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Physical Review A

Relaxation process of quantum system: Stochastic Liouville equation and initial correlation

Masashi Ban, Sachiko Kitajima, Fumiaki Shibata

Published Aug 23, 2010• 24 citations
Publisher

Abstract

Time evolution of a quantum system which is influenced by a stochastically fluctuating environment is studied by means of the stochastic Liouville equation. The two different types of the stochastic Liouville equation and their relation are discussed. The stochastic Liouville equation is shown to be derived from the quantum master equation of the Lindblad under certain conditions. Relaxation processes of single and bipartite quantum systems which are initially correlated with a stochastic environment are investigated. It is shown the possibility that the stochastic fluctuation can create coherence and entanglement of a quantum system with the assistance of the initial correlation. The results are examined in the pure dephasing processes of qubits, which are caused by the nonstationary Gauss-Markov process and two-state jump Markov process.

DOI
10.1103/physreva.82.022111

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Time evolution of a quantum system which is influenced by a stochastically fluctuating environment is studied by means of the stochastic Liouville equation. The two different types of the stochastic Liouville equation and their relation are discussed. The stochastic Liouville equation is shown to be derived from the quantum master equation of the Lindblad under certain conditions. Relaxation processes of single and bipartite quantum systems which are initially correlated with a stochastic environment are investigated. It is shown the possibility that the stochastic fluctuation can create coherence and entanglement of a quantum system with the assistance of the initial correlation. The results are examined in the pure dephasing processes of qubits, which are caused by the nonstationary Gauss-Markov process and two-state jump Markov process.
The reduced time evolution of a single qubit under the influence of a classical environment is studied by means of the stochastic Liouville equation. The qubit initially correlates with the environment, which is created by a preparation process of an initial state. The phase fluctuation caused by the environment is characterized by means of the Gauss-Markov process and the two-state-jump Markov process. It is shown that depending on values of the parameters, the initial correlation enhances or reduces the decoherence of the qubit. In the case of the slow modulation, the effect of the initial preparation of the qubit becomes significant and it cannot be ignored in the reduced time evolution of the qubit.
The time evolution of the trace distance between quantum states of a qubit which is placed under the influence of stochastic dephasing is investigated within the framework of the stochastic Liouville equation. When stochastic dephasing is subject to the homogeneous Gauss-Markov process, the trace distance is exactly calculated in the presence of the initial correlation between the qubit and the stochastic process, where the stochastic process is inevitably a nonstationary process. It is found that even the initial correlation with the classical environment can make the trace distance greater than the initial value if stochastic dephasing causes the slow modulation of the qubit.
We propose the generalized stochastic Liouville equation to investigate the coherent dynamics in single molecule systems coupled to environments which exhibit both nonstationary and non-Markovian features. The generalized stochastic Liouville equation contains a generalized memory kernel associated with both the intrinsic system Hamiltonian and the non-Markovian features of the environmental noise, which returns to the well-known framework established by Kubo in the limit case that the environmental noise is stationary and memoryless. The coherence of the quantum system can be derived by means of the generalized stochastic Liouville equation in two separated averages with no need to consider the statistical characteristics of the environmental noise. We express the exact analytical expressions of the coherence of the single molecule systems induced by the nonstationary and non-Markovian RTN, and the analytical results of the system coherence are in well consistent with that derived by means of some other theoretical approaches.
The kinetics of relaxation in quantum systems induced by anomalous fluctuating noise is studied in detail. In this study two processes are considered, as examples: (1) relaxation in two-level system caused by external noise with slowly decaying correlation function P (t) ~ (wt)^{-alpha}, where 0 < alpha < 1, and (2) anomalous-diffusion controlled radical pair recombination in which relaxation results from the reaction with slowly fluctuating reaction rate whose fluctuations are governed by subdiffusive relative motion of radicals in a potential well. Analysis of these two processes is made within continuous time random walk approach (CTRWA). Rigorous CTRWA-treatment of the processes under study results in the non-Markovian stochastic Liouville equation (SLE) for the density matrix of the systems. This SLE predicts important specific features of relaxation kinetics of quantum systems in the presence of the above mentioned anomalous noise.
Decoherence of qubits under the influence of random phase fluctuations is studied by means of the stochastic Liouville equation, where the phase fluctuations are characterized by means of the Gauss-Markov process and the two-state-jump Markov process. We investigate how an observation of a stochastic variable during the time evolution influences the decoherence of the qubit system. We examine the decays of distinguishability and entanglement of qubit states. It is shown that the decoherence of qubits can be suppressed by the observation of the stochastic variable. It is also found that the suppression of the decoherence does not depend on the observed value.
Time evolutions of quantum correlation, classical correlation and total correlation of two qubits are investigated when the two qubits are placed under the influence of classical phase fluctuations with correlation. Stochastic variables that describe the phase fluctuations are correlated and subject to the stationary Gauss–Markov process. The model includes the local and global dephasing models. It is shown that the quantum correlation measured by the quantum discord is increased by the correlation between the stochastic variables in the initial time region while the classical correlation and the total correlation are not. Furthermore the entanglement, the optimal fidelity of the quantum teleportation and the violation of the Bell inequality are investigated.