Oscillations of rotating trapped Bose-Einstein condensates

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 {Ω}1 on the rotation rate (measured in units of the trap frequency ω₀). 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 γ(3-Ω²)1-3Ω², and are stable otherwise. Thus, a dilute Bose gas, whose equation of state is polytropic with {γ}=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 A₂Ω² with {Ω}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.