Spontaneously coherent orbital coupling of counterrotating exciton polaritons in annular perovskite microcavities

Exciton-polariton condensation is regarded as a spontaneous macroscopic quantum phenomenon with phase ordering and collective coherence. By engineering artificial annular potential landscapes in halide perovskite semiconductor microcavities, we experimentally and theoretically demonstrate the room-temperature spontaneous formation of a coherent superposition of exciton-polariton orbital states with symmetric petal-shaped patterns in real space, resulting from symmetry breaking due to the anisotropic effective potential of the birefringent perovskite crystals. The lobe numbers of such petal-shaped polariton condensates can be precisely controlled by tuning the annular potential geometry. These petal-shaped condensates form in multiple orbital states, carrying locked alternating π phase shifts and vortex–antivortex superposition cores, arising from the coupling of counterrotating exciton-polaritons in the confined circular waveguide. Our geometrically patterned microcavity exhibits promise for realizing room-temperature topological polaritonic devices and optical polaritonic switches based on periodic annular potentials.


Dispersion simulation and fluorescence image
Fig. S1a represents the theoretical momentum-space polaritonic dispersion of the ring with a diameter of 3 m, in good agreement with the experimental data of Fig. 1c of the text. The polaritonic energy separates into multiple discrete coupled orbital modes (COMs). This is obtained from a Fourier transform of the wavefunction ( , , ) into the energy domain in reciprocal space ( , , ). Theoretically, the reciprocal-space dispersion is given by = | ( , , = 0)| 2 , which is a function of and E at = 0.
The fluorescence image of the micro-rings was obtained through an Olympus microscope, where the sample was illuminated by an Olympus U-HGLGPS lamp. In with PMMA is brighter than the exposed areas without PMMA. Thus, one can speculate that the Q-factor inside the ring structure is higher than that of the outside. Consequently, we take into account a different local lifetime inside and outside the potential well in our theoretical model. eigenvalues from 2.95 eV to 3.05 eV possess higher imaginary eigenvalues than other states, which implies massive polariton condensates will occupy in such states. In Fig.   S3, the states above the dashed line possess higher gain, which indicates where the condensation occurs.

FIG. S3
The relationship between the imaginary part of eigenvalue (Im{E}) and the real part of eigenvalues (Re{E}).

Polariton condensates of the 5 m ring
As the mode energy increases, for the COMs | 0,6 ⟩ to | 0,11 ⟩, the number of lobes sequentially increases. When the energy of condensates reaches certain energy, a superposition state of higher-order COMs with a radial index of p = 1 appears, forming a double-annulus-shaped pattern of Fig. S4h. It is worth mentioning that the patterns of | 0,6 ⟩ and | 0,7 ⟩ modes are a little asymmetric, and some lobes are displaced. The reason is the external perturbation or potential symmetry breaking (i.e., a defect or a mutation in structure), resulting in the two energy degenerate modes slightly separating in energy. Nevertheless, the coupling of condensate flows can still stably form the standing-wave pattern throughout the entire annulus, and the pattern symmetry is affected by the defect, which is highly sensitive to defects and potential disorder.
FIG. S4 (a) Momentum-space dispersions of polariton condensates of a 5 m micro ring at = 0 in annular potential well above the threshold at room temperature. The gray dashed lines represent the energy selections of the real-space imaging of condensates. (b-h) Experimental real-space images of petal-shaped polariton condensates with different azimuthal index l (from 6 to11) and radial index p = 1 in the micro ring with a diameter of 5 m, corresponding to the gray dashed lines in (a).

Polariton condensates of the 10 m ring
The geometric parameters of the annular potential determine the number of lobes.
The larger the diameter of the ring, the more the number of lobes, and the larger the azimuthal and radial index of observed condensates, as shown in Fig. S5. The observed petal-states with a single ring are originated from the coherent coupling of two pure orbital modes with zero radial and opposite azimuthal indices, and the relationship between the number of lobes and azimuthal index is = 2 .
FIG. S5 (a-d) Experimental real-space images of petal-shaped polariton condensates with different azimuthal index l (l=16, 20, 21, and 22) and radial index p = 0 in the micro ring with a diameter of 10 m.

Degenerate antisymmetric pattern
Each energy level of condensate states possesses two orthogonally polarized and energetically degenerate antisymmetric patterns. Two counterpropagating exciton polariton flows are excited simultaneously, resulting in the observed standing wave patterns. The geometry in Fig. S6b is orthogonal with Fig. S6a. The phase distributions of both Fig. S6c and d display a stable alternating  phase jump between neighboring lobes, as well as a vortex-antivortex superposition localized in the core of the annulus.
FIG. S6 (a, b) Experimental real-space images and (c, d) the phase of two-fold degenerate petal-shaped polariton condensates in the annulus with a diameter of 3 m. The azimuthal index is six (p = 0, l = 6). The insets are the distribution of geometric symmetry for the lobes of the petal-shaped condensates. The orange balls in the insets of (a) and (b) represent the lobes.

