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Modern Physics Letters B

NEW METHOD FOR TREATING RELAXATION PROCESSES: DAMPED HARMONIC OSCILLATOR AND THE JAYNES-CUMMINGS MODEL WITH CAVITY LOSSES

JOSIP SEKE

Published Aug 30, 1996• 0 citations
Publisher

Abstract

In the present paper a new technique for treating relaxation processes, based on the author’s self-consistent projection-operator method, is developed. This new technique, which yields (instead of a master equation for a reduced density operator) equations of motion for probability amplitudes for a reduced set of states of the total system, opens a new way for treating relaxation processes. The applicability of the new method is demonstrated in the case of a damped harmonic oscillator and in that of the Jaynes-Cummings model with cavity losses.

DOI
10.1142/s0217984996001115

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In the present paper a new technique for treating relaxation processes, based on the author’s self-consistent projection-operator method, is developed. This new technique, which yields (instead of a master equation for a reduced density operator) equations of motion for probability amplitudes for a reduced set of states of the total system, opens a new way for treating relaxation processes. The applicability of the new method is demonstrated in the case of a damped harmonic oscillator and in that of the Jaynes-Cummings model with cavity losses.
In the preceding paper (arXiv: 1103.0329 [quant-ph]) we treated the Jaynes–Cummings model with dissipation and gave an approximate solution to the master equation for the density operator under the general setting by making use of the Zassenhaus expansion. However, to obtain a compact form of the approximate solution (which is in general complicated infinite series) is very hard when an initial condition is given. To overcome this difficulty we develop another approach and obtain a compact approximate solution when some initial condition is given.
We derive a new expansion of the Heisenberg equation of motion based on the projection operator method proposed by Shibata, Hashitsume and Shingū. In their projection operator method, a certain restriction is imposed on the initial state. As a result, one cannot prepare arbitrary initial states, for example a coherent state, to calculate the time development of quantum systems. In this paper, we generalize the projection operator method by relaxing this restriction. We explain our method in the case of a Hamiltonian both with and without explicit time dependence. Furthermore, we apply it to an exactly solvable model called the damped harmonic oscillator model and confirm the validity of our method.