Nonequilibrium Quantum Transport Physics in Nanosystems: Foundation of Computational Nonequilibrium Physics in Nanoscience and Nanotechnology

Contents......Page 10
Preface......Page 8
Overview of Quantum Mechanical Techniques......Page 24
1. Quantum Mechanics: Perspectives......Page 26
1.1 Wave Mechanics of Particles: Schrödinger Wave Function......Page 30
1.1.1 Some Algebraic Relations of Q and P......Page 33
1.1.2 Deterministic Schrödinger Wave Equation......Page 34
1.1.3 Isotopic Wavefunction and Many-Body Wavefunction......Page 35
1.1.3.1 Decoupling of Isotopic Degrees of Freedom......Page 36
1.2 Generator of Position Eigenstates......Page 37
1.4 Non-Hermitian Canonical Variables......Page 41
1.4.1 Left and Right Eigenvectors of Non-Hermitian Operators......Page 42
1.5 Coherent State Formulation as a Mixed q-p Representation......Page 44
2.1.1 The Complex Canonical Variables......Page 46
2.1.3 Second-quantization of the Schrödinger-Like Equation......Page 48
3. The Linear Chain of Atoms Coupled by Harmonic Forces......Page 49
3.1.1 Creation and Annihilation Operator for a Coupled Linear Chain of Atoms......Page 50
4.1 Elementary Lattice Dynamics: The Linear Chain......Page 53
4.1.1 Quantization of the Vibrational Mode: Phonons......Page 58
4.2 Lattice Vibrations in Three Dimensions......Page 59
4.3 Normal Coordinates in Three Dimensions......Page 60
4.3.1 Acoustic and Optic Modes......Page 65
4.3.2 Frequency Distribution of Normal Modes......Page 67
4.5 Hamiltonian in Terms of Normal Coordinates......Page 69
4.6 Phonons in Three Dimensions......Page 71
5.1 Maxwell Equations......Page 73
5.2 The ElectromagneticWave Equations......Page 74
5.2.1 A Single ElectromagneticWave Equation......Page 75
5.3 Covariant Formulation of Electrodynamics......Page 77
5.4 Complex Dynamical Variables......Page 79
6.2 Second Quantization of the Classical φ and φ......Page 85
6.3 Biorthogonal Bases......Page 88
6.4 Coherent State Bases......Page 89
7. Coherent States Formulation of Quantum Mechanics......Page 91
7.1 Non-Orthogonality of Coherent States......Page 95
7.3 Generation of Coherent States......Page 96
7.4 Displacement Operator......Page 98
7.5 Linear Dependence of Coherent States......Page 99
7.6 General Completeness Relation for States Generated by the Displacement Operator......Page 100
7.7 Coordinate Representation of a Coherent State......Page 101
7.8 The Power of Coherent State Representation and the Virtue of Over-Completeness......Page 102
8. Density-Matrix Operator and Quasi-Probability Density......Page 105
8.1 Diagonal Representation of Density-Matrix Operator......Page 106
8.2 Procedures for Determining σ (α)......Page 107
9.1 General Operators......Page 110
9.2 Boson Annihilation and Creation Operators, Ordering......Page 114
9.2.1 Traces of Function of Boson Operators......Page 118
9.3 Characteristic Functions and Distribution Functions......Page 121
9.3.1 TheWigner Distribution Function......Page 123
9.3.1.1 Q-function and P-Function......Page 127
9.3.2 The Husimi Distribution Function......Page 128
9.4 Generalized Coherent States and Squeezing......Page 132
9.5.1 Algebra within Ordered Products......Page 136
9.5.2 Integration within Ordered Products in Quantized Classical Field .......Page 137
9.5.3 Evaluation of Integral of Some Important Mapping Operators......Page 138
9.5.4 Symplectic Transformation and Symplectic Group......Page 139
9.5.4.1 Quadrature States......Page 142
9.5.5 Complex Form of Symplectic Transformation Matrix......Page 143
10.1.1 Wannier Function and Bloch Function......Page 147
10.1.2 Lattice Weyl-Wigner Formulation of Energy-Band Dynamics......Page 148
10.2 Application to Calculation of Magnetic Susceptibility......Page 154
11. The Effective Hamiltonian......Page 158
11.1 Two-Body E.ective Hamiltonian......Page 159
