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Physical Review A

Coupled microcavities with dissipation

Hashem Zoubi, Meir Orenstien, Amiram Ron

Published Aug 3, 2000• 43 citations
Publisher

Abstract

We analyze a system of two coupled cavities, where one of them is leaky, namely, is interacting with the environment. We derive the master equation for the system, and show that the relaxation term, in the case of strongly coupled cavities, is not simply the standard one, which is obtained by neglecting the interaction between the two cavities, when calculating the effect of the bath. The relaxation term is calculated here by first diagonalizing the Hamiltonian of the coupled modes of the cavities, and then it is properly evaluated by eliminating the degrees of freedom of the reservoir, maintaining only the energy-conserving terms. Only by following this prescription one can guarantee that the coupled system approaches thermal equilibrium. We study both the steady-state solutions of the master equation, and the time evolution of the system from given initial conditions.

DOI
10.1103/physreva.62.033801

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We investigate how to model Markovian evolution of coupled harmonic oscillators, each of them interacting with a local environment. When the coupling between the oscillators is weak, dissipation may be modeled using local Lindblad terms for each of the oscillators in the master equation, as is commonly done. When the coupling between oscillators is strong, this model may become invalid. We derive a master equation for two coupled harmonic oscillators that are subject to individual heat baths modeled by a collection of harmonic oscillators and show that this master equation in general contains nonlocal Lindblad terms. We compare the resulting time evolution with that obtained for dissipation through local Lindblad terms for each individual oscillator and show that the evolution is different in the two cases. In particular, the two descriptions give different predictions for the steady state and for the entanglement between strongly coupled oscillators. This shows that when describing strongly coupled harmonic oscillators, one must take great care in how dissipation is modeled and that a description using local Lindblad terms may fail. This may be particularly relevant when attempting to generate entangled states of strongly coupled quantum systems.
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A master equation for a quantum system coupled to a heat bath with stochastically fluctuating energy levels is derived by making use of the ensemble averaging and the averaging with respect to a stochastic process in the bath. Relaxation terms are determined in the Born approximation with respect to the system-bath interaction and the damping parameters related to a relaxation kernel are specified. In parallel with the spectral strength of the bath, the damping parameters determine the transient times for the Markovian description creating the physical origin of the slippage [A. Suarez, R. Sibey, and I. Oppenheim, J. Chem. Phys. 97, 5101 (1992)]. The influence of energy fluctuations of the bath is analyzed for a two-level system including the solution of the corresponding non-Markovian equation for the level population difference. The conditions for the formation of Boltzmann's thermal ratio between steady-state populations are evaluated as well.
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