Exact master equation and quantum decoherence of two coupled harmonic oscillators in a general environment

Abstract

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 [Hu, Paz and 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 towards the construction of a theoretical framework for macroscopic quantum phenomena.

Non-Markovian decoherence theory for a double-dot charge qubit

“…The dissipation-fluctuation equations of motion (30) show that only the solution of v(t) depends on the fermi distribution function in the reservoirs. In other words, only Γ β (t) sensitively depends on the bias.…”

“…The middle two plots in Figs. 18 are the fitting errors by varying Γ L,R but fixing (µ − E, d L,R ) = (10, 5)µeV (the non-Markovian regime) and (30,25) µeV (the Markovian limit). Again we see that in the non-Markovian regime, the best fitting is a sub-exponential decay with s < 1 for all the values of Γ L,R except for some very small Γ L,R (< ∆/2) which indeed enters the Markovian limit where the fitting function becomes a simple exponential decay function.…”

“…The bottom two plots in Figs. 18 are the fitting errors by varying d L,R but fixed (µ − E, Γ L,R ) = (10,25)µeV (the non-Markovian regime) and (30,5) µeV (the Markovian limit). It tells that in the non-Markovian regime, when d L,R < ∆ (the strong non-Markovian regime), the best fitting is still given by a sub-exponential decay (s < 1), while for d L,R > ∆ the system transits to the Markovian, the fitting function becomes again a simple exponential decay function.…”

“…[24]. Very recently, such an exact master equation is further extended to the system of two entangled optical fields and two entangled harmonic oscillators for the study of non-Markovian entanglement dynamics in quantum information processing [29,30]. Nevertheless, using the influence functional theory to derive the exact master equation has been largely focused on the bosonic type of environments, up to date.…”

Quantum Correlations in Qutrit-Qutrit Systems under Local Quantum Noise Channels

“…However, in certain scenarios such approximations are not justified and one needs to go beyond perturbation theory. It is clear that, due to the general complexity of the problem to be studied, exact solutions exist only for simple open quantum systems models such as the well-known Jaynes-Cummings model, quantum Brownian motion model, and certain pure dephasing models [38][39][40]. For the deeper understanding of quantum phenomenons, it is desirable to investigate the behaviours of the quantum properties under the action of decoherence.…”