydberg states. They propose that in DR, initial electron capture occurs into high n (n = 6 or 7) states with low vibrational excitation followed by coupling to lower n states with higher vibrational

Dissociative Recombination of H3þ Ions with Electrons

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0

excitation. The coupling eventually leads to the 2s2A1 state, which is predissociated mostly by the upper 2pE 0 state. They propose that if electron capture involving a single vibrational quantum is most impor­ 0 tant, the 6s2A1 (1, 00) and 7p2E 0 (0, 11) are important for DR at electron energies just above the lowest vibrational level of H3þ. By propagating a wave packet from 7pE 0 (0, 11), they find that the predissociation involves the intermediate states 5p2E 0 , 4s2A10 , 3s2A10 , 3p2E 0 , 2s2A10 , and finally DR via 2pE 0 . However, the precise identification of these states requires greater accuracy in the quantum chemical determinations of their posi­ tions and widths and would be a valuable contribution. Note that the 0 2s2A1 state and those for n > 2 are not included in the calculations of Greene, Kokoouline, and coworkers or those of Jungen and Pratt (2009) and should be considered for future work. Future theoretical studies should explore the role of the ‘ = 0 and 2 partial waves. The work of Tashiro and Kato (2002) indicates that the ‘ = 0 wave may be more important than ‘ = 2. The calculations of Greene, Kokoouline, and coworkers and Jungen and Pratt (2009) treated only ‘ = 1. The inclusion of the ‘ = 0, 2 partial waves may account for some of the differences between theory and experiment. The JT coupling explored by Greene and coworkers is probably the dominant coupling that drives DR. But other derivative couplings that have been identified in prior calculations (Schneider & Orel, 1999; Schneider et al., 2000; Tashiro & Kato, 2002) need to be included in future three-dimensional calculations. The failure of the adiabatic hyperspherical approach for H2Dþ and D2Hþ leads one to ask if it is entirely adequate for H3þ. Instead of calculating vibrational energies to determine the accuracy of this approach, it may be more meaningful to compare the values of S matrix elements resulting from the adiabatic hyperspherical approach to elements calculated by relaxing this approach.

5. HISTORY OF EXPERIMENTAL H 3 þ RECOMBINATION STUDIES As may be seen in Figure 10, the measured recombination coefficients have varied considerably over the years. While all afterglow measurements carried out before 1973 probably refer to mixtures of H3þ and H5þ ions (and impurity ions), the recombining H3þ ions were clearly identi­ fied by mass analysis in the microwave afterglow studies by Leu et al. (1973). The measured recombination rates were very similar to those found for many other ions and nothing unusual was noted. Subsequent studies used either an inclined-beam (Peart & Dolder, 1974) or single-pass merged-beam (Auerbach et al., 1977) measured recombination cross

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α (H3+) [10–7 cm3/s] at 300 K Leu et al., 1973 2.5 SA 2

1.5

1

Macdonald et al., 1984 Adams et al., SA 1984 FALP

Greene, Kokoouline

Theory

Amano, 1988 IR absorption

Gougousi et al., 1995 FALP Larsson 1993 Canosa et Jensen 2001 Smith& al., 1992 McCall 2004 Španel FALP Storagering 1993 FALP Hus et al., 1988 Laube et MB

al., 1998

FALP

0.5

Glosik 2009 AISA, FALP

High H2

Low

H2

0 Figure 10 H3þ recombination coefficients inferred from different types of experiments, at electron temperatures near 300 K

sections over a wider range of energies, confirming the afterglow data within about a factor of two. Macdonald et al. (1984) extended the microwave measurements to higher electron temperatures up to 5000 K by microwave heating of the plasma electrons. While the measured 300 K rate coefficients were somewhat smaller than those of Leu et al. (1973), the temperature dependence was quite close to that expected from the merged-beam results. Not much attention was paid at that time to a theoretical argument by Kulander and Guest (1979) that the usual curve-crossing DR mechanism would not be applicable in the case of H3þ. The situation changed when Michels and Hobbs (1984) again calculated one-dimensional potentialenergy curves of H3þ and showed that the ionic ground-state curve of H3þ in the lowest vibrational states does not intersect a repulsive curve leading to neutral products. However, suitable curve crossings, were found for H3þ ions in the third or higher vibrational states. Hence, Michels and Hobbs suggested that the experimental data referred to vibrationally excited H3þ ions. Their argument was seemingly strength­ ened by new experimental data of Adams et al. (1984), who used their new “Flowing Afterglow Langmuir Probe” (FALP) technique to study the recombination of H3þ. They noticed that the initial electron-density decay was quite fast, compatible with a recombination coefficient near 107 cm3/s, but also that it changed in the later afterglow to a slower decay indicating a much smaller ( 1012 cm3. At [H2] = 1  1011 cm3 (see Figure 11) graphs of the same kind show that the asymptotic value is never approached on the time scale of the experiment (about 40 ms). The authors’ method of recovering the recombination coefficients employed a linear extrapolation (sometimes done approximately on a logarithmic graph) toward 1/ne ! 0. The procedure returns a much smaller and incorrect value of the recombination coefficient. The asymptotic value approached in the limit ne ! 0 should have been used, but in practice this value cannot be obtained at low [H2], even by curve-fitting, with any reasonable degree of precision, since diffusion becomes the dominant loss in the late afterglow. Another way of illustrating the cause of the problem is to examine the evolution of the ion composition during the afterglow, an example of which

−(ne′/ne2+VD /ne) (cm3/s)

1.E−07

1.E−08

1.E−09 0.0E+00

5.0E−10

1.0E−09

1.5E−09

2.0E−09

1/ne (cm3) Figure 11 Numerical simulation of an afterglow in an helium/argon/hydrogen mixture at a hydrogen concentration of 1  1011 cm3 for an assumed H3þ recombination coefficient of 1  107 cm3/s. The arrow indicates the extrapolation to 1/ne = 0, from which a far smaller recombination coefficient is obtained

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Density (cm−3)

1.E+10

1.E+09

1.E+08 0.00

0.02

0.04

0.06

Afterglow time (seconds) Figure 12 Numerical simulation of an afterglow in a helium/argon/hydrogen mixture at a hydrogen concentration of 1  1011 cm3 for an assumed H3þ recombination coefficient of 1  107 cm3/s. The lines indicate the evolution of density of electrons (thick line), Arþ ions (dotted line), ArHþ ions (dashed line), and H3þ ions (dash-dotted line)

is given in Figure 12. Even at an afterglow time of 40 ms, H3þ accounts for only 1/3 of all ions, which makes it impossible to obtain accurate H3þ recombination coefficients. The authors did carry out simultaneous mass spectrometric observations that seemed to indicate that the plasma was dominated by H3þ ions. However, the mass spectrometer samples ions from a region near the wall of the plasma container where the electron density and recombination loss of H3þ is lower, and hence the relative abundance of this ion is higher than it is in the center of the plasma. We conclude that the observations of very low recombination rates at low [H2] are probably in error and that consequently there is no need to search for explanations in terms of H3þ recombination mechanisms. In reality, the situation may be more complicated. A slower increase of the recombination coefficient with H2 concentration is consistently observed at much higher [H2] and this effect must have a different origin (see Section 6.3). While attempting to fit some of the published data samples, we also noticed that better fits were obtained when the model H2 con­ centration was reduced to values below the stated concentrations. This may indicate that a fraction of the H2 was dissociated during the dis­ charge phase of the experiment in the stationary afterglow experiments. Some dissociation of H2 can also occur in flowing afterglow measure­ ments due to metastable argon atoms that enter the recombination region. These remarks are speculative. It may be worthwhile to conduct some experiments to clear up such questions.

Dissociative Recombination of H3þ Ions with Electrons

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A further observation of very low recombination rates was made by Adams et al. (1984) in an afterglow experiment. This observation had a great impact and for a while since it was believed to provide evidence that H3þ ions in their vibrational ground state recombined only slowly. Their experiments showed that the electron density in an H3þ afterglow initially decayed quite fast (indicating a recombination rate of about 1.3  107 cm3/s) but then decayed much more slowly. No such effects were found when the plasma contained O2þ ions. At the time when the experiments were done, it was believed that H3þ in the vibrational ground state recombined very slowly. Hence the experimenters drew the natural conclusion that the initial decay was due to vibrationally excited H3þ and that the later slower decay was due to ground state ions. The lowest 300 K recombination coefficient derived in a later repeti­ tion of this experiment by Smith and Sˇ panˇel (1993) was 3  108 cm3/s, lower by a factor of 2.3 than the storage ring value. The authors believed that this value referred to a mixture of v = 0 and v = 1 ions. However, the accuracy of this value must be regarded as questionable. It was obtained by fitting the observed decay to a model that has too many adjustable parameters, the relative abundance of the two (or possibly three) states, two recombination coefficients, the quenching coefficient from the higher to the lower state, an estimated impurity concentration, and a diffusion rate. Also, the deviation of the decay curve from that corresponding to a simple (single-ion) decay is actually very small (only a few %), which makes it difficult to determine several coefficients by curve-fitting. While a good fit to the data was obtained, it does not necessarily result in a unique value of the recombination coefficient in the late afterglow. We constructed a simple numerical model similar to the one used by the authors and found that equally good fits could be obtained for higher recombination rate coefficients (up to about 6  108 cm3/s) in the late afterglow. If one simply fits the 1/ne(t) graph in the paper by a straight line, one obtains an upper limit of the recombination coefficient in the later afterglow of about 8  108 cm3/s. While the data show that there is indeed something “unusual” about the decay curve, the low inferred value of
(v = 0,1) = 3  107 cm3/s is not sufficiently accurate to be considered a challenge to the storage ring data. In an attempt to reduce the vibrational state to v = 0, Smith and Sˇ panˇ el carried out a second set of measurements in which they used Krþ ions to produce H3þ and, using a different fitting procedure, arrived at an even lower estimated value
(H3þ,v = 0)  (1—2)  107 cm3/s. However, the authors also found evi­ dence that the plasma contained both H3þ and KrHþ in apparent chemi­ cal equilibrium, and it is not at all obvious which of the two ions was responsible for the observed recombination loss. Similar observations of a reduced recombination rate in the later after­ glow were later made in flowing-afterglow measurements by Gougousi

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et al. (1995). Given the considerable uncertainty of the data analysis, the lowest values are not in conflict with the storage ring value at 300 K. Those authors attempted to explain their observations by a three-body mechanism in which ambient electrons induce l-mixing in the autoioniz­ ing states. The model may have contained a kernel of truth, but it relied on unrealistically long lifetimes, taken from a merged-beam experiment, that are not supported by either theory or other measurements. The explanation for the observed faster decay at early afterglow times may actually be that proposed by Smith and Sˇ panˇ el, but in somewhat modified form. We now know from theory (Fonseca dos Santos et al., 2007) that vibrational excitation enhances recombination, even for low vibrational states. Unfortunately, there are no direct measurements of such rates that would help to put this conjecture on a firmer basis. We conclude in this section that there are no afterglow measurements that give strong support for H3þ (v = 0) recombination coefficients sig­ nificantly smaller than those found in storage rings. Those afterglow measurements, in which the state of the ion was identified by spectro­ scopy, consistently yielded higher values. We now turn our attention to the question why many afterglow measurements have yielded higher recombination rates. 6.2 Afterglow Measurements That Yielded High Recombination Coefficients Most afterglow measurements, provided sufficient H2 was present in the gas mixture, yielded recombination coefficients that were higher by factors of 2—3 than those found in the storage ring experiments. The extensive compilation of data presented in the recent paper by Glosik et al. (2009a) shows quite clearly that the observed rate coefficients tend to increase with increasing neutral density (largely helium), which suggests that the recombination is enhanced in the presence of third bodies. The problem is that the conventional three-body collisionalradiative recombination mechanisms for atomic ions, in which either neutrals or electrons act as stabilizing agents, are far too slow to explain the observed three-body rate coefficients. In the next section we will explore more efficient third-body-assisted recombination mechanisms. 6.3 Third-Body Stabilized Recombination of H3þ There are several possible mechanisms that could make third-body effects on recombination more efficient in the case of molecular ions that recombine indirectly via intermediate resonant states that involve capture into high Rydberg orbitals. High Rydberg states are easily

