Spin-orbit-coupled exciton-polariton condensates in lead halide perovskites.

Spin-orbit coupling (SOC) is responsible for a range of spintronic and topological processes in condensed matter. Here, we show photonic analogs of SOCs in exciton-polaritons and their condensates in microcavities composed of birefringent lead halide perovskite single crystals. The presence of crystalline anisotropy coupled with splitting in the optical cavity of the transverse electric and transverse magnetic modes gives rise to a non-Abelian gauge field, which can be described by the Rashba-Dresselhaus Hamiltonian near the degenerate points of the two polarization modes. With increasing density, the exciton-polaritons with pseudospin textures undergo phase transitions to competing condensates with orthogonal polarizations. Unlike their pure photonic counterparts, these exciton-polaritons and condensates inherit nonlinearity from their excitonic components and may serve as quantum simulators of many-body SOC processes.

INTRODUCTION

Spin-orbit coupling (SOC) of electrons is responsible for a number of quantum phenomena in solids, including spin Hall effect () and topological insulators (). These SOC phenomena also emerge in photonic systems (–). In planar microcavities, the natural splitting between transverse electric (TE) and transverse magnetic (TM) modes behaves as a winding in-plane magnetic field on the photon spin and results in the photonic spin Hall effect (–). If in-plane optical anisotropy is present to break rotational symmetry (), there is an effectively constant magnetic field which, in combination with , leads to a non-Abelian gauge field for photons or exciton-polaritons near the so-called diabolical point, where the fields cancel (). There have been increasing interests in recent years to introduce SOC directly via engineering a photonic system to exhibit optical analogs of SOC and topological effects from pseudo-magnetic fields and/or pseudospins (–). While these photonic systems have been demonstrated for synthetic Rashba-Dresselhaus Hamiltonians (, , –), an exciting prospect is forming exciton-polariton condensates (, ) under such an effective field to simulate a number of phenomena in SOC quantum fluids (), in analogy to what have been demonstrated in SOC Bose-Einstein condensates (BECs) in cold atoms (–). Here, we demonstrate spin-polarized exciton-polaritons and condensates in microcavities composed of single-crystal lead halide perovskites (LHPs) () known for strong light-matter coupling in microcavities (–), including topological photonic structures (, ). We take advantage of low-symmetry phases of LHPs with strong optical anisotropy, which is tunable by composition and/or temperature (), and measure the spin textures produced by + using polarization resolved Fourier space photoluminescence (FS-PL) imaging. In the present study, we focus on two regions in the in-plane momentum space (k∥): (i) the region near the so-called diabolic point (kdb) where and cancel each other and the Hamiltonian is of the Rashba-Dresselhaus form; and (ii) the region near k = 0 where condensates form from the spin-polarized exciton-polaritons as their density reaches a threshold. The present study will not probe propagating condensates near kdb; this will be subject of future studies.

In a planar distributed Bragg reflector (DBR) cavity with the presence of TE-TM splitting and optical anisotropy, the Hamiltonian describing the eigenstates in a circular polarization basis is given by (, )where the degenerate diagonal terms are cavity photon modes. k∥ is in-plane momentum with propagation angle φ, = (k∥ cos φ, k∥ sin φ); E0 is the mode energy at k∥ = 0 and m the cavity reduced mass for a given mode; βk∥2 and α represent the strength of TE-TM splitting and optical anisotropy, respectively; and φ0 is the fixed in-plane angle of with respect to the optical axis. As a 2 × 2 Hermitian Hamiltonian, Eq. 1 can be decomposed into a linear combination of Pauli matrices, which can be viewed as Zeeman interaction between an effective magnetic field and the pseudospin of a photon (text S1) ()where is the 2 × 2 unit matrix, is a vector of Pauli Matrices, and is the effective magnetic field given by the sum of and . The pseudospins of the eigenstates are either aligned or antialigned with the effective field. The k∥-dependent effective field acting on the photon pseudospin can be physically interpreted as a photonic SOC around the diabolic points.

