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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Impossibility of naively generalizing squeezed coherent states

Robert A. Fisher, Michael Martin Nieto, Vernon D. Sandberg

Published Mar 15, 1984• 203 citations

Abstract

Pertinent properties of the unitary operators that create coherent states and squeezed coherent states are discussed. We show that certain generalizations of squeezed coherent states do not exist. This is accomplished by demonstrating that for the generalized squeeze operators ${U}_{k}=\mathrm{exp}(i{A}_{k})=\mathrm{exp}[{z}_{k}{({a}^{\ifmmode\dagger\else\textdagger\fi{}})}^{k}+i{h}_{k\ensuremath{-}1}\ensuremath{-}{({z}_{k})}^{*}{a}^{k}]$, $〈0|{U}_{k}|0〉$ diverges, $k>2$. This implies that $|0〉$ is not an analytic vector of ${A}_{k}$ for all $k>2$, where ${h}_{k\ensuremath{-}1}$ is a Hermitian polynomial in $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ up to powers of ($k\ensuremath{-}1$).

DOI: 10.1103/physrevd.29.1107

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Journal of the Optical Society of America B•Oct 1, 1987•58 citations

Pertinent properties of the unitary operators that create coherent states and squeezed coherent states are discussed. We show that certain generalizations of squeezed coherent states do not exist. This is accomplished by demonstrating that for the generalized squeeze operators

{U}_{k}=\mathrm{exp}(i{A}_{k})=\mathrm{exp}[{z}_{k}{({a}^{\ifmmode\dagger\else\textdagger\fi{}})}^{k}+i{h}_{k\ensuremath{-}1}\ensuremath{-}{({z}_{k})}^{*}{a}^{k}]

,

〈0|{U}_{k}|0〉

diverges,

k>2

. This implies that

|0〉

is not an analytic vector of

{A}_{k}

for all

k>2

, where

{h}_{k\ensuremath{-}1}

is a Hermitian polynomial in

a

and

{a}^{\ifmmode\dagger\else\textdagger\fi{}}

up to powers of (

k\ensuremath{-}1

).

A set of generalized squeezed–coherent states for the finite oscillator is obtained. These states are given as linear combinations of the mode eigenstates with amplitudes determined by matrix elements of exponentials in the generators. These matrix elements are given in the ( N + 1)-dimensional basis of the finite oscillator eigenstates and are seen to involve 3 × 3 matrix multi-orthogonal polynomials Q n ( k ) in a discrete variable k which have the Krawtchouk and vector-orthogonal polynomials as their building blocks. The algebraic setting allows for the characterization of these polynomials and the computation of mean values in the squeezed–coherent states. In the limit where N goes to infinity and the discrete oscillator approaches the standard harmonic oscillator, the polynomials tend to 2 × 2 matrix orthogonal polynomials and the squeezed–coherent states tend to those of the standard oscillator.

We first review the properties of the harmonic-oscillator coherent states which can be equivalently defined as (a) a specific subset of the $x\ensuremath{-}p$ minimum-uncertainty states, (b) eigenstates of the annihilation operator, or (c) states created by a certain unitary exponential displacement operator. Then we present a new method for finding coherent states for particles in general potentials. Its basis is the desire to find those states which most nearly follow the classical motion, but it is most nearly a generalization of the minimum-uncertainty method. The properties of these states are discussed in detail. Next we show that the annihilation operator and displacement operator methods, as heretofore defined, cannot be applied to general potentials (whose eigenvalues are not equally spaced). We define a generalization of these methods but show that the states so defined are not, in general, equivalent to the minimum-uncertainty coherent states. We discuss a number of properties of our coherent states and the procedures we have used.

The multiphoton squeezed states defined in this paper are generalizations of the conventional coherent (Glauber) and squeezed (Yuen) states previously discussed by many authors. We define multiphoton generalizations of the latter by a unified class of states that includes the Holstein–Primakoff realizations of SU(2) and SU(1, 1) as well as the standard harmonic oscillator coherent states (Weyl–Heisenberg group) and squeezed states in a general framework that allows also non-Hermitian realizations. We determine the squeezing properties of these states in a unified formalism and study numerically their dependence on the parameter classifying the states.