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.
We report extensive numerical studies of the dynamics of a classical particle in an anharmonic one-dimensional potential while it is harmonically coupled to three different many-particle systems. The studies address the comparatively simpler dynamical problem when the energy of the anharmonic oscillator is sufficiently low. The first model is one in which the many-particle system is a chain of harmonic oscillators that have nearest-neighbor harmonic interactions. The anharmonic oscillator is connected to every harmonic oscillator via harmonic springs. The second model is identical to the first except that each oscillator is subjected to an additional one-body harmonic potential. The third model is identical to the second except that the nearest-neighbor couplings between the individual harmonic oscillators are absent, i.e., the oscillators are decoupled with respect to one another. In the first model we find that the lowest frequency of the anharmonic oscillator, while possessing a lower bound, increases as $\ensuremath{\omega}\ensuremath{\propto}\sqrt{\mathrm{NK}},$ where N is the number of oscillators in the bath and K is the force constant describing the strength of interaction between the anharmonic oscillator and the interacting harmonic many-body system. In the second model we find aspects of dynamical correlations as seen in the canonical ensemble and a dominant frequency that increases as NK. We recover the dynamical correlations of the anharmonic oscillator as obtained within the framework of a canonical ensemble [S. Sen, R. S. Sinkovits, and S. Chakravarti, Phys. Rev. Lett. 77, 4855 (1996)] in the third model for sufficiently many harmonic oscillators. We comment briefly on an alternate way to correctly model heat baths for numerical studies on the dynamics of physical systems using canonical ensembles.
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.