Phase diagram of incoherently driven strongly correlated photonic lattices

We explore theoretically the nonequilibrium photonic phases of an array of coupled cavities in presence of incoherent driving and dissipation. In particular, we consider a Hubbard model system where each site is a Kerr nonlinear resonator coupled to a two-level emitter, which is pumped incoherently. Within a Gutzwiller mean-field approach, we determine the steady-state phase diagram of such a system. We find that, at a critical value of the intercavity photon hopping rate, a second-order nonequilibrium phase transition associated with the spontaneous breaking of the $\text{U}\left(1\right)$ symmetry occurs. The transition from an incompressible Mott-like photon fluid to a coherent delocalized phase is driven by commensurability effects and not by the competition between photon hopping and optical nonlinearity. The essence of the mean-field predictions is corroborated by finite-size simulations obtained with matrix product operators and corner-space renormalization methods.

Figure 1

A sketch of the considered photonic system, consisting of a lattice of coupled nonlinear cavities. Each lattice site is a cavity coupled to a two-level system, which is incoherently pumped at a rate ${\mathrm{\Gamma }}_p$. ${\mathrm{\Omega }}_R$ is the coherent coupling rate (vacuum Rabi frequency) between the cavity mode and the two-level emitter (with frequencies ${\omega }_c$ and ${\omega }_{at}$, respectively), while $U$ is the photon-photon Kerr on-site interaction. The coupling with the environment produces incoherent photon leakage and atomic relaxation at rates ${\mathrm{\Gamma }}_l$ and $\gamma$, respectively. Photons can hop between neighboring sites at a rate $J$.

Figure 2

Steady-state observables for a single-cavity system. The steady-state value of the photon number $n$, photon number fluctuations $\mathrm{\Delta }n$, purity $\mathcal{P}$, and one-photon Fock state population ${\mathrm{\Pi }}_1$ are plotted as a function of the nonlinearity $U/{\mathrm{\Gamma }}_p$ for different values of the cavity dissipation rate ${\mathrm{\Gamma }}_l/{\mathrm{\Gamma }}_p$, as indicated in the legend. The other parameters are ${\omega }_{at}={\omega }_c, {\mathrm{\Omega }}_R/{\mathrm{\Gamma }}_p={10}^{-1}$, and $\gamma /{\mathrm{\Gamma }}_p={10}^{-4}$.

Figure 3

The order parameter $|\langle \widehat{a}\rangle |$ (top panel) and the number of photons $n$, its variance $\mathrm{\Delta }n$, and the compressibility $\mathcal{K}$ (bottom panel) of the steady state of Eq. (11) as a function of $zJ/{\mathrm{\Gamma }}_p$ in the hard-core limit ($U/J=+\infty$). Here ${\mathrm{\Gamma }}_l/{\mathrm{\Gamma }}_p=\gamma /{\mathrm{\Gamma }}_p={10}^{-3}, {\mathrm{\Omega }}_R/{\mathrm{\Gamma }}_p={10}^{-1}$, and ${\omega }_{at}={\omega }_c$. The dashed vertical line signals the predicted critical value of $J$ (see Appendix pp2).

Figure 4

Contour plot of the steady-state Wigner distribution $W\left(\alpha \right)$ in the Mott-like (top panel) and coherent (bottom panel) phases. Each contour denotes a variation of 0.05 of the value of $W\left(\alpha \right)$. The black dashed contour encircles the region with $W\left(\alpha \right)<0$. The parameters are set as in Fig. 3.

