Steady-State Phases and Tunneling-Induced Instabilities in the Driven Dissipative Bose-Hubbard Model

We determine the steady-state phases of a driven-dissipative Bose-Hubbard model, describing, e.g., an array of coherently pumped nonlinear cavities with a finite photon lifetime. Within a mean-field master equation approach using exact quantum solutions for the one-site problem, we show that the system exhibits a tunneling-induced transition between monostable and bistable phases. We characterize the corresponding quantum correlations, highlighting the essential differences with respect to the equilibrium case. We also find collective excitations with a flat energy-momentum dispersion over the entire Brillouin zone that trigger modulational instabilities at specific wave vectors.

Figure 1

Top: Sketch of a square photonic lattice made of nonlinear cavities coupled by tunneling. The system is pumped coherently by a homogeneous laser field at normal incidence. Bottom: Number of mean-field solutions and their stability plotted as a function of $J/\Delta \omega$ and $U/\Delta \omega$, for $F/\Delta \omega =0.4$; $\gamma /\Delta \omega =0.2$, and $\Delta \omega >0$. Light blue (top-left and bottom-right part labeled with a “1”): monostable region, only one solution to Eq. (4). Dark blue part (labeled “2”): bistable region, two solutions to Eq. (4). Yellow part (central region labeled with “1”) has only one stable phase out of two existing solutions. Red part (label “0”): only one solution, which is unstable. $A$, $A^{\prime }$, $B$, and $B^{\prime }$ are points on the edge of the unstable zone whose excitation spectrum is presented in Figs. 4, 5.

Figure 2

Photon occupation number as a function of $J/\Delta \omega$ and $U/\Delta \omega$ for $F/\Delta \omega =0.4$ and $\gamma /\Delta \omega =0.2$. Left panel: results for the boson occupation number in the low-density phase. Right panel: the same quantity but for the high-density phase. For the sake of clarity, the maximal value of the color scale in the high-density phase has been set to 3, but the density is higher than 10 at high $J$ and low $U$. Notice that the monostable region is the same for both panels.

Figure 3

Second-order correlation function $g^2(0)$ as a function of $J/\Delta \omega$ and $U/\Delta \omega$. Same parameters as in Fig. 2. Left panel for the low-density phase. Right panel for the high-density phase.

Figure 4

Energy-momentum dispersion of elementary excitations for points $A$ (upper panel) and $A^{\prime }$ (lower panel), indicated in Fig. 1. Real and imaginary part of the low-energy branches (in units of $\gamma$) are plotted vs $\mathbf{k}$. $\Gamma =(0,0)$, $M=(\pi /a,\pi /a)$, $X=(\pi /a,0)$ are special points in the Brillouin zone of the squared photonic lattice. Thick blue lines depict branches with a flat real part over the entire Brillouin zone, while the imaginary part is strongly dispersive with a resonance around specific wave vectors.

Figure 5

Excitation dispersions for points $B$ (upper panel) and $B^{\prime }$ (lower panel). Same conditions as in Fig. 4.