Nucleation, Growth, and Stabilization of Bose-Einstein Condensate Vortex Lattices

We give a simple unified theory of vortex nucleation and vortex lattice formation in which (i) the thermal cloud plays a crucial role; (ii) the basic process is due to growth of the condensate from a rotating thermal cloud, which provides gain for certain surface modes of nonzero angular momentum; (iii) vortices appear initially as the result of a linear interference between the far fields of the condensate and the surface modes; and (iv) the description applies from the initiation process up to the final stabilization of the lattice. We have simulated the growth of vortex lattices from a rotating thermal cloud, and their production using a rotating trap.

Figure 1

Condensate density during formation of vortex lattice from a rotating thermal cloud. Vortices are marked by $+$ or $-$: (a) $t=29.28t_0$; (b) $t=37.70t_0$; (c) $t=52.40t_0$; (d) $t=187.24t_0$. Condensate wave function seeded as described in text. Parameters are $\alpha =0.65\omega$, $\mu =12\hslash \omega$, $\gamma =0.1$, $u=1000u_0$, and initial ${\mu }_C=12.7\hslash \omega$. Units are time $t_0=1/\omega$, distance $r_0=\sqrt{\hslash /2m\omega }$, and collisional strength $u_0=\hslash \omega r_0^2$.

Figure 2

Time evolution of condensate quantities for the case of Fig. 1. (a) $⟨L_z⟩$ and norm; (b) $⟨{\mu }_{\mathrm{l}\mathrm{o}\mathrm{c}}⟩$ and ${\sigma }_{\mu }$ (in units of $\hslash \omega$).

Figure 3

(a) Evolution of angular momentum occupation probabilities $P_l$ for $l=1$ to $30$ for the case of Fig. 1. (b) Comparison of gain coefficients of $l$ components obtained from the simulation in Fig. 1 (solid line) with the predicted gain coefficients $G=\gamma (\alpha l-{ϵ}_{0,l}/\hslash )$ (dotted line).

Figure 4

Evolution of angular momentum occupation probabilities $P_l$ for (a) rotating elliptical trap with no thermal cloud ($\gamma =0$) and (b) rotating elliptical trap and corotating thermal cloud. Only the even $l$ states are shown. Parameters as in Fig. 1 except trap parameters ${\omega }_x^2=1.05{\omega }^2$, ${\omega }_y^2=1.15{\omega }^2$, and $\alpha =\Omega =0.65\omega$.