Microscopic Observation of Pauli Blocking in Degenerate Fermionic Lattice Gases

The Pauli exclusion principle is one of the most fundamental manifestations of quantum statistics. Here, we report on its local observation in a spin-polarized degenerate gas of fermions in an optical lattice. We probe the gas with single-site resolution using a new generation quantum gas microscope avoiding the common problem of light induced losses. In the band insulating regime, we measure a strong local suppression of particle number fluctuations and a low local entropy per atom. Our work opens a new avenue for studying quantum correlations in fermionic quantum matter both in and out of equilibrium.

Figure 1

Experimental setup. (a) Geometry of the physics lattice. A vertical standing wave of $3\mu \mathrm{m}$ spacing is produced by focusing two beams with an aspheric lens (L) onto the atoms from the side. The horizontal lattice with a spacing of $1.15\mu \mathrm{m}$ is formed by two pairs of beams focused through the imaging objective (Obj) beneath the glass cell (GC). (b) Pinning lattice beams viewed from the side. Two beams propagate close to the imaging objective in the $x\text{−}z$ plane, while the third beam is in the $y$ direction. All beams are retroreflected, forming a 532 nm lattice. (c) Pinning lattice oversampling and Raman beam geometry viewed from top. For imaging, the atoms are loaded from the physics lattice (red dots) to the pinning lattice (gray dots). The Raman beams (red, R1 and R2) and repump beam (blue, RP) propagate in the $x\text{−}y$ plane. For the pinning lattice, we show the sites within $\pm 250\mathrm{nm}$ from one $x\text{−}y$ plane of the physics lattice. In the experiment the relative position between pinning and physics lattice sites is not controlled. (d) Fluorescence image of $N=763(10)$ ${}^6\mathrm{Li}$ atoms acquired over 1 s of Raman sideband cooling. The black box marks the region where we evaluated the fluorescence statistics shown in Fig. 2.

Figure 2

Lattice reconstruction. Statistics of fluorescence levels obtained from the amplitude of local Gaussian fits per site. On 50 images, we evaluated the region marked in black in Fig. 1 with an average filling of 85%. The peaks in the histogram of $n=0$ and $n=1$ atoms are clearly separated (black lines are Gaussian fits). The few counts to the right are due to double occupancies [21]. The dashed lines mark the thresholds for identifying zero, one, and two atoms per site. Inset: Close-up of an exemplary fluorescence image with the results of the reconstruction shown below. Single atoms are marked with blue dots, red pentagons indicate double occupancies.

Figure 3

Local statistical analysis. We analyze the local, single-site resolved occupation statistics of 425 images. Each pixel corresponds to a physics lattice site in the 2D histograms shown in the top row. Bottom row: horizontal cuts averaged over the vertical three sites indicated by the transparent box above. The shaded regions correspond to the systematic uncertainty in the local population assignment during reconstruction. Statistical error bars represent one standard deviation. (a) Average density. Pauli blocking results in a large region of approximately uniform density around the center. (b) Normalized atom number fluctuations. The band insulator manifests itself in the suppression of fluctuations most strongly in the region of approximately flat density in (a). In the wings of the cloud, Poissonian statistics is recovered. (c) Entropy per atom. The lowest entropies of $0.3k_B$ are found in a ring around the center. The entropy increases towards the center and the edge due to the increase of double occupancies and holes, respectively. The red line is the result for the entropy, when assuming four instead of two accessible single particle states per site.

Figure 4

Absence of parity projection. (a) Overall fluorescence level. We observe a constant total fluorescence level while increasing the density by trap compression. Insets: Representative images for two different compression strengths. Approximately unity filling is observed for weak compression, whereas for stronger confinement a dense core of multiply occupied sites is visible. (b) Probability of empty (red) and doubly (green) occupied sites versus compression strength. The number of holes decreases despite an increase in the number of doubly occupied sites. The shaded area corresponds to the systematic uncertainty in the local population assignment during reconstruction. Inset: Close-up of the central region marked by the black box in (a), demonstrating the localization of high fluorescence regions to individual lattice sites. Error bars are one standard deviation of the mean.