Nanophysics: Coherence and Transport (eBook)

Lecturers 12
Seminar speakers 14
Participants 16
Preface 20
Contents 24
Fundamental aspects of electron correlations and quantum transport in one-dimensional systems 34
Introduction 38
Non-Fermi liquid features of Fermi liquids: 1D physics in higher dimensions 41
Long-range effective interaction 49
1D kinematics in higher dimensions 54
Infrared catastrophe 57
1D 57
2D 59
Dzyaloshinskii-Larkin solution of the Tomonaga-Luttinger model 61
Hamiltonian, anomalous commutators, and conservation laws 61
Reducible and irreducible vertices 64
Ward identities 65
Effective interaction 67
Dyson equation for the Green’s function 70
Solution for the case g2 = g4 72
Physical properties 75
Momentum distribution 75
Tunneling density of states 77
Renormalization group for interacting fermions 78
Single impurity in a 1D system: scattering theory for interacting fermions 83
First-order interaction correction to the transmission coefficient 84
Hartree interaction 87
Exchange 89
Renormalization group 90
Electrons with spins 91
Comparison of bulk and edge tunneling exponents 94
Bosonization solution 95
Spinless fermions 95
Bosonized Hamiltonian 95
Bosonization of fermionic operators 98
Attractive interaction 100
Lagrangian formulation 101
Correlation functions 104
Fermions with spin 107
Tunneling density of states 109
Transport in quantum wires 110
Conductivity and conductance 110
Galilean invariance 110
Kubo formula for conductivity 111
Drude conductivity 112
Landauer conductivity 112
Dissipation in a contactless measurement 114
Conductance of a wire attached to reservoirs 115
Inhomogeneous Luttinger-liquid model 116
Elastic-string analogy 116
Kubo formula for a wire attached to reservoirs 119
Experiment 121
Spin component of the conductance 122
Thermal conductance: Fabry-Perrot resonances of plasmons 125
Polarization bubble for small q in arbitrary dimensionality 127
Polarization bubble in 1D 128
Small q 128
q near 2kF 130
Some details of bosonization procedure 131
Anomalous commutators 131
Bosonic operators 134
Commutation relations for bosonic fields phi and nu 134
Problem with backscattering 135
References 138
Impurity in the Tomonaga-Luttinger model: A functional integral approach 142
Introduction 146
Functional integral representation 147
The effective action for the Tomonaga-Luttinger Model 149
The bosonized action for free electrons 150
Gauging out the interaction 152
Tunnelling density of states near a single impurity 155
Jacobian of the gauge transformation 158
References 160
Novel phenomena in double layer two-dimensional electron systems 162
Introduction 166
Overview of physics in the quantum hall regime 167
Basics 167
Quantized hall effects 169
Integer QHE 170
Fractional QHE 173
Composite fermions 175
Double layer systems 176
Coulomb drag between parallel 2D electron gases 179
Basic concept 179
Experimental 181
Elementary theory of Coulomb drag 182
Comparison between theory and experiment 184
Tunneling between parallel two-dimensional electron gases 186
Ideal 2D-2D tunneling 187
Lifetime broadening 189
2D-2D tunneling in a perpendicular magnetic field 190
Strongly-coupled bilayer 2D electron systems and excitonic superfluidity 193
Introduction 193
Quantum hall ferromagnetism 195
Tunneling and interlayer phase coherence at nuT = 1 198
Excitonic superfluidity at nuT = 1 201
Detecting excitonic superfluidity 203
Conclusions 207
References 207
Many–body theory of non–equilibrium systems 210
Introduction 214
Motivation and outline 214
Closed time contour 215
Free boson systems 217
Partition function 217
Green functions 220
Keldysh rotation 222
Keldysh action and causality 224
Free bosonic fields 226
Collisions and kinetic equation 227
Interactions 227
Saddle point equations 229
Dyson equation 231
Self-energy 232
Kinetic term 234
Collision integral 235
