Appendix F: classical, then quantum electromagnetic field. Complex field observable and single-photon field amplitude. Vacuum and Fock states. Single photon state and its polarization properties, quadrature operators for a single-mode field, and its description in phase–space. Heisenberg inequality for rotated quadratures. Vacuum and coherent states have unavoidable phase-independent quantum fluctuations (standard quantum noise). Squeezed states have reduced fluctuations in one of the quadratures. Finally, the appendix considers the measurement of photon coincidence and their characterizatioin in terms of the intensity correlation function g2, and, in particular, the photon bunching effect in thermal states and antibunching effect in single and twin photon states.

This chapter studies linear circuits built from capacitors, inductors, and waveguides. It shows how the excitations of these circuits are quantized and can be described as collections of quantum harmonic oscillators. It discusses the quantum states and quantum operations that are accessible by means of these circuits and external microwave drives. We show how to create coherent states, how microwave resonators decay and decohere, how to amplify and measure the quantum state of a resonator, and what states (e.g., Fock states, individual photons) require other, non-Gaussian means to be produced and detected.

This chapter introduces the basic theoretical tools for handling many-body quantum systems. Starting from second quantized operators, we discuss how it is possible to describe the composite wavefunction of multi-particle systems, and discuss representations in various bases. The algebra of Fock states is described for single and multi-mode systems, and how they relate to the eigenstates of the Schrodinger equation. Finally, we describe how interactions between particles can be introduced in a general way, and then describe the most common type of interaction in cold atom systems, the s-wave interaction