Collective Excitations of a Strongly Correlated Nonequilibrium Photon Fluid across the Insulator-Superfluid Phase Transition

We develop a Gutzwiller theory for the nonequilibrium steady states of a strongly interacting photon fluid driven by a non-Markovian incoherent pump. In particular, we explore the collective modes of the system across the out-of-equilibrium insulator-superfluid transition of the system, characterizing the diffusive Goldstone mode in the superfluid phase and the excitation of particles and holes in the insulating one. Observable features in the pump-and-probe optical response of the system are highlighted. Our predictions are experimentally accessible to state-of-the-art circuit-QED devices and open the way for the study of novel driven-dissipative many-body scenarios with no counterparts at equilibrium.

Figure 1

Top panel: sketch of the driven-dissipative system under consideration. (a) Mean-field phase diagram of the NESS for ${\mathrm{\Gamma }}_l/{\mathrm{\Gamma }}_p=50\gamma /{\mathrm{\Gamma }}_p=5\times {10}^{-2}$. The solid (dashed) green line corresponds to the horizontal cut at $\mathrm{\Omega }/{\mathrm{\Gamma }}_p=1.6\times {10}^{-1}$ ($5\times {10}^{-1}$) shown in panel (b) [panel (c)]. The white point marks the tip of the SFP lobe, while the white dashed lines enclose the region of hole superfluidity. (b) Mean-field average density $n_0$ [red], order parameter $|{\psi }_0|$ [blue] and purity $P$ [black] across the IP-SFP transition at constant $\mathrm{\Omega }/{\mathrm{\Gamma }}_p=1.6\times {10}^{-1}$ for the same parameters of panel (a). The green and blue dots highlight the critical point $J_c$ and the hopping scale $J_m$, respectively. (c) The equivalent of panel (b) for a stronger pumping $\mathrm{\Omega }/{\mathrm{\Gamma }}_p=5\times {10}^{-1}$.

Figure 2

(a)–(a^′) Excitation spectrum of the IP for $\mathrm{\Omega }/{\mathrm{\Gamma }}_p=5\times {10}^{-1}$ [see Fig. 1] and increasing $zJ/{\mathrm{\Gamma }}_p$. The gray dot in panel (a) pinpoints the critical lasing frequency ${\omega }_{*}$. (b)–(b^′) Excitation spectrum of the SFP for $\mathrm{\Omega }/{\mathrm{\Gamma }}_p=3\times {10}^{-1}$ and two values of $zJ/{\mathrm{\Gamma }}_p$ below/above the antiadiabatic crossover at $J=J_m$ (solid-dashed). The letters $G$, $A$, and $D$ indicate the (standard) Goldstone, amplitude and $D$-mode branches. Here, the excitation energy is calculated in the rotating frame of the lasing frequency ${\omega }_0$. In panels (a^′)–(b^′), the gray solid line specifies the cavity loss rate ${\mathrm{\Gamma }}_l/{\mathrm{\Gamma }}_p$. All panels refer to a $d=2$ array.

Figure 3

(a)–(c) DOS in the IP for the same parameters of Figs. 2 and increasing values of $zJ/{\mathrm{\Gamma }}_p$ (from top to bottom) towards the critical point. (a^′)–(c^′) The related reflectivity spectrum. The white horizontal lines indicate the effective chemical potential ${\omega }_{*}$. (d) Transmittivity in the SFP corresponding to the excitation spectra in Figs. 2 for $zJ/{\mathrm{\Gamma }}_p=10$ (antiadiabatic regime).