Nonequilibrium gas-liquid transition in the driven-dissipative photonic lattice

We study the nonequilibrium steady state of the driven-dissipative Bose-Hubbard model with Kerr nonlinearity. Employing a mean-field decoupling for the intercavity hopping $J$, we find that the steep crossover between low and high photon-density states inherited from the single cavity transforms into a gas–liquid bistability at large cavity-coupling $J$. We formulate a van der Waals–like gas–liquid phenomenology for this nonequilibrium setting and determine the relevant phase diagrams, including a new type of diagram where a lobe-shaped boundary separates smooth crossovers from sharp, hysteretic transitions. Calculating quantum trajectories for a one-dimensional system, we provide insights into the microscopic origin of the bistability.

Figure 1

Mean-field phase diagram of an array of nonlinear cavities with interaction $U$ and loss $\kappa$, pumped with amplitude $f$ at a frequency ${\omega }_d$ detuned from the cavity frequency ${\omega }_c$ by $\mathrm{\Delta }={\omega }_d-{\omega }_c$, see top-left inset. Photons tunnel to neighboring cavities with amplitude $J$. The photon density $n$ at the 4-photon resonance $1+2\mathrm{\Delta }/U=4$ is shown as a function of the dimensionless parameters $f/U$ and $J/U$ for small dissipation $\kappa =U/20$. The smooth gas–liquid crossover at small $J/U$ exhibits bimodality (BM region, yellow lines) in the photon number distribution, and gives way to a hysteretic transition at $J_c\approx 0.18U$ (dot), opening a coexistence region of gas and liquid at $J>J_c$ (stripes; colors refer to densities in gas and liquid). The resulting underdriven liquid and overdriven gas phases terminate at the spinodal lines (white), which smoothly extend the lines bounding the bimodal region at small $J$. The stars mark the location of the quantum trajectory results in Fig. 4.

Figure 2

Mean-field phase diagram displaying the nature of the gas–liquid transition in the driven-dissipative photonic lattice. Plotting the dimensionless detuning $2\mathrm{\Delta }/U$ versus hopping $J/U$ at small dissipation $\kappa =U/20$, we show the boundary separating smooth from hysteretic gas–liquid transitions as driven by increasing the pump amplitude $f$. Distinct lobes appear between successive $m$-photon resonances of the individual cavities, i.e., when $1+2\mathrm{\Delta }/U=m$ assumes an integer value, thus featuring a similar commensuration effect as the equilibrium Bose-Hubbard model. Note however that the critical line separating smooth from hysteretic transitions does not extend to $J=0$ in correspondence with the resonances. Going to small $\mathrm{\Delta }/U$ or very small dissipation $\kappa$, instabilities show up in the mean-field analysis, see also Refs. [32, 52]. Numerical errors are of order the size of the points.

Figure 3

(a) Density $n$ and (b) second-order coherence $g^{\left(2\right)}$ as a function of drive detuning $2\mathrm{\Delta }/U$ and drive strength $f/U$ for a single cavity as obtained from the exact solution [6] of Eq. (2) with $J=0$ and $\kappa =U/20$. The modulated gray lines labeled “BM” encompass the bimodal regime. The inset displays the photon number distribution $p_k$ at the two bars marked in the main panel. The white crosses mark the onset $f_{\times }^{\mathrm{sc}}$ of the liquid phase as defined by the condition of unit compressibility $K=1$. The correlator $g^{\left(2\right)}$ illustrates the phases' coherent nature, while the crossover is characterized by superbunching, see also Ref. [34]. The bottom inset displays the density $n$ and correlator $g^{\left(2\right)}$ evaluated at fixed detunings $2\mathrm{\Delta }/U=3,3.5$ (solid, dashed).

Figure 4

Selected quantum trajectories for a 1D cavity array with six sites. The panels (a)–(c) show the photon density in color scale as a function of time ($t\kappa$) and position (lattice site $j$) at fixed drive strength $f/U=0.35$ and for increasing hopping $J/U$ as marked with the stars in the phase diagram of Fig. 1. At small hopping $J\ll J_c$, (a) the trajectories of different sites are uncorrelated, while for $J>J_c$, the entire cavity array switches collectively between gas and liquid states within the coexistence region of the mean-field diagram, see (c). The panels (d)–(f) show trajectories for a single lattice site $j=4$ as a function of time, as taken from the respective top panels (a)–(c). The vertical red bars indicate the photon emissions from the lattice. (d) For $J\ll J_c$ each individual cavity displays intermittency [47] (see also text) at random times, yielding a constant photon emission from the array. (f) For $J>J_c$ the array behaves as a coherent supercavity and a collective intermittency is restored. The horizontal arrows mark the gas and liquid mean-field values, showing that collective switching in panel (c) indeed occurs between the mean-field densities. Panels (a), (b), (c): $J/U=0.01,0.1,0.5$. Other parameters are chosen as in Fig. 1. Convergence of the quantum trajectory results in the photon truncation parameter (cutoff) is illustrated in Appendix pp2.

Figure 5

Convergence plot of the average photon density $n$ [see Eq. (B1)] in the steady state for various values of hopping strengths $J/U$ as calculated with quantum trajectories. The average density $n$ is shown as a function of the photon number truncation parameter $n_{\mathrm{cutoff}}$ for a lattice of $N=5$ sites with PBC. The vertical bars denote one standard deviation (see text). Other parameters as in Fig. 4 of the main text. The vertical arrow indicates the cutoff value $n_{\mathrm{cutoff}}=6$ used in Fig. 4 of the main text, for an array of $N=6$ sites with PBC.

Figure 6

(a) Collective time spent in the liquid phase $\tau$ [see Eq. (C1) and main text] in the steady state as calculated with quantum trajectories (QT) as a function of the size $N$ of a one-dimensional array with PBC. The inset displays a sample trajectory for $N=8$ exhibiting three separate periods where the system dwells in the liquid phase (see also text). To extract ${\tau }_{s,r}$ (see text), an arbitrary threshold $n=0.4$ is used to separate gas and liquid phases and density fluctuations on a scale smaller than $dt=2/\kappa$ are neglected. We checked that the trend in the results (exponential-like increase of $\tau$ with $N$) is invariant with respect to these choices. The trajectory results are shown for a set of parameters within the coexistence region of the mean-field (MF), $f/U=0.15$, $J/U=0.6$, as indicated by the white star in (b). The detuning value $\mathrm{\Delta }={\omega }_d-{\omega }_c$ (detuning between the drive frequency and the cavity frequency) is set to $1+2\mathrm{\Delta }/U=1.55$. For this choice of detuning, convergence of the quantum trajectory results in the photon truncation parameter $n_{\mathrm{cutoff}}$ is achieved for $n_{\mathrm{cutoff}}=3$, much lower than $n_{\mathrm{cutoff}}=6$ required for the detuning value $1+2\mathrm{\Delta }/U=4$ used in Figs. 1 and 4 in the main text. The choice of a lower detuning $\mathrm{\Delta }/U$ with respect to Fig. 4 in the main text thus allows us to study larger system sizes. The dissipation strength is chosen as $\kappa /U=20$, as in Figs. 1 to 4 in the main text.