Controllable Quantum States: Mesoscopic Superconductivity & Spintronics (MS+S2006), Procedings of the International Symposium, NTT Basic Res Labatories, Japan 27 February - 2 Marc

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MS+S2006

CONTROLLABLE QUANTUM Mesoscopic

STATES

Superconductivity and S p i n t r o n i c s

This page intentionally left blank

NTT Basic Research Laboratories, Japan

27 February-2 March 2006

MS+S2M6

CONTROLLABLE QUANTUM Mesoscopic

STATES

Superconductivity and S p i n t r o n i c s

Proceedings of the International Symposium

Editors

Hideaki

Takayanagi

Tokyo University of Science, Japan

Junsaku

Nitta

Tohoku University Japan

Hayato

Nakano

NTT Basic Research Laboratories, Japan

'World Scientific NEW JERSEY • LONDON

• SINGAPORE • BEIJING • SHANGHAI

• H O N G K O N G • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

CONTROLLABLE QUANTUM STATES Mesoscopic Superconductivity and Spintronics (MS+S2006) Proceedings of the International Symposium Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-281-461-6 ISBN-10 981-281-461-2

Printed in Singapore.

Benjamin - Controllable Quantum.pmd

1

8/12/2008, 1:40 PM

v

PREFACE

This issue contains the Proceedings of the fourth International Symposium on Mesoscopic Superconductivity and Spintronics (MS+S2006) which was held from February 27th to March 2nd 2006 at NTT Atsugi R&D Center. The first International Symposium on Mesoscopic Superconductivity 2000 (MS2000) was held in March 2000. The main topic of the first symposium was the Andreev-reflection physics at superconductor/semiconductor and superconductor/normal metal interfaces. The scope of the second symposium was extended to include Spintronics, and it was decided to name the second International Symposium on Mesoscopic Superconductivity and Spintronics 2002 (MS+S2002). The third symposium on Mesoscopic Superconductivity and Spintronics 2004 (MS+S2004) was held to highlight a rapidly growing field of quantum computations by adding the subtitle “In the light of quantum computation”. The leading scientists of these research fields participated in the first MS200, the second MS+S2002, and third MS+S2004. From the MS+S2004, the quantum coherence and manipulation have become important and common topics in the fields of Mesoscopic Superconductivity and Spintronics. The fourth MS+S2006 symposium was organized since many researchers looked forward to the discussions of the progress in these fields. The extensively discussed topics in the fourth MS+S2006 were “Control and readout of quantum states in superconducting qubits” and “Spin coherence and manipulation in nano-scale semiconductors” in addition to “Novel phenomena in mesoscopic superconductors”. We believe that the MS+S symposium series have played an important role in the progress of mesoscopic superconductivity, spintronics, and quantum computations. A total of 131 papers were presented at the symposium, including 26 invited talks. The number of participants was 194; 141 from Japan and 53 from 18 foreign countries. This Proceedings contain 64 papers out of those presented at the symposium. We would like to thank all reviewers for their careful reading of the submitted papers. It is our hope that the Proceedings will be useful for many researchers interested in mesoscopic superconductivity, spintronics, and quantum computations. Finally, we would like to thank all participants for their fruitful and exciting discussion throughout the symposium. The symposium was sponsored by JST (Japan Science and Technology) and NTT Basic Research Laboratories. The Organizing Committee would like to express its sincere gratitude to them for their support.

June 2008 Hideaki Takayanagi (Tokyo University of Science, MANA-NIMS ) Junsaku Nitta (Tohoku University) Hayato Nakano (NTT Basic Research Laboratories)

INTERNATIONAL SYMPOSIUM ON MESOSCOPIC SUPERCONDUCTIVITY AND SPINTRONICS (MS+S2006) Date : February 27 (Mon.) – March 2 (Thu.), 2006 Site : NTT Basic Research Laboratories, Atsugi-shi, Kanagawa 243-0198, JAPAN

Organizing Committee H. Takayanagi J. Nitta T. Akazaki Y. Nakamura H. Nakano S. Nomura K. Semba Y. Shimazu H. Tamura M. Ueda

(NTT Basic Research Laboratories), Chair (Tohoku University), Co-Chair (NTT Basic Research Laboratories) (NEC Corporation) (NTT Basic Research Laboratories) (Tsukuba University) (NTT Basic Research Laboratories) (Yokohama National University) (NTT Basic Research Laboratories) (Tokyo Institute of Technology)

International Advisory Committee A. Andreev T. Claeson J. Clarke K. Kajimura J. Mooij A. Leggett E. Rashba H. Sakaki A. Andreev T. Claeson

(P. L. Kapiza Institute of Physical Problems) (Chalmers University of Technology) (University of California) (Japan Society for the Promotion of Machine Industry) (Delft University of Technology) (University of Illinois at Urbana-Champaign) (University of Buffalo) (University of Tokyo) (P. L. Kapiza Institute of Physical Problems) (Chalmers University of Technology)

Sponsors Japan Science and Technology Agency (JST) NTT Basic Research Laboratories

vii

CONTENTS Preface

v

Mesoscopic Effects in Superconductors

1

Tunneling measurements of charge imbalance of non-equilibrium superconductors R. Yagi

3

Influence of magnetic impurities on Josephson current in SNS junctions T. Yokoyama

9

Nonlinear response and observable signatures of equilibrium entanglement A. M. Zagoskin

15

Stimulated Raman adiabatic passage with a Cooper pair box Giuseppe Falci

21

Crossed Andreev reflection-induced giant negative magnetoresistance Francesco Giazotto

27

Quantum Modulation of Superconducting Junctions

33

Adiabatic pumping through a Josephson weak link Fabio Taddei

35

Squeezing of superconducting qubits Kazutomu Shiokawa

41

Detection of Berry’s phases in flux qubits with coherent pulses D. N. Zheng

47

Probing entanglement in the system of coupled Josephson qubits A. S. Kiyko

53

Josephson junction with tunable damping using quasi-particle injection Ryuta Yagi

59

Macroscopic quantum coherence in rf-SQUIDs Alexey V. Ustinov

65

Bloch oscillations in a Josephson circuit D. Esteve

71

Manipulation of magnetization in nonequilibrium superconducting nanostructures F. Giazotto

77

viii Superconducting Qubits

83

Decoherence and Rabi oscillations in a qubit coupled to a quantum two-level system Sahel Ashhab

85

Phase-coupled flux qubits: CNOT operation, controllable coupling and entanglement Mun Dae Kim

91

Characteristics of a switchable superconducting flux transformer with a DC-SQUID Yoshihiro Shimazu

97

Characterization of adiabatic noise in charge-based coherent nanodevices E. Paladino

103

Unconventional Superconductors

109

Threshold temperatures of zero-bias conductance peak and zero-bias conductance dip in diffusive normal metal/superconductor junctions Iduru Shigeta

111

Tunneling conductance in 2DEG/S junctions in the presence of Rashba spin-orbit coupling T. Yokoyama

118

Theory of charge transport in diffusive ferromagnet/p-wave superconductor junctions T. Yokoyama

124

Theory of enhanced proximity effect by the exchange field in FS bilayers T. Yokoyama

130

Theory of Josephson effect in diffusive d-wave junctions T. Yokoyama

136

Quantum dissipation due to the zero energy bound states in high-Tc superconductor junctions Shiro Kawabata

143

Spin-polarized heat transport in ferromagnet/unconventional superconductor junctions T. Yokoyama

149

Little-Parks oscillations in chiral p-wave superconducting rings Mitsuaki Takigawa

155

ix Theoretical study of synergy effect between proximity effect and Andreev interface resonant states in triplet p-wave superconductors Yasunari Tanuma

161

Theory of proximity effect in unconventional superconductor junctions Y. Tanaka

168

Quantum Information

175

Analyzing the effectiveness of the quantum repeater Kenichiro Furuta

177

Architecture-dependent execution time of Shor’s algorithm Rodney Van Meter

183

Quantum Dots and Kondo Effects

189

Coulomb blockade properties of 4-gated quantum dot Shinichi Amaha

191

Order-N electronic structure calculation of n-type GaAs quantum dots Shintaro Nomura

197

Transport through double-dots coupled to normal and superconducting leads Yoichi Tanaka

203

A study of the quantum dot in application to terahertz single photon counting Vladimir Antonov

209

Electron transport through laterally coupled double quantum dots T. Kubo

215

Dephasing in Kondo systems: comparison between theory and experiment F. Mallet

221

Kondo effect in quantum dots coupled with noncollinear ferromagnetic leads Daisuke Matsubayashi

227

Non-crossing approximation study of multi-orbital Kondo effect in quantum dot systems Tomoko Kita

233

Theoretical study of electronic states and spin operation in coupled quantum dots Mikio Eto

239

Spin correlation in a double quantum dot-quantum wire coupled system S. Sasaki

245

x Kondo-assisted transport through a multiorbital quantum dot Rui Sakano

251

Spin decay in a quantum dot coupled to a quantum point contact Massoud Borhani

256

Quantum Wires, Low-Dimensional Electrons

263

Control of the electron density and electric field with front and back gates Masumi Yamaguchi

265

Effect of the array distance on the magnetization configuration of submicron-sized ferromagnetic rings Tetsuya Miyawaki

271

A wide GaAs/GaAlAs quantum well simultaneously containing two dimensional electrons and holes Ane Jensen

277

Simulation of the photon-spin quantum state transfer process Yoshiaki Rikitake

282

Magnetotransport in two-dimensional electron gases on cylindrical surface Friedland Klaus-Juergen

288

Full counting statistics for a single-electron transistor at intermediate conductance Yasuhiro Utsumi

295

Creation of spin-polarized current using quantum point contacts and its detection Mikio Eto

301

Density dependent electron effective mass in a back-gated quantum well S. Nomura

307

The supersymmetric sigma formula and metal-insulator transition in diluted magnetic semiconductors I. Kanazawa

312

Spin-photovoltaic effect in quantum wires A. Fedorov

318

Quantum Interference

325

Nonequilibrium transport in Aharonov-Bohm interferometer with electron-phonon interaction Akiko Ueda

327

xi Fano resonance and its breakdown in AB ring embedded with a molecule Shigeo Fujimoto, Yuhei Natsume

333

Quantum resonance above a barrier in the presence of dissipation Kohkichi Konno

339

Ensemble averaging in metallic quantum networks F. Mallet

345

Coherence and Order in Exotic Materials

351

Progress towards an electronic array on liquid helium David Rees

353

Measuring noise and cross correlations at high frequencies in nanophysics T. Martin

359

Single wall carbon nanotube weak links K. Grove-Rasmussen

365

Optical preparation of nuclear spins coupled to a localized electron spin Guido Burkard

371

Topological effects in charge density wave dynamics Toru Matsuura

377

Studies on nanoscale charge-density-wave systems: fabrication technique and transport phenomena Katsuhiko Inagaki

383

Anisotropic behavior of hysteresis induced by the in-plane field in the ν = 2/3 quantum Hall state Kazuki Iwata

389

Phase diagram of the ν = 2 bilayer quantum Hall state Akira Fukuda

395

Trapped Ions (Special Talk)

401

Quantum computation with trapped ions Hartmut Häffner

403

List of Participants

409

Mesoscopic Effects in Superconductors

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yagi-tex

33

TUNNELING MEASUREMENTS OF CHARGE IMBALANCE OF NON-EQUILIBRIUM SUPERCONDUCTORS R. YAGI, K. UTSUNOMIYA, K. TSUBOI, T. KUBOTA, Y. TERAO AND Y. IKEBUCHI Graduate School of Advanced Sciences of Matter (ADSM), Hiroshima University, Higashi-Hiroshima, 739-8530, Japan

We have observed excess current due to charge imbalance in the voltage-current characteristics of a superconductor-insulator-normal (SIN) tunnel junction connected to a non-equilibrium superconductor. It was found that that the excess current was unchanged against the bias voltage as expected from the theory of charge imbalance. The estimated excess current approximately agreed with the estimation from one-dimensional diffusion model of charge imbalance transport.

1. Introduction Charge accumulation effect is one of the most interesting phenomena in non-equilibrium superconductivity. In normal metal, the numbers of electron and hole excitations are identical because electron excitation inevitably excites holes. This is a consequence of strict charge neutrality. On the other hand, in superconductor, those numbers can be different owing to the superconducting condensate moving without dissipation to satisfy the charge neutrality. This is called a charge imbalance effect discovered theoretically and experimentally by Clarke and Tinkham1-3. Since the discovery, many experimental and theoretical studies have been done in 1970s and 1980s4-6. Past experimental studies have focused on the chemical potential difference between quasi-particle system and superconducting condensate, which is associated with generation of the charge imbalance. The signal voltage to be measured is, in general, so small that they must use superconducting interferometer device (SQUID) voltmeter to detect it. Recent development of experimental technology has realized the experiments that could not have done in the past. In this study we show direct measurement of charge imbalance with voltage-current (V-I) characteristic measurements. The character of the quasi-particles is determined by relative magnitude of the wave vectors: if |k| is larger than the Fermi wave vector kF, it is electron like and in the opposite case, it is hole like. Charge imbalance is defined by the net charge density of quasiparticle system as ∞

∫

Q* = 2 N (0) f > ( E ) − f < ( E )dE , ∆

(1)

August 8, 2008

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yagi-tex

44

where f>(E) and f 2. The expressions like (1), arising in the lowest-order perturbation theory, are quadratic in qubit operators and therefore catch the absence of 2-tangled components, but are insensitive to the higher-order entanglement. This is why the appearance of globally entangled eigenstates (| ↑↓↑↓ ± | ↓↑↓↑) in 16 could be only inferred from the post mortem analysis. In order to obtain information about (N > 2)-tanglement without reverse-engineering of the Hamiltonian, one requires N -point correlators. They can be obtained from the nonlinear response of a multiqubit system. 3. Equilibrium multipartite entanglement and irreducible higher-order correlators Consider first an N -point correlator of single-qubit (spin) operators Sk (tk ), acting on qubit k at the moment tk , CN ({k, tk }, |p) = p|S1 (t1 )S2 (t2 ) . . . SN (tN )|p, k = 1. . . . N,

(2)

where |p is some eigenstate of the Hamiltonian. The sequence of operators in (2) generally flips N spins in the state |p. Therefore the correlator should be suppressed unless |p is a superposition of components differing by N spins. To check this intuition, let us first define the ”equilibrium N -tanglement”. We will say that the system of M > N qubits (M denoting the set of all qubits) is not

July 18, 2008

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17

N-tangled, if (1) all its eigenstates can be presented as products of states defined on (p) qubit clusters Ωj of less than N qubits each, (p) |pΩ(p) , Ωj = M, (3) |p = (p)

Ωj

j

j

and (2) the clusters do not overlap. The latter condition means that, if there exists (p) (p) an eigenstate |p such that qubits a, b ∈ Ωs , while qubit c ∈ / Ωs , then there is no (q) (q) (q) / Ωr , while b, c ∈ Ωr . such eigenstate |q and cluster Ωr , that a ∈ Together, the conditions (1) and (2) ensure that in equilibrium the qubit system can be uniquely split into mutually non-entangled clusters, M = j Ωj , where Ωj = (p)

maxp (Ωj ), each containing less than N qubits. This is a reasonable definition of the absence of equilibrium N-partite entanglement. In the presence of N -tanglement, at least one cluster contains at least N entangled qubits. This definition is actually the (restricted by the plausible non-overlapping condition) k-producibility recently introduced in 18 (with k = N ). Returning to (2), one can easily express the correlator as a sum over the internal labels p2 , p3 , . . . pN of the correlators (p1 ≡ p) cN ({k, |pk }) = p1 |S1 |p2 p2 |S2 |p3 . . . pN |SN |p1 , k = 1. . . . N,

(4)

with the appropriate energy denominators, similar to (1). Now consider irreducible correlators, that is, the ones where no two states |pi , |pj , i = j coincide. (It is obvious from the definition (4) that the correlator is cyclic invariant. In the presence of coinciding states it can be further reduced to a product of cyclic invariant terms.) For such a correlator cirr N ({k, p}) there are two possibilities: (A) all N qubits belong to the same entangled cluster Ω, or (B) they belong to several clusters. In case (A) the states of the system can be written as |n = |nΩ ⊗ |nΩ¯ etc, ¯ is the complement of Ω. We therefore obtain where Ω ¯ ¯ ¯ (5) cirr N ({k, p}) = (· · · )δp1 p2 (Ω)δp2 p3 (Ω) · · · δpN p1 (Ω); (the Kronecker symbol δpq (Ω) means that the states p and q coincide on the set Ω). We see from (5), that the states of the qubits outside Ω must be the same, and can only differ on Ω. Therefore cirr N is generally nonzero, if there are at least log2 N qubits in Ω (which is not a serious limitation). In case (B) the string of indices S = j1 j2 . . . jN , labeling the qubit operators, consists of 1 < R ≤ N substrings, S = s1 s2 . . . sR . In each substring the indices belong to the qubits in the same entangled cluster Ω1 , Ω2 , . . . ΩR . The only restriction is that clusters corresponding to adjacent index substrings, e.g. Ωr and Ωr±1 , must be different, or the substrings would merge. r r , and jN + 1 ≡ j1r+1 ) The rth substring (the indices in which run from j1r to jN r r ¯ r (the complement yields the product of the Kronecker symbols, ensuring that on Ω of Ωr ) r +1 ≡ |p r+1 . |pj1r = |pj1r +1 = · · · = |pjN j r 1

(6)

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18

Let us call the substrings mutual, if they contain qubit indices from the same entangled cluster; a substring, which has no mutuals on S, is called unique. Then we define a joint (disjoint) correlator as the one, the index string of which does (does not) contain at least one unique substring. For a joint correlator with the unique substring su : S = s1 s2 . . . su−1 su su+1 . . . sR−1 sR , we obtain ¯ 1 · · · δp u−1 p u Ω ¯ u−1 δp u p u+1 Ω ¯u cjoint = (· · · ) δp1 pj2 Ω j j N j j 1 1 1 1 1 ¯ u+1 · · · δp p1 Ω ¯R = ×δp u+1 p u+2 Ω jR j1 j1 1 ¯1 ∩ Ω ¯u ¯2 · · · ∩ Ω ¯ u−1 δp u p u+1 Ω (· · · ) δp1 pju Ω j 1 1 j1 ¯ u+1 ∩ Ω ¯ u+2 · · · ∩ Ω ¯R = ×δpju+1 p1 Ω 1 ¯u ∪ Ω ¯1 · · · ∩ Ω ¯ u−1 ∩ Ω ¯ u+1 · · · ∩ Ω ¯R . (7) (· · · ) δpju p u+1 Ω 1

j1

¯u ∪ Since su is unique, that is, Ωu ∩ Ω1,...,u−1,u+1,...,R = ∅, the set Ω ¯ ¯ ¯ ¯ Ω1 · · · ∩ Ωu−1 ∩ Ωu+1 · · · ∩ ΩR = M. In other words, the states |pj u+1 and |pj1u 1 coincide on all clusters. Therefore joint correlators (7) cannot be irreducible, sice they contain at least two coinciding pairs of internal labels. ¯1 · · · ∩ Ω ¯R = ¯u ∪ Ω For a disjoint correlator this does not hold, since then Ω ¯ u ⊂ M. (The only exception is the case when s1 and sR , the substrings at the Ω beginning and the end of S, are mutual, and don’t have other mutuals. Then, from the cyclic invariance of the irreducible correlator, these substrings effectively merge into a unique substring.) We found that certain irreducible N-point correlators (correlators with repeating qubit indices and disjoint correlators) do not disappear in the absence of N-partite entanglement. Now we demonstrate that their contribution to the correlators of extensive variables nevertheless disappears in the thermodynamical limit, which allows to use the latter as directly observable entanglement signatures. The global response functions of the system contain the irreducible correlators M of average operators, i.e. operators of the type S = M −1 j=1 Sj : CNirr;p1 p2 ...pN = M −N

M

cirr N (k1 , k2 , . . . , kN ; p1 , p2 , . . . , pN ).

(8)

k1 ,k2 ,...,kN =1

Assume that both the number of qubits M and the number of entangled clusters Q exceeds N . Then we shall see that as M, Q → ∞ the contributions to (8) from the correlators with repeating indices and disjoint correlators disappear as O(1/M ), O(1/Q) respectively. Repeating indices. The number of all possible combinations of N indices, each running from 1 to M , is M N . The number of combinations of different indices is M (M − 1)...(M − N + 1), and the number of combinations with repeating indices is M N − M (M − 1)...(M − N + 1) = O(M N −1 ). Therefore in the expression (8) the contribution from the correlators with repeating indices scales as O(1/M ).

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19

N −1 Disjoint correlators. There are ways to split a string of N inR−1 dices in R nonzero substrings. For each substring there are Q entangled clusters we can choose from, and there are on average (M/Q)N/R possible combinations of indices within the cluster. Therefore the total number of different combina N N −1 N tions of indices, counted this way, is (M/Q) QR . On the other R=1 R−1 hand, the number corresponding to joint correlators is no less than of combinations

N N −1 N R−1 (M/Q) (Q−(R−1)). Therefore the number of disjoint comQ R=1 R−1

N N −1 R N binations does not exceed M Q − QR−1 (Q − (R − 1)) = R=1 Q R−1

N N − 1 N M d Q dQ QR−1 =M N Q−1 (N − 1)(1 + Q−1 )N −2 , and the R=1 Q R−1 disjoint contribution to the average irreducible correlator (8) asymptotically disappears at least as fast as O(1/Q) ∼ O(1/M ). Therefore CNirr can be used as an N -tanglement indicator: it is nonzero in the limit M, Q → ∞ only if the system contains a finite proportion of entangled N -qubit (or larger) clusters.

4. Nonlinear susceptibility and equlibrium multipartite entanglement The correlators like (8) enter the expressions for (nonlinear) susceptibility. Therefore the latter could be used as directly observable entanglement signatures. The unperturbed Hamiltonian of M flux qubits and the perturbation due to their coupling to an external field are M M z z 1 x z z z Jij σi σj ; H1 (t) = −h(t) Δj σj + εj σj − λj σj + λxj σjx . (9) H0 = − 2 j=1 i<j j=1

The equilibrium entanglement we are probing is created by the unperturbed Hamiltonian, H0 . Iterating the equation of motion for the density matrix, ρ(t), one obtains the standard expansion of the density matrix and the expansion for

corresponding the magnetic moment of the system, μ(t) = tr ρ(t) j σj . In particular, the 2M (0) 2M quadratic susceptibility, χzz (ω, ω ) = n=1 ρn p,q=1 Cn;pq fn;pq (ω, ω ) , is ex z z z pressed through the the weight factors Cn;pq = ijk n|σi |pp|σj |qq|σk |n ≡ 1 z z z n|μ |pp|μ |qq|μ |n, and the formfactors fn;pq (ω, ω ) = ω+ω −(Eq −Ep )+i0 ×

En +Eq −2Ep En +Ep −2Eq − (ω −(Eq −En )+i0)(ω+ω −(Ep −En )+i0) (ω −(En −Ep )+i0)(ω+ω −(En −Eq )+i0) . In the experimentally relevant low-frequency limit, χn0 = limω→0 pq Cnpq fn;pq (ω, ω),

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20

where χn0 =3

means that terms with Ep = Eq = En are excluded, we find:

|μz nq |2 (μz nn − μz qq )

(Eq − En

q

)2

−6

p

(Cnpq ) ≡ χn0A + χn0B . (10) (E − E )(E − E ) p n q n q 0>

2

0

0

0

15

2

L

>

R

>

0

1 0 0

5

10

15

20

25

1

1.5

2

ϕ

Phase difference ϕ Figure 1. (a) Potential energy of an rf-SQUID with βL =4.5 and Ic = 10 μA for external magnetic fluxes of Φext =0, 0.5, and 0.8 Φ0 (from left to right, each curve offset by ϕ=6 for better visibility), normalized to EJ = Ic Φ0 /2π. (b) Zoom into the shallow well showing the discrete energy levels and the wavefunctions of the corresponding quantum states |0, |1 and |2.

flux quantum with 2e being the Cooper pair charge, and βL ≡ 2πLq IC /Φ0 . For βL ≈ 2π, the potential has the form of a double well which is tilted proportional to the external magnetic flux, see Fig. 1(a). States associated to two neighboring minima, like the states |L and |R in Fig. 1(a), correspond to opposite directions of circulating current. 3. Quantum processes in the rf-SQUID As indicated in Fig. 1(b), the confinement of the phase variable within one potential well leads to the formation of discrete energy states (energy quantization). The two states of lowest energy in the shallow potential well can then be used as the logical states |0 and |1 for quantum computation. Since the depth of the potential well depends on the strength of the applied magnetic field, it is possible to tune the energy difference separating these states in situ. Transitions between the states can be driven by resonant photon absorption and emission, which is experimentally realized by inducing alternating currents of microwave frequencies in the rf-SQUID loop. The anharmonicity of the potential hereby assures that the transition frequencies between adjacent energy levels are different enough to avoid undesired population of higher levels5 . To distinguish in which state the qubit is, one makes use of quantum tunneling through the potential barrier separating the wells. The rate of this process depends exponentially on the barrier height, and therefore tunneling from the excited state occurs much faster than from the ground state. Whether tunneling occurred is determined by monitoring the magnetic flux through the qubit loop. This is realized by measuring the maximum supercurrent of a dc-SQUID which is inductively coupled to the rf-SQUID loop, as illustrated in Fig. 2(a).

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67

(a)

I

Control flux bias

L Φ X

X

Φext

Rb

Qubit rf-SQUID

sq

X

μw

I

Ib

Vsq

Readout dc-SQUID

(b) DC-pulse, measurement

Φ prepare 0

reset

I sq

μW-pulse sequence

reset

freeze

time

readout

0

1

0

>

>

time

Figure 2. (a) Schematic of the phase qubit circuit. Dc- and microwave currents control the flux Φ in the qubit loop by means of an on-chip superconducting coil. For readout, this flux is measured by a dc-SQUID which is coupled inductively to the qubit. (b) Timing profile of the magnetic flux Φ applied to the qubit (top) and the current through the readout SQUID (bottom). The latter is ramped up from zero only after completion of the qubit operation.

4. Qubit operation The qubit is operated in a way illustrated in Fig. 2(b). The qubit state is first reset to the left well by switching the applied magnetic flux to zero. Then, the flux is increased to Φ ≈ Φ0 until the left well becomes shallow enough to contain only a small number of energy levels. After the microwave pulse was applied, a small readout flux pulse of nanosecond length is applied for generating an additional small tilt of the potential, which causes immediate tunneling only if the qubit is in its excited state. The applied magnetic field is then reduced in order to increase the barrier height, avoiding further inter-well transitions. Finally, a current-ramp measurement of the critical current of the readout dc-SQUID decides wether the qubit is in the left or right potential well, and hence wether it was in state |0 or |1 at the time when the readout pulse was applied. 5. Sample Our sample has been fabricated at a commercial foundry provided by Hypres4 using standard Nb/Al-AlOx /Nb-trilayer circuits of 30 A/cm2 current density and minimum 3 × 3 μm2 Josephson junction size. The critical current of the smallest

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68

junction of Ic ≈ 2.7 μA together with the requirement βL = 2πLIC /Φ0 ≈ 4 to 5 determines suitable loop inductances L ≈ 400 to 500 pH. In our sample, following Ref.5 , this inductance is realized by a two-turn coil of about 70 μm diameter. The mutual inductance between the qubit and readout dc-SQUID, M ≈ 15 pH, has been chosen large enough to permit single-shot readout of the qubit state. Measurements were performed in a dilution refrigerator at a base temperature of 20 mK. 6. Rabi oscillations Coherent qubit operation can be demonstrated by observing Rabi flopping of the qubit state. This mechanism leads to oscillating state populations during the application of a resonant microwave field, which effectively couples states |0 and |1. The strength of this coupling grows with the rf-field amplitude, which in consequence also determines the Rabi oscillation frequency fRabi . Experimentally, this oscillation is made visible in the time domain by applying a microwave pulse of varying duration which is followed by the dc-readout pulse. Figure 3(a) shows Rabi oscillation in the probability P (|1) to measure the excited qubit state for three values of applied microwave power and a frequency of frf = 14.2 GHz. We find the oscillation amplitude to decay exponentially with a characteristic time of typically 6 ns. Particularly, for certain values of flux bias, we observed an additional (a)

(b)

P( 1 )

0.8

Rabi frequency [MHz]

0.6

600

Pμw = 1 dBm

0.4

500 0.8

400

0.6 0.4

300

Pμw = -4 dBm

0.2

200

0.6

100 0.4

0

Pμw = -7 dBm

0.2

0

5

10

0 15

1

2

3

4

5

6

microwave amplitude [ mW ]

microwave pulse duration [ns]

Figure 3. (a) Rabi oscillation in the probability P (|1) to measure the excited qubit state. Each curve corresponds to the indicated microwave power P . (b) Measured dependence of the oscillation frequency on microwave amplitude (dots). For small amplitudes, saturation of the Rabi frequency detuning Δfrf of the rf frequency from the resonance between the levels. fRabi indicates residual

Line is a fit to fRabi =

2 + αP , where α is the fitting parameter. Δfrf

July 23, 2008

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69

modulation (beating) in the oscillation amplitude. We attribute this behavior to coherent coupling of the qubit to parasitic two-level fluctuators, which were proven to exist in the silicon oxide used to isolate the junction electrodes6 . In Fig. 3(b), we plot the measured Rabi frequency on microwave amplitude, which displays a linear dependence at high amplitudes as expected for a two-level quantum system. 7. Inhomogeneous broadening An obvious source of decoherence are fluctuations in the qubit parameters between individual repetitions of a measurement, which are then averaged to obtain the statistical state probability. This effect can be distinguished from other decoherence mechanisms by spectroscopic measurements, in which a resonant enhancement of the probability to find the qubit in the excited state occurs in the form of a Lorentzian peak. The width of this peak is increased by fluctuations in the energy level spacing, an effect called inhomogeneous broadening a . According to theory7 , the full width at half maximum of the resonance peak in the strong driving limit is given by 1 T1 1 , + ωRabi Δf ≈ π T2 T2 where T1 is the energy relaxation time, T2 is an intrinsic dephasing time and T2 is the contribution to dephasing due to inhomogeneous broadening. Plotting the linewidth vs. microwave amplitude, T2 results from a linear extrapolation to zero microwave amplitude. Figure 4 shows the result of such an experiment. The extrapolated resonance width of Δf =60 MHz for this data corresponds to T2 = 1/(π Δf ) ≈ 5.3 ns, in close agreement with the lifetime of Rabi oscillations observed in this sample. This suggests that coherence in our qubits is limited by the same mechanisms which give rise to inhomogeneous broadening. Its origin can be fluctuations of the junction critical current as well as magnetic field instabilities due to motion of trapped vortices. 8. Conclusion In conclusion, we described operation of a commercially-fabricated rf-SQUID in the coherent quantum regime as a phase qubit. The rich variety of quantum effects accessible experimentally in superconducting circuits based on Josephson junctions bears great potential in the study of origins and characteristics of decoherence in macroscopic objects and the possibility of realizing solid-state quantum computing.

a This term originates in NMR experiments, where an inhomogeneous magnetic field leads to different energy separations in spatially separated nuclei, which constitute the individual qubits.