11.2 Effective Hamiltonian in Second Quantization......Page 160
11.3 Effective Non-Hermitian Hamiltonian in a Magnetic Field......Page 164
12.1 Evolution Operator and Sumover Trajectories......Page 169
12.2.1 Bose Systems......Page 171
12.2.2 Path Integral for Fermion Systems......Page 172
13. Gauge Theory and Geometric Phase in Quantum Systems......Page 180
13.1 Directional (Covariant) Derivative on Curve Spaces......Page 181
13.2 Parallel Transport in Curvilinear Space......Page 182
13.3 Parallel Transport Around Closed Curve......Page 183
13.4 Generalization to Quantum Mechanics......Page 186
13.5 Born-Oppenheimer Approximation......Page 189
14.1 The Fiber Bundle Concept......Page 193
14.2 Generalizations of Berry’s Geometric Phase in Quantum Physics......Page 196
14.3 Geometric Phase inMany-Body Systems......Page 197
14.3.1 Localized Disturbances of the Ground State of 2+1-D Many-Body Systems......Page 199
14.3.2 Reconstructing Statistical Quantum Fields in Many-Body Physics......Page 202
14.3.2.1 Bosonization......Page 204
15.1 Classical Gauge Theory......Page 205
15.2 The Yang-Mills Lagrangian for the Gauge Field......Page 209
15.4 Quantization of Gauge Theories......Page 210
16.1 Feynman Diagrams......Page 212
16.2 The Birth of String Theory......Page 213
16.3 Need for Extra Dimensions in String Theory......Page 214
16.4 Nanoelectronics and String Theory......Page 215
Mesoscopic Physics......Page 218
17.1 Introduction......Page 220
17.2 Mesoscopic Quantum Transport......Page 221
17.3 Electrical Resistance Due to a Quantum Scattering Event......Page 222
17.4 The Multichannel Conductance Formula......Page 227
17.5 Quantum Interference in Small-Ring Structures......Page 229
17.6 Generalized Four-Probe Conductance Formula......Page 232
17.6.1 Two-Probe Conductance Formula......Page 234
17.6.2 Three-Probe Conductance Formula: Model of Inelastic Scatterers......Page 235
17.6.3 Weakly-Coupled Voltage Probes: Barrier Point Contacts......Page 236
17.6.4 The Landauer Four-Probe Conductance Limit......Page 237
18. Model of an Inelastic Scatterer with Complete Randomization......Page 239
18.1 Conductance Formula for a Sample Containing an Inelastic Scatterer between Two Elastic Scatterers......Page 243
18.2 Quantum Coherence in a Chain of Elastic and Inelastic Scatterers......Page 247
19. Other Applications of Landauer-Büttiker Counting Argument......Page 251
19.1 Integral and Fractional Quantum Hall Effect......Page 252
19.3 Persistent Currents in Small Normal-Metal Loop......Page 253
19.5 Mesoscopic Thermal Noise and Excess Noise......Page 254
19.6 High-Frequency Behavior......Page 255
20.1 Phenomena Associated with the Quantization of Charge......Page 256
21.1 Correlation Functions......Page 260
21.2 Integral Equations of Mesoscopic Physics......Page 263
21.3 Tight-Binding Recursive Technique......Page 267
21.3.1 Tight-Binding Expression for the Current......Page 268
21.3.3 Mesoscopic Transport Along a Linear Atomic Chain......Page 273
21.3.5 Current Formula in the Presence of Real Phonon Scatterings......Page 277
22. Numerical Matrix-Equation Technique in Steady-State Quantum Transport......Page 281
22.1 Kinetic Equation at Low Temperatures......Page 282
22.2 Kinetic Equation at Higher Temperatures and Arbitrary Bias......Page 285
22.3 Relation with Multiple-Probe Büttiker Current Formula......Page 286
23. Alternative Derivation of Büttiker Multiple-Probe Current Formula......Page 291
Heterostructure Quantum Devices: Nanoelectronics......Page 294
24.1 Introduction......Page 296
24.2 Nanodevices......Page 299
24.3 Vertical vs Lateral Transport in Nanotransistor Designs......Page 303
24.4.1 Vertical Transport Designs......Page 304
24.4.2 Lateral TransportDesigns......Page 312
24.4.3 GaAs/AlGaAs MODFET-Based Nanotransistors......Page 315
25.1 Introduction......Page 317
