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arXiv:2108.03128 · Proceedings of the Steklov Institute of Mathematics

Derivation of the Redfield Quantum Master Equation and Corrections to It by the Bogoliubov Method

A. S. Trushechkin

Published Jul 1, 2021• 18 citations
arXivHTMLPDF

Abstract

Abstract Following the ideas N. N. Bogoliubov used to derive the classical and quantum nonlinear kinetic equations, we derive the Redfield quantum linear master equation, which is widely used in the theory of open quantum systems. This method allows one to calculate the joint state of a system and a reservoir at every instant of time, as well as to obtain quantum master equations as autonomous differential equations in an arbitrary order of perturbation theory. We prove that under certain conditions the expressions for the corrections of all orders are well defined. We also discuss the question of whether the quantum dynamics described by the derived quantum master equations is Markovian.

DOI
10.1134/s008154382102022x

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Abstract Following the ideas N. N. Bogoliubov used to derive the classical and quantum nonlinear kinetic equations, we derive the Redfield quantum linear master equation, which is widely used in the theory of open quantum systems. This method allows one to calculate the joint state of a system and a reservoir at every instant of time, as well as to obtain quantum master equations as autonomous differential equations in an arbitrary order of perturbation theory. We prove that under certain conditions the expressions for the corrections of all orders are well defined. We also discuss the question of whether the quantum dynamics described by the derived quantum master equations is Markovian.
The Redfield equation describes the dynamics of a quantum system weakly coupled to one or more reservoirs and is widely used in theory of open quantum system. However, the assumption of weak system-reservoir coupling is often not fully adequate and higher-order corrections to the Redfield equation with respect to the system-bath coupling is required. Here we propose a general method of derivation of higher-order corrections to the Redfield quantum master equation based on the hierarchical equations of motion (HEOM). Also we derive conditions of validity of the Redfield equation as well as the additional secular approximation for it.
A quantum system that interacts with an environment generally undergoes nonunitary evolution described by a non-Markovian or Markovian master equation. In this paper, we construct the non-Markovian Redfield master equation for a quantum scalar field that interacts with a second field through a bilinear or nonlinear interaction on a Minkowski background. We use the resulting master equation to set up coupled differential equations that can be solved to obtain the equal-time two-point function of the system field. We show how the equations simplify under various approximations including the Markovian limit and argue that the Redfield equation-based solution provides a perturbative resummation to the standard second-order Dyson series result. For the bilinear interaction, we explicitly show that the Redfield solution is closer to the exact solution compared to the perturbation theory-based one. Further, the environment correlation function is oscillatory and nondecaying in this case, making the Markovian master equation a poor approximation. For the nonlinear interaction, on the other hand, the environment correlation function is sharply peaked and the Redfield solution matches that obtained using a Markovian master equation in the late-time limit.
Data of the numerical solution of the time-dependent Schrödinger equation of a system containing one spin-1/2 particle interacting with a bath of up to 32 spin-1/2 particles is used to construct a Markovian quantum master equation describing the dynamics of the system spin. The procedure of obtaining this quantum master equation, which takes the form of a Bloch equation with time-independent coefficients, accounts for all non-Markovian effects inasmuch the general structure of the quantum master equation allows. Our simulation results show that, with a few rather exotic exceptions, the Bloch-type equation with time-independent coefficients provides a simple and accurate description of the dynamics of a spin-1/2 particle in contact with a thermal bath. A calculation of the coefficients that appear in the Redfield master equation in the Markovian limit shows that this perturbatively derived equation quantitatively differs from the numerically estimated Markovian master equation, the results of which agree very well with the solution of the time-dependent Schrödinger equation.
Based on the non-equilibrium density matrix method and using the Bogolyubov’s approach for the decoupling of many-particle distribution functions, a closed set of kinetic equations is derived for the state occupancies of closed and open quantum systems. It is shown that the transformation of a non-Markovian master equation into a Markovian one becomes possible at a high-frequency stochastic field alternating the energy levels of a quantum system. At a weak interaction between quantum subsystems composing the open quantum system, the set of linear balance-like kinetic equations for multistate occupancies of the system is transformed into a set of nonlinear kinetic equations for the occupancies of separate quantum subsystems.
We put forth a new class of quantum master equations that correctly reproduce the asymptotic state of an open quantum system beyond the infinitesimally weak system-bath coupling limit. Our method is based on incorporating the knowledge of the reduced steady state into its dynamics. The correction not only steers the reduced system toward a correct steady state but also improves the accuracy of the dynamics, thereby refining the archetypal Born-Markov weak-coupling second-order master equations. In case of equilibrium, we use the exact mean-force Gibbs state to correct the Redfield quantum master equation. By benchmarking it with the exact solution of the damped harmonic oscillator, we show that our method also helps correct the long-standing issue of positivity violation, albeit without complete positivity. Our method of a canonically consistent quantum master equation opens a new perspective in the theory of open quantum systems leading to a reduced density matrix accurate beyond the commonly used Redfield and Lindblad equations, while retaining the same conceptual and numerical complexity.
We develop a Markovian master equation in the Lindblad form that enables the efficient study of a wide range of open quantum many-body systems that would be inaccessible with existing methods. The validity of the master equation is based entirely on properties of the bath and the system-bath coupling, without any requirements on the level structure within the system itself. The master equation is derived using a Markov approximation that is distinct from that used in earlier approaches. We provide a rigorous bound for the error induced by this Markov approximation; the error is controlled by a dimensionless combination of intrinsic correlation and relaxation timescales of the bath. Our master equation is accurate on the same level of approximation as the Bloch-Redfield equation. In contrast to the Bloch-Redfield approach, our approach ensures preservation of the positivity of the density matrix. As a result, our method is robust, and can be solved efficiently using stochastic evolution of pure states (rather than density matrices). We discuss how our method can be applied to static or driven quantum many-body systems, and illustrate its power through numerical simulation of a spin chain that would be challenging to treat by existing methods.