Chapter
Appendix A2.1: Theory of elastic collisions and single scattering
A2.1.1 Asymptotic form of the solutions
A2.1.2 Scattering cross section and scattered flux
A2.1.3 Optical theorem Shadow term
Scattering matrix and optical theorem
A2.1.4 Born approximation
Appendix A2.2: Reciprocity theorem
Appendix A2.3: Light scattering
A2.3.1 Classical Rayleigh scattering
A2.3.3 Atom–photon scattering in the dipole approximation
Non-resonant Rayleigh scattering
3.1.1 Green's function for the Schrödinger equation
Some properties of Green’s functions
Green’s function and density of states
3.1.2 Green's function for the Helmholtz equation
3.2 Multiple scattering expansion
3.3 Average Green's function and average density of states
Appendix A3.1: Short range correlations
4 Probability of quantum diffusion
4.3 Drude--Boltzmann approximation
4.4 Diffuson or ladder approximation
Diffuson and reciprocity theorem
Diffuson and anisotropic collisions
4.5 The Diffuson at the diffusion approximation
4.6 Coherent propagation: the Cooperon
Appendix A4.1: Diffuson and Cooperon in reciprocal space
A4.1.1 Collisionless probability P0(q, ω)
Appendix A4.2: Hikami boxes and Diffuson crossings
A4.2.2 Normalization of the probability and renormalization of the diffusion coefficient
