Universal Random Matrix Behavior of a Fermionic Quantum Gas

Abstract

The pursuit of universal governing principles is a foundational endeavor in physics, driving breakthroughs from thermodynamics to general relativity and quantum mechanics. In 1951, Wigner introduced the concept of a statistical description of energy levels of heavy atoms, which led to the rise of Random Matrix Theory (RMT) in physics. The theory successfully captured spectral properties across a wide range of atomic systems, circumventing the complexities of quantum many-body interactions. Rooted in the fundamental principles of stochasticity and symmetry, RMT has since found applications and revealed universal laws in diverse physical contexts, from quantum field theory to disordered systems and wireless communications. A particularly compelling application arises in describing the mathematical structure of the many-body wavefunction of non-interacting Fermi gases, which underpins a complex spatial organization driven by Pauli's exclusion principle. However, experimental validation of the counting statistics predicted in such systems has remained elusive. Here, we probe at the single-atom level ultracold atomic Fermi gases made of two interacting spin states, obtaining direct access to their counting statistics in situ. Our measurements show that, while the system is strongly attractive, each spin-component is extremely well described by RMT predictions based on Fredholm determinants. Our results constitutes the first experimental validation of the Fermi-sphere point process through the lens of RMT, and establishes its relevance for strongly-interacting systems.

Figures (6)

• Figure 1: Same-Spin Counting Statistics in an Interacting Fermi Gas.(a) Random matrix theory (RMT) allows to describe universal statistical behaviors in complex systems, from the spectra of localized systems or atomic nuclei to financial trends, by means of several random matrix ensembles. (b) Continuum gas microscopy images give direct access to the number of fermions in a disk of radius $R$. (c) Measured probability to find $N$ atoms ($P_N$, $0\leq N \leq 5$) in a disk of variable adimensional radius $k_{\rm F}R$ (circles). The measurement is performed on one spin component of an attractive two-component Fermi gas of reduced temperature $T/T_{\rm F}=0.15(1)$. The standard deviation of the data, obtained from an average over 100 disk locations, is smaller than the marker size. Solid lines show the theoretical prediction from Fredholm determinant calculations at the experimentally measured temperature. Dotted lines correspond to Poisson statistics, describing a classical (uncorrelated) gas.
• Figure 2: Hole probability and nearest neighbor spacing distribution in a 2D Fermi gas. Both quantities are extracted from samples of reduced temperature $T/T_{\rm F} = 0.15(1)$. Data is compared to theoretical predictions for $T= 0$ (blue dashed line) and $T/T_{\rm F} = 0.15$ (solid blue line), and the high-temperature Poisson behavior (dotted blue line). (a) Experimentally measured hole probability $P_0$ as function of $k_{\rm F}\cdot R$ (circles), shown in semi-logarithmic (main panel) and linear (inset) scale. Error bars represent the standard deviation of measurements taken across different probe disk locations. Also shown is the 1D asymptotic behavior $P_0 (r\rightarrow \infty) \propto e^{-r^2/2}$ (black dash-dotted line). (b) The measured nearest-neighbour spacing distribution is represented as a light-blue histogram and blue circles, and compared to the predicted histogram for the same binning. The short range measurement extracted from the density-density correlation function Eq. \ref{['approx_nns']} is shown in white circles, and the dash-dotted line shows a cubic fit to the short-range behavior of the theoretical prediction at $T=0$.
• Figure 3: Temperature crossover of the Counting Statistics.(a-f) From top to bottom: data corresponding to samples of reduced temperatures $T/T_{\rm F}= [0.15(1),0.30(2), 0.53(3), 0.77(5), 2.1(2), \sim 20]$ and vertical ground state populations $p_0=[97.4(9)\%, 92.3(2)\%, 87.2(3)\%, 83.3(4)\%, 72.5(1.2)\%, \sim 23\%]$. Shown are the probability to find 0 (a), 1 (b), 2 (c), 3 (d), and 4 (e) atoms in a circular probe volume of variable radius $k_{\rm F}R$, as well as the nearest-neighbor spacing distribution (blue disks) as function of the scaled spacing $s$(f), including the short-range behavior (white disks) obtained from the measured density-density correlation function Eq. \ref{['approx_nns']} (see text). Like in figure \ref{['fig:fig2']} b, the measured NNS distributions are obtained from histograms of measured spacings, but we only show the mean probability of each bin for legibility. Dashed lines show the zero temperature predictions and dotted lines the high-temperature Poissonian behavior. In the $2^{\rm nd}$ to $5^{\rm th}$ rows, solid lines present finite temperature numerical results, calculated independently using the experimentally determined reduced temperatures $T/T_{\rm F}$ and vertical state populations $p_\nu$. The data in the first row corresponds to samples with $\eta=3.7(2)$, and were already displayed in Figs. \ref{['fig:fig1']} and \ref{['fig:fig2']}.
• Figure S1: Additional hole probabilities and nearest neighbor spacing distributions for 2D Fermi gases. (a-b) Experimental measurement of the hole probability $P_0$ (blue markers) as a function of $k_{\rm F}R$ for samples at interaction strength $\eta = 2.1(2)$ (a) and $\eta = 7.8(5)$ (b), respectively, with reduced temperatures $T/T_{\rm F}=0.11(1)$ and $T/T_{\rm F}=0.17(3)$, presented in semi-logarithmic (main panels) and linear (insets) scales. Dashed (resp. solid) blue lines are theoretical predictions at zero temperature (resp. experimentally measured temperatures). Additional curves show the high-temperature Poisson behavior (dotted blue lines) and the zero temperature asymptotic behaviors in 1D (black dot-dashed lines). (c-d) Histograms (light-blue), combined with data points (blue circles), of the nearest-neighbour spacing distributions $p(s)$ for $\eta = 2.1(2)$ (c) and $\eta = 7.8(5)$ (d). The corresponding finite temperature predictions are shown with blue solid lines and histograms using similar binning. We present the classical Poisson behavior (blue dotted curves) and the zero temperature prediction for 2D fermions (blue dashed curve). The short range measurements extracted from the density-density correlation functions (Eq. \ref{['approx_nns']}) is shown as white circles. The dash-dotted lines are a cubic fit of the zero temperature prediction at short range.
• Figure S2: Additional counting statistics. Data points are the probabilities $P_N$ to find $N=[0,1,2,3,4]$ atoms in a circular probe volume of variable radius $k_{\rm F}R$, as well as the nearest-neighbor spacing distribution $p(s)$ as function of the scaled spacing $s$ for $\eta = 2.1(2)$ (first row) and $\eta = 7.8(5)$ (second row), corresponding to $T/T_{\rm F}=0.11(1)$ and $T/T_{\rm F}=0.17(3)$ respectively. Dashed lines show the zero temperature predictions and dotted lines the high-temperature Poissonian behavior. The short-range behavior (white disks) obtained from the measured density-density correlation function Eq. \ref{['approx_nns']} (see main text).