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70

(a)

(b)

> P( |1 )

Resonance width Δ f [GHz]

0.4

1.6

-6 dBm

0.3

1.2

0.2

0.4 0.3

0.8

0.2

0.1 0.4

0.1

0 -26 dBm

1.4

1.6

2 1.8 2.2 external flux [a.u.]

2.4

0

0

0

0.5

1

0

1.5

0.1

0.2

2

microwave amplitude [mW

0.3

2.5 1/2

0.4

3

]

(a) Spectroscopic resonance peaks in the excited state population probability vs. applied external flux. Both width and amplitude of the Lorentzian peak increase with increasing microwave power. (b) Full width at half maximum of the resonance peaks shows a linear dependence on microwave amplitude. For small amplitudes, saturation of the width above zero is observed due to inhomogeneous resonance broadening.

Figure 4.

References 1. J.R. Friedman, V. Patel, W. Chen, S.K. Tolpygo and J.E. Lukens, Nature 406, 43 (2000). 2. C.H. van der Wal, A.C.J. ter Haar, F.K. Wilhelm, R.N. Schouten, C.J. Harmans, T.P. Orlando, Seth Lloyd, J.E. Mooij, Science 290, 773 (2000). 3. J.E. Mooij, T.P. Orlando, L. Levitov, L. Tian, C.H. van der Wal, and S. Lloyd, Science 285, 1036 (1999); I. Chiorescu, Y. Nakamura, C.J. Harmans, and J.E. Mooij, Science 299, 1869 (2003); B.L.T. Plourde, T.L. Robertson, P.A. Reichardt, T. Hime, S. Linzen, C.-E. Wu, and J. Clarke, Phys. Rev. B 72, 060506(R) (2005). 4. Hypres Inc., Elmsford, NY, USA. 5. R.W. Simmonds, K.M. Lang, D.A. Hite, D.P. Pappas, and J.M. Martinis, Phys. Rev. Lett. 93, 077003 (2004); K. B. Cooper, M. Steffen, R. McDermott, R. W. Simmonds, S. Oh, D. A. Hite, D. P. Pappas, and J.M. Martinis, Phys. Rev. Lett. 93, 180401 (2004). 6. J.M. Martinis, K.B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. Osborn, K. Cicak, S. Oh, D.P. Pappas. R.W. Simmonds and C.C. Yu, Phys. Rev. Lett. 95, 210503 (2005). 7. T. L. Robertson, Fundamentals of Flux-based Quantum Computing, Ph.D. Thesis, University of California, Berkeley (2005).

August 8, 2008

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71 71

BLOCH OSCILLATIONS IN A JOSEPHSON CIRCUIT N. BOULANT, G. ITHIER, F. NGUYEN, P. BERTET, H. POTHIER, D. VION, C. URBINA, AND D. ESTEVE Quantronics, Service de Physique de l’Etat Condensé, URA 2464 CEA-Saclay, 91191 Gif-sur-Yvette, France

Bloch oscillations predicted to occur in current-biased single Josephson junctions have eluded direct observation up to now. Here, we demonstrate similar Bloch oscillations in a slightly richer Josephson circuit, the quantronium. The quantronium is a Bloch transistor with two small junctions in series, defining an island, in parallel with a larger junction. In the ground state, the microwave impedance of the device is modulated 2e periodically with the charge on the gate capacitor coupled to the transistor island. When a current I flows across this capacitor, the impedance modulation occurs at the Bloch frequency f = I /(2e ) , which yields Bloch sidebands in the spectrum of a reflected continuous microwave signal. We have measured this spectrum, and compared it to predictions based on a simple model for the circuit. We discuss the interest of this experiment for metrology and for mesoscopic physics.

1. Bloch oscillations in Josephson junctions The phenomenon of Bloch oscillations [1], was first considered for a quantum particle moving in a periodic potential and subject to a constant driving force (see [2] for a review). When the particle stays in the first Bloch band, its quasi-momentum changes linearly with time until it reaches the boundary of the first Brillouin zone, where it is Bragg-reflected to the symmetric opposite band-edge and increases again. The velocity of the particle oscillates during the motion. This phenomenon pertains to many physical situations where a quantum system with a periodic potential is subject to a constant drive [2]. Experimental evidence of Bloch oscillations can be found in particular for electrons in solid state superlattices [3], and for ultracold atoms in optical lattices [4]. Bloch oscillations are also predicted to occur in a single Josephson junction biased by a dc current I [5]. The variables describing the system are the charge Q on the junction capacitance and the phase difference δ across the junction. They form a set of conjugated variables: [Q, δ ] = i (2e) , with e the electron charge. The Hamiltonian writes:

H = Qˆ 2 /(2C ) − E J cos(δˆ ) − I δˆ /(2e) ,

(1.1)

with C the junction capacitance, and E J the Josephson energy. The Bloch oscillations should manifest here as a periodic oscillation of the voltage across the junction at the Bloch frequency f = I /(2e) . The locking of these oscillations by an external microwave signal applied to the junction is predicted to induce voltage steps at constant current in the current-voltage characteristic. These steps would be dual of Shapiro steps in voltage

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boulant

72 72 biased Josephson junctions, and provide an appealing solution for the metrology of currents. This experiment is however difficult because a source impedance large compared to the resistance quantum RQ = h /(2e) 2 6.5 k Ω is required over a wide frequency range. Although a Bloch-nose feature related to Bloch oscillations and an effect of microwaves have been observed [6-7], the locking regime has never been reached. In this work, we have performed a related experiment which shows a phenomenon analogous to Bloch oscillations in a multi-junction Josephson circuit, the quantronium [8], which is a recently developed quantum bit circuit.

Ng

δ

V

∆n Carrier

0

1/f

t

S (f) 2∆nf

L (Ng,δ=π) 0

1

2

3

4

5

N

ν

Fig (1). When a triangular gate charge modulation (top left) is applied to a quantronium (top right), its inductance varies periodically, which modulates the reflection coefficient of the device at microwave frequencies close to the plasma resonance of the larger junction on the right. When the extrema of the gate sweep correspond to integer or half-integer values of N g , the modulation of the reflection factor is the same as in the case of a linear sweep of the gate charge (continuing dashed line). The spectrum of the reflected signal then presents sidebands shifted from the carrier by multiples of the Bloch frequency 2 ∆nf . For an arbitrary periodic sweep of the gate charge, other sidebands are predicted at all frequency shifts kf .

2. Bloch oscillations with the quantronium The quantronium circuit, schematically shown in Fig. (1), can be figured as a Bloch transistor (i.e. two small junctions in series) in parallel with a larger junction. The circuit is connected to a microwave transmission line used for probing its impedance. The transistor Hamiltonian is controlled by the reduced charge on the gate electrode N g = C g V /(2e) , with C g the gate capacitance, and by the reduced flux δ = φ / ϕ 0 , with φ the flux threading the loop, and ϕ 0 = / 2e :

Hˆ = EC ( Nˆ − N g ) 2 − E J cos(δ / 2)cos θˆ

(1.2)

Here, the conjugate variables are the island pair number N and phase θ . All the circuit properties vary periodically with N g (period 1) and with δ (period 2π ). We consider here the case of a quantronium in which the large junction is loaded by an on-chip capacitor in order to place its plasma resonance frequency in the convenient 1 − 2GHz frequency range. Assuming the circuit stays in the ground state of Hamiltonian (1.2) with

August 8, 2008

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boulant

73 energy ε ( N g , δ ) , the effective inductance of the transistor L = ϕ 02 /(∂ 2ε / ∂δ 2 ) varies with N g as shown in Fig. (1), which modifies the reflection factor of the circuit. The calculations were performed for the parameters of the sample: EC 0.5 K, E J 1.0 K . If the gate charge N g could increase linearly in time, i.e. in the case of a constant gate current I , the circuit properties, such as the island potential or the impedance, would be modulated at the Bloch frequency f B = I / 2e . When a cw signal is injected on the microwave line, the spectrum of the reflected signal is expected to develop sidebands shifted from the carrier by the Bloch frequency. Harmonics are also expected since the modulation of the reflection coefficient is not purely sinusoidal. Of course it is impossible to maintain a constant current through the gate capacitor for a long time. It is nevertheless possible to carry out an experiment showing the Bloch oscillations predicted for a constant current, as now explained. Let us consider the case of a triangular modulation with frequency f and span ∆n of N g between two exactly integer or half-integer values, as shown in Fig. (1). In this situation, the modulation of the inductance, and thus of the reflection coefficient, is expected to be exactly the same as in the case of a constant gate current i = 2∆n f (2 e ) because the inductance modulation is periodic, and symmetric around integer and half-integer values of N g . The corresponding Bloch frequency is f B = 2∆nf . With the inductance deduced from the ground-state energy [8], one easily calculates for an arbitrary periodic gate modulation the Fourier series giving the reflection factor

R(t ) = ∑ k rk exp(2iπ kft ) .

(1.3)

The coefficients rk provide the reflected amplitudes at all frequencies kf shifted from the carrier frequency. The Bloch sidebands shifted by ± f B dominate when the triangular gate modulation is properly tuned, as predicted by the simple physical picture presented above. The spectrum S (ν ) calculated following this procedure is in excellent agreement with a full calculation of the reflected signal in the linear excitation regime of the circuit.

3. Observation of Bloch oscillations A quantronium sample, with an Al/AlOx/Al on-chip capacitor, was fabricated using electron-beam lithography, placed in a sample-holder fitted with microwave lines, and cooled down to 30 mK. The gate was connected to a 250 MHz bandwidth rf line, and the quantronium was connected both to a microwave injection line and to a measuring line through a circulator. The injected microwave power at the circuit level was chosen to maintain the phase dynamics in the harmonic oscillator regime. The reflected signal was sent through two decoupling circulators to a cryogenic amplifier at 4.2 K, and was finally amplified at room temperature. The effective gain of the measuring line was 76 dB. The signal was then either demodulated with the input cw signal in order to directly observe Bloch oscillations, which is possible at low Bloch frequencies, or sent to a spectrum analyzer. A series of spectra showing the Bloch lines is shown in Fig. (2) for a properly

August 8, 2008

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boulant

74 74 tuned triangular gate modulation with ∆n = 10 and zero offset at different sweep frequencies. The Bloch lines have a narrow sub-Hz linewidth. We have checked the selection rules for the offset and the amplitude predicted from the triangular modulation patterns mimicking a linear evolution of the gate charge. When the offset is detuned, other non-Bloch sidebands appear, as calculated using Eq. (1.3). This experiment also revealed that the reflected signal randomly alternates between two values corresponding to gate charges shifted by one electron. This phenomenon is due to spurious quasiparticle states lying in the superconducting gap. This la ck of robustness of the parity, commonly observed in Cooper pair devices, may be further enhanced when the gate charge is swept fast and over a wide range.

-30

S(ν) dBm -50

f 4 kHz 3 kHz 2 kHz 1 kHz 0.5 kHz

-70 -90 160992 0

161000 0 ν (kHz)

161008 0

Fig (2). Spectrum of the reflected signal showing Bloch lines when a triangular gate charge modulation with zero offset and amplitude ∆n = 10 is applied to the gate, for different frequencies f. Curves have been shifted vertically for clarity. The arrow indicates the predicted position of the Bloch lines at fC ± 2 ∆n f , with fC = 1.61GHz the carrier frequency. The second Bloch harmonic is visible on some traces.

The dependence of the sideband amplitude on ∆n is compared to predictions based on Eq. (1.3) on Fig. (3), for three frequencies of the triangular modulation. The agreement with the spectrum calculated with the inductance modulation deduced from the sample parameters is satisfactory at low frequency. The amplitude of an even sideband is maximum when it corresponds to a Bloch line. The dependence on ∆n was checked up to k=40. The agreement strongly degrades when the sweep frequency increases f ≥ 1MHz . This discrepancy might arise from parity changes during the sweeps and from Zener transitions. High frequency operation of the quantronium should indeed be ultimately limited by Zener tunneling towards upper Bloch bands. The gap with the first excited band normally vanishes at the points {N g = 1/ 2 mod(1), δ = π } . In order to avoid this zero gap point, an asymmetry was introduced between the transistor junctions, yielding a designed gap frequency of 2 GHz. This allowed to operate the device close to δ = π where the gate-charge modulation of the inductance is maximum. The calculated Zener transition rate for this gap does not account however for the observed sweep frequency dependence of the spectrum in the present experiment.

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boulant

75

4. Perspectives for metrology and for mesoscopic physics The observation of Bloch oscillations in a Josephson circuit is a first step towards their use for the metrology of currents. A significant effort is presently devoted to close the quantum metrology triangle, which relates time, voltage and current units. The production of a current directly related to a frequency, or reversely, would allow to check consistency with the Josephson and Quantum Hall Effects. The currents presently produced by single electron pumps, in the pA range, are however too small compared to the currents needed in Quantum Hall Effect experiments, even when using cryogenic current comparators based on superconducting transformers. Metrological current sources in the 100 pA are clearly needed to improve the accuracy, presently limited at about 10 −6 .

experiment

f = 0.1 MHz f = 1 MHz f = 10 MHz

-60

theory

sideband at 2f

-80 -100

0

1

2

3

4

level (dBm)

5

6 0

1

2

3

4

5

6

2

3

4

5

6

2

3

4

5

6

2

3

4

5

6

sideband at 4f

-60 -80 -100

0

1

2

3

4

5

6 0

1

sideband at 6f

-60 -80 -100

0

1

2

3

4

5

6 0

1

sideband at 8f

-60 -80 -100

0

1

2

3

∆n

4

5

6 0

1

∆n

Fig (3). Comparaison of the measured sideband amplitude dependence (left) with the predicted one (right), for the even sidebands at three different sweep frequencies. A good agreement is found only at the lowest frequency, with a maximum when the sideband corresponds to a Bloch line. The amplitude dependence departs from the predicted one for sweep frequencies ≥ 1 MHz .

The demonstration of Bloch oscillations at larger frequencies than achieved in this work, with the injection of a dc current, is thus an important goal. In order to inject such a current, the gate capacitor has to be replaced by an impedance large compared to the resistance quantum over a wide frequency range. Large chromium resistors, and linear arrays of junctions have already been used to achieve this goal, and the successful

August 8, 2008

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boulant

76 76 observation of a Bloch nose in the current-voltage characteristics of a small Josephson junction already proves that the Bloch oscillation regime is attainable [7]. The phenomenon of quantum phase slippage in an ultrasmall superconducting wire was also recently proposed to reach this regime [9]. Our experiment is also related to the electron counting experiment [10], which probes the voltage oscillations of a small electrode in a tunnel junction array with a radiofrequency single electron transistor (RFSET). In our experiment, the passage of a Cooper pair through the device induces a cyclic evolution of the microwave reflection coefficient. This time dependence results in Bloch lines in the spectrum, but its direct observation in the time domain is also possible, and was performed in the present experiment up to Bloch frequencies of a few kHz. Counting the number of periods completed in a given time, which could be achieved by processing the demodulation quadrature signals, would provide a direct measure of the current. The transposition of the injected current into the frequency domain may also be useful in mesoscopic physics since the shape of the Bloch lines is determined by the fluctuations of the injected current. The measured spectrum is thus related to statistical properties of the injected current, such as the third moment of its Full Counting Statistics. In conclusion, we have observed Bloch oscillations in the quantronium circuit by microwave reflectometry when suitable signals are applied to the gate electrode of the device. The Bloch sidebands observed in the spectrum vary as predicted theoretically with the amplitude and offset of the gate modulation at small sweep frequency.

Acknowledgments We acknowledge M. Devoret for discussions and for providing us with a sample for preliminary experiments, H. Mooij, D. Haviland, H. Grabert, and F. Hekking for discussions, and P.F. Orfila, P. Sénat, and M. Juignet for technical help. This work was supported by the european project Eurosqip.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

F. Bloch, Z. Phys. 52, 555 (1928). T. Hartmann et al., New J. Phys. 6 2 (2004). C. Waschke et al., Phys. Rev. Lett. 70 3319 (1993). M.G. Raizen, C. Salomon and Q. Niu, Phys. Today 50 30 (1997). K.K. Likharev and A.B. Zorin, J. Low. Temp. Phys. 59, 347 (1985). L.S. Kuzmin and D.B. Haviland, Phys. Rev. Lett. 67, 2890 (1991). M. Watanabe and D.B. Haviland, Phys. Rev. B 67, 094505 (2003). D. Vion et al., Science 296, 886 (2002). J.E. Mooij and Yu. V. Nazarov, cond-mat/0511535. J. Bylander, T. Duty, and P. Delsing, Nature 434, 361 (2005).

July 21, 2008

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Giazottoproceeding2

77

MANIPULATION OF MAGNETIZATION IN NONEQUILIBRIUM SUPERCONDUCTING NANOSTRUCTURES∗

F. GIAZOTTO, F. TADDEI, AND F. BELTRAM NEST CNR-INFM and Scuola Normale Superiore, Pisa, I-56126, Italy E-mail: [email protected] R. FAZIO NEST CNR-INFM and Scuola Normale Superiore, Pisa, I-56126, Italy and International School for Advanced Studies (SISSA), Trieste, I-34014, Italy

Electrostatic control of the magnetization of a normal mesoscopic conductor is analyzed in a hybrid superconductor-normal-superconductor system. This effect stems from the interplay between the non-equilibrium condition in the normal region and the Zeeman splitting of the quasiparticle density of states of the superconductor subjected to a static in-plane magnetic field. Unexpected spin-dependent effects such as magnetization suppression, diamagnetic-like response of the susceptibility as well as spin-polarized current generation are the most remarkable features presented. The impact of scattering events is evaluated and let us show that this effect is compatible with realistic material properties and fabrication techniques.

1. Introduction The interplay between out-of-equilibrium transport and superconductivity was recently successfully exploited in a number of systems in order to implement Josephson transistors 1,2,3 , π junctions 4 and electron microrefrigerators 5 , just to mention a few relevant examples. In this work we explore its potential in the area of magnetism 6 and spintronics 7 and present a novel approach to control the magnetization and spin-dependent properties of a mesoscopic normal conductor 8 . In particular, we show that manipulation of the (nonequilibrium) distribution of a normal metal through an applied voltage can lead to the control of a number of spin-dependent phenomena. The key ingredients are superconductor electrodes (with energy gap Δ) and a weak external magnetic field. As we shall argue, the interplay between Zeeman splitting and nonequilibrium yields dramatic consequences on quasiparticle dynamics stemming from the peculiar shape of the superconductor DOS whose energy gap compares well with magnetic fields readily accessible experimentally. ∗ This

work was supported in part by MIUR under FIRB “Nanotechnologies and Nanodevices for Information Society”, contract RBNE01FSWY and by RTN-Spintronics.

July 21, 2008

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Giazottoproceeding2

78 →

H RI

(a)

N

S

RI

S

VC

(b)

N

S

S

VC Figure 1. Scheme of the structure investigated. An in-plane static magnetic field H is applied across the whole SINIS system ((a), a-type setup) or localized at the S electrodes ((b), b-type setup). A finite voltage bias VC drives the normal metal out-of-equilibrium allowing to control its magnetization. The N wire is assumed quasi-one-dimensional.

2. Operation and discussion Let us consider the system sketched in Fig. 1. It consists of two superconducting reservoirs (S) connected by a mesoscopic normal metal wire (N) through tunnel contacts (I) of resistance RI . The structure is biased at a voltage VC and in the presence of a static in-plane magnetic field H, applied either across the whole structure (Fig. 1(a), in the following referred to as a-type setup) or localized at the superconductors (Fig. 1(b), b-type setup). For the sake of simplicity let us assume a symmetric structure (a resistance asymmetry would not change the overall physical picture). As for the superconductors we focus on conventional low criticaltemperature thin (< 10 nm) films. In this case the effect of H on the electron spin becomes dominant and, assuming negligible spin-orbit interaction 9 , the superconductor DOS per spin is BCS-like but shifted by the Zeeman energy (EH = μB H), NσS (ε) = NFN |Re[(ε + σEH )/2 (ε + σEH )2 − Δ2 ]| 10 , where ε is the quasiparticle excitation energy measured from the Fermi energy (εF ), NFN is the DOS in the normal state at εF (2 spin directions), μB is the Bohr magneton, and σ = ±1 refers to spin parallel(antiparallel) to the field. At a finite bias VC , in the presence of H and in the limit of negligible inelastic collisions, the steady-state distribution functions in the metal wire are spin-dependent and are given by 11 fσ (ε, VC , H) =

NσL F L + NσR F R , NσL + NσR

(1)

where F L(R) = f0 (ε±eVC /2), NσL = NσS (ε+eVC /2), NσR = NσS (ε−eVC /2), f0 (ε) is the Fermi distribution at lattice temperature T and e is the electron charge. Owing to the nonequilibrium regime driven by the applied electric field, the quasiparticle distributions corresponding to different spin species behave differently, f+(−) being shifted towards lower(higher) energy. The magnetic properties of the N region are entirely determined by its (spin-dependent) quasiparticle distribution functions.

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Giazottoproceeding2

79

μB H/Δ = 0.4

6

6

T = 0.1Tc

0 0

(a)

ε

0.9

+

0.6 0.4

0.1 1

2

eVC /Δ

m-3) -1

(JT

T = 0.1Tc

ε

+

_

0.4 0.5

1.0

eVC /Δ

0.4

1.5

μ-

μ+

= Pauli

0.3

(b)

ε

2μB H N(ε)

2.0 = 0

+

εF

N(ε)

_

=0

+

_

-1

0

eVC /Δ

1

2 μB H

N(ε)

= - Pauli

(c)

0.0 -2

N(ε)

μμ+

0

0.2

2 μB H

ε

(b’)

0.2

-12 0.0

μB H/Δ

εF

_

3

μB H/Δ = 0.1 -6

Nonequilibrium

(a’)

0.8

3

0.2 0.1

0 0

T /Tc = 0.95 Equilibrium

(JT-1m-3)

0.3

(JT

-1

m-3)

12

-0.5 -1.0

χ/χPauli

2

Figure 2. (a) Magnetization density M vs bias voltage VC at T = 0.1 Tc for different magnetic fields (H) for a-type setup (see Fig. 1(a)). Inset: M vs VC for different temperatures at H = 0.2 Δ/μB . (b) The same as in (a) for b-type setup. (a’) Schematic diagrams of the N region density of states and quasiparticle occupation both at equilibrium (left) and nonequilibrium (right) for atype setup. (b’) The same as in (a’) for b-type setup. (c) Contour plot of the normalized magnetic susceptibility χ/χP auli vs VC and H at T = 0.1 Tc for b-type setup.

The magnetization density in the wire is indeed given by M(VC , H) = μB dε [N+N (ε)f+ (ε) − N−N (ε)f− (ε)],

(2)

where NσN (ε) = 12 N N (εF + ε + σμB H) and N N (ε) is the N region DOS in the absence of magnetic field. The function M(VC , H) vs VC is displayed in Fig. 2(a,b) for different magnetic-field values. We assumed a silver (Ag) N region (with NFN = 1.03 × 1047 J−1 m−3 ) at temperature T = 0.1 Tc , where Tc = (1.76 kB )−1 Δ = 1.196 K is the critical temperature of bulk aluminum (Al, the material forming the S regions) and kB is the Boltzmann constant. When H is applied across the whole SINIS structure (a-type setup), M decreases upon increasing VC starting from its equilibrium value MP auli = μ2B NFN H typical of a Pauli paramagnet 6 (see Fig. 2(a)). M shows a complete suppression for VC Δ/e, i.e. the N region is demagnetized. The inset of Fig. 2(a) shows how M(VC ) is weakly dependent on the lattice temperature up to T = 0.4Tc owing to the BCS Δ(T ) dependence together with the temperature-induced broadening of f0 (ε). Conversely, when the magnetic field is localized at the S electrodes (b-type setup) a negative magnetization is induced in the wire (see Fig. 2(b)). Note that M is antiparallel to H. Therefore, the N region behaves as a “diamagnet”. For eVC Δ the wire susceptibility χ (shown in Fig. 1(c) at T = 0.1 Tc ) reaches the Pauli value but with opposite sign χ = ∂M/∂H = −μ2B NFN = −χP auli . This gives rise to a sort of “artificial” Pauli diamagnetism. Insight into the physical origin of this superconductivity-controlled magnetism

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80

can be qualitatively gained by considering the (zero-temperature) steady-state DOS diagrams of Fig. 2(a’,b’), where the normal metal is described by parabolic subbands typical of a free electronlike paramagnetic conductor such as silver. At equilibrium the occupation of quasiparticle states is identical for both spin species leading to M = MP auli and M = 0 for a-type and b-type setups, respectively. When VC = 0, electron distributions for the two spin populations are characterized by distinct chemical potentials μσ . Since μ+ < μ− the occupation of spin states antiparallel to the magnetic field is favored with respect to the parallel one, owing to the opposite energy shift of the superconductor spin-dependent DOS in the external magnetic field. This leads to a reduction of M for the a-type setup and to negative magnetization for the b-type setup. In particular, at VC ∼ Δ/e the chemical potential separation is δμ = μ+ − μ− ∼ −2μB H for both setups. This shows a full electrostatic control of the magnetization, a unique feature of the present system. The experimental accessibility of this operational principle must be carefully assessed. Electrons in metals experience both elastic and inelastic collisions. The latter drive the system to equilibrium and can be expected to hinder the observation of the phenomena discussed here. Our analysis will show a remarkable robustness of these effects. At low temperatures (typically below 1 K) electron-electron scattering 12 , and scattering with magnetic impurities 13,14 are the dominant sources of inelastic collisions 15,14,16 . The effect of electron-electron scattering due to direct Coulomb interaction on the spin-dependent distributions can be accounted for by solving a pair of coupled stationary kinetic equations. This can be done by generalizing the method outlined in Ref. 17 to a spin-dependent system.√The strength of the screened electron-electron interaction 18 is given by Kcoll = (L/ 2)(RI /RK ) Δ/D 19,20,17 , where RK = h/2e2 , L is length and D the wire diffusion constant. We stress that Kcoll is linear both in the wire length and in the tunnel contact resistance. We analyzed quantitatively a realistic Ag/Al SINIS microstructure 5 with L = 1 μm, wire cross-section A = 0.2 × 0.02 μm2, and RI = 103 Ω. Figure 3(a) illustrates the effect of electron-electron scattering. We solved the kinetic equations with H = 0.4 Δ/μB , VC = 1.8 Δ/e and T = 0.1 Tc for several Kcoll values from negligible (Kcoll = 0, square), to moderate (Kcoll = 1, circle) and extreme (Kcoll = 100, solid line) 18,19 . As expected, electron-electron interactions have virtually no impact. By increasing the strength of Coulomb interaction the quasiparticle distribution of each spin species relaxes toward spin-dependent Fermi functions still characterized by different chemical potentials (a similar effect is expected in the presence of interaction with the lattice phonons 21 ). As a result the nonequilibrium magnetization in the normal wire here presented is virtually unaffected. The situation drastically changes if we assume the presence of magnetic impurities in the N region, due to the resulting spin-flip processes. Above the Kondo temperature (TK ), the distribution functions can be calculated including in the kinetic equations an additional term derived by generalizing the theory developed by G¨ oppert and Grabert in Ref. 22 . It is noteworthy to mention that its strength

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81

coll

fσ (ε)

VC = 1.8 Δ/e H = 0.4 Δ/μB T = 0.1Tc

(a) -0.3

ε /Δ

0.3

0.4

0 -0.8

0.5

cm

0 0.1 1 20

0.0

ε /Δ

0.0

0.4

0.4

ε /Δ

0.8

1 -4

-1

fσ (ε)

0 0.1 1

(c) -0.4

-0.4

0

1

0 -0.8

cm(ppm)

VC = 1.8 Δ/e H = 0.4 Δ/μB T = 0.1Tc

(b)

-0.4

-3

-0.5

(JT m )

0

1

0 1 100

fσ (ε)

1

0.8

-8

cm= 0

0.001

T = 0.1Tc H = 0.4 Δ/μB (d)

-12 0

1

0.1 0.01

eVC /Δ

2

3

Figure 3. (a) Spin-dependent distribution functions fσ (ε) vs energy ε calculated for three Kcoll values at H = 0.4 Δ/μB , VC = 1.8 Δ/e and T = 0.1 Tc . Solid(dashed) lines correspond to antiparallel(parallel) spin species. (b) fσ (ε) vs ε calculated for various cm values at H = 0.4 Δ/μB , VC = 1.8 Δ/e and T = 0.1 Tc for a-type setup. (c) The same as in (b) for b-type setup. (d) Magnetization density M vs VC at H = 0.4 Δ/μB and T = 0.1 Tc for different magnetic impurity concentration. Open circles refer to b-type setup, filled triangles to a-type setup for cm = 0.001. The latter were shifted by −MP auli . Data in (b)-(d) were obtained assuming D = 0.02 m2 s−1 , TK = 40 mK and S = 12 .

turns out to be proportional, apart from the electron and magnetic impurity spin coupling constant, to the total number of magnetic impurities present within the wire volume (i.e., to the product cm LA, with cm the impurity concentration) and RI . The resulting distribution functions relative to the a-type setup are shown in Fig. 3(b) at H = 0.4 Δ/μB , VC = 1.8 Δ/e and T = 0.1 Tc ≈ 120 mK for various cm values expressed in parts per million (ppm). We assumed D = 0.02 m2 s−1 (typical of high-purity Ag), magnetic impurities with spin S = 12 , and TK = 40 mK (as appropriate, for example, for Mn impurities in Ag) 24 . By increasing cm , examination of the figure immediately shows that spin-dependent distributions are marginally affected even for impurity concentrations as large as 20 ppm. This shows that in the a-type setup the nonequilibrium M is relatively insensitive to large amounts of magnetic impurities. Figure 3(c) shows the fσ (ε) calculated for various cm values for the b-type setup. In such a case, by contrast, the spin-dependent distribution functions tend to merge for much lower values of cm thus suppressing the induced magnetization. In the presence of a magnetic field across the N region (a-type setup) impurity spins tend to polarize yielding a suppression of spin-flip relaxation processes for the field intensities of interest here 13,22,23,14 . This does not occur in the b-type setup and makes magnetic impurities more effective in mixing spins. The full behavior of M(VC ) for b-type setup at T = 0.1 Tc and H = 0.4 Δ/μB is

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displayed in Fig. 3(d) for several cm values (open circles). For comparison, M(VC ) for a-type setup (filled triangles) is shown at low impurity concentration. We wish to underline the robustness of the induced magnetization, M being suppressed only for rather large concentrations: the latter can in fact be limited to less than 0.01 ppm in currently available high-purity metals 24 . 3. Conclusions In conclusion, we have presented a scheme to control the magnetic properties of a mesoscopic metal. Magnetism suppression as well as artificial Pauli diamagnetism can be accessed in metal-superconductor microstructures thus making available a number of characteristics of much relevance in light of possible applications: (1) Full-electrostatic control of magnetization over complex nanostructured metallic arrays for enhanced performance and optimized device geometries; (2) reduced power dissipation (10−14 ÷ 10−11 W depending on the control voltage) owing to the very small driving currents intrinsic to SIN junctions; (3) high magnetization switching frequencies up to 1011 Hz 17 ; (4) ease of fabrication that can take advantage of the well-established metal-based tunnel junction technology. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

A.F. Morpurgo, T. M. Klapwijk, and B. J. van Wees, Appl. Phys. Lett. 72, 966 (1998). A. M. Savin et al., Appl. Phys. Lett. 84, 4179 (2004). F. Giazotto et al., Appl. Phys. Lett. 83, 2877 (2003). J. J. A. Baselmans et al., Nature 397, 43 (1999). J. P. Pekola et al., Phys. Rev. Lett. 92, 056804 (2004). K. Yosida, Theory of Magnetism, Springer Series in Solid-State Sciences, Vol. 122 (Springer-Verlag, Berlin, 1996). I. Zutic, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). F. Giazotto, F. Taddei, R. Fazio, and F. Beltram, Phys. Rev. Lett. 95, 066804 (2005). P. M. Tedrow and R. Meservey, Phys. Rev. Lett. 27, 919 (1971). R. Meservey, P. M. Tedrow, and P. Fulde, Phys. Rev. Lett. 25, 1270 (1970). D. R. Heslinga and T. M. Klapwijk, Phys. Rev. B47, 5157 (1993). B. L. Altshuler and A. G. Aronov, in Electron-Electron Interactions in Disordered Systems, edited by A. L. Efros and M. Pollak (Elsevier, Amsterdam, 1985). A. Kaminski and L. I. Glazman, Phys. Rev. Lett. 86, 2400 (2001). A. Anthore et al., Phys. Rev. Lett. 90, 076806 (2003). H. Pothier et al., Phys. Rev. Lett. 79, 3490 (1997). K. E. Nagaev, Phys. Rev. B52, 4740 (1995). F. Giazotto et al., Phys. Rev. Lett. 92, 137001 (2004). B. L. Altshuler and A. G. Aronov, Zh. Eksp. Teor. Fiz. 75, 1610 (1978) [Sov. Phys. JETP 48, 812 (1978)]. A. Kamenev and A. Andreev, Phys. Rev. B60, 2218 (1999). B. Huard et al., Solid State Commun. 131, 599 (2004). M. L. Roukes et al., Phys. Rev. Lett. 55, 422 (1985). G. G¨ oppert and H. Grabert, Phys. Rev. B68, 193301 (2003). G. G¨ oppert et al., Phys. Rev. B66, 195328 (2002). F. Pierre et. al., Phys. Rev. B68, 085413 (2003).

