Crescent States in Charge-Imbalanced Polariton Condensates

We study two-dimensional charge-imbalanced electron-hole systems embedded in an optical microcavity. We find that strong coupling to photons favors states with pairing at zero or small center-of-mass momentum, leading to a condensed state with spontaneously broken time-reversal and rotational symmetry and unpaired carriers that occupy an anisotropic crescent-shaped sliver of momentum space. The crescent state is favored at moderate charge imbalance, while a Fulde–Ferrel–Larkin–Ovchinnikov-like state—with pairing at large center-of-mass momentum—occurs instead at strong imbalance. The crescent state stability results from long-range Coulomb interactions in combination with extremely long-range photon-mediated interactions.

Figure 1

(a) Semiconductor quantum well embedded in a planar microcavity with net charge tuned by a gate voltage between the bottom mirror and the grounded semiconductor. (b) Occupied bands with finite excitation and charge. (c) Typical anisotropic crescent state represented by the electron occupation. This occupation reaches one at low temperatures inside the yellow crescent-shaped region. (d) $k_y=0$ momentum space slice of (c) showing both occupations and electron-hole coherence. Inside the Fermi surface [yellow in (c)], both conduction and valence bands are occupied so coherence vanishes. Elsewhere only one state is occupied. Parameters (as described in the text) are target charge density $n_0=8.125\times {10}^{-2}{a}_{\mathrm{B}}^{-2}$, excitation chemical potential ${\mu }_{\mathrm{ex}}=E_G+E_{\mathrm{B}}$, temperature $k_{B}T=0.04E_{\mathrm{B}}$, photon cutoff frequency ${\omega }_0=3.06E_{\mathrm{B}}$, matter-light coupling momentum cutoff $\kappa =2.5a_{\mathrm{B}}^{-1}$, matter-light coupling $g_0=0.8E_{\mathrm{B}}{a}_{\mathrm{B}}$, mass ratio $m_e/m_h=1$, $ϵ=1$, and capacitive energy $\alpha =800E_{\mathrm{B}}{a}_{\mathrm{B}}^2$.

Figure 2

Electron occupation $⟨{\widehat{e}}_{\mathbf{Q}/2+\mathbf{k}}^{†}{\widehat{e}}_{\mathbf{Q}/2+\mathbf{k}}⟩$ for various imbalance values $n_0{a}_{\mathrm{B}}^2$: (a) 0, (b) $6.25\times {10}^{-3}$, (c) $1.875\times {10}^{-2}$, (d) 0.125, (e) 0.1875, (f) 0.25. Labels on each panel indicate the phases as described in the text. The values of $Q{a}_{\mathrm{B}}$ are (c) $0.5\times {10}^{-6}$, (d) $0.5\times {10}^{-5}$, (f) 1.05, and zero for panels (a),(b),(e). Other parameters are as in Fig. 1.

Figure 3

Evolution of state with target charge density $n_0$ at $k_{B}T=0.04E_{\mathrm{B}}$ (left) and with temperature $T$ at $n_0{a}_{\mathrm{B}}^2=0.075$ (right); other parameters as in Fig. 1. Top panels show excitonic density (black; left axis) and charge imbalance (blue; right axis). The dashed blue line shows $n_0$. Bottom panels show anisotropy (black; left axis) and photon density ${\varphi }^2{a}_{\mathrm{B}}^2$ (blue; right axis).

Figure 4

Phase diagrams. (a) as a function of charge density $n_0$ and photon cutoff frequency ${\omega }_0$ at $k_{B}T=0.04E_{\mathrm{B}}$. (b) as a function of charge density $n_0$ and temperature $T$ at ${\omega }_0=3.06E_{\mathrm{B}}$. The dashed lines indicate ${\omega }_0=3.06E_{\mathrm{B}}$ (left) and $k_{B}T=0.04E_{\mathrm{B}}$, respectively. All other parameters are as in Fig. 1.