Dissociative Recombination of H3þ Ions with Electrons

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perturbed by neighboring particles, in particular the electronic angular momentum can be altered by l-mixing collisions and the decay by dis­ sociation strongly depends on the electronic angular momentum. The resonant states that play a role in the binary recombination are primarily those in which the ion core is vibrationally excited by the JT interaction. Here the relevant electronic states have relatively low principal quantum numbers (around 6—8), and low angular momentum which makes predissociation fairly efficient compared to autoionization. If one assumes that all captured electrons predissociate, as is done in some simplified treatments (Jungen & Pratt, 2009), then electron capture is the rate limit­ ing step and any additional third-body stabilization mechanism will have no effect. On the other hand, if autoionization is not negligible, then l-mixing by third-body interactions may lead to states that are no longer capable of autoionization, but can be stabilized by further collisions. That would enhance recombination. A mechanism of this kind was once proposed (Gougousi et al., 1995) to explain H3þ recombination at a time when the binary recombination mechanism was not as well established as it is now. The l-mixing due to electrons was thought to be the most important ingredient. In hindsight, the proposed mechanism employed unrealistically long resonance lifetimes, which were based on experimen­ tal observations in merged-beam experiments. A different mechanism for a more efficient third-body-assisted recom­ bination process has recently been proposed by Glosik et al. (2009a,b). It shares some features (like l-mixing) with the model of Gougousi et al., but it focuses on resonant states formed by capture into rotationally excited core states, which form Rydberg states with higher principal quantum number (n = 40—80) and invokes l-mixing due to ambient neu­ tral atoms (helium in particular). These states do not usually contribute much to recombination since they tend to decay quickly by autoioniza­ tion, but they can have fairly long lifetimes (e.g.>10ps) and are thus good candidates for l-mixing. If one now had a further mechanism that stabilizes the population of these Rydberg molecules, i.e., renders them incapable of reverting to an autoionizing state, the overall recombination rate would be enhanced and the neutral density would be one controlling factor in the recombination in the afterglow plasma. Using theoretically calculated life­ times of the initially formed autoionizing states and estimates of the l-mixing efficiency due to helium atoms, and assuming that a large number of Rydberg states (principal quantum numbers from 40 to 100) contribute, the authors succeeded in deriving a three-body rate coefficient that comes close to the experimental value. However, the assumptions underlying his model are not realistic: Firstly, the authors’ estimate assumes a very high l-mixing efficiency of the helium atoms, that is appropriate only for small principal quantum numbers, while theoretical calculations (Hickman, 1978, 1979) show that the efficiency of l-mixing due to helium falls off

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rapidly with principal quantum number as n2.7 for n > 15. Secondly, if one invokes l-mixing due to helium atoms as the rate limiting step, one should also consider l-mixing by electrons, which is known to be faster by many orders of magnitude than that due to helium atoms, and its effi­ ciency rises with the fifth power of the principal quantum number (Dutta et al., 2001). In the range n = 40—80, in typical afterglows with ionization fractions of about 3  107, l-mixing by electrons would be more efficient by factors from 10 to 104 than by helium atoms. Thus, one would also expect a very efficient electron-assisted recombination process, for which, however, there is little experimental evidence. The relevant rate coefficients will be discussed later. The third problem is that this model leaves unan­ swered the question how the l-mixed states are eventually stabilized. Collisional stabilization by stepwise n-reducing collision with either elec­ trons or atoms may occur, but the efficiency of such collisions is not expected to be higher than for atomic systems such that the overall process is not likely to be faster than collisional radiative recombination of atomic systems. The model also has a more basic deficiency. It focuses on the lifetime of the initially formed rotational autoionizing resonances in low l-states and then assumes that higher l-states are exclusively populated by l-mixing. A more complete model should include three-body capture of electrons into all l-states by rotationally excited H3þ ions and its inverse, collisional ionization. For high n-states (with binding energies below 4 KT) colli­ sional ionization occurs on a time scale that is much shorter than the time scale of recombination in an afterglow plasma such that an equilibrium population of l-mixed states is always present. Any additional l-mixing mechanism hence is of no consequence. We will now consider a third mechanism for an efficient three-body mechanism that is an extension of the collisional dissociative process of Collins (1965), who realized that three-body capture of electrons into high Rydberg states of molecules can sometimes lead to predissociating states. Hence, the slow collisional and radiative descent from high-n to low-n states, the only stabilization route open to atomic systems, can be bypassed thus enhancing the overall recombination rate. Collins only treated a hypothetical model system with a single dissociative state, and did not consider effects of orbital angular momentum on the rate of predissociation, which should be included in a fuller treatment. In our model, we also invoke l-mixing but in the direction from high to low angular momenta and stabilization by predissociation of low l-states. We assume that in the plasma an equilibrium population of high Ryd­ berg states, denoted by H3, is maintained by three-body capture and its inverse, collisional ionization, i.e.,  e þ Hþ 3 þ M $ H3 ðnÞ þ M

ð21Þ

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and that the equilibrium constant K(n) of this reaction is approximately given by the Saha equilibrium ½H3 ðnÞ ¼ KðnÞ ¼ n2 3th eEn = kT ; ½Hþ 3 ne

ð22Þ

where n is the principal quantum number and th is the thermal de Broglie wavelength of the electrons at temperature T, i.e., th ¼

h2 ð2me kT Þ 1 = 2

ð23Þ

and En is the ionization potential of the Rydberg state. The assumption is made that three-body capture populates all l and magnetic substates ml evenly. This seems justified since the inverse process, collisional ioniza­ tion, depends only weakly on angular momentum of the Rydberg state. In the traditional theory of collisional radiative recombination of atomic ions, one now considers the departures from the thermal equilibrium due to the downwards collisional and radiative cascading transitions (Stevefelt et al., 1975). Under the conditions of the afterglow experiments discussed here (electron densities

ð25Þ

The factor of 2.7, which arises for decaying signals, is not usually presented in discussions of EDM sensitivity; it would not be present when T2 >> T or when the signal is constant. Assuming detector noise, backgrounds and similar pffiffiffiffi effects are small in the counting experiment, we expect 1=<S> ¼ 1= N, and d »

h pffiffiffiffi ðcount-rate limitedÞ: AET2 N

ð26Þ

For the neutron EDM measurement described in Harris et al. (1999), a single run N = 13,000 neutrons, A = 0.5, and T2 = 130 s. By eliminating the effects of magnetic field noise with the 199Hg comagnetometer, the expected statistical precision was achieved. For the 199Hg EDM experi­ ment, frequency precision of !  2p  109 Hz was reported for T2  100 s and nearly 100% modulation of the intensity of 254 nm light due to Fara­ day rotation. The measurement of the vapor polarization using Faraday

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rotation with light detuned from both the F = 1/2 and F = 3/2 absorption lines has minimal effect on T2 and thus < S > is effectively the total number of photons counted during the measurement, which would be estimated to be 3  1013 or an average detected power of about 0.2 mW. For a non-counting experiment, for example, an NMR-like measure­ ment for which S0 is nominally 0, S1 is proportional to the polarization or magnetization of the sample, and S is not dominated by counting noise. For example, in the noble-gas maser (Section 2.5), Johnson noise in the room-temperature pick-up coils was the dominant source of white-phase noise for the magnetometer species, while the free-running component of the two-species maser was subject to white-frequency noise due to the fluctuating magnetization associated withpffiffiffi maser instabilities. In the case of white-phase noise, S is proportional to B, where B is the bandwidth of the measurement, that is, B/1/T (T2 is effectively infinite) and thus ! / T  3 = 2

ðwhite phase noiseÞ: ð27Þ pffiffiffiffi For white frequency noise, S grows with T as the phase makes a random walk away from the frequency-noise-free phase. In this case ! / T  1 = 2

ðwhite frequency noiseÞ:

ð28Þ

Studies of ! and related quantities such as the Allan deviation are used to isolate the noise sources and to guide improvements to the experi­ ments as shown in Figure 4 in Section 5.1. Also, see discussions in Stoner et al. (1996) and Bear et al. (1998). 4.2 Systematic Effects and False EDM Signals There is no single set of systematic effects that dominate all experiments; however, most experiments completed or anticipated must address the following: • leakage currents that change when the electric field is changed • geometric phases that arise as the polarized spins evolve in space and time • ~ ~ E effects
even in storage cells and bottles where h~ i » 0 but h~   ~ Ei¼0: 6 One general approach is to make use of two or multiple species or transitions insensitive or less sensitive to the P-odd and T-odd EDM effects. This is generally called a comagnetometer. An ideal comagnetometer would average only the magnetic field, in exactly the same space, with exactly the same trajectories, over exactly the same time, with exactly the same T2, as the species considered most sensitive to P-odd and T-odd effects. This is diffi­ cult. For example, in the neutron-EDM experiment with 199Hg comagnet­ ometer, the UCN velocity distribution is quite different from that of the

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room temperature 199Hg used as the comagnetometer. Consequently, the average position of the two ensembles due to gravity was slightly different, and the two species averaged the storage cell magnetic field and ~ ~ E effects differently (Barabanov, 2006; Commins, 1991; Pendlebury et al., 2004). In the molecular EDM approaches described in Section 5.3, the comagnetometer is effected by using sums and differences of frequencies that separate sensitiv­ ity to dE and B. This has the advantage that the accumulated signal averages the same space for the EDM and comagnetometer measurements and could be ideal provided the effective T2’s are the same. We also mention the adoption of “blind analysis” techniques (Klein & Roodman, 2005). For example, in the most recent 199Hg measurement, much of the data were analyzed with a hidden offset to the EDM signal. The motivation for blind analyses in experiments with systematic errors comparable or larger than the statistical error is to eliminate influences that could enter the analyst’s approach because of bias for a specific outcome, for example, an EDM consistent with zero. In EDM measure­ ments, the frequency or EDM signal is determined under varying condi­ tions, for example, a magnetic field reversal motivated by the expectation that an EDM signal will not change; however, the manner in which the two experimental results are combined may be determined by foreseeing the result. There is nothing intrinsically wrong with this in general, and many important systematic effects have been discovered by understand­ ing the results of such correlations; however, we can expect blind ana­ lyses to become more common in future measurements.

5. CONTEMPORARY EXPERIMENTS In this section, we discuss current and planned experiments that are not direct extensions of the earlier efforts discussed in Section 2. The one effort that, in a way, spans the gap between these new endeavors and the ones described previously involves the paramagnetic molecule YbF (Hudson et al., 2002) and is described in Section 5.3. Contemporary experiments are generally focused on very large enhancements of T-odd and P-odd effects in specially selected systems and on applications of recently developed technologies, specifically cold atoms/molecules and condensed systems, or a combination of enhancements and new technologies. Experiments in systems investigated in earlier times, that is, cesium and xenon, are underway using cold-atom techniques and polarized liquid xenon, respectively. Molecular beam experiments with paramagnetic molecules continue to generate significant enthusiasm due to the very large effective enhancement of T-odd and P-odd effects (de and CS). Enhancements of the Schiff moment of diamagnetic atoms with nuclei that have octupole deformation have motivated experiments

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with radioactive atoms. We also describe efforts to measure the electron EDM in solid systems where electric effects would induce detectable magnetization or magnetic effects would induce detectable electric fields. Storage ring techniques similar to those applied for measurement of the muon EDM are contemplated for measurement of the proton and deuteron EDM. 5.1 Liquid Noble Gases The EDM of 129Xe is primarily sensitive to the same phenomenological P-odd and T-odd effects as 199Hg but with about an order of magnitude less sensitivity; moreover, the most recently published results for the two species differ in sensitivity to the atomic EDM by two orders of magni­ tude. Nevertheless, there is significant interest in improving the sensitiv­ ity to the EDM of 129Xe for two reasons: techniques using laser polarized xenon to produce a polarized liquid have the potential for many orders of magnitude improvement in signal-to-noise, and 129Xe appears to be an attractive magnetometer for next generation neutron-EDM efforts (Atchison et al., 2005). The current result, dXe < 6  1027, should be improved by one or two orders of magnitude in order to be useful as a neutron-EDM comagnetometer. Polarization by spin-exchange can be used to produce significant volumes of polarized 129Xe (Zeng et al., 1985) which can be condensed to a liquid and used over a broad temperature range down to the triple point of 161 K (Sauer et al., 1997). In the liquid state, the polarization lifetime can be T1 > 10 min at lower temperatures, and T2 is comparable to T1 in uni­ form magnetic fields and spherical containers. With 129Xe polarization of 10%, the magnetization of the liquid can be 10—100 times larger than in the gas-phase measurements, and a number of interesting but potentially confounding effects can arise due to the self-interaction of the precessing ensemble. For example, non-exponential decay (Romalis & Ledbetter, 2001) and amplification (Ledbetter et al., 2005) of the transverse magneti­ zation have been observed in development work for a liquid-129Xe EDM experiment. Cross-relaxation with the spin-3/2 isotope has been observed and depends on the 129Xe enrichment of the sample (Gatzke et al., 1993). Another potential advantages of the liquid-xenon system is the high dielectric strength that would allow higher electric fields; however, the EDM of an atom of liquid Xe has been shown to be suppressed by about 40% due to screening effects (Ravaine & Derevianko, 2004). One experiment underway envisions the use of liquid-helium-cooled SQUIDs to monitor the precessing magnetic moment of a sample of liquid xenon contained within a sapphire cell with E =75 kV/cm (Romalis & Ledbetter, 2001). The applied magnetic field would also be monitored or feedback-stabilized by one or more SQUIDs. In this case the signal