RESULTS AND DISCUSSIONS

We directly grow single-crystal microplates of MAPbBr3 [MA (methylammonium)] or CsPbBr3 perovskites in an empty cavity formed by two laminated DBRs (Fig. 1, A and B). The natural thickness gradient of the microcavities allows us to tune the cavity resonances across the exciton resonance (Eex). The as-grown single crystals exhibit 10 to 100 μm in lateral dimensions and a few micrometers in thickness. Additional optical images (fig. S1), powder x-ray diffraction (fig. S2), atomic force microscopy (fig. S3), and reflectance spectra (fig. S4) of representative samples are provided in the Supplementary Materials. At room temperature, CsPbBr3 is in the birefringent orthorhombic structure, whereas MAPbBr3 is in the isotropic cubic structure (), and this difference results in distinctive dispersions with pseudospin textures of the resulting polaritons.

Fig. 1.

Spin textures in the anisotropic CsPbBr3 perovskite microcavities.

(A) Schematic (left) and optical image of a CsPbBr3 microcavity formed by two DBRs. The in-plane momentum is ; θ is the polar angle of emission and λ is the emission wavelength. ϕ is the azimuthal angle in the plane. (B) Optical image of a CsPbBr3 single crystal grown in the microcavity channel. (C) Constant-energy cross section at emission wavelength 565 nm from an isotropic MAPbBr3 microcavity, showing a circular ring. (D) Constant-energy cross section at 574.5 nm from an anisotropic CsPbBr3 microcavity showing two offset circular modes. k and k are aligned with the ⟨100⟩ directions of the (pseudo-)cubic perovskite structures (ϕ = 0°). (E) Dispersions along the k and k in the CsPbBr3 microcavity, showing degeneracy at particular k (marked by *) and anisotropic mode splitting at ∥ = 0. The dashed curves are theoretical SOC exciton-polaritons from Eq. 4. (F) The momentum distribution of the effective magnetic field or gauge field and the Stokes polarization vectors parallel or antiparallel to . kdb is the wave vector where the effective magnetic field vanishes. (G1), (H1), and (I1) show cuts of experimental dispersions at energies below (553.0 nm), at (548.9 nm), and above (544.7 nm) the diabolical points, respectively. (G2), (H2), and (I2) are the corresponding experimental spin textures, and (G3), (H3), and (I3) are the corresponding theoretical spin textures from the Rashba-Dresselhaus Hamiltonian. (J) The energy dispersion intersecting the diabolical point in the direction perpendicular to kdb, showing two offset parabolas.

We perform FS-PL imaging on the microcavities (fig. S5) to obtain dispersion in parallel momentum, = (k, k). Constant-energy cross sections in the energy-momentum space are shown in Fig. 1 (C and D, respectively) for MAPbBr3 and CsPbBr3 and in movies S1 and S2). The effective cavity lengths support multiple longitudinal Fabry-Pérot modes. The modes of cubic MAPbBr3 are perfectly concentric in k (Fig. 1C). By contrast, optical anisotropy in CsPbBr3 splits each mode into two ellipses that are degenerate at certain momentum points. The dispersion of the isotropic MAPbBr3 microcavity is characterized by two orthogonal polarization modes degenerate at k = 0 but split at higher k (fig. S6), indicating the presence of the winding only. By comparison, in-plane anisotropy in CsPbBr3 leads to a clear mode splitting at = 0 due to (Fig. 1E). As the cubic MAPbBr3 undergoes phase transitions at lower temperature, the dispersion markedly changes (fig. S7). The tetragonal phase exhibits a small mode splitting due to a finite anisotropy, whereas the orthorhombic phase shows much larger = 0 splitting that is even larger than those in CsPbBr3.

The high quality factor (Q = 970 ± 100, derived from the polariton linewidth below condensation threshold; fig. S8) of our microcavities ensures sufficient coupling of the excitons to the photonic modes, forming SOC polaritons. Diagonalization of the Hamiltonian (Eq. 1) gives two φ-dependent SOC photonic modes ()

When the SOC cavity photons are strongly coupled to an exciton resonance with Rabi splitting Ω, and neglecting the difference in the exciton energies of the two orthogonal polarizations (fig. S9), we obtain two spin-split lower polariton branches (LPBs) and two spin-split upper polariton branches, which are given by

Because the effective field depends exclusively on the momentum vector, not on the energy, the spin texture of each photon mode is preserved in the polariton mode (fig. S10 and text S2).

