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

TL;DR

This paper introduces a time-dependent perturbation theory for open quantum systems to analyze measurement-induced effects in dispersive qubit readout, providing a new effective map that captures dephasing and Stark shifts.

Contribution

The authors develop a novel time-dependent Schrieffer-Wolff-Lindblad perturbation theory for driven open quantum systems, enabling accurate effective descriptions of measurement-induced effects.

Findings

Effective map describes measurement-induced dephasing and Stark shift.

Good agreement with numerical Lindbladian spectrum calculations.

Extends previous results by Gambetta et al. to a time-dependent framework.

Abstract

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.