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Virtues and Limitations of Markovian Master Equations with a Time-Dependent Generator

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Abstract

A Markovian master equation with time-dependent generator is constructed that respects basic constraints of quantum mechanics, in particular the von Neumann conditions. For the case of a two-level system, Bloch equations with time-dependent parameters are obtained. Necessary conditions on the latter are formulated. By employing a time-local counterpart of the Nakajima–Zwanzig equation, we establish a relation with unitary dynamics. We also discuss the relation with the weak-coupling limit. On the basis of a uniqueness theorem, a standard form for the generator of time-local master equations is proposed. The Jaynes–Cummings model with atomic damping is solved. The solution explicitly demonstrates that reduced dynamics can be described by time-local master equations only on a finite time interval. This limitation is caused by divergencies in the generator. A limit of maximum entropy is presented that corroborates the foregoing statements. A second limiting case demonstrates that divergencies may even occur for small perturbations of the weak-coupling regime.

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Authors and Affiliations

  1. Instituut voor Theoretische Fysica, Universiteit van Amsterdam, NL-1018 XE, Amsterdam, The Netherlands

    A. J. van Wonderen

  2. Physikalisch-Chemisches Institut der Universität Zürich, CH-8057, Zürich, Switzerland

    K. Lendi

Authors
  1. A. J. van Wonderen
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  2. K. Lendi
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van Wonderen, A.J., Lendi, K. Virtues and Limitations of Markovian Master Equations with a Time-Dependent Generator. Journal of Statistical Physics 100, 633–658 (2000). https://doi.org/10.1023/A:1018671424739

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