The excitonic and photonic fractions depend on the cavity detuning δ = E0 − Eex. When |δ| > Ω, the polariton modes are more photon-like for the momentum space in our imaging range, and the energy dispersion can be described by Eq. 3, as shown by dashed curves on the lower-energy modes in Fig. 1E. For these modes, the polariton emission strongly increases toward Eex at high k because of the relaxation bottleneck (). Along k Fig. 1E (left), the two anisotropic modes cross at , i.e., the diabolical points (kdb) where and cancel each other. Along k (Fig. 1E, right), the two fields add up, and the energy splitting increases with k. We reproduce the two dispersions using Eq. 3 with E0 = 2132.4 ± 0.6 meV, = 2.24 ± 0.08 meV μm−2, β = 0.34 ± 0.03 meV μm−2, and α = 7.4 ± 0.8 meV. The two modes are orthogonally polarized (figs. S11 and S12), and the pseudospins switch across the diabolical points.

As shown in Fig. 1F, the addition of (green arrows) to (orange arrows) breaks the rotational symmetry of the TE-TM doublets (), separating the TE-TM of 4π winding into a pair of two-dimensional (2D) monopoles of 2π winding. The effective magnetic field vanishes at the angles corresponding to φ = ϕ/2, and at wave vectors , giving rise to the diabolical points. In the region near these points, the Hamiltonian (Eq. 1) can be rewritten (text S3) as a Rashba-Dresselhaus Hamiltonian ()where − . Note that it is within this context that the effective magnetic field produces a non-Abelian gauge field, whereas in general it is an effective magnetic field. For our spin-split polariton modes where the diabolic points are within the measurable k∥ range, we compare the experimental pseudospin textures with predictions from the Hamiltonian (Eq. 1). We map the Stokes vectors of the two polariton eigenstates and plot their -dependent phases ϕStokes = tan−1(S2/S1), where S1 and S2 are the Stokes parameters on the Poincaré sphere (text S4). We also note that the dispersions near the diabolic point resemble the behavior near the Dirac point in graphene, with an additional slope. The Hamiltonian in (Eq. 5) is similar to the tight binding Hamiltonian of graphene, except with an additional “twist,” which reflects the fact that the gauge field can be written as a divergence, a curl, or some combination of the two, depending on the crystal angle (ϕ) (see text S3).

We show constant-energy cuts of PL intensity (Fig. 1, G1, H1, and I1), phase (Fig. 1, G2, H2, and I2), and theoretical phase (Fig. 1, G3, H3, and I3) at energies below (Fig. 1G), at (Fig. 1H), and above (Fig. 1I) the diabolical points, where > , = − , and < , respectively. In all three regions, the experimentally retrieved pseudospin textures agree with the theoretical predictions from Hamiltonian (Eq. 1). In the XY-dominant regime, a nearly constant magnetic field generates Zeeman splitting, giving rise to two modes polarized along ϕ ~ ϕ and ~, respectively. In the TE-TM dominant regime, the two modes show the expected 4π phase winding. At the energy of the diabolical points, exchanging the interior and exterior degenerate eigenstates across the diabolical points gives rise to two circular rings with exactly 2π phase windings. One can observe the pseudospin points to opposite directions at each side of the points, which agrees with the Rashba-Dresselhaus field. Moreover, the Rashba energy structure is confirmed in Fig. 1J, which displays the measured dispersion intersecting the diabolical point in the direction perpendicular to db. The two spin-split parabolas can be well described by the theoretically predicted (the solid lines in Fig. 1J). The anisotropic splitting from in MAPbBr3 in the low-temperature orthorhombic phase (fig. S13) is even stronger than that in CsPbBr3; both are two orders of magnitude higher than those in GaAs and CdTe microcavities (, , ). The magnitudes of Rashba-Dresselhaus splitting of the exciton-polaritons in our LHP microcavities are at levels previously only seen for pure photonic modes (, ). Unlike the reported SOC photonic modes that have no matter components and therefore no nonlinearity (, ), the SOC exciton-polaritons in our LHP microcavities are highly nonlinear and undergo phase transitions to competing condensates, as we establish below.

