Parity oscillations and photon correlation functions in the $Z_2/U(1)$ Dicke model at a finite number of atoms or qubits

In this work, by using the strong coupling expansion and exact diagonization (ED), we study the $Z_2/U(1)$ Dicke model with independent rotating wave (RW) coupling $g$ and counter-rotating wave (CRW) coupling $g^{\prime}$ at a finite $N$. This model includes the four standard quantum optics model: Rabi, Dicke, Jaynes-Cummings ( JC ) and Tavis-Cummings (TC) model as its various special limits. We show that in the super-radiant phase, the system's energy levels are grouped into doublets with even and odd parity. Any anisotropy $β=g/g^{\prime} \neq 1$ leads to the oscillation of parities in both the ground and excited doublets as the atom-photon coupling strength increases. The oscillations will be pushed to the infinite coupling strength in the isotropic $Z_2$ limit $β=1$. We find nearly perfect agreements between the strong coupling expansion and the ED in the super-radiant regime. We also compute the photon correlation functions, squeezing spectrum, number correlation functions which can be measured by various standard optical techniques.

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