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

This book presents the first comprehensive treatment of discrete phase-space quantum mechanics and the lattice Weyl-Wigner formulation of energy band dynamics, by the originator of these theoretical techniques. Also included is the author's quantum superfield theoretical technique for nonequilibrium quantum physics, without the awkward use of artificial time contour employed in previous formulations of nonequilibrium physics. These two main quantum theoretical techniques combine to yield general and exact quantum transport equations in phase-space, appropriate for nonlinear open systems, including excitation-pairing dynamics. The derivation of Landauer and Landauer-Buttiker formulas in mesoscopic physics from the general quantum transport equations is also treated. New emerging nanodevices for digital and communication applications are discussed in the light of the quantum-transport physics equations, and an in-depth treatment of the physics of resonant tunneling devices is given. Extension of discrete phase-space quantum mechanics on finite fields is briefly discussed for completeness, together with its relevance to quantum computing. In addition, quantum information theory is covered in an effort to shed more light on the foundation of quantum dynamics, along with selected topics on nonequilibrium nanosystems in quantum biology. 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

this Book Presents The First Comprehensive Treatment Of Discrete Phase-space Quantum Mechanics And The Lattice Weyl-wigner Formulation Of Energy Band Dynamics, By The Originator Of These Theoretical Techniques. The Author's Quantum Superfield Theoretical Formulation Of Nonequilibrium Quantum Physics Is Given In Real Time, Without The Awkward Use Of Artificial Time Contour Employed In Previous Formulations. These Two Main Quantum Theoretical Techniques Combine To Yield General (including Quasiparticle-pairing Dynamics) And Exact Quantum Transport Equations In Phase-space, Appropriate For Nanodevices. The Derivation Of Transport Formulas In Mesoscopic Physics From The General Quantum Transport Equations Is Also Treated. Pioneering Nanodevices Are Discussed In The Light Of The Quantum-transport Physics Equations, And An In-depth Treatment Of The Physics Of Resonant Tunneling Devices Is Given. Operator Hilbert-space Methods And Quantum Tomography Are Discussed. Discrete Phase-space Quantum Mechanics On Finite Fields Is Treated For Completeness And By Virtue Of Its Relevance To Quantum Computing. The Phenomenological Treatment Of Evolution Superoperator And Measurements Is Given To Help Clarify The General Quantum Transport Theory. Quantum Computing And Information Theory Is Covered To Demonstrate The Foundational Aspects Of Discrete Quantum Dynamics, Particularly In Deriving A Complete Set Of Multiparticle Entangled Basis States.

"This book presents the first comprehensive treatment of discrete phase-space quantum mechanics and the lattice Weyl-Wigner formulation of energy band dynamics, by the originator of these theoretical techniques. The author's quantum superfield theoretical formulation of nonequilibrium quantum physics is given in real time, without the awkward use of artificial time contour employed in previous formulations. These two main quantum theoretical techniques combine to yield general (including quasiparticle-pairing dynamics) and exact quantum transport equations in phase-space, appropriate for nanodevices. The derivation of transport formulas in mesoscopic physics from the general quantum transport equations is also treated. Pioneering nanodevices are discussed in the light of the quantum-transport physics equations, and an in-depth treatment of the physics of resonant tunneling devices is given. Operator Hilbert-space methods and quantum tomography are discussed. Discrete phase-space quantum mechanics on finite fields is treated for completeness and by virtue of its relevance to quantum computing. The phenomenological treatment of evolution superoperator and measurements is given to help clarify the general quantum transport theory. Quantum computing and information theory is covered to demonstrate the foundational aspects of discrete quantum dynamics, particularly in deriving a complete set of multiparticle entangled basis states."--Jacket