Signatures of the superfluid-insulator phase transition in laser-driven dissipative nonlinear cavity arrays

We analyze the nonequilibrium dynamics of a gas of interacting photons in an array of coupled dissipative nonlinear cavities when driven by a pulsed external coherent field. Using a mean-field approach, we show that the response of the system is strongly sensitive to the underlying (equilibrium) quantum phase transition from a Mott insulator to a superfluid state at commensurate filling. We find that the coherence of the cavity emission after a quantum quench can be used to determine the phase diagram of an optical many-body system even in the presence of dissipation.

Figure 1

Schematic representation of a possible realization of the proposed system, employing a two-dimensional array of photonic crystal defect cavities [9]. Photons in the cavities have a finite lifetime, which requires losses to be compensated by some external pump, in our case a coherent pulsed laser field. Nonlinear processes in each cavity lead to an effective Bose-Hubbard repulsion between the cavity photons [1]. Neighboring cavities are tunnel coupled to each other by spatial overlap of the cavity mode profiles.

Figure 2

Time evolution of the zero-time-delay second-order correlation function and of the rescaled order parameter (inset), after a pulse that projects the cavities into a Fock state with one photon per cavity. The rescaled order parameter $\mathrm{\psi ̄}(t)=|\psi (t)|/\sqrt{n(t)}$ compensates for the decay of the photon number in the cavities and is related to the coherent fraction of the emission ($\kappa ={10}^{-2}\hspace{0.5em}U$, $\mathit{zJ}=3.0\hspace{0.5em}U$, $\Delta =0$).

Figure 3

Time average of $g_2(t)$ (main panel) and integral of $|\psi (t)|$ (inset), in the time interval $1.0\hspace{0.5em}{\kappa }^{-1}<t<2.0\hspace{0.5em}{\kappa }^{-1}$ for different values of the damping: $\kappa =2.0\times {10}^{-2}\hspace{0.5em}U$ (empty circles), $1.0\times {10}^{-2}\hspace{0.5em}U$ (filled circles), and $0.5\times {10}^{-2}\hspace{0.5em}U$ (empty triangles); $\Delta =0$. The vertical dashed line marks the critical value of the Mott-to-superfluid transition for the equilibrium Bose-Hubbard model [$(\mathit{zJ}/U){}_c~0.17$].

Figure 4

Time average of $g_2(t)$, for $\kappa ={10}^{-2}\hspace{0.5em}U$. The population ${\rho }_0$ of the vacuum in the initial state is ${\rho }_0=0.02$ for the filled symbols and ${\rho }_0=0.2$ for the empty symbols [$\mathit{zJ}/U=0.1$ (triangles) and $\mathit{zJ}/U=1.0$ (circles), below and above the critical value; $\Delta =0$].

Figure 5

Optimized pulse envelopes for the preparation of a Fock state with one photon per cavity, with a fidelity $\mathcal{F}\simeq 99\%$. The dashed line is the pulse ${\varepsilon }^{(0)}(t)$ for $J=0$. The solid line is the pulse ${\varepsilon }^{(1)}(t)$, detuned by $\Delta =43.25\hspace{0.5em}\kappa$, with $\mathit{zJ}/U=0.5$. The duration $T_{\mathrm{p}}=0.1/\kappa$ of the pulse is much shorter than the photon lifetime ($\kappa ={10}^{-2}\hspace{0.5em}U$ in the plot).