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arXiv:quant-ph/0105065 · Physical Review A

Self-consistent non-Markovian theory of a quantum-state evolution for quantum-information processing

Doyeol Ahn, Jinhyoung Lee, M. S. Kim, Sung Woo Hwang

Published Jul 15, 2002• 46 citations
arXivHTMLPDF

Abstract

We study non-Markovian decoherence phenomena by employing projection-operator formalism when a quantum system (a quantum bit or a register of quantum bits) is coupled to a reservoir. By projecting out the degree of freedom of the reservoir, we derive a non-Markovian master equation for the system, which is reduced to a Lindblad master equation in Markovian limit, and obtain the operator sum representation for the time evolution. It is found that the system is decohered slower in the non-Markovian reservoir than the Markovian because the quantum information of the system is memorized in the non-Markovian reservoir. We discuss the potential importance of non-Markovian reservoirs for quantum-information processing.

DOI
10.1103/physreva.66.012302

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We study non-Markovian decoherence phenomena by employing projection-operator formalism when a quantum system (a quantum bit or a register of quantum bits) is coupled to a reservoir. By projecting out the degree of freedom of the reservoir, we derive a non-Markovian master equation for the system, which is reduced to a Lindblad master equation in Markovian limit, and obtain the operator sum representation for the time evolution. It is found that the system is decohered slower in the non-Markovian reservoir than the Markovian because the quantum information of the system is memorized in the non-Markovian reservoir. We discuss the potential importance of non-Markovian reservoirs for quantum-information processing.
We analyze the appearance of non-Markovian effects in the dynamics of a bipartite system coupled to a reservoir, which can be described within a class of non-Markovian equations given by a generalized Lindblad structure. A master equation is derived, which we term the quantum Bloch-Boltzmann equation, describing both motional and internal states of a test particle in a quantum framework. When due to the preparation of the system or to decoherence effects one of the two degrees of freedom is amenable to a classical treatment and not resolved in the final measurement, though relevant for the interaction with the reservoir, non-Markovian behaviors such as stretched exponential or power law decay of coherences can occur.
We present a non-Markovian master equation for a qubit interacting with a general reservoir, which is derived according to the Nakajima—Zwanzig and the time convolutionless projection operator technique. The non-Markovian solutions and Markovian solution of dynamical decay of a qubit are compared. The results indicate the validity of non-Markovian approach in different coupling regimes and also show that the Markovian master equation may not precisely describe the dynamics of an open quantum system in some situation. The non-Markovian solutions may be effective for many qubits independently interacting with the heated reservoirs.
This thesis investigates two thematic lines of research, both underpinned by non-Markovian system-reservoir interactions in quantum optics. The overarching focus is on modelling the open system dynamics in a non-perturbative fashion, broadly on - though not restricted to - instances when the environment is structured. A theory is developed by means of enlarging the open system over environmental degrees of freedom to include memory effects in its dynamics. This is achieved using an established technique that involves mapping a bosonic environment onto a 1D chain of harmonic oscillators. Within this setting, we apply a Heisenberg equation-of-motion approach to derive an exact set coupled differential equations for the open system and a single auxiliary oscillator of the chain. The combined equations are shown to have their interpretation rooted in a quantum Markov stochastic process. Including the auxiliary chain oscillator as part of the original system then enables us to obtain an exact master equation for the enlarged system, avoiding any need for the Born-Markov approximations. Our method is valid for a dissipative two-state system, with cases of multiple excitations and added driving discussed. Separately, we apply the framework of quantum Darwinism to an atom-cavity system, and, subsequently, to a more general multiple-environment model. In both cases, the time-dependent spread of correlations between the open system and fractions of the environment is analysed during the course of the decoherence process. The degree to which information is redundant across different fractions is checked to infer the emergence of classicality. In the second case, we go further and present a decomposition of information in terms of its quantum and classical correlations. A quantitative measure of redundancy is also studied with regard to its ability to witness non-Markovian behaviour. Besides fundamental interest, our results have application to quantum information processing and quantum technologies, keeping in mind the potential beneficial use of non-Markovian effects in reservoir engineering.
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In this review we discuss recent progress in the theory of open quantum systems based on non-MarkovianNon-Markovian quantum state diffusionQuantum state diffusion equation and master equationsMaster equation. In particular, we show that an exact master equation for an open quantum system consisting of a few qubits can be explicitly constructed by using the corresponding non-Markovian quantum state diffusion equation. The exact master equation arises naturally from the quantum decoherence dynamics of qubit systems collectively interacting with a colored noise. We illustrate our general theoretical formalism by the explicit construction of a three-qubit system coupled to a non-Markovian bosonic environment. This exact qubit master equation accurately characterizes the time evolution of the qubit system in various parameter domains, and paves the way for investigation of the memory effect of an open quantum system in a non-Markovian regime without any approximation.