This chapter presents the theoretical framework that allows us to describe evolutions in the general case using Kraus operators as the main tool. It considers in detail the phenomenon of decoherence and gives examples of such maps. It shows that any evolution can be considered as unitary by going in a larger Hilbert space. The Lindblad equation for the evolution of the density matrix appears as a particular case of evolution in the short memory or Markov approximation. Up-jumps and down-jumps are also described within this framework using cavity damping, spontaneous emission, and the shelving technique as examples.

Appendix J: mechanical effects of light on matter. The appendix first derives the two forces exerted by a light beam on an atom: the radiation pressure force and the dipole force. Appropriate combinations of beams lead to a friction force that slows the atoms (Doppler cooling), and to a trapping force in the so-called magneto-optics trap (MOT). One then considers the forces exerted on ions, leading to trapping in a suitable geometry of electrodes and fields. Two configurations are used, named Paul and Penning traps. In addition, it is possible to cool the ions to their ground motional state using sideband cooling. It is also possible to trap and cool macroscopic nano-objects, such as microdiscs, membranes, toroids, etc. in a resonant optical cavity.

This chapter focuses on the propagation of vortex beams inside a GRIN medium. After an overview in Section 8.1 of polarization-related topics such as the Stokes vector and the Poincare sphere, the concept of a phase singularity is discussed in Section 8.2. This concept is used to form specific combinations of the modes that act as vortices with different state of polarizations. In Section 8.3, we discuss the techniques used for generating different types of vortex beams. Section 8.4 shows that a vortex beam also exhibits the self-imaging property during its propagation inside a GRIN medium. The impact of random mode coupling is also discussed in this section. Vortex-based applications of GRIN fibers are covered in Section 8.5.

This chapter provides an introduction to the subject known as gradient-index optics. In Section 1.1, we present a historical perspective on this subject before introducing the essential concepts needed in later chapters. Section 1.2 is devoted to various types of refractive-index profiles employed for making gradient index devices, with particular emphasis to the parabolic index profile because of its practical importance. In Section 1.3, we discuss the relevant properties of such devices such as optical losses, chromatic dispersion, and intensity dependence of the refractive index occurring at high power levels. The focus of Section 1.4 is on the materials and the techniques used for fabricating gradient-index devices in the form of a rod or a thin fiber

This chapter focuses on the effects of loss or gain in a graded-index (GRIN) medium. In Section 6.1, we discuss the impact of losses on the modes of such a medium. Section 6.2 considers the mechanisms used for providing optical gain inside a GRIN medium. Section 6.3 is devoted to Raman amplifiers and Raman lasers, built with GRIN fibers and pumped suitably to provide optical gain. Parametric amplifiers are discussed in Section 6.4, together with the phase matching required for four-wave mixing to occur. The focus of Section 6.5 is on amplifiers and lasers made by doping a GRIN fiber with rare-earth ions. Section 6.6 includes the nonlinear effects and describes the formation of spatial solitons and similaritons inside an active GRIN medium.

This chapter is devoted to the study of dispersive effects that affect short pulses inside a graded-index fiber. An equation governing the evolution of optical pulses inside a GRIN medium is found in Section 4.1. The dispersion parameters appearing in this equation change, depending on which mode is being considered. Section 4.2 focuses on the distortion of optical pulses resulting from differential group delay and group velocity dispersion. Section 4.3 deals with the effects of linear coupling among the modes, occurring because of random variations in the core’s shape and size along a fiber’s length. A non-modal approach is developed in Section 4.4 for the propagation of short optical pulses inside a GRIN medium. The focus of Section 4.5 is on the applications where optical pulses are sent through a GRIN rod or fiber

Appendix H: treats the interaction between a light beam and a linear optical medium. This first part considers the propagation of a light beam in a sample of two-level atoms using a semiclassical approach, calculates the index of refraction of the medium and its gain when there is population inversion, and losses when the ground state is populated. It then treats in a full quantum way linear attenuation or amplification, for which the "3dB penalty" on the signal-to-noise ratio is derived from basic quantum principles. Finally, it considers the input–output relation for the two input modes of a linear beamplitter, an important example of a symplectic map.

This chapter focuses on the impact of partial coherence on the propagation of optical beams inside a GRIN medium. Section 11.1 introduces the basic coherence-related concepts needed to understand the later material. Section 11.2 uses the evolution of cross-spectral density to study whether periodic self-imaging, an intrinsic property of a GRIN medium, is affected by partial coherence of an incoming beam. Section 11.3 employs the Gaussian-Schell model to discuss how the optical spectrum, the spectral intensity, and the degree of coherence associated with a Gaussian beam change with the beam’s propagation inside a GRIN medium. The focus of Section 11.4 is on Gaussian beams that are only partially polarized. The concept of the polarization matrix is used to study how the degree of polarization evolves when such a partially coherent Gaussian beam is transmitted through a GRIN medium

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.