The first step towards quantum theory was a response to a problem that could not be addressed by the concepts and methods of classical physics: the radiation from black bodies.

One of the most important concepts of classical mechanics is that of a closed system. A closed system is loosely defined as a system whose components interact only with each other, and it is characterized by phase space volume conservation and energy conservation – see Section 1.2.

We saw in Chapter 2 that Born’s statistical interpretation of the wave function was one of the building blocks of quantum theory. According to Born’s interpretation, the wave function of a particle at a given moment of time defined a probability density $|\psi (x){|}^2$ with respect to the position $x$. This result is generalized to state vectors of an Hilbert space and to general observables through the following procedure.

In Chapter 1, we presented the fundamental principles of classical physics, and then we motivated and presented the fundamental principles of quantum physics. The two sets of principles are summarized and compared in Table 10.1.

Chapter devoted to the basic quantum properties of entanglement and separability. Introduces the Schmidt decomposition for pure states and the positive partial transpose criterion for mixed states as entanglement witnesses. Introduces the famous Einstein–Podolsky–Rosen paradox and its implementation in terms of qubits, then the Bell inequality, quickly reviewing the experimental demonstrations that quantum mechanics violates this inequality. Gives examples of the use of entanglement in a quantum algorithm to accelerate an information task, namely a database search (Grover algorithm) and the possibility of teleportation of a quantum state.

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.

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.

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.

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.