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arXiv:cond-mat/0109172 · Physical Review Letters

Dynamical Instability of a Condensate Induced by a Rotating Thermal Gas

JAMES E. JUN. WILLIAMS, E. Zaremba, B. Jackson, Tetsuro Nikuni, Adam Griffin

Published Jan 30, 2002• 45 citations
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Abstract

We study surface modes of the condensate in the presence of a rotating thermal cloud in an axisymmetric trap. By considering collisions that transfer atoms between the condensate and the noncondensate, we find that m>0 modes, which rotate in the same sense as the thermal cloud, damp less strongly than m<0 modes, where m is the polarity of the excitation. We show that above a critical angular rotation frequency, equivalent to the Landau stability criterion, m>0 modes become dynamically unstable, leading to the possibility of vortex nucleation. We also generalize our stability analysis to treat the case where the stationary state of the condensate already possesses a single vortex.

DOI
10.1103/physrevlett.88.070401

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We study surface modes of the condensate in the presence of a rotating thermal cloud in an axisymmetric trap. By considering collisions that transfer atoms between the condensate and the noncondensate, we find that m>0 modes, which rotate in the same sense as the thermal cloud, damp less strongly than m<0 modes, where m is the polarity of the excitation. We show that above a critical angular rotation frequency, equivalent to the Landau stability criterion, m>0 modes become dynamically unstable, leading to the possibility of vortex nucleation. We also generalize our stability analysis to treat the case where the stationary state of the condensate already possesses a single vortex.
We give a simple unified theory of vortex nucleation and vortex lattice formation in which (i) the thermal cloud plays a crucial role; (ii) the basic process is due to growth of the condensate from a rotating thermal cloud, which provides gain for certain surface modes of nonzero angular momentum; (iii) vortices appear initially as the result of a linear interference between the far fields of the condensate and the surface modes; and (iv) the description applies from the initiation process up to the final stabilization of the lattice. We have simulated the growth of vortex lattices from a rotating thermal cloud, and their production using a rotating trap.
The dissipative dynamics of a vortex in a finite temperature trapped Bose-Einstein condensate are shown to be governed by a {\em diffusive instability}. In the weakly interacting regime we find a cross-over from instability to metastability of the vortex, determined only by the relative rotation rate of the vortex and thermal cloud. Even in the thermodynamically stable regime there is a finite lifetime arising from diffusive instability. We find the mean exit time for the vortex from the condensate, and show that near the critical frequency for thermodynamic stability the lifetime has a universal form which is independent of the condensate size and the vortex precession frequency. In the thermodynamically stable regime the steady-state two-time position correlation predicts `thermal bunching', giving an increased probability of positional coincidence for short times.
We study the response of a trapped Bose-Einstein condensate to a sudden turn on of a rotating drive by numerically solving the two-dimensional Gross-Pitaevskii equation. A weakly anisotropic rotating potential excites a quadrupole shape oscillation and its time evolution is analyzed by a quasiparticle projection method. In a quadrupolar resonant regime, which depends on the trap anisotropy, simple periodic oscillations in surface-mode populations disappear and the system exhibits stochastic dynamics. In the presence of the phenomenological dissipation, an initially irrotational condensate is found to undergo damped elliptic deformation followed by unstable surface ripple excitations, some of which develop into quantized vortices that eventually form a lattice. Recent experimental results on the vortex nucleation should be explained not only by the dynamical instability but also by the Landau instability; the latter is necessary for the vortices to penetrate into the condensate.
We consider a large condensate in a rotating anisotropic harmonic trap. Using the method of matched asymptotic expansions, we derive the velocity of an element of a vortex line as a function of the local gradient of the trap potential, the line curvature, and the angular velocity of the trap rotation. This velocity yields small-amplitude normal modes of the vortex for two-dimensional (2D) and 3D condensates. For an axisymmetric trap, the motion of the vortex line is a superposition of plane-polarized standing-wave modes. In a 2D condensate, the planar normal modes are degenerate, and their superposition can result in helical traveling waves, which differs from a 3D condensate. Including the effects of trap rotation allows us to find the angular velocity that makes the vortex locally stable. For a cigar-shaped condensate, the vortex curvature makes a significant contribution to the frequency of the lowest unstable normal mode; furthermore, additional modes with negative frequencies appear. As a result, it is considerably more difficult to stabilize a central vortex in a cigar-shaped condensate than in a disk-shaped one. Normal modes with imaginary frequencies can occur for a nonaxisymmetric condensate (in both 2D and 3D). In connection with recent JILA experiments, we consider the motion of a straight vortex line in a slightly nonspherical condensate. The vortex line changes its orientation in space at the rate proportional to the degree of trap anisotropy and can exhibit periodic recurrences.
Using a focused laser beam we stir a 87Rb Bose-Einstein condensate confined in a magnetic trap. We observe that the steady states of the condensate correspond to an elliptic cloud, stationary in the rotating frame. These steady states depend nonlinearly on the stirring parameters (amplitude and frequency), and various solutions can be reached experimentally depending on the path followed in this parameter space. These states can be dynamically unstable and we observe that such instabilities lead to vortex nucleation in the condensate.
A second-order expansion of the Gross-Pitaevskii equation in the interaction parameter determines the thermodynamic critical angular velocity ${\ensuremath{\Omega}}_{c}$ for the creation of a vortex in a small axisymmetric condensate. Similarly, a second-order expansion of the Bogoliubov equations determines the (negative) frequency ${\ensuremath{\omega}}_{a}$ of the anomalous mode. Although ${\ensuremath{\Omega}}_{c}=\ensuremath{-}{\ensuremath{\omega}}_{a}$ through first order, the second-order contributions ensure that the absolute value $|{\ensuremath{\omega}}_{a}|$ is always smaller than the critical angular velocity ${\ensuremath{\Omega}}_{c}.$ With increasing external rotation $\ensuremath{\Omega},$ the dynamical instability of the condensate with a vortex disappears at ${\ensuremath{\Omega}}^{*}=|{\ensuremath{\omega}}_{a}|,$ whereas the vortex state becomes energetically stable at the larger value ${\ensuremath{\Omega}}_{c}.$ 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 ${\ensuremath{\Omega}}_{m}.$ A variational calculation yields ${\ensuremath{\Omega}}_{m}=|{\ensuremath{\omega}}_{a}|$ to first order (hence ${\ensuremath{\Omega}}_{m}$ also coincides with the critical angular velocity ${\ensuremath{\Omega}}_{c}$ 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.