  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 for 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.
