Appendix C: treatment of composite systems with the help of the tensor product of Hilbert spaces and partial trace operation. No-cloning theorem.

Experimental chapter devoted to quantum observables endowed with continuously varying quantum fluctuations, such as position and momentum, quadrature operators, or phase and amplitude of electromagnetic fields . It shows that one can manipulate this quantum noise by generating squeezed states of light, always within the limits imposed by the Heisenberg inequality, and create strong correlations between these observables to conditionally generate quantum states having intensity quantum fluctuations below the "shot noise" limit imposed by the existence of vacuum fluctuations. Describes an experiment dealing with macroscopic mechanical oscillators displaying motional squeezing below the zero point fluctuations, and another one dealing with macroscopic superconducting exhibiting a whole spectrum of strongly nonclassical states, generated by using the strong anharmonicity of the Josephson potential.

The focus of this chapter is on focusing and self-imaging of optical beams occurring in a graded-index rod. Section 3.1 provides a geometrical-optics perspective and shows why optical rays follow a curved path inside a GRIN medium. The modes of such a medium are used in Section 3.2 to find a propagation kernel and use it discuss the phenomenon of self-imaging. Section 3.3 is devoted to studying how a GRIN rod can be used as a flat lens to focus an incoming optical beam. Imaging characteristics of such a lens are also considered in this section. Several important applications of GRIN devices are discussed in Section 3.4.

Considerable effort has been directed toward developing new types of artificial materials known now as photonic crystals and metamaterials. Even though the initial focus was not on creating a spatially varying refractive index, it was soon realized that such materials can be fabricated with an index gradient in one or more dimensions. In this chapter, we focus on the novel GRIN devices whose design is based on photonic crystals and metamaterials. Section 10.1 introduces the basic concepts needed to understand the physics behind these two types of materials. Section 10.2 is devoted to GRIN structures based on the concept of photonic crystals. Metamaterials designed with an index gradient are discussed in Section 10.3. The focus of Section 10.4 is on a subgroup of metamaterials, known as metasurfaces, which contain nanoscale objects made with dielectric or metallic materials and are thinner than the wavelength of radiation they are intended for.

Experimental chapter presenting different implementations of bipartite systems, made of two subsystems having interacted in the past and developing strong correlations: optical parametric interaction responsible for the generation of highly correlated twin photons and light beams, cascaded spontaneous emission, collectively oscillating trapped ions, macroscopic atomic samples, cavity-mediated correlated SQUIDtransmons.

historical context, aim of the book, instructions for use

This chapter presents the theoretical framework, based on Gleason’s theorem, allowing to describegeneralized measurements, in addition to von Neumann measurements, in terms of POVMs, probability operators and post-measurement operators. It mentions the Naimark theorem according to which a generalized measurement is a von Neumann measurement if one describes it in a Hilbert space of higher dimension. Examples of generalized measurements are given: imperfect measurements, simultaneous measurements of noncommuting operators. It presents the Zurek model that accounts for the decoherence process occurring in a measurement and shows that the quantum measurement process, including state collapse, is not a physical evolution. Finally, it studies the case of successive measurements using Bayes statistics in which the state collapse appears as an updating of the information about a system, and the fundamental property of repeatability of quantum mechanics.

Appendix E: free, then harmonically bound, massive quantum particle. Lowering and rising operators, displacement operator, number states, coherent states, zero point fluctuations.

Appendix A: basic postulates of quantum mechanics, valid for isolated systems and perfect measurements, and direct implications, such as superposition principle and time reversibility.

Presents the basic postulates of quantum mechanics in terms of the density matrix instead of the usual state vector formalism in the case of an isolated system. Extends it to the case of open systems with the help of the reduced density matrix formalism, and to the case of an imperfect state preparation described by a statistical mixture. Introduces the concept of quantum state purity to characterize the degree of mixture of the state, and shows that one can always "purify" a density matrix by going into a Hilbert space of larger dimension.

Appendix G: interaction between a monochromatic field and two-level atom. The problem is treated first in the case of a classical field and a quantum two-level system (semiclassical approach): It is characterised by a rotation of the Bloch vector (Rabi ocillation) and allows us to generate any qubit state by applying a field of well-controlled duration and amplitude. One then includes spontaneous emission to the model, and finally obtains the set of Bloch equations that are used in many different problems of light–matter interaction. One then considers the full quantum case of cavity quantum electrodynamics (CQED), where the field is single mode and fully quantum: this is the Jaynes–Cummings Hamiltonian approach, which is fully solvable when one negelcts spontaneous emission: quantum oscillations and revivals are predicted. Damping is then introduced in the model, and two regimes of strong and weak couplings are predicted in this case.

Experimental chapter that presents experimental devices that allow us to detect individual quantum systems and observe quantum jumps occurring at random times. Described: superconducting single photon detectors, detection of arrays of ions and atoms, the shelving technique that allows us to measure the quantum state of the single atom, state selective field ionization of single Rydberg atoms, detection of single molecules on a surface by confocal microscopy, articial atoms in circuit quantum electrodynamics (cQED)

Appendix B: generalized postulates of quantum mechanics, valid for open systems and general measurements.

Theoretical chapter devoted to the domain of parameter estimation by quantum measurements. It first details the implications of the Heisenberg inequality and gives the expression of the quantum Cramér–Rao bound, a limit that is optimized over all data processing strategies and measurements on a given parameter-dependent quantum state. Measurement optimization over different quantum states and different optical modes in which the quantum state is defined is also discussed in various situations. The chapter then focusses on the measurement-induced perturbation bydiscussing the Heisenberg microscope and the Ozawa inequality, then on different implementations of quantum nondemolition (QND) measurements using the crossed-Kerr effect in quantum optics and opto-mechanics.

The focus in this chapter is on intensity-dependent changes in the refractive index of a GRIN medium, responsible for the Kerr effect. In Section 5.1, we consider self-focusing of an optical beam inside a GRIN medium. Pulsed beams are considered in Section 5.2, where we derive a nonlinear propagation equation and discuss the phenomena of self- and cross-phase modulations. Section 5.3 is devoted to modulation instability and the formation of multimode solitons. Intermodal nonlinear effects are considered in Section 5.4 with emphasis on four-wave mixing and stimulated Raman scattering. Nonlinear applications discussed in Section 5.5 include supercontinuuum generation, spatial beam cleanup, and second harmonic generation.