Nonlinear microwave photon-occupancy of a driven resonator strongly coupled to a transmon qubit

We measure photon-occupancy in a thin-film superconducting lumped element resonator coupled to a transmon qubit at 20 mK and find a nonlinear dependence on the applied microwave power. The transmon-resonator system was operated in the strong dispersive regime, where the ac Stark shift (2χ) due to a single microwave photon present in the resonator was larger than the linewidth (Γ) of the qubit transition. When the resonator was coherently driven at 5.474325 GHz, the transition spectrum of the transmon at 4.982 GHz revealed well-resolved peaks, each corresponding to an individual photon number-state of the resonator. From the relative peak-heights we obtain the occupancy of the photon-states and the average photon-occupancy bar{n} of the resonator. We observed a nonlinear variation of bar{n} with the applied drive power P_rf for bar{n} < 5 and compare our results to numerical simulations of the system-bath master equation in the steady state, as well as to a semi-classical model for the resonator that includes the Jaynes-Cummings interaction between the transmon and the resonator. We find good quantitative agreement using both models and analysis reveals that the nonlinear behavior is principally due to shifts in the resonant frequency caused by a qubit-induced Jaynes-Cummings nonlinearity.