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Access tube C -axis sapphire Conductive coating +HV (50 kV)

Mx

Bz

Mz

18 mm

Machined hemisphere (φ 10 mm) −HV 18 mm 10−3 Slope = −1.5

σy

10−4 10−5 10−6 10−7 10−1

Data Simulation w/ white noise

100

101 sec

E-field reversal half-period, t Figure 4 Top: Schematic diagram of proposed liquid-xenon EDM experiment using low temperature SQUIDs and the three-SQUID coil configurations designed to be sensitive to transverse magnetization, Bz field and longitudinal magnetization. Bottom: The frequency precision as a function of T (see Equation (24)); for measurement time T > 2 s, frequency noise dominates and breaks the slope of T3/2. Courtesy of M. Romalis

S (Equation (22)) is the magnetic field due to the precessing-polarized 129 Xe liquid, and S is the SQUID noise for the measurement averaged pffiffiffi over ffi time T. At a frequency of 1 Hz it is anticipated that S »0:5 fT= T, while the field due to the 129Xe may provide S1 = 1 nT. For a pair of measurements with T = 10 s and |E| = 75 kV/cm, d  1027e-cm. Fluctuations of the applied magnetic field at the position of the liquid xenon would need to be monitored or stabilized at the 2 fT level, which is well within the noise capability of the SQUIDs. This approach does rely on the external SQUIDs to monitor all magnetic field variations including leakage currents, which is not the same as a comagnetometer occupying exactly the same volume; however, the approach is similar to the 199Hg measurement in the sense that magnetometers (in this case SQUIDs) monitor magnetic fields in a geometry that surrounds the measurement region. A schematic is shown in Figure 4.

Permanent EDMs of Atoms and Molecules

Top

155

Side Plug 2 mm

Electrodes

B0 Liquid Xe droplet

Xe B0 supply tube

MicroPick-up structured coils plate at −115 °C

Figure 5 Schematic diagram of proposed liquid-xenon EDM experiment using rotating electric fields. The filled circles indicate the liquid-xenon spheres, and the open circles are electrodes. Courtesy of P. Fierlinger

Another approach using polarized liquid xenon is under development (Fierlinger, 2010). In this scenario, illustrated in Figure 5, the liquid xenon is contained in an array of small spheres of 600 mm diameter. The xenon condenses into wells etched into a glass plate from a highly polarized gas injected above the plate. The central row of three spheres is surrounded by a set of electrodes (open circles) that apply the electric field differently to the different xenon samples. An EDM would lead to precession into or out of the plane. The resulting magnetization misalignment for each sphere would be measured by a nearby superconducting pick-up coil coupled to a low-temperature SQUID. Low noise in the superconducting coil and SQUID-based detection system, high 129Xe polarization and long T2 are anticipated. With somewhat conservative estimates of T2 = 100 s, E = 10 kV/cm, 10% polarization of the xenon and 100 simultaneous mea­ surements in an array of 10  10 spheres, d  2  1027 e-cm for a 100 s measurement. Leakage currents that flow around a sphere when the electric field is applied could add or subtract to ~ B0 and thus generate a false EDM signal. The spheres with no applied electric field are within a few millimeters of the spheres with applied electric field and are also sensitive to such an effect so that correlations could allow a false effect to be separated. The high sensitivity predicted for the polarized-liquid-xenon EDM experiments arises from the potentially very large signal-to-noise ratio of SQUID magnetometers and superconducting pick-up coils, large elec­ tric fields and narrow lines (corresponding to a long T2). It will require correspondingly low frequency noise to realize this sensitivity, and the liquid itself is potentially the greatest source of frequency noise due to the large magnetization. Such effects are mitigated in a spherical sample, and the manufacturing specifications of the sapphire sphere or the actual shape of the liquid drop wetted on the glass surface may ultimately limit these experiments. Systematic effects due to motion of the spins

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(~ ~ E and geometric-phase effects) should be significantly suppressed compared to gas-phase 199Hg. Leakage-current-false-EDM effects would be monitored by external magnetometers. While difficult, these experi­ ments show great promise. 5.2 Cold Alkali-metal Atoms: Fountains and Lattices The EDM measurements in cesium and thallium are motivated by the Z3 enhancement of sensitivity to the electron EDM and to the scalar coupling of the electron to the nucleus. The experimental sensitivity was limited most importantly by the short coherence time (T2) of cesium in a cell and short or observation time (T) for the thallium beam. For thallium, effects due mainly to the geometric phase and related ~ ~ E effects were the dominant systematic limitations. Laser cooling and trapping of the hea­ viest alkali-metal species, cesium and francium, offer promising new directions and several approaches are being pursued. Cesium atomic fountain clocks based on launching atoms from a laser cooled or trapped sample have moved to the forefront of time keeping. Narrow line widths (T2  1 s) are attained as the atoms move up and then down through a resonance region. While the 133Cs atomic frequency stan­ dard uses the DmF = 0 transition, which is insensitive to magnetic fields in first order, an EDM measurement must use DmF  1. From Equation (24), with T = 1 s, T2 >> T and N = 106, the expected uncertainty on ! is expected to be about !  103 Hz, which is consistent with observations of the Allan variance representing the short-term instability of cesium fountain clocks (!/!  1013 for ! = 2p  9.2 GHz (Weyers et al., 2009)). For an EDM measurement with an electric field of 100 kV/cm, which may be feasible, each 1 s shot would have a sensitivity of 6  1024 e-cm, comparable to the sensitivities of both the cesium and thallium measure­ ments presented in Table 1. Thus, significant improvement is possible, and a demonstration experiment with about 1000 atoms per shot and E = 60 kV/cm was reported as a measurement of de by Amini et al. (2006). The result can be interpreted as dCs = (0.57 + 1.6)  1020 e-cm (the authors use e = 114 for cesium). The major limitation in this demonstration was the necessity to map out the entire resonance-line shape spectrum, which is the combination of transitions among the nine F = 4 hyperfine sub-levels complicated by inhomogeneities of the applied magnetic field in the reso­ nance region. This subjected the measurement to slow magnetic field drifts that would be monitored or compensated in the final experiment. If these problems are taken into hand, the statistical sensitivity could be significantly improved with several orders of magnitude more atoms, higher electric field, and duty-factor improvements; however, the major systematic effect due to ~ ~ E was about 2  1022 e-cm (Amini et al., 2006). This could ultimately limit the sensitivity of a single species fountain measurement.

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As discussed in Section 2.3, the EDM of a paramagnetic atom is most sensitive to the electron EDM and the T-odd and P-odd scalar coupling of electrons to the nucleus, which scale approximately as Z3. For francium, e (from Equation (2)) is in the range 900—1200 (Ginges & Flambaum, 2004), and kCs should be similarly enhanced. Francium can be produced in significant quantities in isotope-separator rare-isotope production facilities, and 210Fr has been produced, laser cooled, and trapped in a magneto-optical trap (MOT) at Stony Brook, New York (Gomez et al., 2006). The experiment has been moved to the isotope-separator facility (ISAC) at TRIUMF in Vancouver, Canada. Francium isotopes have halflives of 20 min (for 212Fr) or less, and any experiment would need to be “online,” that is, the EDM apparatus would be at the site of the rareisotope production facility. The development of the cesium-fountain EDM measurement may lead the way to a future program with francium at a future facility such as the Facility for Rare Isotope Beams (FRIB) at Michigan State University. The fountain concept allows line widths on the order of 1 Hz, limited by the time for the cold atoms with vertical velocity of a few m/s to rise and fall about 1 m. Another idea being pursued by D. Weiss and collaborators is to stop and cool alkali-metal atoms in optical molasses near the apogee of their trajectory and trap them in an optical lattice formed in a build-up cavity (Fang & Weiss, 2009). Storage times in the lattice could be many seconds. The lattice would be loaded with multiple launches, filling sites that extend over 5—10 cm, and 108 or more atoms could be used for the EDM measure­ ment. After loading the lattice, the atoms would be optically pumped to maximum polarization and then the population transferred to the mF = 0 state by a series of microwave pulses. A large electric field (e.g., 150 kV/cm) would define the quantization axis in nominally zero magnetic field, and the energies would be proportional to m2F due to the parity-allowed inter­ action. In another planned innovation, a Ramsey separated-field approach would be used with the free-precession interval initiated by pulses that transfer atoms to a superposition of mF = F and mF = F states and termi­ nated by a set of pulses coherent with the initial pulses. The relative populations transferred back to the mF = 0 state would be probed by optical fluorescence that could be imaged with about 1 mm spatial resolution. With the large size of the lattice, the superposition of stretched levels (mF = + F) would amplify the sensitivity by a factor of F relative to experiments that monitor DmF = 1 transitions (Xu & Heinzen, 1999). In a measurement time T = 3 s, and N = 2  108, an EDM sensitivity of 6.5  1026 e-cm for the cesium atom is expected. The optical lattice can also trap rubidium, which would be used as a comagnetometer. Cold atom techniques have been developed over the past 25 years by a large number of groups, and appear to be able to provide statistical power and ways to monitor systematic effects. The continuing advances

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in technology of fountain clocks provides encouragement, though an EDM measurement provides a set of distinctly separate systematic effects. An EDM measurement in a lattice would allow measurements with two or more species, thus providing a comagnetometer to monitor leakage-current and other effects. 5.3 Paramagnetic Molecules Molecular beam EDM experiments with TIF are discussed in Section 2, where it is pointed out that the electric field along the interatomic axis is generally much larger than the external field required to align that axis. For a heavy polar paramagnetic molecule the effective internal electric field interacting with the unpaired electron can be volts per angstrom, for example, for YbF, Eint  2.6  1010 V/cm (Mosyagin et al., 1998) when the molecule is fully polarized. Thus, e  25,000 in the case of YbF (Hudson et al., 2002). Alternatively, this can be viewed as a strong polarizability due to the mixing of rotational states of opposite parity; T-odd and P-odd spin-axis interactions would result in a component of ~ J along ~ E and a ~ resonance-frequency shift proportional to E. For -doublet systems, the electric polarizability of the molecule is so large that relatively small laboratory electric fields can be used, greatly mitigating the leakage-cur­ rent and motional ð~ ~ EÞ effects. For a molecular beam measurement, the observation time T (from Equation (24)) is limited to the flight-time through the apparatus. This can be significantly increased by slowing the beam along its direction of motion (e.g., longitudinal cooling). The observed number of molecules can also be significantly increased and the flight-path made longer with transverse cooling. In general, the large enhancements due to the internal electric field are balanced by the rela­ tively low numbers of molecules synthesized and observed; however, significant improvements over the thallium-EDM measurement are expected in several systems including YbF (Hudson et al., 2002), metastable PbO (Bickman et al., 2009), WC (Lee et al., 2009), and ThO (Vutha et al., 2009) as well as a trapped molecular ion such as HfFþ (Meyer et al., 2006). Hinds and collaborators have developed an experiment using 174YbF, which is illustrated in Figure 6. The molecular beam is cooled to the electronic, vibrational, and rotational ground state. The hyperfine split­ ting of the F = 0 and F = 1 levels formed by the combination of 19F nuclear spin (1/2) and the unpaired electron spin is 170 MHz. The (F = 1, mF = 0) state is lowered in energy relative to the mF = +1 states due to the electric field. The EDM signal is the splitting of the mF = 1 and mF = þ1 states. A small magnetic field is applied so that both ~ E and ~ B can be reversed with respect to the laboratory coordinate system. The magnetic field must be well aligned with the electric field to avoid a false-EDM signal due to a motional field
the transverse components are estimated to be 10 mG.

