PHYSICAL REVIEW A 90, 042110 (2014) Time-driven quantum master equations and their compatibility with the fluctuation dissipation theorem J´ erˆ ome Flakowski, Maksym Osmanov, David Taj, * and Hans Christian ¨ Ottinger Polymer Physics, Department of Materials, Eidgen¨ ossische Technische Hochschule Z¨ urich, CH-8093 Z¨ urich, Switzerland (Received 16 June 2014; published 17 October 2014) 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. ¨ Ottinger, Europhys. Lett. 94, 10006 (2011)]. DOI: 10.1103/PhysRevA.90.042110 PACS number(s): 03.65.Yz, 05.70.Ln, 02.50.Ga I. INTRODUCTION Quantum Markovian master equations [1–3] have been shown to be of paramount importance in modeling decoher- ence and energy relaxation for open quantum systems for over 50 years [4–8]. Surely the most commonly found and adopted master equations are of linear type, in which case they are often required to have a Lindblad form for positivity requirements [3]. One of the reasons for this is that the Davies master equation (DME), which is of Lindblad type, can be obtained as the weak-coupling limit of a Hamiltonian model [9] and is also linked to the celebrated Fermi golden rule [10,11]. However, there has been much dispute around the physical consistency of a Lindblad dynamics for quantum dissipative master equations in the past [12–17], which is still ongoing at present [18–22]. In particular, focus was directed towards the linear response under a time-dependent external perturbation, where the Lindblad dynamics was argued to fail in providing the fluctuation dissipation theorem (FDT) with Kubo canonical correlations [23,24]. This led to the Lindblad master equation being ruled out as a physically acceptable dynamics. The claimed failure of the FDT led some authors to consider radically new alternative master equations [12,13]. In par- ticular, the weak-coupling limit procedure was considerably revised by Grabert in his book [12]. Therein [Chap. 5.4, Eq. (5.4.48)]. Grabert claimed to have discovered a different linear master equation (GME), able to properly capture the physics of the weak coupling. The GME was argued not to be of the DME type and to be compatible with the FDT. Also, an important regime different from that of weak coupling was explored: the Brownian motion regime [13]. This was shown to lead to yet another kind of master equation, acting non- linearly on the density matrix. The nonlinear Grabert master * david.taj@mat.ethz.ch equation was indeed originally proposed by Grabert in [13] and then independently rediscovered and further generalized with the nonlinear thermodynamic master equation (NLTME) in the case of a nonequilibrium bath [19,25]. As opposed to the DME, the latter equation was argued to be consistent with the FDT [13]. The claimed failure of the FDT for the DME is however at variance with a number of recent works (see, e.g., [22,26,27]), and naturally demands to be clarified. This is the main motivation for the present work. In this paper we study the linear response to a time-driven perturbation of a quantum master equation, for a state initially at thermal equilibrium. We present two simple criteria on the perturbation, reported in Eqs. (17) and (19), which guarantee that the dynamics, possibly even nonlinear, is compatible with the FDT under Kubo-structured correlations. We use those criteria to analyze the validity of the FDT theorem for the following three dynamics: the DME, GME, and NLTME. Our criteria for the perturbation are sufficient, but not necessary, to obtain the FDT and model adiabatic drivings for dissipative quantum master equations. Our main result is that the DME is indeed compatible with the FDT, and we argue that the claimed flaw of the DME lies in an inappropriate choice of the perturbation scheme, which modifies only the Hamiltonian part of the irreversible dynamics. To further confirm this, and this is our second result, we show that the GME coincides with the DME. This supports our main result, as Grabert himself in [12] was able to propose a perturbation scheme for the GME which is compatible with the FDT. More directly, we also offer a natural perturbation scheme for the DME which fulfills our criteria on the perturbation, thus proving the FDT for the DME. As a third result, we propose a natural perturbation scheme for the NLTME, which fulfills our criteria and thus guarantees the FDT for the NLTME, in agreement with what was first proven by Grabert in [13]. This shows that our criteria on the perturbation schemes are able to account for potentially 1050-2947/2014/90(4)/042110(10) 042110-1 ©2014 American Physical Society