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Physical Review A

Computable form of the Born-Markov master equation for open multilevel quantum systems

Xian-Ting Liang

Published Feb 19, 2019• 4 citations
Publisher

Abstract

The Redfield and Lindblad forms of the master equations are obtained under not only the Born and Markov approximations but also the secular approximation. In this paper, we propose a computable form of the master equation for open multilevel quantum systems beyond the secular approximation. The proposed form can retain some dynamical effects that are lost due to use of the secular approximation. Two kinds of open multilevel quantum systems are investigated by using this method, and the results are compared to the ones obtained from the Redfield equation and an exactly numerical path integral approach. It is shown that the results obtained from this computable form match those from the exact numerical path integral approach better than results from the Redfield equation. The normal-diffusion coefficient, momentum-damping coefficient, anomalous-diffusion coefficient, etc. in the master equation are also reevaluated.

DOI
10.1103/physreva.99.022118

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The Redfield and Lindblad forms of the master equations are obtained under not only the Born and Markov approximations but also the secular approximation. In this paper, we propose a computable form of the master equation for open multilevel quantum systems beyond the secular approximation. The proposed form can retain some dynamical effects that are lost due to use of the secular approximation. Two kinds of open multilevel quantum systems are investigated by using this method, and the results are compared to the ones obtained from the Redfield equation and an exactly numerical path integral approach. It is shown that the results obtained from this computable form match those from the exact numerical path integral approach better than results from the Redfield equation. The normal-diffusion coefficient, momentum-damping coefficient, anomalous-diffusion coefficient, etc. in the master equation are also reevaluated.
In this paper we derive the Lindblad and Redfield forms with and without secular approximation from the Born-Markov master equation for open quantum systems. The spectral correlation tensor of bath (the Fourier transform of the bath correlation function) and then the coefficients in the two forms of the master equation are reevaluated according to the scheme in Ref.[Phys. Rev. A 99, 022118 (2019)]. They are complex numbers rather than the real numbers getting from traditional simplified methods. The dynamics of two models [one is an open three-level quantum system model, and the other is the model of phycoerythrin 545 (PE545) of modeling a photosynthesis reaction center] are studied by using the obtained equations. The non-secular Lindblad and Redfield equations with the complex coefficients predict almost the same dynamical results from the Born-Markov master equation. However, the results obtained from the traditional Lindblad and Redfield equations deviate the actual dynamics of the open quantum systems.
Abstract In this paper, we derive the Lindblad and Redfield forms of the master equation based on the Born–Markov master equation with and without the secular approximation for open multi-level quantum systems. The coefficients of the equations are re-evaluated according to the scheme in [(2019), Phys. Rev. A 99 , 022118]. They are complex numbers rather than the real numbers obtained from traditional simplified methods. The dynamics of two models (one is an open three-level quantum system model, and the other is the model of phycoerythrin 545 (PE545) in a photosynthesis system) are studied. It is shown that the secular approximation and the simplified real coefficients may cause a small distortion of the dynamics in some environments, but a large distortion of the dynamics in others. These effects are discussed and characterized by studying the dynamics of nontrivial instances of multi-level systems in the presence of dissipation.
Abstract Quantum master equations are widely used to describe the dynamics of open quantum systems. All these different master equations rely on specific approximations that may or may not be justified. Starting from a microscopic model, applying the standard Born and Markov approximations results in the Redfield equation that does not guarantee to preserve positivity. The latter is typically achieved by additionally applying the secular approximation resulting in a quantum master equation in Lindblad form. There are other ways to obtain an equation in Lindblad form, one of which is the recently proposed universal Lindblad equation (ULE). It has been shown that it is in the same equivalence class of approximations as the Redfield master equation although avoiding the heuristic secular approximation (Nathan and Rudner 2020 Phys. Rev. B 102 115109). In this work, we prove that the solutions of the master equation obtained by applying the secular approximation are also obtained by an approximation of the same order as the one performed to obtain the Redfield equation. We hereby provide a mathematical justification for the secular approximation. We show that the result of applying the secular approximation is obtained naturally by applying a self-consistency argument. This shows that the resulting master equation is also in the same equivalence class of approximations as the Redfield master equation and the ULE. We furthermore give numerical considerations about the accuracy of the secular master equation compared to the ULE.
The Markovian dynamics of open quantum systems is typically described through Lindblad equations, which are derived from the Redfield equation via the full secular approximation. The latter neglects the rotating terms in the master equation corresponding to pairs of jump operators with different Bohr frequencies. However, for many physical systems this approximation breaks down, and thus a more accurate treatment of the slowly rotating terms is required. Indeed, more precise physical results can be obtained by performing the partial secular approximation, which takes into account the relevant time scale associated with each pair of jump operators and compares it with the time scale arising from the system-environment coupling. In this work, we introduce a general code for performing the partial secular approximation in the Redfield equation for structured open quantum systems. The code can be applied to a generic Hamiltonian of any multipartite system coupled to bosonic baths. Moreover, it can also reproduce the unified master equation, which captures the same physical behavior as the Redfield equation under the partial secular approximation, but is mathematically guaranteed to generate a completely positive dynamical map. Finally, the code can compute both the local and global version of the master equation for the same physical problem. We illustrate the code by studying the steady-state heat flow in a structured open quantum system composed of two superconducting qubits, each coupled to a bosonic mode, which in turn interacts with a thermal bath. The results in this work can be employed for the numerical study of a wide range of complex open quantum systems.
Abstract We introduce a systematic approximation for an efficient evaluation of Born–Markov master equations for steady state transport studies in open quantum systems out of equilibrium: the energy resolved master equation approach. The master equation is formulated in the eigenbasis of the open quantum system and build successively by including eigenstates with increasing grandcanonical energies. In order to quantify convergence of the approximate scheme we introduce quality factors to check preservation of trace, positivity and hermiticity. Furthermore, we discuss different types of master equations that go beyond the commonly used secular approximation in order to resolve coherences between quasi-degenerate states. For the discussion of complete positivity we introduce a canonical Redfield–Bloch master equation and compare it to a previously derived master equations in Lindblad form with and without using the secular approximation. The approximate scheme is benchmarked for a six orbital quantum system which shows destructive quantum interference under the application of a bias voltage. The energy resolved master equation approach presented here makes quantum transport calculations in many-body quantum systems numerically accessible also beyond six orbitals with a full Hilbert space of the order of ∼106.
We present the Born-Markov approximated Redfield quantum master equation (RQME) description for an open system of noninteracting particles (bosons or fermions) on an arbitrary lattice of NNN sites in any dimension and weakly connected to multiple reservoirs at different temperatures and chemical potentials. The RQME can be reduced to the Lindblad equation, of various forms, by making further approximations. By studying the N=2N=2N=2 case, we show that RQME gives results which agree with exact analytical results for steady-state properties and with exact numerics for time-dependent properties over a wide range of parameters. In comparison, the Lindblad equations have a limited domain of validity in nonequilibrium. We conclude that it is indeed justified to use microscopically derived full RQME to go beyond the limitations of Lindblad equations in out-of-equilibrium systems. We also derive closed-form analytical results for out-of-equilibrium time dynamics of two-point correlation functions. These results explicitly show the approach to steady state and thermalization. These results are experimentally relevant for cold atoms, cavity QED, and far-from-equilibrium quantum dot experiments.