The model Hamiltonian in eq. 1 is a conservative two degree of freedom system. Hence, a useful way of visualizing the changes in the phase space with varying cavity frequency is via the technique of Poincaré surface of sections (SOS). Usually, the SOS for conservative systems is com-puted at a given energy and varying coupling strength or vice versa. However, in the present work, the initial polaritonic wavepacket has a spread of energies with an average ⟨H⟩ that depends on the cavity frequency. Moreover, the effective cavity-system coupling H SC in eq. 4b also depends on the cavity frequency. Thus, for a fixed coupling strength η, the SOS with varying ω c can exhibit dramatic changes. In what follows, we show the SOS at a total energy E ≡ ⟨H⟩ since the average total energy is nearly constant for the time scales of interest.