  We study the ground state phase diagrams of two-photon Dicke, the
one-dimensional Jaynes-Cummings-Hubbard (JCH), and Rabi-Hubbard (RH) models
using mean field, perturbation, quantum Monte Carlo (QMC), and density matrix
renormalization group (DMRG) methods. We first compare mean field predictions
for the phase diagram of the Dicke model with exact QMC results and find
excellent agreement. The phase diagram of the JCH model is then shown to
exhibit a single Mott insulator lobe with two excitons per site, a superfluid
(SF, superradiant) phase and a large region of instability where the
Hamiltonian becomes unbounded. Unlike the one-photon model, there are no higher
Mott lobes. Also unlike the one-photon case, the SF phases above and below the
Mott are surprisingly different: Below the Mott, the SF is that of photon {\it
pairs} as opposed to above the Mott where it is SF of simple photons. The mean
field phase diagram of the RH model predicts a transition from a normal to a
superradiant phase but none is found with QMC.
