Direct measurement of polariton-polariton interaction strength in the Thomas-Fermi regime of exciton-polariton condensation

Bosonic condensates of exciton polaritons (light-matter quasiparticles in a semiconductor) provide a solid-state platform for studies of nonequilibrium quantum systems with a spontaneous macroscopic coherence. These driven, dissipative condensates typically coexist and interact with an incoherent reservoir, which undermines measurements of key parameters of the condensate. Here, we overcome this limitation by creating a high-density exciton-polariton condensate in an optically induced box trap. In this so-called Thomas-Fermi regime, the condensate is fully separated from the reservoir and its behavior is dominated by interparticle interactions. We use this regime to directly measure the polariton-polariton interaction strength, and reduce the existing uncertainty in its value from four orders of magnitude to within three times the theoretical prediction. The Thomas-Fermi regime has previously been demonstrated only in ultracold atomic gases in thermal equilibrium. In a nonequilibrium exciton-polariton system, this regime offers a novel opportunity to study interaction-driven effects unmasked by an incoherent reservoir.

Figure 1

High-density single-mode condensation. (a) Pump power series of total photoluminescence intensity (blue) and ground-state density of polaritons (orange). $P_{th}$ is the pump threshold for condensation. (b)–(f) Real-space distribution and the corresponding (g)–(k) dispersion (spatially filtered emission from inside the trap) of polaritons at increasing pump power corresponding to labeled points in (a). The white overlay in (b) depicts the excitation profile, while the dashed lines in (g)–(k) correspond to the lower polariton dispersion. The photon-exciton detuning is $\mathrm{\Delta }=+2\mathrm{meV}$ and the ring diameter is $D=25\mu \mathrm{m}$. Typical data for negative (photonic) detunings are presented in Sec. A of the Supplemental Material [47].

Figure 2

Dependence of condensation on detuning. (a) and (b) Spatially resolved energy measurements of multimode condensation of highly photonic polaritons $\left(\mathrm{\Delta }=-18\mathrm{meV}\right)$ showing occupation of multiple excited states both at (a) just above threshold and (b) at the highest available pump power. (c) and (d) Ground-state condensation of near-resonance polaritons $\left(\mathrm{\Delta }=0\mathrm{meV}\right)$ showing a small initial blueshift (c) just above threshold and an efficient reservoir depletion at (d) higher pump power as evidenced by the bottom of the energy tail, which is also the bottom of the trapping potential, being lower than the initial blueshift. (e) and (f) Ground-state condensation of highly excitonic polaritons $\left(\mathrm{\Delta }=+20\mathrm{meV}\right)$ showing a (e) large initial blueshift due to tight trapping at threshold and (f) inefficient reservoir depletion evidenced by the higher bottom of the energy tail. The ring pump in (a) and (b) $\left(D=20\mu \mathrm{m}\right)$ and (e) and (f) $\left(D=30\mu \mathrm{m}\right)$ was created using metal masks, while the $D=40\mu \mathrm{m}$ trap in (c) and (d) was created using an axicon lens. Dashed lines are the energy minima of polaritons in the low-density limit $E_{\mathrm{LP}}^0$. The intensity is plotted in logarithmic scale to accentuate relaxation of the energy to the bottom of the trap.

Figure 3

Transition to a flat-top condensate profile in a circular box potential. (a)–(e) Spatial profile (solid lines) of the polariton density in the ground state extracted from single-shot real-space images for $\mathrm{\Delta }=+2\mathrm{meV}$ and $D=30\mu \mathrm{m}$ (a) below and (b)–(e) above condensation threshold, at the respective pump powers: $0.17P_{\mathrm{th}},1.2P_{\mathrm{th}},2.1P_{\mathrm{th}},7.7P_{\mathrm{th}},20P_{\mathrm{th}}$. (f) and (h) Numerically calculated spatial densities of the reservoir $n_{\mathrm{R}}$ (red) and polaritons $n$ (blue) corresponding to the excitation conditions in (b) and (e), respectively. The condensate density in (f) is magnified by a factor of 5. The corresponding real-space density distributions found numerically and measured experimentally are shown in (g) and (i). See Appendix pp2 for details of the numerical modeling.

Figure 4

Measurement of the polariton-polariton energy blueshift. (a) Simulated blueshift (blue points with the line as a guide to the eye) as a function of peak density for a polariton condensate in a fixed $D=20\mu \mathrm{m}$ circular box potential for $g=0.105\mu \mathrm{eV}\mu {\mathrm{m}}^2$. Red solid line corresponds to the linear Thomas-Fermi regime due to polariton-polariton interactions. (Inset) Normalized spatial profiles corresponding to the starred data points. (b) Experimental blueshift-density curves for condensates in optically induced circular traps of various diameters $D$ (measured in $\mu \mathrm{m})$ at the detuning of $\mathrm{\Delta }=+0.7\mathrm{meV}$ showing a linear Thomas-Fermi regime followed by a further increase in blueshift at high densities. Dashed line is the linear fit to the data points that correspond to a flat-top (or steep-edge) condensate profile, which is consistently achieved for trapping diameters $D\ge 29\mu \mathrm{m}$. The overall behavior is independent of the trap sizes for $D\ge 37\mu \mathrm{m}$. (Inset) Normalized spatial profiles of the condensate density corresponding to the marked data points in the linear regime for $D=37\mu \mathrm{m}$.

Figure 5

Measurement of the polariton-polariton interaction strength. (a) Values of the polariton-polariton interaction strength extracted from the blueshift measurement shown in Fig. 4 at different detunings $\mathrm{\Delta }$ relative to the Rabi splitting, $\hslash \mathrm{\Omega }$, and a constant trap diameter of $D=40\mu \mathrm{m}$. Solid curve is the theoretically predicted detuning dependence given by Eq. (2). Upper dashed curve is the theoretical prediction of the polariton-polariton interaction in the Born approximation taking into account the 3D nature of an exciton (see text). The error bars reflect the systematic error of $14\%$ due to density calibration (see Supplemental Material [47], Sec. C) and the random error arising from density inhomogenieties of the “flat-top” condensate (see Appendix pp1). The contribution of the detuning gradient to the error of this measurement is negligible. (b) Comparison of the previously reported values for the polariton-polariton interaction strength and this work (black data points).