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Communications in Theoretical Physics

The Exact Solution, Berry's Phase and Coherent State for the Driven Generalized Time-Dependent Harmonic Oscillator

Jing‐Bo Xu, Tiezheng Qian, Xiao-Chun Gao

Published Oct 1, 1991• 0 citations
Publisher

Abstract

The Lewis' invariant and exact solution for the driven generalized time-dependent harmonic oscillator is found by making use of the Lewis-Riesenfeld quantum theory. Then, the adiabatic asymptotic limit of the exact solution is discussed and the Berry's phase for this system is obtained. We then proceed to use the exact solution to construct the coherent state and calculate the corresponding exact classical phase angle. This phase angle can give the Hannay's angle in the adjabatic limit. The relation between the exact Lewis' phase and the corresponding classical phase angle is discussed.

DOI
10.1088/0253-6102/16/3/311

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The Lewis' invariant and exact solution for the driven generalized time-dependent harmonic oscillator is found by making use of the Lewis-Riesenfeld quantum theory. Then, the adiabatic asymptotic limit of the exact solution is discussed and the Berry's phase for this system is obtained. We then proceed to use the exact solution to construct the coherent state and calculate the corresponding exact classical phase angle. This phase angle can give the Hannay's angle in the adjabatic limit. The relation between the exact Lewis' phase and the corresponding classical phase angle is discussed.
The most general time-dependent Hamiltonian for a harmonic oscillator is both linear and quadratic in the coordinate and the canonical momentum. It describes in general a harmonic oscillator with mass, spring “constant,” and friction (or antifriction) “constant,” all of which are time dependent, that is acted on by a time-dependent force. A generalized Hannay angle, which is gauge invariant, is defined by making a distinction between the Hamiltonian and the energy. The generalized Hannay angle is the classical counterpart of the generalized Berry phase in quantum theory. When friction is present the generalized Hannay angle is nonzero. If the Hamiltonian is (incorrectly) chosen to be the energy, the generalized Hannay angle is different. Nevertheless, in the adiabatic case the same total angle is obtained.
In a recent paper [Y. C. Ge and M. S. Child, Phys. Rev. Lett. 78, 2507 (1997)], by using a Gaussian wave function, Ge and Child presented a nonadiabatic relation between the quantum Berry phase and the classical Hannay angle for the time-dependent harmonic oscillator. In this paper, we present a perspective for this relation without the use of a trial wave function. In particular, an exact explicit formula for the cyclic evolution over the period T in the parameter space of action invariant is obtained; the $\ensuremath{-}(n+1/2)$ relation between the quantum geometric angle and the Hannay angle is rigorously established.
In this work we present a simple and elegant approach to study the adiabatic and nonadiabatic evolution of a generalized damped harmonic oscillator which is described by the generalized Caldirola–Kanai Hamiltonian, in both classical and quantum contexts. Based on time-dependent dynamical invariants, we find that the geometric phase acquired when the damped oscillator evolves adiabatically in time provides a direct connection between the classical Hannay’s angle and the quantum Berry’s phase. In addition, we solve the time-dependent Schrödinger equation for this system and calculate various quantum properties of the damped generalized harmonic one, such as coherent states, expectation values of the position and momentum operators, their quantum fluctuations and the associated uncertainty product.
By making use of linear and quadratic invariants and the invariant operator formulation of Lewis and Riesenfeld, the complete exact solutions of the Schrödinger equation for the generalized time-dependent harmonic oscillator are obtained. It is shown that the general solution of the system under consideration contains both the discrete and continuous spectrum. The connection between linear and quadratic invariants and their corresponding eigenstates via time-dependent auxiliary equations is also established.