Two-Dimensional Homogeneous Fermi Gases

We report on the experimental realization of homogeneous two-dimensional (2D) Fermi gases trapped in a box potential. In contrast to harmonically trapped gases, these homogeneous 2D systems are ideally suited to probe local as well as nonlocal properties of strongly interacting many-body systems. As a first benchmark experiment, we use a local probe to measure the density of a noninteracting 2D Fermi gas as a function of the chemical potential and find excellent agreement with the corresponding equation of state. We then perform matter wave focusing to extract the momentum distribution of the system and directly observe Pauli blocking in a near unity occupation of momentum states. Finally, we measure the momentum distribution of an interacting homogeneous 2D gas in the crossover between attractively interacting fermions and bosonic dimers.

Figure 1

Sketch of the experimental setup: The atoms are loaded from a highly elliptic red detuned optical trap (red) into a single nodal plane of a blue detuned optical lattice (light green) which is formed by two laser beams ($\lambda =532\mathrm{nm}$) intersecting under an opening angle of $\theta =10.4°$ (a),(b). The radial confinement is provided by a ring-shaped repulsive potential (dark green) whose diameter $D$ can be adjusted between 50 and $200\mu \mathrm{m}$ (c). Panels (d)–(f) show averaged (20–50 images) in situ density profiles and the respective central line cuts at different stages of the preparation of a strongly interacting homogeneous Fermi gas at $B=830\mathrm{G}$: After evaporation in the elliptic trap (d), the outer, high-entropy region of the cloud is cut away by the repulsive ring potential (e). After further evaporation, the radial magnetic confinement is ramped down to spill the atoms outside the ring, the atoms are transferred into the lattice, and we obtain a homogeneous 2D gas (f).

Figure 2

Density EOS for noninteracting homogeneous 2D Fermi gases: The EOS is mapped out for different densities and temperatures by imprinting a repulsive potential step onto the atoms. This causes a density depletion $\mathrm{\Delta }n$ in the center of the cloud (a),(b). Measuring this density depletion $\mathrm{\Delta }n$ as a function of the step height directly yields the density EOS of the system. By fitting the data with the EOS of a noninteracting Fermi gas, we extract the temperature $T$ and chemical potential ${\mu }_0$ for each data set. The higher $T/T_F$ for the data set having the lowest density in the outer ring (red squares) is most likely due to a reduced evaporation efficiency. Using the fit results for $T$ and $\mu$ to rescale the data and plotting the dimensionless quantity $n_{2\mathrm{D},\uparrow }{\lambda }_{\mathrm{dB}}^2$ causes the different data sets to collapse onto a single curve (c). The data show excellent agreement with the prediction for a noninteracting 2D Fermi gas (solid purple line).

Figure 3

Momentum distribution of a noninteracting 2D Fermi gas: To measure the momentum distribution, we switch off the confining ring potential and let the gas evolve in a weak harmonic potential. A free time evolution $t$ for a quarter of the trap period $\tau$ performs a rotation in phase space by 90° as sketched in (d),(e), causing the momentum distribution of the gas to be mapped into real space. Averaged images (51–62 realizations) and corresponding azimuthal averages of the density and momentum distribution are shown in (a),(b) and (g),(h), respectively. After a free time evolution of half a trap period, the in situ density distribution is mapped back to real space (c); the azimuthal averages at $t=0$ (red triangles) and $t=\tau /2$ (blue dots) are almost identical (i). A diagonal cut through the momentum distribution (b) reveals the occupation $f(k)$ of the system (j), which shows close to unity occupation around $k=0$ and drops off at the Fermi wave vector $k_F=(1.93\pm 0.02)\mu {\mathrm{m}}^{-1}$ (gray dash-dotted lines). A fit with a Fermi distribution (red dashed line) yields a temperature of $T/T_F=0.31\pm 0.02$.

Figure 4

Saturation in the occupation of momentum states. The occupation $f(k)$ of noninteracting Fermi gases with in situ densities of $\overline{n}=0.24$ (light blue hexagons), 0.38 (dark blue stars), and $0.50\mu {\mathrm{m}}^{-2}$ (red diamonds) is shown in (a). For low momenta, we find a near unity occupation that is independent of the in situ density (see the inset), which is direct evidence of Pauli blocking. The Fermi wave vectors deduced from Fermi fits to the distribution agree well with the $k_{F,\overline{n}}=\sqrt{4\pi {\overline{n}}_{2\mathrm{D},\uparrow }}$ (vertical lines) calculated from the in situ density. An image of the momentum distribution of an attractively interacting Fermi gas is shown in panel (b) and a cut through the distribution in panel (c). The Fermi momentum $k_{F,\overline{n}}$ for a noninteracting gas with equal density is indicated by the red circle (b) and the vertical red line (c), respectively.