Particle in contact with an environment 237
Quantum dissipative action 237
Saddle–point equation 239
Classical limit 239
Langevin equations 240
Martin–Siggia–Rose 241
Thermal activation 242
Fokker-Planck equation 244
From Matsubara to Keldysh 246
Dissipative chains and membranes 247
Fermions 249
Free fermion Keldysh action 249
Keldysh rotation 252
External fields and sources 254
Tunneling current 257
Interactions 258
Kinetic equation 260
Disordered fermionic systems 264
Disorder averaging 264
Non–linear sigma–model 266
Usadel equation 269
Fluctuations 270
Spectral statistics 273
Gaussian integration 275
Single particle quantum mechanics 276
References 278
Non-linear quantum coherence effects in driven mesoscopic systems 280
Introduction 284
Weak Anderson localization in disordered systems 284
Drude approximation 285
Beyond Drude approximation 286
Weak localization correction 288
Non-linear response to a time-dependent perturbation 291
General structure of nonlinear response function 292
Approximation of single photon absorption/emission 294
Quantum rectification by a mesoscopic ring 295
Diffusion in the energy space 301
Quantum correction to absorption rate 305
Weak dynamic localization and no-dephasing points 309
Conclusion and open questions 313
References 314
Noise in mesoscopic physics 316
Introduction 320
Poissonian noise 322
The wave packet approach 324
Generalization to the multi–channel case 327
Scattering approach based on operator averages 328
Average current 329
Noise and noise correlations 332
Zero frequency noise in a two terminal conductor 332
General case 332
Transition between the two noise regimes 333
Noise reduction in various systems 334
Double barrier structures 334
Noise in a diffusive conductor 335
Noise reduction in chaotic cavities 336
Noise correlations at zero frequency 337
General considerations 337
Noise correlations in a Y–shaped structure 338
Finite frequency noise 340
Which correlator is measured? 340
Noise measurement scenarios 341
Finite frequency noise in point contacts 343
Noise in normal metal-superconducting junctions 343
Bogolubov transformation and Andreev current 344
Noise in normal metal–superconductor junctions 347
Noise in a single NS junction 349
Below gap regime 349
Diffusive NS junctions 350
Near and above gap regime 352
Hanbury-Brown and Twiss experiment with a superconducting source of electrons 354
S–matrix for the beam splitter 355
Sub-gap regime 357
Near and above gap regime 358
Noise and entanglement 359
Filtering spin/energy in superconducting forks 360
Tunneling approach to entanglement 362
Bell inequalities with electrons 363
Noise in Luttinger liquids 368
Edge states in the fractional quantum Hall effect 370
Transport between two quantum Hall edges 373
Keldysh digest for tunneling 375
Backscattering current 377
Poissonian noise in the quantum Hall effect 379
Effective charges in quantum wires 383
Conclusions 385
References 388
Higher moments of noise 394
Introduction 398
The probability distribution P(i) 399
A simple model for a tunnel junction 399
Noise in Fourier space 400
Consequences 401
Effect of the environment 401
Imperfect voltage bias 402
dc current: dynamical Coulomb blockade 403
The second moment 403
The third moment 403
Effect of an external fluctuating voltage 404
Voltage vs. current fluctuations 404
Imperfect thermalization 405
Principle of the experiment 406
Possible methods 406
Experimental setup 407
Experimental results 408
Third moment vs. voltage and temperature 408
Effect of the detection bandwidth 409
Effect of the environment 410
Perspectives 410
Quantum regime 410
Noise thermal impedance 412
Conclusion 414
References 414
Electron subgap transport in hybrid systems combining superconductors with normal or ferromagnetic metals 416
Introduction 420
NS junctions in the clean limit 421
Single particle tunnelling in a tunnel junction 421
