Abstract
We show that the nonlinear response of a driven circuit quantum electrodynamics setup displays antiresonant multiphoton transitions, as recently observed in a transmon qubit device. By including photon leaking, we explain the lineshape by a perturbative and a semiclassical analysis. We derive a bistable semiclassical quasienergy surface whose lowest quasienergy eigenstate is squeezed, allowing to define a squeezing-dependent local effective temperature. We study the escape dynamics out of the metastable state and find signatures of dynamical tunneling, similar as for the quantum Duffing oscillator.