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arXiv:2403.04488 · Physical Review A

Dynamics of the nonequilibrium spin-boson model: A benchmark of master equations and their validity

Gerardo Suárez, Marcin Łobejko, Michał Horodecki

Published Oct 25, 2024• 7 citations
arXivHTMLPDF

Abstract

In recent years, there has been tremendous focus on identifying whether effective descriptions of open quantum systems such as master equations can accurately describe the dynamics of open quantum systems. One particular question is whether they provide the correct steady state in the long time limit. The transient regime is also of interest. Description of evolution by various master equations---some of them being not completely positive---is benchmarked against exact solutions [see, e.g., R. Hartmann and W. T. Strunz, Phys. Rev. A 101, 012103 (2020)]. An important property of true evolution is its non-Markovian features, which are not captured by the simplest completely positive master equations. In this paper we consider a non-Markovian yet completely positive evolution (known as refined weak coupling or cumulant equation) for the spin-boson model with an overdamped Drude-Lorentz spectral density and arbitrary coupling. We benchmark it against a numerically exact solution, as well as against other master equations, for different coupling strengths and temperatures. We find the cumulant to be a better description in the weak-coupling regime where it is supposed to be valid. For the examples considered it shows superiority at moderate and strong couplings in the low-temperature regime for all examples considered. In the high-temperature regime, however, its advantage vanishes. This indicates that the cumulant equation is a good candidate for simulations at weak to moderate coupling and low temperature. Our calculations are greatly facilitated due to our concise formulation of the cumulant equation by means of representation of the density matrix in the SU(N) basis.

