The on-site eigenstates of the Jaynes-Cummings Hamiltonian h JC i are labelled by the polariton number n and upper/lower branch index σ = ± . The mixed bo- son ( n, n − 1) - qubit ( g, e ) states define upper and lower polariton states leads to Mott lobes in the quantum phase diagram [2]. with the mixing angle tan θ$_{n}$ = 2 $_{g}$√$_{n/}$$_{(}$$_{δ}$ + 2 χ$_{n}$ ), χ$_{n}$ = √ g $^{2}$n + δ $^{2}$/ 4 and the detuning parameter δ = ω$_{c}$ − ω$_{x}$ . The corresponding eigenvalues are | n + 〉 = sin θ$_{n}$ | n , g 〉 + cos θ$_{n}$ | ( n − 1) , e 〉 , n −〉 = cos θ$_{n}$ n , g 〉 − sin θ$_{n}$ ( n 1) , e , (2) ϵ σ n = − ( µ − ω$_{c}$ ) n − δ/ 2 + σ χ$_{n}$ , σ = ± . (3) The zero polariton state | 0 〉 ≡ | 0 −〉 = | 0 , g 〉 is a special case with ϵ$_{0}$ ≡ ϵ − 0 = 0. A convenient starting point for our slave-boson ap- proach is the polariton representation [13] of the boson operator a$_{i}$ = ∑ nσν f σν n P ν † in − $_{1}$P σ in in terms of standard algebra operators P σ † in = | nσ 〉 ii 〈 0 | and matrix elements f σν n = 〈 n − 1 ν | a | n σ 〉 with f σν n = ( √$_{n}$$_{+}$ σ ν √$_{n}$ − 1 ) / 2 for n > 1 ( f σ − 1 = 1 $_{/}$√$_{2) at zero detuning (}$$_{δ}$ = 0). In this new basis the JCHM becomes H = ∑ i ∞ ∑ n $_{=0}$∑ σ ϵ σ $_{n}$P σ † in P σ in (4) − J ∑ 〈 ij 〉 ∑ n,n ′ $_{=1}$∑ σ,σ ′ ν,ν ′ f σσ ′ n f νν ′ n ′ P σ † in P σ ′ in − $_{1}$P ν $^{′}$† jn ′ − $_{1}$P ν jn ′ . The polariton operators obey bosonic commutation rela- tions if the constraint ∑ nσ P σ † in P σ in = 1 (5) is fullfilled at each site i . In [12] we have shown that the presence of the upper polariton branch ( σ = +) leads to additional high energy conversion modes in the Mott phase with small spectral weight and bandwidth; we ne- glect the upper branch as well as particle conversion tun- neling from now on and drop the branch index σ . In order to calculate the phase boundary and static observables in the superfluid phase near a Mott lobe with filling n , we restrict the Hilbert space to states with n and n ± 1 bosons and make a Gutzwiller Ansatz for the ground-state wave function where we also dropped the index n , i.e., P † iα ≡ P † in + $_{α}$, ϵ σ n + α ≡ ϵ$_{α}$ , and f σν n + α ≡ f$_{α}$ . This yields the variational energy ϵ$_{var}$ = 〈 ψ | H | ψ 〉 , The operator G † creates a new vacuum state, i.e., the mean-field ground state | ψ 〉 = ∏ i G † i | 0 〉 , and E † 1 $_{i}$, E † 2 i are orthogonal operators creating excitations above the ground-state. We express the Hamiltonian in terms of these new operators and eliminate G$_{i}$ by using the constraint (5) in the restricted Hilbert space G$_{i}$ ≈ √ 1 − E † 1 $_{i}$E$_{1}$$_{i}$ − E † 2 $_{i}$E$_{2}$$_{i}$ . Expanding the square root ev- erywhere in the Hamiltonian to quadratic order in E ( † ) (1 , 2) i yields, after a Fourier transformation, an effec- tive quadratic Hamiltonian H$_{eff}$ = ϵ$_{var}$ + ∑ k E † $_{k}$h$_{eff}$$_{,}$$_{k}$ E $_{k}$, where E = ( E$_{1}$$_{k}$, E$_{2}$$_{k}$, E † 1 − $_{k}$, E † 2 − $_{k}$) T and h$_{eff}$$_{,}$$_{k}$ is a 4 × 4 In order to find the excitations we follow the proce- dure outlined in [16] for the Bose-Hubbard model and define a new set of operators R † = ( G † $_{i}$, E † 1 $_{i}$, E † 2 $_{i}$)$^{T}$, which is obtained from the original polariton basis P † = ( P † i $_{0}$, P † i − $_{1}$, P † i $_{1}$) T via a unitary transformation R † = T P † with which has to be minimized with respect to the variational parameters θ and χ . The lobe boundaries as shown in Fig. 1 are determined by the vanishing of the order pa- rameter φ$_{c}$ = 〈 ψ | a | ψ 〉 = sin( θ ) [ f$_{0}$ sin( χ ) + f$_{1}$ cos( χ )] 2 / 2. Our results for the quantum phase diagram agree exactly with those in [12, 13]. Here, we point out that the size of all Mott lobes with filling factor n > 1 decrease for any finite detuning | δ | > 0, while the size of the lowest Mott lobe ( n = 1) increases steadily as the system is tuned through the resonance ( δ = 0). We explain this special behavior when discussing the nature of excitations close to the lobes below. FIG. 1: Phase diagram for the JCHM displaying the low- est three Mott lobes with n = 1 , 2 , 3. Dotted lines represent the critical hopping strength’s J c /g , where chemical potential and detuning are chosen such as to fullfill particle-hole sym- metry. A finite detuning | δ | > 0 decreases the critical hopping strength J c /g for n > 1, but the lowest Mott lobe ( n = 1) steadily increases when tuning through the resonance ( δ = 0). 2 T =   cos( θ ) sin( θ ) cos( χ ) sin( θ ) sin( χ ) − sin( θ ) cos( θ ) cos( χ ) cos( θ ) sin( χ ) 0 − sin( χ ) cos( χ )   . (7) | ψ 〉 = ∏ i [ cos( θ ) P † i $_{0}$+ sin( θ )(sin( χ ) P † i − $_{1}$+cos( χ ) P † i $_{1}$) ] | 0 〉 ϵ$_{var}$ = ϵ$_{0}$ cos( θ ) 2 + sin( θ ) 2 ( ϵ$_{−}$$_{1}$ sin( χ ) 2 + ϵ$_{1}$ cos( χ ) $^{2}$) − Jz/ 4 sin(2 θ ) $^{2}$( f$_{0}$ cos( χ ) + f$_{−}$$_{1}$ sin( χ )) 2 , (6)