Chapter
3.1.2 Interpretation of the wave function
3.2 Expectation values in quantum mechanics
3.2.2 Commutation relations
3.2.3 Canonical quantization
3.2.4 Representations of position and momentum operators
4 Central concepts in stationary quantum theory
4.1 Stationary Schrödinger equation
4.2 One-dimensional Schrödinger equation
4.3 Classification of stationary eigenstates
4.3.1 Propagating solutions
4.3.2 Tunneling solutions
4.4 Generic classification of energy eigenstates
5 Central concepts in measurement theory
5.2.2 Central eigenvalue problems
5.3.1 Heisenberg uncertainty principle
5.3.2 Schrödinger and Heisenberg picture
6 Wigner's phase-space representation
6.1.1 Averages in phase space
6.1.2 Quantum properties of the Wigner function
6.1.3 Negativity of the Wigner function
6.2 Wigner-function dynamics
6.4 Feasibility of quantum-dynamical computations
7 Hamiltonian formulation of classical electrodynamics
7.2 Hamiltonian for classical electrodynamics
7.2.1 Functional derivative
7.2.2 Electromagnetic-field Hamiltonian
7.3 Hamilton equations for light–matter system
7.3.1 Classical particle equations
7.3.2 Classical equations for the electromagnetic field
7.4 Generalized system Hamiltonian
8 System Hamiltonian of classical electrodynamics
8.1 Elimination of the scalar potential
8.2 Coulomb and Lorentz gauge
8.2.1 Scalar-potential elimination in the Coulomb gauge
8.2.2 Scalar-potential elimination in the Lorentz gauge
8.3 Transversal and longitudinal fields
8.4 Mode expansion of the electromagnetic field
8.4.1 Modes with periodic boundary conditions
8.4.2 Real-valued mode expansion
9 System Hamiltonian in the generalized Coulomb gauge
9.1 Separation of electronic and ionic motion
9.2 Inclusion of the ionic polarizability
9.2.1 Generalized Coulomb gauge
9.3 Generalized Coulomb potential
9.3.2 Generalized Coulomb potential
9.4 Generalized light-mode functions
9.4.1 Transmission and reflection of light modes
9.4.2 Boundary conditions
9.4.3 Fresnel coefficients for s- and p-polarized modes
9.4.4 Transfer-matrix solutions for generalized modes
10 Quantization of light and matter
10.1 Canonical quantization
10.1.1 Toward semiconductor quantum optics
10.1.2 Real vs. auxiliary quantization space
10.2 Second quantization of light
10.2.1 Unitary transformations
10.2.2 Complex-valued modes
10.3 Eigenstates of quantized modes
10.3.1 Explicit representation of operators
10.3.2 Properties of creation and annihilation operators
10.3.4 Fock states in x space
10.4 Elementary properties of Fock states
10.4.1 Quantum statistics in terms of Fock states
10.4.2 Vacuum-field fluctuations
11 Quasiparticles in semiconductors
11.1 Second-quantization formalism
11.1.1 Fermion many-body states
11.1.2 Fermion creation and annihilation operators
11.1.3 Fermions in second quantization
11.1.4 Pragmatic formulation of second quantization
11.2 System Hamiltonian of solids
11.2.1 Second quantization of system Hamiltonian
11.2.2 Second quantization of lattice vibrations
12 Band structure of solids
12.1 Electrons in the periodic lattice potential
12.1.2 Two-band approximation
12.2 Systems with reduced effective dimensionality
12.2.1 Quasi two-, one-, and zero-dimensional systems
12.2.2 Electron density of states
13 Interactions in semiconductors
13.1 Many-body Hamiltonian
13.2 Light–matter interaction
13.2.1 Separation of length scales
13.2.2 Light–matter-coupling integrals
13.2.3 Inner products within k · p theory
13.2.4 Light–matter interaction in k · p theory
13.3 Phonon–carrier interaction
13.5 Complete system Hamiltonian in different dimensions
13.5.1 Quantum-well system Hamiltonian
13.5.2 Quantum-wire system Hamiltonian
13.5.3 Quantum-dot system Hamiltonian
14 Generic quantum dynamics
14.1 Dynamics of elementary operators
14.1.1 Evaluation strategy
14.1.2 Quantum dynamics of free quasiparticles
