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Proceedings of the Steklov Institute of Mathematics

An Example of Explicit Generators of Local and Nonlocal Quantum Master Equations

Alexander Evgen’evich Teretenkov

Published Jul 1, 2021• 3 citations
Publisher

Abstract

A simple example of completely positive dynamics is considered for which both the generator of a nonlocal integro-differential equation leading to such dynamics and the time-local generator can be calculated explicitly. It is demonstrated that introducing a small parameter in this simple example allows one to reproduce some nonperturbative phenomena that occur in more realistic models. In addition, a special case of fermionic dynamics is considered, and it is shown that one can actually find families of moments whose dynamics is linear but satisfies non-Markovian equations.

DOI
10.1134/s0081543821020218

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A simple example of completely positive dynamics is considered for which both the generator of a nonlocal integro-differential equation leading to such dynamics and the time-local generator can be calculated explicitly. It is demonstrated that introducing a small parameter in this simple example allows one to reproduce some nonperturbative phenomena that occur in more realistic models. In addition, a special case of fermionic dynamics is considered, and it is shown that one can actually find families of moments whose dynamics is linear but satisfies non-Markovian equations.
We analyze non-Markovian evolution of open quantum systems. It is shown that any dynamical map representing the evolution of such a system may be described either by a nonlocal master equation with a memory kernel or equivalently by an equation which is local in time. These two descriptions are complementary: if one is simple, the other is quite involved, or even singular, and vice versa. The price one pays for the local approach is that the corresponding generator keeps the memory about the starting point "t{0}." This is the very essence of non-Markovianity. Interestingly, this generator might be highly singular; nevertheless, the corresponding dynamics is perfectly regular. Remarkably, the singularities of the generator may lead to interesting physical phenomena such as the revival of coherence or sudden death and revival of entanglement.
Master equations are typically adopted to describe the dynamics of open quantum systems. Such equations are either in integro-differential or in time-local form, with the latter class more frequently adopted due to the simpler numerical methods developed to obtain the corresponding solution. Here we show that any time-local master equation with positive rates in the generator, i.e., any completely positive divisible quantum process, admits a microscopic model whose reduced dynamics is well described by the given equation.
Abstract Master equations are a useful tool to describe the evolution of open quantum systems. In order to characterize the mathematical features and the physical origin of the dynamics, it is often useful to consider different kinds of master equations for the same system. Here, we derive an exact connection between the time-local and the integro-differential descriptions, focusing on the class of commutative dynamics. The use of the damping-basis formalism allows us to devise a general procedure to go from one master equation to the other and vice versa, by working with functions of time and their Laplace transforms only. We further analyze the Lindbladian form of the time-local and the integro-differential master equations, where we account for the appearance of different sets of Lindbladian operators. In addition, we investigate a Redfield-like approximation, that transforms the exact integro-differential equation into a time-local one by means of a coarse graining in time. Besides relating the structure of the resulting master equation to those associated with the exact dynamics, we study the effects of the approximation on Markovianity. In particular, we show that, against expectation, the coarse graining in time can possibly introduce memory effects, leading to a violation of a divisibility property of the dynamics.
Two widely used but distinct approaches to the dynamics of open quantum systems are the Nakajima-Zwanzig and time-convolutionless quantum master equation, respectively.Although both describe identical quantum evolutions with strong memory effects, the first uses a time-nonlocal memory kernel K, whereas the second achieves the same using a time-local generator G.Here we show that the two are connected by a simple yet general fixed-point relation: G ¼ K½G.This allows one to extract nontrivial relations between the two completely different ways of computing the time evolution and combine their strengths.We first discuss the stationary generator, which enables a Markov approximation that is both nonperturbative and completely positive for a large class of evolutions.We show that this generator is not equal to the low-frequency limit of the memory kernel, but additionally "samples" it at nonzero characteristic frequencies.This clarifies the subtle roles of frequency dependence and semigroup factorization in existing Markov approximation strategies.Second, we prove that the fixed-point equation sums up the time-domain gradient or Moyal expansion for the time-nonlocal quantum master equation, providing nonperturbative insight into the generation of memory effects.Finally, we show that the fixed-point relation enables a direct iterative numerical computation of both the stationary and the transient generator from a given memory kernel.For the transient generator this produces nonsemigroup approximations which are constrained to be both initially and asymptotically accurate at each iteration step.
Two widely used but distinct approaches to the dynamics of open quantum systems are the Nakajima-Zwanzig and time-convolutionless quantum master equation, respectively.Although both describe identical quantum evolutions with strong memory effects, the first uses a time-nonlocal memory kernel K, whereas the second achieves the same using a time-local generator G.Here we show that the two are connected by a simple yet general fixed-point relation: G ¼ K½G.This allows one to extract nontrivial relations between the two completely different ways of computing the time evolution and combine their strengths.We first discuss the stationary generator, which enables a Markov approximation that is both nonperturbative and completely positive for a large class of evolutions.We show that this generator is not equal to the low-frequency limit of the memory kernel, but additionally "samples" it at nonzero characteristic frequencies.This clarifies the subtle roles of frequency dependence and semigroup factorization in existing Markov approximation strategies.Second, we prove that the fixed-point equation sums up the time-domain gradient or Moyal expansion for the time-nonlocal quantum master equation, providing nonperturbative insight into the generation of memory effects.Finally, we show that the fixed-point relation enables a direct iterative numerical computation of both the stationary and the transient generator from a given memory kernel.For the transient generator this produces nonsemigroup approximations which are constrained to be both initially and asymptotically accurate at each iteration step.