# Open Quantum Optical Systems: extended program

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

- 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

- 4.1. Interaction between light and a single atom

- 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

- 5.1. Open optical cavities

- 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

- 6.1. General properties of master equations and quantum Langevin equations

- 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

- 8.1. Effective theories in closed systems