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arXiv:quant-ph/0410161 · Open Systems & Information Dynamics

Description of Quantum Dynamics of Open Systems Based on Collision-Like Models

Mário Ziman, Peter Štelmachovič, Vladimír Bužek

Published Mar 1, 2005• 112 citations
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Abstract

Master equations in the Lindblad form describe evolution of open quantum systems that are completely positive and simultaneously have a semigroup property. We analyze the possibility to derive this type of master equations from an intrinsically discrete dynamics that is modelled as a sequence of collisions between a given quantum system (a qubit) with particles that form the environment. In order to illustrate our approach we analyze in detail how the process of an exponential decay and the process of decoherence can be derived from a collision-like model in which particular collisions are described by SWAP and controlled-NOT interactions, respectively.

DOI
10.1007/s11080-005-0488-0

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Master equations in the Lindblad form describe evolution of open quantum systems that are completely positive and simultaneously have a semigroup property. We analyze the possibility to derive this type of master equations from an intrinsically discrete dynamics that is modelled as a sequence of collisions between a given quantum system (a qubit) with particles that form the environment. In order to illustrate our approach we analyze in detail how the process of an exponential decay and the process of decoherence can be derived from a collision-like model in which particular collisions are described by SWAP and controlled-NOT interactions, respectively.
A simple collision model is employed to introduce elementary concepts of open system dynamics of quantum systems. In particular, within the framework of collision models we introduce the quantum analogue of thermalization process called quantum homogenization and simulate quantum decoherence processes. These dynamics are driven by partial swaps and controlled unitary collisions, respectively. We show that collision models can be used to prepare multipartite entangled states. Partial swap dynamics generates W-type of entanglement saturating the CKW inequalities, whereas the decoherence collision models creates GHZ-type of entangled states. The considered evolution of a system in a sequence of collisions is described by a discrete semigroup E_1,...,E_n. Interpolating this discrete points within the set of quantum channels we derive for both processes the corresponding Lindblad master equations. In particular, we argue that collision models can be used as simulators of arbitrary Markovian dynamics, however, the inverse is not true.
We present the probability-preserving description of the decaying particle within the framework of quantum mechanics of open systems, taking into account the superselection rule prohibiting the superposition of the particle and vacuum. In our approach the evolution of the system is given by a family of completely positive trace-preserving maps forming a one-parameter dynamical semigroup. We give the Kraus representation for the general evolution of such systems, which allows one to write the evolution for systems with two or more particles. Moreover, we show that the decay of the particle can be regarded as a Markov process by finding explicitly the master equation in the Lindblad form. We also show that there are remarkable restrictions on the possible strength of decoherence.
We investigate decoherence channels that are modeled as a sequence of collisions of a quantum system (e.g., a qubit) with particles (e.g., qubits) of the environment. We show that collisions induce decoherence when a bipartite interaction between the system qubit and an environment (reservoir) qubit is described by the controlled-UUU unitary transformation (gate). We characterize decoherence channels and in the case of a qubit we specify the most general decoherence channel and derive a corresponding master equation. Finally, we analyze entanglement that is generated during the process of decoherence between the system and its environment.
We derive the general form of a master equation describing the reduced time evolution of a sequence of subsystems "propagating" in an environment which can be described as a sequence of subenvironments. The interaction between subsystems and subenvironments is described in terms of a collision model, with the irreversible dynamics of the subenvironments between collisions explicitly taken into account. In the weak coupling regime, we show that the collisional model produces a correlated Markovian evolution for the joint density matrix of the multipartite system. The associated Lindblad superoperator contains pairwise terms describing cross correlation between the different subsystems. Such a model can describe a broad range of physical situations, ranging from quantum channels with memory to photon propagation in concatenated quantum optical systems.
The majority of quantum open system models in the literature are simplistic in the sense that they only explicitly account for that part of the environment that directly interacts with the system of interest. A quantum open system with an open environment is examined using the projection operator method in the weak coupling limit. The openness of environment is modelled by nonunitary evolution of the Lindblad form. Under certain conditions, the resulting master equation for the system is insensitive to the initial state of the environment and to initial entanglements between the system and environment for time scales greater than the environment relaxation timescales. For the particular case of an environment consisting of a harmonic oscillator bath, the resulting master equations are demonstrated to have the algebraic form for completely positive evolution. The open environment model is illustrated for the particular case of a system linearly coupled to an oscillator bath.
The Lindblad master equation is widely used to describe the reduced dynamics of open quantum systems under Markovian assumptions. Here, we investigate its ability to reproduce the reduced evolution emerging from a microscopic many-body model in which two interacting two-level subsystems are embedded in a larger environment and evolve under fully unitary dynamics. The exact evolution exhibits a clear separation of timescales. At short times, decoherence arises from environmentally induced dephasing, leading to a Gaussian suppression of coherences and a quadratic decay of purity. At intermediate times, collective decoherence channels saturate and a slower, still Gaussian, decay driven by relative environmental fluctuations dominates. At later times the system settles in a complete decohered state. The first two behaviors cannot be reproduced by a Lindblad dynamics with constant coefficients, which always results in an exponential decay: Our work provides a simple example of the breakdown of the effective description relevant in many realistic settings.