Fundamental Problems of Mesoscopic Physics Interactions and Decoherence NATO Science Series

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Series II: Mathematics, Physics and Chemistry – Vol. 154 Fundamental Problems of Mesoscopic Physics Interactions and Decoherence

edited by

Igor V. Lerner University of Birmingham, Birmingham, United Kingdom

Boris L. Altshuler Princeton University and NEC Research Institute, Princeton, New Jersey, U.S.A. and Yuval Gefen The Weizmann Institute of Science, Rehovot, Israel

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Preface xi

Part I Decoherence and Dephasing 1 Electron Dephasing in Mesoscopic Metal Wires 3 Norman O. Birge and F. Pierre 1 Introduction 3 2 τφ in pure Au and Ag samples 5 3 Possible explanations for saturation of τφ 7 4 Aharonov-Bohm experiments in large Cu rings 10 5 Conclusions 13 2 Decoherence effects in the Josephson current of a Cooper pair shuttle 17 Alessandro Romito and Rosario Fazio 1 Introduction 17 2 The model 18 3 Results 20 4 Cooper pair shuttle with SQUID loops 29 3 Dephasing in disordered metals with superconductive grains 33 M. A. Skvortsov, A. I. Larkin, M. V. Feigel’man 1 Introduction 33 2 Description of the formalism 36 3 Dynamics of the phase 37 4 Phase transition 38 5 Cooperon self-energy 40 6 Dephasing time 42 7 Magnetoresistance 44 8 Discussion 45 4 Decoherence in Disordered Conductors at Low Temperatures: the effect 49 of Soft Local Excitations Y. Imry, Z. Ovadyahu and A. Schiller 1 Introduction 50

v vi FUNDAMENTAL PROBLEMS OF MESOSCOPIC PHYSICS

2 The vanishing of the dephasing rate as T →0: theory 51 3 Experimental results 55 4 A tunnelling model for loosely bound heavy impurities 58 5 Conclusions 61 5 Quantum precursor of shuttle instability 65 D. Fedorets, L. Y. Gorelik, R. I. Shekhter and M. Jonson 1 Introduction 65 2 Theoretical model 66 3 Analysis and results 69 6 Dephasing and dynamic localization in quantum dots 75 V.E.Kravtsov 1 Introduction 75 2 Weak dynamic localization 78 3 Role of electron interaction on dynamic localization in closed quan- tum dots. 91 4 Dynamic localization in an open quantum dot and the shape of the Coulomb blockade peak. 93 5 Summary 97 7 Mesoscopic Aharonov-Bohm oscillations in metallic rings 99 T. Ludwig and A. D. Mirlin 1 Introduction 100 2 Low-temperature limit: fully coherent sample 100 3 Dephasing by electron-electron interaction 104 4 Summary 112 8 Influence Functional for Decoherence of Interacting Electrons in Disor- 115 dered Conductors J. von Delft 1 Introduction 115 2 Main results of influence functional approach 118 3 Origin of the Pauli factor 121 4 Calculating τϕ `a la GZ 122 5 Dyson equation and Cooperon self energy 125 6 Vertex contributions 127 7 Discussion and summary 130 Appendix 131 1 Outline of derivation of influence functional 132 2 Cooperon self energy before disorder averaging 135 3 Thermal averaging 136 Contents vii

