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

The notion of f-oscillators generalizing q-oscillators is introduced. For classical and quantum cases, an interpretation of the f-oscillator is provided as corresponding to a special nonlinearity of vibration for which the frequency of oscillation depends on the energy. The f-coherent states (nonlinear coherent states) generalizing q-coherent states are constructed. Applied to quantum optics, photon distribution function, photon number means, and dispersions are calculated for the f-coherent states as well as the Wigner function and Q-function. As an example, it is shown how this nonlinearity may affect the Planck distribution formula.

https://iopscience.iop.org/article/10.1088/0031-8949/55/5/004/pdf

f-oscillators and nonlinear coherent states

Publisher Summary This chapter discusses the quantum interference, superposition states of light and the nonclassical effects. The superposition principle is at the heart of quantum mechanics. The coherent state forms a position-momentum, phase-space patch of minimum area, and is the quantum analog of the classical point in phase space. The chapter discusses how quantum interference between coherent states results in nonclassical effects, and associates quantum coherences between coherent states with quantum interference in the phase space. Schrodinger's argument is based on stability and invariance of Gaussian wavepackets of an isolated quantum harmonic oscillator. The effect of coarsening is identical to the influence of reservoirs on quantum interference. The chapter describes several methods proposed for the generation of quantum-mechanical superposition states of light. The methods for the detection of schrodinger cats include homodyne detection, DAP-QND detection scheme and optical homodyne tomography method. These methods provide a complete quantum mechanical description of the measured mode and therefore, can be applied for characterization of quantum-mechanical superposition states, providing these fields are directly measureable.

I: Quantum Interference, Superposition States of Light, and Nonclassical Effects

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.

In this paper, we present a method to generate continuous-variable-type entangled states between photons and atoms in atomic Bose-Einstein condensate (BEC). The proposed method involves an atomic BEC with three internal states, a weak quantized probe laser, and a strong classical coupling laser, which form a three-level $\ensuremath{\Lambda}$-shaped BEC system. We consider a situation where the BEC is in electromagnetically induced transparency with the coupling laser being much stronger than the probe laser. In this case, the upper and intermediate levels are unpopulated, so that their adiabatic elimination enables an effective two-mode model involving only the atomic field at the lowest internal level and the quantized probe laser field. Atom-photon quantum entanglement is created through laser-atom and interatomic interactions, and two-photon detuning. We show how to generate atom-photon entangled coherent states and entangled states between photon (atom) coherent states and atom-(photon-) macroscopic quantum superposition (MQS) states, and between photon-MQS and atom-MQS states.

https://arxiv.org/pdf/quant-ph/0402031.pdf

Generation of atom-photon entangled states in atomic Bose-Einstein condensate via electromagnetically induced transparency

Properties of the generalized coherent states of the electromagnetic field, introduced by Titulaer and Glauber, are investigated. It is shown that, in contradistinction to photon-counting measurements, the average value of the electromagnetic field and the electron scattering amplitudes depend on the detailed form of these states.

Z. Bialynicka—Birula

Papers

Properties of the generalized coherent state