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Physical Review

Quantum Theory of a Randomly Modulated Harmonic Oscillator

B. Crosignani, Paolo Di Porto, Salvatore Solimeno

Published Oct 25, 1969• 12 citations
Publisher

Abstract

The time evolution of a randomly modulated quantum harmonic oscillator is studied by introducing the master equation for the reduced density operator $s(t)$. The antinormally ordered representation is adopted for $s(t)$, and one is thus confronted with the problem of solving the partial differential equation obeyed by the reduced matrix element $\overline{s}$ of $s(t)$. The two limiting cases of short- and long-range correlations of the frequency fluctuations are treated in detail.

DOI
10.1103/physrev.186.1342

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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's 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 is also solved by using the Wang-Uhlenbeck method of transforming it into a linearized partial differential equation for the Wigner function, subject to either the Gaussian type or the δδδ-function type of initial conditions. The Wigner functions which we obtain are two-dimensional Gaussians with different widths.