159

Probe fluorescence (V)

Permanent EDMs of Atoms and Molecules

Probe PMT 1.0 0.8

Lens

0.6 Probe beam

0.4 0.2

Lens

B

0.0 1.8

2.0

2.2

2.4

RF out Mirror

2.6 E-field plates

Time after Q-switch (ms)

Pump beam

RF in

Yb target mF = −1 0

Skimmer

F=1

170 MHz

Ablation laser

Valve

+1

F=0

Normalized signal (arb. units)

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 − 200

−100

0

100

200

Applied B-field (nT) Figure 6 Schematic diagram of YbF molecular beam experiment. The fluorescence signal, shown in the upper left, measured by the probe photomultiplier tube (PMT) represents the time-of-flight of molecules after the Q-switched ablation-laser pulse. The bottom graph shows the phase of the superposition of the mF = 1 and mF = 1 levels as a function of B, which is probed by the population of the F = 0 state after the second p-pulse. Courtesy of E. Hinds and J. Hudson

The experiment uses lasers to prepare the molecules in the F = 0 state followed by a p-pulse in the region of combined DC electric field (3.3 kV/cm) and 170 MHz RF magnetic field along the x-direction,

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Tim Chupp

which produces the superposition of mF = 1 and mF = þ1 levels. The superposition evolves as the molecules travel through the region of combined electric and magnetic field and then the superposition is probed by a second p-pulse and laser fluorescence to detect the F = 0 population. The pump and probe RF pulses are not in phase, thus the measurement is not a Ramsey separated-field experiment. A combination of reversals of electric and magnetic fields as well as a calibration mag­ netic field are used to determine the EDM signal. The most recent result is (Hudson et al., 2002) de ¼ ð0:2 – 3:2Þ  10  26 e-cm:

ð29Þ

False-EDM signals arising from leakage currents and the magnetic field misalignment are estimated to be two or more orders of magnitude smaller than the statistical error. Refinements to the apparatus include a pulsed supersonic beam, rather than an effusive thermal beam; more efficient cou­ pling of the RF allowing shorter p-pulses, which improves polarization con­ trol and reduces some systematic effects; and a more uniform electric field that reduces geometric phase effects and problems from transverse electric fields. The signal-to-noise ratio for the fluorescence probe and the super­ position signal as a function of magnetic field are shown in Figure 6, and demonstrate the prospect for significantly improving the EDM sensitivity. Experiments in -doublets, first suggested by Kozlov and DeMille (2002) for metastable PbO and by Cornell and coworkers for HfFþ and ThFþ (Meyer et al., 2006), are also being pursued using tungsten-carbide (WC) by Leanhardt and collaborators (Lee et al., 2009) and using thorium­ oxide (ThO) by DeMille, Doyle, Gabrielse, and collaborators (Vutha et al., 2009). The relevant levels of WC are shown in Figure 7. The ΔEup C

μ elE lab

− μB

+deE int

+μB

−deE int E eff

~ ~

~ ~

E lab W

m = −1

B

m=0

m = +1

W ~ ~

E eff C

−μ elE lab −μB

~ ~ −deE int +μB +deE int ΔEdown

Figure 7 Molecular levels of tungsten carbide (WC) in the presence of applied magnetic and electric fields

Permanent EDMs of Atoms and Molecules

161

electronic ground state has structure 3D1 with triplet spin  = 1, orbital angular momentum L = 2, and total angular momentum  = 1. In the absence of an external electric field, the separate m levels are paritydoublets, and in the presence of an applied external electric field, the molecule is polarized and each m level splits into two states with the splitting proportional to |m|, as shown. Any magnetic field, including that due to a leakage current or motional field would split the m = 1 and m = þ1 levels in opposite directions for both molecular orientations while an EDM couples to the internal field. Thus, the energy shifts would also depend on the molecular orientation with respect to the external field. The two energy splittings of the m = 1 and m = þ1 states are DEup ¼ 2dEint þ 2 B

and DEdown ¼ 2dEint þ 2 B

ð30Þ

The difference DEupDEdown = 4deEint measures the EDM and the sum DEup + DEdown = 4 B is the comagnetometer. This built-in comagnet­ ometer is an extremely important feature of these systems, and provides additional motivation as the experimenters develop the techniques and improve the rates. An additional advantage of 21/2 and 3D1 molecular states may be the near cancellation of the orbital L = 2 and spin-triplet  = 1 magnetic moments resulting in a very small that further mitigates leakage current and motional field effects. 5.4 Rare Atoms Rare or radioactive atoms provide attractive systems due to enhancements of atomic or nuclear polarizability. For atomic radium, enhanced atomic polarizability arises due to the near degeneracy of the two electrons in the spin-triplet 7s7p and 7s6d states, which are separated by about 5 cm1 or 103 eV. Thus, a laboratory electric field polarizes the atoms and results in a large internal electric field that interacts with the triplet electrons. The resulting enhanced sensitivity to the election EDM is estimated to be e  5000 (Dzuba et al., 2000). All radium isotopes are radioactive and can be extracted only from sources or isotope production facilities. The isotope 226 Ra (t1/2 = 1600 years) is a daughter in the decay chain of 238U, and 225Ra (t1/2=14.9 days) and can be extracted from a source of 229Th. Sources of 1

Ci and a few millicuries, respectively, that provide samples of about 1014—15 atoms are quite practical. Alternatively, isotopes can be produced at an accelerator facility, for example, at the Kernfysisch Versneller Insti­ tuut in Groningen, The Netherlands, where a program is under develop­ ment to cool and trap radium in a MOT for an EDM measurement (Jungmann, 2007). For diamagnetic atoms, including ground-state radium, the nuclear Schiff moment may be enhanced by near degeneracy of nuclear levels in a manner very similar to the molecular enhancements. For nuclei, the

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analogue to the polar molecules is characterized by the combination of quadrupole and octupole deformation that arises due to the nature of nuclear forces. Octupole deformed nuclei are reflection symmetric so that the states of definite parity are even and odd combinations of the two mirror-image configurations with very small splitting and a resulting large polarizability. Thus, the large intrinsic dipole moment and Schiff moment in the body-fixed frame of the nucleus is polarized along the total angular momentum ~ J by T-odd and P-odd interactions (Auerbach et al., 1996; Engel et al., 2000; Haxton & Henley, 1983; Spevak et al., 1997). The Schiff moment that would result can be expressed in terms of quadrupole and octupole deformation parameters, b2 and b3 respectively, as (Spevak et al., 1997) S » 0:05e

2 23 ZA2 = 3 r30  ; Eþ  E

ð31Þ

where Eþ and E— are the energies of opposite parity states, and  is the strength of the effective T-odd and P-odd interaction of the nucleons (nn, np, or pp are components of the different isospin combinations). For a core density  the potential can be characterized as (Auerbach et al., 1996) G X! ! pffiffiffi i  r: ð32Þ V TP ¼  2mp 2 i In Table 2, we list several cases with potentially large octupole enhance­ ments, which were selected because of work underway to develop techni­ ques for EDM measurements. There are several interesting cases not shown, most notably, 229Pa (protactinium), which has an exceptionally small splitting compared to the scale of nuclear binding energies, that is, dE  0.22 keV, and a Schiff moment that may be 10,000—30,000 times as large as that of 199Hg.

Table 2 Predicted Schiff moments (S) and atomic EDMs (dA) for a generic CP-violating coupling h based on the work of Spevak et al. (1997) and Dzuba et al. (2002). 223

223

225

23.2m 7/2 37 — 1000 3300

11.4d 3/2 170 50.2 400 —3400

14.9d 1/2 47 55.2 300 —2550

Rn

t1/2 I dE th (keV) dE exp (keV) 108S/ (e-fm3) 1025dA/ (e-cm)

Ra

Ra

223

Fr

22m 3/2 75 160.5 500 2800

129

Xe

199

Hg

1/2

1/2

1.75 0.66

—1.4 3.9

 dE = Eþ—E— is the splitting of low-lying-opposite-parity levels measured or predicted by Spevak et al. (1997) using a Woods-Saxon potential.

Permanent EDMs of Atoms and Molecules

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An experiment with 225Ra is underway in the group of Z.T. Lu and collaborators at Argonne National Lab in the United States (Guest et al., 2007). Radium from a source is heated to form a beam and then Zeeman slowed and trapped in a MOT operating on the 1S0—3P0 transition. One of the interesting discoveries of this work was that the MOT repumping, which is required to pump atoms out of the 3D1 to the state, could be provided by blackbody radiation, which transferred atoms from the 3D1 to the 3P0 state. Cold atoms would then be transferred from the MOT to a red-detuned far-off-resonance trap (FORT) at the focus of a laser beam provided by an erbium-fiber laser. The FORT trapped atoms will remain at the laser focus as the focusing optics are physically translated by about 1 m to place the 225Ra sample in the measurement apparatus within a multilayer mu-metal magnetic shield. An EDM measurement would take place in the FORT trap. Nuclear polarization would be optically pumped with circularly polarized light from the 7s2 ground state to the 7s7p 1P1 level with F = 1/2, followed by a Ramsey-type measurement. It is antici­ pated that N = 104 atoms can be trapped in high vacuum so that the atomstorage lifetime will allow a measurement interval of T = 100 s. The high vacuum can allow an electric field as large as 100 kV/cm. The near-term goal is a measurement of the atomic EDM with sensitivity of 1026 e-cm. From Table 2 it appears that 225Ra is about 600 times more sensitive to T-odd and P-odd nuclear interactions than 199Hg. More specifically, the isoscalar, isovector, and isotensor contributions given in Equation (11) for 225 Rn (Dobaczewski & Engel, 2005) and 199Hg (deJesus & Engel, 2005) using the SkO model are e-fm3 1CP þ 0:018g2CP Þ; S199Hg ¼ gNN ð0:01 g0CP þ 0:074g S225Ra ¼

gNN ð1:5 g0CP

þ

6:0 g1CP



4:0g2CP Þ:

ð33Þ ð34Þ

The uncertainty of the coefficients due to the nuclear-force models may be a factor of two; however, a measurement of the EDM of 225Rn at the level of 1026 e-cm would be comparable to the current 199Hg result. The red-detuned FORT is not species specific, in contrast to the Zeeman slower and MOT, so it will be possible to trap more than one species in the FORT to use as a comagnetometer. This would require an additional Zeeman slower and additional MOT lasers, but it will likely be essential in the long term. Alternatively, a different isotope of radium with no octupole deformation could be loaded into the FORT. Frequency shifts that arise in the FORT due to light shifts, particularly residual circular polarization, and due to parity mixing induced by the static electric field, may present challenges. The static field effects are linear in the magnitude of ~ E and could produce a false EDM. These have been worked out for mercury by Romalis and Fortson (1999), and certain orientations of the fields can mitigate the effects. Longer-term

164

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improvements will require more atoms, which may be produced by a more active source or by producing isotopes at an online facility accel­ erator facility such as TRIUMF or FRIB. Table 2 also shows significant potential enhancement of the 223Rn Schiff moment and atomic EDM. This enhancement is somewhat less certain than for 225Ra due to the lack of data on the nuclear level structure and complications in calculating the enhancements (Engel, 2010); how­ ever, the possibility of adapting techniques used for the xenon—helium comparison has led us to develop a program of nuclear structure mea­ surements and the development of techniques for a radon EDM experi­ ment. Due to the short half-life of 23 min, the experiment must be “online,” that is, interfaced to the production-facility beam line. In an experiment being set up at TRIUMF in Vancouver, Canada, the radon is produced by spallation from 500 MeV protons incident on a target of thorium or uranium. Radon evaporates from the target, is ionized, accel­ erated to 30 keV, and transported using magnetic and electrostatic optics to be imbedded in a foil target. After a collection interval of about two half-lives, the foil can be heated and the radon transferred to a measure­ ment cell by freezing to an intermediate cold-finger at liquid-nitrogen temperature and then carried into the cell with nitrogen transfer gas (Nuss-Warren et al., 2004). Once the radon is in the cell, the cell is closed off isolating the nitrogen-radon mixture. The cell also contains rubidium, which is optically pumped, and the radon is polarized by spin exchange. The nitrogen serves several purposes: it is the transfer gas, it assists optical pumping by suppressing radiation trapping (Chupp & Coulter, 1985), and allows higher electric fields for the EDM measurement. Polarization studies with 209Rn (Kitano et al., 1988; Tardiff et al., 2008) and 223Rn (Kitano et al., 1988) have shown that gamma-ray anisotropies can be used to monitor the nuclear polarization. Figure 8 shows data on the temperature dependence of the polarization measured by monitoring the angular dependence of the 337 keV gamma-ray emitted when 209Rn (t1/2= 28.5 m) decays to 209At. The data were taken at the Stony Brook Nuclear Structure Lab with the radon collection and optical pumping apparatus installed on the beam line developed for the original francium trapping studies (Gomez et al., 2006). The isotope 209Fr (t1/2=50 s) was produced by the reaction 16O þ 197Au ! 209Fr þ 4n with an 16O beam energy of about 91 MeV. The francium in the gold target was evaporated and surface ionized, accelerated to 5 keV and transported to the collection foil. Collection times of about 1 h allowed the 209Rn from 209Fr decay to build up to near the maximum. The foil was isolated and heated to drive off the radon (Warner et al., 2005), which was then transferred to the optical pumping cell. The temperature dependence of the radon nuclear polarization shown in Figure 8 illustrates the combination of several competing processes compiled in the rate equation:

Permanent EDMs of Atoms and Molecules

165

Uncoated cell 1.05

R

1 0.95 0.9

Coated cell 1.05

R

1 0.95 0.9 140

160 180 200 Cell temperature (°C)

220

Figure 8 Temperature dependence of the polarization of 209Rn produced by spin exchange with optically pumped rubidium. R is the laser-on/laser-off ratio of 0/90 ratio of count rates for the 337 keV gamma ray from decay of the an excited state in 209 At populated by decay of polarized 209Rn

dPRn PRn ¼ kSE ½RbðPRb  PRn Þ  ; dt T1

ð35Þ

where PRb is the rubidium polarization, [Rb] is the rubidium number density, T1 is the polarization relaxation time, and kSE = hSEvi is the rate constant for spin-exchange. The cross section has been calculated by Walker (1989) with the result SE = 2.5  1021 cm2 so that kSE  9  1017 cm2/s for binary collisions. The equilibrium radon polarization is PRn ¼

kSE ½Rb PRb : kSE ½Rb þ 1=T1

ð36Þ

The factors PRb, [Rb], and T1 are all strongly temperature dependent. The rubidium density increases exponentially with temperature by a factor of about 1.6 for every 10C, and [Rb]  1015 cm3 at 200C; the rubidium polarization produced by laser optical pumping decreases at high tem­ perature due to electron-spin relaxation processes for Rb—Rb and Rb—N2 collisions; and the relaxation rate depends on the residence time of atoms at the wall, which is expected to decrease with increasing temperature: 1 ¼ Gð1ÞeT0 = T ; T1

ð37Þ

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where G(1) is the infinite temperature rate and kT0 is a typical binding energy of atoms at the wall. These depend on the wall material and coating. Cells without and with coating were investigated. The coated cell used OTS or octadecyltrichlorosilane, a coating developed for our xenon—helium EDM experiment (Rosenberry & Chupp, 2001). By using Walker’s spin-exchange cross section, the data could be interpreted to estimate T1  30 s (Tardiff et al., 2008), which is also the limit on T2. With N = 1010 atoms, 108 decays could be observed by monitoring the betaasymmetry in each 30 s measurement; with E = 10 kV/cm, an EDM sensitivity of d  2  1024 e-cm is possible with each measurement, and 1026 e-cm could be achieved in about 2 weeks. With the Schiff moment enhanced by 800 or more, this would also be comparable to the sensitivity of 199Hg. The spin-exchange optical pumping technique offers the ability to polarize any odd-A noble gas species, and thus one or more comagnetometer species could be monitored in the measurement cell and discriminated by precession frequency or nuclear transition. More than one species may be useful to monitor effects of higher order than magnetic dipole. It is also attractive to consider the possibility of a laser experiment that would directly optically pump and probe the radon and take advantage of higher production rates that may be avail­ able at future facilities such as FRIB. A single-photon transition from the ground state would require 178 nm light, which is currently not practi­ cal. Two-photon techniques are also possible and currently being explored. Finally we note that with nuclear spin  1, nuclear moments of higher order, including a T-odd, P-odd magnetic quadrupole moment, could induce an atomic EDM. An interesting hybrid of rare-atom and molecular beam techniques has been suggested using, for example, RaO. In such a molecule, the large internal field along the interatomic axis combined with the enhanced Schiff moment, for example 225Ra, leads to an estimated sensi­ tivity to T-odd/P-odd interactions 500 times greater than TIF (Flambaum, 2008). From Equations (9) and (14), the Schiff-moment contributions to the EDMs are dSTlF ¼ 7:4  10  14 STl cm=fm3 and dSHg ¼ 2:8  10  17 STl cm=fm3 . The sensitivity of the most recent TlF measurements was d = 3  1023 e-cm (Cho et al., 1991). A molecular EDM measurement in RaO at this level would be 10 times more sensitive to T-odd and P-odd NN interactions compared to the 199Hg measurement. Experiments with rare-radioactive atoms, francium and radium, and isotopes, 223Ra and 225Rn, provide significant new technical challenges and significantly lower statistical sensitivity than stable-atom experi­ ments. Several approaches are motivated by enhancements of two or more orders of magnitude, which would balance the loss of four or more orders of magnitude in detection rate. These experiments represent

Permanent EDMs of Atoms and Molecules

167

new applications of cold-atom techniques and spin-exchange optical pumping that may have broader applications. 5.5 Other Directions We finally mention two other directions that are not strictly atomic or molecular EDM measurements, but are important to the landscape that lies ahead: solid-state systems, in which the EDMs of a sea of electrons is measured through the induced electric or magnetic effects, and ion sto­ rage rings. In solid-state systems, the EDM of the unpaired electrons is detectable either through the magnetic field produced when the electron EDMs are EÞ or through the electric aligned by the strong internal electric field ð~ Bind ~ field induced when the electron magnetic moments are polarized by a ! BÞ (Buhmann, 2002; Liu & Lamoreaux, 2004; strong magnetic field ð E ind ~ Shapiro, 1968; Sushkov et al., 2009, 2010). For example, in PbTiO3, a ferro­ electric crystal, sensitivity to the electron EDM is enhanced due to the large number of electrons in the solid and due to the strong internal electric field in a cooled crystal (Mukhamedjanov & Sushkov, 2005). A similar measure­ ment in gadolinium—gallium garnet is underway (Liu & Lamoreaux, 2004). Another approach using ferromagnetic gadolinium—iron garnet would detect the electric field produced by the electron EDMs aligned with the magnetically polarized spins (Heidenreich et al., 2005). A charged particle EDM is defined as the displacement of the center of charge with respect to the center of mass, and can be detected for ions contained by electric and magnetic fields, for example, in a storage ring similar to that used for the muon magnetic-moment and EDM measurements (Bennett et al., 2009). A charged particle in a storage ring is guided by the magnetic field normal to the plane of the ring and addi­ tional electromagnetic fields to constrain the particle. For a particle with a magnetic moment aligned with the momentum at some time, the spin will precess with respect to the momentum in the plane of the ring at a rate that depends on the anomalous magnetic moment and the velocity. An EDM will also lead to a torque that causes spin precession that is out-of-phase with the magnetic moment precession and leads to a spin component that is perpendicular to the plane of the ring. Thus, if the spin direction is measured by detectors that distinguish alignment above the plane of the ring from alignment below the plane of the ring, the anomalous magnetic moment would be measured by the average of the up-down detectors, and the EDM would be measured by the difference of the up-down detectors. A similar approach is proposed for a deuteron EDM measurement (Orlov et al., 2006). The deuteron measurement would measure the EDM of the bare nucleus and thus the Schiff shielding of external electric fields by atomic electrons is not a factor.

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6. CONCLUSION The search for EDMs has been underway since the first neutron measure­ ment over 50 years ago. The potential for discovery of an EDM in any system is strong motivation to continue, but in order to fully parameter­ ize P-odd and T-odd physics and thus explore CP violation, experiments in several different systems will be necessary. Unfortunately, high-energy theory is not a guide to how sensitive future experiments need to be, but theory does provide strong motivation to expect new sources of CP violation along with the Standard Model parameter
QCD. Several new experimental approaches in several new systems are under­ way that may push the limits currently set by the neutron, 199Hg, and thallium. We like to say that there are many orders of magnitude to explore before we reach the Standard Model predictions for EDMs, and thus there are many orders of magnitude of opportunity to discover new physics; however, we have learned a great deal by constraining new physics with the limits already set. It is tempting but inappropriate to anticipate where the break­ throughs will be
but it is safe to expect that the next 50 years might bring the discovery of the first EDM and an era of precision measurements that will set the parameters of CP violation in the Standard Model and beyond.

ACKNOWLEDGMENTS Many have helped in developing the perspectives, the ideas, and the details presented here. The many collaborators as well as colleagues actively working on EDM experiments and related theory who have provided input and updates on their efforts for this chapter and recent review talks include Naftali Auerbach, John Behr, Dave deMille, Jon Engel, Peter Fierlinger, Norval Fortson, Clark Griffith, Ed Hinds, Klaus Jungmann, Gordon Kane, Aaron Leanhardt, Wolfgang Lorenzon, Z.T. Lu, Matt Pearson, Aaron Pierce, Michael Ramsey-Musolf, Michael Romalis, Gene Sprouse, Carl Svensson, and Eric Tardiff.

REFERENCES Abel, S., Khalil, S., Lebedev, O. (2002). Additional stringy sources for electric dipole moments. Physical Review Letters, 89, 121601. Altarev, I. S., Borisov, Y. V., Borovikova, N. V., Brandin, A. B., Egorov, A. I., Ezhov, V. F., et al. (1981). A new upper limit on the electric dipole moment of the neutron. Physics Letters B, 102, 13—16. Altarev, I. S., Borisov, Y. V., Brandin, A. B., Egorov, A. I., Ezhov, V. F., Ivanov, S. N., et al. (1980). A search for the electric dipole moment of the neutron using ultracold neutrons. Nuclear Physics A, 341, 269—283. Amini, J.M., Munger Jr, C.T., & Gould, H. (2006). Demonstration of a cold atom fountain electron electric dipole moment experiment. arXiv/physics, 0602011.

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CHAPTER

5

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation Paul R. Berman and George W. Ford Michigan Center for Theoretical Physics, Physics Department, University of Michigan, 450 Church Street, Ann Arbor, MI 48109-1040, USA

Contents

Introduction Excited State Time Evolution 2.1 General Solution 2.2 Spectral Weight Function F1(!k) 2.3 Spectral Weight Function F2(!k) 2.4 Spectral Weight Function F3(!k) 2.5 Spectral Weight Function F4(!k) 2.6 Spectral Weight Function F5(!k) 2.7 Spectral Weight Function F6(!k) 3. Spectrum and Unitarity 3.1 Spectral Weight Functions F1(!k) and F2(!k) 3.2 Spectral Weight Functions F3(!k) and F4(!k) 3.3 Spectral Weight Functions F5(!k) and F6(!k) 4. Discussion Acknowledgments References

Abstract

The theory of spontaneous emission presented by Weisskopf

and Wigner [Weisskopf, V., & Wigner, E. (1930) Z. Phyz. 63,

54-73] provides an excellent approximation to the actual decay

atoms undergo on optically allowed transitions. However, the

theory cannot be rigorously correct since, when applied in a

consistent fashion that maintains unitarity, it requires that

negative frequencies be permitted in the emitted spectrum. To

avoid such problems, a better treatment is needed. Starting

1. 2.

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with a model Hamiltonian that approximates the decay of a discrete state into the continuum within the rotating-wave approximation, we show explicitly that unitarity is respected for different classes of spectral densities of the vacuum field, physical or not. When the spectral density is bounded from below or above, there is always deviation from exponential decay. In this limit, it is possible for there to be some nonzero probability for the discrete state amplitude to be nonvanishing as the time goes to infinity, even if the spectral density overlaps the transition frequency. The discrete state amplitude is expressed in terms of a finite integral that can be evaluated numerically plus contributions from at most two poles on the imaginary axis that result in terms that do not decay with time. Although the time evolution of the excited state amplitude must be calculated numerically, analytic expressions can be obtained for the spectrum of the spontaneously emitted radiation.

