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arXiv:quant-ph/0412020 · Journal of Physics A Mathematical and General

Non-Markovian master equations from entanglement with stationary unobserved degrees of freedom

Adrián A Budini, Henning Schomerus

Published Oct 5, 2005• 28 citations
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Abstract

We deduce a class of non-Markovian completely positive master equations which describe a system in a composite bipartite environment, consisting of a Markovian reservoir and additional stationary unobserved degrees of freedom that modulate the dissipative coupling. The entanglement-induced memory effects can persist for arbitrary long times and affect the relaxation to equilibrium, as well as induce corrections to the quantum-regression theorem. By considering the extra degrees of freedom as a discrete manifold of energy levels, strong non-exponential behaviour can arise, such as for example power law and stretched exponential decays.

DOI
10.1088/0305-4470/38/42/006

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We deduce a class of non-Markovian completely positive master equations which describe a system in a composite bipartite environment, consisting of a Markovian reservoir and additional stationary unobserved degrees of freedom that modulate the dissipative coupling. The entanglement-induced memory effects can persist for arbitrary long times and affect the relaxation to equilibrium, as well as induce corrections to the quantum-regression theorem. By considering the extra degrees of freedom as a discrete manifold of energy levels, strong non-exponential behaviour can arise, such as for example power law and stretched exponential decays.
We deduce a class of non-Markovian completely positive master equations which describe a system in a composite bipartite environment, consisting of a Markovian reservoir and additional stationary unobserved degrees of freedom that modulate the dissipative coupling. The entanglement-induced memory effects can persist for arbitrary long times and affect the relaxation to equilibrium, as well as induce corrections to the quantum-regression theorem. By considering the extra degrees of freedom as a discrete manifold of energy levels, strong non-exponential behaviour can arise, such as for example power law and stretched exponential decays. PACS numbers: 03.65.Yz, 42.50.Lc, 03.65.Ta, 05.40.−a 1.
We analyze the appearance of non-Markovian effects in the dynamics of a bipartite system coupled to a reservoir, which can be described within a class of non-Markovian equations given by a generalized Lindblad structure. A master equation is derived, which we term the quantum Bloch-Boltzmann equation, describing both motional and internal states of a test particle in a quantum framework. When due to the preparation of the system or to decoherence effects one of the two degrees of freedom is amenable to a classical treatment and not resolved in the final measurement, though relevant for the interaction with the reservoir, non-Markovian behaviors such as stretched exponential or power law decay of coherences can occur.
In this paper we derive an extra class of non-Markovian master equations where the system state is written as a sum of auxiliary matrixes whose evolution involve Lindblad contributions with local coupling between all of them, resembling the structure of a classical rate equation. The system dynamics may develop strong nonlocal effects such as the dependence of the stationary properties with the system initialization. These equations are derived from alternative microscopic interactions, such as complex environments described in a generalized Born-Markov approximation and tripartite system-environment interactions, where extra unobserved degrees of freedom mediates the entanglement between the system and a Markovian reservoir. Conditions that guarantee the completely positive condition of the solution map are found. Quantum stochastic processes that recover the system dynamics in average are formulated. We exemplify our results by analyzing the dynamical action of nontrivial structured dephasing and depolarizing reservoirs over a single qubit.
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Quantum memory effects can be induced even when the (time-dependent) dynamical degrees of freedom associated to the environment are not affected at all by the open system during their joint coupled evolution. In this paper, based on a completely positive bipartite representation of the system-environment dynamics, we found the more general interactions that lead to this class of quantum non-Markovian ``casual bystander'' environments. General properties of the resulting dynamics are studied with a focus on the system-environment correlations, a collisional measurement-based representation, and the quantum regression hypothesis. Memory effects are also characterized through an operational approach, which in turn allows one to detect when the studied properties apply. Single and multipartite qubits dynamics support and exemplify the developed results.