25.2 Time-Dependent Nonequilibrium Green’s Function ´......Page 318
25.2.1 Electron-Electron Interaction via Exchange of Phonons......Page 323
25.3 Intrinsic Bistability of RTD......Page 324
25.4 Quantum Inductance and Equivalent Circuit Model for RTD .......Page 328
25.4.1 Transient Switching Behavior and Small-Signal Response of RTD fromthe QDF Approach......Page 334
25.4.2.1 Linear Response......Page 336
25.4.2.2 Nonlinear Response......Page 339
26.1 Lattice Wigner Function and Band Structure Effects......Page 341
26.2 Coherent and Incoherent Particle Tunneling Trajectories......Page 342
27.1.1 Intrinsic Behavior of Double-Barrier Structures......Page 347
27.1.2 The Physical Picture......Page 348
27.1.3 Analysis of a RTD Memory or Memdiode......Page 349
27.1.4 Two-State I-V and Two Charge States .......Page 354
28.1 Type I RTD High-FrequencyOperation......Page 356
28.2 Type II RTD High-FrequencyOperation......Page 358
28.3 Regional Block Renormalization: Type-I RTD......Page 361
28.3.1 Estimation of Jc 2 and Jc 1......Page 362
28.3.2 Elimination of Fast-Relaxing Variable for Type-I RTD......Page 363
28.4 Regional Block Renormalization: Type-II RTD .......Page 364
28.5.1 Type-I RTD......Page 366
28.5.1.1 Tunneling Matrix Elements......Page 367
28.5.1.2 Elimination of O.-Diagonal Elements of the Density-Matrix......Page 370
28.5.2 Type-II RTD......Page 373
28.6 Stability Analysis......Page 374
28.7 Numerical Results......Page 375
28.8 Perturbation Theory and Limit Cycle Solutions......Page 376
General Theory of Nonequilibrium Quantum Physics......Page 382
29.1 Introduction......Page 384
29.2 Quantum Dynamics in Liouville Space......Page 386
30. Super-Green’s Functions......Page 395
30.1 Connected Diagrams: Correlation Function K......Page 402
30.2 Self-Consistent Equations for GQDF......Page 403
30.2.2 Closure Problem and Renormalization Procedure......Page 404
30.2.3 Iterative Equations for the Vertex Functions......Page 407
31. Quantum Transport Equations of Particle Systems......Page 410
31.1 General QuantumTransport Equations......Page 413
31.2 Transport Equations and Lattice Weyl Transformation......Page 415
32. Generalized Bloch Equations......Page 419
32.1 Generalized Bloch Equations in QuantumOptics......Page 420
32.2 The Bloch Vector Representation......Page 424
32.4 Atomic Energy and DipoleMoment......Page 426
32.6 Transformation to Rotating Frame......Page 429
32.7.1 The Rabi Problem......Page 431
32.7.2 Response to Light Pulse......Page 433
32.7.3 Self-Induced Transparency......Page 434
33. Generalized Coherent-Wave Theory......Page 438
33.1 The Tight-Binding Limit......Page 441
33.1.1 Flat Band Case......Page 442
34. Impact Ionization and Zener Effect......Page 444
34.1 Coulomb Pair Potential . for Impact Ionization and Auger Recombination......Page 445
34.2 Pair Potential . due to Zener Effect......Page 447
35. Quantum Transport Equations in Phase Space......Page 449
35.2.1 Resonant Tunneling Diode (RTD)......Page 452
36. QSFT of Second-Quantized Classical Fields: Phonons......Page 454
36.1 Liouvillian Space Phonon Dynamics......Page 456
36.2 The Phonon Super-Green’s Function......Page 458
36.3 Transport Equation for the Phonon Super-Correlation Function......Page 461
36.4 Phonon Transport Equations in Phase Space......Page 462
36.5 The Phonon Boltzmann Equation......Page 465
Operator Space Methods and Quantum Tomography......Page 468
37.1 The Density Operator in Operator Vector Space......Page 470
37.2 Formulation in Terms of Translation Operators......Page 473
37.2.1 Weyl Transformof GPMOperator......Page 475
37.2.2 Weyl Transform of the GPM Eigenstate Projector......Page 477
37.3.1 . (p, q) in Terms of Intersecting Lines at Point (p, q)......Page 479
38.1 The Quasi-Probability Distribution and Radon Transform.......Page 483
