  Photon-based strongly-correlated lattice models like the Jaynes-Cummings and
Rabi lattices differ from their more conventional relatives like the
Bose-Hubbard model by the presence of an additional tunable parameter: the
frequency detuning between the pseudo-spin degree of freedom and the harmonic
mode frequency on each site. Whenever this detuning is large compared to
relevant coupling strengths, the system is said to be in the dispersive regime.
The physics of this regime is well-understood at the level of a single
Jaynes-Cummings or Rabi site. Here, we extend the theoretical description of
the dispersive regime to lattices with many sites, for both strong and
ultra-strong coupling. We discuss the nature and spatial range of the resulting
qubit-qubit and photon-photon coupling, demonstrate the emergence of photon-
pairing and squeezing, and illustrate our results by exact diagonalization of
the Rabi dimer.
