Supplementary Information: Room-temperature exceptional-point-driven polariton lasing from perovskite metasurface

In order to plot the dispersion surface, we calculate isofrequencies of perovskite PCS at different frequencies (energies), shown in Fig. Determination of the optical gain in the EPs model.. In the range of wavenumbers —$k_{x}/k_{0}$— and —$k_{y}/k_{0}$— ¡ 0.4 we observe two modes, corresponding to the TE and TM polarizations. TE mode has an electric field codirect with the PCS combs, and TM has a magnetic field co-directed with this direction. In the work, we study only TE mode, as it has stronger electric field localization in the material and therefore is strongly coupled to the exciton resonance. Moreover, in this geometry, only the TE mode has the EPs in the region of optical polariton gain. However, it is possible to achieve EPs in TM geometry with the variation of the sample thickness. We extract the isofreqency curves of TE mode at each energy and assemble them in the one dispersion surface, shown in the main text. It should be noted, that the stripe of the intersection points, which are identified as EPs in the experiment, is observed in calculated FMM isofrequency at 2.257 eV. The calculated isofrequency of the ASE sample in the spectral region of the ASE emission with measured Fourier plane is shown in Fig. Determination of the optical gain in the EPs model.. It is shown, that in the experiment the ASE emission is strongly broadened in comparison with the calculation. It is because of the ASE broad spectrum, which contains a broad region of the wavenumbers. Nevertheless, calculated isofreqency in the ASE spectral center well correspond to the experimental observations.

According to the two-coupled oscillator model the polariton state consists of the exciton and photon with their weights in the wavefunction, depending on the wavevector $k_{x}$. The exciton and photon fractions is labeled as $|X_{k}|$ and $|C_{k}|$ respectively and based on the coupling coefficient $g$ can be calculated as [hopfield1958theory]

$$|X_{k}|^{2}=\frac{1}{2}\left(1+\frac{|E_{X}-E_{C}|}{\sqrt{(E_{X}-E_{C})^{2}+g^{2}}}\right),\\ |C_{k}|^{2}=\frac{1}{2}\left(1-\frac{|E_{X}-E_{C}|}{\sqrt{(E_{X}-E_{C})^{2}+g^{2}}}\right)$$

(1)

The optical gain in the polariton system appears due to the polariton accumulation in the particular state on the polariton branch, where there is an extremum between the scattering probability and the lifetime.[zhang2022electric, shan2022brightening] However, calculating the optical gain is a challenging task, and therefore we roughly estimate it from the experimental results in order to show qualitatively the origin of EPs. As two of our samples: the ASE sample and the lasing sample are very similar in terms of the mode dispersions, we extract the ASE spectrum, measured from the ASE sample (Fig. Determination of the optical gain in the EPs model.a). The spectral region of the ASE is in the range of the leaky modes intersection of the lasing sample (Fig. Determination of the optical gain in the EPs model.b). We assume, that in the lasing sample the spectral region of the polariton accumulation can be considered the same as in the ASE sample. As the spectral position of the mode is connected with $k_{x}/k_{0}$ we calculate the gain profile in terms of the $k_{x}/k_{0}$, as $\beta(k_{x})=G_{0}\cdot ASE(k_{x})$, where $ASE(k_{x})$ is normalized ASE spectrum, extracted from the experimental data and $G_{0}$ is the amplitude of the gain. As was described in the main text, we subtract gain profile $\beta(k_{x})$ from the full optical losses: $\red{\gamma}=\alpha(k_{x})-\beta(k_{x})$, where $\alpha(k_{x})$ is determined as linewidths, extracted from the PL, measured under 6 $\mu$J/cm² of pump fluence, shown in Fig.Determination of the optical gain in the EPs model.c.

$$\hat{H}(k_{x})=\pmatrix{E}_{+}(k_{x})&U\\ UE_{-}(k_{x})-i\pmatrix{\red}{\gamma}_{nr}(k_{x})+\red{\gamma}_{r}&\red{\gamma}_{r}\\ \red{\gamma}_{r}\red{\gamma}_{nr}(k_{x})+\red{\gamma}_{r}$$

(2)

With varying of the parameter $G_{0}$ we calculate real and imaginary parts of the hamiltonian eigenvalues as a function of $k_{x}/k_{0}$, shown as dashed blue lines in Fig. Determination of the optical gain in the EPs model.. We put $U$ = 0, as we did not observe the splitting between two modes at the intersection point. To observe the appearance of the EPs we plot the eigenvalues difference of the real and imaginary parts ($\Delta E$, $\Delta\red{\gamma}$), shown in Fig.Determination of the optical gain in the EPs model.. In the linear regime, where $G_{0}$ = 0 we observe the splitting between eigenvalues (Fig. Determination of the optical gain in the EPs model.a). With increasing $G_{0}$ we observe that the difference between the real and imaginary parts of eigenvalues starts to decrease (Fig. Determination of the optical gain in the EPs model.b). At the critical value of $G_{0}$, equal to 0.00117, we observe the degeneracy of the eigenvalues, at some particular values of $k_{x}/k_{0}$, which are supposed to be the appearance of the EPs (Fig. Determination of the optical gain in the EPs model.c). Also, EPs are considered a very sensitive state to the environment, and as we also observe, if we increase $G_{0}$ more, the degeneracy disappears (Fig. Determination of the optical gain in the EPs model.d). However, due to the nature of the polariton lasing, we assume, that in the experiment the gain never exceeds the critical value. When the gain achieves the critical value, the EPs polariton condensation state appears there. It means, that all new polaritons in the state promptly recombine, emitting coherent photons. In this polariton system, as described in this work, EPs can be achieved through nonlinear processes of the polariton relaxation, which results in the polariton lasing.