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arXiv:cond-mat/0007394 · Physical Review A

Oscillations of rotating trapped Bose-Einstein condensates

Armen Sedrakian, Ira Wasserman

Published May 10, 2001• 17 citations
arXivHTMLPDF

Abstract

The tensor-virial method is applied for a study of oscillation modes of uniformly rotating Bose-Einstein condensed gases, whose rigid-body rotation is supported by an vortex array. The second-order virial equations are derived in the hydrodynamic regime for an arbitrary external harmonic trapping potential assuming that the condensate is a superfluid at zero temperature. The axisymmetric equilibrium shape of the condensate is determined as a function of the deformation of the trap; its domain of stability is bounded by the constraint \ensuremath{\Omega}1 on the rotation rate (measured in units of the trap frequency ${\ensuremath{\omega}}_{0}).$ The oscillations of the axisymmetric condensate are stable with respect to the transverse-shear and toroidal modes of oscillations, corresponding to the $l=2,|m|=1,2$ surface deformations. The eigenfrequencies of the modes are real and represent undamped oscillations. The condensate is also stable against quasiradial pulsation modes $(l=2,m=0),$ and its oscillations are undamped, if the superflow is assumed incompressible. In the compressible case we find that for a polytropic equation of state, the quasiradial oscillations are unstable when $\ensuremath{\gamma}(3\ensuremath{-}{\ensuremath{\Omega}}^{2})1\ensuremath{-}3{\ensuremath{\Omega}}^{2},$ and are stable otherwise. Thus, a dilute Bose gas, whose equation of state is polytropic with \ensuremath{\gamma}=2 to leading order in the diluteness parameter, is stable irrespective of the rotation rate. In nonaxisymmetric traps, the equilibrium constrains the (dimensionless) deformation in the plane orthogonal to the rotation to the domain ${A}_{2}{\ensuremath{\Omega}}^{2}$ with \ensuremath{\Omega}1. The second-harmonic-oscillation modes in nonaxisymmetric traps separate into two classes that have even or odd parity with respect to the direction of the rotation axis. Numerical solutions show that these modes are stable in the parameter domain where equilibrium figures exist.

