Open Quantum Optical Systems: extended program

From Institute for Theoretical Physics II / University of Erlangen-Nuremberg

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1. Review of classical mechanics, Hilbert spaces, and quantum mechanics
2. Quantization of the electromagnetic field as a collection of harmonic oscillators
2.1. Light as an electromagnetic wave
2.2. Quasi-1D approximation and relation to a collection of harmonic oscillators
2.3. The classical one-dimensional harmonic oscillator
2.4. The quantum one-dimensional harmonic oscillator: number states, energy quantization, and quadrature eigenstates
2.5. Visualising quantum states in phase space: The Wigner function and Gaussian states
2.6. Coherent states
2.6.1. Formal definition
2.6.2. Phase-space description
2.6.3. Bridge between quantum and classical physics
2.7. Squeezed states
2.7.1. Definition and relevance
2.7.2. Minimum-uncertainty squeezed states
2.8. Thermal states
2.9. General single-mode Gaussian states
2.10. Quantized expression of the electromagnetic field
2.11. Other technologically relevant systems captured by the harmonic oscillator model
2.11.1. Linear superconducting circuits
2.11.2. Motion of mesoscopic and macroscopic objects
2.11.3. Motion of trapped atoms, ions, and molecules
2.11.4. Polarized atomic ensembles
2.11.5. Excitons in semiconductors
3. Quantum theory of atoms and the two-level approximation
3.1. Atomic energy spectrum
3.2. Two-level approximation: Pauli pseudo-spin operators, atomic states, and visualization in the Bloch sphere
3.3. Basic two-level dynamics as rotations in the Bloch sphere
3.4. Other systems that act as "artificial" or "engineered" atoms
3.4.1. Nonlinear superconducting circuits
3.4.2. Trapped hard-core bosonic atoms
3.4.3. Confined electrons: quantum dots
3.4.4. Defects in solid state: nitrogen-vacancy centers in diamond
4. Light-matter interaction
4.1. Interaction between light and a single atom
4.1.1. Hamiltonian in the dipolar approximation
4.1.2. The rotating-wave approximation
4.1.3. Single-mode approximation: the Jaynes-Cummings model
4.1.4. Coherent light: Rabi oscillations, collapses, and revivals
4.2. Light in a nonlinear dielectric
4.2.1. Maxwell equations in the presence of nonlinear response
4.2.2. Basic second-order processes: frequency conversion
4.2.3. Hamiltonian under the independent dipole approximation
4.2.4. Down-conversion of an undepleted pump: Bogoliubov diagonalization and unstable Hamiltonians
5. Quantum optics in open systems
5.1. Open optical cavities
5.1.1. The open cavity model
5.1.2. Heisenberg picture approach: the quantum Langevin equation
5.1.3. Schrödinger picture approach: the master equation
5.1.4. Relation of the model parameters to physical parameters
5.2. Incoherent atomic processes
5.2.1. An atom in free space: spontaneous emission and Lamb shift
5.2.2. Spontaneous emission in a modified electromagnetic environment
5.2.3. Dephasing
5.2.4. Visualisation of spontaneous emission and dephasing in the Bloch sphere
5.3. Some paradigmatic open models
5.3.1. The degenerate parametric oscillator
5.3.2. The Kerr resonator
5.3.3. The laser model
6. Numerical and analytical techniques for open quantum-optical systems
6.1. General properties of master equations and quantum Langevin equations
6.1.1. Steady states in time-independent problems
6.1.2. Example: Driven cavity in a thermal environment
6.2. Superspace approach
6.2.1. The master equation in superspace
6.2.2. Example: Steady state of the single-atom laser
6.3. The method of quantum-state trajectories
6.3.1. The stochastic Schrödinger equation
6.3.2. Example: Quantum jumps in resonance fluorescence
6.4. Phase-space techniques in bosonic systems
6.4.1. Phase-space representations
6.4.2. Dynamical equations in phase space
6.4.3. Fokker-Planck equations and stochastic Langevin equations
6.4.4. Example: The degenerate parametric oscillator
6.5. Lowest order approximate techniques in bosonic systems: a case study on degenerate parametric oscillation
6.5.1. Classical limit: nonlinear dynamical systems
6.5.2. Small quantum fluctuations: Gaussian quantum noise around the classical limit
6.6. Lowest order approximate techniques in the presence of atoms: a case study on the single-atom laser
7. Detection of the output field
7.1. The output field
7.2. Quantum regression theorem
7.3. Ideal detection: An intuitive picture of photodetection and homodyne detection
7.4. Photodetection
7.4.1. The photocurrent and its power spectrum
7.4.2. Bunching and antibunching
7.4.3. Example: Driven cavity in a thermal environment
7.4.4. Example: Resonance fluorescence
7.5. Homodyne detection
7.5.1. The noise spectrum and squeezing
7.5.2. Example: The degenerate optical parametric oscillator
8. Effective Hamiltonians and Liouvillians: elimination of spurious degrees of freedom
8.1. Effective theories in closed systems
8.1.1. Projection-operator technique
8.1.2. Example: effective motional optical potentials on a detuned atom
8.2. Effective theories in open systems
8.2.1. Projection-superoperator technique
8.2.2. Example: sideband cooling with a driven cavity mode