Part II Entanglement and Qubits 9 Low-frequency noise as a source of dephasing of a qubit 141 Y. M. Galperin, B. L. Altshuler and D. V. Shantsev 1 Introduction and model 142 2 Results for a single fluctuator 150 3 Summation over many fluctuators 154 4 Simulations 158 5 Comparison with the noise in the random frequency deviation 162 6 Applicability range of the model 162 7 Conclusions 163 10 Entanglement production in a chaotic quantum dot 167 C.W.J. Beenakker, M. Kindermann, C.M. Marcus, A. Yacoby 1 Introduction 167 2 Relation between entanglement and transmission eigenvalues 169 3 Statistics of the concurrence 171 4 Relation between Bell parameter and concurrence 171 5 Relation between noise correlator and concurrence 172 6 Bell inequality without tunneling assumption 173 7 Conclusion 175 11 Creation and detection of mobile and non-local spin-entangled electrons 179 P. Recher, D. S. Saraga and D. Loss 1 Sources of mobile spin-entangled electrons 179 2 Superconductor-based electron spin-entanglers 180 3 Triple dot entangler 189 4 Detection of spin-entanglement 193 5 Electron-holes entanglers without interaction 197 6 Summary 198 12 Berezinskii-Kosterlitz-Thouless transition in Josephson junction arrays 203 L. Capriotti, A. Cuccoli and A. Fubini, V. Tognetti and R. Vaia 1 Introduction 204 2 The model 205 3 Numerical simulations 206 4 Results 208 5 The phase diagram 212 6 Summary 214 Appendix: PIMC in the Fourier space for JJA 215

Part III Interactions in Normal and Superconducting Systems 13 Quantum coherent transport and superconductivity in carbon nanotubes 219 M. Ferrier, A. Kasumov, R. Deblock, M. Kociak, S. Gueron, B. Reulet and H. Bouchiat 1 Introduction 219 2 Proximity induced superconductivity in carbon nanotubes as a probe of quantum transport 221 3 Intrinsic superconductivity in ropes of SWNT on normal contacts 226 4 Conclusion 235 viii FUNDAMENTAL PROBLEMS OF MESOSCOPIC PHYSICS 14 Quantum Hall ferromagnets, cooperative transport anisotropy, and the 239 random field Ising model J. T. Chalker, D. G. Polyakov, F. Evers, A. D. Mirlin, and P. Wolfle¨ 1 Introduction 240 2 Domain formation and the random field Ising model 242 3 Transport anisotropy arising from domain formation 244 4 Transport along domain walls 247 5 Concluding remarks 249 15 Exotic proximity effects in superconductor/ferromagnet structure 251 F.S.Bergeret, A.F. Volkov and K.B. Efetov 1 Introduction 252 2 The condensate function in a F/S/F sandwich 254 3 Josephson current in a F/S/F/S/F structure 259 4 Effect of spin-orbit interaction 262 5 Induced ferromagnetism in the superconductor 263 6 Summary 270 16 Transport in Luttinger Liquids 275 T. Giamarchi, T. Nattermann and P. Le Doussal 1 Introduction 275 2 Model 276 3 Transport at intermediate temperatures 277 4 Creep 278 5 Variable range hopping 280 6 Open issues 282 17 Interaction effects on counting statistics and the transmission distribution 285 M. Kindermann, Yuli V. Nazarov 18 Variable–range Hopping in One–dimensional Systems 295 J. Prior, M. Ortuno˜ and A. M. Somoza 1 Introduction 295 2 Model 297 3 Model with geometrical fluctuations only 298 4 Hopping matrix elements 301 5 Quantum model 304 6 Fluctuations 305 7 Summary and conclusions 306 19 On the Electron-Electron Interactions in Two Dimensions 309 V. M. Pudalov, M. Gershenson and H. Kojima 1 Introduction 309 2 Renormalized spin susceptibility 311 3 Effective mass and g-factor 318 4 Summary 324 Contents ix 20 Correlations and spin in transport through quantum dots 329 M. Sassetti, F. Cavaliere, A. Braggio and B. Kramer 1 Introduction 329 2 The model 331 3 The characteristic energy scales 333 4 Sequential transport 335 5 Results 339 6 Discussion and conclusion 345 21 Interactions in high-mobility 2D electron and hole systems 349 E. A. Galaktionov, A. K. Savchenko, S. S. Safonov, Y. Y. Proskuryakov, L. Li, M. Pepper, M. Y. Simmons, D. A. Ritchie, E. H. Linfield, and Z. D. Kvon 1 Introduction 349 2 Ballistic regime of electron-electron interaction 351 3 Interaction effects in a 2D hole gas in GaAs 353 4 Electron-electron interaction in the ballistic regime in a 2DEG in Si 361 5 Interaction effects in the ballistic regime in a 2DEG in GaAs. Long- range fluctuation potential. 362 σ 6 Comparison of F0 (rsψ) in different 2D systems 364 7 Conclusion 367