Superconducting Qubits

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July 21, 2008

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brl˙conf˙proceeding

85

DECOHERENCE AND RABI OSCILLATIONS IN A QUBIT COUPLED TO A QUANTUM TWO-LEVEL SYSTEM

S. ASHHAB1, J. R. JOHANSSON1 AND FRANCO NORI1,2 1

Frontier Research System, The Institute of Physical and Chemical Research (RIKEN) Wako-shi, Saitama, Japan 2 Center for Theoretical Physics, CSCS, Department of Physics, University of Michigan Ann Arbor, Michigan, USA

In this paper we review some of our recent results on the problem of a qubit coupled to a quantum two-level system. We consider both the decoherence dynamics and the qubit’s response to an oscillating external field.

1. Introduction Significant advances in the field of superconductor-based quantum information processing have been made in recent years1 . However, one of the major problems that need to be treated before a quantum computer can be realized is the problem of decoherence. Recent experiments on the sources of qubit decoherence saw evidence that the qubit was strongly coupled to quantum two-level systems (TLSs) with long decoherence times2 . Furthermore, it is well known that the qubit decoherence dynamics can depend on the exact nature of the noise causing the decoherence. Therefore, an environment comprised of a large number of TLSs that are all weakly coupled to the qubit will generally cause non-Markovian decoherence dynamics in the qubit. The two above observations comprise our main motivation to study the decoherence dynamics of a qubit coupled to a quantum TLS. A related problem in the context of the present study is that of Rabi oscillations in a qubit coupled to a TLS. That problem is of great importance because of the ubiquitous use of Rabi oscillations as a qubit manipulation technique. We perform a systematic analysis with the aim of understanding various aspects of this phenomenon and seeking useful applications of it. Note that the results of this analysis are also relevant to the problem of Rabi oscillations in a qubit that is interacting with other surrounding qubits. This paper is organized as follows: in Sec. 2 we introduce the model system and Hamiltonian. In Sec. 3 we analyze the problem of qubit decoherence in the absence of an external driving field. In Sec. 4 we discuss the Rabi-oscillation dynamics of the qubit-TLS system. We finally conclude our discussion in Sec. 5.

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2. Model system and Hamiltonian The model system that we shall study in this paper is comprised of a qubit that can generally be driven by an harmonically oscillating external field, a quantum TLS and their weakly-coupled environment3 . The Hamiltonian of the system is given by: ˆ ˆ q (t) + H ˆ TLS + H Î + H ˆ Env , H(t) =H

(1)

ˆ q and H ˆ TLS are the qubit and TLS Hamiltonians, respectively, H ˆ I describes where H ˆ the coupling between the qubit and the TLS, and HEnv describes all the degrees of freedom in the environment and their coupling to the qubit and TLS. The (generally time-dependent) qubit Hamiltonian is given by: ˆ q (t) = − Eq sin θq σ ˆx(q) + cos θq σ ˆz(q) + F cos(ωt) sin θf σ ˆx(q) + cos θf σ ˆz(q) , (2) H 2 (q)

where Eq and θq are the adjustable control parameters of the qubit, σ ˆα are the Pauli spin matrices of the qubit, F and ω are the amplitude (in energy units) and frequency, respectively, of the driving field, and θf is an angle that describes the orientation of the external field relative to the qubit σ ˆz axis. We assume that the TLS is not coupled to the external driving field, and its Hamiltonian is given by: ˆ TLS = − ETLS sin θTLS σ (3) H ˆx(TLS) + cos θTLS σ ˆz(TLS) , 2 where the parameters and operators are defined similarly to those of the qubit, except that the parameters are uncontrollable. The qubit-TLS interaction Hamiltonian is given by: Î = − λ σ H ˆ (q) ⊗ σ ˆz(TLS) , 2 z

(4)

where λ is the (uncontrollable) qubit-TLS coupling strength. Note that, with an ˆ I .3 appropriate basis transformation, this is a rather general form for H 3. Qubit decoherence in the absence of a driving field We start by studying the effects of a single quantum TLS on the qubit decoherence ˆ Env is small enough that its effect on the dynamics dynamics. We shall assume that H of the qubit+TLS system can be treated within the framework of the Markovian Bloch-Redfield master equation approach. The quantity that we need to study is therefore the 4 × 4 density matrix describing the qubit-TLS combined system. Following standard procedures we can write a master equation that describes the time-evolution of that density matrix. We shall not include that master equation explicitly here. Once we find the dynamics of the combined system, we can trace out the TLS degree of freedom to find the dynamics of the reduced 2 × 2 density matrix describing the qubit alone. From that dynamics we can infer the effect of the TLS on the qubit decoherence dynamics.

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3.1. Analytic results for the weak-coupling limit If we take the limit where λ is much smaller than any other energy scale in the problem6 , and we take the TLS decoherence rates to be substantially larger than those of the qubit, we can perform a perturbative calculation on the master equation and obtain the following approximate expressions for the leading-order corrections (q) (q) to be added to the background relaxation and dephasing rates, Γ1 and Γ2 ( = 1): (q)

δΓ1 ≈ (q)

δΓ2 ≈

(TLS)

(q)

(TLS)

(q)

(q)

Γ2 + Γ2 − Γ1 1 2 2 λ sin θq sin2 θTLS 2 2 (TLS) (q) (q) 2 Γ2 + Γ2 − Γ1 + (Eq − ETLS ) Γ2 − Γ2 1 2 2 λ sin θq sin2 θTLS . 2 4 (TLS) (q) 2 Γ2 − Γ2 + (Eq − ETLS )

(5)

The above expressions can be considered a generalization of the well-known results of the traditional weak-coupling approximation. Those results are obtained if we take the qubit decoherence rates to be much smaller than those of the TLS. We shall discuss shortly, however, that our expressions have a wider range of validity. 3.2. Numerical solution of the master equation Given the large number of parameters that can be varied, we restrict ourselves to certain special cases that we find most interesting to analyze6 . Since the TLS effects on the qubit dynamics are largest when the two are resonant with each other, we set Eq = ETLS . We are therefore left with the background decoherence rates and the coupling strength as free parameters that we can vary in order to study the different possible types of behaviour in the qubit dynamics. We first consider the weak-coupling regimes. Here we only discuss the relaxation dynamics (see Ref.4 for full analysis). Figure 1 shows the relative correction to the qubit relaxation rate as a function of time for three different sets of parameters differing by the relation between the qubit and TLS decoherence rates, maintaining (q) (q) (TLS) (TLS) /Γ1 = 2. As a general simple rule, which is the relation Γ2 /Γ1 = Γ2 inspired by Fig. 1(a), we find that the relaxation rate starts at its unperturbed value and follows an exponential decay function with a characteristic time given by (TLS) (q) (q) + Γ2 − Γ1 )−1 , after which it saturates at a steady-state value given by (Γ2 Eq. (5) (with Eq = ETLS ): dPex (t)/dt (q) (q) (TLS) (q) (q) ≈ −Γ1 − δΓ1 1 − exp − Γ2 + Γ2 − Γ1 t . (6) Pex (t) − Pex (∞) It turns out that all the curves shown in Fig. 1 agree very well with Eq. (6). In the limit when the TLS decoherence rates are much larger than those of the qubit, the qubit decoherence rate saturates quickly to a value that includes the correction given in Eq. (5). In the opposite limit, i.e. when the TLS decoherence rates are much smaller than those of the qubit, the contribution of the TLS to the qubit relaxation dynamics is a Gaussian decay function.

July 21, 2008

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88 −2

x 10

−2

x 10

6

1.2

4

0 0

0.5

(q)

Γ1 t

1

1.5

−1

1

0.8

1

(q)

0.4

x 10

(c) δΓ(q)/Γ(q)

(q)

δΓ1 /Γ1

δΓ(q) /Γ(q) 1 1

0.8

1.2

(b)

(a)

2 0 0

0.5

Γ(q)t

1

1.5

1

0.4

0 0

0.5

(q)

Γ1 t

1

1.5

Figure 1. Relative corrections to qubit relaxation rate as a function of scaled time in the case of (TLS) (q) (a) strongly, (b) moderately and (c) weakly dissipative TLS. The ratio Γ1 /Γ1 is 10 in (a), 1.5 (q)

in (b) and 0.1 in (c). The solid, dashed, dotted and dash-dotted lines correspond to λ/Γ1 0.3, 0.6 and 0.9, respectively. θq = π/3 and θTLS = 3π/8.

= 0,

In the strong-coupling regime corresponding to large values of λ, one cannot simply speak of a TLS contribution to qubit decoherence. We therefore do not discuss that case here. Instead, we discuss the transition from weak to strong coupling. We use the criterion of visible deviations in the qubit dynamics from exponential decay as a measure of how strongly coupled a TLS is. The results of our calculations can be summarized as follows: a given TLS can be considered to interact weakly with the qubit if the coupling strength λ is smaller than the largest background decoherence rate in the problem. We have also checked the boundary beyond which the numerical results disagree with our analytic expressions given in Eq. (5), and we found that the boundary is similar to the one given above. That result confirms the statement made in Sec. 3.2 that our analytic expressions describing the contribution of the TLS to the decoherence rates have a wider range of validity than those of the traditional weak-coupling approximation. 4. Dynamics under the influence of a driving field We now include the oscillating external field in the qubit Hamiltonian (Eq. 2). Furthermore, since decoherence does not have any qualitative effect on the main ideas discussed here, we neglect decoherence completely for most of this section. 4.1. Intuitive picture If we take the experimentally relevant limit λ Eq , we easily find the energy levels to be given by: λcc ETLS + Eq 1 λcc 2 − ; E2,3 = ∓ , (7) E1,4 = ∓ (ETLS − Eq ) + λ2ss + 2 2 2 2 where λcc = λ cos θq cos θTLS , and λss = λ sin θq sin θTLS . If a qubit with energy splitting Eq is driven by a harmonically oscillating field with a frequency ω close to its energy splitting as described by Eq. (2), one obtains

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89

the well-known Rabi oscillation peak in the frequency domain with on-resonance Rabi frequency Ω0 = F | sin(θf − θq )|/2 and full g ↔ e conversion probability on resonance. Note that the width of the Rabi peak in the frequency domain is also given by Ω0 . We can now combine the above arguments as follows: The driving field tries to flip the state of the qubit alone, with a typical time scale of Ω−1 0 , whereas the TLS . Therefore if Ω0 λss , can respond to the qubit dynamics on a time scale of λ−1 ss we expect the TLS to have a negligible effect on the Rabi oscillations. If, on the other hand, Ω0 is comparable to or smaller than λss , the driving field becomes a probe of the four-level spectrum of the combined qubit-TLS system. 4.2. Numerical results (q)

In our numerical analysis, we focus on the quantity P↑,max , which is defined as the maximum probability for the qubit to be found in the excited state between (q) times t = 0 and t = 20π/Ω0 . Figure 2 shows P↑,max as a function of detuning (δω ≡ ω − Eq ) for different values of coupling strength λ. In addition to the splitting of the Rabi peak into two peaks, we see an additional sharp peak at zero detuning and some additional dips. The peak can be explained as a two-photon transition where both qubit and TLS are excited from their ground states to their excited states (note that Eq = ETLS ). The dips can be explained as “accidental” suppressions of the oscillation amplitude when one energy splitting in the four-level spectrum is a multiple of another energy splitting in the spectrum. 1

1 (a)

0.8 0.6

0.8 0.6

0.2

0

0

0 δω/Ω0

2

↑,max

0.4

0.2 −2

P(q)

P(q)

0.4

(c)

0.8 0.6

↑,max

↑,max

P(q)

1

(b)

0.4 0.2

−2

0

δω/Ω0

2

4

0

−2

0

2 4 δω/Ω

6

0

(q)

Figure 2. Maximum qubit excitation probability P↑,max between t = 0 and t = 20π/Ω0 for λ/Ω0 = 0.5 (a), 2 (b) and 5 (c). θq = π/4, and θTLS = π/6.

4.3. Experimental considerations In the early experiments on phase qubits coupled to TLSs2 , the qubit relaxation (q) rate Γ1 (∼40 MHz) was comparable to the splitting between the two Rabi peaks λss (∼20-70 MHz). The constraint that Ω0 cannot be reduced to values much lower than the decoherence rate made the strong-coupling regime, where Ω0 λss , inaccessible. Although the intermediate-coupling regime was accessible, observation of the additional features in Fig. 2 discussed above would have required a time at

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least comparable to the qubit relaxation time. With the new qubit design7 , the qubit relaxation time has been increased by a factor of 20. Therefore, all the effects that were discussed above should be observable. We finally consider how our results can be applied to the problem of characterizing an environment comprised of TLSs. Since measurement of the locations of the three peaks in Fig. 2 provides complete information about the four-level spectrum, both λcc and λss can be extracted from such results. One can therefore obtain the distribution of values of both ETLS and θTLS for all the TLSs in the environment. Note that in some cases, e.g. a phase qubit coupled to the TLSs through the operator of charge across the junction, we find that θq = π/2, and therefore λcc vanishes for all the TLSs. In that case the two-photon peak would always appear at the midpoint (to a good approximation) between the two main Rabi peaks. Although that would prevent the determination of the values of ETLS and θTLS separately, it would provide information about the qubit-TLS coupling mechanism. 5. Conclusion We have studied the problem of a qubit that is coupled to an uncontrollable twolevel system and a background environment. We have derived analytic expressions describing the contribution of a quantum TLS to the qubit decoherence dynamics, and we have used numerical calculations to test the validity of those expressions. Our results can be considered a generalization of the well-known results of the traditional weak-coupling approximation. Furthermore, our results concerning the qubit’s response to an oscillating external field can be used in experimental attempts to characterize the TLSs surrounding a qubit, which can then be used to reduce or eliminate the TLS’s detrimental effects on the qubit operation. Acknowledgments This work was supported in part by the NSA and ARDA under AFOSR contract number F49620-02-1-0334; and also by the NSF grant No. EIA-0130383. One of us (S. A.) was supported by a fellowship from JSPS. References See e.g. J. Q. You and F. Nori, Phys. Today 58 (11), 42 (2005). R. W. Simmonds et al., Phys. Rev. Lett. 93, 077003 (2004). For a more detailed discussion of our assumptions, see Refs.4,5 . S. Ashhab, J. R. Johansson, and F. Nori, cond-mat/0512677. S. Ashhab, J. R. Johansson, and F. Nori, cond-mat/0602577. Note that in this paper we shall only consider the zero temperature case. For a treatment of the finite temperature case, see Ref.4 . Furthermore, we take the energy splitting, which is the largest energy scale in the problem, to be much larger than all other energy scales, such that its exact value does not affect any of our results. 7. J. M. Martinis et al., Phys. Rev. Lett. 95, 210503 (2005).

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

July 21, 2008

12:39


mdkim˙mss06˙rev

91

PHASE-COUPLED FLUX QUBITS: CNOT OPERATION, CONTROLLABLE COUPLING AND ENTANGLEMENT

MUN DAE KIM Korea Institute for Advanced Study, Seoul 130-722, Korea E-mail: [email protected]

Coupling and entanglement of two flux qubits are studied. The three-Josephson junction qubits with threading fluxes are coupled by a connecting loop interrupted by Josephson junctions, but not by mutual inductance. By coupling the phase differences of the flux qubits using the connecting loop instead of the weak induced fluxes, present scheme offers the advantages of a large and tunable level splitting in implementing the controlled-NOT (CNOT) operation. We obtain the coupling strength as a function of the coupling energy of the Josephson junctions in connecting loop. The Josephson junction in the connecting loop can be replaced by dc-SQUID loop and, by varying the magnetic fluxes threading the dc-SQUID’s, we can control the coupling strength of the coupled qubits. We also suggest a scheme to achieve the maximum entanglement of two phase-coupled flux qubits. Bell states are obtained at the ground and excited state of the coupled qubits system, as the energy levels are split and the two-qubit tunnelling channel is opened.

1. Introduction The qubits using Josephson junction device have been proposed as promising candidates of quantum computer owing to the advantages of relatively long decoherence time and possible scalability. These devices are named as the charge, phase and flux qubit. The flux qubit has a long decoherence time compared to the charge qubit that could be affected severely by the background charge fluctuation. The magnetic background, on the other hand, is relatively clean and stable. The flux qubits have shown quantum superposition between two eigenstates.1 The observation of Rabi oscillation has been reported recently for one using three-junction qubit.2 Therefore, the feasibility of the flux qubit as a practical quantum computer increases as time goes by. One of universal logic gates is composed of the controlled-NOT operation and special single-qubit rotations. However, previous coupling schemes which use mutual inductance between nearest neighbor qubits3,4 cannot give strong coupling strength enough to implement two-qubit coherent oscillations and CNOT gate operation. Hence we propose a design of two-qubit coupling connected by a loop carrying a persistent current (Fig. 1) whose coupling can be as large as the coupling energy of the Josephson junctions.5

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2. Phase-Coupled Flux Qubits The boundary condition of a qubit loop becomes approximately 2πn + 2πft − (φ1 + φ2 + φ3 ) = 0,

(1)

where the total flux ft is the sum of external and induced one, ft = f +find = f + LΦs0I , where f ≡ Φext /Φ0 with the external flux Φext and the superconducting unit flux quantum Φ0 = h/2e and Ls is the self-inductance of the qubit loop. Thus we can obtain the relation for the current in the qubit loop given by φR(L)1 + φR(L)2 + φR(L)3 Φ0 nR(L) + fR(L) − IR(L) = − . (2) Ls 2π From this expression we can get for the effective potential Ueff (φ), R(L) Ueff (φR(L)1 , φR(L)2 , φR(L)3 )

=

3

EJi (1 − cos φR(L)i ) + EnR(L) ,

(3)

i=1

where EJi = EJRi = EJLi is the Josephson coupling energy and the induced energy R(L) 2 = IR(L) /2Ls can be represented as5,6 En φR(L)1 + φR(L)2 + φR(L)3 2 Φ20 R(L) En = . (4) nR(L) + fR(L) − 2Ls 2π In order to obtain the effective potential of connecting loop, we use the periodic boundary condition over the circumference of connecting loop with self inductance Ls and obtain the relation for the current in connecting loop such that Φ0 φ + φR + φR1 − φL1 I = − r − L , (5) Ls 2π where r is an integer. The effective potential of connecting loop with the Josephson coupling energy EJ can be represented by Ueff (φL , φR , φR1 , φL1 ) = En + EJ (1 − cos φL ) + EJ (1 − cos φR ),

(6)

Figure 1. Coupled flux qubits with penetrating fluxes of opposite directions, where φ’s are phase differences across the Josephson junctions with the Josephson coupling energies EJ i and EJ in qubit loops and connecting loop, respectively. Here, for example, we show the state, |↑↓, out of four current states. Thick lines show the diamagnetic (paramagnetic) current of left (right) qubit and thin line the current in connecting loop.

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mdkim˙mss06˙rev

93

where En

2 Φ20 φL + φR + φR1 − φL1 = . r− 2Ls 2π

(7)

tot The total effective potential, Ueff , of the coupled qubit system is given by tot L R Ueff (φ) = Ueff (φL1 , φL2 , φL3 ) + Ueff (φR1 , φR2 , φR3 ) + Ueff (φL , φR , φR1 , φL1 ), (8)

where φ = (φL1 , φL2 , φL3 , φR1 , φR2 , φR3 , φL , φR ). The local minima of the total tot (φ) effective potential can be obtained by minimizing the effective potential Ueff tot with respect to φi , i.e., ∂Ueff /∂φi = 0, which results in the current relations at the Josephson junctions as follows: 2πEJ1 sin φR1 Φ0 2πEJ1 −I + IL + sin φL1 Φ0 2πEJi sin φR(L)i IR(L) + Φ0 2πEJ I + sin φR(L) Φ0 I + IR +

= 0,

(9)

= 0,

(10)

= 0,

(11)

= 0,

(12)

where i=2,3. From these coupled equations, we can calculate four energy levels, Ess , corresponding to the two-qubit current states and show the lowest energy level as a function of the external fluxes in Fig. 2(a), where s, s = 1/2 and −1/2 stand for

(a)

(b)

Figure 2. (a) Ground state energy diagram for coupled current qubits. A,B,C, and D denote the resonance lines where two energy levels of 1st excited as well as ground states of two-qubit states |s, s are degenerated. The external flux difference, δfg , between the straight resonance lines originates from the coupling energy. We consider EJ = EJ . (b) Time evolutions of the coupled qubits with t1 = t2 =0.8GHz for EJ = EJ , fR = 0.457 and fL = 0.4837 (upper panel) and EJ = 0.005EJ , fR = 0.4995 and fL = 0.4793 (lower panel). The solid (dashed) lines display the probabilities that the target (control) qubits occupy the corresponding states.

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paramagnetic (spin up, ↑) and diamagnetic (spin down, ↓) current state, respectively. The Hamiltonian for the coupled qubits then can be written as H= (Ess |s, s s, s | + t1 |s, s −s, s | + t2 |s, s s, −s |) , (13) s,s =± 12

where t1 (t2 ) is the single-qubit tunnelling matrix element related with the left (right) qubit. The charging energies of the Josephson junctions in the qubit loop arising from the quantum fluctuation of the Josephson junction phase give rise to the tunnelling between the qubit current states. Figure 2(b) shows the CNOT gate operation. Writing the two-qubit state as |Ψ(t) = c1 | ↓↓+c2 | ↑↓+c3 | ↓↑+c4 | ↑↑ with the initial condition, c1 = c2 = 1 and odinger c3 = c4 = 0, at t = 0, the time evolution of this state is obtained by the Schr¨ equation, HΨ(t) = i∂Ψ(t)/∂t, with the Hamiltonian in Eq. (13). At time Ωt ≈ π with Ω ≡ 2t1 = 2t2 we can see that for the strong coupling case in the upper panel the state | ↑↓ evolves into the state | ↑↑, while the states, | ↓↓ and | ↓↑ do not respond to this operation, which is the CNOT gate operation. However, for the weak coupling strength in the lower panel which corresponds to the usual inductive coupling, the states, | ↓↓ and | ↓↑, also evolve during the operation, which means that weak coupling cannot effectively implement discriminating operation. 3. Controllable Coupling and Entanglement of Phase-Coupled Flux Qubits In real quantum computing the coupling strength should be controllable as well as strong. In Fig. 3(a) the coupling between two qubits can be tunable by varying the control fluxes threading the dc-SQUID’s in the connecting loop,7 where the periodic boundary conditions become φL(R)1 + φL(R)2 + φL(R)3 = 2π(nL(R) + ft,L(R) ),

(14)

φ1

(15)

+

−φ1

φ3 +

= 2π(r + find )+ φ2 = 2π(fL + p),

(φL1 − φR1 ), −

φ3

+

φ4

=

−2π(fR

+ q).

(16)

If we introduce a rotated coordinate such as φp ≡ (φL3 + φR3 )/2, φm ≡ (φL3 − φR3 )/2, φp ≡ (φ1 + φ3 )/2 and φm ≡ (φ1 − φ3 )/2, the energies of two qubit loops, Uqubit , and connecting loop, Uconn , can be given by Uqubit (φm , φp ) = 2EJ1 cos 2φp cos 2φm − 4EJ cos φp cos φm + 2EJ1 + 4EJ , (17) ˜ cos(φ + πf ) cos 2φm , Uconn(φm , φm ) = 4EJ − 4E (18) J m ˜ ≡ E cos πf and f = f = f . where E J J L R Since the induced energy can be negligible, the total effective potential Ueff is given by the sum of the energies in Eqs. (17) and (18) such as Ueff (φm , φp , φm ) = Uqubit (φm , φp ) + Uconn (φm , φm ) and we can calculate the energies, Ess , of the coupled two-qubit states. Then, since the coupling constant J of the coupled qubits

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can be represented as J = (1/4)(E↓↑ + E↑↓ − E↓↓ − E↑↑ ),6 we get J=

EJ2 EJ cos πf EJ1 EJ1 + 2EJ cos πf

(19)

as shown in Fig. 3(b). For f = 0, J is of the order of EJ so that we can obtain sufficiently strong coupling by using Josephson junctions with high coupling energy. By increasing f we can control the coupling strength from the maximum value to zero at f = 0.5. Here the directions of control fluxes, fL and fR , threading the dcSQUID’s in the connecting loop play the key role for achieving controllable coupling. If two control fluxes are in opposite directions, they can be cancelled. However, if two control fluxes are in the same direction or there is only one dc-SQUID loop in the connecting loop, the qubit states become disturbed so that the two-qubit Hamiltonian cannot be described solely by the change of coupling strength. Another important key to quantum information science is the quantum entanglement but, for flux qubits, direct entanglements have not yet been observed. The main reason, we think, is the weak coupling strength between two inductively coupled qubits.8 In this study we show how we can obtain maximal entanglement and thus Bell states using phase-coupled flux qubits. We, for simplicity, consider the coupled qubits in Fig. 1 with one Josephson junction instead of two in connecting loop.9 Then the effective potential Ueff (φˆp , φˆm ) is given by Ueff (φp , φm ) = 2EJ1 (1 + cos 2φp cos 2φm ) + 4EJ (1 − cos φp cos φm ) + EJ (1 − cos 4φm ).

(20)

As increasing the coupling strength, we have found that the energy levels of different spin states are lifted and thus two-qubit tunnelling processes become dominant over the single qubit tunnelling. Thus we introduced the two-qubit tunnelling Hamilto-

(a)

(b)

Figure 3. (a) Phase-coupled flux qubits with a connecting loop interrupted by two dc-SQUID’s. Gray squares denote Josephson junctions with Josephson coupling energy EJ i for qubit loops and EJ for connecting loop. (b)Energies of coupled qubit states and coupling constant J in Eq. (19), where the coupled qubits have two dc-SQUID’s with control fluxes in opposite directions and we set EJ = 0.1EJ .

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Figure 4.

Concurrences for various EJ with fm = 0.

nian in addition to Hamiltonian in Eq. (13) as follows, H2 = (t3 |s, s−s, −s| + t4 |s, −s−s, s|),

(21)

s=± 12

where t3 (t4 ) is two-qubit tunnelling matrix element related with coupled qubits of the same (different) spins. Then the entanglement of the eigenstates can be obtained by calculating the concurrence C(ρ) = max{0, λ1 − λ2 − λ3 − λ4 }, where λ s are √ √ ρ˜ ρ ρ in decreasing order, ρ is the eigenvalues of the Hermitian matrix R ≡ the density matrix of an eigenstate of the total Hamiltonian H and ρ˜ is defined as ρ˜ ≡ (σy ⊗ σy )ρ∗ (σy ⊗ σy ).10 In Fig. 4 we can see that maximal entanglements can be observed for strong coupling but, for weak coupling case of EJ /EJ = 0.005 which corresponds to the inductive coupling, we can expect just partial entanglement. We actually calculated the coefficients of eigenstates and found that the Bell states, √ |Φ+ = √12 (| ↓↓ + | ↑↑) and |Φ− = (1/ 2)(| ↓↓ − | ↑↑), are formed at ground state and 1st excited states at the co-resonance point fR = fL = 0.5, repectively. We believe that present coupling scheme is experimentally achievable and thus experimental implementation is now invoked. References 1. J. E. Mooij et al., Science 285, 1036 (1999); Caspar H. van der Wal et al., Science 290, 773 (2000). 2. I. Chiorescu et al., Science 299, 1869 (2003). 3. J. B. Majer et al., Phys. Rev. Lett. 94, 090501 (2005). 4. Y.-x. Liu, L. F. Wei, J. S. Tsai, and F. Nori, Phys. Rev. Lett. 96, 067003 (2006); P. Bertet, C. J. P. M. Harmans, J. E. Mooij, Phys. Rev. B 73, 064512 (2006); B. L. T. Plourde et al., Phys. Rev. B 70, 140501(R) (2004). 5. M. D. Kim and J. Hong, Phys. Rev. B 70, 184525 (2004). 6. M. D. Kim, D. Shin, and J. Hong, Phys. Rev. B 68, 134513 (2003). 7. M. D. Kim, Phys. Rev. B 74, 184501 (2006). 8. A. Izmalkov et al.,Phys. Rev. Lett. 93, 037003 (2004). 9. M. D. Kim and S. Y. Cho, Phys. Rev. B, to be published. 10. W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).

