Time-driven quantum master equations and their compatibility with the fluctuation dissipation theorem

We contribute to a long-standing debate on the supposed failure of the fluctuation dissipation theorem (FDT) for the Davies master equation (DME), an important class of Lindblad quantum master equations, describing time-driven quantum systems weakly coupled to a heat bath. First we propose two simple and natural criteria on the driving which guarantee compatibility with the FDT. We show through our setting that, contrary to what is often stated in the literature, the DME is fully compatible with the FDT. We thus argue that the cause of the dispute lies in the adopted perturbation scheme, rather than in the Lindblad character of the master equation itself. We confirm our statement by proving that the Grabert master equation, first proposed by Grabert [Projection Operator Techniques in Nonequilibrium Statistical Mechanics (Springer, Berlin, 1982)] as an alternative linear dynamics fulfilling the FDT, is nothing else than the incriminated DME. Our criteria for the FDT can also be used in the analysis of the nonlinear thermodynamical master equation, first obtained in the Brownian motion limit [H. Grabert, Z. Phys. B 49, 161 (1982)] and later independently rediscovered and generalized on purely thermodynamic grounds [H. C. Öttinger, Europhys. Lett. 94, 10006 (2011)].

Figure 1

Pictorial representation of the link between the different sets of examined master equations. Davies master equations (DMEs) are Lindblad equations often encountered in the weak-coupling-limit procedure and are defined in Eqs. (31), (35), and (37). The Grabert master equation (GME) is Eq. (5.4.48) of [12] [here reported in Eqs. (40) and (41)], and it is a linear master equation argued therein to fulfill the FDT, unlike the DME. The nonlinear thermodynamic master equation (NLTME) was first found by Grabert as arising from the Brownian motion limit, and is here reported in Eq. (54), and was then independently rediscovered and generalized out of equilibrium in [25].