Realization of a Floquet-Engineered Moat Band for Ultracold Atoms

We experimentally engineer a moatlike dispersion in a system of weakly interacting bosons. By periodically modulating the amplitude of a checkerboard optical lattice, the two lowest isolated bands are hybridized such that the single particle energy displays a continuum of nearly degenerate minima that lie along a circle in reciprocal space. The moatlike structure is confirmed by observing a zero group velocity at nonzero quasimomentum and we directly observe the effect of the modified dispersion on the trajectory of the center of mass position of the condensate. We measure the lifetime of condensates loaded into different moat bands at different quasimomenta and compare to theoretical predictions based on a linear stability analysis of Bogoliubov excitations. We find that the condensate decay increases rapidly as the quasimomentum is decreased below the radius of the moat minimum, and argue that such dynamical instability is characteristic of moatlike dispersions, including spin-orbit coupled systems. The ground state of strongly interacting bosons in such degenerate energy landscapes is expected to be highly correlated, and our work represents a step toward realizing fractional quantum Hall-like states of bosons in an optical lattice.

Figure 1

Generation of a moat band via amplitude modulation. (a) Lowest bare energy bands of the checkerboard lattice, plotted versus quasimomentum $\overrightarrow{q}=(q_x,q_y)$ over the first Brillouin zone. (b) Floquet-Bloch band with nearly degenerate moatlike minima, resulting from hybridization of the bands in (a). (c) Cross section along $q_y=0$ of the quasienergy spectrum $ϵ(\overrightarrow{q})$ showing both dressed bands. Left: adiabatic preparation of a condensate, initially occupying the bare ground band, into the upper band by sweeping the drive frequency down, starting above resonance. Right: preparation into the lower band by sweeping the frequency up. (d) Peak-to-peak variation of $ϵ$ along the moat, $\mathrm{\Delta }{ϵ}_{\mathrm{MOAT}}$, and moat flatness $\mathcal{F}$ versus detuning of the drive. The moat radii are indicated in the upper axis.

Figure 2

Dressed band spacing, bare band admixture, and group velocity profiles of the Floquet bands as a function of quasimomentum $q$, measured along a line passing through the center of the Brillouin zone (BZ1). (a) Spacing $\mathrm{\Delta }ϵ$ between the Floquet bands at two detunings, as measured from quench dynamics. Solid lines indicate calculated spacings. (For $\delta =-550\mathrm{Hz}$ only, ${\alpha }_m=0.18$.) (b) Measured bare excited band admixture for Floquet states in the moat band. The solid (dashed) lines are the admixture calculated with (without) mean-field interactions. Marker shapes correspond to different modulation detunings. The inset shows an example of a single-shot two-dimensional admixture profile within BZ1, calculated from absorption images of heated clouds like those shown in (c), where the dashed lines outline BZ1. (d) Time-averaged group velocity $⟨v_g{⟩}_T$ for Floquet states in the moat band. The solid and dashed colored lines are calculated numerically assuming interacting and noninteracting particles, respectively (see Ref. [23]). The dashed (dotted) gray line indicates the group velocity associated with the bare lattice (free-particle) dispersion.

Figure 3

Time evolution of quasimomentum in different moat bands for the same initial conditions. (a) Predicted quasimomentum trajectories, plotted on top of the respective quasienergy spectrum. Yellow dots indicate the initial value of $\overrightarrow{q}$. Blue, black, and red curves correspond to moats with radii $0k_L$ (no modulation), $0.23k_L$, and $0.30k_L$. (b) Position after time of flight, $(x_{\mathrm{TOF}},y_{\mathrm{TOF}})$, scaled by $L_{\mathrm{TOF}}=\hslash k_L/m{T}_{\mathrm{TOF}}$, for different times held in the effective moat band. Experimental data taken under conditions that yield dressed bands similar to those shown in (a). Blue points were obtained in the absence of modulation. Modulation frequencies for the black and red points are $f_m=2.90$ and $f_m=2.65\mathrm{kHz}$, respectively. Time elapsed for blue, black, and red points is 130, 26, and 21 ms, respectively. Dashed lines are the predicted position after time of flight for condensates held in the bands shown in (a).

Figure 4

Instabilities of a condensate in the dressed bands. (a) Observed condensate fraction decay rates for BECs in lower and upper bands. For each band, we measured rates for $|\overrightarrow{q}|=0k_L$ and $|\overrightarrow{q}|\simeq 0.2k_L$. The solid (dashed) vertical gray line indicates the frequency at which the moat radius $q_{\min }$ is $0.2k_L$ ($0k_L$). (b) Calculation of the growth rate of the most unstable mode, based on a Floquet-Bogoliubov linear stability analysis for conditions similar to those in (a). The instability rates calculated for $|\overrightarrow{q}|=0.2k_L$ including only contributions from the single BEC-occupied band are shown with open symbols. The unoccupied band induces increased decay for $|\overrightarrow{q}|$ near resonance.