14.1.3 Photon-operator dynamics
14.1.4 Macroscopic matter-response operators
14.1.5 Phonon and carrier dynamics
14.2 Formal properties of light
14.2.1 Quantized wave equation
14.3 Formal properties of general operators
14.3.1 Operator hierarchy problem
14.3.2 BBGKY hierarchy problem
15 Cluster-expansion representation of the quantum dynamics
15.1 Singlet factorization
15.1.1 Expectation values of a Slater-determinant state
15.1.2 Hartree–Fock approximation and singlet factorization
15.2.1 Boson and Fermion factorizations
15.2.2 Most relevant singlet–doublet factorizations
15.3 Quantum dynamics of expectation values
15.4 Quantum dynamics of correlations
15.5 Scattering in terms of correlations
16 Simple many-body systems
16.1.1 Electron–hole system
16.1.2 Separation of relative and center-of-mass motion
16.2 Hydrogen-like eigenstates
16.2.1 Low-dimensional systems
16.2.2 Numerical solutions of bound and unbound states
16.3.1 Momentum-matrix elements
16.3.2 Long-wave-length limit for the  · p̂ interaction
17 Hierarchy problem for dipole systems
17.1 Quantum dynamics in the  · p̂ picture
17.1.2 Time scale for the center-of-mass motion
17.2 Light–matter coupling
17.3.1 Dipole-emission dynamics
17.3.2 Emission of planar dipoles
17.4 Quantum dynamics in the Ê · x̂ picture
17.4.1 Göppert-Mayer transformation
17.4.2 Dipole self-energy
17.4.3 System Hamiltonian
18 Two-level approximation for optical transitions
18.1 Classical optics in atomic systems
18.1.1 Separation of relative and center-of-mass motion
18.1.2 Formal aspects of the optical excitation
18.1.3 Two-level approximation
18.1.4 Rotating-wave approximation (RWA)
18.2 Two-level system solutions
18.2.1 Analytic solution of the two-level system
18.2.2 Bloch-vector representation
18.2.4 Pulse area and Rabi flopping
18.2.5 Square-pulse excitation
19 Self-consistent extension of the two-level approach
19.1 Spatial coupling between light and two-level system
19.1.1 Center-of-mass distribution in optical coupling
19.1.2 Optical Bloch equations
19.1.3 Angle parametrization of Bloch vector
19.2 Maxwell-optical Bloch equations
19.2.1 Radiative decay of the atomic dipole
19.2.2 Radiative decay of planar dipoles
19.3 Optical Bloch equations with radiative coupling
20 Dissipative extension of the two-level approach
20.1 Spin representation of optical excitations
20.2 Dynamics of Pauli spin matrices
20.3 Phenomenological dephasing
20.3.1 Dephasing-induced effects
20.3.2 Dephasing and radiative decay
20.4 Coupling between reservoir and two-level system
20.4.1 Master-equation description of dephasing
20.4.2 Master-equation for two-level system
21 Quantum-optical extension of the two-level approach
21.1 Quantum-optical system Hamiltonian
21.1.1 Reduction to two-level system
21.1.2 Rotating-wave approximation
21.1.4 Quantum-optical hierarchy problem
21.2 Jaynes–Cummings model
21.2.2 Interacting Jaynes–Cummings states
21.2.3 Jaynes–Cummings ladder
22 Quantum dynamics of two-level system
22.1 Formal quantum dynamics
22.1.1 Wave-function dynamics
22.1.2 Dynamics of density matrices
22.2 Quantum Rabi flopping
22.2.1 Observation of quantum rungs
22.2.2 Semiclassical interpretation
22.3.1 Displacement operator
22.4 Quantum-optical response to superposition states
22.4.1 Collapses and revivals of excitations
22.4.2 Origin of collapses and revivals
22.4.3 Quantum-statistical modifications
23 Spectroscopy and quantum-optical correlations
23.1 Quantum-optical spectroscopy
23.2 Quantum-statistical representations
23.2.2 Quantum-statistical pluralism
23.3.1 Thermal fluctuations
23.3.2 Quantum-optical spectroscopy with thermal source
23.4 Cluster-expansion dynamics
23.4.1 Identification of correlated clusters
23.4.2 Beyond Maxwell-optical Bloch equations
23.4.3 General singlet–doublet dynamics
23.4.4 Luminescence equations for two-level system
