Mott-Insulating and Glassy Phases of Polaritons in 1D Arrays of Coupled Cavities

By means of analytical and numerical methods we analyze the phase diagram of polaritons in one-dimensional coupled cavities. We locate the phase boundary, discuss the behavior of the polariton compressibility and visibility fringes across the critical point, and find a nontrivial scaling of the phase boundary as a function of the number of atoms inside each cavity. We also predict the emergence of a polaritonic glassy phase when the number of atoms fluctuates from cavity to cavity.

Figure 1

Upper panels: Phase diagram for the Hamiltonian model I, with $N=1$, 2 atoms inside each cavity at $ϵ=\omega$. Lower panels: System compressibility $\kappa$ for the first lobe (i.e., $\rho =1$), for different system sizes $L$, with $N=1$ (left) and $N=2$ (right).

Figure 2

Upper panels: Phase diagram of model II. The detuning parameters are set to zero and $g/\Omega =1$. Lower panels: System compressibility $\kappa$ for the first lobe, for different system sizes $L$, with $N=1$ (left) and $N=2$ (right).

Figure 3

Upper graphs: Critical hopping $t^{*}$ in the first lobe as a function of the number $N$ of atoms inside each cavity, for the two different models with all the detunings set to zero. Dashed black lines indicate a behavior $t^{*}\sim 1/\sqrt{N}$ and are plotted as guidelines. Lower graph: Critical hopping $t^{*}$ for model I with $N=10$ atoms per cavity [23], as a function of the relative detuning $\Delta$. Blue squares have been evaluated with the DMRG. Red data are estimates obtained from the effective on-site polaritonic repulsion $U_{\mathrm{eff}}$ at zero hopping, with filling $\rho =1$.

Figure 4

Photonic visibility $\mathcal{V}$ in the first insulating lobe ($\rho =1$) as a function of the hopping strength $t/\beta$, for model I. Here we considered the case $N=1$, $\Delta =0$. Insets: The photon number correlation function $\mathcal{S}(k)$ for $L=48$ in the deep insulating regime ($t/\beta =0.01$), and at the edge of the superfluid regime ($t/\beta =0.20$).

Figure 5

Relative standard deviation of the effective on-site interaction strength $U_{\mathrm{eff}}$ averaged over $L={10}^4$ cavities, as a function of the atom number fluctuations $\delta N$; from left to right $⟨N⟩=1$, 10, 100, 1000. Dashed line indicates an effective variation of the interaction strength equal to the standard deviation of a random interaction uniformly distributed in the interval $U_{\mathrm{eff}}(i)/⟨U_{\mathrm{eff}}⟩\in [-ϵ,+ϵ]$ with $ϵ=0.25$. In the inset an example of the variation on the on-site effective interaction is shown; $⟨N⟩=100$, $\delta N=20$.