Winding-number dependence of Bose-Einstein condensates in a ring-shaped lattice

We study the winding-number dependence of the stationary states of a Bose-Einstein condensate in a ring-shaped lattice. The system is obtained by confining atoms in a toroidal trap with equally spaced radial barriers. We calculate the energy and angular momentum as functions of the winding number and the barrier height for two quite distinct particle numbers. In both cases we observe two clearly differentiated regimes. For low barriers, metastable vortex states are obtained up to a maximum winding number that depends on the particle number and barrier height. In this regime, the angular momentum and energy show, respectively, almost linear and quadratic dependences on the winding number. For large barrier heights, on the other hand, stationary states are obtained up to a maximum winding number that depends only on the number of lattice sites, whereas energy and angular momentum are shown to be sinusoidal functions of the winding number.

Figure 1

Left panel: equipotential curves for the external trap $V_{\text{trap}}(x,y)/\hslash {\omega }_r$ with $V_b/\hslash {\omega }_r=80$. Right panel: effective potential barrier $V_{\text{trap}}(x,0)$ along the positive $x$ axis for several values of the parameter $V_b/\hslash {\omega }_r$.

Figure 2

Density isocontours for several values of the barrier parameter. From left to right: $V_b/\hslash {\omega }_r=0,10,$ and $80$. The number of particles corresponds to $N={10}^5$.

Figure 3

Phase of the order parameter $\varphi$ as a function of the angular coordinate for $r_0=6\hspace{0.5em}{l}_r$, a winding number $n=4$, and two values of the barrier height $V_b/\hslash {\omega }_r=60$ (left panel) and $V_b/\hslash {\omega }_r=120$ (right panel). The number of particles is $N={10}^5$.

Figure 4

Excitation energy per particle $E_n^{\prime }$ (top panel), and angular momentum per particle $L_z=\langle {\mathrm{L̂}}_z\rangle$ (bottom panel), as functions of the winding number for $N={10}^3$ particles and $V_b/\hslash {\omega }_r=0$ (circles), 3 (stars), and 5 (triangles). The solid lines correspond to Eq. (8) (top) and Eq. (10) (bottom).

Figure 5

Same as Fig. 4 for the condensate of $N={10}^5$ particles with barrier heights $V_b/\hslash {\omega }_r=0$ (circles), 10 (squares), 40 (stars), and 60 (triangles).

Figure 6

Snapshots of the phase distribution in the condensate of ${10}^5$ particles with a barrier height $V_b/\hslash {\omega }_r=40$ and an initially imprinted winding number $n=9$. Panels (a) to (d) correspond to ${\omega }_{r}t=0$, 0.374, 0.796, and 3.183, respectively. White dots in panels (b) to (d) denote vortices with topological charge $m=+1$, while black dots in (c) and (d) denote antivortices with $m=-1$. The dashed line circles in (b) and (d) correspond to circulations yielding initial $n=9$ and final $n^{\prime }=-7$ winding numbers, respectively.

Figure 7

Excitation energy per particle $E_n^{\prime }$ (top panel) and angular momentum per particle $L_z=\langle {\mathrm{L̂}}_z\rangle$ (bottom panel), in units of their maximum values $E_{\text{max}}$ and $L_{\text{max}}$, respectively, as functions of the winding number for the condensate of $N={10}^3$ particles, with barrier heights $V_b/\hslash {\omega }_r=10$ (crosses) and 40 (squares). The solid lines represent the functions: $[1-\cos (2\pi n/N_c)]/2$ (top panel) and $\sin (2\pi n/N_c)$ (bottom panel).

Figure 8

Same as Fig. 7 for the condensate of $N={10}^5$ particles with barrier heights $V_b/\hslash {\omega }_r=80$ (crosses) and 120 (squares).

Figure 9

Phase of the order parameter as a function of the angular coordinate for $r_0=6\hspace{0.5em}{l}_r$ (left panels), and phase distribution in the $x$-$y$ plane (right panels), in the condensate of ${10}^5$ particles with a barrier height $V_b/\hslash {\omega }_r=80$. The imprinted winding number is $n=8$ (top panels) and $n=9$ (bottom panels). The long (blue) vertical arrow in the left-bottom panel shows a phase difference between neighboring sites larger than $\pi$, which amounts to the smaller phase difference denoted by the short (red) arrow. Note also the slightly negative phase slope within each site, $\partial \varphi /\partial \theta <0$, which yields the final winding number $n^{\prime }=9-N_c=-7$ and the corresponding negative angular momentum, as seen in the bottom panel of Fig. 8.

Figure 10

Snapshot of the phase distribution in the condensate of ${10}^5$ particles with a barrier height of $V_b/\hslash {\omega }_r=80$ and an initially imprinted winding number $n=9$. Real-time dissipative evolution at $t=0.2387\hspace{0.5em}{\omega }_r^{-1}$ showing vorticities with a topological charge $m=+1$ spread along each barrier. Such a spreading may be appreciated by following the circulation along the rectangular contour enclosing one of the barriers.

Figure 11

Ground-state chemical potential $\mu$ of the condensate containing ${10}^5$ particles (squares) and ${10}^3$ particles (triangles) as a function of the barrier height $V_b$. The solid line corresponds to the effective potential barrier minima of Fig. 1. All quantities are given in units of $\hslash {\omega }_r$.