Experimental Realization of a Fermionic Spin-Momentum Lattice

We experimentally realize a spin-momentum lattice with a homogeneously trapped Fermi gas. The lattice is created via cyclically rotated atom-laser couplings between three bare atomic spin states, and are such that they form a triangular lattice in a synthetic spin-momentum space. We demonstrate the lattice and explore its dynamics with spin- and momentum-resolved absorption imaging. This platform will provide new opportunities for synthetic spin systems and the engineering of topological bands. In particular, the use of three spin states in two spatial dimensions would allow the simulation of synthetic magnetic fields of high spatial uniformity, which would lead to ultranarrow Chern bands that support robust fractional quantum Hall states.

Figure 1

Experiment schematic and coupling details. (a) A degenerate Fermi gas (DFG) in a crossed optical dipole trap (ODT) is exposed to cyclic Raman couplings between three internal spin states; see text for details. A beam ${\omega }_{\text{L}\mathrm{ift}}$ provides a nonlinear energy splitting between the states. (b) States $X$, $Y$, $Z$ are connected among themselves through spin-momentum exchange in units of $\hslash {\delta \mathbf{k}}_{ij}=\hslash ({\mathbf{k}}_i-{\mathbf{k}}_j)$. With a single set of beams, no net phase pickup is possible, denoted by the 0. (c) The couplings form a lattice in momentum space. Atoms encircling plaquettes labeled by $\alpha$, $\beta$, $\gamma$ can pick up a net phase. The link color indicates the frequency set in (d) to which the beams belong. (d) Details of the resonant couplings between the three internal states labeled $X$, $Y$, and $Z$, which represent the nuclear angular momentum projections $m_F=-9/2,-7/2,-5/2$, respectively, in the ${{}^{1}S}_0(F=9/2)$ ground state. We circularly exchange the roles of the frequencies colored red (${\omega }_i$) in the blue (${\omega }_i^{\prime }$) and green (${\omega }_i^{\prime \prime }$) coupling sets.

Figure 2

Demonstration of the spin-momentum lattice. A DFG is exposed to the nine-beam coupling scheme, with the ${\mathbf{k}}_1$ triplet frequency swept in order to populate more lattice sites; see text for details. The sweep progress is indicated in units of recoil energy $E_R$. The top row shows spin-unresolved momentum-space images, in which the lattice is most clearly evident. The second row shows the predicted dynamics from simulations (Sim.). Subsequent rows show spin-resolved images. In each row, we draw a link connecting each lattice site with a color corresponding to the involved beams, as detailed in Fig. 1. In the top row, we encircle and label each new lattice site as it becomes apparent during the sweep, and in the spin-resolved rows the expected populations are also encircled. As a guide to the eye, the links are also drawn across all rows. Each image is an average of $\approx 100$ experimental runs taken at 12 ms time of flight. The peak atomic density is indicated in the top right of each image, in units of $N$ atoms per $\mu {\mathrm{m}}^2$.

Figure 3

Building the spin-momentum lattice; all images taken at a common sweep time of $12E_R$. The drawn links and circles are as described in Fig. 2, with the second row showing the predicted dynamics (Sim.). Each removed frequency corresponds to a missing link in the full lattice setup, allowing the exploration of quasi-1D SOC in the first column, to 2D SOC in the second column, culminating in the full lattice shown in the last column. The multiply-colored links in the last two columns indicate two SM lattices simultaneously overlaid: a two-state configuration, in which the experiment began with equal populations of $X$, $Y$.

Figure 4

Spin-resolved imaging process. (a) Spin-blast Rabi frequency calibrated by electron shelving of the $m_F=-7/2$ ($Y$) state. The fitted single-photon Rabi frequency is $\mathrm{\Omega }=335(1)\mathrm{kHz}$. (b) Averaged atom shot of the SM lattice at sweep time $8E_R$. (c) Spin-blast beam resonant with $m_F=-7/2$ applied to the SM lattice. (d) Subtraction of images (b) and (c), removing the common-mode background and revealing the locations of $m_F=-7/2$ atoms. A region of negative optical density forms at the locations of the blast-scattered atoms.