Drive-induced nonlinearities of cavity modes coupled to a transmon ancilla

High-Q microwave cavity modes coupled to transmon ancillas provide a hardware-efficient platform for quantum computing. Due to the coupling, the cavity modes inherit finite nonlinearity from the transmons. We theoretically and experimentally investigate how an off-resonant drive on the transmon modifies the nonlinearity of the cavity modes in qualitatively different ways, depending on the interrelation among cavity-transmon detuning, drive-transmon detuning and transmon anharmonicity. For weak transmon-cavity coupling, the cavity Kerr nonlinearity relates to the third-order nonlinear susceptibility function of the driven transmon. This susceptibility function provides an efficient way of computing the cavity Kerr particularly for systems with many cavity modes controlled by a single transmon. It also serves as a diagnostic tool for identifying undesired drive-induced multiphoton resonances. Lastly, we show that an off-resonant transmon drive with a judiciously chosen amplitude can be used to cancel the cavity self-Kerr or the inter-cavity cross-Kerr. This provides a way of dynamically correcting the cavity Kerr nonlinearity during bosonic operations and quantum error correction protocols that rely on the cavity modes being linear.