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arXiv:0905.1020 · Physical Review A

Completely positive Markovian quantum dynamics in the weak-coupling limit

David Taj, Fausto Rossi

Published Nov 13, 2008• 15 citations
arXivHTMLPDF

Abstract

We obtain exponential decay laws for solutions of density-matrix master equations in the weak-coupling limit: after comparing with results already present in the literature and developing the necessary techniques, we study the crucial aspect of complete positivity under fairly general conditions. We propose a time average that guarantees complete positivity and approximates, in Markovian fashion, the exact dynamics for a plethora of physical applications, no matter which are the spectral properties of the subsystem, or its dimensions. We shall comment on some interesting examples, such as a quantum version of the celebrated Fermi's ``golden rule'' and some recently proposed entangling projections.

DOI
10.1103/physreva.78.052113

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We obtain exponential decay laws for solutions of density-matrix master equations in the weak-coupling limit: after comparing with results already present in the literature and developing the necessary techniques, we study the crucial aspect of complete positivity under fairly general conditions. We propose a time average that guarantees complete positivity and approximates, in Markovian fashion, the exact dynamics for a plethora of physical applications, no matter which are the spectral properties of the subsystem, or its dimensions. We shall comment on some interesting examples, such as a quantum version of the celebrated Fermi's ``golden rule'' and some recently proposed entangling projections.
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In a recent work we have found a contraction semigroup able to correctly approximate a projected and perturbed one-parameter group of isometries in a generic Banach space, in the limit of weak-coupling. Here we study its generator by specializing to W∗W^*W∗-algebras: after defining a Physical Subsystem in terms of a completely positive projecting conditional expectation, we find that it generates a Quantum Dynamical Semigroup. As a consequence of uniqueness and strong generality (well defined dynamics, irrespective of the Physical Subsystem spectral properties or dimensions), its generator deserves to be referred as "the" Quantum Fokker-Planck Equation. We then provide important examples of the limit dynamics, one of which constitutes a new Quantum generalization of the celebrated Fermi Golden Rule.
The hierarchy equations of motion provide an elegant formalism for the description of non-Markovian quantum dynamics, which has successfully been applied to a broad variety of physical problems ranging from light-harvesting complexes to resonant tunnelling junctions. When these equations are derived from microscopic models of system and environment, then they typically expand to an infinite series of coupled differential equations. The truncation of this series is accompanied by an approximation of the dynamics and can give rise to a violation of complete positivity. The goal of this thesis is to investigate under what conditions a set of hierarchy equations of motion induces completely positive dynamics. Such conditions are in particular crucial for a phenomenological understanding of the equations of motion. Several approaches to this problem are proposed and compared for classical systems before a generalisation to quantum mechanical problems is carried out. The main result is a concise set of operator inequalities that are sufficient for complete positivity and that can be implemented efficiently in terms of a semi-definite program. The applicability of this technique is demonstrated by various explicit examples. For truncated hierarchy equations for which a violation of complete positivity is inevitable the framework is extended such that the degree of this violation can be estimated and the validity of the truncation can be gauged. In particular in the presence of initial system-environment correlations a dynamical process can also be considered physically reasonable, when the time evolution is not completely positive and it is shown how hierarchy equations that describe such scenarios can be identified. Eventually, the framework is altered such that it gives rise to conditions on hierarchy equations that are sufficient not only for completely positive but even Markovian dynamics.
By modeling the interaction of an open quantum system with its environment through a natural generalization of the classical concept of continuous time random walk, we derive and characterize a class of non-Markovian master equations whose solution is a completely positive map. The structure of these master equations is associated with a random renewal process where each event consist in the application of a superoperator over a density matrix. Strong nonexponential decay arise by choosing different statistics of the renewal process. As examples we analyze the stochastic and averaged dynamics of simple systems that admit an analytical solution. The problem of positivity in quantum master equations induced by memory effects [S. M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001)] is clarified in this context.
Quantum systems coupled to environments exhibit intricate dynamics. The master equation gives a Markov approximation of the dynamics, allowing for analytic and numerical treatments. It is ubiquitous in theoretical and applied quantum sciences. The accuracy of the master equation approximation was so far proven in the regime where time must not exceed an upper bound depending on the system-environment interaction strength (weak coupling regime). Here, we show that the Markov approximation is valid for fixed coupling strength and for all times. We also construct a new approximate markovian dynamics -- a completely positive, trace preserving semigroup -- which is asymptotically in time exact, to all orders in the coupling.
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.