DOI
10.1103/physreva.110.042428

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In recent years, there has been tremendous focus on identifying whether effective descriptions of open quantum systems such as master equations can accurately describe the dynamics of open quantum systems. One particular question is whether they provide the correct steady state in the long time limit. The transient regime is also of interest. Description of evolution by various master equations---some of them being not completely positive---is benchmarked against exact solutions [see, e.g., R. Hartmann and W. T. Strunz, Phys. Rev. A 101, 012103 (2020)]. An important property of true evolution is its non-Markovian features, which are not captured by the simplest completely positive master equations. In this paper we consider a non-Markovian yet completely positive evolution (known as refined weak coupling or cumulant equation) for the spin-boson model with an overdamped Drude-Lorentz spectral density and arbitrary coupling. We benchmark it against a numerically exact solution, as well as against other master equations, for different coupling strengths and temperatures. We find the cumulant to be a better description in the weak-coupling regime where it is supposed to be valid. For the examples considered it shows superiority at moderate and strong couplings in the low-temperature regime for all examples considered. In the high-temperature regime, however, its advantage vanishes. This indicates that the cumulant equation is a good candidate for simulations at weak to moderate coupling and low temperature. Our calculations are greatly facilitated due to our concise formulation of the cumulant equation by means of representation of the density matrix in the SU(N) basis.
The spin-boson model is a widely used model for understanding the properties of a two-level open quantum system. Accurately describing its dynamics often requires going beyond the weak system-environment coupling approximation. However, calculating the higher-order generators of such a dynamics, with a system-environment coupling that is not too weak, has been known to be challenging, both numerically and analytically. This work presents the analytical derivation of the complete fourth-order time-convolutionless (TCL) generator for a generic spin-boson model, accurate up to 4th order in the system-environment coupling parameter, under the assumption that the environmental spectral density is an odd function of frequency. In the case of a semiconductor double-quantum-dot system, our results reveal corrections to the dynamics that may become physically significant in some parameter regimes. Furthermore, we report that the widely used second-order TCL master equation tends to overestimate the non-Markovianity of a dynamics over a large parameter regime. Within the regime of its applicability, our results provide a computational advantage over numerically exact techniques. The accuracy of the fourth-order TCL generator is rigorously benchmarked against specialized analytical calculations done for the Ohmic spectral density with Drude cutoff and against the numerically exact Hierarchical Equations of Motion technique.
Open quantum systems are ubiquitous, however, their description beyond ultraweak system-bath coupling is challenging. The standard Redfield master equation can violate positivity. Recently, Nathan and Rudner proposed an alternative equation, not suffering from this problem. The authors compare here both equations to the exact solution of a simple model. The Redfield equation performs noticeably better for transient dynamics. For the steady state it is slightly better for high temperatures, but fails at low temperatures, where the Nathan-Rudner equation is still accurate.
We outline a non-perturbative approach for simulating the behavior of open quantum systems interacting with a bosonic environment defined by a generalized spectral density function. The method is based on replacing the environment by a set of damped harmonic oscillators - the pseudomodes - thereby forming an enlarged open system whose dynamics is governed by a Markovian master equation. Each pseudomode is connected to one of the poles of the spectral density when analytically continued to the lower-half complex frequency plane. Here, we extend previous results to a completely generic class of open system models, and discuss how our framework can be used as a powerful and versatile tool for analyzing non-Markovian open system dynamics. The effectiveness of the method is demonstrated on the spin-boson model by accurately benchmarking its predictions against numerically exact results.
We study the properties of a refined weak-coupling limit that preserves complete positivity in order to describe non-Markovian dynamics in the spin-boson model. With this tool, we show the system presents a rich non-Markovian phenomenology. This implies a dynamical difference between entanglement and coherence: the latter undergoes revivals, whereas the former not, despite the induced dynamics being fully incoherent. In addition, the evolution presents ``quasieternal'' non-Markovianity, becoming nondivisible at any time period where the system evolves qualitatively. Furthermore, the method allows for an exact derivation of a master equation that accounts for a reversible energy exchange between system and environment. Specifically, this is obtained in the form of a time-dependent Lamb shift term.
Quantum master equations are commonly used to model the dynamics of open quantum systems, but their accuracy is rarely compared with the analytical solution of exactly solvable models. In this work, we perform such a comparison for the damped Jaynes-Cummings model of a qubit in a leaky cavity, for which an analytical solution is available in the one-excitation subspace. We consider the non-Markovian time-convolutionless master equation up to the second (Redfield) and fourth orders as well as three types of Markovian master equations: the coarse-grained, cumulant, and standard rotating-wave approximation (RWA) Lindblad equations. We compare the exact solution to these master equations for three different spectral densities: impulse, Ohmic, and triangular. We demonstrate that the coarse-grained master equation outperforms the standard RWA-based Lindblad master equation for weak coupling or high qubit frequency (relative to the spectral density high-frequency cutoff ${\ensuremath{\omega}}_{c}$), where the Markovian approximation is valid. In the presence of non-Markovian effects characterized by oscillatory, nondecaying behavior, the time-convolutionless approximation closely matches the exact solution for short evolution times (in units of ${\ensuremath{\omega}}_{c}^{\ensuremath{-}1}$) even outside the regime of validity of the Markovian approximations. For long evolution times, all master equations perform poorly, as quantified in terms of the trace-norm distance from the exact solution. The fourth-order time-convolutionless master equation achieves the top performance in all cases. Our results highlight the need for reliable approximation methods to describe open-system quantum dynamics beyond the short-time limit.
The famous Davies-GKSL secular Markovian master equation is tremendously successful in approximating the evolution of open quantum systems in terms of just a few parameters. However, the fully secular Davies-GKSL equation fails to accurately describe timescales short enough, i.e., comparable to the inverse of differences of frequencies present in the system of interest. A complementary approach that works well for short times but is not suitable after this short interval is known as the quasisecular master equation. Still, both approaches fail to have any faithful dynamics in the intermediate-time interval. Simultaneously, descriptions of dynamics that apply to the aforementioned "gray zone" often are computationally much more complex than master equations or are mathematically not well-structured. The filtered approximation (FA) to the refined weak-coupling limit has the simplistic spirit of the Davies-GKSL equation and allows capturing the dynamics in the intermediate-time regime. At the same time, our non-Markovian equation yields completely positive dynamics. We exemplify the performance of the FA equation in the cases of the spin-boson system and qutrit-boson system in which two distant timescales appear.