Universal dissipators in master equations for weakly driven systems

We present analytical results in the time-convolutionless-master-equation description of weakly driven open quantum systems. More specifically, we treat in parallel two common pictures used to describe quantum systems interacting with their surrounding environment, i.e. the stochastic Hamiltonian description, where the environment is implicitly included in the fluctuating internal parameters of the system, or the explicit inclusion of the environment via the projection operator method. Utilizing these two different frameworks, we show that the dissipator characterising the reduced system dissipative dynamics, determined up to second order in the noise strength or system-bath coupling with the cumulant expansion method, is universal. This means that it assumes the same form, irrespective of the drive term, when this is weak. We discuss thoroughly the assumptions on which this treatment relies and its limitations. Furthermore, we consider higher orders in the cumulant expansion and discuss their role. In particular, the third order extension leads to a renormalization of the drive. As a relevant example, we apply this formalism to the problem of dephasing in driven qubits.