  We demonstrate that a highly excited quantum electromagnetic mode strongly
interacting with a single qubit exhibits several distinct resonances in
addition to the Bloch-Siegert resonance condition that arises in the
interaction with classical radiation. The resonance phenomena are associated
with non-classical effects: Collapse of Rabi oscillations, splitting of the
photon wave packet, and field-qubit entanglement. The analysis is based on a
semiclassical phase-space flow of 2-by-2 matrices and becomes exact when the
photon number tends to infinity. The flow equations are solved perturbatively
and numerically, showing that the resonant detuning of the field and qubit
frequencies lie on two branches in the leading order, that further split into
distinct curves in higher orders.
