Matter-wave Fourier optics with a strongly interacting two-dimensional Fermi gas

We demonstrate and characterize an experimental technique to directly image the momentum distribution of a strongly interacting two-dimensional quantum gas with high momentum resolution. We apply the principles of Fourier optics to investigate three main operations on the expanding gas: focusing, collimation, and magnification. We focus the gas in the radial plane using a harmonic confining potential and thus gain access to the momentum distribution. We pulse a different harmonic potential to stop the rapid axial expansion, which allows us to image the momentum distribution with high resolution. Additionally, we propose a method to magnify the mapped momentum distribution to access interesting momentum scales. All these techniques can be applied to a wide range of experiments, and in particular to study many-body phases of quantum gases.

Figure 1

Mapping between momentum and position space in phase-space representation. The dashed lines illustrate the elliptic phase-space trajectories of noninteracting particles in a harmonic potential. The phase-space distributions $n(x,p)$ at $t=0$ and $t=T_{\text{exp}}/4$ are shown in red and green.

Figure 2

Experimentally determined in situ density distribution (a) and the momentum distribution (b) of a 2D gas at a magnetic field of $B=692\mathrm{G}$, where $a_{3\text{D}}\simeq 1000a_{\text{Bohr}}$. The momentum distribution of the gas is obtained from the mapping to a density distribution using the focusing technique described in Sec. 3. At sufficiently low temperatures, we observe an enhanced occupation of low momentum states, which is not apparent in the in situ density distribution. A detailed investigation of this phenomenon will be reported elsewhere [16].

Figure 3

(a) Schematic of the axial expansion with and without the collimation pulse. The initial cloud (gray) expands strongly in the axial direction when the collimation pulse is not applied (red). With the collimation pulse, however, the axial expansion is significantly slowed down (blue). (b) Measurement of axial width as a function of TOF with (blue) and without (red) a collimation pulse. In this case, the duration of the collimation pulse is $\Delta t_{\text{col}}=0.5\mathrm{ms}$.

Figure 4

Magnifying the momentum distribution. Phase-space trajectory of a particle traveling in a combination of anticonfining and confining potentials (gray-shaded). By letting the system expand first in an anticonfining potential, a given value of $p$ is mapped to a larger value of $x$, thus leading to an effective magnification. The colors represent the hyperfine levels of ${}^6\mathrm{Li}$.

Figure 5

(a) Experimentally obtained axial width and the corresponding theoretical prediction for a cloud expanding in the magnetic potential at $692\mathrm{G}$. (b) Average number of scattering events per particle in the sample as a function of the elapsed expansion time. After a quick initial rise of $N_{\text{sc}}$, the number of scattering events saturates and reaches about $0.1$ for expansion times exceeding $20\mathrm{ms}$.