For polaritons with E0 close to Eex, we observe deviations from parabolic dispersions at high k and avoided crossings with the exciton resonance, as seen for both MAPbBr3 (fig. S6) and the CsPbBr3 microcavities (Fig. 1D and fig. S14 at different δ values). The dispersions can be reproduced by the LPBs in Eq. 4 with E0 = 2296.4 ± 0.6 meV, = 1.08 ± 0.05 meV μm−2, β = 0.10 ± 0.01 meV μm−2, α = 10 ± 1 meV, and Ω = 12 ± 2 meV, giving rise to diabolical points at k ~ 7.1 μm−1. The two polariton modes are orthogonally polarized (figs. S11 and S12), in agreement with a report by Bao et al. () on a CsPbBr3 microcavity. Note that these authors did not observe diabolic points, spin textures, or phase transition into competing condensates.

Polariton formation becomes more obvious at lower temperatures, shown here by dispersions from a CsPbBr3 microcavity at T = 77, 120, 200, and 290 K, respectively (Figs. 2, A1 to A4, and fig. S15). The dispersions of the LPBs closest to Eex are flattened at large ∥ because of avoided crossing. Our modeling with Eq. 4 yields Ω = 25 ± 3, 25 ± 3, 18 ± 2, and 16 ± 2 meV for this cavity at 77, 120, 180, and 290 K, respectively. The strong coupling is further supported by a clear cavity detuning effect seen in fig. S16, which shows a series of dispersions for positive, resonant, and negative detuning. The polariton modes inherit the spin textures from the cavity photon modes, as shown by the dispersions at 77 K of horizonal (Fig. 2B), vertical (Fig. 2B), and 45° (Fig. 2B) polarizations. Polarization angular dependences of the two polariton modes near Eex (Fig. 2B) confirm that the two spin-split polariton modes have orthogonal polarizations.

Fig. 2.

Temperature-dependent and polarization-resolved dispersions in CsPbBr3 microcavities.

(A) Dispersion of a CsPbBr3 microcavity measured at 77 K (A1), 120 K (A2), 180 K (A3), and 290 K (A4). Dashed lines are the expected optical cavity dispersions; solid lines are the modeled polariton dispersions. The corresponding Rabi splittings are 25 ± 3, 25 ± 3, 18 ± 2, and 16 ± 2 meV, respectively. (B) Polarization-resolved dispersions of a CsPbBr3 microcavity at 77 K. The polarization is (B1) horizontal, (B2) vertical, and (B3) diagonal, with respect to the direction of the entrance slit in front of a spectrometer. (B4) Polarization-resolved PL emission of the two anisotropic modes ~538 nm at k∥ = 0, showing that the two modes are mutually orthogonal and linearly polarized.

The coupling constant Ω obtained here is more than a factor of 2 smaller than the Ω = 60 meV reported by Su et al. () for a room temperature CsPbBr3 microcavity. In addition to differences in cavity structure and crystal growth procedures between our work and that of Su et al., the different Ω values may likely come from the different Eex values used. We use Eex = 2.353 ± 0.005 eV in our analysis. By examining the dispersions at different cavity detuning in our wedged samples, we consistently observe the avoided crossing at Eex = 2.353 ± 0.005 eV. Note that, unlike the nearly constant PL peak energy, the absorption peak energy blue-shifts with increasing temperature (), as is confirmed in our reflectance spectra (fig. S4). This blue shift is also accompanied by notable peak broadening with temperature, giving rise to large uncertainty in the determination of Eex at room temperature. If we use the same Eex = 2.407 eV as in (), our fitting yields Ω = 50 ± 5 meV, in closer agreement with Su et al. ().

