Vortex Rings and Lieb Modes in a Cylindrical Bose-Einstein Condensate

We present a calculation of a solitary wave propagating along a cylindrical Bose-Einstein trap. For sufficiently strong couplings of experimental interest, it is found to be a hybrid of a one-dimensional (1D) soliton and a three-dimensional vortex ring. The calculated energy-momentum dispersion exhibits characteristics similar to those of a mode proposed sometime ago by Lieb within a 1D model, as well as some rotonlike features.

Figure 1

Radial dependence of the solitary wave function at $v=c/2$ for various positive values of $\xi$ in steps of $\delta \xi =0.1$. The result for negative $\xi$ is obtained through the symmetry relations $\mathrm{R}\mathrm{e}\Psi (\rho ,\xi )=\mathrm{R}\mathrm{e}\Psi (\rho ,-\xi )$ and $\mathrm{I}\mathrm{m}\Psi (\rho ,\xi )=-\mathrm{I}\mathrm{m}\Psi (\rho ,-\xi )$. Distances are measured in units of $a_{\perp }$.

Figure 2

Radial dependence of the local particle density $n=|\Psi {|}^2$ for the four special cases discussed in the text. Conventions are the same as in Fig. 1.

Figure 3

Energy $E$ in units of ${\gamma }_{\perp }(\hslash {\omega }_{\perp })$ versus impulse $Q$ in units of ${\gamma }_{\perp }(\hslash /a_{\perp })$. The solid line corresponds to the main sequence of solitary waves discussed in the text, and the dashed line to the auxiliary sequence that contains a black soliton (point IV).

Figure 4

Contour levels of the local particle density $n$ in a plane that contains the $z$ axis and cuts across the cylindrical trap. The complete 3D picture may be envisaged by simple revolution around the $z$ axis. Regions with high particle density are bright while regions with zero density are black. The four special cases considered are the same as in Fig. 2.