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arXiv:cond-mat/0408093 · Journal of Experimental and Theoretical Physics

The non-Markovian stochastic Liouville equation and anomalous quantum relaxation kinetics

A. I. Shushin

Published Oct 1, 2005• 6 citations
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Abstract

We analyze within the continuous-time random walk approach, the kinetics of phase and population relaxation in quantum systems induced by noise with the anomalously slowly decaying correlation function P ( t ) ∝ ( wt ) -α , where 0 < α < 1. The relaxation kinetics is shown to be anomalously slow. Moreover, for α < 1, in the limit of a short characteristic time of fluctuations w -1 , the kinetics is independent of w . As α → 1, the relaxation regime changes from the static limit to narrowing of fluctuation. Simple analytical expressions are obtained that describe the specific features of the kinetics.

DOI
10.1134/1.2131926

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We analyze within the continuous-time random walk approach, the kinetics of phase and population relaxation in quantum systems induced by noise with the anomalously slowly decaying correlation function P ( t ) ∝ ( wt ) -α , where 0 < α < 1. The relaxation kinetics is shown to be anomalously slow. Moreover, for α < 1, in the limit of a short characteristic time of fluctuations w -1 , the kinetics is independent of w . As α → 1, the relaxation regime changes from the static limit to narrowing of fluctuation. Simple analytical expressions are obtained that describe the specific features of the kinetics.
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.
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.
Half century has past since the pioneering works of Anderson and Kubo on the stochastic theory of spectral line shape were published in J. Phys. Soc. Jpn. 9 (1954) 316 and 935, respectively. In this review, we give an overview and extension of the stochastic Liouville equation focusing on its theoretical background and applications to help further the development of their works. With the aid of path integral formalism, we derive the stochastic Liouville equation for density matrices of a system. We then cast the equation into the hierarchy of equations which can be solved analytically or computationally in a nonperturbative manner including the effect of a colored noise. We elucidate the applications of the stochastic theory from the unified theoretical basis to analyze the dynamics of a system as probed by experiments. We illustrate this as a review of several experimental examples including NMR, dielectric relaxation, Mössbauer spectroscopy, neutron scattering, and linear and nonlinear laser spectroscopies. Following the summary of the advantage and limitation of the stochastic theory, we then derive a quantum Fokker–Planck equation and a quantum master equation from a system–bath Hamiltonian with a suitable spectral distribution producing a nearly Markovian random perturbation. By introducing auxiliary parameters that play a role as stochastic variables in an expression for reduced density matrix, we obtain the stochastic Liouville equation including temperature correction terms. The auxiliary parameters may also be interpreted as a random noise that allows us to derive a quantum Langevin equation for non-Markovian noise at any temperature. The results afford a basis for clarifying the relationship between the stochastic and dynamical approaches. Analytical as well as numerical calculations are given as examples and discussed.
It is shown in this paper that the quantum master equation can be mapped to a modified continuous time random walk (CTRW) if the relaxation term is composed of transitions over a set of states. When the Hamiltonian is time-independent and transitions are between its eigenlevels, such a modified CTRW reduces to the Markovian walk equivalent to the Pauli master equation. On the other hand, the memory in such a modified CTRW is composed of a temporal factor and the probability determined by the Liouville flow when the relaxation term is reduced as a special dephasing term.
We address the dynamics of continuous-time quantum walks on one-dimensional disordered lattices inducing dynamical noise in the system. Noise is described as time-dependent fluctuations of the tunneling amplitudes between adjacent sites, and attention is focused on non-Gaussian telegraph noise, going beyond the usual assumption of fast Gaussian noise. We observe the emergence of two different dynamical behaviors for the walker, corresponding to two opposite noise regimes: slow noise (i.e., strong coupling with the environment) confines the walker into few lattice nodes, while fast noise (weak coupling) induces a transition between quantum and classical diffusion over the lattice. A phase transition between the two dynamical regimes may be observed by tuning the ratio between the autocorrelation time of the noise and the coupling between the walker and the external environment generating the noise. We also address the non-Markovianity of the quantum map by assessing its memory effects, as well as evaluating the information backflow to the system. Our results suggest that the non-Markovian character of the evolution is linked to the dynamical behavior in the slow noise regime, and that fast noise induces a Markovian dynamics for the walker.
We analyze the effect of a classical noise into the entanglement dynamics between two particles, initially entangled, subject to continuous time quantum walks in a one-dimensional lattice. The noise is modeled by randomizing the transition amplitudes from one site to another. Both Markovian and non-Markovian environments are considered. For the Markov regime an exponential decay of the initial quantum correlation is found, while the loss of coherence of the quantum state increases monotonically with time up to a saturation value depending upon the degrees of freedom of the system. For the non-Markov regime the presence or absence of entanglement revival and entanglement sudden death phenomena is found or deduced depending on the peculiar characteristics of the noise. Our results indicate that the entanglement dynamics in the non-Markovian regime is affected by the persistence of the memory effects of the environment and by its intrinsic features.