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arXiv:1507.02529 · Physical Review A

Transition from non-Markovian to Markovian dynamics for generic environments

Nephtalí Garrido, T. Gorin, Carlos Pineda

Published Jan 15, 2016• 4 citations
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Abstract

Using random matrices, we study the reduced dynamics of a two-level system interacting with a generic environment. In the weak-coupling limit, the result can be obtained directly from known results for purity decay, and result in Markovian dynamics. We then focus on the case of strong coupling, when the dynamics is known to be non-Markovian. In this regime, the coupling dominates over the local parts of the Hamiltonian, and thus we treat the latter as a perturbation of the former. With the help of a linear response approximation, this allows us to obtain an analytical description of the reduced dynamics. Finally, we find a transition from non-Markovian to Markovian dynamics at a point where the coupling and the local Hamiltonian are comparable in size.

DOI
10.1103/physreva.93.012113

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Using random matrices, we study the reduced dynamics of a two-level system interacting with a generic environment. In the weak-coupling limit, the result can be obtained directly from known results for purity decay, and result in Markovian dynamics. We then focus on the case of strong coupling, when the dynamics is known to be non-Markovian. In this regime, the coupling dominates over the local parts of the Hamiltonian, and thus we treat the latter as a perturbation of the former. With the help of a linear response approximation, this allows us to obtain an analytical description of the reduced dynamics. Finally, we find a transition from non-Markovian to Markovian dynamics at a point where the coupling and the local Hamiltonian are comparable in size.
We study the open dynamics of a quantum two-level system coupled to an environment modeled by random matrices. Using the quantum channel formalism, we investigate different quantum Markovianity measures and criteria. A thorough analysis of the whole parameter space reveals a wide range of different regimes, ranging from strongly non-Markovian to Markovian dynamics. In contrast to analytical models, all non-Markovianity measures and criteria have to be applied to data with fluctuations and statistical uncertainties. We discuss the practical usefulness of the different approaches.
We study a driven two-state system interacting with a generic structured environment. We outline the derivation of the time-local microscopic non-Markovian master equation, in the limit of weak coupling between the system and the reservoir, and we derive its analytic solution for general reservoir spectra in the regime of validity of the secular approximation. We also consider the non-Markovian master equation without the secular approximation and study the effect of nonsecular terms on the system dynamics for two classes of reservoir spectra: the Ohmic and the Lorentzian reservoir. Finally, we derive the analytic conditions for complete positivity of the dynamical map, with and without the secular approximation. Interestingly, the complete positivity conditions have a transparent physical interpretation in terms of the characteristic time scales of phase diffusion and relaxation processes.
As the memory effect may be helpful in quantum information processing, non-Markovian dynamics plays an important role in the description of many-body open systems. Among these topics, the system consisting of independent qubits interacting with several coupled environments is of particular interest. In this paper, we study the exact non-Markovian dynamics of two independent qubits. Each of the qubits interacts individually with its environment, and these two environments coupled with each other. We investigate the non-Markovianity measure of the system for the whole parameter regime without the rotating-wave approximation (RWA) and compare the results with that under the RWA. We find that the non-Markovianity measure for two qubits manifests a transition from a non-Markovian to Markovian regime regardless of the coupling strength between the environments. The physical origin of this transition is revealed, and a possible observation of the prediction in superconducting quantum interference devices is discussed.
The question, whether an open system dynamics is Markovian or non-Markovian can be answered by studying the direction of the information flow in the dynamics. In Markovian dynamics, information must always flow from the system to the environment. If the environment is interacting with only one of the subsystems of a bipartite system, the dynamics of the entanglement in the bipartite system can be used to identify the direction of information flow. Here we study the dynamics of a two-level system interacting with an environment, which is also a heat bath, and consists of a large number of two-level quantum systems. Our model can be seen as a close approximation to the `spin bath' model at low temperatures. We analyze the Markovian nature of the dynamics, as we change the coupling between the system and the environment. We find the Kraus operators of the dynamics for certain classes of couplings. We show that any form of time-independent or time-polynomial coupling gives rise to non-Markovianity. Also, we witness non-Markovianity for certain parameter values of time-exponential coupling. Moreover, we study the transition from non-Markovian to Markovian dynamics as we change the value of coupling strength.
The question, whether an open system dynamics is Markovian or non-Markovian can be answered by studying the direction of the information flow in the dynamics. In Markovian dynamics, information must always flow from the system to the environment. If the environment is interacting with only one of the subsystems of a bipartite system, the dynamics of the entanglement in the bipartite system can be used to identify the direction of information flow. Here we study the dynamics of a two-level system interacting with an environment, which is also a heat bath, and consists of a large number of two-level quantum systems. Our model can be seen as a close approximation to the 'spin bath' model at low temperatures. We analyze the Markovian nature of the dynamics, as we change the coupling between the system and the environment. We find the Kraus operators of the dynamics for certain classes of couplings. We show that any form of time-independent or time-polynomial coupling gives rise to non-Markovianity. Also, we witness non-Markovianity for certain parameter values of time-exponential coupling. Moreover, we study the transition from non-Markovian to Markovian dynamics as we change the value of coupling strength.
We investigate the dynamics of a two-level system in the presence of an overall environment composed of two layers. The first layer is just one single-mode cavity which decays to memoryless reservoir while the second layer is the two coupled single-mode cavities which decay to memoryless or memory-keeping reservoirs. In the weak-coupling regime between the qubit and the first-layer environment, our attention is focused on the effects of the coupling in the hierarchical environment on the non-Markovian speedup dynamics behavior of the system. We show that, by controlling the coupling in the second-layer environment, the multiple dynamics crossovers from Markovian to non-Markovian and from no-speedup to speedup can be realized. This results hold independently on the nature of the second-layer environment. Differently, we find that how the coupling between the two layers affects the non-Markovian speedup dynamics behavior depends on the nature of the second-layer environment.