  We study the quantum dynamics of a two-level system interacting with a
quantized harmonic oscillator in the deep strong coupling regime (DSC) of the
Jaynes-Cummings model, that is, when the coupling strength g is comparable or
larger than the oscillator frequency w (g/w > 1). In this case, the
rotating-wave approximation cannot be applied or treated perturbatively in
general. We propose an intuitive and predictive physical frame to describe the
DSC regime where photon number wavepackets bounce back and forth along parity
chains of the Hilbert space, while producing collapse and revivals of the
initial population. We exemplify our physical frame with numerical and
analytical considerations in the qubit population, photon statistics, and
Wigner phase space.