Having established the SOC exciton-polariton modes with spin textures, we now turn to their phase transitions into competing condensates. Figure 3 (A to C) shows PL spectra at 77 K as functions of pump fluence (P) and the calculated exciton density (nex; text S4) for CsPbBr3 microcavities with increasing detuning δ = E0 – Eex = +1, −15, and −40 meV, respectively. For the most resonant cavity, δ = 1 meV (Fig. 3A), we observe two thresholds in the appearance of sharp PL peaks assigned to lasing, at Pth1 = 5.0 ± 0.3 μJ cm−2 and Pth2 = 53 ± 3 μJ cm−2, corresponding to nex ~ 1 × 1017 and ~1 × 1018 cm−3, respectively. With increased detuning, δ = −15 meV (Fig. 3B), the two lasing thresholds up-shift to Pth1 = 8.6 ± 0.4 μJ cm−2 and Pth2 = 84 ± 4 μJ cm−2. For the largest detuning, δ = −40 meV (Fig. 3C), there is mainly one threshold that is slightly different for the two SOC modes, Pth1_a = 10.6 ± 0.5 μJ cm−2 and Pth1_b = 13.2 ± 0.7 μJ cm−2 at ~537 and ~540 nm, respectively. The insets in Fig. 3 (A to C) show representative spectra, black, blue, and red (or green), for P < Pth1, Pth1 < P < Pth2 (or Pth1_a < P < Pth1_b), and P > Pth2 (or P > Pth1_b), respectively. For P < Pth1, the broad spectrum (525 to 550 nm) is spontaneous emission, and the linewidth of the polariton is 0.55 nm. For δ = +1 or −15 meV, a sharp lasing peak in one or both SOC modes at ~533 nm appears with a two–orders of magnitude decrease in full width at half maximum (FWHM) to 0.21 ± 0.01 nm, corresponding to an effective quality factor of Q = 2500. When P > Pth2, two additional lasing peaks emerge at the lower-energy cavity modes (~547 nm). For the largest detuned cavity, δ = −40 meV, the second threshold for the red-shifted lasing modes is not observed within the fluence range. Instead, we observe the lower SOC mode appearing slightly delayed and increasing after the higher SOC mode has saturated.

Fig. 3.

Condensation of SOC exciton-polaritons in CsPbBr3 microcavities at 77 K.

(A to C) 2D pseudo-color plots of PL spectra as functions of pump fluence (P; left axis) or exciton density (nex; right axis) of three CsPbBr3 microcavities with cavity detuning δ = (A) +1, (B) −15, and (C) −40 meV, respectively. The golden-dashed line in (A) marks the Mott density. Insets in (A) to (C) show normalized PL spectra at three pump fluences: (A) P = 4.0 (black), 5.7 (blue), and 108 (red) μJ cm−2; (B) P = 1.6 (black), 9.7 (blue), and 170 (red) μJ cm−2; (C) P = 6.5 (black), 13 (blue), and 97 (green) μJ cm−2. (D) Integrated PL intensity (left axis) as a function of P in a log-log scale for the main lasing peaks (blue solid triangles, 531 to 536 nm; red solid circles, 545 to 552 nm), showing the two-threshold behavior for the δ = −15 meV cavity. Also shown are the FWHMs (right axis) of the lasing peaks at ~535 nm (blue open triangles) and ~550 nm (red open circles). a.u., arbitrary units. (E) Total PL intensity (525 to 555 nm) (gray squares, left axis) and lasing peak positions (open circles, right axis) as a function of P for the δ = −15 meV cavity. (F) P dependences of integrated PL intensities (left axis) of the two lasing peaks (blue solid triangles, 534 to 538 nm; green solid circles, 538 to 541 nm) and corresponding peak FWHMs (right axis; blue open triangles, ~537 nm; and green open circles, ~540 nm) for the δ = −40 meV cavity.

