Thermodynamic consistency of quantum master equations

Abstract

Starting from a microscopic system-baths description, we derive the general conditions for a time-local quantum master equation (QME) to satisfy the first and second laws of thermodynamics at the fluctuating level. Using counting statistics, we show that the fluctuating second law can be rephrased as a generalized quantum detailed balance condition (GQDB), i.e., a symmetry of the time-local generators which ensures the validity of the fluctuation theorem. When requiring in addition a strict system-bath energy conservation, the GQDB reduces to the usual notion of detailed balance which characterizes QMEs with Gibbsian steady states. However, if energy conservation is only required on average, QMEs with non-Gibbsian steady states can still maintain a certain level of thermodynamic consistency. Applying our theory to commonly used QMEs, we show that the Redfield equation breaks the GQDB, and that some recently derived approximation schemes based on the Redfield equation (which hold beyond the secular approximation and allow one to derive a QME of Lindblad form) satisfy the GQDB and the average first law. We find that performing the secular approximation is the only way to ensure the first and second laws at the fluctuating level.

Methods and conversations in (post)modern thermodynamics

“…However, it is possible to relax the strict energy conservation condition (10.63) and to maintain a thermodynamic consistency on average, provided that the generalized detailed balance condition (10.71) is satisfied. This is for instance the case in the weak coupling limit, as illustrated in [181].…”

“…For the rest of this section, we assume that the system Hamiltonian ĤA is time-independent. The condition (10.71) is called the generalized quantum detailed balance condition [181].…”

“…Remark: In practice, many quantum master equations do not meet the strict energy conservation condition (10.80) or the generalized quantum detailed balance condition (10.71). In fact, in order to satisfy both these conditions, it is necessary to perform the secular approximation (we refer to the section 3.3 in [179] for the definition of the secular approximation); see [181] for a detailed proof. However, it is possible to relax the strict energy conservation condition (10.63) and to maintain a thermodynamic consistency on average, provided that the generalized detailed balance condition (10.71) is satisfied.…”

“…derivation, see the supplemental material in[181].To alleviate the notations, we dropped the t dependence of ρ0,λ B A (t). The term Ĥ′ 0,λ B (t) is the sum of the system Hamiltonian ĤA and of a Lamb shift contribution Ĥ0,λ BLS-a shift in the system's energies induced by the interaction with the bath -and D 0,λ B (t) describes the dissipation.…”

Unification of the first law of quantum thermodynamics

“…To illuminate the relation between the two we derive the semi-classical power from the autonomous definition. We begin the derivation by applying the semi-classical prescription, equations (34) and (35), in the interaction picture relative to the bare control Hamiltonian to calculate the semi-classical power. In this picture the effective Hamiltonian H(L) (t) = e i ĤCt/ℏ Ĥ(L) e −i ĤCt/ℏ , which together with equation (20)…”