Quantum Gas Microscopy of Fermions in the Continuum

Abstract

Microscopically probing quantum many-body systems by resolving their constituent particles is essential for understanding quantum matter. In most physical systems, distinguishing individual particles, such as electrons in solids, or neutrons and quarks in neutron stars, is impossible. Atom-based quantum simulators offer a unique platform that enables the imaging of each particle in a many-body system. Until now, however, this capability has been limited to quantum systems in discretized space such as optical lattices and tweezers, where spatial degrees of freedom are quantized. Here, we introduce a novel method for imaging atomic quantum many-body systems in the continuum, allowing for in situ resolution of every particle. We demonstrate the capabilities of our approach on a two-dimensional atomic Fermi gas. We probe the density correlation functions, resolving their full spatial functional form, and reveal the shape of the Fermi hole arising from Pauli exclusion as a function of temperature. Our method opens the door to probing strongly-correlated quantum gases in the continuum with unprecedented spatial resolution, providing in situ access to spatially resolved correlation functions of arbitrarily high order across the entire system.

Figures (5)

• Figure 1: Probing in-situ correlations in the continuum.(a) Single-atom image of a non-interacting Fermi gas of $N = 331$ atoms. (b) Extraction of the $g_2$ correlation function from the experimental images. Identified atoms are marked by blue circles. For each atom, we count the number of surrounding atoms at a distance between $r$ and $r+\delta r$, as indicated by the dashed circles. (c) Extraction of the $g_3$ correlation function. We identify all pairs with certain distance $r_{12}$. For each of these pairs, we register the position $\mathbf{r}_3 =(x_3, y_3)$ of each surrounding atom, relative to the center of mass of the pair. (d) Example of measured density-density correlation function $g_2(r)$ (blue circles), zero temperature prediction (grey line), and theoretical fit yielding $T/T_{\rm F} = 0.47(7)$ (blue line, with shaded area indicating uncertainty). (e) Experimental $g_3$ for $r_{12} = 4.1k_{\rm F}^{-1}$ with the third atom positioned along the interparticle axis of the pair ($\mathbf{r}_{12}$), and theory at $T/T_{\rm F} = 0.47(7)$ (solid line and shaded area). Error bars show the standard error of the mean.
• Figure 2: Three-body correlation functions and Wick analysis.(a) Schematic showing the coordinate system definitions (top) and the region of $g_3$ plotted in the sub-panels of panel (b) (bottom). (b) Density plots and central cuts of the $g_3$ correlation function for a non-interacting Fermi gas. Each sub-panel consists of a 2D density plot of $g_3$ and a central cut ($y_3 = 0$, data points) similar to Fig. 1(e). For the respective panels we have (left to right, top to bottom): $k_{\rm F}r_{12} = 11.7, 9.7, 8.7, 7.6, 6.6, 5.6, 5.1, 4.1, 3.6, 3.1, 2.6, 2.0, 1.5$ and $1.0$. (c) Average $g_3$-values as a function of $k_{\rm F}r_{12}$ for two limiting cases, as illustrated by the inset. The dark blue data points show $\lim_{r_3\rightarrow\infty}g_3(r_{12},x_3,y_3)$. The light blue data points are taken from the center of each trace in panel (b), where $x_3 = y_3 \approx 0$. Solid lines in panels (b) and (c) show theory at the temperature obtained from the $g_2$ measurement, with shaded areas indicating uncertainties. (d) Coherence function $g_1(r)$ extracted from the $g_2-$function through Wick's theorem. (e) Comparison between the directly extracted $g_3$ correlation function from panel (b) (blue data points) and the $g_3-$function obtained through application of Wick's theorem on the extracted $g_1$ from panel (d) (purple data points). (f) Visual representation of Wick's theorem applied to the $g_3$ correlation function. Blue (white) circles represent creation (annihilation) operators. Black lines represent the different contractions that allow $g_3$ to be expressed in terms of $g_1$ correlation functions.
• Figure 3: Full characterization of a quasi-2D non-interacting Fermi gas.(a) Schematic representation of the two motional state populations in the vertical harmonic trap potential. Atoms in the $\nu = 0$ and 1 states are represented by the light and dark blue particles, respectively. (b) Top row: Experimental $g_2$-correlation functions for $N = 71, 142$ and $325$ atoms (data points)--labeled 1 through 3, respectively--with theoretical fits using $T$ as the sole fitting parameter (blue curves). The shaded regions represent fitting uncertainties. Grey curves show the $g_2$ of a purely two-dimensional gas at $T = 0$. Error bars represent the standard error of the mean. Insets show the distribution of the vertical level population obtained from the fit. Bottom row: $\nu = 0$ contribution to the $g_2$-correlation function for the respective top panels with insets showing the $g_1$-coherence function obtained from a Wick decomposition. (c - e) Temperatures ($T$), reduced temperatures of the $\nu = 0$ contribution ($T/T_{\rm F,0} = T/(T_{\rm F} p_0)$) and $p_1$-values for systems prepared at different Fermi energies ($E_{\rm F}$). Dashed lines are guides to the eye. Dark and light blue shaded areas indicate statistical and systematic uncertainties, respectively. (f) Value of $g_2^\mathrm{tot}$ (dark blue) and $g_2^{\lbrace 0\rbrace}$ (light blue) at a distance of one lattice site ($a_{\rm L}$). The grey dashed line shows the value of $g_2(a_{\rm L})$ in the pure-2D zero-temperature limit. Blue dashed lines and shaded areas serve as guides to the eye.
• Figure 4: Temperature dependence of the correlation functions.(a) Two-body correlation function at several reduced temperatures. (b) Central slices ($y_3 = 0)$ of the three-body correlation function for $k_{\rm F} r_{12} = 4$ using the same reduced temperatures as in panel (a).
• Figure 5: Three-body correlations in a quasi-2D non-interacting Fermi gas.(a) Each row shows central cuts of the $g_3$ reduced density correlations for samples prepared at different Fermi energies $E_{\rm F}$, labeled 1 through 4. Each column corresponds to approximately the same value of $k_{\rm F} r_{12} =$ 1.7, 2.8, 3.6, 4.5 and 5.8. In each row, the solid lines are theoretical predictions resulting from a joint fit of all experimental cuts (25 for each preparation) with the temperature as a single free parameter. (b) Temperatures obtained from theory fits of the $g_2$ data (dark blue) and $g_3$ data (light blue). Dark (light) blue shaded areas show statistical (systematic) errors in the temperature obtained from the fit to the two-body density correlation function, as shown in Fig. \ref{['fig:g2_data']} of the main text. (c) First excited $z$-level populations obtained from the Fermi-Dirac distributions using the temperatures in panel (b) for both the $g_2$ (dark blue) and $g_3$ (light blue) fits, with shaded areas corresponding to those shown in panel (b). The dashed lines and shaded areas serve to guide the eye.