August 8, 2008

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CHARACTERISTICS OF A SWITCHABLE SUPERCONDUCTING FLUX TRANSFORMER WITH A DC-SQUID Y. SHIMAZU† AND T. NIIZEKI Yokohama National University, Hodogaya-ku, Yokohama 240-8501, Japan CREST, Japan Science and Technology Agency, Kawaguchi, 332-0012, Japan

We have investigated the flux transfer characteristics of a switchable flux transformer comprising a superconducting loop and a DC-SQUID. This system can be used to couple multiple flux qubits with a controllable coupling strength. Its characteristics were measured by using a flux input coil and a DC-SQUID for readout coupled to the transformer loop in a dilution refrigerator. The observed characteristics were in good agreement with the calculation results. We have demonstrated the reversal of the slope of the characteristics and complete switching-off of the transformer, which are useful features for its application as a controllable coupler.

1. Introduction Superconducting qubits are a promising candidate for the implementation of a scalable quantum computer [1]. Coupling of qubits is necessary for constructing a multiple qubit gate. A flux qubit, which is a superconducting loop interrupted by ultrasmall Josephson junctions [2], can be coupled inductively by means of the flux generated by the circulating currents. Previous experiments on coupled flux qubits employed fixed coupling through mutual inductances [3]. It is very desirable for the coupling to be switchable with a fast switching time in order to realize efficient operation on multiple qubits. Mooij et al. have presented a scheme for a switchable flux transformer to meet this requirement [2]. This transformer is a closed superconducting loop that contains two Josephson junctions in parallel, the structure of a DC-SQUID. The coupling strength can be varied by changing the magnetic flux in the SQUID loop, which is applied by the current in the control coil adjacent to the SQUID loop. We have investigated this original scheme of the switchable flux transformer theoretically and experimentally. A switchable flux transformer using a DC-SQUID with a different configuration has recently been studied by Castellano et al. [4]. Instead of using a control coil, we injected a current in the segment of the DC-SQUID to change the effective magnetic flux in the SQUID loop. This method is advantageous in that the control current can be small owing to the large kinetic inductance associated with a superconducting wire [5], thereby minimizing the influence of the control current on the qubits. The observed flux transfer characteristics will be compared with the calculation results. We note that for the circuit that is the same as the

†

E-mail: [email protected]

97

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98 98

Figure 1. Schematic of the switchable flux transformer. The crosses represent Josephson junctions. An input coil and a DC-SQUID for readout are coupled through mutual inductances Min and Mout in order to measure the characteristics of the transformer. The transformer involves a controlling DC-SQUID. The flux transfer characteristic is controlled by the magnetic flux Φc in the SQUID loop, which is varied by the control current Icont injected in the loop segment of the SQUID. The magnetic flux produced by the circulating current J is detected by the readout SQUID.

present system under investigation, a quantum superposition of two magnetic flux-states was experimentally demonstrated [6]. 2. Theoretical analysis and calculation results Figure 1 shows a schematic of the switchable flux transformer. An input coil and a DCSQUID for readout, which are used to measure the characteristics of the transformer, are also shown. We will present experimental results for the sample described by this schematic. The macroscopic variables that describe the switchable flux transformer are the fluxes Φ in the main loop (inductance L) and Φc in the controlling SQUID loop (inductance l) [7]. The externally applied fluxes for the loops, Φx and Φcx, are given by the currents Icoil and Icont in the input coil and control line, respectively. We assume that the critical currents I0 of the junctions in the SQUID are equal. The 2D potential describing the system is given by U (ϕ , ϕc ) =

Φ 02  1 1 ϕc ϕc  . 2 2  (ϕ − ϕ x ) + γ (ϕc − ϕcx ) − β 0 cos(ϕ + ) cos( )  4π 2 L  2 2 2 2 

(1)

where γ = L / l , β 0 = 4π I 0 L / Φ 0 , ϕ = 2πΦ / Φ 0 , ϕ x = 2πΦ x / Φ 0 , ϕc = 2πΦ c / Φ 0 , ϕcx = 2πΦcx / Φ0 , and Φ0 is the flux quantum. In the case of γ >> 1 similar to our sample, ϕc is frozen to the equilibrium value ϕcx. Then, the effective 1D potential for ϕ is given by

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shimazu

99 U (ϕ ) =

Φ 02  1 ϕcx ϕ  2 ) cos( cx )  .  (ϕ − ϕ x ) − β 0 cos(ϕ + 4π 2 L  2 2 2 

(2)

In the classical ground state of the system, ϕ is fixed as the minimum of this 1D potential. The circulating current in the main loop is given by J=

Φ0 (ϕ − ϕ x ) . 2π L

(3)

The equation that determines the equilibrium value of J as a function of ϕx and ϕcx is expressed as follows: j + β 0 sin( j + ϕ x +

ϕcx 2

) cos

ϕcx 2

= 0,

(4)

where j = 2π LJ / Φ0 . This equation implies that the current J is a periodic function of Φx and Φcx with a period of Φ0. The variation in J can be detected by the readout DC-SQUID coupled to the main loop of the transformer. The overall flux transfer function is determined by the response of J to Icoil. This is also a periodic function of Icont, which is proportional to Φcx, and can be calculated using Eq. (4).

Figure 2. Calculated circulating currents in the transformer main loop as a function of the normalized magnetic flux fx externally applied to the loop. The results for various values of fcx, which is the normalized magnetic flux in the control SQUID loop, are compared. The curves are offset for better visibility. The range of fcx is (a) from 0.2 to 1.8 and (b) from 0.47 to 0.53.

Figures 2 (a) and (b) indicate the calculated circulating currents j in the classical ground state as a function of the normalized input flux fx for various values of the control flux fcx, where fx = Φx/Φ0 and fcx = Φcx/Φ0. The phase of the oscillation of j as a function of fx gradually changes with increasing fcx until fcx is a half-integer. It suddenly jumps as fcx crosses the half-integer value, at which point j is zero and the input flux is not

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100 100 transferred. When fcx is in the vicinity of the half-integer value, the amplitude of the oscillation is small, and the phase change is π as fcx crosses the half-integer value, as shown in Fig. 2 (b). 3. Experimental results and discussion The sample shown in Fig. 1 was fabricated by e-beam lithography. All the superconducting wires and Josephson junctions were made of Al. The junctions were fabricated using the shadow deposition technique [8]; their areas were about 0.06 (µm)2. The critical current of each junction in the DC-SQUID was found to be about 500 nA. The area of the readout SQUID loop is 5700 (µm)2. The readout SQUID was made relatively large in order to obtain a large output flux signal by increasing the coupling between the readout SQUID and the transformer. The experiment was carried out in a dilution refrigerator at a base temperature of 25 mK. We measured the switching current ISW of the readout DC-SQUID as a function of the input current Icoil, the control current Icont, and an external magnetic field. It should be noted that the change in ISW is proportional to the change in the circulating current in the transformer loop.

Figure 3. The switching current ISW of the readout DC-SQUID as a function of the control current Icont. The current Icoil in the input coil was zero, and the external magnetic field was fixed. ISW is modulated periodically with a period of Φ0 with respect to the flux in the SQUID loop of the transformer.

Figure 3 shows the switching current ISW as a function of Icont at a fixed magnetic field with Icoil = 0, while the dependence of ISW on Icoil for various control currents is presented in Figs. 4 (a) and (b). The periodic modulations shown in these figures exhibit the periodic behavior of the circulating current in the transformer as a function of both Φx and Φcx, which was explained in the previous section. The linear background in the figures is due to the spurious coupling of the input coil and control line to the readout SQUID. The opposite tendencies of the backgrounds of Figs. 3 and 4 can be attributed to the directions of the currents Icoil and Icont shown in Fig. 1.

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101 101

Figure 4. The switching current ISW of the readout SQUID as a function of Icoil for various control currents Icont. The periodicity of the signal corresponds to the flux change of Φ0 in the main loop of the transformer. The range of Icont is (a) from -5 to 7 µA and (b) from -5 to -3 µA. The abrupt change in the phase of the oscillation is observed for Icont between -4 and -3 µA and that between 6 and 7 µA.

From the period of the modulation, the mutual inductance between the input coil and transformer main loop is estimated to be 180 pH, while that between the transformer SQUID loop and control current line is determined to be 220 pH. These mutual inductances reasonably agree with the values estimated from the geometry of the sample and the kinetic inductance. It should be noted that the mutual inductance between the transformer SQUID loop and control current line, which share the same line of length 40 µm, is dominated by the kinetic inductance associated with the superconducting wire. We have verified the estimate of the kinetic inductance associated with the sample by directly measuring the effective mutual inductance in a separate sample [9]. In Fig. 4 (a) the gradual change and sudden jump of the phase of the oscillation with increasing Icont is indicated, which is in agreement with the features shown in Fig. 2 (a). Discontinuities of the circulating current, as shown in the calculation, are not observed experimentally. This can be attributed to the thermal excitation and quantum rounding. The experimental data for the case when the normalized magnetic flux in the control SQUID loop fcx is near the half-integer value is shown in Fig. 4 (b). The decrease in the amplitude of the oscillation and the change in the phase, which agree with the calculation shown in Fig. 2 (b), are also observed in this figure. In the system under investigation, the sign of the slope of the flux transfer characteristics can be inverted at a particular value of Icoil by varing Icont, as clearly shown in Fig. 4 (b). This is an attractive feature of the present system because complete switching-off of the transformer, that is, zero response for a small finite change in the input flux, can be achieved at a particular operation point. In contrast, for a similar

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102 102 switchable flux transformer, shown in Fig. 5 (a), complete switching-off is not realized since the slope of the flux transfer characteristics is always positive, as shown in Fig. 5 (b) [4]. This system can be applied to couple not only multiple qubits but also a single qubit to a readout SQUID. The influence of the SQUID readout on the coherence in a flux qubit might be investigated by means of a controllable coupling between the qubit and the readout SQUID.

Figure 5. (a) Schematic of the switchable flux transformer with a different configuration. (b) Calculated circulating currents in the transformer main loop as a function of fx (normalized magnetic flux externally applied to the main loop) for various applied flux Φcx for the control SQUID. The curves are for fcx = Φcx/Φ0 = 0, 0.2, 0.4, 0.46, 0.48, 0.5, 0.52, 0.54, 0.6, 0.8, and 1.

References 1. Y. Makhlin, G. Schön and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). 2. J. E. Mooij, T.P. Orlando, L. Levitov, Lin Tian, Caspar H. van der Wal and Seth Lloyd, Science 285, 1036 (1999). 3. J. B. Majer, F. G. Paauw, A. C. J. ter Haar, C. J. P. M. Harmans and J. E. Mooij, Phys. Rev. Lett. 94, 090501 (2005). 4. M. G. Castellano, F. Chiarello, R. Leoni, D. Simeone, G. Torrioli, C. Cosmelli and P. Carelli, Appl. Phys. Lett. 86, 152504 (2005). 5. J. B. Majer, J.R. Butcher and J. E. Mooij, Appl. Phys. Lett. 80, 3638 (2002). 6. J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo and J. E. Lukens, Nature 406, 43 (2000). 7. S. Han, J. Lapointe and J. E. Lukens, Phys. Rev. Lett. 63, 1712 (1989). 8. G. J. Dolan, Appl. Phys. Lett. 31, 337 (1977). 9. Y. Shimazu and T. Niizeki, Physica C in press.

July 21, 2008

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CHARACTERIZATION OF ADIABATIC NOISE IN CHARGE-BASED COHERENT NANODEVICES

A. D’ARRIGO, G. FALCI, A. MASTELLONE AND E. PALADINO∗ MATIS CNR -INFM, Catania (Italy) & Dipartimento di Metodologie Fisiche e Chimiche (DMFCI), Universit` a di Catania. Viale A. Doria 6, 95125 Catania (Italy)

Low-frequency noise, often with 1/f spectrum, has been recognized as the main mechanism of decoherence in present-day solid state coherent nanodevices. The responsible degrees of freedom are almost static during the coherent time evolution of the device leading to effects analogous to inhomogeneous broadening in NMR. Here we present a characterization of the effects of adiabatic noise exploiting the tunability of nanodevices.

1. Introduction Over the last years impressive progress has been achieved in quantum control of coherent nanodevices. Many early proposals of implementation of quantum bits with superconductors and semiconductors have been demonstrated to exhibit coherent properties in the time domain 1−5 . Solid state nanodevices are influenced by noise sources with broadband spectrum and a variety of statistical features. However they often operate in regimes of limited sensitivity to details of the noise nature, either because of limited control on protocols (responsible for instance of inhomogeneous broadening) or because protocols effectively decouple part of noise sources (echo protocol). This induces to develop approximation schemes including systematically only the relevant information about noise, focusing on the effects of the environment on the controlled dynamics rather than on the specific nature of the noise sources. This is the appropriate point of view in the perspective of using nanodevices as processors for quantum information. The following classes of noise can be identified and studied with “ad hoc” tools: quantum noise responsible for spontaneous decay, adiabatic noise whose main effect is analogous to inhomogeneous broadening in NMR and strongly coupled noise, producing the analogue of uncontrollable chemical shifts 6 . In this paper we focus on low frequency noise which has been recognized as the main mechanism of decoherence for present-day experiments 1,2,7 . It originates from microscopic degrees of freedom in the device with dynamics slow compared to its typical frequencies Ω0 , acting as adiabatic uncontrollable extra driving fields 6 . Examples are charged switching impurities in the oxides or in the substrates 8,9 , or trapped flux tubes in persistent current qubits or nuclear magnetic moments for spin ∗ e-mail:

[email protected]

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qubits. Often low frequency noise has 1/f spectrum 10 , the corresponding degrees of freedom being almost static during the coherent time evolution of the device. Here we discuss a way to characterize the effects of low-frequency adiabatic noise, which can be achieved by exploiting the tunability of nanodevices. Since they may work at different operating points also the resilience of the device to external noise is tunable and may be expressed in a very transparent way in terms of response functions of the system to external perturbations. In this work we will explicitly work out the case of adiabatic charge noise in the Cooper Pair Box (CPB). 2. Cooper Pair Box based nanodevices Design of charge devices presents at least an island of metallic or semiconducting material, connected to the rest of the system by tunnel junctions. If the system is superconducting, the excess charge in the island may be controlled at the level of a single Cooper pair. The one-island circuit is the CPB described by 11 HBOX = EC (ˆ q − qx )2 − EJ cos ϕˆ

[ϕ, ˆ qˆ] = i ,

(1)

qˆ is the excess number of Cooper pairs in the island which may change due to Josephson tunneling, e±iϕ |q = |q ± 1. EC is the charging energy, preventing states with large q − qx to enter the dynamics, and EJ is the Josephson energy. If the dynamics involves only the two lowest energy states, the CPB implements a qubit. The computational states may be superpositions of two or more charge states |q, depending on the ratio EJ /EC . For charge qubits EJ /EC 1 1 , whereas for charge-phase qubits EJ /EC ∼ 1 2 . Control is efficiently operated with electric fields, via a gate electrode capacitively coupled to the island modulating the control parameter qx (t) = Cg Vg (t)/(2e) a . The spectrum of the device depends periodically on the operating point fixed by qx . By operating at the symmetry point qx = 1/2 where the splitting has a minimum, the decoherence time is maximized. The environment couples to the device via the same ports allowing driving 12 . At the quantum level sources of charge noise are accounted for by adding to ˜ = qˆX ˆ + HR . Here HR describes the reserEq.(1) the environment Hamiltonian H ˆ is an environment operator, whose fluctuations correspond to voir alone and X ˆ ↔ −2EC δqx . The environment is characterfluctuations of the gate charge X ized statistically, the gross information being contained in the power spectrum ∞ ˆ X(0) ˆ ˆ X(t) ˆ + X(0) . In solid state devices S(ω) extends S(ω) = −∞ dt eiωt 21 X(t) over several decades, often exhibiting 1/ω behavior at low frequencies. We may ˆ f , where X ˆ f describes fast modes of the environment (frequencies ˆ →X ˆs + X split X ≥ Ω0 ), responsible for quantum noise whose main effect at a low enough temperˆ s corresponds to degrees of freedom slow atures is spontaneous decay. Instead X on the scale of the system dynamics, responsible for adiabatic noise which can be ˆ s → X(t) of frequencies modeled as an additional stray adiabatic classical field X a Resorting

to various SQUID geometries it is also possible to modulate EJ . Since coupling via the EJ port is less effective we consider it as a fixed quantity.

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ω < γM 2 subspace is not equivalent to estimating Ω0 (qx ) by truncating the Hamiltonian to the two lowest states at the optimal point. Discrepancies are more evident with increasing EJ , thus variations of qx , or equivalently fluctuations of X, determine leakage from the Hilbert space spanned by the qubit computational states at the optimal point. Due to the smooth variation of Ω0 (qx ) with gate charge fluctuations, Eq.(3) with the perturbative expansion of Ω0 (qx ) is a good approximation of the exact expression obtained numerically by diagonalization (in a basis with Ns 1 charge states) and direct evaluation of D(qx , t). Dephasing time: The dephasing time τ ∗ (D(τ ∗ ) = −1) depends on the bias qx via θ and Ω0 (qx ). We notice that the only parameter characterizing the environment is σX , whose actual value has a weak dependence on the total duration of the experiment. Figures 1 and 2 show also a comparison between Eq.(3) and the result obtained by numerically integrating Eq.(2) with the exact band-shape. Eq.(3) is √ ∗ −1 2 accurate close to the optimal point c ≈ 0 where [στ (qx )] ≈ s σ/ 2. Away from it small discrepancies are observed with increasing values of the stray polarization distribution width σ. Close to the “pure dephasing” point √ s → 0 the dephasing rate does not depend on σ and reads [στ ∗ (qx )]−1 → |c|/ 2. Because of the band bending by approaching s = 0, this is a region of greater sensitivity to the width σ. Discrepancies with increasing σ when qx = 1/2 are due to contributions to the integral both from the operating frequency Ω0 (qx ) and from the “most stationary” frequency Ω0 ( 12 ), the latter being suppressed for smaller σ. However in this regime Eq.(3) is still a good interpolation formula. This agreement is even better in the c

2

δΩ(qx , X) = c(qx )X +s2 (qx ) 2ΩX(q 0

x)

; c(qx ) = q11 −q00 ;

s2 (qx ) 2

2 − = 2q10

m≥2

2 q1m Ω0 Em −E1

q2 Ω − E 0m−E0 . m

0

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(b)

(√2)/στ*

3

0.1

(c) 2.5

Ω(qx)

(√2)/στ*

0.1

0.01 0

(a)

2

1.5

0.2

0.4

0.6

qx

0.8

1 0

1

0.2

0.4

-0.15

qx

0.6

0.8

1

-0.1

-0.05

0

c

0.05

0.1

0.01 0.15

√ Figure 2. Dephasing rate 2/στ ∗ as a function of qx (a), √ and of c (b) for charge-phase qubits EJ /EC = 1.27 and widths σX /EC = 0.06 (gray), 0.06 2 (black). Continuous line result from diagonalizing exactly HBOX in a Ns = 14 subspace and numerical integration of D(qx , t); dots are obtained by inserting s(qx ) and c(qx ) given in footnote c (parameters obtained diagonalizing HBOX ) into Eq.(3); squares follow the approximate Ω(qx ) resulting from truncating the Hamiltonian to the lowest eigenstates at the optimal point and exact numerical integration of D(qx , t). (c) Exact level splittings and approximate Ω0 (qx ) obtained as in Fig.(1).

charge-phase regime, since the splitting Ω0 (qx ) is smoother, Fig.2. Therefore the series expansion analysis is valid at least for devices exhibiting coherent dynam√ ics 15 . We stress that the simple result [στ ∗ (qx )]−1 = |c|/ 2 differs in two ways from the standard result for quantum noise obtained from the Master Equation 13 . In this latter case the dephasing rate is proportional to the power spectrum at low √ frequencies S(0), whereas for slow noise it is proportional to σX ∼ S. Moreover for quantum noise 1/τ ∗ ∝ c2 therefore the effects of slow and high frequency noise can be to a certain extent discriminated. Indeed, both c(qx ) and s(qx ) have a clear physical interpretation, c = q11 − q00 (qij = φi | qˆ |φj ) representing the charge sensitivity of the device and s2 (qx ) being directly related to the quantum capacitance of the CPB 14 . Interestingly enough we found that characterization of adiabatic noise involves operating regimes where leakage from the two-state Hilbert space of the computational degree of freedom has to be taken into account. Line-shapes: Some characteristic features of low frequency noise may also be inferred from the analysis in frequency domain 7 . Close to the optimal point the asymmetric line-shape peaked at Ω(qx ) is indicative of the form of the distribution p(x), this information being progressively washed out with increasing qx , as shown in Fig.3(a). This is easily understood from the Fourier transform of ρ(t) in the static path approximation, close to the optimal point for Γf → 0, √ p(−a+k 2(w − 1)), where w = ω/EJ , a = 2EC (qx −1/2)/EJ . ρ˜(w) ∝ Θ(w−1) w−1 k=±1

At the optimal point charge noise can only increase the effective level splitting, leading to the high frequency tail 7 . The static path approximation (SPA) has been proven 6,7 to quantitatively explain the observed initial reduction of the signal in the experiment of Ref 2 . On the other side the behavior of the line-shape close to Ω0 (qx ) also reflects the signal decay at longer time scales, thus it is sensitive also to the dynamics of faster degrees of freedom in the environment. These include higher frequency components in the spectrum both slow compared to Ω0 (qx ), like

July 21, 2008

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paladino-revised

108 8e+03

2e+05

6e+03

qx=0.40

4e+04

(b)

(a) 1e+05

2e+04

qx=0.42

4e+03

qx=0.44

8e+04

0 1

qx=0.46

2e+03

qx=0.48

1.00005 1.0001

4e+04

qx=0.50

0

0 1

1.005

1.01

1.015

ω/Δ

1.02

1.025

1

1.00005

1.0001

ω/Δ

Figure 3. Real part of the Fourier transform of ρ(t) (Δ = 1.03 · 1011 rad/s, σ = 1.92 · 10−2 ): (a) close to the optimal point qx = 1/2 (black), for Γf /2Δ = 6 · 10−5 (curves for the indicated values of qx are shifted upwards for visibility). (b) static path approximation Eq.(3) with s = 1, and c = a(1/2) = 0 (black), and stochastic Schr¨ odinger simulations for 1/f noise with γmin /Δ = 10−11 and γmax /Δ = 10−5 (gray) or γmax /Δ = 10−3 (dashed). Inset: convolution with a Lorentzian due to relaxation with Γf /2Δ = 10−5 .

tails of 1/f noise with ω ≈ 106 rad/s Ω0 (qx ), and those responsible for relaxation ω ≈ Ω0 (qx ). Dynamical impurities responsible for 1/f noise broaden the line and slightly shift the maximum to higher frequencies (gray and dashed lines in Fig.(3)(b)). Convolution with a Lorentzian due to relaxation leads to a similar qualitative effect. In experiments where typically both “slow dynamical” fluctuators and quantum noise are present, we expect that the corresponding effects might be hardly distinguishable, whereas the observed asymmetric line-shapes are clear signatures of adiabatic noise components. References 1. Y. Nakamura, Yu. A. Pashkin and J. S. Tsai, Nature 398, 786 (1999). 2. D. Vion et al., Science 296, 886 (2002). 3. Y. Yu et al., Science 296, 889 (2002); J. M. Martinis et al., Phys. Rev. Lett. 89, 117901 (2002); I. Chiorescu et al., Science 299, 1869 (2003); T. Duty et al., Phys. Rev. B 69, 140503(R) (2004); O. Astafiev et al., Phys. Rev. B 69, 180507(R) (2004). 4. T. Hayashi et al., Phys. Rev. Lett. 91, 226804 (2003); J. Gorman, D. G. Hasko and D. A. Williams, Phys. Rev. Lett. 95, 090502 (2005). 5. T. Yamamoto et al., Nature 425, 941 (2003); E. Collin et al., Phys. Rev. Lett. 93, 157005 (2004). 6. G. Falci et al., Phys. Rev. Lett. 94, 167002 (2005). 7. G. Ithier et al., Phys. Rev. B 72, 134519 (2005). 8. E. Paladino et al., Phys. Rev. Lett. 88, 228304 (2002); E. Paladino, L. Faoro and G. Falci, Adv. Solid State Phys. 43, 747 (2003). 9. M. Galperin et al., Phys. Rev. Lett. 96, 097009 (2006). 10. M. B. Weissmann, Rev. Mod. Phys. 60, 537 (1988). 11. Y. Makhlin, G. Sh¨ on and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). 12. G. Falci, E. Paladino, R. Fazio, in “Quantum Phenomena of Mesoscopic Systems”, V. Tognetti and B. Altshuler eds., IOS Press, 2003. 13. C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Atom-Photon Interactions, Wiley-Interscience, New York (1993). 14. T. Duty et al., Phys. Rev. Lett. 95, 206807 (2005). 15. G. Falci et al., in preparation.

Unconventional Superconductors

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THRESHOLD TEMPERATURE OF ZERO-BIAS CONDUCTANCE PEAK AND ZERO-BIAS CONDUCTANCE DIP IN DIFFUSIVE NORMAL METAL / SUPERCONDUCTOR JUNCTIONS

IDURU SHIGETA1,∗, TAKEHITO YOKOYAMA2, YASUHIRO ASANO3 , FUSAO ICHIKAWA4 , AND YUKIO TANAKA2,5 1

Department of General Education, Kumamoto National College of Technology, Kumamoto 861-1102, Japan 2 Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan 3 Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan 4 Department of Physics, Kumamoto University, Kumamoto 860-8555, Japan 5 CREST, Japan Science and Technology Agency (JST), Nagoya 464-8603, Japan

We have studied how zero-bias conductance dip (ZBCD) and zero-bias conductance peak (ZBCP) are smeared by increasing temperature in the diffusive normal metal (DN) / s-wave or d-wave superconductor junctions using the theory based on the KeldyshNambu quasiclassical Green’s function formalism. Tunneling conductance is calculated by changing the magnitudes of the resistance in DN, the Thouless energy in DN, the transparency of the insulating barrier, and the angle between the normal to the interface and the crystal axis of d-wave superconductors. We present a threshold temperature from a possible observation of the ZBCD and ZBCP in line shapes of tunneling conductance.

1. Introduction The low-energy transport in the mesoscopic system is governed by Andreev reflection.1 For diffusive normal metal / superconductor (DN/S) junctions, the phase coherence between incoming electrons and outgoing Andreev holes persists in DN at a mesoscopic length scale. These results in strong interference effects on the probability of Andreev reflection through the proximity effect. The theory based on the Keldysh-Nambu quasiclassical Green’s function formalism has recently explained that the interplay between diffusive and interface scattering produces a wide variety of line shapes of the tunneling conductance, not only for d-wave junctions, but also for s-wave junctions: zero-bias conductance dip (ZBCD), zero-bias conductance peak (ZBCP), gap-like, and rounded bottom structures.2–7 Here, we focus on the temperature dependence of the ZBCD and ZBCP due to the proximity effect, and discuss the thermal broadening effect on line shapes of the tunneling conductance. Then, we give a threshold temperature TTh , which is defined by disappearing temperature of the structure of the ZBCD or ZBCP. ∗ Present

address: Department of Physics, Kagoshima University, Kagoshima 890-0065, Japan.

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2. Formulation We consider a junction consisting of a normal reservoir (N) and a superconducting one connected by a quasi-one-dimensional DN conductor with a resistance Rd and a length L, much larger than the mean-free path . The interface between the DN conductor and the superconductor electrode has a resistance Rb , while the N/DN interface has zero resistance. The positions of the N/DN interface and the DN/S interface are denoted as x = −L and x = 0, respectively. The scattering zone (x = 0) is modeled as an insulating δ-function barrier with the transparency T (φ) = 4 cos2 φ/(4 cos2 φ + Z 2 ), where Z is a dimensionless constant, and φ is the injection angle measured from the interface normal to the junction. The Thouless energy ETh can be expressed by ETh = D/L2 with the diffusive constant D in the DN and the length L of the DN. Here, ETh is determined by the properties of a normal metal as a counterelectrode in tunnel junctions. In the present paper, we adapt the units with kB = = 1. From the retarded or advanced component of the Usadel equation,8 the spatial dependence of θ(x) in the DN is determined by D

∂ 2 θ(x) + 2iE sin [θ(x)] = 0. ∂x2

(1)

Taking account of the Nazarov’s boundary condition at the DN/S interface,3 we obtain L ∂θ(x) F = . (2) Rd ∂x x=0− Rb The average over the angle of injected particles at the interface is defined by π/2 π/2 F = F cos φ dφ T (φ) cos φ dφ, −π/2

(3)

−π/2

and F =

2T (φ) [Γ1 cos θ0 − Γ2 sin θ0 ] , [2 − T (φ)] Γ3 + T (φ) [Γ2 cos θ0 + Γ1 sin θ0 ]

(4)

Γ1 = f + + f − , Γ2 = g + + g − , Γ3 = 1 + g + g − + f + f − , (5) 2 with θ0 = θ(x = 0− ), f± = Δ± (φ) Δ± (φ) − E 2 and g± = E E 2 − Δ2± (φ). E denotes the energy of the quasiparticles measured from the Fermi energy EF . Δ+ (φ) (or Δ− (φ)) is the pair potential felt by the outgoing (or incoming) quasiparticles. The total resistance R in the superconducting state is written by Rd 0 dx Rb + R= , (6) Ib0 L −L cosh2 θIm (x) where Ib0 is obtained from the Keldysh component of the matrix current Iˇ and is a complex function of g± , f± , θ0 , and T (φ).6 By using the relations of σS (E) = 1/R

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and σN (E) = 1/(Rd + Rb ), the normalized tunneling conductance σT (eV ) in finite temperatures is given by ∞ E + eV σS (E) sech2 dE 2T −∞ σT (eV ) = ∞ . (7) E + eV σN (E) sech2 dE 2T −∞ For the equations, we assume Δ(T ) and Δ(T ) cos[2(φ ± α)] as temperature dependent pair potentials of s-wave and d-wave superconductors, where Δ(T ) is the maximum amplitude of the pair potential at a temperature T , Δ0 = Δ(0), and α denotes the angle between the normal to the interface and the crystal axis of d-wave superconductors. It is presumed that Δ(T ) changes on temperatures according to the Bardeen-Cooper-Schrieffer (BCS) theory. 3. Results 3.1. Line shapes of tunneling conductance Figure 1 shows temperature dependence of line shapes of tunneling conductance σT (eV ), which is normalized by its value in the normal state for s-wave superconductors. Here, we choose high transparent junctions with Z = 0, Rd /Rb = 1. The Thouless energy is ETh /Δ0 = 10−4 in Fig. 1(a), ETh /Δ0 = 10−3 in Fig. 1(b), ETh /Δ0 = 10−2 in Fig. 1(c), and ETh /Δ0 = 10−1 in Fig. 1(d), respectively. As shown in Fig. 1, the tunneling conductance has a ZBCD enhancement at T /Tc = 0, 1.5 -4

Z=0 Rd/Rb=1 ETh/Δ0=10

-3


1.3 T/Tc=0.3 T/Tc=0.2 T/Tc=0.1 T/Tc=0.05 T/Tc=0.0

㩷

σT(eV/Δ0)

1.4

1.2 1.1

(a) 㩷

(b)

s-wave

1.5 -1

-2

㩷

1.3 㩷

σT(eV/Δ0)



1.4

1.2 1.1 1.0

(c) -0.4

(d) -0.2

0.0

eV/Δ0

0.2

0.4

-0.4

-0.2

0.0

0.2

0.4

eV/Δ0

Figure 1. Temperature dependence of the normalized tunneling conductance for s-wave superconductor with Z = 0 and Rd /Rb = 1. (a) ETh /Δ0 = 10−4 , (b) ETh /Δ0 = 10−3 , (c) ETh /Δ0 = 10−2 and (d) ETh /Δ0 = 10−1 , respectively.

