Quantum Optics

Part I: Interaction between the quantum atom and the classical electromagnetic field - Parametrization of the two-level system (Bloch sphere)
- Electric dipole interaction and Hamiltonian
- Solution of the Schrödinger equation in the rotation frame

- Equivalent rotation on the Bloch sphere Tutorials (TD1): Ramsey fringes

- Evolution of the density matrix
- Optical Bloch equation
- Ad hoc introduction of damping in the optical Bloch equation
- Approximation in the case of fast damping coherence and derivation of the stimulated emission. Application to lasers.
- Approximation in the case of a dilute gas (opening for the cold atoms course)

Part II: Quantization of the electromagnetic field - Historical introduction

Tutorials (TD2): Einstein models (1905 and 1916)
- Decomposition of the field in spatial and frequency modes
- Mode decomposition of the classical Hamiltonian of the field - Quantization of the Hamiltonian: â and â+ operators
- Photon number basis and vacuum state
- Change of mode decomposition
- Momentum operator
- Quadrature operators, commutation relations.
- Wave function of photon number states
- Displacement operator
- Coherent states, notion of shot noise

Tutorials (TD3): Properties of coherent states
- Definition and basic properties of the Wigner function
- Examples of Wigner functions for various quantum states
- Thermal state
- Squeezing operator. Effect on the quadratures and on the Wigner function. Squeezing of coherent states

Part III: Experiments in quantum optics
- Quantum model of the beam splitter, examples of various states interfering

Tutorials (TD4): Example of a multimode quantum state: the EPR pair
- Shot noise and the beam-splitter; Squeezing and the beam splitter: the attenuation process in quantum optics
- Phase independent amplification
- Photo detection process: an interferometric point of view