Quantum Droplets of Light in Semiconductor Microcavities

Abstract

Quantum droplets are dilute self-bound configurations of bosons that result from the balance between a mean-field attraction and a repulsion induced by quantum fluctuations. Such droplets have been successfully realized in cold atomic gases and represent a signature of their quantum nature. Here, we predict the existence of a similar droplet phase in a solid-state system, involving polaritons formed from the strong coupling between excitons (bound electron-hole pairs) and photons in a semiconductor microcavity. We consider a spin mixture of exciton-polaritons near a biexciton Feshbach resonance, which allows one to tune the interspecies interactions to be attractive and comparable in magnitude to the intraspecies repulsion. We find that self-bound quantum droplets are achievable for realistic parameters in atomically thin semiconductors, and that they can be detected via their excitation spectrum and spatial profile. This exotic phase could potentially lead to polariton condensation at lower thresholds and it opens an alternative avenue to achieve the long-sought quantum polaritonic regime.

Figures (7)

• Figure 1: (a) Schematic illustration of the energy scales for one (bottom) and two (top) particles. The hybridization of an exciton (X) and a photon (C) yields upper (U) and lower (L) polariton quasiparticles. Two $\uparrow$ and $\downarrow$ lower polaritons repel (attract) each other when their energy $2 \epsilon_{\rm LP}$ is above (below) the energy $-\varepsilon_{\rm B}$ of the biexciton (X$\uparrow$-X$\downarrow$). (b) Two-lower-polariton (green) and two-photon (black) energies as a function of detuning $\delta_0$. The blue point denotes the Feshbach resonance that occurs when $2 \epsilon_{\rm LP} = -\varepsilon_{\rm B}$. (c) Detuning dependence of the intra- (blue) and inter-species (red) exciton-exciton interaction coefficients. The magenta (orange) square is an example of repulsive (attractive) interspecies interactions. The black curve shows $\delta g = g + g_{\uparrow \downarrow}$, and the yellow region denotes the necessary condition $\delta g < 0$ for the appearance of quantum droplets. We use parameters inspired by recent experiments with MoSe$_2$ monolayers: $\Omega_{\rm R} = 28$ meV, $\varepsilon_{\rm X} = 470$ meV Dufferwiel2015, $\varepsilon_{\rm B} = 20$ meV Hao2017, and exciton mass $m_{\rm X} = 1.14 m_0$Kylanpaa2015 (with $m_0$ the bare electron mass).
• Figure 2: (a-d) Thermodynamic potential as a function of exciton density and pairing field for 4 values of the chemical potential $\delta \mu = \mu-\epsilon_{\rm LP}$: $0$ (a), $-0.10$ meV (b), $-0.22$ meV (c), $-0.40$ meV (d). The white dot-dashed contour denotes the zero pressure line, i.e., the zeros of $\Omega$, while the green dashed and red solid curves correspond to the conditions $\partial_{\alpha}\Omega = 0$ and $\partial_{\Phi}\Omega = 0$, respectively. The filled dots in panels (a) and (b) are the stationary points satisfying both conditions. The empty dot in panel (c) indicates the quantum droplet which appears as a first order transition, signaling phase coexistence with the vacuum (white-red point). Thermodynamic potential as a function of pairing field (e) and exciton density (f) for the same values of $\delta\mu$, obtained by evaluating $\Omega$ along the red solid curve of the corresponding panels (a-d). The detuning is fixed at $\delta_0 = 0$, while all the other system parameters are the same as in Fig. \ref{['fig:Fig1']}.
• Figure 3: (a) Phase diagram of detuning versus exciton density in the polariton condensate. The blue solid line bounds the region (QD) where quantum droplets can form, and it corresponds to the saturation density of the quantum droplet in phase coexistence with the vacuum. Outside this region, the system is in a superfluid miscible phase (SF). Quantum droplets only occur in the detuning range $-19.97$ meV $<\delta_0<9.6$ meV, where $\delta g\le 0$ for our parameters. (b) Absorption spectrum, featuring the four Bogoliubov branches $E_{-,\mathbf{k}}^{a/b}$ (bottom panel) and $E_{+,\mathbf{k}}^{a/b}$ (top panel), which includes a linear gapless mode. The LP and UP dispersions are plotted as white dashed lines. The cavity linewidth has been fixed to $\Gamma=0.025$ meV. (c) Droplet density profiles, obtained from a Gross-Pitaevskii (GP)-like approach (see text), for $\delta_0 = 0.0$ (black, solid line), $-3.0$ (blue dashed) and $-6.0$ meV (red dash-dotted). System parameters are as in Fig. \ref{['fig:Fig1']}.
• Figure S1: (a) Phase diagram of cavity detuning vs. pairing field. The blue solid line represents the droplet pairing fields and denotes a first order phase transition between the quantum droplet (QD) phase and a miscible superfluid (SF) phase. (b) Detuning vs. chemical potential $\delta \mu = \mu - \epsilon_{\rm LP}$. The chemical potentials in the droplet phase ($\delta \mu_{\rm D}$) form the blue solid line which separates the vacuum from the SF phase.
• Figure S2: Speed of sound extracted from a low-momentum expansion of the gapless branch $E_{-,k}^a$. We show this in (a) as a function of cavity detuning and in (b) as a function of the droplet saturation density.