  We study photon/phonon statistics of a qubit-plasmon-phonon hybrid system in
the ultrastrong coupling regime. The introduced qubit coupling causes parity
conserving and non-conserving situations. We employ an analytic approximation
approach for the parity conserving case to reveal the statistical behaviors of
photons and phonons. It indicates that both photons and phonons show strong
antibunching at the same frequency. Even though the bunching properties of
photons/phonons occupy the dominant regions of the considered frequencies,
phonons tend to weakly antibunching within the photonic strong-bunching area.
In contrast, one can find that the configurations of correlation functions for
both photons and phonons in the parity conserving case are squeezed towards the
central frequency by parity breaking, which directly triggers the reverse
statistical behaviors for the different parties at the low-frequency regions
and the strong bunching properties at other frequency regions. The
photon-phonon cross-correlation function also demonstrates similar
parity-induced differences, indicating that the non-conserving parity induces
the photon-phonon bunching behavior. We finally analyze the delayed
second-order correlation function with different driving frequencies, which
illustrates striking oscillations revealing the occurrence of simultaneous
multiple excitations.
