Supersolid light in a semiconductor microcavity

Abstract

Supersolidity - simultaneous superfluid flow and crystalline order - has been realized in quantum atomic systems but remains unexplored in purely photonic platforms operating at weak light-matter coupling. We predict a supersolid phase of light in a plasma-filled optical microcavity, where photons acquire effective mass and interact via nonlocal, plasma-mediated nonlinearities. By deriving a Gross-Pitaevskii equation with a tunable photon-photon interaction kernel, we show that under coherent driving the cavity light field can spontaneously crystallize into a supersolid lattice via modulational instability. Crucially, this supersolid arises from a weak photon-electron coupling enabled by virtual electronic transitions, and it does not require hybrid polariton formation. Using doped semiconductor microcavities, we identify feasible conditions (electron densities $\sim 10^{10}- 10^{11}~\mathrm{cm}^{-2}$ and optical intensities $\sim 10^{2}-10^{4}~\mathrm{W/cm}^{2}$) for experimental realization. This work establishes plasmonic cavities as a platform for correlated photonic matter with emergent quantum order.

Figures (10)

• Figure 1: Time evolution of the intracavity normalized field amplitude and phase for driven--dissipative conditions. The phase is indicated by te color of the 3D curve at each point, while the field amplitude is shown both in the $z$-coordinate and in the ground level in each plot. The top row shows a manifestation of superfluid behavior, where an initial homegenous field with large phase fluctuations is seen to recover global phase coherence while remaining homogeneous. The middle row illustrates the case of supersolid formation, where an initial homogeneous field breaks into a periodic lattice structure and develops global phase coherence. The bottom shows the case of a homogeneous initial condition that crystalizes while maintains large phase fluctuations, leading to a solid. Times are measured given in units of $M/\hbar k_F^2$. Simulation parameters: $(\alpha, \kappa, \Gamma)$ = $(0.5, 0.1, 0.1)$, $(1, 0.1, 0.5)$ and $(10, 0.1, 0.5)$, respectively for top, middle and bottom rows, $N_0 = \Gamma^2/\kappa^2$ and $\Omega_p = \alpha g_0 N_0$ for all cases, and $g(\mathbf k)$ of a degenerate plasma with drift velocity $v_0 = 0.45 v_F$.
• Figure 1: Photon--photon interaction potential $g(\mathbf k)$ mediated by a degenerate plasma at zero temperature and moving at drift velocity $\mathbf v_0 = v_0 \mathbf e_x$. Panel (a) shows $g(k_x,0)$ for different drift velocities, while panels (b) and (c) show $g(\mathbf k)$ for both isotropic and anisotropic cases. For $v_0 = 0.45$, $g(\mathbf k)$ attains negative values in a small region along the $k_x$ direction. All velocities are given in units of the Fermi velocity, while $g(\mathbf k)$ is normalized to $G_0 = e^4 m/ (2\pi \hbar m \epsilon_0 D_0 \omega_0^3)$.
• Figure 2: Phase diagrams with the steady-state phase of the driven-dissipative GP equation as a function of dimensioless interaction strength $\alpha$ and (left panel) varying pump rate $|\Gamma|$ for fixed loss rate $\kappa = 0.1$, or (right panel) varying loss rate for fixed $|\Gamma| = 0.2$ (all parameters are given in dimensionless units). Each representative case of Fig. \ref{['fig_phase_d']} is marked with (a) superfluid, (b) supersolid and (c) solid. The phase boundaries were extracted by evaluating two order parameters, $\lim_{x\rightarrow\infty} g^{(1)}(x)$ and $\max_{\mathbf k \neq 0} S(\mathbf k)$, where $g^{(1)}(x)$ is the first-order coherence function (averaged over $y$) and $S(\mathbf k)$ is the structure factor, and determining where they become nonzero (in practice, above a small threshold)
• Figure 2: Dispersion relation of Eq. \ref{['dispRel_conservative']} for $v_0 = 0.45 v_F$ and several values of the coupling constant along $k_y = 0$. The spectrum goes from quadratic (noninteracting limit) to a roton-like form with a finite-$k$ minimum. Instability sets in when the roton gap closes around $\mathbf k_\ast = (k_\ast,0)$, driven by increasing photon-photon interactions.
• Figure 3: Ground-state characterization of the conservative GP equation as a function for varying interaction strength $\alpha$ and $v_0 = 0.45 v_F$. Each column corresponds to a stationary solution obtained via imaginary-time evolution. Top row: real-space density profiles $|E(\mathbf r)|^2$, normalized to the maximum value across the three panels. Second row: phase distribution. Third row: first-order coherence $g^{(1)}(x) = \int dy \, g^{(1)}(\mathbf r)$. Bottom row: structure factor $S(\mathbf k)$ in log scale. For weak interactions ($\alpha=0.001$), the ground state is dominated by kinetic energy and approaches a smooth Gaussian profile. At intermediate coupling ($\alpha=0.1$), the condensate becomes a uniform superfluid with global phase coherence. For strong interactions ($\alpha=10$), attractive components of $g(\mathbf k)$ at finite wavevectors lead to density modulation and pronounced Bragg peaks in $S(\mathbf k)$ and oscillatory behavior of $g^{(1)}$ mantaining a finite value over all the entire volume, which signals the emergence of a supersolid.