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Journal of Mathematical Physics

Generalized annihilation operator coherent states

Gautam Kumar Ghosh

Published Mar 1, 1998• 3 citations
Publisher

Abstract

A set of generalized coherent states as eigenstates of the annihilation operator is proposed. These states are analytic functions of a complex variable and admit a resolution of identity with positive measure. Guided by the classical action angle transformation and the correspondence principle a formalism is developed for the construction of the annihilation operator for a given Hamiltonian.

DOI
10.1063/1.532384

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A set of generalized coherent states as eigenstates of the annihilation operator is proposed. These states are analytic functions of a complex variable and admit a resolution of identity with positive measure. Guided by the classical action angle transformation and the correspondence principle a formalism is developed for the construction of the annihilation operator for a given Hamiltonian.
The coherent states for a set of quadratic Hamiltonians in the trap regime are constructed. A matrix technique which allows us to directly identify the creation and annihilation operators will be presented. Then, the coherent states as simultaneous eigenstates of the annihilation operators will be derived, and will be compared with those attained through the displacement operator method. The corresponding wavefunction will be found, and a general procedure for obtaining several mean values involving the canonical operators in these states will be described. The results will be illustrated through the asymmetric Penning trap.
In the present series of papers, the coherent states of Sp(2d,R), corresponding to the positive discrete series irreducible representations 〈λd+n/2,..,λ1+n/2〉, encountered in physical applications, are analyzed in detail with special emphasis on those of Sp(4,R) and Sp(6,R). The present paper discusses the annihilation-operator coherent states, i.e., the eigenstates of the noncompact lowering generators corresponding to complex eigenvalues. These states generalize the coherent states introduced by Barut and Girardello for Sp(2,R), and later on extended by Deenen and Quesne to the Sp(2d,R) irreducible representations of the type 〈(λ+n/2)d〉. When λ1,...,λd are not all equal, it was shown by Deenen and Quesne that the eigenvalues do not completely specify the eigenstates of the noncompact lowering generators. In the present work, their characterization is completed by a set of continuous labels parametrizing the (unitary-operator) coherent states of the maximal compact subgroup U(d). The resulting coherent states are therefore of mixed type, being annihilation-operator coherent states only as regards the noncompact generators. A realization in a subspace of a Bargmann space of analytic functions shows that such coherent states satisfy a unity resolution relation in the representation space of 〈λd +n/2,...,λ1+n/2〉, and therefore may be used as a continuous basis in such space. The analytic functions and the differential operators representing the representation space discrete bases and the Sp(2d,R) generators, respectively, are found in explicit form. It is concluded that the annihilation-operator coherent state representation provides the mathematical foundation for the use of differentiation operators with respect to the noncompact raising generators in symbolic expressions of the Sp(2d,R) generators. This is to be compared with the habit of replacing a boson annihilation operator by a symbolic differentiation with respect to the corresponding creation operator, accounted for by the Bargmann representation of such operators.
General sets of coherent states and the JaynesndashndashndashCummings model
Quantum dynamics of integrable systems is discussed. Localized wave packets generalizing the conventional coherent states of minimal uncertainty are constructed. The wave packet moves along a certain trajectory and does not change its shape for times of order 1 ¯h. In this Letter we suggest that the definition of a coherent state should be related to the Hamiltonian.The conventional coherent states [1] are defined as eigenstates of the annihilation operator α | z> = z | z> (1) The real and the imaginary parts of the complex variable z can be expressed by the mean values of the position ˆx and momentum ˆp operators in the coherent state. Under the oscillator Hamiltonian evolution Ut Ut|z> = |exp(−iωt)z> (2) where ω is the oscillator ′ √ s frequency. Then, because the state | z> is con-2¯h centrated around x = ω Rez it follows that Ut|z> has its support on the
We introduce a right inverse for the annihilation operator which appears in the factorization of associated hypergeometric-type operators. This makes possible the definition of generalized coherent states for the associated eigenfunctions. These generalized coherent states satisfy the relevant properties, i.e., label continuity, overcompleteness, temporal stability and action identity required to establish a close connection between the quantum and classical formulations of a given physical system.
Based on Gaussian wave packet solutions of the time-dependent Schrödinger equation, a generalization of the conventional creation and annihilation operators and the corresponding coherent states can be obtained. This generalization includes systems where also the width of the coherent states is time-dependent as they occur for harmonic oscillators with time-dependent frequency or systems in contact with a dissipative environment. The key point is the replacement of the frequency ω 0 that occurs in the usual definition of the creation/annihilation operator by a complex time-dependent function that fulfils a nonlinear Riccati equation. This equation and its solutions depend on the system under consideration and on the (complex) initial conditions. Formal similarities also exist with supersymmetric quantum mechanics. The generalized creation and annihilation operators also allow to construct exact analytic solutions of the free motion Schrödinger equation in terms of Hermite polynomials with time-dependent variable.