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

A nonpolynomial one-dimensional quantum potential representing an oscillator, which can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator (harmonic oscillator with a centripetal barrier), is studied. First the general case, that depends on a parameter a, is considered and then a particular case is studied with great detail. It is proven that it is Schr?dinger solvable and then the wavefunctions ?n and the energies En of the bound states are explicitly obtained. Finally, it is proven that the solutions determine a family of orthogonal polynomials related to the Hermite polynomials and such that: (i) every is a linear combination of three Hermite polynomials and (ii) they are orthogonal with respect to a new measure obtained by modifying the classic Hermite measure.

http://iopscience.iop.org/article/10.1088/1751-8113/41/8/085301/pdf

A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator

A nonpolynomial one-dimensional quantum potential representing an oscillator, that can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator (harmonic oscillator with a centripetal barrier), is studied. First the general case, that depends of a parameter $a$, is considered and then a particular case is studied with great detail. It is proven that it is Schr\"odinger solvable and then the wave functions $\Psi_n$ and the energies $E_n$ of the bound states are explicitly obtained. Finally it is proven that the solutions determine a family of orthogonal polynomials ${\cal P}_n(x)$ related with the Hermite polynomials and such that: (i) Every ${\cal P}_n$ is a linear combination of three Hermite polynomials, and (ii) They are orthogonal with respect to a new measure obtained by modifying the classic Hermite measure.

A quantum exactly solvable non-linear oscillator related with the isotonic oscillator

Dissipation and quantization for composite systems

The properties of a nonlinear oscillator with an additional term kg/x2, characterizing the isotonic oscillator, are studied. The nonlinearity affects to both the kinetic term and the potential and combines two nonlinearities associated to two parameters, κ and kg, in such a way that for κ = 0 all the characteristics of the standard isotonic system are recovered. The first part is devoted to the classical system and the second part to the quantum system. This is a problem of quantization of a system with position-dependent mass of the form m(x) = 1/(1 − κx2), with a κ-dependent non-polynomial rational potential and with an additional isotonic term. The Schrodinger equation is exactly solved and the (κ, kg)-dependent wave functions and bound state energies are explicitly obtained for both κ 0.

/pdf/a-quantum-quasi-harmonic-nonlinear-oscillator-with-an-2wwyk1j4tg.pdf

A quantum quasi-harmonic nonlinear oscillator with an isotonic term

We used the numerical method to study nonclassical properties of the even and odd generalized coherent states and superposition states of generalized coherent states for an isotonic oscillator. The following was shown. (1) The quantum statistical properties of the even and odd generalized coherent states are very different from those of the usual even and odd coherent states, and the Nth-order (N = 2m + 1, m = 0, 1, 2,...) squeezing and sub-Poisson distribution appear alternately for both the even and odd generalized coherent states in some ranges of z = \beta\(2) The weaker the isotonic oscillator potential, the narrower the ranges. (2) The superposition states of generalized coherent states may exhibit the Nth-order squeezing effect too, and this kind of higher-order squeezing effect appears periodically.

http://iopscience.iop.org/article/10.1088/1464-4266/2/6/307/pdf

Nonclassical properties of even and odd generalized coherent states for an isotonic oscillator

Coulomb blockade and quantum fluctuations of mesoscopic inductance coupling circuit

Starting from the equation of motion of a mesoscopic inductance coupling circuit, the quantum fluctuations of charge and current in the circuit are investigated in both the eigenstates of the system and the squeezed vacuum state. The results show that there exist quantum fluctuations of the charge and current in both cases, and the fluctuations in each component circuit are connected.

Quantum Fluctuations in a Mesoscopic Inductance Coupling Circuit

The completeness of the k orthonormalized eigenstates of the operator (âf())k (k≥3) is investigated. We introduce a new kind of higher-order squeezing and an antibunching. The properties of the Mth-order squeezing and the antibunching effect of the k states are studied. The result shows that these states may form a complete Hilbert space, and the Mth-order (M = (n + 1/2)k; n = 0,1,...) squeezing effects exist in all of the k states when k is even. There is an antibunching effect in all of the states. An alternative method for constructing the k states is proposed, and the result shows that all of them can be generated by linear superposition of the time-dependent nonlinear coherent states at different instants.

Quantum statistical properties of orthonormalized eigenstates of the operator (â f (n̂))k

A class of even and odd nonlinear coherent states are introduced. The properties of some related states, including quadrature squeezing, antibunching effect and phase probability distribution, are studied.

Tang-Kun Liu

Papers

Nonclassical properties of even and odd generalized coherent states for an isotonic oscillator

Coulomb blockade and quantum fluctuations of mesoscopic inductance coupling circuit

Quantum Fluctuations in a Mesoscopic Inductance Coupling Circuit

Quantum statistical properties of orthonormalized eigenstates of the operator (â f (n̂))k

A Class of Even and Odd Nonlinear Coherent States and Their Properties