Canonical form of master equations and characterization of non-Markovianity

By Michael J. W. Hall, J. D. Cresser, Li Li, Erika Andersson. 2010

303 citations recorded by the source corpus.

Master equations govern the time evolution of a quantum system interacting with an environment, and may be written in a variety of forms. Time-independent or memoryless master equations, in particular, can be cast in the well-known Lindblad form. Any time-local master equation, Markovian or non-Markovian, may in fact also be written in a Lindblad-like form. A diagonalization procedure results in a unique, and in this sense canonical, representation of the equation, which may be used to fully characterize the non-Markovianity of the time evolution. Recently, several different measures of non-Markovianity have been presented which reflect, to varying degrees, the appearance of negative decoherence rates in the Lindblad-like form of the master equation. We therefore propose using the negative decoherence rates themselves, as they appear in the canonical form of the master equation, to completely characterize non-Markovianity. The advantages of this are especially apparent when more than one decoherence channel is present. We show that a measure proposed by Rivas et al. [Phys. Rev. Lett. 105, 050403 (2010)] is a surprisingly simple function of the canonical decoherence rates, and give an example of a master equation that is non-Markovian for all times t>0, but to which nearly all proposed measures are blind. We also give necessary and sufficient conditions for trace distance and volume measures to witness non-Markovianity, in terms of the Bloch damping matrix.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/2462bb80f39c9aa18636fd932178fca56d2f4dfd

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Citations in the corpus

Concepts of quantum non-Markovianity: A hierarchy (2017, methodology): “These two approaches are in fact hierarchically related: quantum Markovianity in the sense of divisibility implies Markovianity in the sense of decreasing system-state distinguishability, but not vice versa [13, 14, 22].”
Colloquium: Non-Markovian dynamics in open quantum systems (2015, methodology): “Note that this can be the case even if a decoherence rate stays negative at all times as in Hall et al. (2014) and Vacchini et al. (2011).”
Dynamics of non-Markovian open quantum systems (2015, methodology): “Moreover, according to (Breuer et al., 2015; Hall et al., 2014), a time-local master equation is Markovian if an only if the canonical decoherence rates are positive during the whole evolution.”