Strong Coupling Theory for the Jaynes-Cummings-Hubbard Model

We present an analytic strong-coupling approach to the phase diagram and elementary excitations of the Jaynes-Cummings-Hubbard model describing a superfluid-insulator transition of polaritons in an array of coupled QED cavities. In the Mott phase, we find four modes corresponding to particle or hole excitations with lower and upper polaritons, respectively. Simple formulas are derived for the dispersion and spectral weights within a strong-coupling random-phase approximation (RPA). The phase boundary is calculated beyond RPA by including the leading correction due to quantum fluctuations.

Figure 1

Quantum phase diagram for a hypercubic lattice in $D=2$ (left figure) and $D=3$ (right figure). We show two Mott lobes for $n=1$, 2 at zero detuning $\delta =0$. We compare first-order perturbation theory (dashed), RPA (dot-dashed) and quantum fluctuations (solid) with recent results from a quantum Monte Carlo (solid dots) [14] and variational cluster (stars) [17] approach. The two crosses (left figure) connected by the dotted line mark the parameter values used in Fig. 2. The insets show the critical hopping strength $J_c/g$ at the tip of the lobe as a function of the detuning $\delta$ for $n=1$ and $n=2$ calculated within strong-coupling RPA in $D=3$ dimensions.

Figure 2

Particle or hole dispersion of the conventional modes ${\omega }_{(p,h)}^{-}$ for $D=2$ and $n=1$ at zero detuning $\delta =0$. Left figure: Deep inside the Mott insulator (see cross in Fig. 1) for $(\mu -{\omega }_c)/g=-0.78$ and $J/g=0.01$. The inset shows the conventional modes ${\omega }_{(p,h)}^{-}$ together with the conversion mode ${\omega }_p^{+}$. Right figure: At the phase boundary (see cross in Fig. 1) for $(\mu -{\omega }_c)/g=-0.78$ and $J/g=0.04$. The inset shows the bandwidth $W$ of the conventional particle mode ${\omega }_p^{-}$ as a function of detuning $\delta$ at the tip of the lobe.

Figure 3

Particle or hole gaps ${\Delta }_{(p,h)}^{-}$ (left figure) and effective masses $m_{(p,h)}^{-}$ (right figure) of the conventional modes for $D=2$ and $n=1$ at zero detuning $\delta =0$ as a function of the tunneling strength $J/g$ for $(\mu -{\omega }_c)/g=-0.78$ (along the dotted line in Fig. 1).