Observing Spatial Charge and Spin Correlations in a Strongly-Interacting Fermi Gas

Abstract

In this work, we explore two-dimensional attractive Fermi gases at the microscopic level by probing spatial charge and spin correlations in situ. Using atom-resolved continuum quantum gas microscopy, we directly observe fermion pairing and study the evolution of two- and three-point correlation functions as inter-spin attraction is increased. The precision of our measurement allows us to reveal nonlocal anticorrelations in the pair correlation function, fundamentally forbidden by the mean-field result based on Bardeen-Cooper-Schrieffer (BCS) theory but whose existence we confirm in exact auxiliary-field quantum Monte Carlo calculations. We demonstrate that the BCS prediction is critically deficient not only in the superfluid crossover regime but also deep in the weakly attractive side. Guided by our measurements, we find a remarkable relation between two- and three-point correlations that establishes the dominant role of pair-correlations. Finally, leveraging local single-pair losses, we independently characterize the short-range behavior of pair correlations, via the measurement of Tan's Contact, and find excellent agreement with numerical predictions. Our measurements provide an unprecedented microscopic view into two-dimensional Fermi gases and constitute a paradigm shift for future studies of strongly-correlated fermionic matter in the continuum.

Figures (11)

• Figure 1: Single-Charge and Single-Spin imaging of interacting 2D Fermi gases.(a) Phase diagram of the spin-balanced 2D attractive Fermi gas across the BEC-BCS crossover. At sufficiently low temperatures, the system is superfluid for any non-zero attraction. At higher temperatures, it exhibits well-understood Fermi and Bose liquid behaviours, but the state right above the normal-to-superfluid temperature is believed to display a pseudogap behaviour below a certain threshold (black dashed line). (b) The spin-$1/2$ mixture is imaged via atom-resolved quantum gas microscopy giving access to the total density (charge), the spin--$\uparrow$ component, and the spin--$\downarrow$ component. Single-spin images are obtained after removal of the other spin component using a resonant light pulse. Sites occupied by two atoms upon pinning appear empty after imaging due to light-assisted collisions, which we use as an independent probe of short-range correlations. (c) Raw single-charge (top) and single-spin (middle and bottom) experimental images along with their processing. In the shown charge image, fermions are seen to organise by pairs.
• Figure 2: Two-point charge and spin correlations.(a) Charge and pair correlations as a function of the interaction parameter $\eta$ (the values in the labels are rounded for readability). The diamonds at short distance are obtained via pair-loss measurements (see text). The upshoot at short range represents a direct observation of fermion pairing in real space. (b) Close-up of correlations for selected values of $\eta$. Top row: Equal-spin correlations. Middle row: Charge correlations. Bottom row: Pair correlations. Dashed lines: mean-field BCS theory. Solid lines: AFQMC calculations for the 2D gas at our experimental values of $\eta$ and reduced temperature $T/T_{\rm F}$. Dotted lines (top row): two-parameter fit of $g_{\sigma\sigma}$ (see Note1). For all interactions the measured correlations display clear deviations from the BCS prediction. Furthermore, inter-spin correlations display a dip where $g_{\uparrow\downarrow}<1$, in violation of BCS theory, which is also observed in the AFQMC results. In the weakly interacting regime we find excellent agreement between our data and AFQMC. In the crossover region interactions induce the occupation of excited $z$-levels, leading to a reduced contrast in $g_{\uparrow\uparrow}$ and expected differences with AFQMC in 2D.
• Figure 3: Three-point charge and spin correlations.(a) On equilateral triangles and for balanced spin-populations there are enough symmetries to extract all relevant three-point correlation functions from the single-spin and single-charge images. (b) Measured three-point correlations on equilateral triangles for various values of the interaction parameter $\eta$ (rounded for readability). The small difference in the values of $\eta$ with respect to Fig. \ref{['fig:fig2']} is due to the use of a slightly larger region of the cloud to measure $g_3$ (see Note1). Dashed lines: zero-temperature BCS prediction. Solid lines: AFQMC calculations at the same reduced temperatures as in the experiments, showing very good agreement for $\eta\lesssim2$, as well as in the crossover when $k_{\rm F}r\gtrsim2$. The observed discrepancies at short distance are expected due to interaction-induced population of excited $z$--levels, leading to an offset in $g_{\uparrow\uparrow\uparrow}$ and an enhanced rise of $g_{nnn}$ and $g_{\uparrow\uparrow\downarrow}$. The shaded areas are the experimental results obtained by applying Eqs. (\ref{['eq:g3wick1']}--\ref{['eq:g3wick3']}) to the measured two-point correlations, showing very good agreement for all interactions.
• Figure 4: Contact density.(a) Distribution of the atom number for spin $\uparrow$ (red), spin $\downarrow$ (blue) and both spins (purple) in a central subregion of the cloud for $\eta\approx 0.7$, $2.1$, and $7.8$. The solid black line is the sum of the average atom number in state $\uparrow$ and in state $\downarrow$. The dashed purple line is the average atom number in images where no atoms are removed. In grey, we show the expected histogram for $N_{\uparrow} + N_{\downarrow}$ in the absence of light-assisted collisions. For increasing attraction, losses due to double occupancies become larger and yield a direct measure of the probability to find a $\uparrow\downarrow$--pair on the same site. (b) Contact density as a function of $\eta$. The contact density is measured via losses in different quasi-homogeneous subregions of the cloud which correspond to different values of the interaction parameter $\eta$. The grey dashed line is the BCS mean-field prediction $1/(k_{\rm F}a)^2$. The blue dotted line is obtained from Fermi-liquid theory engelbrecht1992. The solid line is the result of AFQMC calculations at $T\, =\, 0$shi2015. Inset: linear scale contact density in the crossover.
• Figure S1: Spin balance.(a) Histogram of the atom number measured in a central region of the cloud when removing $\left|{1}\right\rangle$ (red) and removing $\left|{3}\right\rangle$ (blue) for 3 different interaction strengths. (b) Absolute difference $\Delta g_{\sigma\sigma} = g_{\uparrow\uparrow} - g_{\downarrow\downarrow}$ of the corresponding correlation functions.