Seventy five years ago, three remarkable papers by Schr¨ odinger, Kennard and Darwin were published. They were devoted to the evolution of Gaussian wave packets for an oscillator, a free particle and a particle moving in uniform constant electric and magnetic fields. From the contemporary point of view, these packets can be considered as prototypes of the coherent and squeezed states, which are, in a sense, the cornerstones of modern quantum optics. Moreover, these states are frequently used in many other areas, from solid state physics to cosmology. This paper gives a review of studies performed in the field of so-called ‘nonclassical states’ (squeezed states are their simplest representatives) over the past seventy five years, both in quantum optics and in other branches of quantum physics. My starting point is to elucidate who introduced different concepts, notions and terms, when, and what were the initial motivations of the authors. Many new references have been found which enlarge the ‘standard citation package’ used by some authors, recovering many undeservedly forgotten (or unnoticed) papers and names. Since it is practically impossible to cite several thousand publications, I have tried to include mainly references to papers introducing new types of quantum states and studying their properties, omitting many publications devoted to applications and to the methods of generation and experimental schemes, which can be found in other well known reviews. I also mainly concentrate on the initial period, which terminated approximately at the border between the end of the 1980s and the beginning of the 1990s, when several fundamental experiments on the generation of squeezed states were performed and the first conferences devoted to squeezed and ‘nonclassical’ states commenced. The 1990s are described in a more ‘squeezed’ manner: I have confined myself to references to papers where some new concepts have been introduced, and to the most recent reviews or papers with extensive bibliographical lists.

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'Nonclassical' states in quantum optics: a 'squeezed' review of the first 75 years

It is noted that the study of a quantum algebra sup,q(2), with two independent deformation parameters (p, q), leads to a '(p, q)-oscillator' realization for it. The analysis is extended to the (p, q)-analogues of su(1, 1), osp(2/1) and the centreless Virasoro algebra. The standard single-parameter (q) deformations are obtained in the limit p=q.

https://iopscience.iop.org/article/10.1088/0305-4470/24/13/002/pdf

A (p, q)-oscillator realization of two-parameter quantum algebras

Quantum groups and their applications in nuclear physics

We construct a new family of q-deformed coherent states |zq, where 0 < q < 1. These states are normalizable on the whole complex plane and continuous in their label z. They allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated Stieltjes power-moment problem and expressed in terms of one of the two Jacksons's q-exponentials. They also permit exact evaluation of matrix elements of physically-relevant operators. We use this to show that the photon number statistics for the states is sub-Poissonian and that they exhibit quadrature squeezing as well as an enhanced signal-to-quantum noise ratio over the conventional coherent state value. Finally, we establish that they are the eigenstates of some deformed boson annihilation operator and study some of their characteristics in deformed quantum optics.

https://iopscience.iop.org/article/10.1088/0305-4470/35/43/316/pdf

New q-deformed coherent states with an explicitly known resolution of unity

We show that the single-mode parafermionic type systems possess supersymmetry, which is based on the symmetry of characteristic functions of the parafermions related to the generalized deformed oscillator of Daskaloyannis et al. The supersymmetry is realized in both unbroken and spontaneously broken phases. As in the case of parabosonic supersymmetry observed recently by one of the authors, the form of the associated superalgebra depends on the order of the parafermion and can be linear or nonlinear in the Hamiltonian. The list of supersymmetric parafermionic systems includes usual parafermions, finite-dimensional q-deformed oscillator, q-deformed parafermionic oscillator and parafermionic oscillator with internal Z2 structure.

http://cds.cern.ch/record/388078/files/9905149.pdf

Supersymmetry of Parafermions

Remarks on the q-quantization

Quantum friction and the Caldirola-Montaldi procedure

An exact solution of Schrodinger equation is presented for the problem of a harmonic oscillator of mass varying with time and coordinates according to \begin{aligned} m(q,t)=m_{0}/\text{e}^{-\nu t}-k_{0}^{2}q^{2}. \end{aligned} This case, compared with known types of equivalent interactions, leads again to a new one that relates the damped harmonic oscillator with velocity-depentent interaction problem.

Exactly Soluble Harmonic Oscillator for a Particular Form of Time and Coordinates-Dependent Mass

Some remarks on the discrete-time Schrödinger equation

In this paper the procedure of Caldirola and Montaldi is applied in the Heisenberg representation. It is also proved here that the introduction of the parameter τ contributes to the specification of a quantum-dissipative system, as exactly happens in the Schrodinger representation. As an example the case of the harmonic oscillator as well as that of the damped harmonic oscillator are examined.

V. Zisis

Papers

Remarks on the q-quantization

Quantum friction and the Caldirola-Montaldi procedure

Exactly Soluble Harmonic Oscillator for a Particular Form of Time and Coordinates-Dependent Mass

Some remarks on the discrete-time Schrödinger equation

Heisenberg equations of motion in the Caldirola-Montaldi procedure