A4.2.3 Crossing of two Diffusons
Quantitative description of two-Diffuson crossings
Appendix A4.3: Anisotropic collisions and transport mean free path
Appendix A4.4: Correlation of diagonal Green’s functions
Appendix A4.5: Other correlation functions
A4.5.1 Correlations of Green's functions
A4.5.3 Correlations of wave functions
5 Properties of the diffusion equation
5.2 Heat kernel and recurrence time
5.2.1 Heat kernel – probability of return to the origin
5.4 Diffusion in a periodic box
5.5 Diffusion in finite systems
5.5.1 Diffusion time and Thouless energy
5.5.2 Boundary conditions for the diffusion equation
5.5.3 Finite volume and "zero mode"
5.5.4 Diffusion in an anisotropic domain
5.6 One-dimensional diffusion
5.6.1 The ring: periodic boundary conditions
5.6.2 Absorbing boundaries: connected wire
5.6.3 Reflecting boundaries: isolated wire
Appendix A5.1: Validity of the diffusion approximation in an infinite medium
Appendix A5.2: Radiative transfer equation
A5.2.3 Boundary conditions
A5.2.4 Slab illuminated by an extended source
A5.2.5 Semi-infinite medium illuminated by a collimated beam
Appendix A5.3: Multiple scattering in a finite medium
A5.3.1 Multiple scattering in a half-space: the Milne problem
A5.3.2 Diffusion in a finite medium
Appendix A5.4: Spectral determinant
Spectral determinant and density of states
Example: diffusion in the infinite plane
Example: topological invariant
Appendix A5.5: Diffusion in a domain of arbitrary shape –Weyl expansion
Appendix A5.6: Diffusion on graphs
A5.6.1 Spectral determinant on a graph
A5.6.3 Thermodynamics, transport and spectral determinant
6.1 Dephasing and multiple scattering
6.1.2 Mechanisms for dephasing: introduction
6.2 Magnetic field and the Cooperon
6.3 Probability of return to the origin in a uniform magnetic field
6.4 Probability of return to the origin for an Aharonov–Bohm flux
6.5 Spin-orbit coupling and magnetic impurities
6.5.1 Transition amplitude and effective interaction potential
6.5.2 Total scattering time
6.5.6 The diffusion probability
6.6 Polarization of electromagnetic waves
6.6.1 Elastic mean free path
6.6.3 Classical intensity
6.6.4 Coherent backscattering
6.7 Dephasing and motion of scatterers
6.7.1 General expression for the phase shift
6.7.2 Dephasing associated with the Brownian motion of the scatterers
6.8 Dephasing or decoherence?
Appendix A6.1: Aharonov–Bohm effect in an infinite plane
Spectral determinant and Aharonov–Bohm effect
Appendix A6.2: Functional representation of the diffusion equation
A6.2.1 Functional representation
A6.2.2 Brownian motion and magnetic field
Lévy law of algebraic areas and uniform field
Distribution of winding numbers in a ring
Aharonov–Bohm flux in the plane: the Edwards problem
Appendix A6.3: The Cooperon in a time-dependent field
Appendix A6.4: Spin-orbit coupling and magnetic impurities, a heuristic point of view