Figure 5

Results of the Gutzwiller mean-field stability analysis in the hard-core regime ($U/J=+\infty$). The yellow area denotes the region where $\langle \widehat{a}\rangle =0$ is stable, while the light blue area is the region where the solution $\langle \widehat{a}\rangle =0$ is unstable and the symmetry is broken. When not varying them, we fixed the parameters as ${\mathrm{\Gamma }}_p=1, {\mathrm{\Omega }}_R={10}^{-1}, {\mathrm{\Gamma }}_l=\gamma ={10}^{-3}, {\omega }_{at}={\omega }_c$. The solid lines are the predictions for the critical hopping rate given by Eq. (17), which well approximates the phase boundary in the ${\mathrm{\Gamma }}_{em}^0/{\mathrm{\Gamma }}_l\gg 1$ limit [see Eq. (16)]. The dashed horizontal lines denote the ${\mathrm{\Gamma }}_{em}^0/{\mathrm{\Gamma }}_l=1$ threshold. The condition ${\mathrm{\Gamma }}_{em}^0/{\mathrm{\Gamma }}_l>1$ is necessary in order to have a significant population in the symmetry-broken phase.

Figure 6

Mean-field steady-state phase diagram in the $U/{\mathrm{\Gamma }}_p-zJ/{\mathrm{\Gamma }}_p$ plane. The white area corresponds to the region of the parameters for which $|\langle \widehat{a}\rangle |=0$, while in the dotted region the $U\left(1\right)$ symmetry is spontaneously broken and the steady state exhibits a limit cycle ($|\langle \widehat{a}\rangle |\ne 0$). Here $\gamma /{\mathrm{\Gamma }}_p={\mathrm{\Gamma }}_l/{\mathrm{\Gamma }}_p={10}^{-2}, {\mathrm{\Omega }}_R/{\mathrm{\Gamma }}_p={10}^{-1}$, and ${\omega }_{at}={\omega }_c$. The dashed vertical line denotes the critical hopping rate predicted by the Gutzwiller stability analysis $z{J}_c^{\mathrm{HC}}/{\mathrm{\Gamma }}_p=1.51$.

Figure 7

Top: the average photon density $n$ in the steady state (left panel) and its variance $\mathrm{\Delta }{n}^2$ (right panel) as a function of $J/{\mathrm{\Gamma }}_p$. Bottom: steady-state value of the compressibility $\mathcal{K}$ (left panel) and of the entropy (right panel) as a function of $J/{\mathrm{\Gamma }}_p$ in the hard-core limit ($U/J=+\infty$). The various curves are for different sizes, as indicated in the legend. The solid horizontal lines are the single-cavity values ($J=0$) of the quantity under consideration. The dashed vertical lines denote the critical hopping rates predicted by the Gutzwiller mean-field theory. The parameters are set as ${\omega }_{at}={\omega }_c-zJ, {\mathrm{\Gamma }}_l/{\mathrm{\Gamma }}_p=\gamma /{\mathrm{\Gamma }}_p={10}^{-3}, {\mathrm{\Omega }}_R/{\mathrm{\Gamma }}_p={10}^{-1}$. For the largest size considered ($M=8$), the convergence of the considered quantities has been achieved with 3000 states in the corner space (the full Hilbert space has a dimension equal to $4^8=65536$).

Figure 8

Derivative of the entropy with respect to the hopping rate $\partial S/\partial (zJ/{\mathrm{\Gamma }}_p)$ for different system sizes as indicated in the legend. The dashed vertical line denotes the critical hopping rate predicted by the Gutzwiller mean-field theory. The other parameters are set as in Fig. 7.

Figure 9

Top: spatial decay of the correlation function $c(i,i+r)$ [as defined in Eq. (18)] with the distance $r$ for $M=20, {\omega }_{at}={\omega }_c$ and different values of $zJ/{\mathrm{\Gamma }}_p$, as indicated in the legend. Correlators have been chosen in a symmetric way with respect to the center of the chain. The other parameters are set as in Fig. 7. Bottom: the correlation length $\lambda$ obtained fitting $c(i,i+r)$ with an exponentially decaying function. The solid line is the scaling predicted by Eq. (20). For all the values of the parameters considered, convergence has been achieved with a bond link dimension $\chi =50$ for the MPO algorithm.