1. INTRODUCTION The problem of spontaneous emission plays a central role in atomic phy­ sics. To a very good approximation, the probability for an isolated atom to remain in an excited state decays exponentially in time. Moreover, the spectrum emitted by an atom in decaying from its first excited state to its ground state is approximately a Lorentzian centered at the atomic transi­ tion frequency. In fact, this characterization of atomic decay is exactly that predicted in the landmark paper by Weisskopf and Wigner (1930). To arrive at their results, Weisskopf and Wigner made two critical assumptions. They assumed that the density of states for the vacuum radiation field could be evaluated at the atomic transition frequency and they extended the fre­ quency integration over vacuum field modes to minus infinity in calculat­ ing the excited state transition amplitude. The Weisskopf-Wigner approximation works exceptionally well. As far as we know, there are no experimental results on isolated atoms in free space for which deviation from exponential decay has been detected. Despite its success, however, a theory based on the Weisskopf-Wigner approximation cannot be rigor­ ously correct since it admits negative frequencies in the spontaneous emis­ sion spectrum. In this chapter we examine the problem of spontaneous emission without invoking the Weisskopf-Wigner approximation, provid­ ing a unified treatment of both the decay of the initial state amplitude and the spectrum of the emitted radiation. An atom in an excited state decays as a result of its interaction with the vacuum field. To describe this interaction, we must solve coupled ampli­ tude equations of the form

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rffiffiffiffiffi Z1  b_ 2 ¼ i d!k f ð!k Þe  i ð!k  !0 Þ t b1k ; 

ð1aÞ

rffiffiffiffiffi  b_ 1k ¼ i f ð!k Þei ð!k  !0 Þ t b2 ; 

ð1bÞ

1

where b2 is the probability amplitude for the atom to be in excited state |2i with no photons in the radiation field, b1k is the probability amplitude for the atom to be in its ground state |1i with a photon having frequency !k = kc in the field, 2 is the excited state decay rate in a Markovian limit, !0 is the atomic transition frequency, and f(!k) is a dimensionless real function that reflects the frequency dependence of the density of states of the vacuum radiation field. The rotating-wave approximation (RWA) is implicit in Equa­ tions (1a) and (1b), since terms involving the simultaneous emission of a photon and excitation of the atom have been neglected. Equations (1a) and (1b) are, in some sense, generic equations that correspond to the decay of a discrete state into a continuum. All angular averages have been absorbed into these equations and the transition from a sum over discrete vacuum field modes has been converted to an integral over these modes. For example, if the atom-field interaction in dipole approximation is taken to be ep  A/m (e is the magnitude of the electron charge, p is the momentum operator for the electron of a one-electron atom, m is the electron mass, and A is the vector potential of the vacuum radiation field), then f ð!k Þ ¼

 pffiffiffiffiffiffiffiffiffiffiffiffiffi !k =!0 0

!k  0 : !k < 0

ð2Þ

If the atom-field interaction is taken to be er  E (r is the position operator of the electron in the atom and E is the vacuum electric field), then 

f ð!k Þ ¼

ð!k = !0 Þ 3 = 2 0

!k  0 !k < 0

ð3Þ

Equations (1a) and (1b) are written in an interaction representation, with the state vector given by jcðtÞi ¼ b2 ðtÞe  i!0 t j2; 0iþ

Z1

d!k b1k ðtÞe  i!k t j1; !k i;

ð4Þ

1

where the ket |2; 0i corresponds to the atom in its excited state with no photons in the field and ket |1; !ki to the atom in its ground state and a photon of frequency !k in the field. The ket normalization is such that h1; ! k0 j1; !k i¼ð!k  ! k0 Þ while the initial condition is b2 ð0Þ ¼ 1; bflkgðtÞ ¼ 0;

ð5Þ

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that is, the atom is prepared in its excited state. The goal is to calculate the subsequent time evolution of the excited state, as well as the state ampli­ tude b1k(t). The quantity Sð!k Þ ¼ lim jb1k ðtÞj2

ð6Þ

t!1

represents the spectrum of the spontaneously emitted radiation. Note that |b1k(t)|2 is actually a probability density, having units of !1 k . It is easy to show from Equations (1a) and (1b) that 0 d@jb2 ðtÞj þ

Z1

2

1 jb1k ðtÞj d!k A 2

1

¼ 0;

dt

ð7Þ

which is a statement of conservation of probability. Thus, the equations are unitary, since, in fact, they can be derived from a Hermitian Hamiltonian, Z1 H ¼ h!0 j2; 0ih2; 0jþ h

d!k !k j1; !k ih1; !k j 1

rffiffiffiffiffi Z1  þ h d!k f ð!k Þ½j2; 0ih1; !k jþj1; !k ih2; 0j: 

ð8Þ

1

However, if one is not careful, one can run into inconsistencies in dealing with Equations (1a) and (1b). For example, in the Weisskopf-Wigner approximation, the excited state amplitude decays exponentially, b2 ðtÞ ¼ e  t :

ð9Þ

If this result is substituted into Equation (1b), we find rffiffiffiffiffi Z1  b1k ð1Þ ¼ i f ð!k Þ dt ei ð!k  !0 Þ t b2 ðtÞ  0

rffiffiffiffiffi  f ð!k Þ ¼ i :    ið!k  !0 Þ

ð10Þ

As a consequence, Sð!k Þ ¼ jb1k ð1Þj2 ¼

 Fð!k Þ ;   2 þ ð!k  !0 Þ 2

ð11Þ

where Fð!k Þ ¼ ½f ð!k Þ 2

ð12Þ

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

179

will be referred to as the spectral weight function associated with the atom-field interaction. For f(!k) given by Equation (2) or (3), the integral Z1 S¼

Z1 d!k jb1k ðtÞj2

d!k Sð!k Þ ¼ lim 1

t!1

ð13Þ

1

diverges, a result that clearly violates unitarity. Even if a cutoff is intro­ duced into Equation (13), unitarity is still violated since S is not equal to unity. This result implies that, for the density of states associated with Equa­ tions (2) and (3), the decay cannot be purely exponential. In other words, if the correct expression for b2(t) is used in Equation (10), Equation (7) must hold. The fact that the decay is not exponential is well known (Knight & Milonni, 1976; Seke & Herfort, 1989). Approximate expressions for the excited state amplitude that have been derived indicate that there are deviations from exponential decay around t=0 and for t >> 1. In general, however, these treatments do not give an expression that is valid for all time, nor do they examine the final spectrum to see if unitarity is respected. Thus, it may prove instructive to examine Equations (1a) and (1b) for different classes of spectral weight functions F(!k) to see how they affect both the excited state decay and the emitted spectrum. We would like to stress that Equations (1a) and (1b) may not correspond to physical reality for specific choices of F(!k). For example, in the RWA, it is known (Ber­ man, 2004; Milonni et al., 2004) that Equations (1a) and (1b) lead to acausal emission. That is, the signal is nonvanishing at any distance R from the atom for t > 0. Generalizing the results to include non-RWA terms (Berman, 2004; Milonni et al., 2004) removes this embarrassment and leads to a nonvanishing signal only for R £ ct. Spectral densities of the form (2) and (3) are also know to lead to divergences in level shifts that depend at least linearly on an imposed cutoff frequency. The cutoff appears naturally if the dipole approximation is not made and retarda­ tion effects are included. In addition, there are problems associated with the assumption that the atom is excited instantaneously at t=0. A more realistic excitation protocol leads to a natural cutoff in F(!k) that is of the order of inverse of the rise time of the excitation pulse (Berman, 2005). All of this is very interesting, but is not of concern here. What we want to show explicitly is that Equations (1a) and (1b) lead to unitary results for different classes of spectral weight functions F(!k) and to examine the nature of the excited state evolution and the emitted spectrum for different F(!k). In doing so, we explain why the Weisskopf-Wigner approximation leads to unitary results when applied in a consistent fashion. We are also able to follow the transition from near-Markovian behavior when the spectral range F(!k) is a broad function that overlaps

180

Paul R. Berman and George W. Ford

the transition frequency !0 to no decay at all when the spectral function does not overlap !0. It turns out that, even when F(!k) completely encom­ pass !0, there can be a small, but finite, probability to remain in state 2 as t  1 for specific choices of F(!k). Moreover, for very long times, the asymptotic form for the excited state amplitude can exhibit somewhat unusual behavior, in that it does not vary solely as an inverse power of the time (Knight & Milonni, 1976; Seke & Herfort, 1989). While the time evolution of the excited state amplitude must be calculated numerically in most cases, it turns out, rather remarkably, that it is possible to obtain a very general analytic expression for the emitted spectrum. It should be noted that the transition from exponential decay to Rabi oscillations (no decay) was studied by Cohen-Tannoudji and Avan (1977) as a function of coupling strength (vacuum-induced level shifts) using graphical methods (Cohen-Tannoudji et al., 1994). To illustrate the physical concepts, we consider the following spectral weight functions: F1 ð!k Þ ¼ 1 F2 ð!k Þ ¼

!2w ð!k  !0 Þ 2 þ !2w 

F3 ð!k Þ ¼ 

ð14bÞ

1 0  !k  !c 0 otherwise

exp½ð!k  !0 Þ=!c  !k 0 !k  !k =!k 0  !k  0 F5 ð!k Þ ¼ otherwise 0  ð!k =!0 Þexp½ð!k  !0 Þ=!c  F6 ð!k Þ ¼ 0 F4 ð!k Þ ¼

ð14aÞ

ð14cÞ 0 0 is a cutoff frequency and !w > 0 is the frequency width of the Lorentzian spectral weight function. The spectral weight function F1(!k) corresponds to the Weiss­ kopf-Wigner (1930) approximation and leads to pure, simple exponen­ tial decay. Spectral weight function F2(!k) is chosen to illustrate that the early time dependence of the excited state amplitude has quadratic time Z 1 dependence whenever Fð!k Þd!k exists. Moreover, it allows us to see 1

how the emitted spectrum can split into a doublet for  < !0. With F3(!k),  we move to spectral weight functions that are identically zero for !k < 0, a necessary condition when one considers spontaneous decay. For such spectral weight functions, one must include a cutoff at large !k to avoid

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

181

infinite level shifts. We shall see that the sharp cutoffs at !k=0 and !k=!c in F3(!k) lead to small, but finite, contributions to b2(1) even if !c >> !0 >> . The contribution to b2(1) from the cutoff at !k=!c is removed when the high frequency cutoff is made smoothly, as with the spectral weight function F4(!k). The function F5(!k) is chosen to simulate the ep  A/m interaction. The linear dependence on !k in F5(!k) eliminates the contribu­ tion to b2(1) from the region about !k=0, provided the line shift is less than !0. The line shift in this case has a component that depends linearly on !c and allows us to examine a type of phase transition that occurs when the line shift is equal to !0. The spectral weight function F6(!k) corresponds to the ep  A/m interaction with a smooth cutoff at (!k - !0  !c) and removes all contributions to b2(1), provided the line shift is less than !0. For F4(!k) and F6(!k), it is assumed that !c >> !0. All the spectral weight functions are chosen such that F(!0) = 1. Any contributions to b2(1) are extremely small when !c >> !0 >> , provided the shifts are less than !0. When such contributions are neglected, one can obtain the excited state amplitude in the form of a single integral that can be evaluated numerically. The integral form for the excited state amplitude enables one to obtain asymptotic expressions valid for large t. In Section 2, we derive expressions for the excited state amplitude for each of the spectral densities (14). In Section 3, we calculate jb1k ðtÞj2 , as well as the spectrum S(!k) of the emitted radiation. It is shown explicitly that the conservation of probability, Equation (7), holds for each of the spectral densities given in Equations (14). A summary and discus­ sion of the results is given in Section 4.

