  We develop a time-dependent Schrieffer-Wolff-Lindblad perturbation theory to
study effective interactions for driven open quantum systems. The starting
point of our analysis is a given Lindblad equation, based on which we obtain an
effective (averaged) map that describes the renormalization of both the
Hamiltonian and collapse operators due to the drive. As a case study, we apply
this method to the dispersive readout of a transmon qubit and derive an
effective disperive map that describes measurement-induced dephasing and Stark
shift for the transmon. The effective map we derive is completely positive and
trace-preserving under adiabatic resonator response. To benchmark our method,
we demonstrate good agreement with a numerical computation of the effective
rates via the Lindbladian spectrum. Our results are also in agreement with, and
extend upon, an earlier derivation of such effects by Gambetta et al. (Phys.
Rev. A 74, 042318 (2006)) using the positive P-representation for the resonator
field.