A two-threshold lasing behavior has been considered as strong evidence for exciton-polariton condensation: The first is attributed to stimulated scattering to form the condensates, also called polariton lasing (, ), and the second, at density above the Mott threshold, to stimulated emission from electron-hole (e-h) plasmas (–). Polariton lasing requires the system to remain in strong coupling. With increasing nex, many-body screening reduces the exciton binding energy, causing the system to undergo Mott transition to an e-h plasma with a reduction of the oscillator strength and resulting in weak coupling to photons (). The Mott density in single-crystal CsPbBr3 or MAPbBr3 is nMott ~ 8 × 1017 cm−3 (, ). Thus, Pth1 is far below nMott, while Pth2 is within a factor of 2 to 4 (depending on detuning) above nMott. The second lasing threshold may be related to the Bardeen-Cooper-Schrieffer (BCS) polariton lasing, a mechanism predicted theoretically (–) and more recently demonstrated experimentally in a GaAs quantum well microcavity (). In the BCS mechanism, the e-h pair is no longer tightly bound as in the exciton but Coulomb correlated to form a BCS-like pair () in the so-called nondegenerate e-h plasma (). As the exciton density further increases and the e-h pair is no longer Coulomb correlated, the mechanism transitions to photonic lasing, i.e., stimulated emission from a degenerate e-h plasma (). The gain mechanisms of both BCS polariton lasing and photonic lasing follow fermionic statistics, as opposed to bosonic statistics in the BEC-type polariton lasing (). However, unambiguously distinguishing BCS polariton lasing from photonic lasing above the second threshold is challenging (), and we leave this as a subject for future experimental and theoretical investigation.

In conventional semiconductor microcavities, stimulated emission from e-h plasmas usually comes from the same cavity mode that forms the polariton condensates. However, the multiple polariton modes of our samples give rise to distinct transition behavior, i.e., the stimulated emission occurs in the next lower-energy cavity mode. This interpretation is supported by analysis of P-dependent lasing peak positions, FWHM, and dispersions. We focus on the cavity with δ = −15 meV (see fig. S17 for δ = +1 and −40 meV) and plot the P-dependent individual lasing peak intensities and FWHMs in Fig. 3D and peak positions (λP) and total PL intensity in Fig. 3E. Above Pth1, the intensity of polariton lasing at ~533 nm rises rapidly by over two orders of magnitude in a small P window (1 to 4 × Pth1), followed by a slow rise and a plateau. Above Pth2, the red-shifted stimulated emission lasing peaks at ~548 nm rise rapidly, by a similar rate as that of the polariton lasing peak. The nonlinear increases in lasing intensity above both thresholds are accompanied by blue shifts with increasing P (Fig. 3E). Above Pth1, repulsive polariton-polariton interaction induces the blue shift (), while above Pth2, the blue shift can be attributed to cavity mode renormalization as a result of carrier density–dependent reduction in the refractive index (). The effect of a Mott transition is also evident in FWHM (open symbols in Fig. 3D). While the FWHM of the polariton lasing peak increases with P, the rate of increase accelerates above nMott and is similar to the rate of increase in FWHM for the stimulated emission lasing peak above Pth2. The appearance of the additional lasing peaks that are strongly red-shifted and mode hopped to the next lower-lying cavity modes is consistent with the stimulated emission from e-h plasmas involving the emission of plasmons (). An alternative interpretation of the two-threshold behavior is multimode polariton lasing, where polariton lasing may switch modes with increasing excitation density, giving rise to distinct thresholds due to the different relaxation dynamics (). However, our observation of the second threshold being above nMott and the likely transition into weak coupling disfavors this interpretation. For comparison, we also show in fig. S18 results for a room temperature CsPbBr3 microcavity, where only one lasing threshold near nMott is observed, suggesting the transition to weak coupling across the threshold. Note that the FWHMs of the lower-energy polariton lasing peaks also increases dramatically when the exciton density exceeds nMott (Fig. 3, D and F) and are likely indicative of transition to stimulated emission.

Further evidence for polariton condensation comes from power-dependent dispersions (Fig. 4). At P ~ 0.5Pth1 (Fig. 4A), emissions from the two high-energy spin-split polariton modes closest to Eex are the strongest at || = 0 and decrease monotonically with increasing ||. By contrast, emissions from the lower-energy polaritons with large detuning are confined to the bottleneck region at high ∥. When P is above Pth1 (Fig. 4A), emission occurs exclusively at || = 0, accompanied by a stepwise rise in intensity, a marked peak narrowing, and a blue shift with increasing P (see Fig. 3), as expected for exciton-polariton condensation. When P is increased to ~Pth2 (Fig. 4A) the lasing peak near the excitonic resonance is broadened and further blue-shifted. Additional red-shifted lasing peaks emerge and attributed to the photonic lasing mechanism accompanied by plasmon emission.

