Polaritonic excitations and Bose-Einstein condensation in the Rabi lattice model

The unitary transformation method is used to study the properties of the polaritonic states and Bose-Einstein condensation in the Rabi lattice model, where the on-site two-level systems (TLSs) are coupled with the intersite hopping photons. It is shown that the counter-rotating coupling (CRC) between TLS and photon, which breaks down the conservation of the excitation numbers, may induce a long-range Ising-like interaction among intersite TLSs. We have shown that the coupled TLSs and photons are hybridized to form the polaritonic states, and the corresponding ground state and polaritonic excitation spectra are calculated by diagonalizing our transformed Hamiltonian. When the photon hopping J is weak, the excitation spectrum is gapped. But the gap decreases with increasing J, and at critical value J = Jc the gap becomes zero at the I' point and the ground state becomes instable. Thus, Jc is the phase transition point where the polaritons are condensed at the I' point to form the delocalized superradiant phase. The results show that the larger detuning between TLS and photon favors the disorder insulator phase, which requires the stronger Jc to get the long-range order phase. The phase diagram of the delocalized superradiant phase transition has been obtained where there are no Mott lobes since the CRC breaks down the number conservation. The ground state and the excitation spectra of the delocalized superradiant phase are also calculated. It is shown that in the delocalized superradiant state the order parameter is the nonzero ground-state average of the photon annihilation operator. The polaritonic excitation has a finite excitation gap because the CRC breaks down the number conservation but leaves only a discrete symmetry, the parity conservation.