Appendix I: propagation of a light beam in a nonlinear parametric medium, inducing a medium-assisted energy transfer between the input beam and the generation of signal and idler beams, hence the name three-wave mixing given to this phenomenon, which is first treated classically, then in a fully quantum way. One finds that, as in the case of fluorescence by spontaneous emission, the phenomenon of spontaneous parametric down conversion (or parametric fluorescence) requires a full quantum treatment, whereas parametric gain can be calculated semiclassically. It gives rise to entangled signal and idler photons as well as twin beams when one inserts the nonlinear medium in a resonant optical cavity (optical parametric oscillator) and to squeezing when the signal and idler modes are identical.

This chapter focuses on photonic analog of the spin-orbit coupling of electrons occurring inside a graded index medium. Section 9.1 describes two physical mechanisms that can produce changes in the state of polarization of an optical beam. The vectorial form of the wave equation is solved in Section 9.2 to introduce a path-dependent geometrical phase. The photonic analog of the spin-orbit coupling and its implications are also discussed in this section. Section 9.3 considers how the scalar LPlm modes change when the coupling term is taken into account. We treat this term first as a perturbation and then obtain the exact vector modes of a GRIN medium. A quantum approach is used in Section 9.4 to discuss various polarization-dependent effects.

Propagation of electromagnetic waves inside a GRIN medium is studied in this chapter. Section 2.1 starts with Maxwell’s equations and uses them to derive a wave equation in the frequency domain. A mode based technique is used in Section 2.2 for solving the wave equation for a GRIN device fabricated with a parabolic index profile. The properties of both the Hermite’Gauss and the Laguerre-Gauss modes are discussed. Section 2.3 is devoted to other power-law index profiles and employs the Wentzel-Kramers Brillouin method to discuss the properties of modes supported by them. We discuss in Section 2.4 the relative efficiency with which different modes are excited by an optical beam incident on a GRIN medium. The intermodal dispersive effects that become important for pulsed beams are also covered. Section 2.5 describes several non-modal techniques that can be used for studying wave propagation in GRIN media.

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.

Appendix D: two-level quantum mechanical systems, or qubits. Description in terms of Bloch vector. Poincaré sphere. Expression of purity. Projection noise in an energy measurement. Description of a set of N coherently driven qubits by a collective Bloch vector.

The focus of this chapter is on longitudinal variations of the refractive index and how such variations affect the propagation of light inside a GRIN medium. Section 7.1 describes the ray-optics and wave-optics techniques that can be used for this purpose. Section 7.2 focuses on tapered GRIN fibers and describes the impact of tapering on the periodic self-imaging for a few different tapering profiles. The analogy between a GRIN medium and a harmonic oscillator is exploited in Section 7.3 by employing several quantum-physics techniques for solving the GRIN problem. Section 7.4 is devoted to the case of periodic variations in the refractive index that are induced by changing the core’s radius of a GRIN fiber along its length in a periodic fashion.

Experimental chapter that presents examples of quantum processes concerning single quantum systems, i.e. sequences comprising a state preparation part, an evolution or propagation part due to the interaction with the outer world, and a detection part. The whole sequence is repeated and its successive results stored. The examples concern quantum control of trapped ions and microwave photonsinteracting in a nondestructive way with Rydberg state cavities. It also presents "boson sampling" of photons placed in a multimode linear interferometer, a system likely to exhibit "quantum advantage," atoms trapped in an optical lattice, a promising platform for quantum simulation of complex systems, generation of "Schrödinger cats" in superconducting circuits.

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