Fig. 4.

Competing polariton condensates with orthogonal polarizations at 77 K.

(A) Angle-resolved PL spectra of a CsPbBr3 microcavity measured at (A1) ~ 0.5Pth1, (A2) ~ 2Pth1, and (A3) ~ Pth2. The cavity detunings are −18 and −98 meV for the higher-energy (~534 nm) and lower-energy (~550 nm) polariton modes, respectively. (B) Angle-resolved PL spectra of a CsPbBr3 microcavity with the cavity detuning of −35 meV measured above Pth1with the PL polarization along horizontal (B1), vertical (B2), and diagonal (B3) direction with respect to the entrance slit of the spectrometer. The ⟨100⟩ axis of pseudo-cubic perovskite structure is aligned parallel to the entrance slit of the spectrometer. Dashed curves are the expected optical cavity dispersions; solid curves are the model polariton dispersions.

DISCUSSION

While the fields do not determine the condensation dynamics, they are responsible for the pseudospins of the exciton-polaritons evolve as they undergo Bose scattering down the LBP and undergo phase transitions to the condensates. Below the condensation threshold, and both contribute to give the spin textures at finite ||, as exemplified near the diabolic point in Fig. 1H. As the system approaches condensation, population builds up around || = 0 and the spin texture evolves to one dominantly resulting from , i.e., two orthogonally polarized polaritons illustrated in Fig. 1G. This is most obvious when the magnitude of detuning allows polariton condensation into a pair of spin-split modes with similar intensities (Fig. 4B). Compared to conventional semiconductor microcavities, the orders of magnitude larger anisotropy in our samples can give rise to two competing condensates with different energies. The polarizations of the two condensates are determined by , not the stochastic condensate polarizations in conventional microcavity exciton-polariton condensates (, ). The competing condensation dynamics can be understood from the balance between thermodynamics and kinetics. The lower-energy anisotropic mode is thermodynamically favored, while the higher-energy anisotropic mode is kinetically favored. The kinetic argument can be understood from the smaller curvature along the path toward minimum for the lower-energy mode than that for the higher-energy one, leading to a more efficient scattering pathway toward || = 0 in the former (). While the competition between the two spin-polarized modes is always present because of the thermodynamic and kinetic factors, we find that modest detuning (δ~20 to 40 meV) makes this competition most obvious in the simultaneous presence of both modes. Too small a detuning minimizes the effective field (discussion in text S1), and too large a detuning diminishes the difference in scattering pathways, leading to preferential condensation in the thermodynamically favored mode.

Our discovery of competing exciton-polariton condensates in anisotropic LHP microcavities with artificial and tunable gauge fields offers exciting opportunities for the exploration and simulation of many-body SOC physics in the quantum fluid phase. While we demonstrate the gauge field and the interesting Rashba-Dresselhaus Hamiltonian for exciton-polaritons at finite || near the diabolic point (Fig. 1), condensation (Figs. 2 to 4) occurs at || = 0 where = 0 and the effective magnetic field results exclusively from . As an exciting future research direction, we may systematically tune the relative amplitude of and, thus, the gauge field and experimentally access momentum space away from || = 0, particularly near the diabolic point. This is possible from off-normal and resonant optical excitation to launch propagating exciton-polaritons and condensates. Theory predicts that, for these propagating SOC condensates in a finite k range above or below the diabolic point, the gauge field can result in dynamic instability, spin-textured phase separation, and strip formation in the quantum fluid (). Our finding provides a starting point for the experimental exploration of a range of phenomena in SOC quantum fluids in a solid state and high-temperature model system, as done previously in cold atom–based SOC-BECs (–).

