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

Generalized Coherent States

David Stoler

Published Oct 15, 1971• 188 citations
Publisher

Abstract

Using an operator technique similar to one we have used in a previous paper, we study the generalized coherent states which were introduced by Titulaer and Glauber and discussed recently by Bialynicka-Birula. Using this technique, we provide an alternative derivation of the most general coherent state and then derive a number of interesting properties, some of which do not seem to be in the literature. For example, we show that the generalized coherent states are not minimum-uncertainty packets.

DOI
10.1103/physrevd.4.2309

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The study of coherent states (CS) for a quantum mechanical system has \nreceived a lot of attention. The definition, applications, generalizations of \nsuch states have been the subject of work by researchers. A common starting \npoint of all these approaches is the observation of properties of the original \nCS for the harmonic oscillator. It is well-known that they are described \nequivalently as (a) eigenstates of the usual annihilation operator, (b) from \na displacement operator acting on a fundamental state and (c) as minimum \nuncertainty states. What we observe in the different generalizations proposed \nis that the preceding definitions are no longer equivalent and only some of \nthe properties of the harmonic oscillator CS are preserved. \nIn this thesis we propose to study a new class of coherent states and \nits properties. We note that in one example our CS coincide with the ones \nproposed by Glauber where a set of three requirements for such states has \nbeen imposed. The set of our generalized coherent states remains invariant \nunder the corresponding time evolution and this property is called temporal \nstability. Secondly, there is no state which is orthogonal to all coherent states (the coherent states form a total set). The third property is that we \nget all coherent states by acting on one of these states [¿fiducial vector¿] with \noperators. They are highly non-classical states, in the sense that in general, \ntheir Bargmann functions have zeros which are related to negative regions of \ntheir Wigner functions. Examples of these coherent states with Bargmann \nfunction that involve the Gamma and also the Riemann ¿ functions are represented. \nThe zeros of these Bargmann functions and the paths of the zeros \nduring time evolution are also studied.

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Certain superposition (linear combination) of energy eigenfunctions of linear harmonic oscillators are called coherent states because they are of significance in optics for the representation of coherent light waves. A hallmark of coherent states is that the variances (squares of the uncertainties) of x and px are constant in time and further their product becomes the minimum value allowed by the Heisenberg uncertainty principle. On the other hand, some linear combinations of harmonic oscillator energy functions give rise to squeezed states . A curious property of squeezed states is that the variances of x and px oscillate in time 180o out of phase with one another with the frequency twice of the oscillator. The wave packet of a coherent state possesses a minimum uncertainty. 〈x〉 and 〈px〉 have the same oscillatory forms as in the classical case. The coherent states have also been called minimum uncertainty coherent states, the Schrödinger coherent states or the Glauber coherent states [1-6]. The squeezed states are sometimes referred to in the literature as two-photon coherent states and generalized coherent states.

CRC Press eBooks•Dec 10, 2014•0 citations
Using an operator technique similar to one we have used in a previous paper, we study the generalized coherent states which were introduced by Titulaer and Glauber and discussed recently by Bialynicka-Birula. Using this technique, we provide an alternative derivation of the most general coherent state and then derive a number of interesting properties, some of which do not seem to be in the literature. For example, we show that the generalized coherent states are not minimum-uncertainty packets.
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.
Both the coherent states and also the squeezed states of the harmonic oscillator have long been understood from the three classical points of view: the 1) displacement operator, 2) annihilation- (or ladder-) operator, and minimum-uncertainty methods. For general systems, there is the same understanding except for ladder-operator and displacement-operator squeezed states. After reviewing the known concepts, I propose a method for obtaining generalized minimum-uncertainty squeezed states, give examples, and relate it to known concepts. I comment on the remaining concept, that of general displacement-operator squeezed states.
The study of coherent states (CS) for a quantum mechanical system has \nreceived a lot of attention. The definition, applications, generalizations of \nsuch states have been the subject of work by researchers. A common starting \npoint of all these approaches is the observation of properties of the original \nCS for the harmonic oscillator. It is well-known that they are described \nequivalently as (a) eigenstates of the usual annihilation operator, (b) from \na displacement operator acting on a fundamental state and (c) as minimum \nuncertainty states. What we observe in the different generalizations proposed \nis that the preceding definitions are no longer equivalent and only some of \nthe properties of the harmonic oscillator CS are preserved. \nIn this thesis we propose to study a new class of coherent states and \nits properties. We note that in one example our CS coincide with the ones \nproposed by Glauber where a set of three requirements for such states has \nbeen imposed. The set of our generalized coherent states remains invariant \nunder the corresponding time evolution and this property is called temporal \nstability. Secondly, there is no state which is orthogonal to all coherent states (the coherent states form a total set). The third property is that we \nget all coherent states by acting on one of these states [¿fiducial vector¿] with \noperators. They are highly non-classical states, in the sense that in general, \ntheir Bargmann functions have zeros which are related to negative regions of \ntheir Wigner functions. Examples of these coherent states with Bargmann \nfunction that involve the Gamma and also the Riemann ¿ functions are represented. \nThe zeros of these Bargmann functions and the paths of the zeros \nduring time evolution are also studied.
Certain superposition (linear combination) of energy eigenfunctions of linear harmonic oscillators are called coherent states because they are of significance in optics for the representation of coherent light waves. A hallmark of coherent states is that the variances (squares of the uncertainties) of x and px are constant in time and further their product becomes the minimum value allowed by the Heisenberg uncertainty principle. On the other hand, some linear combinations of harmonic oscillator energy functions give rise to squeezed states . A curious property of squeezed states is that the variances of x and px oscillate in time 180o out of phase with one another with the frequency twice of the oscillator. The wave packet of a coherent state possesses a minimum uncertainty. 〈x〉 and 〈px〉 have the same oscillatory forms as in the classical case. The coherent states have also been called minimum uncertainty coherent states, the Schrödinger coherent states or the Glauber coherent states [1-6]. The squeezed states are sometimes referred to in the literature as two-photon coherent states and generalized coherent states.