0 0.1 0.2 0.3 0 1 2 3 0 0.2 0.4 0 0.5 1 1.5 2 J/J$_{c0}$ 0 0.2 0.4 g , c s /g 0 1 2 3 0 0.2 0.4 0 1 2 3 0 0.2 0.4 0 1 2 3 0 0.2 0.4 (a)  c0 (b)  c0 c$_{s}$ c$_{s}$ (i) (ii) (iii) (iv) FIG. 2: Spectral gaps and sound velocity as a function of tunneling strength J/J c 0 at zero detuning δ/g = 0 and for (a) µ = µ c 0 where µ c 0 denotes the critical chemical po- tential at the tip of the lobe with critical hopping strength J c = J c 0 (top figure) and (b) away from the tip at µ = 1 . 2 µ c 0 with J c = 0 . 566 J c 0 (bottom figure). Shown are the gaps of particle (dashed) and hole (solid) modes in the Mott phase ( J < J c ) and the gaps of the Amplitude mode (dashed) and the sound velocity of the Goldstone mode (solid) in the super- fluid phase ( J > J c ). The insets show the corresponding ex- citation spectra at i) J = 0 . 5 J c 0 ii) J = J c 0 iii) J = 1 . 5 J c 0 iv) J = 0 . 566 J c 0 . At the phase boundary, the particle and hole mode of the Mott phase are identical with the Goldstone and Amplitude modes of the superfluid phase. At the tip of the lobe ii), where the polariton density can remain constant dur- ing the superfluid-insulator transition, the Amplitude mode becomes gapless and linear (its mass vanishes). The sound velocity of the Goldstone mode remains non-zero, confirming a special point in the phase diagram with dynamical critical exponent z = 1. Away from the tip iv), the Amplitude mode remains gapped and the Goldstone mode becomes quadratic with a vanishing sound velocity corresponding to a generic dynamical critical exponent z = 2. This confirms the change of the universality class along the phase boundary found in [12, 13], but put into question in [10]. matrix. The sum over k runs over the first Brioullin zone. The effective Hamiltonian can be diagonalized by a bosonic Bogoliubov transformation [20] yielding with ϵ$_{fluct}$ a fluctuation-generated correction of the ground-state energy and d † α k creating excitations with energy ϵ$_{α}$ ( k ). The analytic expressions for the two modes ϵ$_{α}$ ( k ) are rather lengthy and will be presented elsewhere. They depend on the variational parameters θ and χ . In order to calculate the excitations at different points in H$_{eff}$ = ϵ$_{var}$ + ϵ$_{fluct}$ + ∑ α =1 , $_{2}$∑ k ϵ$_{α}$ ( k ) d † α $_{k}$d$_{α}$$_{k}$ (8) the phase diagram, we first determine the opt imal pa- rameters θ$_{opt}$ and χ$_{opt}$ from a minimization o f (6) and insert them into the corresponding algebraic ex pressions for the spectra ϵ$_{α}$ ( k ). The results are sum m arized in Fig. 2. In the Mott phase, where θ$_{opt}$ = 0, w e find two gapped modes corresponding to particle and ho le excita- tions of lower polaritons. Our results agree exac tly with the ones obtained from a strong-coupling RPA [12]. In the superfluid phase we obtain a gapless Goldsto ne mode which is linear for small k describing the prop agation of phonons with a characteristic velocity c$_{s}$ and a gapped Amplitude mode, which describes local density fluctua- tions between normal fluid and condensate. Its e xistence has been predicted for cold atoms in an optical lattice as described by the BHM [16]. In cavity QED systems temporal and spatial correla- tion functions are experimentally accessible and provide an important tool to characterize the nature of different phases. In particular, the sound mode is a key s ignature of superfluidity and an important quantity whic h signals the presence of interactions in a condensate. I n Fig. 3, we study the dependence of the sound velocity c$_{s}$ and the condensate density ρ$_{c}$ = | φ$_{c}$ | 2 on detuning δ/g . In the strong-coupling regime ( J ≈ J$_{c}$ ) the sound ve- locity develops an anomaly vs. detuning at lo w polari- ton densities ρ ≡ 〈 N 〉 /N$_{s}$ (with N$_{s}$ the num ber of lat- tice sites). While for ρ = 2 (and higher) c$_{s}$ decreases for any finite detuning | δ | > 0, it increases steadily for ρ = 1 when tuning through the resonance. To ex- plain this behavior, we define the particle-hole gap in the atomic limit U$_{n}$ ( δ ) = ϵ$_{n}$$_{+1}$ − 2 ϵ$_{n}$ + ϵ$_{n}$$_{−}$$_{1}$ . This provides a useful measure for the effective repulsive interaction between polaritons and its dependence on detuning near the Mott lobes, where the mixing of states with differ- ent polariton numbers is small. For the Bose-Hubbard model U$_{n}$ ( δ ) = U is simply the Hubbard interaction parameter but for the JCHM the particle-hole gap de- pends on filling n and detuning δ/g . For n ≫ 1 we find U$_{n}$ ( δ ) − U$_{n}$ (0) = − (3 / 8 n 5 / $^{2}$)( δ $^{2}$/g ) + O ( δ $^{4}$/g$^{3}$, 1 /n 7 / $^{2}$) and thus U$_{n}$ decreases for any finite detuning | δ | > 0, consistent with a decrease of the sound velocity. How- ever, for n = 1 we find U$_{1}$ ( δ ) − U$_{1}$ (0) = δ/ 2 + O ( δ $^{2}$/g ), i.e., the effective repulsion increases steadily when tuning through the resonance. This causes the anomaly in c$_{s}$ at low densities. The different behavior of U$_{1}$ is rooted in the special nature of the zero-polariton state with ϵ$_{0}$ = 0 independent of detuning. This special behavior is also the reason for the decrease of the condensate fraction ρ$_{c}$/ρ for ρ = 1 (see inset in Fig. 3) and the increase of the size of the lowest Mott lobe (see phase diagram in Fig. 1). At higher densities the condensate fraction increases while the size of the lobes decrease for any fi- nite detuning, consistent with a decrease of the effective repulsion. We conclude that in the strongly-correlated regime both sound velocity c$_{s}$ and condensate fraction ρ$_{c}$/ρ are dominated by the repulsive interaction between 3