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arXiv:1706.09025 · Open Systems & Information Dynamics

Quantum Non-Markovian Piecewise Dynamics from Collision Models

Salvatore Lorenzo, Francesco Ciccarello, G. Massimo Palma, Bassano Vacchini

Published Dec 1, 2017• 37 citations
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Abstract

Recently, a large class of quantum non-Markovian piecewise dynamics for an open quantum system obeying closed evolution equations has been introduced [1]. These dynamics have been defined in terms of a waiting-time distribution between quantum jumps, along with quantum maps describing the effect of jumps and the system evolution between them. Here, we present a quantum collision model with memory, whose reduced dynamics in the continuous-time limit reproduces the above class of non-Markovian piecewise dynamics, thus providing an explicit microscopic realization.

DOI
10.1142/s123016121740011x

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We construct a large class of completely positive and trace-preserving non-Markovian dynamical maps for an open quantum system. These maps arise from a piecewise dynamics for statistical operators characterized by a continuous time evolution interrupted by jumps, randomly distributed in time and described by a quantum channel. The state of the open system is shown to obey a closed evolution equation, given by a master equation with a memory kernel and an inhomogeneous term. The non-Markovianity of the obtained dynamics is explicitly assessed studying the behavior of the distinguishability of two different initial system's states with elapsing time.
Simple, controllable models play an important role in learning how to manipulate and control quantum resources. We focus here on quantum non-Markovianity and model the evolution of open quantum systems by quantum renewal processes. This class of quantum dynamics provides us with a phenomenological approach to characterise dynamics with a variety of non-Markovian behaviours, here described in terms of the trace distance between two reduced states. By adopting a trajectory picture for the open quantum system evolution, we analyse how non-Markovianity is influenced by the constituents defining the quantum renewal process, namely the time-continuous part of the dynamics, the type of jumps and the waiting time distributions. We focus not only on the mere value of the non-Markovianity measure, but also on how different features of the trace distance evolution are altered, including times and number of revivals.
We study the applicability of collisional models for non-Markovian dynamics of open quantum systems. By allowing interactions between the separate environmental degrees of freedom in between collisions we are able to construct a collision model that allows us to study quantum memory effects in open system dynamics. We also discuss the possibility to embed non-Markovian collision model dynamics into Markovian collision model dynamics in an extended state space. As a concrete example we show how, using the proposed class of collision models, we can discretely model non-Markovian amplitude damping of a qubit. In the time-continuous limit, we obtain the well-known results for spontaneous decay of a two-level system into a structured zero-temperature reservoir.
We present a theoretical framework to tackle quantum non-Markovian dynamics based on a microscopic collision model (CM), where the bath consists of a large collection of initially uncorrelated ancillas. Unlike standard memoryless CMs, we endow the bath with memory by introducing interancillary collisions between next system-ancilla interactions. Our model interpolates between a fully Markovian dynamics and the continuous interaction of the system with a single ancilla, i.e., a strongly non-Markovian process. We show that in the continuous limit one can derive a general master equation, which, while keeping such features, is guaranteed to describe an unconditionally completely positive and trace-preserving dynamics. We apply our theory to an atom in a dissipative cavity for a Lorentzian spectral density of bath modes, a dynamics which can be exactly solved. The predicted evolution shows a significant improvement in approaching the exact solution with respect to two well-known memory-kernel master equations.
Simulation methods based on stochastic realizations of state vector evolutions are commonly used tools to solve open quantum system dynamics, both in the Markovian and non-Markovian regimes. Here, we address the question of the waiting time distribution (WTD) of quantum jumps for non-Markovian systems. We generalize Markovian quantum trajectory methods in the sense of deriving an exact analytical WTD for non-Markovian quantum dynamics and show explicitly how to construct this distribution for certain commonly used quantum optical systems.
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.