Introduction 421
Tunnel Hamiltonian 421
Perturbation theory: golden rule 422
Higher order processes 424
Bogoliubov-de Gennes equations 425
BCS Hamiltonian and diagonalization 425
Simple examples 426
NS interface 428
Disordered NIS junctions 432
Introduction 432
Perturbation theory for NIS junction 432
Tunnel Hamiltonian and golden rule 432
Real space representation 435
Disorder averaging 437
Example: quasi-one-dimensional diffusive wire connected to a superconductor 439
Calculation of the spectral current 439
Zero-temperature limit 440
Subgap noise of a superconductor-normal-metal tunnel interface 442
Current fluctuations in NS systems 442
Current noise in tunnel systems 443
Generalized Schottky relation 444
An explicit example: a wire out of equilibrium 446
Tunnelling in a three-terminal system containing ferromagnetic metals 449
Introduction 450
Co-tunnelling and crossed Andreev tunnelling rates 452
Tunnel Hamiltonian 452
Calculation of spin-dependent tunnel rates 453
Spin-dependent conductance matrix 454
Discussion 456
Nonmagnetic probes 456
Spin-polarized probes 457
References 458
Low-temperature transport through a quantum dot 460
Introduction 464
Model of a lateral quantum dot system 466
Thermally-activated conduction 472
Onset of Coulomb blockade oscillations 472
Coulomb blockade peaks at low temperature 474
Activationless transport through a blockaded quantum dot 476
Inelastic co-tunneling 477
Elastic co-tunneling 479
Kondo regime in transport through a quantum dot 481
Effective low-energy Hamiltonian 482
Linear response 486
Weak coupling regime: TK < <
Strong coupling regime: T < <
Beyond linear response 495
Splitting of the Kondo peak in a magnetic field 497
Kondo effect in quantum dots with large spin 501
Concluding remarks 505
References 508
Transport through quantum point contacts 512
Introduction 516
Spin-density-functional calculations 518
The Anderson model 520
Results 521
Current noise 524
Conclusions 524
References 525
Transport at the atomic scale: Atomic and molecular contacts 528
Introduction 532
Parity oscillations in atomic chains 533
Superconducting quantum point contacts 541
The Hamiltonian approach 542
Comparison to experimental results 547
Environmental effects 549
Classical phase diffusion 550
Dynamical Coulomb blockade 554
Single-molecule junctions 555
Concluding remarks 563
References 564
Solid State Quantum Bit Circuits 570
Why solid state quantum bits? 574
From quantum mechanics to quantum machines 574
Quantum processors based on qubits 576
Atom and ion versus solid state qubits 578
Electronic qubits 578
Qubits in semiconductor structures 579
Kane’s proposal: nuclear spins of P impurities in silicon 579
Electron spins in quantum dots 579
Charge states in quantum dots 581
Flying qubits 581
Superconducting qubit circuits 582
Josephson qubits 583
Hamiltonian of Josephson qubit circuits 583
The single Cooper pair box 584
Survey of Cooper pair box experiments 585
How to maintain quantum coherence? 586
Qubit-environment coupling Hamiltonian 587
Relaxation 587
Decoherence: relaxation + dephasing 588
The optimal working point strategy 588
The quantronium circuit 589
Relaxation and dephasing in the quantronium 591
Readout 592
Switching readout 593
AC methods for QND readout 593
Coherent control of the qubit 594
Ultrafast ’DC’ pulses versus resonant microwave pulses 594
NMR-like control of a qubit 595
Probing qubit coherence 596
Relaxation 596
Decoherence during free evolution 598
Decoherence during driven evolution 600
Qubit coupling schemes 600
Tunable versus fixed couplings 601
A tunable coupling element for Josephson qubits 601
Fixed coupling Hamiltonian 602
Control of the interaction mediated by a fixed Hamiltonian 603
Running a simple quantum algorithm 603
Conclusions and perspectives 604
References 605
Abstracts of seminars presented at the School 610