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American Journal of Physics

Quantum solutions of the damped harmonic oscillator

Gerard J Milburn, Daniel F. Walls

Published Dec 1, 1983• 33 citations
Publisher

Abstract

Solutions to the Fokker–Planck equation for a damped harmonic oscillator corresponding to uniquely quantum-mechanical initial states with no classical correspondence are given.
DOI
10.1119/1.13324

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The quantum theory of damping is presented and illustrated by means of a driven damped harmonic oscillator. The theory is formulated in the coherent state representation which illustrates very vividly the nearly classical nature of the problem. In this representation the reduced system density operator equation becomes a Fokker-Planck equation. Green's function solutions are found for the damped oscillator in closed form and as an eigenfunction expansion. In addition, a quantum regression theorem due to Lax is derived in the coherent state representation. The theorem allows two-time averages to be computed from one-time averages.
The Wigner transform of the master equation describing the reduced dynamics of the system, of a harmonic oscillator coupled to an oscillator bath, was obtained by Karrlein and Grabert [Phys. Rev. E 55, 153 (1997)]. It was shown that for some special correlated initial conditions the master equation reduces, in the classical limit, to the corresponding classical Fokker-Planck equation obtained by Adelman [J. Chem Phys. 64, 124 (1976)]. However, for separable initial conditions the Adelman equations were not recovered. We resolve this problem by showing that, for separable initial conditions, the classical Langevin equations are somewhat different from the one considered by Adelman. We obtain the corresponding Fokker-Planck equation and show that they exactly match the classical limit of the evolution of the Wigner function obtained from the master equation for separable initial conditions. We also discuss why thermal initial conditions correspond to Adelman's solution.
Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function, subject to either the Gaussian type or the δδδ-function type of initial conditions, is also solved by using the Wang-Uhlenbeck method. The obtained Wigner functions are two-dimensional Gaussians with different widths.