Moiré superlattice structures in kicked Bose-Einstein condensates

Vortex lattices in rapidly rotating Bose-Einstein condensates lead to a periodic modulation of the superfluid density with a triangular symmetry. Here we show that this symmetry can be combined with an external perturbation in order to create superlattice structures with two or more periodicities. Considering a condensate that is kicked by an optical lattice potential, we find the appearance of transient moiré lattice structures, which can be identified using the kinetic energy spectrum.

Figure 1

(a) Vortex lattice ground state in a harmonic trap with ${\omega }_{\perp }=2\pi {\mathrm{s}}^{-1}$ and rotating at $\mathrm{\Omega }=0.995{\omega }_{\perp }$. This plot shows a condensate with a diameter of approximately $580\mu \text{m}$; (b) Zoom in of vortex lattice at central density; (c) Optical lattice potential with a periodicity matching that of the vortex lattice.

Figure 2

Compressible energy spectrum of a nonrotating condensate directly following a kick. A peak at $k=4\pi /\left(\sqrt{3}{a}_o\right)$ can be seen, which corresponds to the lattice spacing, $a_o$ (indicated by the dashed line), and the smaller, higher-energy peaks can be attributed to higher harmonics between nearest and next-nearest neighbours.

Figure 3

(a) Main peak of the compressible kinetic energy spectrum for a kicking strength of $V_0\approx 1.35\times {10}^{-2}\mu$. It can be seen to revive, and eventually disperse, over a wide range of wave numbers. (b) Condensate densities at several times during the evolution. A pattern matching the optical potential can be observed to appear and disappear several times over the course of the evolution.

Figure 4

Condensate density at $t=1.4\times {10}^{-2}$ s for several optical lattice rotation angles. The cell size of the superlattice structures can be seen to shrink as the angle is increased. The angles for the examples shown are $\left(a\right){\theta }_{\mathrm{\Delta }}=0,\left(b\right){\theta }_{\mathrm{\Delta }}=2\pi /45,\left(c\right){\theta }_{\mathrm{\Delta }}=4\pi /45,\left(d\right){\theta }_{\mathrm{\Delta }}=2\pi /15$.

Figure 5

Condensate density after receiving a kick with ${\theta }_{\mathrm{\Delta }}=\pi /9$. The appearance and disappearance of a moiré structure with wavelength ${\lambda }_M\approx 2.9a$ over a timescale of about 50 ms can be seen.

Figure 6

Size of the resulting moiré superstructures as a function of the relative angle between the vortex and optical lattice. The dashed green line indicates the condensate radius. Inset: The different vectors in $\mathbf{k}$ space of the two lattices, with the optical lattice rotated by an angle ${\theta }_{\mathrm{\Delta }}$. The ${\mathbf{g}}_{l{l}^{\prime }}=|{\mathbf{g}}_l-{\mathbf{g}}_l^{\prime }|$ vectors defining the dominant moiré wavelength are those for which the enclosed angle is smallest.

Figure 7

Compressible kinetic energy spectrum as a function of ${\theta }_{\mathrm{\Delta }}$. All values are time averaged over an interval $t=0$ s to $t=1$ s. The moiré peak corresponding to the lowest wave number can be seen shifting to larger values for increasing angles and similar behavior is visible for the higher-order components.

Figure 8

(a) For a kicking strength of $V_0=5.4\times {10}^{-2}\mu$ for a nonrotating condensate higher-order modes become non-negligible contributions to the compressible kinetic energy spectrum. This leads to a variety of different density structures, with some closeups shown for (b) 24 ms, (c) 36 ms, (d) 56 ms, and (e) 88 ms. Note that the larger structures in these plots are given by the optical lattice constant, which sets the scale.