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Progress of Theoretical Physics

Derivation of Master Equation for Totally Quadratic System in Quantum Brownian Motion

Takashi Okamura

Published Feb 1, 1994• 2 citations
Publisher

Abstract

We consider Quantum Brownian motion of a toy model, which is a harmonic oscillator that interacts with environment consisting of harmonic oscillators through both qQ and pP types of coupling. At first, we formally derive exact master equation following Unruh and Zurek's method. Subsequently we estimate the behavior of the coefficients in the master equation at initial stage and at late time stage under weak coupling approximation. We find that, as the model with qQ coupling only, the system in supraohmic environment is less dissipative and less diffusive beyond the cutoff time scale than the system in ohmic environment. The cause is due to the weakness of spectral densities of environment in infrared sector. We also find that the remarkable feature of the coexistence of qQ and pP coupling is steep dependence of dissipative coefficients on time at initial stage compared with the case of qQ coupling only or pP coupling only. Further we find that the sign of the dissipation coefficient at initial stage can be negative, that is, the system can be anti-dissipative.

DOI
10.1143/ptp.91.219

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We revisit the model of a quantum Brownian oscillator linearly coupled to an environment of quantum oscillators at finite temperature. By introducing a compact and particularly well-suited formulation, we give a rather quick and direct derivation of the master equation and its solutions for general spectral functions and arbitrary temperatures. The flexibility of our approach allows for an immediate generalization to cases with an external force and with an arbitrary number of Brownian oscillators. More importantly, we point out an important mathematical subtlety concerning boundary-value problems for integro-differential equations which led to incorrect master equation coefficients and impacts on the description of nonlocal dissipation effects in all earlier derivations. Furthermore, we provide explicit, exact analytical results for the master equation coefficients and its solutions in a wide variety of cases, including ohmic, sub-ohmic and supra-ohmic environments with a finite cut-off.
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In this paper we derive an exact master equation for two coupled quantum harmonic oscillators interacting via bilinear coupling with a common environment at arbitrary temperature made up of many harmonic oscillators with a general spectral density function. We first show a simple derivation based on the observation that the two harmonic oscillator model can be effectively mapped into that of a single harmonic oscillator in a general environment plus a free harmonic oscillator. Since the exact one harmonic oscillator master equation is available [B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 45, 2843 (1992)], the exact master equation with all its coefficients for this two harmonic oscillator model can be easily deduced from the known results of the single harmonic oscillator case. In the second part we give an influence functional treatment of this model and provide explicit expressions for the evolutionary operator of the reduced density matrix which are useful for the study of decoherence and disentanglement issues. We show three applications of this master equation: on the decoherence and disentanglement of two harmonic oscillators due to their interaction with a common environment under Markovian approximation, and a derivation of the uncertainty principle at finite temperature for a composite object, modeled by two interacting harmonic oscillators. The exact master equation for two, and its generalization to N, harmonic oscillators interacting with a general environment are expected to be useful for the analysis of quantum coherence, entanglement, fluctuations, and dissipation of mesoscopic objects toward the construction of a theoretical framework for macroscopic quantum phenomena.