Adiabatic preparation of vortex lattices

By engineering appropriate artificial gauge potentials, a Bose-Einstein condensate (BEC) can be adiabatically loaded into a current-carrying state that is analogous to a vortex lattice of a rotating uniform Bose gas. We give two explicit, experimentally feasible protocols by which vortex lattices can be smoothly formed from a condensate initially at rest. In the first example we show how this can be achieved by adiabatically loading a uniform BEC into an optical flux lattice, formed from coherent optical coupling of internal states of the atom. In the second example we study a tight-binding model that is continuously manipulated in parameter space such that it smoothly transforms into the Harper-Hofstadter model with $1/3$ flux per plaquette.

Figure 1

A cut through the dispersion of the lowest band in the optical flux lattice along $k_x$ with $k_y=0$ (passing through the minima) for lattice depths $V_0/E_{\mathrm{L}}=0,2,4$ (top). The dressed states of the lowest band are superpositions of spin-$\uparrow$ (shown in blue) and spin-$\downarrow$ (red) states. The dispersion $y$ axis has units of $E_{\mathrm{L}}$. The dispersion (bottom) shown here is for a lattice depth of $V_0/E_{\mathrm{L}}=4$.

Figure 2

(a) Total density and current density for the condensate wave function described in the main text at lattice depth $V_0/E_{\mathrm{L}}=4$ [light (dark) colors correspond to high (low) density]. The large black arrows denote the Bravais lattice vectors ${\mathbf{a}}_1,{\mathbf{a}}_2$ of the flux lattice. In the deep lattice limit, the atoms are localized at the lattice sites marked by the red dots. (b) Density of the analog condensate wave function for the corresponding tight-binding lattice model (large blue dots indicate high density). The arrows indicate the direction of mass currents (all currents have the same magnitude).

Figure 3

As the depth of this optical flux lattice is ramped up, currents appear smoothly in the condensate. We quantify the mass flow by calculating the line integral $I$ [Eq. (7)] over total current density $\mathbf{j}={\mathbf{j}}_{\uparrow }+{\mathbf{j}}_{\downarrow }$ along the contour shown in the inset. We have normalized $I$ with $I_0=\hslash n_0/M$.

Figure 4

(a) Illustration of hopping matrix elements for the tight-binding lattice model in the gauge described in the text. The area enclosed by the red dashed line is a unit cell containing three lattice sites. (b) Flux per plaquette for the lattice shown in (a). When $\alpha =2\pi /3$, the flux through each plaquette is $1/3$ of an elementary flux quantum.

Figure 5

(a) When adiabatically ramping with the hopping matrix elements of Eq. (10), the angle $\theta$ selects between different translations of the vortex lattice. To be adiabatic, coming from $r=1$ (circle), one has to avoid crossing the dashed lines when approaching $r=0$ (center) as shown, for example, by the solid red path. (b) Normalized average current for the square lattice as a function of $1-r$ along $\theta =0$.