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

Time-dependent Schrieffer-Wolff-Lindblad Perturbation Theory: measurement-induced dephasing and second-order Stark shift in dispersive readout