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arXiv:1603.00248 · Physical Review A

Class of exact memory-kernel master equations

Salvatore Lorenzo, Francesco Ciccarello, G. Massimo Palma

Published May 16, 2016• 52 citations
arXivHTMLPDF

Abstract

A well-known situation in which a non-Markovian dynamics of an open quantum system $S$ arises is when this is coherently coupled to an auxiliary system $M$ in contact with a Markovian bath. In such cases, while the joint dynamics of $S\text{\ensuremath{-}}M$ is Markovian and obeys a standard (bipartite) Lindblad-type master equation (ME), this is in general not true for the reduced dynamics of $S$. Furthermore, there are several instances (e.g., the dissipative Jaynes-Cummings model) in which a closed ME for the $S$'s state cannot even be worked out. Here, we find a class of bipartite Lindblad-type MEs such that the reduced ME of $S$ can be derived exactly and in a closed form for any initial product state of $S\text{\ensuremath{-}}M$. We provide a detailed microscopic derivation of our result in terms of a mapping between two collision models.

DOI
10.1103/physreva.93.052111

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A well-known situation in which a non-Markovian dynamics of an open quantum system SSS arises is when this is coherently coupled to an auxiliary system MMM in contact with a Markovian bath. In such cases, while the joint dynamics of S\ensuremath-MS\text{\ensuremath{-}}MS\ensuremath-M is Markovian and obeys a standard (bipartite) Lindblad-type master equation (ME), this is in general not true for the reduced dynamics of SSS. Furthermore, there are several instances (e.g., the dissipative Jaynes-Cummings model) in which a closed ME for the SSS's state cannot even be worked out. Here, we find a class of bipartite Lindblad-type MEs such that the reduced ME of SSS can be derived exactly and in a closed form for any initial product state of S\ensuremath-MS\text{\ensuremath{-}}MS\ensuremath-M. We provide a detailed microscopic derivation of our result in terms of a mapping between two collision models.
We have recently presented a collision-model-based framework to tackle non-Markovian quantum dynamics (Ciccarello et al 2012 arXiv:1207.6554). As a distinctive feature, memory is introduced in a dynamical way by adding extra inter-ancillary collisions to a standard (memoryless) collision model. In this paper, we focus on the case where such intra-bath collisions are described by incoherent partial swap operations. After briefly reviewing the model, we show how to include temperature as an additional parameter by relaxing the assumption that each bath ancilla is initially in a pure state. We also calculate explicitly the dynamical map entailed by the master equation in the paradigmatic instance of a Jaynes–Cummings system–ancilla coupling.
Non-Markovian master equations describe general open quantum systems when no approximation is made. We provide the exact closed master equation for the class of Gaussian, completely positive, trace preserving, non-Markovian dynamics. This very general result allows us to investigate a vast variety of physical systems. We show that the master equation for non-Markovian quantum Brownian motion is a particular case of our general result. Furthermore, we derive the master equation unraveled by a non-Markovian, dissipative stochastic Schrödinger equation, paving the way for the analysis of dissipative non-Markovian collapse models.
Away from equilibrium, the properties of open quantum systems depend on the details of their environment. A microscopic derivation of a master equation (ME) is therefore crucial. Of particular interest are Lindblad-type equations, not only because they provide the most general class of Markovian MEs, but also since they are the starting point for efficient quantum trajectory simulations. Lindblad-type MEs are commonly derived from the Born-Markov-Redfield equation via a rotating-wave approximation (RWA). However the RWA is valid only for ultraweak system-bath coupling and often fails to accurately describe nonequilibrium processes. Here we derive an alternative Lindbladian approximation to the Redfield equation, which does not rely on ultraweak system-bath coupling. Applying it to an extended Hubbard model coupled to Ohmic baths, we show that, especially away from equilibrium, it provides a good approximation in large parameter regimes where the RWA fails.
We provide the exact non-Markovian master equation for a two-level system interacting with a thermal bosonic bath, and we write the solution of such a master equation in terms of the Bloch vector. We show that previous approximated results are particular limits of our exact master equation. We generalize these results to more complex systems involving an arbitrary number of two-level systems coupled to different thermal baths, providing the exact master equations also for these systems. As an example of this general case we derive the master equation for the Jaynes-Cummings model.
We consider the exact reduced dynamics of a two-level system coupled to a bosonic reservoir, further obtaining the exact time-convolutionless and Nakajima-Zwanzig non-Markovian equations of motion. The system considered includes the damped and undamped Jaynes-Cummings model. The result is obtained by exploiting an expression of quantum maps in terms of matrices and a simple relation between the time evolution map and the time-convolutionless generator as well as the Nakajima-Zwanzig memory kernel. This nonperturbative treatment shows that each operator contribution in Lindblad form appearing in the exact time-convolutionless master equation is multiplied by a different time-dependent function. Similarly, in the Nakajima-Zwanzig master equation each such contribution is convoluted with a different memory kernel. It appears that, depending on the state of the environment, the operator structures of the two sets of equations of motion can exhibit important differences.
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.