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arXiv:2104.14390 · Chaos An Interdisciplinary Journal of Nonlinear Science

On the hybrid Davies like generator for quantum dissipation

Dariusz Chruściński

Published Feb 1, 2021• 5 citations
arXivHTMLPDF

Abstract

We provide a class of quantum evolution beyond Markovian semigroup. This class is governed by a hybrid Davies like generator such that dissipation is controlled by a suitable memory kernel and decoherence by standard Gorini-Kossakowski-Lindblad-Sudarshan generator. These two processes commute and both of them commute with the unitary evolution controlled by the systems Hamiltonian. The corresponding memory kernel gives rise to semi-Markov evolution of the diagonal elements of the density matrix. However, the corresponding evolution needs not be completely positive. The role of decoherence generator is to restore complete positivity. Hence, to pose the dynamical problem, one needs two processes generated by classical semi-Markov memory kernel and purely quantum decoherence generator. This scheme is illustrated for a qubit evolution.

DOI
10.1063/5.0036620

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We provide a class of quantum evolution beyond Markovian semigroup. This class is governed by a hybrid Davies like generator such that dissipation is controlled by a suitable memory kernel and decoherence by standard Gorini-Kossakowski-Lindblad-Sudarshan generator. These two processes commute and both of them commute with the unitary evolution controlled by the systems Hamiltonian. The corresponding memory kernel gives rise to semi-Markov evolution of the diagonal elements of the density matrix. However, the corresponding evolution needs not be completely positive. The role of decoherence generator is to restore complete positivity. Hence, to pose the dynamical problem, one needs two processes generated by classical semi-Markov memory kernel and purely quantum decoherence generator. This scheme is illustrated for a qubit evolution.
In this paper we analyze the evolution of generalized Pauli channels governed by the memory kernel master equation. We provide necessary and sufficient conditions for the memory kernel to give rise to legitimate (completely positive and trace-preserving) quantum evolution. In particular, we analyze a class of kernels generating the quantum semi-Markov evolution, which is a natural generalization of the Markovian semigroup. Interestingly, the convex combination of Markovian semigroups goes beyond the semi-Markov case. Our analysis is illustrated with several examples.
We provide a large class of quantum evolutions governed by the memory kernel master equation. This class defines a quantum analog of so-called semi-Markov classical stochastic dynamics. In this paper we provide a precise definition of quantum semi-Markov evolution, and using the appropriate gauge freedom we propose a suitable generalization which contains a majority of examples considered so far in the literature. The key concepts are quantum counterparts of classical waiting time distribution and survival probability---a quantum waiting time operator and a quantum survival operator, respectively. In particular collision models and their generalizations considered recently are special examples of generalized semi-Markov evolution. This approach allows for an interesting generalization of the trajectory description of the quantum dynamics in terms of positive operator-valued measure densities.
Semi-Markov processes represent a well-known and widely used class of random processes in classical probability theory. Here, we develop an extension of this type of non-Markovian dynamics to the quantum regime. This extension is demonstrated to yield quantum master equations with memory kernels which allow the formulation of explicit conditions for the complete positivity of the corresponding quantum dynamical maps, thus leading to important insights into the structural characterization of the non-Markovian quantum dynamics of open systems. Explicit examples are analyzed in detail.
Abstract The convergence in probability of a sequence of iterations of independent random quantum dynamical semigroups to a Markov process describing the evolution of an open quantum system is studied. The statistical properties of the dynamics of open quantum systems with random generators of Markovian evolution are described in terms of the law of large numbers for operator-valued random processes. For compositions of independent random semigroups of completely positive operators, the convergence of mean values to a semigroup described by the Gorini–Kossakowski–Sudarshan–Lindblad equation is established. Moreover, a sequence of random operator-valued functions with values in the set of operators without the infinite divisibility property is shown to converge in probability to an operator-valued function with values in the set of infinitely divisible operators.
We investigate the mathematical structure of entropy-weighted quantum evolution within the framework of open quantum systems. Starting from a system–reservoir interaction model with weak entropy-coupled dynamics, we derive the reduced evolution of the system density matrix under the Born–Markov approximation. The resulting generatoris shown to be of Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) form, ensuring complete positivity and trace preservation. We further demonstrate that the associated dynamical semigroup admits both Stinespring dilation and Kraus operator representations, establishing that entropy-weighted reduced dynamics can be embedded into a fully unitary system environment framework. The entropy operator appearing in the effective description is identified with the positive-semidefinite dissipative structure induced by reservoir corre lation functions.The construction does not modify standard quantum mechanics but provides a parameterization of dissipative generators in terms of entropy-weighted couplings. This establishes a mathematically consis tent bridge between entropy-based dynamical formulations and conventional open quantum system theory.
A well established approach to describe open quantum system dynamics is the celebrated Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation. In this article we aim to shed light on semigroup dynamics that are not of this form. Such dynamics may appear in real physical systems describing asymptotic long-time dynamics. Thus, they correspond to semigroup dynamics applicable to a restricted set of initial states only. As a way of motivating our studies, first we review known results about the single-mode Gaussian Quantum Brownian motion channel and corresponding master equations. Next we discuss a conceptually simpler family of qubit semigroup generators and identify parameter regions of completely positive, merely positive, and non-positive dynamics. We then focus on the latter case to which we will refer to as restricted-positive, as the semigroup generator may well be physically meaningful when applied to a restricted set of admissible initial states.