2. EXCITED STATE TIME EVOLUTION Integrating Equation (1a) formally and substituting the result back into Equation (1b), we find  b_ 2 ¼  

Zt 0

dt 0

Z1

0

d!k Fð!k Þe  i ð!k  !0 Þ ðt  t Þ b2 ðt 0Þ:

ð15Þ

1

Before examining the specific forms of F(!k), we derive some general results. If we assume that Z1

F¼

d!k Fð!k Þ

ð16Þ

1

and 1 !¼ F

Z1 d!k ð!k  !0 ÞFð!k Þ 1

ð17Þ

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Paul R. Berman and George W. Ford

exist, and that D >  is some characteristic maximum value of |!k - !0| that contributes in Equation (15), then for t 1; for even longer times, this dependence can be modified by a factor that is a logarithmic function of !ct. It is possible to get an approximate solution that is valid when the spectral function does not overlap the transition frequency. If we assume 6 0 for 0 £ !k £ !c < !0 and that both !0/ >> 1 and that F(!k) ¼ ð!0  !c Þ= >> 1, we can solve for Z!c jb2 ðtÞj2 ¼ 1 

jb1k ðtÞj2 d!k

ð22Þ

0

using a perturbation theory result for b1k(t), since jb1k ðtÞj > 1. The behavior corresponds to inhomogeneously broadened, off-resonant driving of an atom-field system. A graph of the solution is presented later in this section. 2.1 General Solution Equation (15) can be solved by Laplace transform techniques. With Z1 BðsÞ ¼

e  st b2 ðtÞdt

ð26Þ

1 ; s þ GðsÞ

ð27Þ

0

and b2(0) = 1, one finds BðsÞ ¼

where GðsÞ ¼

¼

   

Z1

Z1 d

d!k Fð!k Þe  i ð!k  !0 Þ  e  s

1

0

Z1 d!k 1

Fð!k Þ : s þ ið!k  !0 Þ

ð28Þ

On taking the inverse transform of Equation (27), one arrives at the formal solution b2 ðtÞ ¼

1 2i

þ Z i1

  i1

est ds ; s þ GðsÞ

ð29Þ

where  is a small positive frequency. It is possible to derive some very general results without specifying the exact form of the spectral weight function. Let us suppose that Fð!k Þ ¼ 0;

!k < !a and !k > !b ;

ð30Þ

and that F(!) is an analytic function for !a < ! < !b. If need be, the limits !a  -1, !b  1 can be taken. With a simple change of variable, Equation (29) can be written as b2 ðtÞ ¼

ei!0 t 2i

þ Z i1

  i1

est ds ; s þ i!0 þ GðsÞ

ð31Þ

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Paul R. Berman and George W. Ford

where  GðsÞ ¼ Gðs þ i!0 Þ ¼ 

Z!b d! !a

Fð!Þ : s þ i!

ð32Þ

In this form, it is apparent that G(s) has a branch cut along the imaginary axis between s = -i!b and s = -i!a. To prove this, consider values of s equal to +x þ iy in the limit that x  0 and evaluate Gd ðyÞ ¼ lim ½Gðx þ iyÞ  Gðx þ iyÞ x!0

¼

 lim  x!0

Z!b d! !a

2xFð!Þ ð! þ y Þ 2 þ x2

Z!b ¼ 2

d!Fð!Þð! þ yÞ;

ð33Þ

!a

where (!) is a Dirac delta function. If Gd(y)  0 the point s = iy is not on the branch cut, but if Gd(y) is finite, the point s = iy is on the branch cut since the value of the function changes as we move across the imaginary axis. Clearly, it follows from Equation (33) that the branch cut exists for values of y between -!b and -!a or, equivalently, between s = -i!b and s = -i!a. To evaluate the inverse Laplace transform (31), one can choose a contour that excludes the branch cut such as the one shown in Figure 1. Using the residue theorem and the fact that the contributions along all the curved segments of the contour go to zero, one then evaluates

– ωa

– ωb

Figure 1 Bromwich contour for the inverse Laplace transform. The full vertical line is displaced by a positive frequency s from the y-axis and the vertical lines around the branch cut are displaced by –", where " is a positive frequency. Ultimately, the limit is taken in which " and s tend to zero. There are at most two poles in the contour, above and below the branch cut, on the imaginary axis. For physically acceptable spectral weight functions such as F3(w k)-F6(w k), w a = 0 since emission must occur at positive frequencies

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation  Z!a

ei!0 t 2

b2 ðtÞ þ

 !b

ei!0 t þ 2 ¼

X

185

eiyt dy iy þ i!0 þ Gð" þ iyÞ

 Z!b

 !a

eiyt dy iy þ i!0 þ Gð" þ iyÞ ð34Þ

Rðp; tÞ;

p

where the sum is over all the residues R(p,t) of the function Wðs; tÞ ¼

ei!0 t est s þ i!0 þ GðsÞ

ð35Þ

at the poles p contained in the contour, and " is a small positive frequency. Using Equation (33), we transform this equation into  Z!a ei!0 t 2FðyÞeiyt dy lim b2 ðtÞ ¼ ½y þ !0  iGð" þ iyÞ½y þ !0  iGð" þ iyÞ 2 "!0  !b X þ Rðp; tÞ:

ð36Þ

p

It remains only to find the poles in the contour shown in Figure 1. It is not too difficult to prove that there are at most two poles and that these poles are located on the imaginary axis. The poles are found as solutions of the equation  s þ i!0 þ GðsÞ ¼ s þ i!0 þ 

Z!b d! !a

Fð!Þ ¼ 0; s þ i!

ð37Þ

with s = x þ iy. Setting the real part of this equation equal to zero, we find  xþ 

Z!b d! !a

xFð!Þ ¼ 0: x2 þ ð! þ y Þ 2

ð38Þ

Since F(!) is a positive analytic function in the range of integration, the only solution of this equation can be x=0; any poles must be on the imaginary axis. Setting x = 0 and taking the imaginary part of Equation (37), we find that the poles exist inside the contour provided y þ !0 ¼

 

Z!b d! !a

Fð!Þ : !þy

ð39Þ

Solutions of Equation (39) must be restricted to values of y >!a and/or y > -!b since points on the branch cut -!b £ y £ -!a are not contained in the

186

Paul R. Berman and George W. Ford

contour. Thus, there can be poles on the imaginary axis that lie outside the branch cut and inside the integration contour shown in Figure 1. Whether the poles exist depend on the specific form of F(!); however, there is a maximum of one pole for y>!a and one pole for y > -!b as can be verified by a simple graphical analysis, given the fact that F(!) > 0. As a consequence, Equation (36) can be rewritten as ei!0 t lim b2 ðtÞ ¼ 2 "!0

 Z!a

 !b

2FðyÞeiyt dy ½y þ !0  iGð" þ iyÞ½y þ !0  iGð" þ iyÞ

2 X þ Rðyj ; tÞ;

ð40Þ

j¼1

where Rðyj ; tÞ ¼ rj ei!0 t eiyj t ;

ð41Þ

[a result that follows from Equation (35)], yj is a solution of  yj þ !0  

and

Z!b d! !a

Fð!Þ ¼ 0; ! þ yj

2 31 Z!b h i1  Fð!Þ 5 >0 ¼ 41 þ d! rj ¼ 1 þ dGðsÞ=dsjs ¼ iyj  ð! þ yj Þ 2

ð42Þ

ð43Þ

!a

is the magnitude of the residue associated with the pole at yj. Equation (40) is pretty remarkable. From the Riemann-Lebesgue lemma (Whittaker & Watson, 1927), the integral term vanishes as t  1. However, the residue terms always give rise to a finite contribution to b2(1), independent of the value of !0, lim b2 ðtÞ ¼

t!1

2 X

rj ei!0 t eiyj t :

ð44Þ

j¼1

As long as the spectral weight function is bounded from above or below, b2(1) does not necessarily go to zero. Similar conclusions have been reached in looking at the decay of a discrete state into a bounded con­ tinuum (Longhi, 2007; Miyamoto, 2005). If !a  -1, !b  1 (as in the Weisskopf-Wigner approximation), the residues from the poles asympto­ tically go to zero and b2(1) is exactly equal to zero. Even for finite !a and infinite !b it is possible that b2(1) = 0, provided that the spectral function does not admit poles for y > -!a, as is the case for the spectral function given in Equation (14f). Equation (40) is a central result of this chapter. Although the residues contribute to b2(1), the contributions are negli­ gibly small provided that (1) !0/ >> 1, (2) the spectral function has a range

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

187

D!k >> , and (3) the transition frequency, including any level shifts, is somewhere in the central region of D!k; in other words, the overlap of the spectral function and the transition frequency is significant. On the other hand, at least one of the poles can contribute dominantly if the spectral function does not overlap the transition frequency. We now proceed to examine the specific spectral weight functions given in Equation (14). 2.2 Spectral Weight Function F1(w k) In this and Section 2.3, it is possible to use Equation (40) to obtain b2(t) by setting !a = -!c, !b = !c, and taking the limit that !c  1. However, for the spectral weight functions (14a) and (14b) it is much simpler to solve Equation (15) directly. This is the method we use. If F1(!k) = 1, it follows from Equation (15) that  b_ 2 ðtÞ ¼  

Zt

dt 0

¼2

0

d!k e  i ð!k  !0 Þ ðt  t Þ b2 ðt 0Þ

1

0

Zt

Z1

dt 0ðt  t 0Þb2 ðt 0Þ ¼ b2 ðtÞ

ð45Þ

0

and b2 ðtÞ ¼ e  t :

ð46Þ

The decay is purely exponential. There is no shift of the transition fre­ quency, owing to the fact that the (nonphysical) spectral weight function F1(!k) = 1(1 < !k < 1) is, in effect, symmetric about !k = !0. Since F1(!k) = 1 corresponds to the Weisskopf-Wigner spectral weight function, we see that the Weisskopf-Wigner approximation leads to purely expo­ nential decay. Moreover, we will see below that, when applied in a consistent fashion, the Weisskopf-Wigner approximation maintains uni­ on t for t0, but tarity. The amplitude b2(t) does not depend quadratically Z a quadratic dependence is guaranteed only if is not the case for F1(!k).

1

1

d!k F1 ð!k Þ exists, which

2.3 Spectral Weight Function F2(w k) Again, it is simplest and most instructive to use Equation (15) to evaluate b2(t). Combining Equations (14b) and (15), we find !w b_ 2 ðtÞ ¼   2

Zt 0

dt 0

Z1

0

d!k 1

e  i ð!k  !0 Þ ðt  t Þ b2 ðt 0Þ ð!k  !0 Þ 2 þ !2w

188

Paul R. Berman and George W. Ford

Zt

0

dt 0e  !w ðt  t Þ b2 ðt 0Þ;

ð47Þ

€b2 ðtÞ ¼ !w b_ 2 ðtÞ  !w b2 ðtÞ:

ð48Þ

¼!w 0

from which it follows that



The initial conditions are b2(0)=1; b2(0)=0. Equation (48) is recognized as the differential equation of a damped harmonic oscillator. As a consequence the decay can be classified as overdamped if !w > 4, underdamped if !w < 4 and critically damped if !w ¼ 4. In all cases, the solution varies as b2 ðtÞ  1 

!w 2 t 2

near t=0, in agreement with Equation (21), since F2 ¼

ð49Þ Z 1 1

d!k F2 ð!k Þ ¼ !w .

The solution of Equation (48) subject to the initial conditions b2(0) = 1; b_ 2 ð0Þ ¼ 0 0

1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " u B 1B 1 4 u 1 C C b2 ðtÞ ¼ B1 þ u Cexp  !w t 1  1  t 4 @ A 2 2 !w 1 !w 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " u C 1 4 1B u 1 C B : þ B1  u Cexp  !w t 1 þ 1  t 4 A 2 !w 2@ 1 !w

ð50Þ

For !w >> 4; b2 ðtÞ&e  t, for !w ¼ 4; b2 ðtÞ ¼ e  2t ð1 þ 2tÞ, and for !w Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

189

with the atomic transition frequency, the excited state amplitude decays to zero for sufficiently long times. It is the overlap of the tail of the distribution with the atomic transition frequency in this limit that allows population to “leak” back to the ground state. 2.4 Spectral Weight Function F3(w k) We now consider the spectral weight function, 

F3 ð!k Þ ¼

1 0  !k  !c ; 0 otherwise

ð53Þ

that vanishes identically for !k < 0. Any physically acceptable spectral function must vanish for !k < 0. Since the spectral weight function given in Equation (53) leads to many features that are characteristic of decay of a discrete state into a bounded continuum, we analyze it in great detail. The early time behavior is given by Equation (21), b2 ðtÞ  1 

since

F3 ¼

Z ! c 0

!c 2 t ; 2

ð54Þ

d!k F3 ð!k Þ ¼ !c . Equation (54) is valid for times

j!c  !0 jt; !0 t 1.3, weight function F3(!k) if 0< !00 /  0 6 0 if !c =!0 < 1:3 (when the upper bound of the spectral weight and b2(1) ¼  function approaches the transition frequency). On the other hand, for !00 < 0 we enter a qualitatively new regime in which the effective transition frequency is negative. In this case, the spectral weight function can never overlap the transition frequency and the excited state amplitude b2(1)6¼0. The excited state amplitude is shown in Figure 7 for !0/=10 and |b2(t )| 1.0

0.9

0.8

0

0.25

0.5

0.75

1.0

γt

Figure 7 Excited state amplitude |b2(t)| for the spectral weight function F5(w k) and w 0 / g =10,w c / g =400(w 00 =2.73)

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

203

!c = ¼ 400ð!00 = ¼ 2:73Þ, exhibiting behavior similar to that shown in Figure 5d. The dominant pole is at y1=2.35 and the magnitude of the residue at this pole is 0.883. There is something like a phase transition that occurs for fixed != < 10 as !c is varied around !c=!p, where  !p ¼ !20 =:

ð106Þ

The value !c=!p corresponds to !00 ¼0. For 2!0 < !c < ! , there is no pole  p for y1 > 0 and the pole with y2 < !c contributes negligibly, b2 ð1Þ& 0. On the other hand, for !c > !p, there is a pole for y1 > 0 that contributes significantly to the excited state population as t  1. When !c > !p one can use Equation (102) to estimate the pole position as y1&

where

!00 ; 1 þ ~ ln ð1  !c =!00 Þ

ð107Þ

 : !0

ð108Þ

~ ¼

  Equation (107) is valid provided ~ ln 1  !c =!00 > !p ; !00 &~ r1 ¼ jb2 ð1Þj

1  1 þ ~ ln 1 þ ~

1

;

ð110Þ

Note that this asymptotic result depends only on ~ and not !c. For !0/ =10 and !c > !0, the major difference of Equations (101) and (113) is in their asymptotic expansions. Since there is no endpoint contribution from y ¼ 1, the only endpoint contribution is from y=0 and the result analo­ gous to Equation (105) is (Knight & Milonni, 1976; Seke & Herfort, 1989) I1 ðtÞ  

ei!0 t e!0 = !c : !0 !0002 t2

ð116Þ

There is no logarithmic time dependence in this case, even in the limit of very long times.

