Three-dimensional dynamics of vortex-lattice formation in Bose-Einstein condensates

We numerically study the dynamics of vortex lattice formation in a rotating cigar-shaped Bose-Einstein condensate. The study is a three-dimensional simulation of the Gross-Pitaevskii equation with a phenomenological dissipation term. The simulations reveal previously unknown dynamical features of the vortex nucleation process, in which the condensate undergoes a strongly turbulent stage and the penetrating vortex lines vibrate rapidly. The vibrations arise from spontaneous excitation of Kelvin waves on the protovortices during a surface wave instability, caused by an inhomogeneity of the condensate density along the elongated axial direction.

Figure 1

(Color online.) Time development of parameters in the 2D and 3D simulations. (a) The deformation parameter $\alpha$. In both plots, the 3D results are the thick lines and the 2D results are thin lines. (b) The angular momentum per atom $⟨L_z⟩$ (solid curve) and the total free energy $F$ (dotted curve). The values of the parameters are $\lambda ={9.2}^{-1}$, $\Omega =0.7{\omega }_{\perp }$, and $u=54760$ for the 3D simulation and $u_{2\mathrm{D}}=440$ for the 2D simulation.

Figure 2

(Color online.) Time development of vortex formation in 3D. (a) The constant surface density ${\mid \Psi \mid }^2=2\times {10}^{-5}$. (b) The column density $\rho (x,y)=\int dz{\mid \Psi (x,y,z)\mid }^2$. (c) The cross-sectional density ${\mid \Psi (x,y,0)\mid }^2$. The times are (i) $95\mathrm{msec}$, (ii) $220\mathrm{msec}$, (iii) $270\mathrm{msec}$, (iv) $365\mathrm{msec}$, (v) $500\mathrm{msec}$. The box dimensions along $x$, $y$, and $z$ axis are $-14.5$ to $+14.5$, $-14.5$ to $+14.5$, and $-72.5$ to $+72.5$, respectively, in units of $b_{\mathrm{ho}}=0.728\mu \mathrm{m}$ ($116\times 116\times 580$ discretized space).

Figure 3

(Color online.) Time development of a surface of constant superfluid flow velocity with $\mid \mathbf{v}\mid =3.2$. (a) Snapshots in 3D. (b) Top views of (a). The times of (i)–(v) are the same as those in Fig. 2.