A6.4.1 Spin-orbit coupling
A6.4.2 Magnetic impurities
Appendix A6.5: Decoherence in multiple scattering of light by cold atoms
A6.5.1 Scattering amplitude and atomic collision time
Atomic collision time and elastic mean free path
A6.5.2 Elementary atomic vertex
Integral equation for the Diffuson
Integral equation for the Cooperon
7.2 Incoherent contribution to conductivity
7.2.1 Drude–Boltzmann approximation
7.2.2 The multiple scattering regime: the Diffuson
7.2.3 Transport time and vertex renormalization
7.3 Cooperon contribution
7.4 The weak localization regime
7.4.1 Dimensionality effect
7.4.2 Finite size conductors
7.4.3 Temperature dependence
7.5 Weak localization in a magnetic field
7.5.1 Negative magnetoresistance
7.5.2 Spin-orbit coupling and magnetic impurities
7.6 Magnetoresistance and Aharonov–Bohm flux
7.6.2 Long cylinder: the Sharvin–Sharvin effect
7.6.3 Remark on the Webb and Sharvin–Sharvin experiments…
7.6.4 The Aharonov--Bohm effect in an infinite plane
Appendix A7.1: Kubo formulae
A7.1.1 Conductivity and dissipation
A7.1.2 Density-density response function
Appendix A7.2: Conductance and transmission
A7.2.1 Introduction: Landauer formula
A7.2.2 From Kubo to Landauer
A7.2.3 Average conductance and transmission
A7.2.4 Boundary conditions and impedance matching
A7.2.5 Weak localization correction in the Landauer formalism
A7.2.6 Landauer formalism for waves Wave guide geometry
Appendix A7.3: Real space description of conductivity
Appendix A7.4: Weak localization correction and anisotropic collisions
8 Coherent backscattering of light
8.2 The geometry of the albedo
8.2.2 Albedo of a diffusive medium
8.3.1 Incoherent albedo: contribution of the Diffuson
8.3.2 The coherent albedo: contribution of the Cooperon
8.4 Time dependence of the albedo and study of the triangular cusp
8.6 Coherent albedo and anisotropic collisions
8.7 The effect of polarization
8.7.1 Depolarization coefficients
8.7.2 Coherent albedo of a polarized wave
8.8.1 The triangular cusp
8.8.2 Decrease of the height of the cone
8.8.3 The role of absorption
8.9 Coherent backscattering at large
8.9.1 Coherent backscattering and the "glory" effect
8.9.2 Coherent backscattering and opposition effect in astrophysics
8.9.3 Coherent backscattering by cold atomic gases
Depolarization of the Diffuson
Reduction of the enhancement factor of the coherent albedo
8.9.4 Coherent backscattering effect in acoustics
9 Diffusing wave spectroscopy
9.2 Dynamic correlations of intensity
9.3 Single scattering: quasi-elastic light scattering
9.4 Multiple scattering: diffusing wave spectroscopy
9.5 Influence of the geometry on the time correlation function
9.5.1 Reflection by a semi-infinite medium
9.5.2 Comparison between…
9.5.3 Reflection from a finite slab
Appendix A9.1: Collective motion of scatterers
10 Spectral properties of disordered metals
10.1.1 Level repulsion and integrability
10.1.2 Energy spectrum of a disordered metal
10.2 Characteristics of spectral correlations
10.3 Poisson distribution
10.4 Universality of spectral correlations: random matrix theory
10.4.1 Level repulsion in 2 × 2 matrices
10.4.2 Distribution of eigenvalues for N × N matrices
10.4.3 Spectral properties of random matrices
Two-point correlation function
Level spacing distribution
10.5 Spectral correlations in the diffusive regime
10.5.1 Two-point correlation function
10.5.3 Free diffusion limit
Appendix A10.1: The GOE-GUE transition
11 Universal conductance fluctuations
11.2 Conductivity fluctuations
Use of the Einstein relation
Calculation from the Kubo formula
11.2.1 Fluctuations of the density of states
11.2.2 Fluctuations of the diffusion coefficient
11.3 Universal conductance fluctuations
11.4 Effect of external parameters
11.4.2 Temperature dependence
11.4.3 Phase coherence and the mesoscopic regime
11.4.4 Magnetic field dependence
11.4.5 Motion of scatterers
11.4.6 Spin-orbit coupling and magnetic impurities
Appendix A11.1: Universal conductance fluctuations and anisotropic collisions
Appendix A11.2: Conductance fluctuations in the Landauer formalism
12 Correlations of speckle patterns
12.1 What is a speckle pattern?
12.2 How to analyze a speckle pattern
12.3 Average transmission coefficient
12.4 Angular correlations of the transmitted light
12.4.1 Short range C(1) correlations
12.4.2 Long range correlations C(2)
12.4.3 Two-crossing contribution and C(3) correlation
12.4.4 Relation with universal conductance fluctuations
12.5 Speckle correlations in the time domain
12.5.1 Time dependent correlations C(1) (t) and C(2) (t)
12.5.2 Time dependent correlation C(3) (t)
12.6 Spectral correlations of speckle patterns
12.7 Distribution function of the transmission coefficients
12.7.1 Rayleigh distribution law
12.7.2 Gaussian distribution of the transmission coefficient…
12.7.3 Gaussian distribution of the electrical conductance
Appendix A12.1: Spatial correlations of light intensity
A12.1.1 Short range correlations
A12.1.2 Long range correlations
13 Interactions and diffusion
13.2 Screened Coulomb interaction
13.3 Hartree–Fock approximation
13.4 Density of states anomaly
13.4.1 Static interaction
13.4.2 Tunnel conductance and density of states anomaly
13.4.3 Dynamically screened interaction
13.4.4 Capacitive effects
13.5 Correction to the conductivity
13.6 Lifetime of a quasiparticle
13.6.1 Introduction: Landau theory and disorder
13.6.2 Lifetime at zero temperature
13.6.3 Quasiparticle lifetime at finite temperature
13.6.4 Quasiparticle lifetime at the Fermi level
13.7.2 Phase coherence in a fluctuating electric field
13.7.3 Phase coherence time in dimension d=1
13.7.4 Phase coherence and quasiparticle relaxation
13.7.5 Phase coherence time in dimensions d=2 and d=3
13.7.6 Measurements of the phase coherence time …
Appendix A13.1: Screened Coulomb potential in confined geometry
Appendix A13.2: Lifetime in the absence of disorder
14 Orbital magnetism and persistent currents
14.2 Free electron gas in a uniform field
14.2.1 A reminder: the case of no disorder
Landau susceptibility and de Haas–van Alphen effect
14.2.2 Average magnetization
14.2.3 Fluctuations of the magnetization
14.3 Effect of Coulomb interaction
14.3.1 Hartree–Fock approximation
14.3.2 Cooper renormalization
14.3.3 Finite temperature
14.4 Persistent current in a ring
14.4.1 Clean one-dimensional ring: periodicity and parity effects
14.5 Diffusive limit and persistent current
14.5.1 Typical current of a disordered ring
14.5.2 Effect of the Coulomb interaction on the average current
14.5.3 Persistent current and spin-orbit coupling
14.5.4 A brief overview of experiments
Experiments involving a large number of isolated rings
Appendix A14.1 Average persistent current in the canonical ensemble
15.1 Density of states and conductance
Classical conductivity and conductance
15.2 Fourier transforms: definitions
15.3 Collisionless probability P0(r, r´, t)
15.4 Probability P(r, r´, t)
Infinite plane (d = 2) in a magnetic field, for…
One-dimensional ring and Aharonov–Bohm flux
15.5 Wigner–Eckart theorem and 3j-symbols
15.8 Temperature dependences
15.9 Characteristic times introduced in this book
Mesoscopic Physics of Electrons and Photons
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