Superfluid Vortex Dynamics on Planar Sectors and Cones

We study the dynamics of vortices formed in a superfluid film adsorbed on the curved two-dimensional surface of a cone. To this aim, we observe that a cone can be unrolled to a sector on a plane with periodic boundary conditions on the straight sides. The sector can then be mapped conformally to the whole plane, leading to the relevant stream function. In this way, we show that a superfluid vortex on the cone precesses uniformly at fixed distance from the apex. The stream function also yields directly the interaction energy of two vortices on the cone. We then study the vortex dynamics on unbounded and bounded cones. In suitable limits, we recover the known results for dynamics on cylinders and planar annuli.Comment: 10 pages, 8 figure

Bose gas: Theory and Experiment

For many years, $^4$He typified Bose-Einstein superfluids, but recent advances in dilute ultra-cold alkali-metal gases have provided new neutral superfluids that are particularly tractable because the system is dilute. This chapter starts with a brief review of the physics of superfluid $^4$He, followed by the basic ideas of Bose-Einstein condensation (BEC), first for an ideal Bose gas and then considering the effect of interparticle interactions, including time-dependent phenomena. Extensions to more exotic condensates include magnetic dipolar gases, mixtures of two components, and spinor condensates that require a focused infrared laser for trapping of all the various hyperfine magnetic states in a particular hyperfine $F$ manifold of $m_F$ states. With an applied rotation, the trapped BECs nucleate quantized vortices. Recent theory and experiment have shown that laser coupling fields can mimic the effect of rotation. The resulting synthetic gauge fields have produced vortices in a nonrotating condensate

Excited states of a static dilute spherical Bose condensate in a trap

The Bogoliubov approximation is used to study the excited states of a dilute gas of $N$ atomic bosons trapped in an isotropic harmonic potential characterized by a frequency $\omega_0$ and an oscillator length $d_0 = \sqrt{\hbar/m\omega_0}$. The self-consistent static Bose condensate has macroscopic occupation number $N_0 \gg 1$, with nonuniform spherical condensate density $n_0(r)$; by assumption, the depletion of the condensate is small ($N' \equiv N - N_0\ll N_0$). The linearized density fluctuation operator $\hat \rho'$ and velocity potential operator $\hat\Phi'$ satisfy coupled equations that embody particle conservation and Bernoulli's theorem. For each angular momentum $l$, introduction of quasiparticle operators yields coupled eigenvalue equations for the excited states; they can be expressed either in terms of Bogoliubov coherence amplitudes $u_l(r)$ and $v_l(r)$ that determine the appropriate linear combinations of particle operators, or in terms of hydrodynamic amplitudes $\rho_l'(r)$ and $\Phi_l'(r)$. The hydrodynamic picture suggests a simple variational approximation for $l >0$ that provides an upper bound for the lowest eigenvalue $\omega_l$ and an estimate for the corresponding zero-temperature occupation number $N_l'$; both expressions closely resemble those for a uniform bulk Bose condensate.Comment: 5 pages, RevTeX, contributed paper accepted for Low Temperature Conference, LT21, August, 199

Rapid rotation of a Bose-Einstein condensate in a harmonic plus quartic trap

A two-dimensional rapidly rotating Bose-Einstein condensate in an anharmonic trap with quadratic and quartic radial confinement is studied analytically with the Thomas-Fermi approximation and numerically with the full time-independent Gross-Pitaevskii equation. The quartic trap potential allows the rotation speed $\Omega$ to exceed the radial harmonic frequency $\omega_\perp$. In the regime $\Omega \gtrsim \omega_\perp$, the condensate contains a dense vortex array (approximated as solid-body rotation for the analytical studies). At a critical angular velocity $\Omega_h$, a central hole appears in the condensate. Numerical studies confirm the predicted value of $\Omega_h$, even for interaction parameters that are not in the Thomas-Fermi limit. The behavior is also investigated at larger angular velocities, where the system is expected to undergo a transition to a giant vortex (with pure irrotational flow).Comment: 14 pages, 5 figure