Virial theorems for vortex states in a confined Bose-Einstein condensate

We derive a class of virial theorems which provide stringent tests of both analytical and numerical calculations of vortex states in a confined Bose-Einstein condensate. In the special case of harmonic confinement we arrive at the somewhat surprising conclusion that the linear moments of the particle density, as well as the linear momentum, must vanish even in the presence of off-center vortices which lack axial or reflection symmetry. Illustrations are provided by some analytical results in the limit of a dilute gas, and by a numerical calculation of a class of single and double vortices at intermediate couplings. The effect of anharmonic confinement is also discussed.

Figure 1

Contour plots of the particle density $n$ for a class of vortex states calculated in a harmonic trap with $\delta =2$. The four characteristic values of the angular momentum $\ell$ correspond to the four branches $AB,B^{\prime }\Gamma ,\Gamma \Delta$, and $\Delta \dots$ in Fig. 2.

Figure 2

Angular frequency $\omega$ (in units of ${\omega }_0$) as a function of angular momentum per particle $\ell$ (in units of $\hslash$) for a class of vortex states calculated in a harmonic trap with $\delta =2$.

Figure 3

Angular frequency $\omega$ as a function of angular momentum per particle $\ell$ for a class of vortex states calculated in an anharmonic trap with $\delta =2$ and $\lambda =1∕4$.

Figure 4

Linear momentum $P$ (in units of $\hslash ∕a_0$), linear moment $R$ (in units of $a_0$), and generalized moment $Q$ (in units of $a_0^3$), as functions of angular momentum per particle $\ell$ (in units of $\hslash$), for single off-center vortices in an anharmonic trap with $\delta =2$ and $\lambda =1∕4$.