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where Tc is the transition temperature of superconductors. At zero temperature, the zero-bias conductance σT (0) is independent of ETh and the ZBCD has a width with E ETh . As a result, the ZBCD width at zero temperature widens gradually with increasing ETh . In high transparent interface, the proximity effect involving the coherent Andreev reflection (CAR) suppresses the tunneling conductance because the DN plays a role of the insulating barrier. In this case the probability of the Andreev reflection decreases around zero-bias voltage. This is the reason that the ZBCD appears in line shapes of tunneling conductance for high transparent junctions around zero temperature. The amplitude of pair potentials Δ(T ) for d-wave superconductors is the same as that for s-wave superconductors. We can choose 0 α π/4. It is known that quasiparticles with π/4 − α < φ < π/4 + α can contribute to the midgap Andreev resonant state (MARS) at the interface and are responsible for the ZBCP enhancement in low transparent junctions. It was shown that the proximity effect and MARS do not coexist in the d-wave symmetry.6 In fact, at α/π = 0, the MARS does not exist while the proximity effect is possible. Thus, we can expect similar results of the d-wave symmetry in the case of α/π = 0 to the s-wave symmetry. Therefore, we choose α/π = 0 for d-wave superconductors. In Fig. 2, we plot line shapes of the tunneling conductance as a function of temperatures for the low transparent junction of d-wave superconductors, where Z = 10, Rd /Rb = 1 and α/π = 0. The Thouless energy is ETh /Δ0 = 10−4 in Fig. 2(a), ETh /Δ0 = 10−3 in Fig. 2(b), ETh /Δ0 = 10−2 in Fig. 2(c), and ETh /Δ0 = 10−1 in Fig. 2(d), respectively. The zero-bias conductance σT (0) is independent of ETh at 1.0 -4

-3

Z=10 Rd/Rb=1 ETh/Δ0=10 α/π=0

Z=10 Rd/Rb=1 ETh/Δ0=10 α/π=0

㩷

0.6 㩷

σT(eV/Δ0)

0.8

0.4 0.2

(a)

(b)

d-wave

㩷

1.0 㩷

Z=10 Rd/Rb=1 ETh/Δ0=10 α/π=0

㩷

-1

Z=10 Rd/Rb=1 ETh/Δ0=10 α/π=0

0.6 T/Tc=0.3 T/Tc=0.2 T/Tc=0.1 T/Tc=0.05 T/Tc=0.0

㩷

σT(eV/Δ0)

0.8

-2

0.4 0.2 0.0

(c) -0.4

(d) -0.2

0.0

eV/Δ0

0.2

0.4

-0.4

-0.2

0.0

0.2

0.4

eV/Δ0

Figure 2. Temperature dependence of the normalized tunneling conductance for d-wave superconductor with Z = 10, Rd /Rb = 1 and α/π = 0. (a) ETh /Δ0 = 10−4 , (b) ETh /Δ0 = 10−3 , (c) ETh /Δ0 = 10−2 and (d) ETh /Δ0 = 10−1 , respectively.

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zero temperature. The ZBCP also has a width with E ETh . Hence, the ZBCP width for d-wave superconductors increases with increasing ETh , as well as the ZBCD width for s-wave superconductors. Contrary to the high transparent junctions, the probability of the Andreev reflection increases around zero-bias voltage in low transparent interface.6 Consequently, the ZBCP appears in line shapes of the tunneling conductance for low transparent junctions around zero temperature. 3.2. Tunneling conductance at zero-bias voltage Figures 3(a) and 3(b) show temperature dependence of the normalized tunneling conductance σT (0) at zero-bias voltage in Figs. 1 and 2, respectively. σT (0)’s have 㩷

㩷

1.2

1.4

(a)

(b) Z=10 Rd/Rb=1 α/π=0

Z=0 Rd/Rb=1

0.6

㩷

㩷

σT(0)

1.2

-1

ETh/Δ0=10

σT(0)

0.9

1.3

-2

ETh/Δ0=10

1.1

-3

s-wave 1.0 0.0

0.2

ETh/Δ0=10

d-wave

0.4

0.6

T/Tc

0.8

1.0

0.2

0.3

-4

ETh/Δ0=10

0.4

0.6

0.8

0.0 1.0

T/Tc

Figure 3. Normalized tunneling conductance σT (0) at zero-bias voltage as a function of T /Tc for DN/S tunnel junctions. (a) Z = 0 and Rd /Rb = 1 for s-wave superconductors. (b) Z = 10, Rd /Rb = 1 and α/π = 0 for d-wave superconductors.

a dip around zero temperature as shown in Fig. 3(a) because the probability of the Andreev reflection decreases for high transparent interfaces. On the other hand, the probability of the Andreev reflection increases for low transparent interfaces. As a consequence, σT (0)’s have a peak around zero temperature as shown in Fig. 3(b). Corresponding to the sharp ZBCD and ZBCP in Figs. 1 and 2, σT (0)’s with small ETh tend to be affected by the thermal broadening and quickly change around zero temperature. Thus, σT (0)’s are a decreasing (or increasing) function of ETh /Δ0 for the high (or low) transparent junction especially in low temperatures except for T /Tc = 0. 3.3. Threshold temperature We can give the threshold temperature TTh on the basis of calculated results for temperature dependence of line shapes of tunneling conductance. TTh is defined by disappearing temperature of the structure of the ZBCD or ZBCP with the width of E ETh . Figure 4 shows the TTh plots as a function of ETh , which are obtained from Figs. 1 and 2. As shown in Fig. 4, we found that TTh increases monotonically with increasing ETh in the small ETh region and has a maximum TTh /Tc 0.2

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TTh/Tc

0.2

0.1

ZBCD ZBCP

0.0 -4

10

-3

10

-2

10

-1

10

0

10

ETh/Δ0 Figure 4. Threshold temperature of the ZBCD for s-wave superconductors and the ZBCP for d-wave superconductors in the DN/S junctions. The solid and broken lines are a guide for eyes.

around ETh /Δ0 = 0.01–0.1 for both of the s-wave and d-wave symmetries. Thereafter, TTh becomes zero above ETh /Δ0 = 1 because the ZBCD and ZBCP do not exist inside the coherent peaks of superconductors. TTh has a maximum in the region of ETh /Δ0 0.01–0.1 also for any other parameter values. The result suggests that it is easy to observe the ZBCD and ZBCP due to the proximity effect on the condition of ETh /Δ0 = 0.01–0.1 also in actual measurements. Therefore, we finally estimate several actual TTh ’s from our results. Using the typical values of D = 45 cm2 /s and L = 1 μm for a Ag film as a DN conductor,9 we calculated the value of ETh /Δ0 = 1.0 × 10−4 and TTh = 4.5 K for hightemperature oxide superconductor Bi2 Sr2 CaCu2 O8+δ from Tc = 90 K and Δ0 = 30 meV. Similarly, we obtained the value of ETh /Δ0 = 2.0 × 10−4 and TTh = 1.4 K for conventional superconductor Nb from Tc = 7.2 K and Δ0 = 1.5 meV. 4. Conclusions We have studied the tunneling conductance in DN/S junctions in finite temperatures, and have given the threshold temperature TTh of the ZBCD and ZBCP, due to the proximity effect. The proximity effect decreases σT (0) for high transparent junctions, while increases for low transparent junctions. The ZBCD and ZBCP have a width of E ETh and TTh /Tc has a maximum at ETh /Δ0 = 0.01–0.1. The facts indicate a possibility of distinguishing electronic transport of DN/S junctions by careful comparison of the present calculations and experimental results.

References 1. 2. 3. 4. 5.

A. F. Andreev, Zh. Eksp. Teor. Fiz. 46, 1823 (1964) [Sov. Phys. JETP 19, 1228 (1964)]. A. F. Volkov, A. Z. Zaitsev and T. M. Klapwijk, Physica C210, 21 (1993). Yu. V. Nazarov, Superlattices Microstruct. 25, 1221 (1999). Y. Tanaka, Yu. V. Nazaro and S. Kashiwaya Phys. Rev. Lett. 90, 167003 (2003). Y. Tanaka, A. A. Golubov and S. Kashiwaya Phys. Rev. B68, 054513 (2003).

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6. Y. Tanaka, Yu. V. Nazarov, A. A. Golubov and S. Kashiwaya, Phys. Rev. B69, 144519 (2004). 7. Y. Tanaka, S. Kashiwaya and T. Yokoyama, Phys. Rev. B71, 094513 (2005). 8. K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970). 9. V. T. Petrashov, V. N. Antonov, P. Delsing and T. Claeson, Phys. Rev. Lett. 74, 5268 (1995).

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TUNNELING CONDUCTANCE IN 2DEG/S JUNCTIONS IN THE PRESENCE OF RASHBA SPIN-ORBIT COUPLING

T. YOKOYAMA, Y. TANAKA AND J. INOUE Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan and CREST Japan Science and Technology Cooporation (JST), Nagoya 464-8603, Japan E-mail: [email protected]

We have studied the tunneling conductance in two dimensional electron gas (2DEG) / insulator / superconductor junctions in the presence of Rashba spin-orbit coupling in 2DEG. It is clarified how the tunneling conductance is influenced by the Rashba spin-orbit coupling and found that for low insulating barrier the tunneling conductance is suppressed by the Rashba spin-orbit coupling while for high insulating barrier the tunneling conductance is slightly enhanced by the Rashba spin-orbit coupling in contrast to ferromagnet / superconductor junctions. The results give a possibility to control the Andreev reflection probability by a gate voltage.

1. Introduction In normal metal / supercunductor (N/S) junctions Andreev reflection (AR)1 is one of the most important process for low energy transport. Taking the AR into account, Blonder, Tinkham and Klapwijk (BTK) proposed the formula for the calculation of the tunneling conductance in N/S junctions2 . This method was extended to the ferromagnet / superconductor (F/S) junctions and used to estimate the spin polarization of the F layer experimentally3,4 . In F/S junctions, AR is suppressed because the retro-reflectivity is broken by the exchange field in the F layer5, which causes many interesting phenomena6,7,8 . Spin dependent transport can be important to fabricate a new device manipulating electron’s spin. This field is called spintronics. Spintronics has recently received much attention because of its potential impact on electric devices and quantum computing. Among recent works, many efforts have been devoted to study the effect of spin-orbit coupling on transport properties of two dimensional electron gas (2DEG)9,10,11,12 . The pioneering work by Datta and Das suggested the way to control the precession of the spins of electrons by the Rashba spin-orbit coupling13 in F/2DEG/F junctions14 . This spin-orbit coupling depends on the applied field and can be tuned by a gate voltage. On the other hand spin dependent transport based only on spin-orbit coupling without ferromagnet is also a hot topic15,16 . The Rashba spin-orbit coupling induces an energy splitting, but the energy splitting doesn’t break the time reversal symmetry unlike an exchange splitting in ferromagnet. Therefore transport properties in 2DEG/S junctions may be quali-

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tatively different from those in F/S junctions. However, in 2DEG/S junctions the effect of Rashba spin-orbit coupling on transport phenomena is not studied well. Recent experimental and theoretical advances in spintronics drives us to attack this problem. In the present paper we calculate the tunneling conductance in 2DEG/S junctions and clarify how Rashba spin-orbit interaction affects it. We think the obtained results are useful for the design of mesoscopic 2DEG/S junctions and for a better understanding of related experiments. In the present paper we confine ourselves to zero temperature. 2. Formulation We consider a ballistic 2DEG / superconductor junctions. The 2DEG/S interface located at x = 0 (along the y-axis) has an infinitely narrow insulating barrier described by the delta function U (x) = U δ(x). The effective Hamiltonian with Rashba spin-orbit coupling is given by ⎛ ⎞ ξk iλk− θ (−x) 0 Δθ (x) ⎜ −iλk+ θ (−x) ⎟ ξk −Δθ (x) 0 ⎟ H=⎜ (1) ⎝ −iλk+ θ (−x) ⎠ 0 −Δθ (x) −ξk −ξk Δθ (x) 0 iλk− θ (−x) 2 2 with k± = kx ± iky , the energy gap Δ, ξk = 2m k − kF2 , Fermi wave number kF , Rashba coupling constant λ, and step function θ(x). Velocity operator in the x-direction is given by17 ⎛ ⎞ ∂ iλ 0 0 mi ∂x θ (−x) ⎜ − iλ θ (−x) ∂ ⎟ ∂H 0 0 mi ∂x ⎟. vx = =⎜ (2) ∂ iλ ⎠ − θ (−x) 0 0 − mi ∂kx ⎝ ∂x iλ ∂ 0 0 θ (−x) − mi ∂x Wave function ψ(x) for x ≤ 0 (2DEG region) is represented using eigenfunctions of the Hamiltonian: ⎡

⎛

⎢ ⎜ ⎢ ⎜ eiky y ⎢ √12 eik1(2) cos θ1(2) x ⎜ ⎣ ⎝ ⎛ ⎜ c1(2) −ik1 cos θ1 x ⎜ e + √ ⎜ 2 ⎝

−i kk1+ 1 1 0 0

⎛

⎞ 0 ⎜ 0 ⎟ ⎟ ⎜ ⎟ ⎟ a√ 1 eik1 cos θ1 x ⎜ k1+ ⎟ + ⎟ + 1(2) 2 ⎝ i k1 ⎠ ⎠ 0 1 ⎛ k ⎞⎤ ⎞ 0 i k2+ ⎜ 2 ⎟⎥ ⎟ ⎟ d√ −ik2 cos θ2 x ⎜ 1 ⎟⎥ e ⎜ ⎟ + 1(2) ⎟⎥ 2 ⎝ 0 ⎠⎦ ⎠ 0

(−) i

k1(2)− k1(2)

⎞

⎛

0 ⎜ 0 b1(2) ik2 cos θ2 x ⎜ √ e ⎜ k2+ 2 ⎝ −i k2 1

⎞ ⎟ ⎟ ⎟ ⎠

mλ 2 for an injection wave with wave number k1(2) where k1 = − mλ + + kF2 , k2 = 2 2 mλ mλ 2 + kF2 and k1(2)± = k1(2) e±iθ1(2) . a1(2) and b1(2) are AR coefficients. 2 + 2

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c1(2) and d1(2) are normal reflection coefficients. θ1(2) is an angle of the wave with wave number k1(2) with respect to the interface normal. Similarly for x ≥ 0 ψ(x) (S region) is given by a linear combination of the eigenfunctions. Note that since the translational symmetry holds for the y-direction, the momenta parallel to the interface are conserved: ky = kF sin θ = k1 sin θ1 = k2 sin θ2 . The wave function follows the boundary conditions17 : ψ (x)|x=+0 = ψ (x)|x=−0 vx ψ (x)|x=+0 − vx ψ (x)|x=−0 =

⎛ 2mU mi 2

10 ⎜0 1 ⎜ ⎝0 0 00

⎞ 0 0 0 0 ⎟ ⎟ ψ (0) −1 0 ⎠ 0 −1

Applying BTK theory to our calculation, we obtain the dimensionless conductance represented in the form: θ 2 2 2 2 σs = N1 −θCC 12 K21 + |a1 | K21 + |b1 | K12 λ21 − |c1 | K21 − |d1 | K12 λ21 cos θdθ π 2 2 2 2 +N2 −2 π Re 12 K12 + |a2 | K21 λ12 + |b2 | K12 − |c2 | K21 λ12 − |d2 | K12 cos θdθ 2

with K12 = 1 + λ12 =

k1 k2 , K21

k1 cos θ1 k2 cos θ2

=1+

k2 k1

λ21 =

k2 cos θ2 k1 cos θ1

and N1 =

1 1+

mλ 2 k1

N2 =

1 1−

mλ 2 k2

.

N1 and N2 are normalized density of states for wave number k1 and k2 respectively. The critical angle θC is defined as cos θC = 2mλ 2k . 1 σN is given by the conductance for normal states, i.e., σS for Δ = 0. We define and Z = 2mU normalized conductance as σT = σS /σN and parameters as β = 2mλ 2k 2k . F F 3. Results First we study the normalized tunneling conduntace σT as a function of bias voltage V in Fig. 1. For Z = 3 where the AR probability is low, σT is almost zero within the enregy gap and independent of β. For Z = 0.3 the dependence of σT on β is also weak. For Z = 0.1 where the AR probability is very high, σT becomes nearly two for β = 0 within the energy gap. It is reduced by the increase of β within the energy gap. This is because the increase of β induces a mismatch of wave number between 2DEG and S, which plays a role of insulating barrier. Next we focus on the dependence of σS on β at zero voltage. Fig. 2 shows the dependence of σS on β at zero voltage for various Z. For Z = 3 it has an exponential dependence on β but its magnitude is small. This is an essentially different feature from the one in F/S junctions where the tunneling conductance is always reduced as increasing exchange field. For Z = 0.1 and Z = 0 it decreases linearly as a function of β.

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Figure 1.

Normalized tunneling conductance.

The effect of the Rashba splitting on conductance is essentially different from that of Zeeman splitting on conductance. This can be explained as follows. The Zeeman splitting gives unbalance of populations of up and down spin electrons. Thus it suppresses the AR where pairs of spin-up and spin-down electrons are transmitted to S. On the other hand, the Rashba splitting never causes such an unbalance. Thus it cannot suppresses the AR, which results in various β dependence of the conductance. 4. Conclusions In the present paper we have studied the tunneling conductance in two dimensional electron gas / insulator / superconductor junctions with Rashba spin-orbit coupling. We have extended the BTK formula and calculated the tunneling conductance. It is found that for low insulating barrier the tunneling conductance is suppressed by the Rashba spin-orbit coupling while for high insulating barrier the tunneling conductance is slightly enhanced by the Rashba spin-orbit coupling in contrast to F/S junctions.

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Figure 2. Tunneling conductance for superconducting states at zero voltage. Here the inequality 0 ≤ σS ≤ 8 holds from the definition.

The results give a possibility to control the AR probability by a gate voltage. We believe that the obtained results are useful for the design of mesoscopic 2DEG/S junctions and for a better understanding of related experiments. Here we have considered the N/S junctions where Rashba spin-orbit coupling exists in the N region. The N/S junctions with Rashba spin-orbit coupling in the S region also give an interesting result as shown in Ref.18 . Acknowledgments The authors appreciate useful and fruitful discussions with A. Golubov. This work was supported by NAREGI Nanoscience Project, the Ministry of Education, Culture, Sports, Science and Technology, Japan, the Core Research for Evolutional Science and Technology (CREST) of the Japan Science and Technology Corporation (JST) and a Grant-in-Aid for the 21st Century COE “Frontiers of Computational Science” . The computational aspect of this work has been performed at the Research Center for Computational Science, Okazaki National Research Institutes and the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo and the Computer Center.

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References 1. A. F. Andreev, Sov. Phys. JETP 19 (1964) 1228. 2. G. E. Blonder, M. Tinkham, T. M. Klapwijk, Phys. Rev. B 25 (1982) 4515. 3. P. M. Tedrow, R. Meservey, Phys. Rev. Lett. 26 (1971) 192 ; Phys. Rev. B 7 (1973) 318 ; R. Meservey, P. M. Tedrow, Phys. Rep. 238 (1994) 173 . 4. S. K. Upadhyay, A. Palanisami, R. N. Louie, R. A. Buhrman Phys. Rev. Lett. 81 (1998) 3247. 5. M.J.M. de Jong, C.W.J. Beenakker, Phys. Rev. Lett. 74 (1995) 1657. 6. A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005). 7. F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys. 77, 1321 (2005). 8. T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev. B 72, 052512 (2005); Phys. Rev. B 73, 094501 (2006). 9. J. E. Hirsch, Phys. Rev. Lett. 83 (1999) 1834. 10. P. Streda, P. Seba, Phys. Rev. Lett. 90 (2003) 256601. 11. J. Schliemann, D. Loss, Phys. Rev. B 68 (2003) 165311. 12. J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, A. H. MacDonald, Phys. Rev. Lett. 92 (2004) 126603. 13. E. I. Rashba, Fiz. Tverd. Tela (Leningrad) 2 (1960) 1224; [Sov. Phys. Solid State 2(1960) 1109 ]; Yu. A. Bychkov, E. I. Rashba, J. Phys. C 17 (1984) 6039. 14. S. Datta, B. Das, Appl. Phys. Lett. 56 (1990) 665. 15. V. M. Edelstein, Solid State Commun. 73 (1990) 233. 16. J. Inoue, G. E. W. Bauer, L. W. Molenkamp, Phys. Rev. B 67 (2003) 033104; 70 (2004) 041303. 17. L. W. Molenkamp, G. Schmidt, G. E. W. Bauer, Phys. Rev. B 64 (2001) 121202. 18. T. Yokoyama, Y. Tanaka, and J. Inoue Phys. Rev. B 72, 220504(R) (2005)

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THEORY OF CHARGE TRANSPORT IN DIFFUSIVE FERROMAGNET/p-WAVE SUPERCONDUCTOR JUNCTIONS

T. YOKOYAMA AND Y. TANAKA Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan and CREST Japan Science and Technology Cooporation (JST), Nagoya 464-8603, Japan E-mail: [email protected] A. A. GOLUBOV Faculty of Science and Technology, University of Twente, 7500 AE, Enschede, The Netherlands E-mail: [email protected] Tunneling conductance and density of states (DOS) in the diffusive ferromagnet (DF) / insulator / p-wave superconductor junctions are calculated by changing the exchange field in DF and the transparencies of the insulating barriers. It is shown that zero-energy peak in DOS and zero bias conductance peak split into two peaks as increasing exchange field.

1. Introduction There is a continuously growing attraction in charge and spin transport in ferromagnet(F) / superconductor(S) junctions. In this junctions Cooper pairs penetrate into the F layer from the S layer and have a nonzero momentum by the exchange field1,2,3 . This property produces many interesting phenomena. For example, a strong enhancement of the proximity effect occurs by the exchange field4 . This resonant proximity effect can influence various physical quantities5 . To study the proximity effect in F/S junctions, a quasiclassical Green functions theory with Nazarov’s boundary condition is often used6,7 . The original Nazarov’s boundary condition is generalized and applied to diffusive normal metal (DN) / unconventional superconductor junctions8,9 . The formation of the midgap Andreev resonant states (MARS) at the interface of unconventional superconductors10 is naturally taken into account in this approach. It was demonstrated that the formation of MARS in DN/p-wave superconductor junctions coexists with the proximity effect, which produces a giant zero bias conductance peak (ZBCP)9 . This theory treat spin independent charge transport. Calculations of tunneling conductance in the presence of the magnetic impurities in DN /S junctions were performed in Ref11,12,13 . Spin dependent transport is one of the most interesting topics in this field and also realized in F/S junctions. In the present paper, we study the tunneling conductance in normal metal / insulator / diffusive ferromagnet

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/ insulator / p-wave superconductor junctions for various parameters such as the heights of the insulating barriers at the interfaces, resistance Rd in DF, the exchange field h in DF, the Thouless energy ET h in DF. We consider the px -wave junctions because in this case proximity effect is strongly enhanced. The conductance σT (eV ) as a function of the bias voltage V is given by σT (eV ) = σS (eV )/σN (eV ) where σS(N ) (eV ) is the tunneling conductance in the superconducting (normal) state at a bias voltage V . In the present paper we confine ourselves to zero temperature and put kB = = 1. 2. Formulation In this section we introduce the model and the formalism. The formulation is the same as the one in Ref.5 except for treating the p-wave superconductors. We consider a junction consisting of normal and p-wave superconducting reservoirs connected by a quasi-one-dimensional diffusive ferromagnet conductor (DF) with a length L much larger than the mean free path. The interface between the DF conductor and the S electrode has a resistance Rb while the DF/N interface has a resistance Rb . The positions of the DF/N interface and the DF/S interface are denoted as x = 0 and x = L, respectively. We model infinitely narrow insulating barriers by the delta function U (x) = Hδ(x − L) + H δ(x). The resulting trans are given by Tm = 4 cos2 φ/(4 cos2 φ + Z 2 ) parencies of the junctions Tm and Tm 2 2 2 and Tm = 4 cos φ/(4 cos φ + Z ), where Z = 2H/vF and Z = 2H /vF are dimensionless constants and φ is the injection angle measured from the interface normal to the junction and vF is Fermi velocity. We apply the quasiclassical Keldysh formalism in the following calculation of the tunneling conductance. The 4 × 4 Green’s functions in N, DF and S are ˇ 1 (x) and G ˇ 2 (x) where the Keldysh component K ˆ 0,1,2 (x) is ˇ 0 (x), G denoted by G ˆ ˆ ˆ ˆ ˆ ˆ given by Ki (x) = Ri (x)fi (x)− fi (x)Ai (x) with retarded component Ri (x), advanced ˆ ∗ (x) using distribution function fî (x)(i = 0, 1, 2). In the component Aî (x) = −R i ˆ 0 (x) = τˆ3 and fˆ0 (x) = fl0 + τˆ3 ft0 . R ˆ 2 (x) is expressed ˆ 0 (x) is expressed by R above, R √ √ ˆ 2 (x) = (gˆ τ3 +f τˆ2 ) with g = / 2 − Δ2 and f = Δ/ Δ2 − 2 , where denotes by R the quasiparticle energy measured from the Fermi energy and fˆ2 (x) = tanh[/(2T)] in thermal equilibrium with temperature T . We put the electrical potential zero in ˇ 1 (x) in DF is determined the S-electrode. In this case the spatial dependence of G by the static Usadel equation 14 . ˇ 1 (x) at the DF/S interface The Nazarov’s generalized boundary condition for G 9 ˇ 1 (x) is given by Ref. . We also use Nazarov’s generalized boundary condition for G at the DF/N interface: ˇ1 L ˇ ∂G −1 )|x=0+ = −Rb (G1 < B > , (1) Rd ∂x B=

ˇ ˇ 1 (0+ )] 2Tm [G0 (0− ), G . ([G ˇ 0 (0− ), G ˇ 1 (0+ )]+ − 2) 4 + Tm

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The average over the various angles of injected particles at the interface is defined as π/2 dφ cos φB(φ) −π/2 () < B(φ) > = π/2 () (φ) cos φ −π/2 dφT ()

with B(φ) = B and T () (φ) = Tm . The resistance of the interface Rb is given by ()

()

Rb = R0 π/2 −π/2

2 dφT () (φ) cos φ

.

()

()−1

= Here R0 is Sharvin resistance, which in three-dimensional case is given by R0 () () e2 kF2 Sc /(4π 2 ), where kF is the Fermi wave-vector and Sc is the constriction area. ˇ 1 (x) as The electric current per one spin is expressed using G ∞ ˇ −L ˇ 1 (x) ∂ G1 (x) )K ], Iel = dTr[τˆ3 (G (2) 8eRd 0 ∂x ˇ1 (x) K ˇ1 (x) where (Gˇ1 (x) ∂ G∂x ) denotes the Keldysh component of (Gˇ1 (x) ∂ G∂x ). In the actual calculation it is convenient to use the standard θ-parameterization where ˆ 1 (x) = τˆ3 cos θ(x) + τˆ1 sin θ(x). The parameter θ(x) ˆ 1 (x) is expressed as R function R is a measure of the proximity effect in DF. Following the method in Ref.5 , the differential resistance R per one spin projection at zero temperature is given by 2Rd L 2Rb dx 2Rb + + (3) R= < Ib0 > L 0 cosh2 θim (x) < Ib1 >

with Ib1 =

2 Λ1 + 2Tm (2 − Tm )Real{cos θ0 } Tm ) + T cos θ |2 | (2 − Tm 0 m

Λ1 = (1+ | cos θ0 |2 + | sin θ0 |2 ) where Ib0 is given in Ref.9 . In the above θim (x), θ0 and θL denote the imaginary part of θ(x), θ(0+ ) and θ(L− ) respectively. Then the total tunneling conductance in the superconducting state σS (eV ) is given by σS (eV ) = ↑,↓ 1/R. The local density of states (DOS) normalized by its normal states value in the DF layer is given by 1 Re cos θ(x). (4) 2 ↑,↓

In the following section, we will discuss the normalized DOS at x = 0 and the normalized tunneling conductance σT (eV ) = σS (eV )/σN (eV ) where σN (eV ) is the tunneling conductance in the normal state given by σN (eV ) = σN = 1/(Rd + Rb + Rb ).

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3. Results Let us first choose relatively large barier Z = Z = 3. In this case a giant ZBCP and zero-energy peak (ZEP) in the DOS appear due to the enhanced proximity effect. Figure 1 (a) and (b) show the tunneling conductance and the corresponding DOS for Z = 3, Rd /Rb = 0.1, Z = 3, Rd /Rb = 0.1, ET h /Δ = 0.01 and various h/Δ. The ZBCP splits into two peaks at eV ∼ h with increasing h/Δ. The corresponding DOS also splits with increasing h/Δ. In (c) and (d), we plot a tunneling conduntance for majority and minority spins respectively. In (c), the spin-resolved ZBCP is shifted to lower energy as increasing exchange field while it is shifted to higher energy as increasing exchange field as shown in (d). Therefore the ZBCP splits into two peaks with increasing h/Δ as shown in (a). Similar discussion about DOS also holds.

Z = 3 Rd / Rb = 0.1 Z ′ = 3 Rd / Rb′ = 0.1 ETh / Δ = 0.01 20

20

(a)

h/ h/ h/

σ

T10

0

-0.1

0

h/

8

S O D

h/ h/

6

Δ Δ Δ

(c )

=0 =0.01 =0.1

σ

h/ h/

10

0

-0.1

ΔΔ Δ

0

=0.01 =0.1

σ

(d )

h/ h/ h/

T

=0 =0.01 =0.1

0.1

eV Δ

20 =0

ΔΔ Δ

=0 =0.01 =0.1

10

(b)

4

h/

T

0.1

eV Δ

10

ΔΔ Δ

2 0

0

0.05

ε

Δ

0.1

0.15

0

-0.1

0

eV Δ

0.1

Figure 1. (a) normalized tunneling conductance and (b) DOS. (c) and (d) spin-resolved tunneling conductance.

