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

Marco Cattaneo*, Gian Luca Giorgi, Sabrina Maniscalco, Roberta Zambrini

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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We address the structure of the Liouvillian superoperator for a broad class of bosonic and fermionic Markovian open systems interacting with stationary environments. We show that the accurate application of the partial secular approximation in the derivation of the Bloch-Redfield master equation naturally induces a symmetry on the superoperator level, which may greatly reduce the complexity of the master equation by decomposing the Liouvillian superoperator into independent blocks. Moreover, we prove that, if the steady state of the system is unique, one single block contains all the information about it, and that this imposes a constraint on the possible steady-state coherences of the unique state, ruling out some of them. To provide some examples, we show how the symmetry appears for two coupled spins interacting with separate baths, as well as for two harmonic oscillators immersed in a common environment. In both cases the standard derivation and solution of the master equation is simplified, as well as the search for the steady state. The block diagonalization does not appear when a local master equation is chosen.

Original languageEnglish
Article number042108
Number of pages15
JournalPhysical Review A
Issue number4
Publication statusPublished - 13 Apr 2020
MoE publication typeA1 Journal article-refereed

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