Observation of Density-Dependent Gauge Fields in a Bose-Einstein Condensate Based on Micromotion Control in a Shaken Two-Dimensional Lattice

We demonstrate a density-dependent gauge field, induced by atomic interactions, for quantum gases. The gauge field results from the synchronous coupling between the interactions and micromotion of the atoms in a modulated two-dimensional optical lattice. As a first step, we show that a coherent shaking of the lattice in two directions can couple the momentum and interactions of atoms and break the fourfold symmetry of the lattice. We then create a full interaction-induced gauge field by modulating the interaction strength in synchrony with the lattice shaking. When a condensate is loaded into this shaken lattice, the gauge field acts to preferentially prepare the system in different quasimomentum ground states depending on the modulation phase. We envision that these interaction-induced fields, created by fine control of micromotion, will provide a stepping stone to model new quantum phenomena within and beyond condensed matter physics.

Figure 1

Atoms in a two-dimensional shaken lattice. (a) A 2D square lattice (orange surface) is shaken by inducing periodic displacements $\delta x$ and $\delta y$ along the $x$ and $y$ axes, respectively, (arrows) with equal amplitude $s$ at frequency $\omega \equiv 2\pi /\tau$, shaking period $\tau$, and relative phase ${\theta }_s$. (b) Shaking above the critical amplitude $s>s_c$ results in a single particle dispersion with four degenerate minima in the ground band at $\mathbf{q}=(+q^{*},+q^{*})$, $(-q^{*},+q^{*})$, $(-q^{*},-q^{*})$, and $(+q^{*},-q^{*})$, denoted, respectively, by red, black, blue, and white dots. (c) The shaking phase ${\theta }_s$ controls the polarization of the lattice displacement. The polarization does not affect the single particle dispersion shown in (b).

Figure 2

Interaction-momentum coupling due to micromotion. (a) Examples of micromotion for linear shaking (${\theta }_s=0°$). Snapshots of the density ${|{\psi }_{\mathbf{q}}(x,y,t)|}^2$ within a single 2D lattice site are shown for two states, $(+q^{*},+q^{*})$ (red) and $(-q^{*},+q^{*})$ (black), within a shaking period $\tau$. (b) As a result of the micromotion, the mean microscopic density $⟨n_{\mathbf{q}}(t)⟩$ oscillates and reaches a maximum when the wave function is most localized, and a minimum when it is most delocalized. Each curve is colored as in Fig. 1; note that the density oscillations of the white state are identical to the plotted black curve. Dashed lines show the averaged densities. (c) Maps of the interaction factor ${\eta }_{\mathbf{q}}$, equal to the time-averaged microscopic density (see text), for different polarizations. The colored dots mark the ground states after accounting for the interaction factor. Note that circular polarization retains the $D4$ symmetry of the single particle dispersion.

Figure 3

Observed coupling of interaction and momentum. (a) Example, reconstructed domain structures (see text) representing the density profiles of atoms in each well, measured after crossing the phase transition with the shaking polarizations indicated on each image. The dashed circles guide the eye to the region containing the condensate. The correspondence between color and pseudospin density (see text) is shown in the upper-right corner of panel (b). (b) The imbalance $D$ (see text) of the atomic populations between the two quasimomentum diagonals characterizes the anisotropy which results from the quasimomentum-dependent interactions for different polarizations. The solid curve is a sinusoidal fit. The orange, dashed curve shows the expected imbalance in the absolute ground state; the star emphasizes that the expected imbalance is $D=0$ for circular shaking (${\theta }_s=90°$).

Figure 4

Density-dependent synthetic field from synchronized shaking and interaction strength modulation. (a) The upper panel plots the mean, microscopic density for circular shaking (${\theta }_s=90°$). Each curve is colored as in Fig. 1. The lower panel shows the modulated interaction strength $g(t)=g_0-g_1\cos (\omega t-{\theta }_g)$. The modulated interactions raise (lower) the energy of quasimomentum states whose density oscillates in phase (out of phase) with the interaction modulation. (b) Modulated interaction factors for ${\theta }_g=90°$ (left) and ${\theta }_g=45°$ (right). (c) Measurement of the average quasimomentum of the condensate ($q^{*}=0.08q_L$) in the presence of the interaction-induced field (circles). Error bars show standard error. The dashed curves show simultaneous, sinusoidal fits, which yield a phase offset of only $4\pm 3°$ from expectations. Simulations using the Gross-Pitaevskii equation [40] (solid magenta curves) agree well with the experiment.