Non-Hermitian trapping of Dirac exciton-polariton condensates in a perovskite metasurface

Abstract

Massless Dirac particles avoid trapping due to their exceptional tunneling properties manifested in the so-called Klein paradox. This conclusion stems from the conservative treatment, but so far, it has not been extended to a non-Hermitian framework. Recently, driven-dissipative bosonic condensation of Dirac exciton-polaritons was demonstrated in metasurface waveguides. Here, we report an experimental observation of spatial binding and energy quantization of Dirac exciton-polaritons in a halide perovskite metasurface. A combination of spatially profiled nonresonant optical excitation and exciton-polariton interaction forms an effective non-Hermitian complex potential responsible for the observed effect. In the case of tightly focused pump spots spanning from 9 to 17~$μ$m, several bound states simultaneously achieve macroscopic occupation, constituting a multi-mode bosonic condensation of exciton-polaritons. Our theoretical analysis based on the driven-dissipative extension of the Dirac equation reveals that the non-Hermitian character of the effective trap allows for confinement even in the case of the gapless Dirac-like photonic dispersion, both above and below the energy of the dispersion crossing.

Figures (3)

• Figure 1: (a) A sketch of the locally pumped MAPbBr$_3$ metasurface and the effective optically induced non-Hermitian potential resulting in trapping and size-quantization of a nonequilibrium polariton condensate. (b) A model of the step-like non-Hermitian potential $U(x)$ (red line) and the condensate density profile (green solid line) for the highest growth rate state with the energy between the pristine $E_0$ (yellow) and blue-shifted $E^{\prime} = E_0 + {\rm Re}(U)$ (cyan) Dirac points of free polariton dispersion shown with dashed crosses. (c) Real-space image of the quantized polariton condensation emission. White dots represent the intensity profile along the $x$-axis, while the yellow line corresponds to the results of the full theoretical model. (d) Exciton-polariton energy dispersion (yellow solid line) emerging from strong coupling of exciton (red dashed line) and photonic waveguide (beige dashed line) modes in the measured angle-resolved emission spectrum under a femtosecond pump below the threshold pump fluence $P < P_{\rm th}$. The blue dashed rectangle illustrates the axes limits in panel (e). (e) Angle-resolved emission spectrum of a quantized polariton condensate above the pump threshold. The cyan and yellow lines schematically show polariton dispersion within and outside the pump region, respectively. (f) The total emission intensity as a function of pump fluence (solid colorized line, left vertical axis). The $Q$-factor, as a ratio of central energy $E_{i}$ and lasing linewidth $\Delta E$, of the dominant emitted mode is shown with circles (right vertical axis). The gray circles describe a broadband emission below threshold. (g) The emission spectra integrated over the whole momentum $k_{\parallel}$ range for various pump fluences, demonstrating evolution of the quantized states. The colors correspond to the pump fluences shown in panel (f).
• Figure 2: Integrated over the angles, measured (a) and numerically modeled (f) emission spectra as a function of pump fluence. Measured angle-resolved emission spectra at pumps of 1.05, 1.5, 2.0, and 4.5 of $P_{\rm th}$ (corresponding to 1.32, 1.94, 2.64, and 5.82 ${\rm mJ}/{\rm cm}^2$ respectively) (b-e) and numerically modeled angle-resolved emission spectra at the corresponding pump strengths (g-j).
• Figure 3: (a) The complex energy diagram of the eigenvalue problem \ref{['eq:eigenValueProblem']}. The parameters are $\alpha/\beta = 4$, $D=0.3\hbar v_g/\Gamma_{\rm p}$, $V=(0.1+0.5i)\Gamma_{\rm p}$, where $\Gamma_{\rm p}$ is the polariton linewidth. The energy level is counted from the Dirac point position $E_0 \approx 2.26$ eV. (b) The imaginary part of the complex eigenenergies versus peak reservoir density $N_0$ measured in units of $\Gamma_{\rm p}/\beta$. The dashed segments correspond to the solutions with exponentially divergent wavefunctions. The thick segments mark the state with the highest growth rate, which dominates in the emission spectrum. (c) Localization efficiency criterion $f_{\rm out}$ describing the fraction of polaritons localized outside the trap. The inset shows polariton densities of the state $A_2$ for the three values of potential height $N_0$ marked with color dots. (d) Reciprocal space cross-sections of quantized levels at $N_0\beta/\Gamma_{\rm p}=5$ (the gray dashed line in panel (a)); cf. with Fig. \ref{['fig1_v1']}e. (e) The phase diagram of the state with the highest growth rate, which dominates in the emission spectrum. The vertical axis shows the ratio between the repulsion $\alpha$ and gain $\beta$ parts of the complex trapping potential \ref{['eq:potential']}. The color shows the real part of the dominating mode eigenenergy measured in units of $\Gamma_{\rm p}$. (d) The phase diagram on the $(D,N_0)$ parameter space. In (e) and (f), all the states are below threshold within the white region, while in the pale-red region, the condensation occurs near the blue-shifted Dirac point as sketched in the inset to panel (f). The dashed horizontal lines indicate the parameters corresponding to panels (a)-(c), relevant to our experimental conditions.