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Physical Review A

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

Maximilian Schlosshauer, Andrew P. Hines, Gerard J Milburn

Published Jul 23, 2008• 1 citation
Publisher

Abstract

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.

DOI
10.1103/physreva.78.016102

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

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Reply to ‘‘Comment on ‘Why quantum mechanics cannot be formulated as a Markov process’’’

Daniel T. Gillespie

In reply to the preceding Comment on an article by D. T. Gillespie [Phys. Rev. A 49, 1607 (1994)], it is emphasized that the most plausible interpretation of the singly conditioned probability function implies that the evolution of the two-state quantum system being considered cannot be characterized as a Markov process on those states over any time interval. And it is pointed out that an alternative interpretation that is offered in the Comment appears to be in conflict with quantum mechanics.

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Comment on “Why quantum mechanics cannot be formulated as a Markov process”

Lucién Hardy, Dipankar Home, Euan J. Squires +1 more

We reply to suggestions made by Gillespie [Phys. Rev. A 49, 1607 (1994)] that our earlier paper [Phys. Rev. A 45, 4267 (1992)] contained mathematical inconsistencies. By use of a concrete model, we demonstrate explicitly that the behavior of a quantum-mechanical two-state oscillator may be simulated by a classical jump process.

Physical Review A•Oct 1, 1997•5 citations
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.
We derive and analyze the Born-Markov master equation for a quantum harmonic oscillator interacting with a bath of independent two-level systems. This hitherto virtually unexplored model plays a fundamental role as one of the four ``canonical'' system-environment models for decoherence and dissipation. To investigate the influence of further couplings of the environmental spins to a dissipative bath, we also derive the master equation for a harmonic oscillator interacting with a single spin coupled to a bosonic bath. Our models are experimentally motivated by quantum-electromechanical systems and micron-scale ion traps. Decoherence and dissipation rates are found to exhibit temperature dependencies significantly different from those in quantum Brownian motion. In particular, the systematic dissipation rate for the central oscillator decreases with increasing temperature and goes to zero at zero temperature, but there also exists a temperature-independent momentum-diffusion (heating) rate.
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.