Figure 2 displays the tunneling conductance and the corresponding DOS for Z = 0, Rd /Rb = 0.1, Z = 0, Rd /Rb = 0.1, ET h /Δ = 0.01 and various h/Δ. The ZBCP and ZEP in the DOS also appear in this case and but thier magnitudes are small compared to those in Fig. 1. This is because the resonant states are formed only for large Z. The ZBCP and ZEP in the DOS split with an increase in h/Δ as in Fig. 1.

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Z = 0 Rd / Rb = 0.1 Z ′ = 0 Rd / Rb′ = 0.1 ETh / Δ = 0.01 4

(a)

h/ h/ h/

σ T

3

Δ Δ Δ

=0 =0.01 =0.1

2

-0.1

0

0.1

eV Δ

10

(b)

8

S O D

6 h/ h/

4

h/

Δ Δ Δ

=0 =0.01 =0.1

2 0

0

0.05

ε Figure 2.

Δ

0.1

0.15

(a) normalized tunneling conductance and (b) DOS.

4. Conclusions In the present paper, based on Nazarov’s boundary condition, we have found a splitting of peak of the conductance and the DOS in N/DF/p-wave superconductor junctions with increasing exchange field. This stems from the fact that the peak for majority (minority) spin is shifted to lower (higher) energy as increasing exchange field.

Acknowledgments The authors appreciate useful and fruitful discussions with Yu. Nazarov and H. Itoh. This work was supported by NAREGI Nanoscience Project, the Ministry of Education, Culture, Sports, Science and Technology, Japan, the Core Research for Evolutional Science and Technology (CREST) of the Japan Science and Technology Corporation (JST) and a Grant-in-Aid for the 21st Century COE “Frontiers of Computational Science” . The computational aspect of this work has been performed at the Research Center for Computational Science, Okazaki National Research Institutes and the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo and the Computer Center.

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References 1. A.I. Buzdin, L.N. Bulaevskii, and S.V. Panyukov, Pis’ma Zh. Eksp. Teor. Phys. 35, 147, (1982) [JETP Lett. 35, 178 (1982)]. 2. A.I. Buzdin and M.Yu. Kupriyanov,, Pis’ma Zh. Eksp. Teor. Phys. 53, 308 (1991) [JETP Lett. 53, 321 (1991)]. 3. E. A. Demler, G. B. Arnold, and M. R. Beasley, Phys. Rev. B 55, 15 174 (1997). 4. T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev. B 72, 052512 (2005). 5. T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev. B 73, 094501 (2006). 6. Yu. V. Nazarov, Superlattices and Microstructuctures 25, 1221 (1999). 7. Y. Tanaka, A. A. Golubov, S. Kashiwaya, Phys. Rev. B 68 (2003) 054513. 8. Y. Tanaka, Yu. V. Nazarov, S. Kashiwaya, Phys. Rev. Lett. 90 (2003) 167003; Y. Tanaka, Yu. V. Nazarov, A. A. Golubov, S. Kashiwaya, Phys. Rev. B 69 (2004) 144519. 9. Y. Tanaka and S. Kashiwaya, Phys. Rev. B 70 012507 (2004); Y. Tanaka, S. Kashiwaya, and T. Yokoyama Phys. Rev. B 71, 094513 (2005). 10. C.R. Hu, Phys. Rev. Lett. 72 (1994) 1526;Y. Tanaka, S. Kashiwaya, Phys. Rev. Lett. 74 (1995) 3451; S. Kashiwaya, Y. Tanaka, M. Koyanagi, K. Kajimura, Phys. Rev. B 53 (1996) 2667; Y. Tanuma, Y. Tanaka, S. Kashiwaya Phys. Rev. B 64 (2001) 214519;S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63 (2000) 1641 and references therein. 11. A.F. Volkov, A.V. Zaitsev and T.M. Klapwijk, Physica C 210, 21 (1993). 12. S. Yip, Phys. Rev. B 52, 15504 (1995). 13. T. Yokoyama, Y. Tanaka, A. A. Golubov, J. Inoue, and Y. Asano, Phys. Rev. B 71, 094506 (2005). 14. K.D. Usadel Phys. Rev. Lett. 25 (1970) 507.

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THEORY OF ENHANCED PROXIMITY EFFECT BY THE EXCHANGE FIELD IN FS BILAYERS

T. YOKOYAMA AND Y. TANAKA Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan and CREST Japan Science and Technology Cooporation (JST), Nagoya 464-8603, Japan E-mail: [email protected] A. A. GOLUBOV Faculty of Science and Technology, University of Twente, 7500 AE, Enschede, The Netherlands E-mail: [email protected] Enhanced proximitiy effect in normal metal / insulator / diffusive ferromagnet / insulator / superconductor (N/I/DF/I/S) junctions is studied for various situations solving the Usadel equation under the Nazarov’s generalized boundary condition. Conductance of the juntion and density of states of the DF are calculated by changing the magnitude of the resistance, Thouless energy and the exchange field in DF. We heve clarified that due to the enhanced proximity effect, a sharp zero bias conductance peak (ZBCP) appears for small Thouless energy while a broad ZBCP appears for large Thouless energy. The magnitude of this ZBCP can exceed its value for normal states in contrast to the ZBCP observed in diffusive normal metal / superconductor junctions. We find structures similar to the conductance in the density of states.

1. Introduction In diffusive ferromagnet / superconductor (DF/S) junctions Cooper pairs penetrate into the DF layer from the S layer and have a nonzero momentum by the exchange field1,2,3 . This property induces many interesting phenomena. The exchange field in the ferromagnet usually breaks Cooper pairs. But for a weak exchange field the pair amplitude can be enhanced and the energy dependent DOS can have a zero-energy peak4 . Although the DOS has been studied extensively4,5,6,7 , the condition for the appearance of the DOS peak was not studied systematically. We studied the conditions for the appearance of such anomaly, i.e., strong enhancement of the proximity effect and found two conditions corresponding to weak proximity effect and strong one8 . Since DOS is a fundamental quantity, this resonant proximity effect can influence various transport phenomena. The purpose of the present paper is to study the influence of the resonant proximity effect by the exchange field on the tunneling conductance and the DOS in DF/ S junctions with Nazarov’s boundary conditions. A weak exchange field compared to the Fermi energy can be realized in ex-

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periments with , e.g., Ni doped Pd7 . Therefore it is interesting to predict new phenomena in DF/S junctions for a future experiment. In the present paper, we study the tunneling conductance in normal metal / insulator / diffusive ferromagnet / insulator / superconductor junctions for various parameters such as resistance Rd in DF, the exchange field h in DF, the Thouless energy ET h in DF. In the present paper we confine ourselves to zero temperature and put kB = = 1. 2. Formulation In this section we introduce the model and the formalism. The formulation is the same as the one in Ref.9 . We consider a junction consisting of normal and superconducting reservoirs connected by a quasi-one-dimensional diffusive ferromagnet conductor (DF) with a length L much larger than the mean free path. The interface between the DF conductor and the S electrode has a resistance Rb while the DF/N interface has a resistance Rb . The positions of the DF/N interface and the DF/S interface are denoted as x = 0 and x = L, respectively. We model infinitely narrow insulating barriers by the delta function U (x) = Hδ(x − L) + H δ(x). The resulting are given by Tm = 4 cos2 φ/(4 cos2 φ + Z 2 ) transparency of the junctions Tm and Tm 2 2 2 and Tm = 4 cos φ/(4 cos φ + Z ), where Z = 2H/vF and Z = 2H /vF are dimensionless constants and φ is the injection angle measured from the interface normal to the junction and vF is Fermi velocity. We apply the quasiclassical Keldysh formalism in the following calculation of the tunneling conductance. The 4 × 4 Green’s functions in N, DF and S are ˇ 1 (x) and G ˇ 2 (x) where the Keldysh component K ˆ 0,1,2 (x) is ˇ 0 (x), G denoted by G ˆ i (x)fî (x)− fî (x)Aî (x) with retarded component R ˆ i (x), advanced ˆ i (x) = R given by K ∗ ˆ ˆ ˆ component Ai (x) = −Ri (x) using distribution function fi (x)(i = 0, 1, 2). In the ˆ 0 (x) = τˆ3 and fˆ0 (x) = fl0 + τˆ3 ft0 . R ˆ 2 (x) is expressed ˆ 0 (x) is expressed by R above, R √ √ 2 2 2 2 ˆ τ3 +f τˆ2 ) with g = / − Δ and f = Δ/ Δ − , where denotes by R2 (x) = (gˆ the quasiparticle energy measured from the Fermi energy and fˆ2 (x) = tanh[/(2T)] in thermal equilibrium with temperature T . We put the electrical potential zero in the S-electrode. ˇ 1 (x) at the DF/S interface The Nazarov’s generalized boundary condition for G 11,12,13 and the one at the DF/N interface is given in Ref.8 . is given in Ref. The resistance of the interface Rb is given by 2 () () Rb = R0 π/2 . () (φ) cos φ −π/2 dφT ()

()−1

= Here R0 is Sharvin resistance, which in three-dimensional case is given by R0 () 2 2 () 2 e kF Sc /(4π ), where kF is the Fermi wave-vector and Sc is the constriction area. ˇ 1 (x) as The electric current per one spin is expressed using G ∞ ˇ −L ˇ 1 (x) ∂ G1 (x) )K ], dTr[τˆ3 (G (1) Iel = 8eRd 0 ∂x

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132 ˇ1 (x) K ˇ1 (x) where (Gˇ1 (x) ∂ G∂x ) denotes the Keldysh component of (Gˇ1 (x) ∂ G∂x ). In the actual calculation it is convenient to use the standard θ-parameterization where ˆ 1 (x) = τˆ3 cos θ(x) + τˆ2 sin θ(x). The parameter ˆ 1 (x) is expressed as R function R θ(x) is a measure of the proximity effect in DF. The spatial dependence of θ(x) is determined by the following equation

D

∂2 θ(x) + 2i( − (+)h) sin[θ(x)] = 0 ∂x2

(2)

with the diffusion constant D in DF for minority (majority) spin where h denotes the exchange field. Note that we assume a weak ferromagnet and neglect the difference of Fermi velocity between majority spin and minority spin. Following the method in Ref.9 , the differential resistance R per one spin projection at zero temperature is given by 2Rd L 2Rb 2Rb dx + + R= (3) < Ib0 > L 0 cosh2 θim (x) < Ib1 > with Ib1 =

2 Tm Λ1 + 2Tm (2 − Tm )Real{cos θ0 } ) + T cos θ |2 | (2 − Tm 0 m

Λ1 = (1+ | cos θ0 |2 + | sin θ0 |2 ) where Ib0 is given in Ref.12 . In the above θim (x), θ0 and θL denote the imaginary part of θ(x), θ(0+ ) and θ(L− ) respectively. Then the total tunneling conductance in the superconducting state σS (eV ) is given by σS (eV ) = ↑,↓ 1/R. The local density of states (DOS) normalized by its normal states value in the DF layer, N , is given by 1 Re cos θ(x). (4) N= 2 ↑,↓

In the following section, we will discuss the normalized DOS at x = 0 and the normalized tunneling conductance σT (eV ) = σS (eV )/σN (eV ) where σN (eV ) is the tunneling conductance in the normal state given by σN (eV ) = σN = 1/(Rd + Rb + Rb ). Below we fix Z = Z = 3 and Rd /Rb = 0.1. 3. Results In this section, we study the influence of the resonant proximity effect on tunneling conductance as well as the DOS in the DF region. The resonant proximity effect is characterized as follows. When the proximity effect is weak (Rd /Rb 1), the condition is given by Rd /Rb ∼ 2h/ET h . When the proximity effect is strong (Rd /Rb 1), the condition is given by ET h ∼ h. We choose Rd /Rb = 0.2 as a typical value to study the weak proximity regime. We also choose Rd /Rb = 4 to

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study the strong one. Let us first study the tunneling conductance. In the following we show that a zero bias conductance peak (ZBCP) appears due to the enhanced proximity effect when resonant condition is satisfied. Figure. 1 shows the tunneling conductance for Z = 3, Z = 3, Rd /Rb = 0.1 and various ET h /Δ with (a) Rd /Rb = 0.2 and h/Δ = 0.1, and (b) Rd /Rb = 4 and h/Δ = 1. In Fig. 1 (a) dip structures at eV ∼ 0, ±h appear at small ET h /Δ. When resonant condition is satisfied (ET h /Δ = 1), a ZBCP emerges and it is suppressed with further increasing ET h /Δ. In Fig. 1 (b) it is shown that the tunneling spectrum is flat at small ET h /Δ. When resonant condition is satisfied (ET h /Δ = 1), a broad ZBCP appears and it disappears by increasing ET h /Δ. The magnitude of the ZBCP exceeds unity. Contrary, in diffusive normal metal / superconductor junctions, the normalized conductance never exceed unity12 . (a)

1.4

Rd / Rb = 0.2 h / Δ = 0.1 E Th /

1.2

E Th / E Th /

1

Δ Δ Δ

=0.2 =1 =2

0.8

σ

T 0.6 (b) 2

Rd / Rb = 4 h / Δ = 1 E Th / E Th / E Th /

Δ Δ Δ

=0.2 =1 =2

1

-1

Figure 1.

0

eV Δ

1

Normalized tunneling conductance.

In Figs. 2 and 3 we show the DOS as a function of ε and x with corresponding parameters to those of Fig. 1 (a) and (b) respectively. We can see the similar structures to those of the conductances. Especially the zero energy peak in the DOS appears when the resonant conditions are satisfied. 4. Conclusion In the present paper we calculated the conductance and the DOS in the DF/S junctions by solving the Usadel equation and found that due to the enhanced proximity effect, a sharp ZBCP appears for small Thouless energy while a broad ZBCP ap-

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(a )

ETh / Δ = 0.2

(b)

ETh / Δ = 1

(c )

ETh / Δ = 2

Figure 2.

Normalized DOS with Rd /Rb = 0.2 and h/Δ = 0.1.

pears for large Thouless energy. The magnitude of this ZBCP can exceed its value for normal states in contrast to the ZBCP observed in diffusive normal metal / superconductor junctions. We found structures similar to the conducance in the density of states. References 1. 2. 3. 4. 5.

A.I. Buzdin, L.N. Bulaevskii, and S.V. Panyukov, JETP Lett. 35, 178 (1982). A.I. Buzdin and M.Yu. Kupriyanov, JETP Lett. 53, 321 (1991). E. A. Demler, G. B. Arnold, and M. R. Beasley, Phys. Rev. B 55, 15 174 (1997). A. A. Golubov, M. Yu. Kupriyanov, and Ya. V. Fominov, JETP Lett. 75, 223 (2002). A. Buzdin, Phys. Rev. B 62, 11 377 (2000).

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135

(a)

ETh / Δ = 0.2

(b)

ETh / Δ = 1

(c )

ETh / Δ = 2

Figure 3.

Normalized DOS with Rd /Rb = 4 and h/Δ = 1.

6. M. Zareyan, W. Belzig, and Yu. V. Nazarov, Phys. Rev. Lett. 86, 308 (2001); Phys. Rev. B 65, 184505 (2002). 7. T. Kontos, M. Aprili, J. Lesueur, and X. Grison, Phys. Rev. Lett. 86, 304 (2001); T. Kontos, M. Aprili, J. Lesueur, X. Grison, and L. Dumoulin, Phys. Rev. Lett. 93, 137001 (2004). 8. T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev. B 72, 052512 (2005). 9. T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev. B 73, 094501 (2006). 10. K.D. Usadel Phys. Rev. Lett. 25 (1970) 507. 11. Yu. V. Nazarov, Superlattices and Microstructuctures 25, 1221 (1999). 12. Y. Tanaka, A. A. Golubov, S. Kashiwaya, Phys. Rev. B 68 (2003) 054513. 13. T. Yokoyama, Y. Tanaka, A. A. Golubov, J. Inoue, and Y. Asano, Phys. Rev. B 71, 094506 (2005).

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THEORY OF JOSEPHSON EFFECT IN DIFFUSIVE d-WAVE JUNCTIONS

T. YOKOYAMA AND Y. TANAKA Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan and CREST Japan Science and Technology Cooporation (JST), Nagoya 464-8603, Japan E-mail: [email protected] A. A. GOLUBOV Faculty of Science and Technology, University of Twente, 7500 AE, Enschede, The Netherlands E-mail: [email protected] Y. ASANO Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan E-mail: [email protected] We study the Josephson effect in d-wave superconductor / diffusive normal metal / dwave superconductor junctions by solving the Usadel equation, changing the magnitude of the resistance and Thouless energy in the diffusive normal metal, the transparencies of the insulating barriers at the interfaces and the angles between the normal to the interfaces and the crystal axis of d-wave superconductors. We find that, in contrast to the case of conventional s-wave junctions, the product of the Josephson current and the normal state resistance is enhanced with the decrease of transparency of the interface. In the presence of midgap Andreev resonant states (MARS), the Josephson current has a nonmonotonic temperature dependence due to the competition between the proximity effect and the MARS.

1. Introduction Since the discovery of Josephson effect1 in superconductor / insulator / superconductor (SIS) junctions, it has been studied in various situations2,3 . In SIS and superconductor / diffusive normal metal / superconductor (S/DN/S) junctions the critical current increases monotonically with decreasing temperature4,5,6 . In S/DN/S junctions Josephson current is carried by Cooper pairs penetrating into the DN as a result of the proximity effect. On the other hand, Josephson effect depends strongly on pairing symmetries of superconductors. In d-wave superconductor / insulator / d-wave superconductor (DID) junctions, nonmonotonic dependence of critical current on temperature7,8,9,10 occurs due to the formation of midgap Andreev resonant states (MARS) at the interface11 . The MARS stem from sign change of pair potentials of d-wave superconductors 12 .

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In order to study the combined effect of the proximity effect and MARS, Tanaka et al. 13,14,15 have recently extended the circuit theory16 to the systems with unconventional superconductors. The circuit theory provides the boundary conditions for the Usadel equations 17 widely used in diffusive superconducting junctions. These boundary conditions generalize the Kupriyanov-Lukichev conditions 5 for an arbitrary type of connector which couples the diffusive nodes. An application of the extended circuit theory to the DN/d-wave superconductor junctions has revealed a strong competition (destructive interference) of the MARS with the proximity effect in DN13,14 . However, this competition has not yet been tested experimentally, thus it’s important to propose the way to verify this prediction. In the present paper, we show that Josephson effect is a suitable tool to observe the above competition. We calculate Josephson current in d-wave superconductor / diffusive normal metal / d-wave superconductor (D/DN/D) junctions, solving the Usadel equations with new boundary conditions derived in Ref.18 . This allows us to study simultaneously the influence of the proximity effect and the formation of MARS on the Josephson current. We find that the competition between the proximity effect and the formation of MARS provides a new mechanism for a nonmonotonic temperature dependence of the maximum Josephson current.

2. Formulation We consider a junction consisting of d-wave superconducting reservoirs (D) connected by a quasi-one-dimensional diffusive conductor (DN) with a resistance Rd and a length L much larger than the mean free path. The DN/D interface located at x = 0 has a resistance Rb , while the DN/D interface located at x = L has a resistance Rb . We model infinitely narrow insulating barriers by the delta function U (x) = Hδ(x − L) + H δ(x). The resulting transparencies of the junctions Tm and 2 are given by Tm = 4 cos2 φ/(4 cos2 φ + Z 2 ) and Tm = 4 cos2 φ/(4 cos2 φ + Z ), Tm where Z = 2H/vF and Z = 2H /vF are dimensionless constants and φ is the injection angle measured from the interface normal to the junction and vF is Fermi velocity. We parameterize the quasiclassical Green’s functions G and F with a function Φω 2,3 : ω Φω , Fω = 2 Gω = 2 ∗ ω + Φω Φ−ω ω + Φω Φ∗−ω

(1)

where ω is the Matsubara frequency. Then Usadel equation reads17 2 πTC ∂ 2 ∂ ξ (2) Gω Φω − Φω = 0 ωGω ∂x ∂x with the coherence length ξ = D/2πTC , the diffusion constant D and the transition temperature TC . We use the boundary conditions in Ref.18

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The Josephson current is given by the expression eIR ∂ ∗ ∂ RT L G2ω ∗ =i Φω Φ−ω − Φ−ω Φω πTC 2Rd TC ω ω 2 ∂x ∂x

(3)

with temperature T and R ≡ Rd + Rb + Rb , respectively. In the following we focus on the IC R value where IC denotes the magnitude of the maximum Josephson current. Below α and β denote the angles between the normal to the interface and the crystal axes of d-wave superconductors for x ≤ 0 and x ≥ L respectively, and ϕ is a phase difference between d-wave superconductors. 3. Results We consider symmetric barriers with Rb = Rb and Z = Z for simplicity. Figure 1 (a) shows IC R value as a function of temperature for Rd /Rb = 1, ET h /Δ(0) = 0.5 and (α, β) = (0, 0) with various Z. For α = β = 0, IC R increases with the decrease of the temperature as in the case of s-wave junction. The magnitude of IC R is enhanced with increasing Z in contrast to the s-wave junctions where IC R is enhanced with decreasing the magnitude of Z as shown in (b). We can explain the increase of IC R with Z by calculating anomalous Green’s functions F which corresponds to the degree of the proximity effect. We choose x = L/2 and ϕ = π/2 with corresponding parameters in (a) at T /TC = 0.1. As the value of Z increases, both the magnitudes of ReF and ImF increase as shown in (c) and (d). This results in the larger magnitude of the IC R. In d-wave junctions, as shown in our previous papers 14 , the magnitude of the proximity effect is reduced with the decrease of the value of Z, i.e., the increase of the transparency of the junctions. In d-wave junctions with α = β = 0, quasiparticles with injection angles φ in π/4 2 ∗ φ+ φ− > < v− , C3 = 2 < v+ v− |φ+ |2 >

C1 =

(2)

where v± = (vx ±ivy )/2 with a Fermi velocity (vx , vy ), and < · · · > indicates the average on p along the Fermi surface. The parameters C1 and C3 represent the anisotropy of Fermi surface. They have finite values when the pairing functions on the Fermi surface have the fourfold symmetric structure. For isotropic Fermi surface, C1 and C3 are taken to be zero. In this paper, we treat C1 ,C2 ,C3 as parameters because detailed forms of φ± have not been established yet. In what follows, we consider that C1 = C3 = 0 and C2 = 1. The GL free energy for the s-wave symmetry is obtained by η− (r) = 0.0 and η+ (r) = η(r) in Eq. (1). In our simulations, we use the TDGL equation ∂ 1 ∂ f˜ η± = − ∗ , ∂t 12 ∂η±

(3)

as shown in Ref. 12. First, we solve the TDGL equation at a fixed magnetic field H at a certain temperature T slightly above Tc . Temperatures is then decreased until sufficient amplitudes of order parameters appears after solving the TDGL equation, which define the superconducting transition temperature Tc (H) at H (0 < H < Hc2 ). The calculations are performed in a two-dimensional square shaped superconductor with area being 25ξ0 ×25ξ0 where ξ0 is the coherence length at T = 0 in the GL theory. Sizes of the square hole are chosen as 5ξ0 ×5ξ0 , 10ξ0 ×10ξ0 , and

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15ξ0 ×15ξ0 . For circular shaped superconductor with diameter being 25ξ0 , hole diameters are 5ξ0 , 10ξ0 , and 15ξ0 . Outside of the open boundary and inside of the hole, we set η+ (r)=η− (r)=0 and B(r)=H with an applied field H. 3. Result 3.1. S-wave (One Component) First, we study the Little Parks oscillation for the s-wave pairing in our formulation. Figures 1(a) and 1(b) show the transition lines Tc (H) in H-T -plane for the square 0 = φ0 /πξ02 . and circular shaped superconducting rings, respectively. We set Hc2 The critical temperature in the absence of a hole is denoted by Tc0 . It is an effect of the finite-size superconductor that Tc (H) is suppressed even at H=0. When H increases along the Tc (H)-line, vortices penetrate into the hole from outside of the superconductor. Throughout this process, spatial structures of the order parameter changes continuously. While the vortices goes across superconductor, Tc (H) takes a local minimum and the phase winding number of order parameter around the hole changes. Figure 1 shows that behavior of the Little-Parks oscillations depends on size of hole when H increases from zero to Hc2 . In order to analyze a period of the oscillations, we show the Fourier component of Tc (H) which is given by F (θH ) = 1 −iH θH 0 T (H )e dH , H =H/Hc2 in Figs. 1(c) and 1(d). The characteristic peaks c 0 appear in F (θH ). Amplitudes of the peaks in circular case in Fig. 1(d) are larger than those in square case in Fig. 1(c). In circular shaped rings, the winding number changes by 4 at each period of oscillations in Tc (H), this number appears in the structure of ArgΔ(r). We note that high symmetry of ring is responsible for this behavior. The winding number can change by one when a hole position deviates from the center of rings. The oscillation behavior depends on the hole size and the winding number around the hole. The shape of rings is very important for the Tc (H) oscillations. 3.2. Chiral P -wave (Two Components) We show the transition line Tc (H) in H-T -plane for the chiral p-wave symmetry in Fig. 2. When H increases along the Tc (H)-line, periodic penetration of vortex into a hole makes oscillations of Tc (H) in one-component case. For the chiral pwave, however, the oscillations of the transition line does not appear. We confirmed that the phase winding number around a hole increases with increasing H. In order to understand why the Little-Parks oscillations do not appear, we plot spatial 0 , η− structures of the two order parameters in Fig. 3 for several H. At H = 0.1Hc2 in (d) and η+ in (a) are dominant and subdominant component, respectively. At 0 , amplitude relation is switched with each other, η+ in (b) and η− in (e) H = 0.2Hc2 are dominant and subdominant component, respectively. With increasing H, two order parameters are continuously transformed and η+ or η− appears alternatively as a dominant component. It is noted that there appears vortex only in the minor

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component, while dominant component does not have vortex. Since amplitudes of order parameters correspond to the condensation energy, the dominant component mainly keeps the condensation energy and, at the same time, the subdominant component has vortices. In this way, order parameters change their phase winding number with saving of the condensation energy. Thus conventional Little-Parks oscillations disappear in two-component superconducting states such as the chiral p-wave symmetry. The vortices appearing in the subdominant component may have fractional flux quantum. 12 4. Summary In summary, solving the two-component TDGL equation on the two dimensional superconductor with a hole, we study the Little-Parks oscillations both in the s-wave and the chiral p-wave pairing symmetries. In the s-wave pairing, the clear oscillation of Tc (H) appears in the H-T phase diagram. The characteristic behavior of the oscillations are sensitive to size and shape of the hole. The chiral p-wave symmetries are described by two components of order parameters (i.e.,η+ and η− ). With increasing magnetic fields, η+ and η− become the dominant component alternately and vortex of the minor component is able to change phase winding numbers without loss of condensation energy. Due to these effects, the conventional Little-Parks oscillation does not appear in the chiral p-wave superconductor. References 1. 2. 3. 4. 5. 6. 7. 8.

9.

10. 11. 12.

W.A. Little and R.D. Parks, Phys. Rev. Lett. 9, 9 (1962). R.P. Groff and R.D. Parks, Phys. Rev. 176, 567 (1968). L. Meyers and R. Meservey, Phys. Rev. B 4, 824 (1971). J. Cayssol, T. Kontos and G Montambaux, Phys. Rev. B 67, 184508 (2003). Y. Maeno, H. Hashimono, K. Yoshida, S. Nishizaki, T. Fujita, J.G. Bednorz and F. Litenberg, Nature 372, 532 (1994). T.M. Rice and M. Sigrist, J. Phys.: Condens Matter 7, L643 (1995). K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z.Q. Mao, Y. Mori and Y. Maeno, Nature 396, 658 (1998). G.M. Luke, Y. Fudamoto, K.M. Kojima, M.I. Larkin, J. Merrin, B. Nachumi, Y.J. Uemura, Y. Maeno, Z.Q. Mao, Y. Mori, H. Nakamura and M. Sigrist, Nature 394, 558 (1998). P.G. Kealey, T.M. Riseman, E.M. Forgan, L.M. Galvin, A.P. Machenzie, S.L. Lee, D.McK. Paul, R. Cubitt, D.F. Agterberg, R. Heeb, Z.Q. Mao and Y. Maeno, Phys. Rev. Lett. 84, 6094 (2000). Y. Hasegawa, K. Machida and M. Ozaki, J. Phys. Soc. Jpn. 69 336 (2000). V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950). M. Ichioka, Y. Matsunaga and K. Machida, Phys. Rev. B 71, 172510 (2005).

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1

1

l/ξ0=5 l/ξ0=10 l/ξ0=15

.9 .8

.8

¼¾

.7 .6

.7 .6

.5

.5

.4

.4

.3

.3

.2

.2

.1

.1

0

0 0

0.4 0.2

´ µ

r/ξ0=5 r/ξ0=10 r/ξ0=15

.9

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

0

.1

.2

.3

.4

¼

l/ξ0=5 l/ξ0=10 l/ξ0=15

0.4 0.2

0

0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1

.5

.6

.7

.8

.9

1

¼

r/ξ0=5 r/ξ0=10 r/ξ0=15

-1 .1

.2

.3

.4

.5

.6

´¾ µ

.7

.8

.9

1

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

´¾ µ

Figure 1. The transition lines Tc (H) of the s-wave symmetry are shown for (a) square shaped ring and (b) circular shaped ring, when diameters of the hole are 5ξ0 , 10ξ0 , and 15ξ0 . Fourier components of Tc (H) in (a) and (b) are shown in (c) and (d), respectively.

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160 1

.8

.8

.6

.5

.5

.4

.4

.3

.3 .2

.1

.7

.6

.2

r/ξ0=5 r/ξ0=10 r/ξ0=15

.9

.7

¼¾

1

l/ξ0=5 l/ξ0=10 l/ξ0=15

.9

.1

0

0 0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

0

.1

.2

.3

.4

.5

¼

.6

.7

.8

.9

1

¼

Figure 2. The transition lines Tc (H) of the chiral p-wave symmetry are shown for (a) square shaped ring and (b) circular shaped ring.

·

¼¾

Figure 3. Spatial structures of order parameters in the circular shaped chiral p-wave rings are shown for several magnetic fields H/Hc2 = 0.1,0.2 and 0.3, where the diameter of the hole is fixed at 5ξ0 and a white ring in (d) corresponds to the superconductor. We present |η+ | and |η− | as density plots, where brighter regions indicate larger amplitudes. The subdominant components are located at the dark region within the superconductor.

