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arXiv:2003.09063 · Quantum

Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation

Dragomir M Davidović

Published Sep 21, 2020• 63 citations
arXivHTMLPDF

Abstract

Here we present a Lindblad master equation that approximates the Redfield equation, a well known master equation derived from first principles, without significantly compromising the range of applicability of the Redfield equation. Instead of full-scale coarse-graining, this approximation only truncates terms in the Redfield equation that average out over a time-scale typical of the quantum system. The first step in this approximation is to properly renormalize the system Hamiltonian, to symmetrize the gains and losses of the state due to the environmental coupling. In the second step, we swap out an arithmetic mean of the spectral density with a geometric one, in these gains and losses, thereby restoring complete positivity. This completely positive approximation, GAME (geometric-arithmetic master equation), is adaptable between its time-independent, time-dependent, and Floquet form. In the exactly solvable, three-level, Jaynes-Cummings model, we find that the error of the approximate state is almost an order of magnitude lower than that obtained by solving the coarse-grained stochastic master equation. As a test-bed, we use a ferromagnetic Heisenberg spin-chain with long-range dipole-dipole coupling between up to 25-spins, and study the differences between various master equations. We find that GAME has the highest accuracy per computational resource.

DOI
10.22331/q-2020-09-21-326

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Here we present a Lindblad master equation that approximates the Redfield equation, a well known master equation derived from first principles, without significantly compromising the range of applicability of the Redfield equation. Instead of full-scale coarse-graining, this approximation only truncates terms in the Redfield equation that average out over a time-scale typical of the quantum system. The first step in this approximation is to properly renormalize the system Hamiltonian, to symmetrize the gains and losses of the state due to the environmental coupling. In the second step, we swap out an arithmetic mean of the spectral density with a geometric one, in these gains and losses, thereby restoring complete positivity. This completely positive approximation, GAME (geometric-arithmetic master equation), is adaptable between its time-independent, time-dependent, and Floquet form. In the exactly solvable, three-level, Jaynes-Cummings model, we find that the error of the approximate state is almost an order of magnitude lower than that obtained by solving the coarse-grained stochastic master equation. As a test-bed, we use a ferromagnetic Heisenberg spin-chain with long-range dipole-dipole coupling between up to 25-spins, and study the differences between various master equations. We find that GAME has the highest accuracy per computational resource.
Markovian master equations are a ubiquitous tool in the study of open quantum systems, but deriving them from first principles involves a series of compromises. On the one hand, the Redfield equation is valid for fast environments (whose correlation function decays much faster than the system relaxation time) regardless of the relative strength of the coupling to the system Hamiltonian, but is notoriously non-completely-positive. On the other hand, the Davies equation preserves complete positivity but is valid only in the ultra-weak coupling limit and for systems with a finite level spacing, which makes it incompatible with arbitrarily fast time-dependent driving. Here we show that a recently derived Markovian coarse-grained master equation (CGME), already known to be completely positive, has a much expanded range of applicability compared to the Davies equation, and moreover, is locally generated and can be generalized to accommodate arbitrarily fast driving. This generalization, which we refer to as the time-dependent CGME, is thus suitable for the analysis of fast operations in gate-model quantum computing, such as quantum error correction and dynamical decoupling. Our derivation proceeds directly from the Redfield equation and allows us to place rigorous error bounds on all three equations: Redfield, Davies, and coarse-grained. Our main result is thus a completely positive Markovian master equation that is a controlled approximation to the true evolution for any time-dependence of the system Hamiltonian, and works for systems with arbitrarily small level spacing. We illustrate this with an analysis showing that dynamical decoupling can extend coherence times even in a strictly Markovian setting.
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.
Abstract Understanding nonsecular dynamics in open quantum systems is addressed here, with emphasis on systems with large numbers of Bohr frequencies, zero temperature, and fast driving. We employ the master equation, which replaces arithmetic averages of the decay rates in the open system, with their geometric averages, and find that it can improve the second order perturbation theory, known as the Redfield equation, while enforcing complete positivity on quantum dynamics. The characteristic frequency scale that governs the approximation is the minimax frequency: the minimum of the maximum system oscillation frequency and the bath relaxation rate; this needs to be larger than the dissipation rate for it to be valid. The concepts are illustrated on the Heisenberg ferromagnetic spin-chain model. To study the accuracy of the approximation, a Hamiltonian is drawn from the Gaussian unitary ensemble, for which we calculate the fourth order time-convolutionless master equation, in the Ohmic bath at zero temperature. Enforcing the geometric average, decreases the trace distance to the exact solution. Dynamical decoupling of a qubit is examined by applying the Redfield and the geometric-arithmetic master equations, in the interaction picture of the time dependent system Hamiltonian, and the results are compared to the exact path integral solution. The geometric-arithmetic approach is significantly simpler and can be super-exponentially faster compared to the Redfield approach.
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 reduced dynamics of an open quantum system obtained from an underlying microscopic Hamiltonian can in general only approximately be described by a time-local master equation. The quality of that approximation depends primarily on the coupling strength and the structure of the environment. Various such master equations have been proposed with different aims. Choosing the most suitable one for a specific system is not straightforward. By focusing on the accuracy of the reduced dynamics we provide a thorough assessment for a selection of methods (Redfield equation, quantum optical master equation, coarse-grained master equation, a related dynamical map approach, and a partial-secular approximation). We consider secondary, here, whether or not an approach guarantees positivity. We use two qubits coupled to a Lorentzian environment in a spin-boson-like fashion modeling a generic situation with various system and bath time scales. We see that, independent of the initial state, the simple Redfield equation with time-dependent coefficients is significantly more accurate than all other methods under consideration. We emphasize that positivity violation in the Redfield equation formalism becomes relevant only in a regime where any of the perturbative master equations considered here are rendered invalid anyway. This implies that the loss of positivity should in fact be welcomed as an important feature: it indicates the breakdown of the weak-coupling assumption. In addition we present the various approaches in a self-contained way and use the behavior of their errors to provide further insight into the range of validity of each method.
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.