Coherence measurements of polaritons in thermal equilibrium reveal a power law for two-dimensional condensates

TL;DR

The study reveals that a spatially homogeneous two‑dimensional polariton gas in thermal equilibrium exhibits a universal power law for the coherent fraction as a function of total density, $n^{3.2}$, observed experimentally and reproduced by a number‑conserving 2D Gross–Pitaevskii model. By combining angle‑resolved BE fits to extract temperature and chemical potential with k‑space interference to quantify coherence, the authors demonstrate true equilibrium behavior in a finite 2D system and connect coherence to correlation area and BKT‑like physics. The results challenge existing analytical predictions by showing a robust, universal scaling across nearly three orders of magnitude in density, and are supported by extensive numerical simulations and cross‑sample reproducibility. This work provides a concrete, experimentally accessible link between coherence, finite‑size 2D Bose gas theory, and universal scaling laws relevant for engineered quantum fluids.

Abstract

We have created a spatially homogeneous polariton condensate in thermal equilibrium, up to very high condensate fraction. Under these conditions, we have measured the coherence as a function of momentum, and determined the total coherent fraction of this boson system from very low density up to density well above the condensation transition. These measurements reveal a consistent power law for the coherent fraction as a function of the total density over nearly three orders of its magnitude. The same power law is seen in numerical simulations solving the two-dimensional Gross-Pitaevskii equation for the equilibrium coherence. This power law has not been predicted by prior analytical theories.

Figures (19)

• Figure 1: Real-space polariton emission. Polariton emission created by a wide area nonresonant pump. The white dashed circle indicates the region where the photoluminescence (PL) is collected.
• Figure 2: Equilibrium distribution of polaritons. Occupation of the lower polariton as a function of energy. The solid lines are best fits to the equilibrium Bose-Einstein distribution in Eq. \ref{['eq:BE']}. The temperature and chemical potential extracted from the fit are shown in Fig. \ref{['fig3']}. The power values from low to high are 0.008, 0.031, 0.132, 0.530, 0.653, 0.821, 0.940, 1.164 and 1.265 times the threshold pump power. The threshold power $P_{\mathrm{th}}$ is defined in the Supplementary Information.
• Figure 3: Extracted temperature and chemical potential.(A) The effective temperature of the polariton gas and (B) the reduced chemical potential obtained from the fits to the Bose-Einstein distribution. The vertical dashed line denotes when the occupation at $E=0$ becomes equal to one, i.e $N(E=0)=1$.
• Figure 4: Interference pattern. The interference pattern in k-space obtained from the experiment (left column) and the numerics (right column) for three different densities, (A)$n = 1.5\;\mathrm{\mu m^{-2}}$, (B)$n = 4.5\;\mathrm{\mu m^{-2}}$ and (C)$n = 9.3\;\mathrm{\mu m^{-2}}$
• Figure 5: Coherent fraction. Black circles: experimentally measured coherent fraction as a function of the total polariton density for a pinhole with an area $A = \pi (6\;\mu m)^{2}$. The quasicondensate fraction is defined in the text. Red triangles: coherent fraction defined the same way, for the numerical simulations. Blue line: $n^{3.2}$ power law. The vertical dashed line denotes the critical density, which is defined as the total density of polaritons at the threshold power $P/P_{\mathrm{th}} =1$, defined in the Supplementary Information.