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161

THEORETICAL STUDY OF SYNERGY EFFECT BETWEEN PROXIMITY EFFECT AND ANDREEV INTERFACE RESONANT STATES IN TRIPLET p-WAVE SUPERCONDUCTORS

Y. TANUMA1∗, Y. TANAKA2 AND S. KASHIWAYA3 1

Institute of Physics, Kanagawa University, Rokkakubashi, Yokohama, 221-8686, Japan 2 Department of Applied Physics, Nagoya University, Nagoya 464-8063, Japan 3 National Institute of Advanced Industrial Science and Technology, Tsukuba, 305-8568, Japan

The interplay between Andreev interface resonant states and proximity effect in normal metal/ triplet p-wave superconductor junctions is investigated within quasiclassical Green’s function methods. We show numerical results for self-consistently determined pair potentials both in the normal metal and triplet p-wave superconductor sides. Moreover, the quasiparticle local density of states at the interface is calculated in detail by changing the transparency of junctions. The resulting local density of states at the interface depends on the transparency of the junctions as well as pairing symmetries of the pair potentials.

1. Introduction Nowadays spin-triple superconductivity has much attention as an important problem in solid states physics. The existence of triplet superconductivity has become promising after a series of experiments in Sr2 RuO4 1 . Tunneling spectroscopy using Andreev interface resonant states (AIRS) is essentially phase sensitive for unconventional superconductors 2,3,4 . The most dramatic effect is the emergence of a zero-bias conductance peak (ZBCP) in tunneling spectra between normal metal and unconventional superconductors. The AIRS, which is originated from the interference effect in the effective pair potential through the reflection at the interface, plays an important role in order to determine the pairing symmetry of unconventional superconductors. One open question is how the resonant states and proximity effect multiplies in normal metal and unconventional superconductor junctions. Actually, the ZBCP reflecting on the existence of the AIRS is observed in several tunneling experiments of Sr2 RuO4 5,6,7 . On the other hands, tunneling experiments in Sr2 RuO4 with Ru-metals by Mao et al. 6 suggest that the overall-line shape of tunneling spectra has a broad ZBCP with a sharp peak. This feature is different from those in high-Tc cuprate junctions where only sharp ZBCP is reported 8,9 . Since these resonant states are expected for singlet d-wave pairing as well as triplet ∗ E-mail:

[email protected]

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pairings 10 , the difficulty may arise in determining the pairing symmetry only from conventional tunneling spectroscopy. This is why we focus on the proximity effect between the normal metal (N) and triplet p-wave superconductor (P) junctions. From theoretical view points, one of the remarkable features of triplet p-wave junctions, which is different from d-wave ones, is that injected quasiparticles perpendicular to the interface form the AIRS 11 . Therefore, the perpendicular injection contributes most dominantly to the tunneling conductance. Beside this, it is revealed that the proximity effect is enhanced by the AIRS in diffusive N/P junctions 12,13,14 . However, it is not clarified how these resonant states near the interface in the P side with broken-time reversal symmetry states (BTRSS) affect on the induced s-wave component by the proximity effect in the N side. The study along this direction may serve as a guide to understand novel interface phenomena expected in Ru/Sr2 RuO4 junctions 15 . For these reasons, it is one of challenging issues to study the proximity effect in the N/P junctions in the presence of the induced pair potential in the N side. 2. Formulation We introduce the generalized Eilenberger equations 16 for spin-triplet superconductors σσ ∂ σσ ˇ α , x) gαβ i|vFx | gαβ (φ, x) = −α iωm τˇ3 + Δ(φ (φ, x) ∂x σσ ˇ β , x) , + βgαβ (φ, x) iωm τˇ3 + Δ(φ (1) ˇ α , x) = Δ(φ

iΔ(φα , x) · σ ˆσ ˆy ˆ0 iˆ σy Δ(φα , x)∗ · σ ˆ ˆ0

,

(2)

where vFx = vF cos θ and τî (i = 1, 2, 3) stand for the x component of the Fermi velocity and the Pauli matrices, respectively. Here ωm = πT (2m + 1) (m: integer) is the Matsubara frequency. The Pauli matrix τˇ3 in Eq.(1) is a 4 × 4 matrix. The σσ ˇ α , x) quasi-classical Green’s function gαβ (φ, x) and 4 × 4 pair potential matrix Δ(φ have different spin dependencies. The triplet pair potential Δ(φα , x), is represented by d-vector as Δ = i(d· σ ˆ )ˆ σy . In the present study, we consider triplet p-wave states ˇ α , x) is reduced to be by choosing dx = dy = 0. The 4 × 4 pair potential matrix Δ(φ ˆ α , x). the 2 × 2 matrix Δ(φ Next, we consider the N/P junctions in order to determine the spatial variations of pair potentials self-consistently. Considering a semi-infinite geometry, where the N is located at x < 0 and the P extended elsewhere, the pair potential in P [N] side will tend to the bulk value [zero] ΔP (φα , ∞) [ΔN (φα , −∞)] at sufficiently large x. For simplicity, since we do not consider spin-flip scattering at the interface, we ↑↑ ↓↓ ↑↓ ↓↑ = gαβ = gαβ and gαβ = gαβ = 0. can put the quasi-classical Green’s functions gαβ Therefore, we only have to deal with the Eilenberger equations constructed out of gαβ . However, we need to pay attention to sign of 21 off-diagonal element as the odd parity in Δ(φα , x) is maintained.

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163 l The resulting quasi-classical Green’s function gˆαα (φα , x) for l(= N,P) regions is given by 17 N i (x)F−N (x) 2iF−N (x) 1 + D− N gˆ−− (φ− , x) = , (3) N N N (x)F N (x) (x) −1 − D− (x)F−N (x) 2iD− 1 − D− − P i (x)F+P (x) 2iF+P (x) 1 + D+ P (φ+ , x) = gˆ++ , (4) P P P (x)F P (x) (x) −1 − D+ (x)F+P (x) 2iD+ 1 − D+ + l l (φ− , x) = −ˆ g++ (−φ+ , x)† . The quantities Dαl (x) and Dαl (x) obey the with gˆ−− following Riccati type equations:

∂ l Dα (x) =α 2ωm Dαl (x) + Δl (φα , x)Dαl (x)2 − Δl (φα , x)∗ , ∂x ∂ |vFx | Fαl (x) = − α 2ωm Fαl (x) − Δl (φα , x)∗ Fαl (x)2 + Δl (φα , x) . ∂x

|vFx |

(5) (6)

Moreover, the boundary condition of the Dαl (0) and Fαl (0) at the interface x = 0 are 18,19,20 F−N = F+P =

P P N − D+ + (1 − R)D+ RD− , P P DP − RD+ ] − (1 − R)D− +

(7)

N N P RD+ − D− + (1 − R)D− , P [D N − RD N ] − (1 − R)D N D N D− + − + −

(8)

N [D P D+ −

with reflection probability R. The pair potentials for both N and P sides are given by 21 π/2 l 1 l l Δ (φ, x) = dφ V l (φ, φα ) [ˆ gαα (φα , x)]12 − [ˆ gαα (φα , x)† ]12 , 4π −π/2 α 0≤m= π/2 , Tφ = 2 2 ( ) Z + 4 cos φ Z + 4 cos2 φ dφT cos φ φ −π/2 (4) where Z and Z denote the barrier parameters at the interfaces. In the present = 0.25Δ0 . calculation, we fix Z = 1,Z = 1, Rd /Rb = 1, R d /Rb = 0.01 and ET h

In the above, g± and f± are given by g± = ε/ 2 − Δ2± f± = Δ± / Δ2± − 2 , respectively. Here, Δ(φ+ ) and Δ(φ− ) are the pair potentials felt by quasiparticles with an injection angle φ and π − φ in US. It is possible to fix the quasiclassical Green’s function in US as g± τˆ3 +f± τˆ2 . The resulting quasiclassical Green’s function τ3 + f τˆ1 for spin for DN can be given by gˆ τ3 + f τˆ2 for spin singlet junctions and gˆ triplet ones with g = cos θ and f = sin θ. Here, we choose a d-wave superconductor with Δ± = Δ0 cos(2φ ∓ 2β) as an example of a singlet superconductor, where β

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denotes the angle between the normal to the interface and the lobe direction of the d-wave pair potential. As an example of a triplet superconductor, we choose a pwave superconductor with Δ± = ±Δ0 cos(φ∓α) where α denotes the angle between the normal to the interface and the lobe direction of the p-wave pair potential. In the above, Δ0 is the magnitude of the maximum value of the pair potentials. In the following, we concentrate on the quasiparticle local density of state (LDOS) ρ(ε) as a function of quasiparticle energy ε. First, we discuss the local density of states and pair amplitude in DN for conventional s-wave superconductor junctions. As shown in Fig. 2, the resulting LDOS has a gap like structure. On the other hand, pair amplitude f (ε) satisfies f (ε) = f ∗ (−ε). Since f (ε) is a pure real number at ε = 0, the resulting amplitude of real part of g(ε) is smaller than unity. Next, we focus on the d-wave pair potential case. We s-wave Electrode

DN

0

L spin singlet s−wave

spin singelt s −wave ρ(ε)

1 f(ε)

1

0

−1 0 −0.3

0

ε/Δ 0

0.3 −0.3

0

ε/Δ 0

0.3

Figure 2. Left panel: LDOS of DN in s-wave superconductor junctions. Right panel: corresponding pair amplitude f (ε). Solid line, real part. Dotted line, imaginary part.

choose β = 0, i.e., dx2 −y2 -wave and β = π/4, i.e., dxy -wave case. For the latter case, quasiparticles feel MARS at the interface of d-wave superconductor independent of the direction of their motions. As shown in Fig.3, for dx2 −y2 -wave case, LDOS has a gap like structure similar to the s-wave superconductor case. However, for dxy -wave case, LDOS becomes always unity. This means the absence of the proximity effect for dxy -wave case. In the present case, MARS and proximity effect compete with each other. Due to this competition, anomalous temperature dependence of the Josephson current emerges in d-wave junctions14 . As regards the ε dependence of the pair amplitude, f (ε) = f ∗ (−ε) is satisfied as in the case of s-wave superconductor. To understand the relation between the MARS and the proximity effect in detail, we then focus on the triplet superconductor junctions. We choose p-wave pairing

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ρ(ε) Electrode

DN

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0

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Figure 3.

0

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LDOS for d-wave superconductor. dx2 −y 2 and dxy pairing.

spin singelt d−wave 1 f (ε) 0

−1 −0.3

0

ε/Δ 0

0.3

Figure 4. The corresponding pair amplitude f (ε) for dx2 −y 2 -wave pairing in Fig. 3. Solid line, real part. Dotted line, imaginary part.

for α = 0, i.e. px -wave pairing, and α = π/2, i.e. py -wave pairing. For the former case, quasiparticles feel MARS at the interface independent of their directions of motions. For px -pairing, LDOS has a zero energy peak (ZEP). On the other hand, for py -wave case, due to the absence of the proximity effect, LDOS becomes unity as shown in Fig. 5. If we choose a general value of α within 0 < α < π/2, the ZEP always emerges 8 . Thus the existence of ZEP of LDOS in DN is a significantly robust features. It is possible to propose an experimental setup which discriminates the spin triplet superconducting state from spin singlet one by the presence or absence of ZEP in DN by using scanning tunneling spectroscopy. It is also interesting to look at the ε dependence of pair amplitude f (ε). The resulting f (ε) satisfies f (ε) = −f ∗ (−ε). Such an unusual energy dependence can not be realized with the even frequency pairing 15 . In such a case, the resulting pair amplitude at ε = 0 becomes a pure imaginary number. This property is related to the emergence of the ZEP in the LDOS.

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-

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0

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spin triplet p−wave

10 f (ε) 0

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Figure 6. The corresponding pair amplitude f (ε) of px -wave pairing in Fig. 5. Solid line, real part. Dotted line, imaginary part.

3. Conclusion In the present paper, we have summarized an activity of theoretical works about unconventional superconductor junctions. For spin singlet d-wave superconductor junctions, mid gap Andreev state competes with the proximity effect. In this case, local density of state of quasiparticle in DN has a gap like structure as a function of ε. The pair potential f (ε) satisfies conventional relation f (ε) = f ∗ (−ε). On the other hand, for spin triplet p-wave junctions, the resulting LDOS has a zero energy peak. At the same time, pair amplitude f (ε) satisfies f (ε) = −f ∗ (−ε). References 1. M. Sigrist and T. M. Rice, Rev. Mod. Phys. 67 (1995) 503. 2. D. J. Van Harlingen, Rev. Mod. Phys. 67 (1995) 515, C.C. Tsuei and J. R. Kirtley, Rev. Mod. Phys. 72, (2000) 969.

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3. 4. 5. 6. 7.

8.

9. 10. 11. 12. 13. 14. 15.

Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, (1995) 3451. S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63, (2000) 1641. Yu. V. Nazarov, Phys. Rev. Lett. 73 (1994) 1420; Yu. V. Nazarov, Superlattices and Microstructuctures 25, (1999) 1221. Y. Tanaka, Yu. V. Nazarov, S. Kashiwaya, Phys. Rev. Lett. 90 (2003) 167003; Y. Tanaka, Yu. V. Nazarov, A. A. Golubov, S. Kashiwaya, Phys. Rev. B 69 (2004) 144519. Y. Tanaka and S. Kashiwaya, Phys. Rev. B 70 (2004)012507; Y. Tanaka, S. Kashiwaya and T. Yokoyama, Phys. Rev. B 71 (2005) 094513, Y. Tanaka, Y. Asano, A. Golubov, S. Kashiwaya, Phys. Rev. B 72 (2005) 14503. M. Kawamura, H. Yaguchi, N. Kikugawa, Y. Maeno, H. Takayanagi, J. Phys. Soc. Jpn. 74, (2005) 531. Ch. W¨ alti, H.R. Ott, Z. Fisk, and J.L. Smith, Phys. Rev. Lett. 84, (2000) 5616. P. M. C. Rourke, M. A. Tanatar, C. S. Turel, J. Berdeklis, C. Petrovic, and J. Y. T. Wei, Phys. Rev. Lett. 94, (2005) 107005. K. Ichimura, S. Higashi, K. Nomura and A. Kawamoto, Synthetic Metals Vol. 153 (2005) 409. K.D. Usadel, Phys. Rev. Lett. 25, 507 (1970). T. Yokoyama, Y. Tanaka, A. A. Golubov and Y. Asano, Phys. Rev. B 73, 140504(R) (2006). Y. Tanaka and Golubov, Phys. Rev. Lett. Y. Tanaka and A. Golubov, Phys. Rev. Lett. 98, 037003 (2007).

Quantum Information


7/24/2008 10:36:09 AM


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ANALYZING THE EFFECTIVENESS OF THE QUANTUM REPEATER

KENICHIRO FURUTA AND HIROFUMI MURATANI Toshiba Corporation 1, Komukai-Toshiba-cho, Saiwai-ku, Kawasaki, 212-8582, JAPAN E-mail: [email protected] The range of quantum key distribution is limited by exponential attenuation of photons in optical fibers. It is believed that a quantum repeater can improve the order of the attenuation and can be useful for extending the communication distance of QKD. It is already shown that quantum repeater is effective when not considering the repeater noise. In this paper, we analyze the effectiveness of the quantum repeater when considering the repeater noise. We point out that there is some threshold of length of EPR pairs, under which quantum repeater protocol is valid and over which quantum repeater is not valid. Besides, this threshold depends on the largeness of the repeater noise. So, it is important to suppress the repeater noise in order that the quantum repeater scheme can really improve the bit rate. We also analyze the effectiveness from the viewpoint of the security and show that QKD is secure even if quantum repeater is used.

1. Introduction There is an everlasting threat that a current practical cryptographic scheme whose security is based on computational assumptions will become insecure due to a future improvement of computers. Therefore, quantum key distribution (QKD), e.g. BB84[2] and B92, has been attracting considerable attention because its security is based only on quantum principles and it is unconditionally secure. Due to exponential attenuation of photons in the channel, the naive QKD is valid only in the range of short distance. It is important to extend the communication distance from a practical viewpoint. So far, three approaches have been proposed to extend the range of QKD: 1.Protocol modification for multiple photon emission: Some protocol modifications[7] which make the scheme robust against the photon number splitting(PNS) attack were proposed. However, modified protocols can extend the range of QKD to only a few times that of the original. Due to this limit, further extensions require introduction of other improvements. 2.Coherent states: Some protocol using coherent states[1] were proposed. It was demonstrated that they are more resistant to noises than the single photon protocols and can achieve high bit rate even in long distance. However, its security has been discussed enthusiastically[6, 8]. In this paper, we do not consider this. 3.Quantum repeater protocol: In order to reduce noises on quantum state transferred through the optical fiber, quantum teleportation is used to send the quantum state by using an EPR pair generated by a quantum repeater protocol. We call such a scheme QKD with quantum repeater. The quantum repeater recursively applies

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entanglement swapping(ES) and entanglement purification protocol(EPP) to shortlength-EPR pairs[4]. It was demonstrated in [4] that a quantum repeater protocol can generate the long-length-EPR pair for the quantum teleportation with high fidelity. At a glance, the quantum repeater seems to be the most promising approach of the three approaches. However, the discussion in [4] seems to assume that noises in quantum memory on checkpoints can be negligible. Therefore, we reexamine the practical possibility of the quantum repeater by evaluating the order of the bit rate with considering the repeater noise. Although the repeater noise is noticed in [5], the evaluation is done without the repeater noise. We also examine the security of QKD with quantum repeater. In Section 2, we review the quantum repeater protocol in [4]. In section 3, we categorize noises that occur in the protocol. In Section 4, we evaluate the bit rate without considering the repeater noise. In Section 5, we evaluate the bit rate with considering the repeater noise. In Section 6, we prove the security of QKD with quantum repeater. In Section 7, we provide a summary of this paper. 2. Quantum repeater protocol The quantum repeater is a scheme which extends the length of an EPR pair with high fidelity. In this section, we review the quantum repeater protocol in [4]. 2.1. Abstract specification We explain an abstract specification of the quantum repeater protocol. The protocol recursively applies ES and EPP to short-length-EPR pairs and finally generates a long-length-EPR pair with high fidelity. Let L be the number of EPR pairs which are linked in a single ES execution, N be the number of checkpoints and n be the depth of the recursive executions. These satisfy a relation, N = Ln . In the channel between the sender A and the receiver B, N − 1 checkpoints, denoted C1 , C2 , · · · , CN −1 , are settled. For convenience, A and B are denoted C0 and CN , respectively. The distance between A and B is denoted as D and the distance between two adjacent checkpoints is denoted as d. That is, D = N d. After completing the above protocol, an EPR pair with high fidelity shared between C0 and CN is obtained. 2.2. Entanglement swapping In ES, partners of two EPR pairs are swapped. Here, we provide an explicit realizations of ES based on local measurement. ES can also be realized based on Bell measurement. First, Controlled NOT gate (CNOT) is applied to photons 2 and 3 of |φ+ 1,2 ⊗ + |φ 3,4 . Next, WH(Walsh-Hadamard) transformation is applied and produces 1 + + − − 2 (|02 |03 ⊗ |φ 1,4 + |02 |13 ⊗ |ψ 1,4 + |12 |03 ⊗ |φ 1,4 + |12 |13 ⊗ |ψ 1,4 ). Then, one of four computational bases of photons 2 and 3 is measured and this measurement maps the state of two photons 1 and 4 into a Bell state. Here, the observed basis of photons 2 and 3 indexes the projected Bell state of photons 1 and 4. In the next step, the projected Bell state of photons 1 and 4 is transformed into |φ+ . For this purpose, the measurement results of photons 2 and 3 are sent from

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Algorithm 1 Quantum repeater 1: Initialization: At each checkpoint Ci , i = 0, · · · , N −1, EPR pairs are generated and one photon of each pair is sent to the next checkpoint Ci+1 . 2: for x = 1 to n do 3: ES: Execute ES in each of the checkpoints CkLx−1 ,(k = 1, 2, · · · , N/Lx−1 ) except CLx , C2Lx , · · · , CN −Lx . Then, the EPR pairs of length Lx can be obtained. 4: EPP: Execute EPP for EPR pairs in each of the checkpoints CLx , C2Lx , · · · , CN −Lx . Then, the EPR pairs of length Lx with high fidelity can be obtained. 5: end for a checkpoint having photons 2 and 3 to ckechpoints having photons 1 and 4 with the classical communication. 2.3. Entanglement purification EPP pulls out an EPR pair of high fidelity from multiple EPR pairs of low fidelity. We consider an EPP which is also considered in [4]. The validity of EPP requires that the fidelity of EPR pairs before the purification should be in a certain range. It is demonstrated as follows. Let F and F be the fidelity of the EPR pairs before EPP and the fidelity of purified EPR pair, respectively. In the case that EPP generates the purified pair from two EPR pairs, F can be expressed in terms of F as follows[3]: F = Φ/Λ, where F¯ = (1 − F )/3, Φ = F 2 + F¯ 2 and Λ = F 2 + 2F F¯ + 5F¯ 2 . (1)

In order that F ≥ F in Eq.(1), F should be in the range of 1/2 ≤ F ≤ 1. 3. Noises Here, we categorize possible noises which occur during an execution of the protocol. 3.1. Noises during the protocol execution Several types of noises can occur during the execution of the quantum repeater protocol. We classify them by their causes. The measurement noise is noises which occur during a measurement of a quantum state. The one-qubit operation noise is noises which occur during a one-qubit operation in the protocol. The two-qubit operation noise is noises which occur during a two-qubit operation in the protocol. The channel noise is noises of a quantum state transferring through the channel. The repeater noise is noises of a quantum state in the repeater devices even in the absence of operation. Here, the one-qubit operation and the two-qubit operation mean a unitary operation on one qubit and a unitary operation on two qubits, respectively. We consider that quantum noises which occur during the classical communication in the execution of EPP or quantum teleportation is an example of the repeater noise. The first three noises were modeled and analyzed in [4]. For the channel noise and the repeater noise, we evaluate with the order, such as exponential or polynomial. We review the models and analyses of the first three noises in the next subsection.

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3.2. Conventional models and analysis We show models of such noises and the modification of Eq.(1) caused by these noises. Let ρ be a density matrix before operations. First, the one-qubit operation noise is ideal 1 are one-qubit modeled as ρ → O1 ρ = p1 O1ideal ρ+ 1−p 2 tr1 ρ⊗I1 , where O1 and O1 operations with and without the noise, respectively, and I1 is the identity operator and p1 is the probability with which the operations are performed without noise. ideal 2 ρ + 1−p The two-qubit operation noise is modeled as ρ → O12 ρ = p2 O12 4 tr12 ρ ⊗ ideal I12 , where O12 and O12 are two-qubit operations with and without the noise, respectively, and I12 is the identity operator and p2 is the probability with which the operations are performed without noise. The measurement noise is modeled as P0η = η|00| + (1 − η)|11|, P1η = η|11| + (1 − η)|00|, where η is the probability with which the measurements are performed correctly and P0η and P1η are POVM |00| and |11|, respectively, with error probability η. Based on the above noise models, the fidelity, FL , after linking L EPR pairs 2 L−1 p p (4η 2 −1) 4F −1 L . Similarly, by ES executions is expressed as FL = 14 + 34 1 2 3 3 based on the above noise models, the change of the fidelity by EPP is expressed as follows: (2) F = {ΘΦ + 2η η¯Ξ + Π}/{ΘΛ + 4(2η η¯Ξ + Π)}, 2

1−p where η¯ = 1 − η, Θ = η 2 + η¯2 , Ξ = F F¯ + F¯ 2 and Π = 8p22 . F and F have three 2 intersections. Let two intersections except 0.25 be Fmin andFmax , where Fmin < Fmax . Then, in order that F ≥ F in Eq.(2), F should be in the range of Fmin ≤ F ≤ Fmax . The range of F where the quantum repeater is valid become narrow as noises become large. 4. Bit rate in absence of repeater noise In [4], it was demonstrated that the required amount of resources of the quantum repeater increases as a polynomial function of the distance between A and B, D. This leads to the conclusion that the bit rate of the quantum repeater decreases as an inverse of a polynomial function of the distance. This result was derived under the condition that only the channel noise, the measurement noise, the one-qubit operation noise and the two-qubit operation noise are considered. Theorem 4.1. Consider QKD with quantum repeater. If only the channel noise, the measurement noise, the one-qubit operation noise and the two-qubit operation noise are considered, there exists a polynomial function p(·) such that the bit rate of the QKD decreases as Ω(p(D)−1 ). Here, g(n) = Ω(f (n)) means ∃c > 0 ∃N ∈ N ∀n > N g(n) ≥ cf (n).

Proof. The bit rate is estimated by considering both the merit of quantum repeater, keeping high fidelity, and the demerit, increase of resource. Let M be the number of EPR pairs consumed by a single execution of EPP. Then, the number of EPR pairs, R, in the whole execution of the quantum repeater, is R = (LM )n = Ln M n = N M n = N (LlogL M )n = N logL M+1 , where N is proportional to the communication distance D, and L and M do not depend on D. Thus, R is a polynomial function of D.

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In this scheme, when not considering the repeater noise, the fidelity of the EPR pair generated by the quantum repeater protocol stays constant even if D increases. So, the bit rate of the QKD decreases as Ω(p(D)−1 ). In contrast to exponential damping in the absence of the quantum repeater, the bit rate of QKD in the presence of the quantum repeater decreases as an inverse of a polynomial function of the communication distance. Although Theorem 4.1 does not indicate whether the exact value of the bit rate is really improved by the quantum repeater, it can be expected to be effective for sufficiently large D. 5. Bit rate in presence of repeater noise We next consider the case the repeater noise is taken into account. In ES and EPP in the quantum repeater protocol, classical communications between repeater devices are needed. In addition, quantum teleportation sends classical information from A to B. During these classical communications, the quantum states in the repeater devices lose their fidelity. We assume the repeater noise as follows: The fidelity of a quantum state in a repeater device decreases exponentially with respect to the time length of a classical communication. We call the assumption of this model the exponential damping assumption. Theorem 5.1. Under the exponential damping assumption, the bit rate of QKD with quantum repeater decreases exponentially with respect to the distance D. Proof. The length of EPR pairs increases as the quantum repeater protocol proceeds. As far as the length is small, the fidelity can be recovered by EPP. However, if the length exceeds a threshold, Dth , then the fidelity becomes lower than Fmin and EPP can’t recover the fidelity any more. The reason is that EPR pairs have to stay in quantum memory on ckeckpoints during classical communication and the time of classical communication becomes large as the length of EPR pairs become large. Thus, time of being affected by the repeater noise get larger. After the fidelity goes under the threshold of EPP, the fidelity continues to decrease as the quantum repeater protocol proceeds. So, after crossing the threshold, ΔFES +ΔFEP P +ΔFRN ≥ ΔFRN , where ΔFES , ΔFEP P and ΔFRN are the fidelity decreases due to ES, EPP and the repeater noise, respectively, during an iteration in the recursive execution of the quantum repeater protocol. The dumping due to the repeater noise is exponential according to the exponential dumping assumption. So, the overall dumping is exponential according to the equation above. Of course, there may be many other quantum repeater protocols. However, in general, our result holds for protocols provided the classical communication whose distance is proportional to that between a sender and a receiver is used in the protocols. 6. Security The security proof can be done with simple idea. Let IE be the amount of eavesdropper’s information and Pcont be the person who controls the quantum repeater unit and Eve be an eavesdropper. The following relationship holds for IE . (IE in

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original QKD)≥ (IE in QKD with quantum repeater, where Pcont is Eve)≥ (IE in QKD with quantum repeater, where Pcont is except Eve). The reason is as follows. For QKD with quantum repeater, Eve can get more information (or equal at least) when he controls repeater unit more than when he does not. So, (IE in QKD with quantum repeater, where Pcont is Eve)≥ (IE in QKD with quantum repeater, where Pcont is except Eve). Thus, it is sufficient to prove the security when repeater unit is controlled by Eve. Here we deal with QKD protocols where operations for quantum repeater protocol can be done within attacks allowed for Eve in original QKD protocol. Unconditionally secure protocols, such as BB84[B84], belong to this category because Eve is allowed to do almost every quantum operations as attacks. So, when repeater unit is controlled by Eve, operations for quantum repeater can be considered as a part of Eve’s attacks allowed in original QKD. Then, (IE in original QKD)≥ (IE in QKD with quantum repeater, where Pcont is Eve). Thus, we can turn the proof of QKD with quantum repeater into the proof of original QKD. 7. Summary We demonstrated that the bit rate of QKD with quantum repeater decreases asymptotically exponentially with respect to the communication distance when the repeater noise is taken into account. This is because EPP can not work when the length of EPR pairs exceed the threshold. In contrast, quantum repeater protocol works when the length of EPR pairs does not exceed the threshold. This threshold depends on the largeness of the repeater noise. So, it is important to suppress the repeater noise in order to enlarge the range where quantum repeater is effective. Besides, we showed abstract of proof that QKD with quantum repeater is secure. References 1. G. Barbosa, E. Corndorf, P. Kumar, and H. Yuen. Secure Communication Using Mesoscopic Coherenet States. Phys. Rev. Lett., Vol. 90, p. 227901, 2003. 2. C.H. Bennett and G. Brassard. Quantum cryptography: Public key distribution and coin tossing. Proc. of IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India (IEEE New York), pp. 175-179, 1984. 3. C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, and W.K. Wootters. Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels. Phys. Rev. Lett., Vol. 76, No. 5, pp. 722-725, 1996. 4. H.J. Briegel, W. D¨ ur, J.I. Cirac, and P. Zoller. Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication. Phys. Rev. Lett., Vol. 81, No. 26, pp. 5932–5935, 1998. 5. L. Childress, J.M. Taylor, A.S. Sørensen, and M.D. Lukin. Fault-tolerant quantum repeaters with minimal physical resources and implementations based on single photon emitters. quant-ph/0502112, 2005. 6. T. Nishioka, T. Hasegawa, H. Ishizuka, K. Imafuku, and H. Imai. How much security does Y-00 protocol provide us? Phys. Lett. A, Vol. 327, pp. 2832, 2004. 7. V. Scarani, A. Acin, G. Ribordy, and N. Gisin. Quantum Cryptography Protocols Robust against Photon Number Splitting Attacks for Weak Laser Pulse Implementations. Phys. Rev. Lett., Vol. 92, No. 5, p. 057901, 2004. 8. H. Yuen, E. Corndorf, G. Barbosa, and P. Kumar. Barbosa et al. Reply:. Phys. Rev. Lett., Vol. 94, No. 4, p. 048902, 2005.

