arXiv:0908.0769 · Physical Review A

Non-Markovian nonstationary completely positive open-quantum-system dynamics

Adrián A. Budini, Paolo Grigolini A

Published Aug 4, 2009• 33 citations

Abstract

By modeling the interaction of a system with an environment through a renewal approach, we demonstrate that completely positive non-Markovian dynamics may develop some unexplored nonstandard statistical properties. The renewal approach is defined by a set of disruptive events, consisting in the action of a completely positive superoperator over the system density matrix. The random time intervals between events are described by an arbitrary waiting-time distribution. We show that, in contrast to the Markovian case, if one performs a system preparation (measurement) at an arbitrary time, the subsequent evolution of the density-matrix evolution is modified. The nonstationary character refers to the absence of an asymptotic master equation even when the preparation is performed at arbitrary long times. In spite this property, we demonstrate that operator expectation values and operators correlations have the same dynamical structure, establishing the validity of a nonstationary quantum regression hypothesis. The nonstationary property of the dynamics is also analyzed through the response of the system to an external weak perturbation.

DOI: 10.1103/physreva.80.022103

Related papers

Stochastic representation of a class of non-Markovian completely positive evolutions

Adrián A. Budini

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.

arXiv:1801.10011Physical Review AApr 21, 2004102 citations

Non-Markovian correlation functions for open quantum systems

Jinshuang Jin, Christian Karlewski, Michael Marthaler

Beyond the conventional quantum regression theorem, a general formula for non-Markovian correlation functions of arbitrary system operators both in the time- and frequency-domain is given. Weapproach the problem by transforming the conventional time-non-local master equation into dispersed time-local equations-of-motion. The validity of our approximations is discussed and we find that the non-Markovian terms have to be included for short times. While calculations of the density matrix at short times suffer from the initial value problem, a correlation function has a well defined initial state. The resulting formula for the non-Markovian correlation function has a simple structure and is as convenient in its application as the conventional quantum regression theorem for the Markovian case. For illustrations, we apply our method to investigate the spectrum of the current fluctuations of interacting quantum dots contacted with two electrodes. The corresponding non- Markovian characteristics are demonstrated.

Jan 1, 20163 citations

Non-Markovian correlation functions for open quantum systems

Jinshuang Jin, Christian Karlewski, Michael Marthaler

Beyond the conventional quantum regression theorem, a general formula for non-Markovian correlation functions of arbitrary system operators both in the time- and frequency-domain is given. We approach the problem by transforming the conventional time-non-local master equation into dispersed time-local equations-of-motion. The validity of our approximations is discussed and we find that the non-Markovian terms have to be included for short times. While calculations of the density matrix at short times suffer from the initial value problem, a correlation function has a well defined initial state. The resulting formula for the non-Markovian correlation function has a simple structure and is as convenient in its application as the conventional quantum regression theorem for the Markovian case. For illustrations, we apply our method to investigate the spectrum of the current fluctuations of interacting quantum dots contacted with two electrodes. The corresponding non-Markovian characteristics are demonstrated.

arXiv:1512.01906New Journal of PhysicsAug 18, 20166 citations

Non-Markovian master equations from piecewise dynamics

Bassano Vacchini

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.

arXiv:1301.2902Physical Review AMar 8, 201361 citations

Embedding non-Markovian quantum collisional models into bipartite Markovian dynamics

Adrián A. Budini

A wide class of non-Markovian completely positive master equations can be formulated on the basis of quantum collisional models. In this phenomenological approach the dynamics of an open quantum system is modeled through an ensemble of stochastic realizations that consist in the application at random times of a (collisional) completely positive transformation over the system state. In this paper, we demonstrate that these kinds of models can be embedded in bipartite Markovian-Lindblad dynamics consisting of the system of interest and an auxiliary one. In contrast with phenomenological formulations, here the stochastic ensemble dynamics an the interevent time interval statistics are obtained from a quantum measurement theory after assuming that the auxiliary system is continuously monitored in time. Models where the system intercollisional dynamics is non-Markovian [B. Vacchini, Phys. Rev. A 87, 030101(R) (2013)] are also obtained from the present approach. The formalism is exemplified through bipartite dynamics that lead to non-Markovian system effects such as an environment-to-system backflow of information.

arXiv:1309.0122Physical Review ASep 25, 201339 citations

Completely-positive non-signalling non-Markovian dynamics

Serhii Kryhin, Vivishek Sudhir

We define non-Markovian quantum dynamics as evolution in which the current state depends on all past states, and completely characterize its structure under the assumptions of complete positivity and non-signalling. The resulting continuous-time dynamics is an integro-differential equation that augments the Gorini-Kossakowski-Sudarshan-Lindblad equation with a memory integral, and is capable of describing the quantum state of systems exposed to noise with any integrable power spectral density with no further approximations. We then establish a formalism to evaluate multi-time correlations of measurement outcomes in this general setting, obviating the need for a regression theorem. As an application, we derive the emission spectrum of a driven two-level system coupled to a non-Markovian bath: the familiar Mollow triplet acquires a frequency-dependent linewidth that encodes the memory of the bath. Our work provides a rigorous yet transparent description of the quantum state of non-Markovian systems, opening the door for state estimation and state-based quantum control beyond the Markovian regime.

quant-pharXiv:2605.03197May 6, 20260 citations