23.5 Quantum optics at the singlet–doublet level
24 General aspects of semiconductor optics
24.1 Semiconductor nanostructures
24.1.1 Homogeneous many-body states
24.1.2 Two-band model and electron–hole picture
24.2 Operator dynamics of solids in optical regime
24.2.1 Dynamics of elementary operators
24.2.2 Explicit operator dynamics
24.3 Cluster-expansion dynamics
24.4 Relevant singlets and doublets
24.5.1 Separation of singlets and doublets
24.5.2 Closed set of singlets
24.5 Dynamics of singlets
25 Introductory semiconductor optics
25.1 Optical Bloch equations
25.1.1 Two-level aspects of semiconductors
25.1.2 Electronic wave function in the singlet analysis
25.2.1 Linear polarization
25.2.2 Weak excitation of densities
25.2.3 Weak square-pulse excitation
25.2.4 Transient polarization vs. stationary density
25.3 Coherent vs. incoherent quantities
25.3.1 Coherent vs. incoherent correlations
25.3.2 Coherence in quantum optics
25.4 Temporal aspects in semiconductor excitations
25.4.1 Rotating-wave approximation
25.4.2 Separation of time scales
25.4.3 Electrical field and dipole interaction
26 Maxwell-semiconductor Bloch equations
26.1 Semiconductor Bloch equations
26.1.1 Maxwell-semiconductor Bloch equations
26.1.2 Coupling to doublet correlations
26.2.1 Left- and right-handed excitonic states
26.2.2 Density-dependent aspects of the 1s resonance
26.3 Semiconductor Bloch equations in the exciton basis
26.4 Linear optical response
26.4.2 Self-consistent optical response
26.4.3 Quantitative measurements and dephasing
26.4.4 Radiative polarization decay
26.5 Excitation-induced dephasing
26.5.1 Density-dependent absorption
27 Coherent vs. incoherent excitons
27.1 General singlet excitations
27.1.2 Many-body state of singlet excitations
27.1.3 Coherent excitonic polarization
27.2.1 Dynamics of exciton correlations
27.2.2 Polarization-to-population transfer
27.3 Electron–hole correlations in the exciton basis
27.3.1 Correlated electron–hole plasma
27.3.2 Energy considerations
28 Semiconductor luminescence equations
28.1 Incoherent photon emission
28.1.1 Photon emission vs. electron–hole recombination
28.1.2 Exciton-correlation dynamics
28.2 Dynamics of photon-assisted correlations
28.2.1 Spontaneous-emission source
28.2.2 Semiconductor luminescence equations
28.3 Analytic investigation of the semiconductor luminescence
28.3.1 Wannier excitons with finite center-of-mass momentum
28.3.2 Elliott formula for luminescence
28.3.3 Plasma vs. population source in photoluminescence
28.4 Excitonic signatures in the semiconductor luminescence
29 Many-body aspects of excitonic luminescence
29.1 Origin of excitonic plasma luminescence
29.1.1 Energy redistribution dynamics
29.1.2 Energy flow between many-body states and photons
29.2 Excitonic plasma luminescence
29.2.1 Radiative recombination of carriers vs. excitons
29.2.2 Hole burning in exciton distributions
29.3 Direct detection of excitons
30 Advanced semiconductor quantum optics
30.1 General singlet–doublet dynamics
30.1.1 Coherent coupling in the Πv,c dynamics
30.1.2 General Πλ,λ' dynamics
30.1.3 General dynamics of carrier doublets
30.1.4 Singlet–doublet correlations and beyond
30.2 Advanced quantum optics in the incoherent regime
30.2.1 Semiconductor luminescence in a cavity
30.2.2 Interference effects in incoherent luminescence
30.2.4 Quantum-dot emission
30.3 Advanced quantum optics in the coherent regime
30.3.1 Squeezing in the resonance fluorescence
30.3.2 Coherent quantum-optical correlations
30.3.3 Quantum-optical spectroscopy
30.3.4 Quantum optics in simple vs. complicated systems
Appendix Conservation laws for the transfer matrix
A.1 Wronskian-induced constraints
A.2 Current-induced constraints
A.3 Explicit conservation laws
Semiconductor Quantum Optics
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