Tuning Photon-Mediated Interactions in a Multimode Cavity: From Supersolid to Insulating Droplets Hosting Phononic Excitations

Ultracold atoms trapped in laser-generated optical lattices serve as a versatile platform for quantum simulations. However, as these lattices are infinitely stiff, they do not allow to emulate phonon degrees of freedom. This restriction can be lifted in emerged optical lattices inside multimode cavities. Motivated by recent experimental progress in multimode cavity QED, we propose a scheme to implement and study supersolid and droplet states with phononlike lattice excitations by coupling a Bose gas to many longitudinal modes of a ring cavity. The interplay between contact collisional and tunable-range cavity-mediated interactions leads to a rich phase diagram, which includes elastic supersolid as well as insulating droplet phases exhibiting roton-type mode softening for a continuous range of momenta across the superradiant phase transition. The nontrivial dynamic response of the system to a local density perturbation further proves the existence of phononlike modes.

Figure 1

(a) Sketch of the system: A BEC is coupled to $N_c$ longitudinal running-wave modes ${\widehat{a}}_{j_{\pm }}$ of a ring cavity and transversely illuminated by $N_c$ lasers with Rabi frequencies ${\mathrm{\Omega }}_j$. (b)–(d) The dimensionless cavity-mediated interaction potential ${\sum }_j{\tilde{\mathcal{D}}}_j(x,x^{\prime })$ as a function of the relative interatomic distance for different number of involved modes: (b) $N_c=1$, (c) $N_c=10$, and (d) $N_c=20$.

Figure 2

Phase diagram of the system. The superfluid fraction $f_s$ (a) and the inverse participation ratio IPR (b) as functions of the contact-interaction strength $N{g}_0/\hslash {\omega }_r{\lambda }_m$ and the number of involved modes $N_c$ related to the effective range of the cavity-mediated interactions. Five regimes are identified in the phase diagrams: rigid supersolid (RSS), elastic supersolid (SS), array of droplets (ADs), single droplet (SD), and a normal phase (NP). The inset in (a) shows the rescaled total photon number ${\tilde{n}}_{\mathrm{ph}}$ in all modes in the same parameter plane. (c) A horizontal cut through the phase diagram at $N{g}_0=15\hslash {\omega }_r{\lambda }_m$ [the brown dashed line in panel (a)] shows the corresponding change in the atomic density $n(x)$ for a varying number of involved modes. (d)–(f) Exemplary density profiles for RSS at $N_c=1$ (d), for ADs at $N_c=4$ (e), and for SS at $N_c=6$ (f). The insets in (b) display typical atomic density profiles for NP ($N{g}_0=15\hslash {\omega }_r{\lambda }_m$, $N_c=13$) and for SD ($N{g}_0=0.5\hslash {\omega }_r{\lambda }_m$, $N_c=20$). The parameters are set to $(\sqrt{N}\eta ,NU,\kappa )=(3.2/\sqrt{N_c},-1,1){\omega }_r$ and ${\lambda }_m=L_{\mathrm{cav}}/m$ with $m=50$ being the lowest mode in the series. Here, ${\mathrm{\Delta }}_{c_j}$ vary from $-10{\omega }_r$ to $-4{\omega }_r$, as shown in the inset of Fig. 3 for $N_c=20$.

Figure 3

Excitation spectrum and real-time dynamics. (a) Excitation spectrum in NP slightly below the threshold with $\sqrt{N}\eta =6.4/\sqrt{N_c}{\omega }_r$ where $N_c=20$ and $N{g}_0=15\hslash {\omega }_r{\lambda }_m$. Here, $q_{\max }$ is the dominant momentum component in each Bogoliubov eigenvector; see the main text. The momentum discretization $\mathrm{\Delta }q$ is related to the system size via $\mathrm{\Delta }q=1/50{\lambda }_m$. The inset shows the distribution of the cavity detunings $\{{\mathrm{\Delta }}_{c_j}\}$. (b) Time evolution of the atomic density $n_{\mathrm{pert}}(x,t)$ after the initial kick of the central droplet with velocity $v=5/{\lambda }_m$. The initial steady state is for the parameters $\sqrt{N}\eta =3.2/\sqrt{N_c}$, $N{g}_0=0.5\hslash {\omega }_r{\lambda }_m$, and $N_c=10$. The other parameters are the same as in Fig. 2.