3. SPECTRUM AND UNITARITY We now turn our attention to the spectrum of the emitted radiation. The probability to find the atom in state 1 with a photon having frequency !k in the field at time t is obtained from Equation (1b) as Sð!k ; tÞ ¼ jb1k ðtÞj2 ¼

t  Z 2  0 Fð!k Þ dt 0ei ð!k  !0 Þ t b2 ðt 0Þ : 

ð117Þ

0

At any time t, conservation of probability requires that Z1 jb2 ðtÞj2 þ

Sð!k ; tÞd!k ¼ jb2 ð0Þj2 ¼ 1:

ð118Þ

1

To obtain an analogous expression related to conservation of energy, we evaluate the expectation value of the Hamiltonian (8) for the state vector (4) and find

206

Paul R. Berman and George W. Ford

Z1 h!0 jb2 ðtÞj2 þ

h!k Sð!k ; tÞd!k þ 2 h 1

rffiffiffiffiffi Z1  Re f ð!k Þei ð!k  !0 Þ t b2 ðtÞb1k ðtÞd!k  1

2

¼ h!0 jb2 ð0Þj ¼ h!0 :

ð119Þ

The third term on the left-hand side of the equation is the expectation value of the atom-field interaction energy and is nonvanishing as t1 if b2(1)6¼0. We have been able to verify that Equations (118) and (119) are respected for each of the spectral weight functions considered in this work. In the case of the spectral weight functions F3 - F6, the calculations must be done numerically. Thus, there is no problem with either con­ servation of probability (unitarity) or conservation of energy if Equations (1a) and (1b) are solved without approximation. To examine the spectral distribution of the emitted radiation, it is necessary to evaluate Sð!k Þ ¼ lim Sð!k ; tÞ: t!1

ð120Þ

Only if b2 ð1Þ  0 does Z1 S¼

Sð!k Þd!k ¼ 1;

ð121Þ

1

otherwise S ¼ 1  jb2 ð1Þj2 : It is possible to obtain a simple analytic expression for S(!k). At first glance, it might seem that such an expression can be derived using  2 Z1     i ð!  ! Þ t Sð!k Þ ¼ Sð!k ; 1Þ ¼ Fð!k Þ dt e k 0 b2 ðtÞ ;   

ð122Þ

which, when combined with Equations (26), (27), and (32) yields  Fð!k ÞjB½ið!k  !0 Þj2     2  1   ; ¼ Fð!k Þ   !k  !0 þ iGði!k Þ 

Sð!k Þ ¼

ð123Þ

where G(s) is given by Equation (32). Equation (123) is valid only if there are no poles inside the contour shown in Figure 1. The problem in its derivation is that the limit t ! 1 was taken directly in the integral, rather than squaring the integral and then taking the limit. If the proper limiting process is carried out, the poles give rise to an additional contribution to Sð!k Þ. A simple example illustrates this point. Consider the function b2 ðtÞ ¼ ae  t þ

N X j¼1

rj ei ðyj þ !0 Þ t ;

ð124Þ

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

207

where a is a complex constant and yj and rj are real constants with rj > 0. The Laplace transform of this function is BðsÞ ¼

N X rj a þ ;  þ s j ¼ 1 s  iðyj þ !0 Þ

ð125Þ

such that B½ið!k  !0 Þ ¼

N X rj a þ :   ið!k  !0 Þ j ¼ 1 ið!k þ yj Þ

ð126Þ

To calculate the spectrum correctly, we start from Equation (1b), 0 1 rffiffiffiffiffi Zt N X  0 0 0 b1k ðtÞ ¼ i f ð!k Þ dt 0 ei ð!k  !0 Þ t @ae  t þ rj ei ðyj þ !0 Þ t A  j¼1 20  3  rffiffiffiffiffi N  t þ i ð!k  !0 Þ t i ð!k þ yj Þ t X a 1  e r ðe  1Þ  j 5 ¼i f ð!k Þ4 þ    ið!k  !0 Þ ið!k þ yj Þ j¼1

ð127Þ

and evaluate

 2  N N ið!k þyj Þt  X X  r e r  a j j  ; Sð!k Þ¼lim jb1k ðtÞj2 ¼ Fð!k Þlim  þ þ t!1 t!1ið!k !0 Þ ið!k þyj Þ j ¼ 1 ið!k þyj Þ   j¼1 ð128Þ

where have set lim e  t ¼ 0. The cross term involving the oscillating t!1 exponential does not contribute as t ! 1 and we are left with 2 3 2   2 N N X X r   r  a j j 7 6  þ Sð!k Þ ¼ Fð!k Þ4 þ 5  ð!k þ yj Þ 2     ið!k  !0 Þ j ¼ 1 ið!k þ yj Þ  j¼1 0 1 N X r2j  2 A ¼ Fð!k Þ@jB½ið!k  !0 Þj þ  ð!k þ yj Þ 2 j¼1 0 1  2 N X r2j    1  þ A; ¼ Fð!k Þ@ ð!k þ yj Þ 2  !k  !0 þ iGði!k Þ  j¼1

ð129Þ

which is the correct expression for the spectrum when there are poles on the imaginary axis that lead to a finite value for b2(1). Equation (123) agrees with this expression only if all the rj vanish, that is, if b2(1)=0. In the limit that t  1, the equation for conservation of probability, Equation (118), reduces to

208

Paul R. Berman and George W. Ford

2 X j¼1

Z1 rj2 þ

Sð!k Þd!k ¼ 1;

ð130Þ

1

2

where jb2 ð1Þj has been evaluated using Equation (44) (recall that rj is the magnitude of the residue from pole yj) and S(!k) is given by Equation (129). In a similar manner, one can show that the analogous equation for conservation of energy, Equation (119) evaluated in the limit that t  1, is !0

2 X j¼1

Z r2 j þ

1

1

!k Sð!k Þd!k  2

 

Z

1

1

Fð!k Þ

2 X

r2 j

j¼1

ð!k þ yj Þ

d!k ¼ !0 :

ð131Þ

If Equation (42) is used, this can be transformed into !0

2 X j¼1

Z1 rj2 þ 1

2 X !k Sð!k Þd!k  2 ð!0 þ yj Þrj2 ¼ !0 :

ð132Þ

j¼1

which applies equally for all spectral weight functions.3 Equation (129) is truly remarkable in that it gives the spectrum of sponta­ neous emission in terms of the Laplace transform of the excited state ampli­ tude and the position of any poles and their residues in the contour integration used to obtain the inverse Laplace transform of B(s). Since we have relatively simple, analytic forms for both G(s) and the magnitudes of the residues rj for each of the spectral weight functions F(!k), the only quantities that must be calculated numerically in Equation (129) are the positions, yj, of the poles. Moreover, finding the positions of the poles is a simple task since it involves only a numerical solution of Equation (39). Thus, in effect, we have derived an analytic expression for the spectrum. We use Equation (129) to evaluate the spectrum S(!k) for the spectral weight functions F3(!k)F6(!k). The spectrum for the spectral weight functions F1(!k) and F2(!k) can be calculated directly using Equation (122) and the analytic expressions we found previously for b2(t); Equation (122) can be used since b2(1)  0. For reference purposes, the probabilities Sð!k ; tÞ ¼ jb1k ðtÞj2 are given for the spectral weight functions F3(!k)F6(!k); the expressions in these equations must be calculated numerically. A word of caution is in order here. Although we calculate and plot the spectrum given by Equation (129), it is not obvious that this spectrum corresponds to an experimentally measurable quantity when 3 The interpretation of the interaction term (third term in Equation 132) remains somewhat of a mystery to us, although an analogous term appears in the off-resonant Jaynes-Cummings model. The interaction term can be substantial. For example, for the spectral weight function F5(!k) with !0/=10 and !c/=400 (yp=2.35, rp=0.883), the first term in Equation (132) equals 7.808, the second term 21.467, and the interaction term–19.269, which sum to 10 as required.

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

209

b2(1) 6¼ 0. It does appear that the second term in Equation (129) gives rise to a far field that would contribute to an experimentally measured spec­ trum, but this feature needs to be analyzed in more detail. Since the expectation value of atom-field interaction energy is nonvanishing for b2(1) 6¼ 0, the field continues to interact with the atom for all times in this limit. 3.1 Spectral Weight Functions F1(w k) and F2(w k) For the spectral weight function F1(!k) given by Equation (14a), it follows from Equations (122) and (46) that Sð!k Þ ¼

1  :  ð!k  !0 Þ 2 þ  2

ð133Þ

The spectrum integrated over !k is equal to unity, S=1. The spectral weight function F1(!k) corresponds to the Weisskopf-Wigner approxima­ tion. We see that the Weisskopf-Wigner approximation leads to a Lor­ entzian spectrum that is consistent with unitarity. Of course, this spectrum is not physical since it extends to negative frequencies. For the spectral weight function F2(!k) given by Equation (14b), the decay is bi-exponential, except in the case of critical damping. It follows from Equations (122), (50), and (121) that Sð!k Þ ¼

 

1  2 ð!k  !0 Þ 2 ð!k  !0 Þ 2 þ   !w

ð134Þ

and S=1. If !w >> 4 the spectrum is approximately the single Lorentzian given by Equation (133). In the opposite limit when !w > 1;  !0

ð146Þ

the spectrum is essentially the same as for the spectral weight function F3(!k), except that there is a linear dependence on !k near !k=0 rather than a sharp resonance near !k=0. The sharp resonance near !k ¼ !c is broadened by a factor of !c =!0 . For !00 = < 0, there is a qualitative change in the spectrum, since the transition frequency is shifted to negative values and does not overlap

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

215

S(ωk)

0.002

0.001

0

25

50

75

100

ωk /γ

Figure 11 Spectrum for the spectral weight function F5(w k) and w 0 / g = 10; w c / g = 400, when w 00 /g < 0

the spectral weight function. The spectrum given by Equation (145) is shown in Figure 11 for !0/=10 and !c/=400. In this case !00 = < 0 and the pole at y1 = ¼ 2:348, obtained as a solution of Equation (102), makes an important contribution to the spectrum,the magnitude of the residue equal to 0.8833. This term would lead to a delta function contribution to the spectrum at !k = ¼ 2:348, but, since the spectrum equals zero for !k = < 0, only the “tail” of the residue term contributes to the spectrum. One finds numerically that jb2 ð1Þj¼0:780 and S ¼ 0:220 ¼ 1  jb2 ð1Þj2 :

ð147Þ

Thus, even though the upper state is shifted below the ground state in energy, unitarity is maintained, as required. For !0/=10 the value of S as a function of !c = is given by 1 the square of the curve in Figure 8. For large !c =; S  1  0:9232 ¼ 0:148. The contributions from the two terms in Equation (145) are approxi­ mately equal. More generally, for !c