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Physical Review A

Comment on “Decoherence and dissipation of a quantum harmonic oscillator coupled to two-level systems”

Dmitri S. Mogilevtsev, Vyacheslav Shatokhin

Published Jul 23, 2008• 2 citations
Publisher

Abstract

In a recent paper by [Schlosshauer et al., Phys. Rev. A 77, 022111 (2008)] a model of a harmonic oscillator coupled to a spin bath was studied using the Born-Markov master equation. We show that the bath correlation function forbids application of the Born-Markov approximation, which renders this master equation incorrect and the conclusions following from its solution physically irrelevant.

DOI
10.1103/physreva.78.016101

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Contrary to the assertion by Mogilevtsev and Shatokhin [preceding paper, Phys. Rev. A 78, 016101 (2008)], we show that the applicability of the Born-Markov master-equation approach in our treatment of the oscillator-spin model depends on the physical situation under study. Heating effects do occur although they may not be accurately captured by second-order perturbation theory inherent in the Born-Markov scheme.

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In a recent paper by [Schlosshauer et al., Phys. Rev. A 77, 022111 (2008)] a model of a harmonic oscillator coupled to a spin bath was studied using the Born-Markov master equation. We show that the bath correlation function forbids application of the Born-Markov approximation, which renders this master equation incorrect and the conclusions following from its solution physically irrelevant.
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We calculate the reduced density matrix for a system of coupled harmonic oscillators in a bosonic heat bath using the Born-Markov approximation and show that the expectation values of position and momentum evolve like classical quantities. We consider the cases where every oscillator is coupled to its own reservoir and where all oscillators are coupled to the same heat bath, pointing out the differences between these two models. In particular, for systems coupled to a common reservoir, we present an uncommon dissipation mechanism, which disappears, when the dynamics of the system are gouverned by the low frequency modes of the environment (Caldeira-Leggett limit). We also address the interesting phenomenon of reservoir-induced interactions.
Markovian quantum master equations are generally thought to satisfy the following three conditions: (i) they describe the dynamics of time scales that are larger than the reservoir correlation time scale, (ii) these stationary solutions give the thermal equilibrium states with a reservoir, and (iii) they can be written in the Lindblad form. In fact, for single systems such as a single harmonic oscillator and a single two-level system, Markovian quantum master equations satisfying all these conditions can be obtained. However, for composite systems, which consist of more than one subsystem coupled with each other, such equations have not been obtained thus far. In this study, we found that we cannot use the Born–Markov approximation for obtaining quantum master equations for composite systems in the strong-coupling regime, no matter how short the reservoir correlation time is.
A model for a quantum register R\mathcal{R}R made of NNN replicas of a ddd-dimensional quantum system (cell) coupled with the environment is studied by means of a Born-Markov master equation (ME). Dissipation and decoherence are discussed in various cases in which a subdecoherent encoding can be rigorously found. For the quantum-bit case (d=2)(d=2)(d=2) we have solved, for small N,N,N, the ME by numerical direct integration and studied, as a function of the coherence length ${\ensuremath{\xi}}_{c}$ of the bath, fidelity and decoherence rates of states of the register. For large enough ${\ensuremath{\xi}}_{c}$ the singlet states of the global su(2) pseudo spin algebra of the register (noiseless at ${\ensuremath{\xi}}_{c}=\ensuremath{\infty})$ are shown to have a much smaller decoherence rates than the rest of the Hilbert space.