38.1.1 The Radon Transform......Page 484
38.2 Line Eigenstates and Line Projection Operators......Page 485
38.2.1 Density Operator in Terms of Line Projectors......Page 488
38.3 Translational Covariance of the Wigner Function......Page 490
38.4 Transformation Properties of the Radon Transform......Page 492
38.5 Intersection of Line Projectors: Mutually Unbiased Basis......Page 494
Discrete Phase Space on Finite Fields......Page 498
39.1 DiscreteWigner Function on Finite Fields......Page 500
39.1.1 Line in Discrete Phase Space: Pure Quantum State......Page 501
39.1.2 Commutation Relation Between Q(λ) and T (q, p)sym......Page 502
39.2 Generalized PauliMatrices......Page 503
39.2.1 Commutation Relations and Products of Yq p......Page 504
39.2.2 Expansion of Operators: Hamiltonian in Terms of Generalized PauliMatrices......Page 506
39.2.3 PauliMatrices......Page 507
39.3 Discrete Fourier Transform and Generalized Hadamard Matrix......Page 509
39.3.1 Eigenfunctions and Eigenvalues of X1, Z1, and Y1,1......Page 510
39.3.2 General Quantum State of a Two-Level System: Bloch Sphere......Page 512
39.3.2.2 Bloch Sphere......Page 514
39.3.3 Exponential Map......Page 516
39.3.3.1 Rotation about an Arbitrary Axis in Real 3-D Space......Page 520
39.3.3.2 Arbitrary Unitary Operator for a Qubit: QuantumControl......Page 521
39.3.4 Density Operator for a Two-Level System: Disordered and Pure States......Page 523
40.1 Tensor Product of Operators......Page 524
40.1.1 Entanglement Due to Interactions......Page 528
40.1.2 The No-Cloning Theorem......Page 529
40.2 Quantum Control......Page 530
40.2.1 Pauli Operators over Power-of-Prime Finite Fields......Page 532
40.2.1.1 Phase Space for a Spin- 1 2 System or Single Qubit......Page 534
40.3 Striations andMutually Unbiased Bases......Page 535
41. Discrete Wigner Distribution Function Construction......Page 540
41.1 Discrete Wigner Function for a Single Qubit......Page 543
41.2 Discrete Phase Space Structure for Two Qubits......Page 551
41.2.1 Striations Construction......Page 552
41.2.2 Binary String Encoding of Points in Discrete Phase Space......Page 554
41.2.3 Construction of Dual Field Basis for Two Qubits......Page 556
41.2.3.1 Commutation Relation......Page 557
41.3.1 Product Hilbert Space for a Two Qubit System......Page 558
41.3.3 Vertical Striation Ray and ‘Position’ Basis......Page 564
41.3.4 Horizontal Striation Ray and ‘Momentum’ Basis......Page 567
41.3.5 Diagonal Striation Ray and ‘Y Y ’ Basis......Page 570
41.3.6 Low-Slope-Striation Ray and ‘Belle’ Basis......Page 573
41.3.7 High-Slope-Striation Ray and ‘Beau’ Basis......Page 575
41.4.1 The Origin in Phase Space, q = 0, p = 0......Page 579
41.4.4 The Point (˘ω, 0)......Page 580
41.4.7 The Point (0, ˘ω)......Page 581
41.4.10 The Point (˘ω, ˘ω)......Page 582
41.4.13 The Point (˘ω, ω)......Page 583
41.4.16 The Point (1, ˘ω)......Page 584
41.5.2 Example 2......Page 585
41.5.3 Example 3......Page 586
41.6 Quantum Nets: Arbitrary Assignment to a ‘Vacuum’ Line......Page 587
41.7 Potential Applications......Page 588
Phenomenological Superoperator of Open Quantum Systems: Generalized Measurements......Page 590
42. Interference and Measurement......Page 592
42.1 ProjectiveMeasurements......Page 594
42.1.2 Effects of Measurements on Entanglement......Page 596
42.1.3 Measurements in Quantum Teleportation......Page 597
43. Quantum Operations on Density Operators......Page 598
43.2.3 Von NeumannMeasurements......Page 599
43.2.4 POVMs......Page 600
44. Generalized Measurements......Page 602
44.1 Distinguishing Quantum States......Page 606
44.2 Utility of POVM......Page 607
45. Phenomenological Density Matrix Evolution......Page 609
45.1 Quantum Channels......Page 611
45.2 Depolarizing Channel......Page 612
45.2.2 Kraus Representation of the Channel......Page 613