MATERIALS AND METHODS

Microcavity fabrication

All chemicals and regents were purchased from Sigma-Aldrich and used as received, unless noted otherwise. CsPbBr3 precursor solution with a concentration of 0.4 M was prepared by dissolving stoichiometric 1:1 CsBr and PbBr2 in dimethyl sulfoxide. MAPbBr3 precursor solution with a concentration of 1.0 M was prepared by dissolving stoichiometric 1:1 MABr and PbBr2 in dimethylformamide. Two types of DBRs were used to fabricate the microcavities studied here. One type of DBR consists of pairs of alternating silicon dioxide and silicon nitride, which were deposited on quartz or silicon wafers on an Oxford PlasmaPro PECVD NPG80 instrument. The center of the stop band was measured at 544 nm with a band width of 108 nm. The other type was custom-produced by Thorlabs (subtractive magenta dichoric filter), and the center of the stop band was measured at 545 nm with a band width of 160 nm. Two opposing DBRs were bonded by epoxy, forming an empty cavity. Perovskite solutions were filled into the cavity through capillary action. The microcavity was then placed in a fused silica tube mounted in a single-zone Lindberg/Blue M tube furnace and connected to vacuum pump. The pressure in the tube was ∼27 mtorr, and the furnace temperature was ~80°C. The solvent in the cavity slowly evaporated to form single-crystal perovskites within a few days. Representative optical images of MAPbBr3 and CsPbBr3 are shown in fig. S1.

Structural characterization

The powder x-ray diffraction data were collected on as-grown samples using a Bruker D8 Advance powder x-ray diffractometer with Cu Kα radiation. The atomic force microscopy measurements were performed using a Bruker Dimension FastScan atomic force microscope.

Spectroscopic characterization

Angle-resolved PL measurements were carried out on a home-built confocal setup (see fig. S5 for the details of the setup). The reflectance spectra of the crystals were measured with the same setup using a white lamp source. Low-temperature PL measurements were performed in a cryostat (Cryo Industries of America, RC102-CFM Microscopy Cryostat with LakeShore 325 Temperature Controller). The cryostat was operated at pressures <10−7 mbar (pumped by a Varian turbo pump) and cooled with flow-through liquid nitrogen. The second harmonic of a Clark-MXR Impulse laser (repetition rate of 0.5 MHz, 250-fs pulses, and 1040 nm), and a white-light signal generated via CaF2 and Impulse laser fundamental, was used to pump a home-built noncollinear optical parametric amplifier to generate 800-nm pulses, which was used to generate 400-nm pulses via second harmonic generation. The beam size is expanded to ensure large-area illumination with a spot size ~25 μm and focused onto the sample with a far-field epifluorescence microscope (Olympus, IX73 inverted microscope) equipped with a ×40 objective with a numerical aperture of 0.6, with correction collar (Olympus LUCPLFLN40X) and a 490-nm long-pass dichroic mirror (Thorlabs, DMPL490R). The emission spectra were collected with a liquid nitrogen–cooled charge-coupled device (Princeton Instruments, PyLoN 400B) coupled to a spectrograph (Princeton Instruments, Acton SP 2300i). We used the Lightfield software suite (Princeton Instruments) and LabVIEW (National Instruments) in data collection.

Carrier density estimation

We estimated the exciton density in photoexcited LHP single crystals using the following equationin which n3D is the exciton density, R is the reflectivity of microcavity, ADBR is the absorbance of top DBR, APVK is the absorbance of the perovskite crystal, P is the average power from the pulsed pump laser, frep is the repetition rate of the pump laser, ℏω is the photon energy of the pump laser, πr2 is the area of the pump spot, and d ~ 1 μm is the thickness of the perovskite crystal (fig. S3). Assuming negligible reflectivity and negligible absorbance of the top DBR, and unity absorbance of the perovskite crystal, the upper limit of the carrier density at the first threshold (~5.0 μJ cm−2) is estimated to be ∼1.0 × 1017 cm−3, and the one at the second threshold (~53 μJ cm−2) is ∼1.1 × 1018 cm−3.

We note that while the carrier density estimations in polariton systems may have large uncertainties, they can serve as an important check of the Mott transition. In our experiment, we observed unambiguous spectral signatures of the Mott transition above the second threshold: (i) a second threshold appears, (ii) the photon lasing hops to the next, lower-energy cavity modes due to plasmon emission, and (iii) the photon lasing transition is accompanied by significant linewidth broadening and apparent blue shift of the lasing peak, which are consistent with the disappearance of a bound excitonic state.