-2 0 2  g 0 0.5 1 ρ$^{c}$ $_{/}$ ρ -2 0 2  g 0 0.5 1  ρ$^{c}$ $_{/}$ ρ -1 0 1 2 /g -2 -1 0 1 2 g 0 1 2 3 4 c s /c s0 FIG. 3: Sound velocity as a function of detuning δ/g for fixed polariton density ρ = 1 (left figure) and ρ = 2 (right figure) near the Mott lobes with J = J c 0 ( δ ) + 0 . 2 J c 0 (0) (solid lines) obtained from slave-boson theory and deep inside the super- fluid phase with J = J c 0 ( δ ) + 5 J c 0 (0) (dashed lines) obtained from an effective Bogoliubov theory. Here, J c 0 ( δ ) denotes the critical hopping strength at the tip of the lobe. The insets show the condensate fraction ρ c /ρ with the same notation as above. polaritons when δ/g is varied. In the weakly interacting regime ( J ≫ J$_{c}$ ), the slave- boson approach is no longer a good approximation since states with different polariton numbers mix strongly. We use an effective Bogoliubov theory instead, originally de- rived for a generalized Dicke model describing localized excitons coupled to photons confined in a single mi- crocavity with a gapped, quadratic photon dispersion ϵ ( k ) = ω$_{0}$ + k $^{2}$/ 2 m with ω$_{0}$ = πc/L and m = π/ ( cL ) ( L being the width of the cavity) [17]. The JCHM maps onto the generalized Dicke model after a Fourier transfor- mation of the bosonic field operator and expanding the lattice dispersion for small k -vectors as ϵ ( k ) ≈ Jz + J k 2 (for a hypercubic lattice with lattice constant a = 1). Using the results in [17] with the formal replacements ω$_{0}$ = ω$_{c}$ − zJ and m = 1 / (2 J ), we can compare our find- ings in the strongly interacting regime with those valid for weak interactions shown as dashed lines in Fig. 3. We observe that the sound anomaly is still present deep in- side the superfluid phase, but slightly suppressed. This is due to the admixture of states with higher polariton numbers n > 1 for ρ = 1. On the other hand, the max- imum of the sound velocity at higher densities ρ = 2 is slightly shifted to positive detuning due to a weak admix- ture of polariton states with n = 0 , 1. On the contrary, the condensate density behaves different in the weakly interacting regime far away from the Mott lobes. It de- creases independent of the polariton density when tuning through the resonance and is thus no longer dominated by the effective repulsion but rather by the internal struc- ture of the polaritons. To understand this we have to con- sider the boson density ρ$_{B}$ = ρ$_{n}$ + ρ$_{c}$ , which consists of a condensate ρ$_{c}$ and a normal part ρ$_{n}$ (depletion). In the weakly interacting regime, the depletion ρ$_{n}$ is small and the condensate density ρ$_{c}$ behaves similar to the density In summary, the slave-boson theory for str ongly corre- lated polaritons presented in this paper pred icts the exis- tence of gapped particle/hole modes in the Mott regime and, besides a gapless, linear Goldstone m ode, the ex- istence of a gapped Amplitude mode in th e superfluid phase. The anomaly of the sound velocity of the Gold- stone mode as a function of detuning and po lariton den- sity is rooted in the special nature of the ze ro polariton state within the Jaynes-Cummings ladder. It is present in the strong as well as weak coupling reg ime and may constitute an interesting experimental signa ture for the composite nature of superfluid lattice polarit ons in future experiments with coupled cavities. We thank H. Tureci and S. 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