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Chinese Physics Letters

Damping in a Squeezed Bath and Its Time Evolution through the Complete Class of Gaussian Quasi-Distributions

M. R. Bazrafkan, Seyed Mahmoud Ashrafi, Fahimeh Naghdi

Published Jul 1, 2014• 1 citation
Publisher

Abstract

By virtue of the thermo-entangled states representation of the density operator and using the dissipative interaction picture, we solve the master equation of a driven damped harmonic oscillator in a squeezed bath. We show that the essential part of the dynamics can be expressed by the convolution of the initial Wigner function with a special kind of normalized Gaussian in the phase space and relate the dynamics with the change of Gaussian ordering of the density operator.

DOI
10.1088/0256-307x/31/7/070303

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By virtue of the thermo-entangled states representation of the density operator and using the dissipative interaction picture, we solve the master equation of a driven damped harmonic oscillator in a squeezed bath. We show that the essential part of the dynamics can be expressed by the convolution of the initial Wigner function with a special kind of normalized Gaussian in the phase space and relate the dynamics with the change of Gaussian ordering of the density operator.
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.
By introducing a fictitious mode to be a counterpart mode of the system mode under review we introduce the entangled state representation ?|, which can arrange master equations of density operators ?(t) in quantum statistics as state-vector evolution equations due to the elegant properties of ?|. In this way many master equations (respectively describing damping oscillator, laser, phase sensitive, and phase diffusion processes with different initial density operators) can be concisely solved. Specially, for a damping process characteristic of the decay constant ? we find that the matrix element of ?(t) at time t in ?| representation is proportional to that of the initial ?0 in the decayed entangled state ?e??t| representation, accompanying with a Gaussian damping factor. Thus we have a new insight about the nature of the dissipative process. We also set up the so-called thermo-entangled state representation of density operators, ? = ?(d2?/?)?|?D(?), which is different from all the previous known representations.