Observation of Pauli Crystals

The Pauli exclusion principle is a fundamental law underpinning the structure of matter. Because of their antisymmetric wave function, no two fermions can occupy the same quantum state. Here, we report on the direct observation of the Pauli principle in a continuous system of up to six particles in the ground state of a two-dimensional harmonic oscillator. To this end, we sample the full many-body wave function by applying a single atom resolved imaging scheme in momentum space. We find so-called Pauli crystals as a manifestation of higher order correlations. In contrast to true crystalline phases, these unique high-order density correlations emerge even without any interactions present. Our work lays the foundation for future studies of correlations in strongly interacting systems of many fermions.

Figure 1

Sketch of the experimental setup. The atoms are trapped in a single site of an attractive one-dimensional optical lattice in the vertical direction superimposed with a tightly focused optical tweezer [(a), top]. The degeneracy of the effectively two-dimensional harmonic confinement leads to the formation of a nontrivial shell structure [(a), bottom]. Binarized image of the $N=6$ closed-shell system taken with a single photon-counting EMCCD camera after a time-of-flight expansion (b). We extract the atom momenta by searching for local maxima in the low-pass-filtered image (c). All momenta are plotted in natural units of the harmonic oscillator confinement. To reveal correlations between the particles we subtract the center of mass motion [(c), arrow 1] and rotate to a common symmetry axis [(c), arrow 2].

Figure 2

Pauli crystal measurements with $N=3$ and $N=6$ particles. A 2D histogram of the measured momenta $p_i$ minus the center of mass momentum $\overline{p}$ leads to the one-particle relative momentum distribution of the $N=3$ system (a). The reduced density in the center is a result of the exact form of the harmonic oscillator eigenfunctions and not a consequence of antisymmetrization. The inset shows a measurement of the momentum point spread function of our imaging setup plotted on the same scale. The strong correlations between the fermions only reveal themselves in the configuration probability densities for $N=3$ (b) and $N=6$ (c), where each experimental run has been rotated separately to a common symmetry axis.

Figure 3

Angle correlations for $N=6$ particles. The angular correlation function $C_2$ is computed after removing the center of mass motion by fixing one particle at ${\varphi }_1=0$ and creating a histogram of the measured angles of the other particles. $C_2(0,\varphi )$ shows 4 maxima in addition to the hole around $\varphi =0$ as expected for the fivefold symmetry of the Pauli crystal (blue points). The measurement agrees well with a Monte Carlo simulation (dashed line). Both the total angular atom distribution $N(\varphi )$ (yellow points) and the correlation function of shuffled data (gray points) show no structure.

Figure 4

Melting the Pauli crystal. (a)–(c) $N=6$ Momentum configuration distributions for different initial state energies. We find that the Pauli crystal melts quickly when we add energy to the initial state by modulating the trap potential. (d) We extract the contrast $C$ by fitting a sine function to the distribution at a fixed momentum $p=2p_0$ [highlighted by the yellow line in (a)]. (e) The contrast reduces with increasing total mean energy of the system. (f) We compare our measurement to a Monte Carlo simulation, where we sample states from a thermal $N$-body density matrix. We follow the simulation procedure discussed in Ref. [14] and include only states up to a maximal excitation energy of $6\hslash {\omega }_r$ to reduce the computational cost. The dashed lines are linear fits to the data.