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arXiv:2102.03063 · Physical Review A

Coarse-graining master equation for periodically driven systems

Ronja Hotz, Gernot Schaller

Published Nov 24, 2021• 8 citations
arXivHTMLPDF

Abstract

We analyze Lindblad-Gorini-Kossakowski-Sudarshan-type generators for selected periodically driven open quantum systems. All these generators can be obtained by temporal coarse-graining procedures, and we compare different coarse-graining schemes. Similarly to undriven systems, we find that a dynamically adapted coarse-graining time, effectively yielding non-Markovian dynamics by interpolating through a series of different but individually Markovian solutions, gives the best results among the different coarse-graining schemes, albeit at the highest computational cost.

DOI
10.1103/physreva.104.052219

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We analyze Lindblad-Gorini-Kossakowski-Sudarshan-type generators for selected periodically driven open quantum systems. All these generators can be obtained by temporal coarse-graining procedures, and we compare different coarse-graining schemes. Similarly to undriven systems, we find that a dynamically adapted coarse-graining time, effectively yielding non-Markovian dynamics by interpolating through a series of different but individually Markovian solutions, gives the best results among the different coarse-graining schemes, albeit at the highest computational cost.
We study the coarse-graining approach to derive a generator for the evolution of an open quantum system over a finite time interval. The approach does not require a secular approximation but nevertheless generally leads to a Lindblad-Gorini-Kossakowski-Sudarshan generator. By combining the formalism with full counting statistics, we can demonstrate a consistent thermodynamic framework, once the switching work required for the coupling and decoupling with the reservoir is included. Particularly, we can write the second law in standard form, with the only difference that heat currents must be defined with respect to the reservoir. We exemplify our findings with simple but pedagogical examples.
Several approximation procedures, such as the full or partial rotating-wave, time-averaging, and geometric-arithmetic approximations, have been proposed to derive Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generators from the Born-Markov quantum master equation (e.g., the Redfield equation). Establishing rigorous error bounds for these approximations is of fundamental and practical importance. However, existing bounds face two major limitations: they are highly specific to individual methods, and, more critically, they diverge in the long-time limit, ensuring the accuracy of the derived GKSL generator only in short-time regimes. In this Letter, we resolve both issues by deriving a unified, rigorous error bound for a general class of approximation methods -- termed temporal coarse graining -- that encompasses all aforementioned schemes. Crucially, our error bound is time-uniform. This guarantees that GKSL generators obtained via temporal coarse graining remain accurate for arbitrarily long times, provided the dissipation timescale is significantly longer than the bath correlation timescale.
Periodically driven open quantum systems are central to quantum thermodynamics and quantum control. These systems are typically described using Floquet-Born-Markov master equations, derived with the use of a full secular approximation, and whose thermodynamic implications are often overlooked. In this context, we show that such a strong secular approximation may lead to unphysical predictions for steady state energy currents. We then demonstrate that a coarse-grained formulation of the master equation can regularize these issues while yielding completely positive dynamics and consistent energy currents. The coarse-graining time has a clear physical interpretation, as it defines the temporal resolution at which a Markovian master equation can describe the evolution of the periodically driven system. We show the consistency and validity of our approach by comparing to an exact non-Markovian simulation in paradigmatic examples: a driven two-level system and a three-level maser coupled to hot and cold thermal reservoirs. Our work provides a practical framework for correctly applying the secular approximation in periodically driven-dissipative systems and for assessing the accuracy of master equations of the GKSL form.
We present a first-principles derivation of the Markovian semigroup master equation without invoking the rotating-wave approximation (RWA). Instead we use a time coarse-graining approach that leaves us with a free time-scale parameter, which we can optimize. Comparing this approach to the standard RWA-based Markovian master equation, we find that significantly better agreement is possible using the coarse-graining approach, for a three-level model coupled to a bath of oscillators, whose exact dynamics we can solve for at zero temperature. The model has the important feature that the RWA has a nontrivial effect on the dynamics of the populations. We show that the two different master equations can exhibit strong qualitative differences for the population of the energy eigenstates even for such a simple model. The RWA-based master equation misses an important feature which the coarse-graining-based scheme does not. By optimizing the coarse-graining time scale the latter scheme can be made to approach the exact solution much more closely than the RWA-based master equation.
We formulate an effective-description framework for the dynamics of open quantum systems by extending the time-coarse-graining formalism to open systems. Our coarse-graining procedure efficiently removes high-frequency processes which are responsible for coherences between lower- and upper-manifold states and are irrelevant when considering only low-energy dynamics. We investigate the regime of validity of the resulting coarse-grained master equation by applying it to multilevel atoms driven by far-detuned lasers. Except for such high-frequency coherences, we find good agreement between the exact and coarse-grained dynamics unless the driving lasers are too strong or the initial high-frequency coherences are sizable.
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.