DOI
10.1103/physreva.63.063605

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The tensor-virial method is applied for a study of oscillation modes of uniformly rotating Bose-Einstein condensed gases, whose rigid-body rotation is supported by an vortex array. The second-order virial equations are derived in the hydrodynamic regime for an arbitrary external harmonic trapping potential assuming that the condensate is a superfluid at zero temperature. The axisymmetric equilibrium shape of the condensate is determined as a function of the deformation of the trap; its domain of stability is bounded by the constraint \ensuremath{\Omega}1 on the rotation rate (measured in units of the trap frequency \ensuremathω0).{\ensuremath{\omega}}_{0}).\ensuremathω0​). The oscillations of the axisymmetric condensate are stable with respect to the transverse-shear and toroidal modes of oscillations, corresponding to the l=2,∣m∣=1,2l=2,|m|=1,2l=2,∣m∣=1,2 surface deformations. The eigenfrequencies of the modes are real and represent undamped oscillations. The condensate is also stable against quasiradial pulsation modes (l=2,m=0),(l=2,m=0),(l=2,m=0), and its oscillations are undamped, if the superflow is assumed incompressible. In the compressible case we find that for a polytropic equation of state, the quasiradial oscillations are unstable when \ensuremathγ(3\ensuremath−\ensuremathΩ2)1\ensuremath−3\ensuremathΩ2,\ensuremath{\gamma}(3\ensuremath{-}{\ensuremath{\Omega}}^{2})1\ensuremath{-}3{\ensuremath{\Omega}}^{2},\ensuremathγ(3\ensuremath−\ensuremathΩ2)1\ensuremath−3\ensuremathΩ2, and are stable otherwise. Thus, a dilute Bose gas, whose equation of state is polytropic with \ensuremath{\gamma}=2 to leading order in the diluteness parameter, is stable irrespective of the rotation rate. In nonaxisymmetric traps, the equilibrium constrains the (dimensionless) deformation in the plane orthogonal to the rotation to the domain A2\ensuremathΩ2{A}_{2}{\ensuremath{\Omega}}^{2}A2​\ensuremathΩ2 with \ensuremath{\Omega}1. The second-harmonic-oscillation modes in nonaxisymmetric traps separate into two classes that have even or odd parity with respect to the direction of the rotation axis. Numerical solutions show that these modes are stable in the parameter domain where equilibrium figures exist.
Based on the method of matched asymptotic expansions and on a time-dependent variational analysis, we study the dynamics of a vortex in a three-dimensional disk-shaped nonaxisymmetric condensate in the Thomas-Fermi limit. Both methods show that a vortex in a trapped Bose-Einstein condensate has formally unstable normal mode(s) with positive normalization and negative frequency, corresponding to a precession of the vortex line around the center of the trap. In a rotating trap, the solution becomes stable above an angular velocity Omega(m), characterizing the onset of metastability with respect to small transverse displacements of the vortex from the central axis.
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.
We derive the finite-temperature oscillation modes of a harmonically confined Bose-Einstein condensed gas undergoing rigid body rotation supported by a vortex lattice in the condensate. The hydrodynamic modes separate into two classes corresponding to center of mass and relative oscillations of the thermal cloud and the condensate. These classes are independent of each other in the case where the thermal cloud is inviscid for all modes studied, except the radial pulsations which couple them because the pressure perturbations of the condensate and the thermal cloud are governed by different adiabatic indices. If the thermal cloud is viscous, the two classes of oscillations are coupled, i.e., each type of motion involves simultaneously mass and entropy currents. The relative oscillations are damped by the mutual friction between the condensate and the thermal cloud mediated by the vortex lattice. The damping is large for the values of the drag-to-lift ratio of the order of unity and becomes increasingly ineffective in either limit of small or large friction. An experimental measurement of a subset of these oscillation modes and their damping can provide information on the values of the phenomenological mutual friction coefficients and the quasiparticle-vortex scattering processes in dilute atomic Bose gases.
The rotational motion of an interacting Bose-Einstein condensate confined by a harmonic trap is investigated by solving the hydrodynamic equations of superfluids, with the irrotationality constraint for the velocity field. We point out the occurrence of an overcritical branch where the system can rotate with angular velocity larger than the oscillator frequencies. We show that in the case of isotropic trapping the system exhibits a bifurcation from an axisymmetric to a triaxial configuration, as a consequence of the interatomic forces. The dynamical stability of the rotational motion with respect to the dipole and quadrupole oscillations is explicitly discussed.
A hydrodynamic description is used to study the normal modes of a vortex in a zero-temperature Bose-Einstein condensate. In the Thomas-Fermi limit, the circulating superfluid velocity far from the vortex core provides a small perturbation that splits the originally degenerate normal modes of a vortex-free condensate. The relative frequency shifts are small in all cases considered (they vanish for the lowest dipole mode with ∣m∣=1|m|=1∣m∣=1), suggesting that the vortex is stable. The Bogoliubov equations serve to verify the existence of helical waves, similar to those of a vortex line in an unbounded weakly interacting Bose gas. In the large-condensate (small-core) limit, the condensate wave function reduces to that of a straight vortex in an unbounded condensate; the corresponding Bogoliubov equations have no bound-state solutions that are uniform along the symmetry axis and that decay exponentially far from the vortex core.
Equilibrium vortex formation in rotating binary Bose gases with a rotating frequency higher than the harmonic trapping frequency is investigated theoretically. We consider the system being evaporatively cooled to form condensates and a combined numerical scheme is applied to ensure the binary system being in an authentic equilibrium state. To keep the system stable against the large centrifugal force of ultrafast rotation, a quartic trapping potential is added to the existing harmonic part. Using the Thomas-Fermi approximation, a critical rotating frequency ${\ensuremath{\Omega}}_{c}$ is derived, which characterizes the structure with or without a central density hole. Vortex structures are studied in detail with rotation frequency both above and below ${\ensuremath{\Omega}}_{c}$ and with respect to the miscible, symmetrically separated, and asymmetrically separated phases in their nonrotating ground-state counterparts.