Observation of critical scaling in the Bose gas universality class

TL;DR

The paper reports the experimental observation of critical scaling in a two-dimensional Bose gas of essentially noninteracting photons that thermalize via radiative contact with a dye reservoir inside a microcavity. By engineering a near-uniform 2D box potential and measuring the momentum distribution to extract the first-order coherence, the authors determine the correlation-length exponent $\nu=0.52(3)$, providing the first experimental test of the Bose gas universality class predictions in 2D. The work demonstrates algebraic growth of correlations approaching the Bose-Einstein condensation transition in a noninteracting system and highlights the role of finite-size effects and trap geometry via a Pöschl-Teller-based density of states analysis. This establishes critical behavior for the ideal Bose gas in 2D and opens avenues to explore nonequilibrium scaling and critical phenomena in photonic quantum gases.

Abstract

Critical exponents characterize the divergent scaling of thermodynamic quantities near phase transitions and allow for the classification of physical systems into universality classes. While quantum gases thermalizing by interparticle interactions fall into the XY model universality class, the ideal Bose gas has been predicted to form a distinct universality class whose signatures have not yet been revealed experimentally. Here, we report the observation of critical scaling in a two-dimensional quantum gas of essentially noninteracting photons, which thermalize by radiative contact to a reservoir of molecules inside a microcavity. By measuring the spatial correlations near the condensation transition, we determine the critical exponent for the correlation length to be $ν= 0.52(3)$. Our results constitute a first experimental test of the long-standing scaling predictions for the Bose gas universality class.

Figures (6)

• Figure 1: $\vert$Critical behavior in a uniform 2D Bose gas. (A) Experimental system and measurement principle. Photons are trapped and thermalize in a dye-filled microcavity formed by a plane and a nanostructured mirror realizing a nearly uniform 2D Bose gas; the imprint on the mirror forms a confining square shaped box potential. The far-field angular intensity distribution, which corresponds to the photon momentum distribution $n_k(k_x,k_y)$, is related to $g^{(1)}(s_x,s_y)$ by a Fourier transform (see text). (B) Top panel: Horizontal cut through one of the experimentally used trap potentials with depth $U_\mathrm{D} = 1.4 k_{\text{B}} T$; the potential energy is calculated from the surface height profile of the mirror. Bottom panel: Zoom-in view on the data in the bottom part of the trap (symbols) shows the slightly softened edges as compared to a rigid box (dashed green line), well described by a Pöschl-Teller potential with $\alpha=1.5$ (solid black line). (C) Surface density of a quantum degenerate photon gas with phase-space density $n\lambda^2 = 4.8(5)$ in a box of $L = 80µm$. (D) First-order correlations $g^{(1)}(s_x)$, for increasing (decreasing) particle number $N$ (reduced temperature $t$). The correlation length $\xi$ (red line; see insets) grows algebraically with a critical exponent $\nu$ and diverges at the phase transition.
• Figure 2: $\vert$Momentum distributions and growth of correlations. (A) Far-field momentum distribution $n_k(k_x,k_y)$ (top, red) and derived coherence $G^{(1)}(s_x,s_y)$ (bottom, blue) at three different particle numbers, for $N = \{8.3, 9.4, 9.5\}\cdot 10^3$ and $L = 60µm$. The corresponding reduced temperatures $t$ are given in the top panels. Below $N_{\text{c}}$ we observe short-range correlations corresponding to a Boltzmann distribution. As $N$ increases, the coherence grows due to the enhanced population at low $k$; dashed lines indicate trap depth wave number $k_{\text{D}}$ (top) and extent of the ground mode intensity which imposes an upper limit to the range of measurable correlations (bottom). All color scales are in logarithmic scale. (B) Radially averaged momentum distributions $n_k(k)$ for the quantum degenerate gas at the same values of $t$ as in (A) exhibit a thermal occupation of the excited states at $300K$ (line, vertically shifted for better visibility). (C) Radially averaged first-order coherence $g^{(1)}(s)$ with fits (lines) in the region indicated by dashed lines and the padding of the data points. The inset shows the correlations after rescaling with the fitted $\xi$, indicating the self-similarity of the correlations for the shown particle numbers.
• Figure 3: $\vert$Critical behavior at the condensation phase transition. (A) Top panels: Occupation in ground (open symbols) and excited (filled symbols) states versus particle number for the investigated system sizes and $\alpha=\mathrm{const.}$; dashed lines indicate the critical points based on the saturation of the population in the excited states as determined by piecewise linear fits. Main panel: Measured correlation length $\xi$ along with theory prediction (lines), where $L$ is treated as a free parameter that is in good agreement with the imprinted structure sizes on the mirror. The critical particle numbers $N_{\mathrm{c},\xi} = \{3512, 6232, 9549, 18946\}$ are determined from the points of divergence (dashed lines). (B) The critical photon numbers extracted with the two methods based on the onset of saturation and the correlation length divergence agree within the error margins for all investigated system sizes. The solid curve with a unity slope is a guide to the eye. (C) Correlation length as a function of phase-space density for the corresponding four system sizes. The data follows the expected exponential growth (solid line) of the infinite 2D Bose gas described in the main text (left of dashed line) until condensation sets in (right of dashed line). The error bars in all panels show standard fitting errors.
• Figure 4: $\vert$Divergence of correlations and critical exponent. (A) Correlation length $\xi$ as a function of reduced temperature $t$, exemplarily shown for $L = 50µm$ in the main panel. The solid line shows a fit (see main text), which yields $\nu= 0.51(4)$. Inset: Scaling the correlation length with the system size, all data sets collapse onto the same curve consistent with a power-law with exponent $\nu\approx 0.52$ (solid line), demonstrating the universal scaling behavior of the photon gas. The filled symbols indicate the data used for fitting $\xi(t)$, while open symbols are not included. (B) The measured critical exponents $\nu$ for all system sizes give consistent results, averaging to $\nu=0.52(3)$ (solid line). (C) Critical exponent $\nu$ against thermalization time $\tau_\mathrm{th}$ for $L = 50µm$. For $\tau_\mathrm{th}$ exceeding the average photon cavity lifetime $\tau_\mathrm{cav}$ (vertical dashed line), the measured critical exponents (green circles) exhibit a deviation from the equilibrium value (solid line).
• Figure S1: $\vert$State space in the Pöschl-Teller potential. (A) Graphical representation of the number of states $\Gamma(\epsilon)$ given in eq. \ref{['eq:S9']}. The number of states corresponds to the blue shaded section of a displaced circle, as obtained from the energy spectrum in the Pöschl-Teller potential. (B)$\Gamma(\epsilon)$ is obtained by integrating the displaced circle.