Nonequilibrium Dynamics of Coupled Qubit-Cavity Arrays

We study the coherence and fluorescence properties of the coherently pumped and dissipative Jaynes-Cummings-Hubbard model describing polaritons in a coupled-cavity array. At weak hopping we find strong signatures of photon blockade similar to single-cavity systems. At strong hopping the state of the photons in the array depends on its size. While the photon blockade persists in a dimer consisting of two coupled cavities, a coherent state forms on an extended lattice, which can be described in terms of a semiclassical model.

Figure 1

Level scheme for the lowest excitations of the JCHM for (a) a single cavity, (b) a dimer model, and (c) a coupled cavity array. Hopping between cavities gives (anti)symmetric superpositions of photon states in (b) and photon bands $ϵ(\mathbf{k})=-J\cos (\mathbf{k})$ in (c). The qubits $Q$ are resonant with the lowest photon states: (a) the cavity mode $C$ (${\omega }_q={\omega }_r$), (b) the symmetric superposition of photon states (${\omega }_q={\omega }_r-J$), and (c) the bottom of the photon band (${\omega }_q={\omega }_r-J$). This gives rise to dressed (polariton) states (LP, UP) in (a) and (b) and polariton bands in (c). In this Letter the laser frequency ${\omega }_l$ is near the lowest excitations in the system (bottom of the polariton band).

Figure 2

Photon field $|\varphi |$ and second-order coherence $g^{(2)}$ for the dimer of two coupled cavities, (a) as a function of resonator detuning ${\delta }_r/g$ at hopping $J/g=0$ (black), 0.02 (red or light gray), 0.05 (green or midgray), and drive strength $f/g=0.005$, and (b) as a function of hopping strength $J/g$ at drive $f/g=0.001$ (black), 0.005 (red or light gray), 0.02 (blue or midgray) when the laser frequency is resonant with the lowest excitation (${\delta }_r=g+J$, see Fig. 1). The dashed line in (a) corresponds to a single cavity ($J=0$) at very low drive $f/g=0.001$. The crosses mark points in (a) where ${\delta }_r=g+J$. Panel (c) shows $g^{(2)}$ as a function of the hopping strength $J/g$ for the same drive strengths as in (b). All dissipation rates are $\kappa =\gamma =0.005g$.

Figure 3

Photon field $|\varphi |$ (upper panels) and second-order coherence $g^{(2)}$ (lower panel) for the array as obtained from mean-field theory for the same parameters as in Fig. 2. Dashed lines in (a) and (b) correspond to the effective spin model in Eq. (4). The solid horizontal lines in (b) are the semiclassical asymptotes for Eq. (5).

Figure 4

Fluorescence spectra $S(\omega )$ for the dimer (upper panels) and for the array (lower panels) at fixed drive strength (a) $f=0.01g$, (c) $f=0.005g$, and fixed hopping (b),(d) $J=10g$. The laser frequency is resonant with the lowest polariton state (${\delta }_r=g+J$, see Fig. 1). Blue crosses and red circles in (c),(d) correspond to the effective models at weak drive in Eq. (4) and strong hopping in Eq. (5), respectively. The decay rates are $\kappa =\gamma =0.001g$ in (a),(b) and $\kappa =\gamma =0.005g$ in (c),(d). The effective model in Eq. (4) was applied to the upper and lower polariton branches separately.