The Setup of Michelson interferometer
The emergence of long-range spatial coherence of polariton condensates can be demonstrated by a Michelson interferometer with a retroreflector in one arm, which allows a centrosymmetric inverting image, as shown in Fig. S7a. In our sample, the real-space image of petal-shaped condensates (p = 0, l = 6) above the threshold is sent into the interferometer, then the signal superimposed with its reverted image is collected.
The interference fringe contrast is the manifestation of phase coherence between points r and -r with respect to the center. In the simulated interferogram image of Fig

Orbital angular momentum Poincaré sphere representation of the degenerate states
The arbitrary state | ⟩ in orbital angular momentum (OAM) Poincaré sphere (PS) representation can be described by left-rotation eigenvector | + ⟩ and right-rotation eigenvector | − ⟩, as following: can create more orbital states on PS.

The influence of linear polarization for petal-like polaritons
In the annular microcavity, we have measured the angle-resolved PLs with the different polarizations ( Fig. S9a and b), which show the energies of horizontal (H) polarization modes are higher than the corresponding vertical (V) polarization modes, due to the V-H linear polarization splitting (~10 meV at zero-momentum). In the theoretical calculation, we take the linear polarization splitting from the birefringence effect into consideration, then introduce the linear polarization splitting strength into Eq. 1 of the main text as follows:

The transition from the strong to the weak coupling regime
To prove the transition from a strong coupling region to a weak coupling region on the annular sample, we have measured the intensity, linewidth, and spatial image of polaritons emission from low pump fluence to ultrahigh pump fluence. Our system is more likely to reach a sufficiently high carrier density in the reservoir for the formation of an electron-hole plasma (EHP) and the appearance of an eventual photonic lasing process in the weak coupling regime. With these additional results, we have brought clear proof that the nonlinear behavior observed in our perovskite samples corresponds to a polariton lasing effect induced by the formation of a polariton condensate, and not to the more classical photonic lasing effect observed in the weak exciton-photon coupling regime. The transition from a strong coupling region (polaritonic lasing) to a weak coupling region (photonic lasing) is supported by the dispersions, spatial images, and the power dependence measurements.
FIG S11. Characterizations of polariton lasing in the annular perovskite microcavity. a, b, the emission intensity and linewidth of the state with OAM of 1, at k = 0 as a function of pump fluence. A typical S shape power-dependent emission relationship suggests the occurrence of polariton lasing with a first threshold Pth of 26.7 J cm -2 , while the emission intensity decreases sharply crossing a pump fluence Pth2 of 310 J cm -2 . The linewidth first slightly increases below threshold Pth and narrows from 2.017 nm to 0.55 nm when crossing the threshold Pth, and then decreases with the further increase of pump fluence. c, Edge emission intensity as a function of pump fluence, extracted from the real space images of the perovskite microcavity. A clear second threshold Pth2 of 310 J cm -2 was observed for the emission in the lateral cavity modes.

Linear polarization splitting in a birefringent perovskite microcavity
At room temperature, our perovskite is known to be orthorhombic with birefringent effects, which has been shown to support anisotropic polaritons before (Nat. Phys. 16, 301 (2020)). Here, we provide more experimental evidence regarding the birefringence effect of this system. We can define the long (Y) and short (X) axes from the optical microscopy image of Fig. S12a. The birefringent behavior in our system could also be recognized from the polarization-resolved polariton dispersions. In the planar microcavity, Fig. S12b shows the zero wavevector emission has energy splitting up to 11 meV under V and H linear polarization, which arises from the birefringent effect of the CsPbBr3 perovskite. The ratio between the two effective masses is around 0.7 from the fitting, which is in good agreement with the previous report. The same energy splitting could also be observed for the crystal X axis under the V and H polarizations.
The curvatures of these four dispersions (YV, YH, XV, XH) are different, indicating the relationship of four effective masses is XV < YV < YH < XH. There are two kinds of splitting in our system. One is the linear polarization splitting which enables V and H linear polarization mode to split in energy. The other one is the polariton dispersion with an elliptic crossed section, i.e., anisotropic effective mass, which is significant to the counterpropagating polariton coupling and their spatial pattern distribution.

The propagation of polaritons in the 10 m annular microcavity
To prove the propagation of polaritons, we have focused the femtosecond pulse laser with the smaller spot (1 m) on the left side of the annulus (diameter: 10 m) to excite the sample (Fig. S13). An unambiguous long-range propagation and a clear petal-like pattern are demonstrated by the spatial image. The coupled modes for a given OAM are populated, polaritons will have propagation around the ring. The petal-like pattern does not depend on the size and position of the excitation spot, the long-range propagation is supported by the experiments of the excitation covering one side of the ring and the whole ring.
FIG S13. Real-space image of the petal-shaped polariton condensate above the threshold, where a pulse laser beam of the diameter of 1 m is focused on the left side of the 10 m ring to excite the sample. The red circle represents the position of the excitation laser.