Coarse-graining in the derivation of Markovian master equations and its significance in quantum thermodynamics

Abstract

The coarse-graining approach to deriving the quantum Markovian master equation is revisited, with close attention given to the underlying approximations. It is further argued that the time interval over which the coarse-graining is performed is a free parameter that can be given a physical measurement-based interpretation. In the case of the damping of composite systems to reservoirs of different temperatures, currently of much interest in the study of quantum thermal machines with regard to the validity of 'local' and 'global' forms of these equations, the coupling of the subsystems leads to a further timescale with respect to which the coarse-graining time interval can be chosen. Different choices lead to different master equations that correspond to the local and global forms. These can be then understood as having different physical interpretations based on the role of the coarse-graining, as well as different limitations in application.

Symmetry and block structure of the Liouvillian superoperator in partial secular approximation

“…( 7) is not fulfilled [37]. Nonetheless, some recent studies have shown that the PSA performed through a suitable coarse-graining does lead to a GKLS master equation [38,39], as can be also found in a previous work which did not mention the PSA [52]. This method of applying the PSA is analogous to the one used in the present paper, based on the condition in Eq.…”

“…in the case of two slightly-detuned spins [37], the Davies' limit cannot be performed, since it corresponds to applying a "full secular approximation" removing all the oscillating terms in the interaction picture dynamics without discriminating which of them are fast and which are slow, instead of a more accurate partial secular approximation. The latter was implicitly suggested by Redfield himself [33], and an extensive study about it has been performed in the very recent past [37][38][39][40][41], in particular showing that applying an accurate partial secular approximation to the microscopic derivation of the master equation allows one to recover the GKLS form [38,39].…”

Transient synchronization in open quantum systems

“…The weight of such terms is expected to vanish for large values of λ. We remark here that such a partial secular approximation still guarantees that the master equation has a Lindblad form [40,43,44]. It is clear from the structure of the master equation that synchronization can only take place if the dissipation rates of the two normal modes η 1 and η 2 are significantly different between each other and the modes themselves have a finite superposition over the two local spins.…”

Going beyond Local and Global approaches for localized thermal dissipation

“…Allowing for mathematical structures that do not comply with such requirement paves the way to a series of inconsistencies that include negative probabilities of measurements outcomes, violation of the uncertainty relation, and ultimately the non-contractive character of the underlying dynamics. Ways to correct or to circumvent the pathology exhibited by the Redfield equation typically relay on the full [8] or the partial [20][21][22][23][24][25] implementation of the secular approximation: a coarsegrain temporal average of the system dynamics which, performed in conjunction with the above mentioned Born and Markov approximations, leads to a more reliable differential equation for the system density matrix known as the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation [5,6].…”