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Open Systems & Information Dynamics

Dynamic Correlations in Open Quantum Systems: The Dephasing Model

V. V. Ignatyuk

Published Jun 1, 2020• 0 citations
Publisher

Abstract

We study the dynamical correlations in open quantum systems on an example of an exactly solvable dephasing model. The system of non-Markovian kinetic equations for the generalized coherence and quasi-temperature is derived up to the 2-nd order in the coupling constant using the generalized quantum master equation [20]. Numerical and analytical solutions of the kinetic equations are obtained. The analytical result shows a redistribution of the real and imaginary parts of the coherence as compared to the exact one. We give a physical interpretation to the quasi-temperature, relating it to the nonequilibrium kinetic energy of the environment.

DOI
10.1142/s1230161220500079

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We study the dynamical correlations in open quantum systems on an example of an exactly solvable dephasing model. The system of non-Markovian kinetic equations for the generalized coherence and quasi-temperature is derived up to the 2-nd order in the coupling constant using the generalized quantum master equation [20]. Numerical and analytical solutions of the kinetic equations are obtained. The analytical result shows a redistribution of the real and imaginary parts of the coherence as compared to the exact one. We give a physical interpretation to the quasi-temperature, relating it to the nonequilibrium kinetic energy of the environment.
We propose a systematic approach to the dynamics of open quantum systems in the framework of Zubarev’s nonequilibrium statistical operator method. The approach is based on the relation between ensemble means of the Hubbard operators and the matrix elements of the reduced statistical operator of an open quantum system. This key relation allows deriving master equations for open systems following a scheme conceptually identical to the scheme used to derive kinetic equations for distribution functions. The advantage of the proposed formalism is that some relevant dynamical correlations between an open system and its environment can be taken into account. To illustrate the method, we derive a non-Markovian master equation containing the contribution of nonequilibrium correlations associated with energy conservation.
We theoretically investigate the non-Markovian dynamical decoherence of a quantum system coupled to nonequilibrium environments with nonstationary statistical properties. We show the time evolution of the decoherence factor in real-imaginary space to study the environment-induced energy renormalization and backaction of coherence which are associated with the unitary and nonunitary parts of the quantum master equation, respectively. It is also shown that the nonequilibrium decoherence dynamics displays a transition between Markovian and non-Markovian and the transition boundary depends on the environmental parameters. The results are helpful for further understanding non-Markovian dynamics and coherence backaction on an open quantum system from environments.
The non-Markovian master equation for open quantum systems is obtained by generalization of the standard Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations is written for the reduced density matrices of the bath ϱB(t) and of the system ϱS(t). A formal solution of the equation for ϱB(t) in the 2nd approximation in interaction yields a specific extra term related to the intrinsic bath dynamics. This term is nonlinear in the reduced density matrix ϱS(t), and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we apply the generalized ZN projection scheme to a simple dephasing model. We study the obtained kinetic equation both in the Markovian approximation and beyond it (for the term related to the intrinsic bath dynamics) and compare the results with the exact ones.
We evaluate the non-Markovian finite-temperature two-time correlation functions (CF's) of system operators of a pure-dephasing spin-boson model in two different ways, one by the direct exact operator technique and the other by the recently derived evolution equations, valid to second order in the system-environment interaction Hamiltonian. This pure-dephasing spin-boson model that is exactly solvable has been extensively studied as a simple decoherence model. However, its exact non-Markovian finite-temperature two-time system operator CF's, to our knowledge, have not been presented in the literature. This may be mainly due to the fact, illustrated in this article, that in contrast to the Markovian case, the time evolution of the reduced density matrix of the system (or the reduced quantum master equation) alone is not sufficient to calculate the two-time system operator CF's of non-Markovian open systems. The two-time CF's obtained using the recently derived evolution equations in the weak system-environment coupling case for this non-Markovian pure-dephasing model happen to be the same as those obtained from the exact evaluation. However, these results significantly differ from the non-Markovian two-time CF's obtained by wrongly directly applying the quantum regression theorem (QRT), a useful procedure to calculate the two-time CF's for weak-coupling Markovian open systems. This demonstrates clearly that the recently derived evolution equations generalize correctly the QRT to non-Markovian finite-temperature cases. It is believed that these evolution equations will have applications in many different branches of physics.
The master equation for an open quantum system is derived in the weak-coupling approximation when the additional dynamical variable—the mean interaction energy—is included into the generic relevant statistical operator. This master equation is nonlocal in time and involves the “quasi-temperature”, which is a non- equilibrium state parameter conjugated thermodynamically to the mean interaction energy of the composite system. The evolution equation for the quasi-temperature is derived using the energy conservation law. Thus long-living dynamical correlations, which are associated with this conservation law and play an important role in transition to the Markovian regime and subsequent equilibration of the system, are properly taken into account.
An open quantum system is a quantum system that interacts with some environment whose degrees of freedom have been coarse grained away. This model describes non-equilibrium processes more general than scattering-matrix formulations. Furthermore, the microscopically-derived environment provides a model of noise, dissipation and decoherence far more general than Markovian (white noise) models. The latter are fully characterized by Lindblad equations and can be motivated phenomenologically. Non-Markovian processes consistently account for backreaction with the environment and can incorporate effects such as finite temperature and spatial correlations. We consider linear systems with bilinear coupling to the environment, or quantum Brownian motion, and nonlinear systems with weak coupling to the environment. For linear systems we provide exact solutions with analytical results for a variety of spectral densities. Furthermore, we point out an important mathematical subtlety which led to incorrect master-equation coefficients in earlier derivations, given nonlocal dissipation. For nonlinear systems we provide perturbative solutions by translating the formalism of canonical perturbation theory into the context of master equations. It is shown that unavoidable degeneracy causes an unfortunate reduction in accuracy between perturbative master equations and their solutions. We also extend the famous theorem of Lindblad, Gorini, Kossakowski and Sudarshan on completely positivity to non-Markovian master equations. Our application is primarily to model atoms interacting via a common electromagnetic field. The electromagnetic field contains correlations in both space and time, which are related to its relativistic (photon-mediated) nature. As such, atoms residing in the same field experience different environmental effects depending upon their relative position and orientation. Our more accurate solutions were necessary to assess sudden death of entanglement at zero temperature. In contrast to previous claims, we found that all initial states of two-level atoms undergo finite-time disentanglement. We were also able to access regimes which cannot be described by Lindblad equations and other simpler methods, such as near resonance. Finally we revisit the infamous Abraham-Lorentz force, wherein a single particle in motion experiences backreaction from the electromagnetic field. This leads to a number of well-known problems including pre-acceleration and runaway solutions. We found a more a more-suitable open-system treatment of the nonrelativistic particle to be perfectly causal and dissipative without any extraneous requirements for finite size of the particle, weak coupling to the field, etc..