Index 371 Preface

The physics of mesoscopic devices came to existence in the mid-eighties when it became apparent that due to the onset of global phase coherence in systems of smaller dimensions, conventional approaches fail to describe sub- micron and nanoscale systems. Such systems with sizes intermediate between macro- and micro (i.e. single-atomic sizes) are now referred to as mesoscopic. Mesoscopic physics remains the focus of intense experimental and theoretical activity for more than 15 years. This diverse field is continually fuelled by rapid advances in materials and nanostructure technology, and in low tempera- ture techniques. A wide variety of new devices extremely promising for major novel directions in technology, including carbon nanotubes, ballistic quantum dots, hybrid mesoscopic junctions made of different type of materials etc, came to existence during the last few years. This, in turn, demands a deep understand- ing of fundamental physical phenomena on mesoscopic scales. As a result, the forefront of fundamental research in condensed matter has been moved to the areas, where that the interplay of electron -electron correlations and quantum interference of phase-coherent electrons scattered by impurities and/or bound- aries is the key to such an understanding. In spite of a substantial recent progress and significant achievements in the understanding of, e.g., physics of quantum dots, quantum wires and nanotubes, a set of extremely important fundamental issues still remains unresolved. One of the most intriguing problems at the heart of mesoscopics is that of dephasing and decoherence at low temperatures. Numerous recent experiments on dephasing in quantum dots and wires have added a new dramatic twist to this problem which had beforehand seemed to be well understood. On the face of it, these experimental results contradict to the most fundamental principles of quantum theory. Our belief is that the situation is not that dramatic and that these new results will be understood within the mainstream theory of disordered electronic systems, as the problem of decoherence in mesoscopics is clearly a part of a wider problem of the electron-electron interaction. A very interesting related direction in mesoscopic physics is investigating the possibility of using normal and superconducting nanodevices for the im- plementation of quantum entanglement and quantum manipulations in scalable

xi xii FUNDAMENTAL PROBLEMS OF MESOSCOPIC PHYSICS systems. Most of the currently proposed implementations of quantum com- puting devices use quantum optical systems, characterised by long decoher- ence times and controllable dynamics. However, their large-scale integration is highly problematic. Mesoscopic normal and superconducting systems, and even more so hybrid and magnetic systems look as very promising possible alternatives. In particular, the employment of spin-related physics appears to be promising. the possibility to control the electron spin degree of freedom, and the relatively long coherence time of this degree of freedom led to the emergence of a new field – spintronics. In general, disorder and/or chaos which are inherent for mesoscopic devices make experimental manifestation of the interactions much richer than in pure bulk systems. The understanding of decoherence as well as other effects of the interactions is crucial for developing future electronic, photonic and spintronic devices, including the element base for quantum computation. In this rapidly changing area, regular meetings of leading researchers in the field play a very important role. The present volume contains review articles written by the key speakers of a joint NATO advanced research workshop – EURESCO conference held in September 2003 in Granada, Spain. The talks have been concentrated on the topics described above, so that the volume is divided into three parts covering these topics. Acknowledgements. We are thankful to the NATO Scientific Affairs Divi- sion and EURESCO (European Research Conference Organisation of the Euro- pean Science Foundation) for providing an excellent opportunity for bringing together leading researchers working in mesoscopic physics. We gratefully acknowledge an additional support by the US Army Research Laboratory - Eu- ropean Research Office. We thank all the officers of EURESCO who helped in the organisation of this meeting, especially Irene Mangion for her patient help at all the stages of preparing the meeting, and Anne Guehl for smooth, helpful and efficient running of all the organisational errands at the meeting venue in Granada.

BORIS ALTSHULER,YUVAL GEFEN AND IGOR LERNER