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

Quantum wave packet revivals

This paper is a direct illustration of a construction of coherent states which has been recently proposed by two of us (JPG and JK). We have chosen the example of a particle trapped in an infinite square-well and also in Poschl-Teller potentials of the trigonometric type. In the construction of the corresponding coherent states, we take advantage of the simplicity of the solutions, which ultimately stems from the fact they share a common SU(1,1) symmetry a la Barut--Girardello. Many properties of these states are then studied, both from mathematical and from physical points of view.

Temporally stable coherent states for infinite well and P\"oschl-Teller potentials

A variety of coherent states of the harmonic oscillator is considered. It is formed by a particular superposition of canonical coherent states. In the simplest case, these superpositions are eigenfunctions of the annihilation operator $A=P(d/dx+x)/\sqrt2$, where $P$ is the parity operator. Such $A$ arises naturally in the $q\to -1$ limit for a symmetry operator of a specific self-similar potential obeying the $q$-Weyl algebra, $AA^\dagger-q^2A^\dagger A=1$. Coherent states for this and other reflectionless potentials whose discrete spectra consist of $N$ geometric series are analyzed. In the harmonic oscillator limit the surviving part of these states takes the form of orthonormal superpositions of $N$ canonical coherent states $|\epsilon^k\alpha\rangle$, $k=0, 1, \dots, N-1$, where $\epsilon$ is a primitive $N$th root of unity, $\epsilon^N=1$. A class of $q$-coherent states related to the bilateral $q$-hypergeometric series and Ramanujan type integrals is described. It includes a curious set of coherent states of the free nonrelativistic particle which is interpreted as a $q$-algebraic system without discrete spectrum. A special degenerate form of the symmetry algebras of self-similar potentials is found to provide a natural $q$-analog of the Floquet theory. Some properties of the factorization method, which is used throughout the paper, are discussed from the differential Galois theory point of view.

https://arxiv.org/pdf/quant-ph/9601030.pdf

Universal superpositions of coherent states and self-similar potentials.

https://escholarship.org/content/qt39c963kn/qt39c963kn.pdf?t=p0uvze

Phase-space representations of wave equations with applications to the eikonal approximation for short-wavelength waves

https://link.springer.com/content/pdf/10.1007/BF02776154.pdf

Coherent states for general potentials

Lie-admissible unification of dissipative Schrödinger equation

Quantum friction and the Caldirola-Montaldi procedure

We discuss some aspects of the time picture of tunneling for open quantum systems described by non-Hermitian (NH) Hamiltonians. The concept of sojourn time for such systems is introduced in the framework of the biorthonormal formalism. Due to the various definitions of probability density in the non-Hermitian case, we get three different sojourn times, two real and one complex. We consider as model of a dissipative NH system the complex, generalized parametric oscillator, for which we derive the exact expressions of the three sojourn times in terms of the Wei-Norman characteristic functions entering the non-unitary evolution operators. The special case of the inverted Caldirola-Kanai oscillator with complex friction parameter is investigated for an initial extended wavepacket. We also discuss the Landau-Zener-like transitions of the NH parametric oscillator, i.e. the dissipative tunneling through a dynamical barrier due to the perturbative effect of the damping.

Non-hermitian tunneling of open quantum systems

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.

V. Papatheou

Papers

Coherent states for general potentials

Lie-admissible unification of dissipative Schrödinger equation

Quantum friction and the Caldirola-Montaldi procedure

Non-hermitian tunneling of open quantum systems

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