Stability of a vortex in a small trapped Bose-Einstein condensate

Abstract

A second-order expansion of the Gross-Pitaevskii equation in the interaction parameter determines the thermodynamic critical angular velocity Ωc for the creation of a vortex in a small axisymmetric condensate. Similarly, a second-order expansion of the Bogoliubov equations determines the (negative) frequency ωa of the anomalous mode. Although Ωc = −ωa through first order, the second-order contributions ensure that the absolute value |ωa| is always smaller than the critical angular velocity Ωc. With increasing external rotation Ω, the dynamical instability of the condensate with a vortex disappears at Ω * = |ωa|, whereas the vortex state becomes energetically stable at the larger value Ωc. Both second-order contributions depend explicitly on the axial anisotropy of the trap. The appearance of a local minimum of the free energy for a vortex at the center determines the metastable angular velocity Ωm. A variational calculation yields Ωm = |ωa| to first order (hence Ωm also coincides with the critical angular velocity Ωc to this order). Qualitatively, the scenario for the onset of stability in the weak-coupling limit is the same as that found in the strong-coupling (Thomas-Fermi) limit.

Vortex stabilization in a small rotating asymmetric Bose-Einstein condensate

“…As is familiar from degenerate perturbation theory, applying an infinitesimal rotation breaks the degeneracy and selects out the circularly polarized helicity states with unnormalized wave functions ψ ± (r, φ) ∝ (x ± iy) exp(− 1 2 r 2 ) = re ±iφ exp(− 1 2 r 2 ). In particular, the + mode rotates in the positive sense defined by the right-hand rule and its frequency ω + = ω ⊥ − Ω decreases with increasing angular velocity as is obvious when viewed in the rotating frame [27].…”

Rotating trapped Bose-Einstein condensates

“…1]. Detailed studies (Linn and Fetter, 1999;Svidzinsky and Fetter, 2000a) show that ω a ≈ −ω ⊥ in the near-ideal limit and ω a ≈ −Ω m in the TF limit, so that ω a remains negative for all coupling strengths.…”

Hartree-Fock Approach for Rotating Bose-Einstein Condensation in One-dimensional Deep Optical Lattice

“…The latter provides the criterion stability of the rotating condensate, it does not necessarily indicate the critical frequency for vortex nucleation. The corresponding thermodynamic rotation rate can be estimated using the relation [36], -We will discuss the required conditions to reach a stable rotating Q2D briefly. The Q2D can be reached when the thermal energy is less than the site spacing energy in the direction, i.e., When the temperature is of the order of this spacing, i.e.…”