Fig. 3: Cancellation of the ac-Stark shift and measurement-induced dephasing.

a Numerical simulation of a two-level qubit spectral density without cancellation (top) and with cancellation (bottom) for various drive amplitudes ε₁/2π from 10 to 60 MHz. The simulation parameters are ω_q/2π = 5.7 GHz, ω_r,1/2π = 7.6 GHz, \({g}^{{\prime} }/2\pi=200\) MHz, and κ/2π = 50 MHz. b Dots: Experimentally measured increased dephasing rate δΓ (top) and qubit frequency shift δω (bottom) as a function of drive power extracted from Ramsey interferometry without cancellation (blue) and with cancellation (red). Error bars are statistical. Solid lines: Numerical simulations of the measurement-induced dephasing and ac-Stark shift. The simulation which accounts exactly for the \(\cos {\hat{\varphi }}_{{{{{{{{\rm{tr}}}}}}}}}\) potential of the Josephson junction is performed using the bare parameters ω_r/2π = 7.66 GHz, E_J/h = 16.83 GHz, E_C/h = 199.7 MHz, g/2π = 140.6 MHz, and κ/2π = 10.1 MHz resulting in dressed parameters that match the experimentally measured ones. The cavity drive frequency is ω₁/2π = 7.6648 GHz.