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We measure photon-occupancy in a thin-film superconducting lumped elementresonator coupled to a transmon qubit at 20$\,$mK and find a nonlineardependence on the applied microwave power. The transmon-resonator system wasoperated in the strong dispersive regime, where the ac Stark shift ($2\chi$)due to a single microwave photon present in the resonator was larger than thelinewidth ($\Gamma$) of the qubit transition. When the resonator was coherentlydriven at $5.474325\,$GHz, the transition spectrum of the transmon at$4.982\,$GHz revealed well-resolved peaks, each corresponding to an individualphoton number-state of the resonator. From the relative peak-heights we obtainthe occupancy of the photon-states and the average photon-occupancy $\bar{n}$of the resonator. We observed a nonlinear variation of $\bar{n}$ with theapplied drive power $P_{rf}$ for $\bar{n} < 5$ and compare our results tonumerical simulations of the system-bath master equation in the steady state,as well as to a semi-classical model for the resonator that includes theJaynes-Cummings interaction between the transmon and the resonator. We findgood quantitative agreement using both models and analysis reveals that thenonlinear behavior is principally due to shifts in the resonant frequencycaused by a qubit-induced Jaynes-Cummings nonlinearity.

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