Modular Dynamical Semigroups for Quantum Dissipative Systems

Abstract: We introduce a class of Markovian quantum master equations, able to describe the dissipative dynamics of a quantum system weakly coupled to one or several heat baths. The dissipative structure is driven by an entropic operator, the so called modular Hamiltonian, which makes it nonlinear. The generated Modular Dynamical Semigroup (MDS) is not, in general, a Quantum Dynamical Semigroup (QDS), whose dynamics is of the popular Lindblad type. The MDS has a robust thermodynamic structure, which guarantees for the positivity of the time evolved state, the correct steady state properties, the positivity of the entropy production, a positive Onsager matrix and Onsager symmetry relations (arising from Green-Kubo formulas). We show that the celebrated Davies generator, obtained through the Born and the secular approximations, generates a MDS. By unravelling the modular structure of the former, we provide a different and genuinely nonlinear MDS, not of QDS type, which is free from the severe spectral restrictions of the Davies generator, while still being supported by a weak coupling limit argument. With respect to the latter, the present work is a substantial extension of \cite{Ottinger2011_GEO,Ottinger2010_TLS_DHO}