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ARCHITECTURE-DEPENDENT EXECUTION TIME OF SHOR’S ALGORITHM

RODNEY VAN METER1∗, KOHEI M. ITOH1 AND THADDEUS D. LADD2 1

Graduate School of Science and Technology, Keio University and CREST-JST 3-14-1 Hiyoshi, Kohoku-ku, Yokohama-shi, Kanagawa 223-8522, Japan 2 Edward L. Ginzton Laboratory Stanford University, Stanford, CA, 94305-4085, USA

We show how the execution time of algorithms on quantum computers depends on the architecture of the quantum computer, the choice of algorithms (including subroutines such as arithmetic), and the “clock speed” of the quantum computer. The primary architectural features of interest are the ability to execute multiple gates concurrently, the number of application-level qubits available, and the interconnection network of qubits. We analyze Shor’s algorithm for factoring large numbers in this context. Our results show that, if arbitrary interconnection of qubits is possible, a machine with an application-level clock speed of as low as one-third of a (possibly encoded) gate per second could factor a 576-bit number in under one month, potentially outperforming a large network of classical computers. For nearest-neighbor-only architectures, a clock speed of around twenty-seven gates per second is required.

1. Introduction Quantum computers are currently being designed that will take advantage of quantum mechanical effects to perform certain computations much faster than can be achieved using current (“classical”) computers 1 . Many technological approaches have been proposed, some of which are being investigated experimentally. DiVincenzo proposed five criteria which must be met by any useful quantum computing technology 2 . In addition to these criteria, a useful quantum computing technology must also support a quantum computer system architecture which can run one or more quantum algorithms in a usefully short time. This observation subsumes into one requirement several issues which, while not strictly necessary to build a quantum computer, will have a strong impact on the possibility of engineering a practical system. These include the importance of gate “clock” speed, support for concurrent gate operations, the total number of application-level qubits supportable, and the complexities of the qubit interconnect network 3 . This paper discusses the impact of these architectural elements on algorithm execution time using the example of Shor’s algorithm for factoring large numbers 4 . Shor’s algorithm ignited much of the current interest in quantum computing because of the improvement in computational class it appears to offer on this important problem. Using Shor’s algorithm, a quantum computer can solve the problem in polynomial time, for a superpolynomial speed∗ e-mail

address: [email protected].

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Figure 1. Scaling of number field sieve (NFS) on classical computers and Shor’s algorithm for factoring on a quantum computer, using BCDP modular exponentiation with various clock rates. Both horizontal and vertical axes are log scale. The horizontal axis is the size of the number being factored.

up. Shor’s algorithm is theoretically important, well defined, and utilizes building blocks (arithmetic, the quantum Fourier transform) with broad applicability, making it ideal for our analysis. On a classical computer, or a collection thereof, the time and computing resources to factor a large number, using the fastest known algorithm, scale superpolynomially in the length of the number (in decimal digits or bits). This algorithm is the generalized number field sieve (NFS) 5 . Its asymptotic computational complexity on large numbers is O(e(nk log

2

n)1/3

)

(1)

where n is the length of the number, in bits, and k = 64 9 log 2. The comparable computational complexity to factor a number N using Shor’s algorithm is dominated by the time to exponentiate a randomly chosen number x, modulo N , for a superposition of all possible exponents. Therefore, efficient arithmetic algorithms for calculating modular exponentiation in the quantum domain are critical. Very often clock speed and other architectural features are ignored as issues in quantum computing devices, assuming that the superpolynomial speed-up will dominate, making the algorithm practical on any experimentally realizable quantum computer. Shor’s algorithm runs in polynomial time, but the details of the polynomial matter: what degree is the polynomial, and what are the constant factors? An immediate comparison of the execution time to factor a number on classical and quantum computers is shown in Figure 1. The performance of Shor’s algorithm on a quantum computer using the Beckman-Chari-Devabhaktuni-Preskill (BCDP) modular exponentiation algorithm 6 is compared to classical computers running the general Number Field Sieve (NFS). The steep curves are for NFS on a set of classical computers. The left curve is extrapolated performance based on a previous world record, factoring a 530-bit number

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in one month, established using 104 PCs and workstations made in 2003 7 . The right curve is speculative performance using 1,000 times as much computing power. This could be 100,000 PCs in 2003, or, based on Moore’s law, 100 PCs in 2018. From these curves it is easy to see that Moore’s law has only a modest effect on our ability to factor large numbers. The shallower curves on the figure are predictions of the performance of a quantum computer running Shor’s algorithm, using the BCDP modular exponentiation routine, which uses 5n qubits to factor an n-bit number, requiring ∼ 54n3 gate times to run the algorithm on large numbers. The four curves are for different clock rates from 1 Hz to 1 GHz. The performance scales linearly with clock speed. Factoring a 576-bit number in one month of calendar time requires a clock rate of 4 kHz. A 1 MHz clock will solve the problem in about three hours. If the clock rate is only 1 Hz, the same factoring problem will take more than three hundred years. The performance of the BCDP modular exponentiation algorithm is almost independent of architecture. However, the performance of most polynomial-time algorithms varies noticeably depending on the system architecture 8,9 . The main objective of this paper is to show how we can improve the execution time shown in Figure 1 by understanding the relationship of architecture and algorithm. 2. Results We have analyzed two separate architectures, still technology independent but with some important features that help us understand performance. The AC (abstract concurrent) architecture is our abstract model, akin to what is commonly used when drawing quantum circuits. It supports arbitrary concurrency and gate operands any distance apart without penalty. The second architecture, NTC (neighbor-only, two-qubit gate, concurrent) , assumes the qubits are laid out in a one-dimensional line, and only neighboring qubits can interact. This is a reasonable description of several important experimental approaches, including a one-dimensional chain of quantum dots 10 , the original Kane proposal 11 , and the all-silicon NMR device 12 . Above the architecture resides the choice of algorithm, especially for basic arithmetic operations. The computational complexity of an algorithm can be calculated for total cost, or for latency or circuit depth, if the dependencies of variables allow multiple parts of a computation to be conducted concurrently. Fundamentally, the computational complexity of quantum modular exponentiation is O(n3 ) 13,6 , that is, the execution cost grows as the cube of the number of qubits. It consists of 2n modular multiplications of n-bit numbers, each of which consists of O(n) additions, each of which requires O(n) operations. However, O(n3 ) operations do not necessarily require O(n3 ) time steps; the circuit depth can be made shallower than O(n3 ) by performing portions of the calculation concurrently. On an abstract machine, we can reduce the running time of each of the three layers (addition, multiplication, exponentiation) to O(log n) time steps by running some of the gates in parallel, giving a total running time of O(log3 n). This requires O(n3 ) qubits and the ability to execute an arbitrary number of gates on separate qubits. Such large numbers of qubits are not expected to be practical for the foreseeable future, so interesting engineering

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lies in optimizing for a given set of architectural constraints. Addition forms the basis of multiplication, and hence of exponentiation. Classically, many forms of adders have been used in computer hardware 14 . The most basic type of adder, variants of which are used in both VBE and BCDP (as well as our algorithm F, below), is the carry-ripple adder, in which the carry portion of the addition is done linearly from the low-order bits to the high-order. This form of adder is O(n) in both circuit depth and complexity; it is the only efficient type for NTC linear architectures, in which the time to propagate the low-order carry is inherently constrained to O(n). When long-distance gates are available, as in AC architectures, the use of faster adders such as conditional-sum, carry-lookahead, or carry-save adders can result in O(log n) latency, though the complexity remains O(n) 15,16,17 . We have composed several algorithm variants, A through F, as well as investigated concurrent and parallel versions of the original Vedral-Barenco-Ekert (VBE) 13 and BCDP algorithms 15 ; only the fastest for our AC and NTC architectures are presented here. Four parameters control the behavior of the algorithm variants and how well they match a particular architecture. These parameters include the choice of type of adder and the amount of space required. Algorithm variant D is tuned for AC using the conditional-sum adder, and F is tuned for NTC using the Cuccaro-Draper-Kutin-Moulton (CDKM) carry-ripple adder 18 . We have optimized the parameter settings for each individual data point, though the differences are just barely visible on our log-log plot. The values reported here for both algorithms are calculated using 2n2 qubits of storage to exponentiate an n-bit number, the largest number of qubits our algorithms can effectively use. The primary characteristics of the algorithms shown in Figure 2 are summarized in Table 1. The table lists the number of multiplication units executing concurrently, the space, measured in number of logical qubits, the concurrency, or number of logical operations taking place at the same time, and the overall circuit depth, or time, measured in gates. Table 1. Composition of our algorithms. algorithm conc. BCDP algorithm D algorithm F

adder BCDP cond. sum CDKM

multipliers (s) 1 ∼ n/4 ∼ n/4

space 5n + 3 2n2 2n2

concurrency 2 ∼ n2 ∼ 3n/4

depth ∼ 54n3 ∼ 9n log22 (n) ∼ 20n2 log2 (n)

Figure 2 shows our results for our faster algorithms. We have kept the 1 Hz and 1 MHz lines for BCDP, and added matching lines for our fastest algorithms on the AC and NTC architectures. For AC, our algorithm D requires a clock rate of only about 0.3 Hz to factor the same 576-bit number in one month. For NTC, using our algorithm F, a clock rate of around 27 Hz is necessary. The graph shows that, for problem sizes larger than 6,000 bits, our algorithm D is one million times faster than the basic BCDP algorithm, and algorithm F is one thousand times faster. For very large n, the latency of D is ∼ 9n log22 (n). The latency of F is ∼ 20n2 log2 (n). This relationship of architecture and algorithm has obvious architectural implications: concurrency is critical, and support for long-distance gates is important.

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1 billion years

one year one month

104 PCs , 20 03

1 thousand years 100 years 10 years

NFS ,

Time to Factor an n-bit Number

1 million years

Sh

, NTC

.F z, alg

r, 1H

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D, AC Hz, alg.

100 seconds

.

arch

P

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z, B

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Shor, 1

, NTC

one day one hour

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.

arch

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, 1M

Shor

. D, AC

MHz, alg

Shor, 1

arch.

one second 100

1000

10000

100000

n (bits)

Figure 2. Scaling of number field sieve (NFS) and Shor’s algorithms for factoring, using faster modular exponentiation algorithms.

3. Discussion A fast clock speed is obviously also important for a fast algorithm; however, it remains an open question whether those quantum computing technologies which feature naturally fast physical quantum gates will have the fastest overall algorithm speed. All quantum computing technologies feature some level of decoherence, requiring resources for quantum error correction 19,20,1 . As an example, quantum computers based on Josephson junctions are likely to have extremely fast single-qubit and two-qubit gates, with a physical clock rate at the gigahertz level, as demonstrated in recent experiments 21 . However, the single-qubit decoherence time is only about 1 µs for the most coherent superconducting qubits 22 . Although “fast,” the difficulty in long-term qubit storage and the needed resources for fault tolerant operation may be quite large, so these implementations might make excellent processors with poor memories. In sharp contrast, NMR-based approaches 11,12 are quite slow, with nuclear-nuclear interactions in the kilohertz range. However, the much longer coherence times of nuclei 23 make the use of NMR-based qubits as memory substantially easier 24 . Ion trap implementations have the benefit of faster single-qubit-gate, two-qubit-gate, and qubit-measurement speeds with longer coherence times, but the added complication of moving ionic qubits from trap to trap physically 25 or exchanging their values optically 26 complicates the picture for the application-level clock rate. New physical proposals for overcoming speed and scalability obstacles continue to be developed, leaving the ultimate hardware limitations on clock speed and its relation to algorithm execution time uncertain. 4. Conclusions We have shown that the actual execution time of Shor’s algorithm is dependent on the important features of concurrent gate execution, available number of qubits, interconnect topology, and clock speed, as well as the critical choice of an architecture-appropriate

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arithmetic algorithm. Our algorithms have shown a speed-up factor ranging from nearly 13,000 for factoring a 576-bit number to one million for a 6,000-bit number. Acknowledgments The authors wish to thank Eisuke Abe, Kevin Binkley, Fumiko Yamaguchi, Seth Lloyd, Kae Nemoto, and W. J. Munro for helpful discussions. References 1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000). 2. D. P. DiVincenzo, Science 270, 255 (1995). 3. R. Van Meter and M. Oskin. J. Emerging Tech. in Comp. Sys., 2(1), Jan. 2006. 4. P. W. Shor, in Proc. 35th Symposium on Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, CA, 1994), pp. 124–134. 5. D. E. Knuth, The Art of Computer Programming, volume 2 / Seminumerical Algorithms, 3rd ed. (Addison-Wesley, Reading, MA, 1998). 6. D. Beckman, A. N. Chari, S. Devabhaktuni, and J. Preskill, Phys. Rev. A 54, 1034 (1996). 7. RSA Security Inc., web page, 2004, http://www.rsasecurity.com/rsalabs/node.asp?id=2096. 8. N. Kunihiro, IEICE Trans. Fundamentals, E88-A(1):105–111, (2005). 9. A. G. Fowler, S. J. Devitt, and L. C. Hollenberg, Quantum Information and Computation 4, 237 (2004). 10. D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). 11. B. E. Kane, Nature 393, 133 (1998). 12. T. D. Ladd et al., Phys. Rev. Lett. 89, 17901 (2002). 13. V. Vedral, A. Barenco, and A. Ekert, Phys. Rev. A 54, 147 (1996). 14. M. D. Ercegovac and T. Lang, Digital Arithmetic (Morgan Kaufmann, San Francisco, CA, 2004). 15. R. Van Meter and K. M. Itoh, Phys. Rev. A 71, 052320 (2005). 16. T. G. Draper, S. A. Kutin, E. M. Rains, and K. M. Svore, A Logarithmic-Depth Quantum CarryLookahead Adder, http://arXiv.org/quant-ph/0406142 (2004). 17. P. Gossett, Quantum Carry-Save Arithmetic, http://arXiv.org/quant-ph/9808061 (1998). 18. S. A. Cuccaro, T. G. Draper, S. A. Kutin, and D. P. Moulton, A new quantum ripple-carry addition circuit, http://arXiv.org/quant-ph/0410184, 2004. 19. A. M. Steane, Phys. Rev. A 68, 042322 (2003). 20. S. J. Devitt, A. G. Fowler, and L. C. Hollenberg, Simulations of Shor’s algorithm with implications to scaling and quantum error correction, http://arXiv.org/quant-ph/0408081, 2004. 21. T. Yamamoto, et al., Nature 425, 941 (2003). 22. D. Vion, et al., Science 296, 886 (2002). 23. T. D. Ladd et al., Phys. Rev. B 71, 014401 (2005). 24. K. M. Itoh, Solid State Comm. 133, 747 (2005). 25. D. Kielpinski, C. Monroe, and D. J. Wineland, Nature 417, 709 (2002). 26. B.B. Blinov, D. L. Moehring, L.-M. Duan, and C. Monroe, Nature 428, 153 (2004). 27. L. M. Duan, Phys. Rev. Lett. 93, 100502 (2004).

Quantum Dots and Kondo Effects


7/24/2008 10:36:53 AM


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COULOMB BLOCKADE PROPERTIES OF 4-GATED QUANTUM DOT SHINICHI AMAHA1†, TSUYOSHI HATANO1, SATOSHI SASAKI2, TOSHIHIRO KUBO1, YASUHIRO TOKURA1, 2, SEIGO TARUCHA1, 3 1

ICORP-JST, 3-1 Morinosato-Wakamiya, Atsugi-shi,Kanagawa, 243-0198, Japan 2NTT BRL, 3-1 Morinosato-Wakamiya, Atsugi-shi,Kanagawa, 243-0198, Japan 3 Graduate School of Applied Phys., Univ. Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

We fabricated a few electron vertical quantum dot (QD) device having four separate gate electrodes and characterized the electronic properties. In this device, geometrical symmetry in the confining potential for the QD, that is, orbital degeneracy for the QD electronic states can be tuned by adjusting the voltages applied to the four gates. It is then necessary to precisely characterize the performance of each gate. We use a nonlinear single electron tunneling spectroscopy technique to characterize the gate performances and apply a capacitance network model for this device to reproduce the observed gate performances.

1. Introduction Circular vertical quantum dots (QDs) are often referred as to “artificial atoms” because of atom-like features, such as shell filling and Hund’s rule [1]. The shell filling is well understood by considering electronic states confined by a two-dimensional (2D) harmonic potential with a high degree of rotational symmetry [2]. This brings about a novel concept to manipulate the degree of shell filling or orbital degeneracy by introducing anisotropy to the 2D harmonic potential in a controlled manner. We use a four-gate tuning technique to manipulate anisotropy in the lateral confinement potential for a vertical QD (See Fig. 1) [3]. The electron number in this QD can be varied one-by-one, starting from zero up to ~20, however, the tunability of the confinement anisotropy is significantly different from device to device, probably because the way of metal electrode attachment to the mesa is different from gate to gate. This causes a variation of gate performance to control the confinement anisotropy among the four gates. Here we introduce a capacitance model to deal with the difference in the gate performance among the four gates, and compare with the experimental data of Coulomb diamonds. We finally discuss the ability of our device to manipulate the degree of shell filling in the QD.

†

Email: [email protected].

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192 192

Fig.1. Schematic diagram of the four-gate vertical QD.

2. Capacitance network model for the quantum dot having four gates Figure 2 shows a capacitance network model for the four-gate QD [4], that is capacitively coupled to the four gate electrodes labeled i=1 to 4 and also connected to the source (S) and drain (D) contact electrodes (source and drain contacts).

Fig.2. Capacitance network of the four-gate QD.

We employ the constant interaction (CI) model and write the total charge Q on the QD as sum of the charges on all capacitors (source, drain and four gates) 4

Q = CS (V − VS ) + C D (V − VD ) + ∑ C gi (V − Vgi ) ,

(1)

i =1

where CS (CD) is the capacitance between the QD and the source (drain) electrode and Cgi is that between the QD and the gate i. VS (VD) is the electronic potential of the source (drain) electrode and Vgi is that of the gate i. The electrochemical potential (N) for the N-electron QD is defined as an energy cost for adding the Nth electron to the (N-1)-electron QD. Using Eq. (1), (N) is presented as

μ

µ ( N ) = ( N − N 0 − 1 / 2)

μ

4 e2 e 4 − ∑ C giVgi ≡ − e ∑ α iVgi + β N , C C i =1 i =1

(2)

α≡

where N0 is electron number in the QD at VS, VD, and Vgi = 0, C=∑Ci+CS+CD, i Cgi/C and βN ≡(N-N0-1/2) e2/C. The condition of Coulomb blockade for the N-th electron with respect to the source-drain voltage, Vsd, can be written as:

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193 193

µ ( N ) < µ0 −

1 1 e Vsd < µ 0 + eVsd < µ ( N + 1) , 2 2

(3)

where µ0 is the Fermi energy of the source and drain electrodes at Vsd=0. The condition (3) is also written as

β N − µ0 +

e Vsd 2

4

< e ∑ α iVgi < β N + i =1

e Vsd e2 . − µ0 − 2 C

(4)

Figure 3 shows the schematic diagram of Coulomb diamonds for sweeping Vg1 with the other gate voltages constant (a) and with the other gate voltages altogether, (Vg1=Vg2=Vg3=Vg4 ≡Vg) (b), respectively. The vertical sizes of the Coulomb diamonds, ∆Vgi , and ∆Vg , for (a), and (b), respectively, are presented as:

 4   ∑ C gi  ∆Vg = C gi ∆Vgi = e .   i =1

(5)

We use Eqs. (5) to analyze the measured Coulomb diamonds, and derive the capacitance ratio for each gate electrode. It is ideal that all gate capacitances are the same in order to create complete circular symmetry in the confining potential. However, this is not the case for real devices, as described bellow.

Fig.3. Schematic diagrams of Coulomb diamonds for the four-gate QD (a) Coulomb blockade region plotted in Vg v.s. Vsd plane with setting all gate voltages the same (Vg=Vg1=Vg2=Vg3=Vg4). (b) Coulomb blockade region plotted in Vg1 v.s. Vsd plane with setting other gate voltages (Vg2, Vg3, Vg4) fixed.

3. Experiments and discussion 3.1. Device fabrication and measurement technique The starting material for device fabrication is a specially designed AlGaAs/GaAs/AlGaAs double barrier structure (DBS) with an n-AlGaAs contact layer above and below. In this DBS the well and contacts are so strongly coupled that the Kondo effect is observed in the conductance measurement at low temperatures. Details of the Kondo effect observed in this device will be discussed elsewhere. We first made a electrode metal pattern on the

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194 194 material surface: a small square (~0.35µm x 0.35µm) with a narrow line extending from each corner outwards, etched the material using the metal pattern as a self-aligned mask to make a cross-shaped mesa (Fig. 1), and evaporated gate metal in the surrounding of the mesa. The line mesa is narrow enough that current only flows in the metal. This current then flows in the cross-shaped mesa downwards and reaches the bottom electrode. We use a standard DC technique to measure the current, I, flowing through the QD in response to a voltage Vsd applied to the bottom electrode. We apply voltage Vgi to each gate electrode Gi independently. The measurement temperature is 1.5K. 3.2. Coulomb diamonds with sweeping voltage for all gates common Figure 4 shows the measured dI/dVsd in the plane of Vsd and Vi (=Vg1=Vg2=Vg3=Vg4), which is the case for Fig. 3(a). A series of Coulomb diamonds are observed in the center for Vg > -1.06 V. However, for the more negative gate voltage there are no more closed Coulomb diamonds observed, indicating N = 0 for Vg < -1.06 V. Note no Coulomb oscillations are observed in this gate voltage range, either. So the number of electrons is increased for each Coulomb diamond one-by-one to the top, starting from N=0. The Coulomb diamond with N=2 is larger than that with N=1 corresponding to the complete shell filling of the1s state. Edges of each Coulomb diamond are not clear due to high sample temperature (~1.5K) and the strong coupling between the well and source drain electrodes.

Fig.4. Measured conductance plotted in source drain voltage (Vsd) v.s. gate voltage applied to all gate electrodes as common (Vg1=Vg2=Vg3=Vg4=Vg).

3.3. Coulomb diamonds with sweeping voltage just to one gate Figure 5 shows the Coulomb diamonds measured in the same as those in Fig. 4 but by sweeping just one gate voltage: Vgi =Vg1 (a), Vg2 (b), Vg3 (c), and Vg4 (d), respectively. In

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195 195 Fig. 4 we see the N=6 Coulomb diamond for Vg1=Vg2=Vg3=Vg4=-0.60V. So we can compare the gate performance for this Coulomb diamond among the four gates in the setup for Fig. 3(b). Actually in Fig. 5 we can compare the N =5 Coulomb diamonds as well. The shape of N=6 Coulomb diamond looks similar among (a) to (d). However, the threshold lines forming this Coulomb diamond have the slopes different among (a) to (d). In Fig.5 the N=6 Coulomb diamond width is evaluated as ∆Vg1= 0.34V, ∆Vg2=0.44V, ∆Vg3=0.28V and ∆Vg4=0.22(V), respectively. In addition, this width is evaluated as ∆Vg=0.097V in Fig.4. Using these values, we derive α1=0.28, α2=0.22, α3=0.34, and α4=0.45, respectively. From these αi (i=1, 2, 3, 4), we can also evaluate capacitance ratio as C1:C2:C3:C4~2.5:2:3:4. So that, only two diamonds are clearly distinguished in (a) and (b), whereas three, and four diamonds are distinguished in (c), and (d), respectively. Thus obtained α value is so significantly different among the four gates, but still in the relevant range for tuning the degree of symmetry with voltages applied to the four gates. The capacitances of gate 1 and 2 are estimated to be slightly lower than that of gate 3 and 4. The Ti/Au metals of gate 1 and 2 electrodes are expected to lower attachment to the cross-shaped mesa than that of gate 3 and 4 due to the tilt on evaporation process of gate metal.

Fig.5. Measured conductance plotted in each gate voltage v.s. source drain voltage by fixed other gate voltages at -0.60V. (a) Vg1 v.s. Vsd (Vg2=Vg3=Vg4=-0.60V). (b) Vg2 v.s. Vsd (Vg1=Vg3=Vg4=-0.60V). (c) Vg3 v.s. Vsd (Vg1=Vg2=Vg4=-0.60V). (d) Vg4 v.s. Vsd (Vg1=Vg2=Vg3=-0.60V).

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196 196 4. Conclusion Gate performance is studied for a few-electron vertical QD having four gates. The capacitance of each gate to the QD is evaluated using a capacitance network model and from measurements of Coulomb diamonds by sweeping only the corresponding gate voltage. The derived capacitance values are significantly different among the four gates but still in the relevant range for tuning the degree of symmetry in the QD confining potential with voltages applied to the four gates. Acknowledgments The authors thank K. Ono, T. Maruyama, W. G. van der Wiel, D.G. Austing, R. Sakano, T. Kita, M. Eto, S. Suga and N. Kawakami for fruitful discussion. Part of this work is financially supported by the Grant-in-Aid for Scientific Research A (No. 40302799) and by CREST-JST, and by IT Program, MEXT and by the DARPA-QUIST program (DAAD19-01-1-0659). References 1. S. Tarucha, D.G. Austing, T. Honda, R.J. van der Hoge and L.P. Kowenhoven, Phys. Rev. Lett. 77, 3613(1996). 2. P. Matagne and J.P. Leburton, Phys. Rev. B 65, 235323(2002). 3. D.G. Austing, T. Honda and S. Tarucha, Jpn. J. Appl. Phys. 36, 4151 (1997). 4. W.G. van der Wiel, S. De Franceschi, J.M. Elzerman, T. Fujisawa, S. Tarucha and L.P. Kouwenhoven, Rev. Mod. Phys. 75, 1 (2003).

July 23, 2008

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ORDER-N ELECTRONIC STRUCTURE CALCULATION OF n-TYPE GaAs QUANTUM DOTS

S. NOMURA Institute of Physics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8571, Japan E-mail: [email protected] T. IITAKA RIKEN (The Institute of Physical and Chemical Research), 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan E-mail: [email protected] A linear scale method for calculating electronic properties of large and complex systems is introduced within a local density approximation. The method is based on the Chebyshev polynomial expansion and the time-dependent method, which is tested in calculating the electronic structure of a model n-type GaAs quantum dot.

1. Introduction Linear scale methods for calculating the electronic structures have been actively investigated in the last decade because of increasing demand for predicting properties of large and complex systems with computational cost linear scale with respect to the system size N . One of such methods is the Chebyshev polynomial expansion method.1 The electron density is obtained by using a matrix representation of the Fermi-operator, which is expanded in the Chebyshev matrix polynomials. With a combination of the Chebyshev polynomial expansion method and time-dependent method2 (CPE-TDM), the electron density of states (DOS) is obtained without calculating eigenenergies and eigenstates. The computational time of CPE-TDM scales as O(N ), as compared with that of the conventional method such as conjugate gradient method (CGM), which grows as O(N 2 ). Thus CPE-TDM enables us to calculate large systems which require prohibitively large computational time by CGM. CPE-TDM was applied to calculate the optical properties of hydrogenated Si nanocrystals containing atoms more than 10,000 within the empirical pseudopotential formalism3 and the ESR spectrum of s = 1/2 antiferromagnet Cu benzoate,4 which have proved the advantage of CPE-TDM. However, CPE-TDM has not been applied to calculation of the electronic structure within a LDA. In this paper, we report on an implementation of CPE-TDM for a calculation of the electronic struc-

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ture of n-type GaAs quantum dot (QD) within a LDA and compare the results with a CGM.

2. Method of calculation The model structure is a 20 nm-wide GaAs quantum well sandwiched by undoped Alx Ga1−x As(x = 0.3) barriers. The electrons are assumed to be supplied from 5 nm-thick Si-doped Alx Ga1−x As layer, located 20 nm above the GaAs quantum well layer. The Fermi-energy (EF ) is taken as the origin of the energy. The Fermi-level pinning model is assumed. The number of the electrons in a QD is not fixed to an integer number and is determined by EF and the potential energy. The lateral confining potential is fixed to a parabolic potential, which may be created by a surface gate structure in experiments. The model Hamiltonian of the system within the LDA is H=

1 p2 + m∗ ω02 r 2 + Vc (r) + VH (r) + Vx (r) ∗ 2m 2

(1)

where m∗ is the effective mass of the electrons, ω0 = 3 meV, and, Vc (r), VH (r), and Vx (r) are the vertical confining potential, the Hartree potential, the exchange potential, respectively. A 3D mesh of 64 × 64 × 8 is used for the calculation of the electron density, and 64 × 64 × 16 is used for the calculation of the potentials. The axis perpendicular to the quantum well layer is taken to be z-direction. The Hamiltonian is discretized in real space by the higher-order finite difference method.5 N A random phase vector as defined by |Φ ≡ n=1 |nξn , where {|n} is a basis set and ξn are a set of random phase variables, is used as an initial state. This was shown to give results with the smallest statistical error.6 Here Φ is a Nx × Ny × Nz column vector for a system defined by a real-space uniform grid of Nx ×Ny ×Nz . The 1 electron density n(r) is extracted by the Fermi operator function f (H) = eβ(H−E F ) +1 as n(r) = |Φ|f (H )|r|2 .

(2)

The Fermi operator is evaluated by the Chebyshev polynomial expansion, f (H)|Φ =

ak (β)Tk (H )|Φ.

(3)

k

Actually the electron density is calculated with jmax sets of |Φ as n(r) = j =jmax | r|f (H )|Φj |2 /jmax , where · stands for statistical average. The j =1 Hartree and exchange potentials are calculated using n(r). The new solution of the potential VHnew (r) is combined with the solution obtained for the previous iteration old (r) + αVHnew (r). Similarly, in order to reduce the statistical by VH (r) = (1 − α)VH fluctuation, n(r) is combined with the density obtained for the previous iteration

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199

by n(r) = (1 − γ)n old (r) + γn new (r). The parameter α is varied between 0.10 to 0.05. The parameter γ is varied between 0.30 and 0.025 for the CPE-TDM method. For the CGM calculations, γ is fixed to 1. The DOS is calculated by a time evolution method as given by 1 (4) ρ(ω) = − Im( Φ|G(ω + iη)|Φ ), π where G(ω + iη) is a real-time Green’s function. The DOS is calculated with kmax sets of |Φ. The energy resolution η is chosen to be 0.5 meV. It should be noted that kmax used for calculating the DOS can be independently chosen from jmax for each self-consistent iteration procedure. 3. Results and discussions Model calculations are performed for a GaAs QD containing about 77 electrons. The number of the self-consistent iterations is fixed to 100 for both the CGM and CPETDM calculations. The potential is converged to |VH (r) − VHnew (r)| < 0.003meV for the CGM calculation. The electron density distributions are shown in Fig. 1 for CPE-TDM with jmax = 64 and for CGM. The calculated electron density distribution reasonably agrees with the result by a CGM within the statistical fluctuations. The Friedel-type spatial oscillations of the electron density7 are reproduced in both the results by the CPE-TDM and CGM.

*+,

*?, A

Controllable Quantum States: Mesoscopic Superconductivity & Spintronics (MS+S2006), Procedings of the International Symposium, NTT Basic Res Labatories, Japan 27 February - 2 Marc

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Spintronics

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