Generation and detection of matter-wave gap vortices in optical lattices

We analyze numerically the process of dynamical generation of spatially localized vortices in Bose-Einstein condensates (BEC’s) with repulsive atomic interactions confined by two-dimensional optical lattices. Akin to bright solitons in a repulsive condensate, these nonlinear localized states exist only within the gaps of the matter-wave band-gap spectrum imposed by the periodicity of the lattice potential. We discuss the complex structure of matter-wave phase singularities associated with different types of stationary gap vortices and suggest two different excitation methods. In one method, the condensate is adiabatically driven to the edge of the Brillouin zone where a vortex phase is subsequently imprinted onto the condensate wave packet. Alternatively, a vortex is created in a condensate confined in a harmonic trap and then is nonadiabatically released into the lattice potential. We find that only the latter method leads to robust and reliable generation of vortices within the gap of the matter-wave band-gap spectrum. Moreover, the nonadiabatic excitation can lead to the formation of broad gap vortices from the initial BEC wave packets with a large number of atoms. These broad vortices are intimately connected to self-trapped nonlinear states of the BEC recently demonstrated in experiments with a one-dimensional optical lattice. Our numerical simulations also confirm the feasibility of a homodyne interferometric detection of broad gap vortices.

Figure 1

Boundaries of the first complete gap (unshaded) of the matter-wave Bloch spectrum in a 2D optical lattice as a function of lattice depth, $V_0$. The top edge of the first band and the bottom edge of the second band (bands are shaded) correspond to the points $M(k_x=1,k_y=1)$ and $X(k_x=1,k_y=0)$ of the first irreducible BZ, respectively. Inset: contour plot of the lattice potential and the first irreducible BZ.

Figure 2

(Color online) Families of the off-site gap vortices at $V_0=4\left(E_L\right)$. Top: dependence of the norm of the condensate wave function, $N={\int }_{-\infty }^{\infty }\mid \psi {\mid }^{2}d\mathbf{r}$, on the chemical potential. The letters (A)–(D) mark families of vortices corresponding to the particular examples (a)–(d) shown below. Solid circles show norm of the vortices shown in (a–d). The actual number of particles for the physical parameters chosen here is obtained by multiplying the norm by the factor $10.13$. Bottom: spatial density and phase distributions.

Figure 3

(Color online) Top: families of on-site gap vortices at $V_0=4\left(E_L\right)$ in the same format as Fig. 2. Bottom: spatial density and phase distributions corresponding to the marked points on the plot above.

Figure 4

(Color online) Particle flow structure for the “narrow” (a) off-site and (b) on-site gap vortices shown in Figs. 2a, 3b, respectively, plotted together with contours of the atomic density profile.

Figure 5

(Color online) Broad-vortex states inside the band gap $(\mu =4.5)$ of a lattice potential with $V_0=4$. (a) Broad vortex with off-site singularity location, containing $1.3\times {10}^3$ atoms. (b) “Nonrectangular” broad vortex containing $2\times {10}^3$ atoms. (c) Broad vortex localized over a large number of lattice sites and containing $1.1\times {10}^4$ atoms.

Figure 6

(Color online) (a) The simple particle circulation of a broad off-site vortex. Almost all circulation occurs at the vortex core, just as in the case of a vortex in a homogeneous condensate. (b) Radial density structure displaying a typical structure of a truncated nonlinear Bloch wave [18].

Figure 7

(Color online) Evolution of an initial BEC wave packet with a vortex phase imprinted onto a nonlinear Bloch state corresponding to the $M$-band edge in an optical lattice with $V_0=4$. The four rows show the snapshots of the $15\mathrm{ms}$ evolution of the condensate with the initial densities (a) below, (b) at critical, (c) slightly above critical, and (d) well above critical peak density required for the formation of a localized “narrow” vortex. The initial (final) atom numbers, initial chemical potential, and initial position of the vortex singularity are (a) $8.7\times {10}^2$ $(7.9\times {10}^2)$, $\mu =1.2$, off site; (b) $8.6\times {10}^2$ $(7.3\times {10}^2)$, $\mu =1.2$, on site; (c) $1.2\times {10}^3$ $(1.06\times {10}^3)$, $\mu =1.6$, off site; (d) $2.6\times {10}^3$ $(2.3\times {10}^3)$, $\mu =2.0$, off site.

Figure 8

(Color online) Nonadiabatic loading of a narrow wave packet into the lattice with $V_0=4.0$ demonstrating dynamical generation of a narrow-gap vortex. Row (a): density (left panel), phase (middle panel), and density cross section (right panel, dotted line) for the initial state at $t=0$ containing $1.07\times {10}^3$ atoms. Solid line on the right panel shows the cross section of the final density profile. Color bar corresponds to the density plots. Row (b): density (left panel), phase (middle panel), and the particle flow (right panel) of the final gap state at $t=100$ ($5\mathrm{ms}$ with $3.0\times {10}^2$ atoms. Color bar corresponds to the phase plots. (c) Relaxation of the chemical potential from a band (shaded) into the gap (unshaded); point $B$ corresponds to the vortex shown in row (b) above.

Figure 9

(Color online) Nonadiabatic loading into the lattice of a broad wave packet shown in (a) at $t=0$ and in (b) at $t=300$, demonstrating dynamical evolution into the gap states. (c) Relaxation of the chemical potential (solid line) from the band (shaded) into the gap (unshaded) for the initial and final states shown above. The dashed line in (c) is for an initial state (not shown) with twice the initial density of the state in (a).

Figure 10

(Color online) Density (left) and phase (right) distribution of a narrow on-site gap vortex $2.5\mathrm{ms}$ after release from the lattice.

Figure 11

(Color online) Interference fringes of the condensates initially containing an off-site broad vortex (left panel) $0.1\mathrm{ms}$ and (right panel) $0.5\mathrm{ms}$ after a Bragg diffraction pulse is applied after $2.5\mathrm{ms}$ of lattice free expansion. The relative velocities of the condensates are (left) $0.58\mathrm{cm}∕\mathrm{s}$ and (right) $0.29\mathrm{cm}∕\mathrm{s}$, respectively.