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Master equation for non-Markovian quantum Brownian motion: The emergence of lateral coherences

Sapphire Lally1,*, Nicholas Werren1,2,*,†, Jim Al-Khalili1, and Andrea Rocco1,2,‡

  • 1Department of Physics, University of Surrey, GU2 7XH Guildford, United Kingdom
  • 2Department of Microbial Sciences, University of Surrey, GU2 7XH Guildford, United Kingdom

  • *These authors contributed equally to this work.
  • †Present address: Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, United Kingdom.
  • ‡a.rocco@surrey.ac.uk
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Phys. Rev. A 105, 012209 – Published 18 January, 2022

DOI: https://doi.org/10.1103/PhysRevA.105.012209

Abstract

Understanding the behavior of a quantum system coupled to its environment is of fundamental interest in the general field of quantum technologies. It also has important repercussions on foundational problems in physics, such as the process of decoherence and the so-called quantum measurement problem. There have been many approaches to explore Markovian and non-Markovian dynamics within the framework of open quantum systems, but the richness of the ensuing dynamics is still not fully understood. In this paper we develop a non-Markovian extension of the standard Caldeira-Leggett model, based on expanding the dynamics of the reduced system at high temperature in inverse powers of the high-frequency cutoff of the Ohmic spectral density of the environment and derive a non-Markovian master equation for the reduced density matrix for the case of a general potential. We also obtain a fully analytical solution in the free particle case. While the short-time behavior of this solution does not diverge substantially from the Markovian behavior, at intermediate times we find a resurgence of coherence, which we name lateral coherence. We identify this with a corresponding transient negative entropy production rate, which is understood to be characteristic of non-Markovian dynamics. We also analyze the positivity of the reduced density matrix and derive the corresponding Fokker-Planck equation in the classical limit.

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Phys. Rev. A 105, 012209– Published 18 January, 2022
  • Received 3 July 2019
  • Revised 5 February 2020
  • Accepted 17 December 2021
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