45.2.3 Relative-State Representation......Page 614
45.3 Phase Damping Channel......Page 616
45.3.2 Kraus Operators......Page 617
45.4 Amplitude-Damping Channel......Page 618
45.4.1 POVMand Unchanging Environment......Page 619
46. Master Equation for the Density Operator......Page 621
46.1 The Lindblad Master Equation......Page 622
46.2.1 Spontaneous Emission......Page 626
46.2.2 Bloch Equations in Magnetic Resonance for Spin 1/2......Page 627
46.3 The PauliMaster Equation......Page 628
46.4 Lindblad Equation for a Damped Harmonic Oscillator......Page 629
46.5 Lindblad Equation for Phase Damped Harmonic Oscillator......Page 631
46.6 Coherent State and Decoherence......Page 633
47. Microscopic Considerations of a Two-Level System Revisited......Page 635
47.1 Quantized Radiation Field......Page 636
47.2 Perturbation Expansion of Density Operator......Page 641
47.2.1 First-order Contribution......Page 643
47.2.2 Resonance Approximation......Page 644
47.2.3 Bloch Equation......Page 645
47.3 Second Order Contribution......Page 647
47.4 Master Equation to Second Order......Page 649
47.4.1 Thermal Reservoir......Page 652
48. Stochastic Meaning of Nonequilibrium Quantum Superfield Theory......Page 657
48.1 Kubo-Martin-Schwinger Condition......Page 659
48.1.1 Mass, Dissipation, and Noise Kernels in Nonequilibrium Quantum Superfield Theory......Page 662
48.2 A Two-State System Interacting with a Heat Bath......Page 664
48.3 Nonequilibrium Quantum Superfield Theory Correlations......Page 667
48.4 Lamb Shift, Dissipation Kernel, and Noise Kernel......Page 673
48.4.1 Comparison with the Master Equation of Sec. 47.4......Page 675
Quantum Computing and Quantum Information: Discrete Phase Space Viewpoint......Page 680
49.1 QuantumTeleportation......Page 682
49.1.1 Unified Teleportation Procedure......Page 687
49.3 Formal Derivation of Entangled Basis States......Page 688
49.3.1 Bell Basis......Page 689
49.3.2 Three-Qubit Entangled Basis......Page 693
49.3.3 A Qubit Teleportation Using Three-Particle Entanglement......Page 695
49.4 Teleportation Using Three-Particle Entanglement and an Ancilla......Page 697
49.5 Two-Qubit Teleportation Using Three-Particle Entanglement......Page 699
50. Superdense Coding......Page 703
50.2 Reduced DensityMatrices......Page 707
50.3 Quantum Channel, Generalized Dense Coding......Page 708
51.1 QuantumFourier Transform......Page 710
51.1.1 Order-Finding Algorithm.......Page 716
51.1.2 Phase Estimation Algorithm......Page 718
51.1.3 Connection Between Root Finding and Phase Estimation......Page 722
51.2 QuantumSearch Algorithm......Page 725
51.3 Discrete Logarithms......Page 727
51.3.1 Quantum Solution......Page 728
51.4 Hidden Subgroup Problem......Page 729
51.4.1 Quantum Hidden Subgroup Algorithm......Page 731
Appendix A Commutation Relation between Components of π (x, t)

and A(x , t)......Page 734
Appendix B Lattice Weyl Transform of One-Particle E.ective Hamiltonian in Magnetic Field......Page 738
Appendix C Second Quantization Operators in Solid-State Band Theory......Page 741
Appendix D Direct Construction of Fermionic Path Integral......Page 747
Appendix E Hot-Electron Green’s Function......Page 753
Appendix F Derivation of Generalized Semiconductor Bloch Equations......Page 755
G.1.1 First-Order Contribution to the Electron Self-Energy .......Page 764
G.1.2 Four-Point Vertex Function to Second Order......Page 765
Appendix H Radon Transformation of Phase Space Functions......Page 799
Appendix I Introduction to Finite Fields......Page 813
I.1.1 GF(9)......Page 816
I.1.2 GF(8)......Page 818
I.2 Constructing Bases of Finite Field......Page 819
I.3 Trace Operation on Elements of Finite Field......Page 821
I.4.1 Construction of Dual Basis......Page 823
I.5 Transformation of Coordinates......Page 825
Bibliography......Page 826
Index......Page 834