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Physical Review

Dynamics of Coherent States

C. L. Mehta, P. Chand, E. C. G. Sudarshan, R. Vedam

Published May 25, 1967• 55 citations
Publisher

Abstract

A system of harmonic oscillators in the presence of interaction, and with an arbitrary number of degrees of freedom, is considered. The most general form of the Hamiltonian is derived under the restriction that the states which are initially coherent remain coherent at all times. The equation of motion for the annihilation operator, obtained by using this Hamiltonian, is solved, and the frequency spectrum of the annihilation operator is discussed. By giving specific examples it is shown that, in general, the annihilation operators (or their eigenvalues) contain positive as well as negative frequency components and hence are not analytic signals. Some special cases are also considered where the annihilation operators are analytic signals.

DOI
10.1103/physrev.157.1198

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Physical Review•Oct 25, 1967•95 citations
A system of harmonic oscillators in the presence of interaction, and with an arbitrary number of degrees of freedom, is considered. The most general form of the Hamiltonian is derived under the restriction that the states which are initially coherent remain coherent at all times. The equation of motion for the annihilation operator, obtained by using this Hamiltonian, is solved, and the frequency spectrum of the annihilation operator is discussed. By giving specific examples it is shown that, in general, the annihilation operators (or their eigenvalues) contain positive as well as negative frequency components and hence are not analytic signals. Some special cases are also considered where the annihilation operators are analytic signals.
The dynamical invariant for a general time-dependent harmonic oscillator is constructed by making use of two linearly independent solutions to the classical equation of motion. In terms of this dynamical invariant we define the time-dependent creation and annihilation operators and relevantly introduce even and odd coherent states for time-dependent harmonic oscillator. The mathematical and quantum statistical properties of these states are discussed in detail. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.
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.
In this study, we discuss the harmonic oscillator with the time-dependent frequency, ω(t), and the mass, M(t), by generalizing the holomorphic coordinates for the harmonic oscillator. In general cases, we solve the Schrödinger equation by reducing it into the Riccati equation and discuss the uncertainties for the quasi-coherent states of the time-dependent harmonic oscillator. In special cases, we find the following results: First, for a time-dependent harmonic oscillator, if [ω(t)M(t)] is constant, then the coherent states will evolve as the coherent states. Second, for the driven harmonic oscillator, the coherent states will evolve as the coherent states with new eigenvalues. Third, we derive quasi-coherent states for the Caldirola–Kanai Hamiltonian and show that the product of uncertainties, ΔxΔp, is larger than minimum value; however, it is constant. We also discuss the classical equations of motion for the system.
This paper is concerned with the dynamics of coupled quantum oscillators in the presence of a classical harmonic driving force. By means of a theorem which describes the evolution of Gaussian distributions of coherent states, the solutions of the equations of motion are presented in a very compact form. Specific applications deal with forced oscillation and parametric amplification. The theoretical models include systems of damping oscillators with a continuous frequency distribution. The approach to thermal equilibrium in the absence of a driving force is discussed as a special case.
The statistical properties of systems of coupled quantum-mechanical harmonic oscillators are analyzed. The Hamiltonian for the system is assumed to be an inhomogeneous quadratic form in the creation and annihilation operators, and is allowed to have an explicit time dependence. The relationship to classical theory is emphasized by expressing pure states in terms of the coherent-state vectors, and density operators by means of the PPP representation and an analogous representation involving the Wigner function. The state which evolves from an initially coherent state of the system is found, and equations governing the time evolution of the Wigner function and the weight function for the PPP representation are derived, in differential and integral form, for arbitrary initial states of the system. The results remain valid for couplings which do not preserve the vacuum state, and for cases in which the time dependence of the coupling parameters gives rise to large-scale amplification of the initial field intensities. The analysis is performed by first treating general linear inhomogeneous canonical transformations on the oscillator variables, and then specializing to the case in which these transformations represent the solutions for the Heisenberg operators in terms of their initial values. The results are illustrated within the context of a model of parametric amplification.