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arXiv:cond-mat/0210111 · Physical Review A

Scissors mode of a rotating Bose-Einstein condensate

Marco Cozzini, S. Stringari, Vincent Bretin, P. Rosenbusch, Jean Dalibard

Published Feb 7, 2003• 17 citations
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Abstract

A scissors mode of a rotating Bose-Einstein condensate is investigated both theoretically and experimentally. The condensate is confined in an axisymmetric harmonic trap, superimposed with a small rotating deformation. For angular velocities larger than ${\ensuremath{\omega}}_{\ensuremath{\perp}}/\sqrt{2},$ where ${\ensuremath{\omega}}_{\ensuremath{\perp}}$ is the radial trap frequency, the frequency of the scissors mode is predicted to vanish like the square root of the deformation, due to the tendency of the system to exhibit spontaneous rotational symmetry breaking. Measurements of the frequency confirm the predictions of theory. Accompanying characteristic oscillations of the internal shape of the condensate are also calculated and observed experimentally.

DOI
10.1103/physreva.67.021602

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A scissors mode of a rotating Bose-Einstein condensate is investigated both theoretically and experimentally. The condensate is confined in an axisymmetric harmonic trap, superimposed with a small rotating deformation. For angular velocities larger than \ensuremathω\ensuremath⊥/2,{\ensuremath{\omega}}_{\ensuremath{\perp}}/\sqrt{2},\ensuremathω\ensuremath⊥​/2​, where \ensuremathω\ensuremath⊥{\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremathω\ensuremath⊥​ is the radial trap frequency, the frequency of the scissors mode is predicted to vanish like the square root of the deformation, due to the tendency of the system to exhibit spontaneous rotational symmetry breaking. Measurements of the frequency confirm the predictions of theory. Accompanying characteristic oscillations of the internal shape of the condensate are also calculated and observed experimentally.
We investigate the oscillation of a dilute atomic gas generated by a sudden rotation of the confining trap (scissors mode). This oscillation reveals the effects of superfluidity exhibited by a Bose-Einstein condensate. The scissors mode is also investigated in a classical gas above Tc{T}_{c}Tc​ in various collisional regimes. The crucial difference with respect to the superfluid case arises from the occurrence of low frequency components, which are responsible for the rigid value of the moment of inertia. Different experimental procedures to excite the scissors mode are discussed.
We show that a Bose-Einstein condensate with a vortex lattice in a rotating anisotropic harmonic potential exhibits a very low frequency scissors mode. The appearance of this mode is due to the SO(2) symmetry breaking introduced by the vortex lattice. The corresponding Goldstone mode in an isotropic trap becomes, in an anisotropic trap, a scissors mode, whose frequency is finite and tends to zero with the trap anisotropy ϵϵϵ, with a generic ϵ\sqrt{ϵ}ϵ​ dependence. We present analytical formulas giving the mode frequency in the low ϵϵϵ limit and we show that the mode frequency for some class of vortex lattices can tend to zero as ϵϵϵ or faster. We demonstrate that the standard classical hydrodynamics approach fails to reproduce this low frequency mode, because it does not contain the discrete structure of the vortex lattice.
We report the observation of the scissors mode of a Bose-Einstein condensed gas of 87Rb atoms in a magnetic trap, which gives direct evidence of superfluidity in this system. The scissors mode of oscillation is excited by a sudden rotation of the anisotropic trapping potential. For a gas above T(c) (normal fluid) we detect the occurrence of oscillations at two frequencies, with the lower frequency corresponding to the rigid body value of the moment of inertia. Well below T(c) the condensate oscillates at a single frequency, without damping, as expected for a superfluid.
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.
We study the scissors modes of a harmonically trapped Bose-Einstein condensate under the influence of a synthetic magnetic field, which induces rigid rotational components in the velocity field. Our investigation reveals that the scissors mode, excited in the plane perpendicular to the synthetic magnetic field, becomes coupled to the quadrupole modes of the condensate, giving rise to typical beating effects. Moreover, the two scissors modes excited in the vertical planes are also coupled together by the synthetic magnetic field, resulting in intriguing gyroscope dynamics. Our analytical results, derived from a spinor hydrodynamic theory, are further validated through numerical simulations of the three-dimensional Gross-Pitaevskii equation. These predictions for the condensates subject to a synthetic magnetic field are experimentally accessible with current cold-atom setups and hold promise for potential applications in quantum sensing.
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∣=1,2l=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 $\ensuremath{\gamma}(3\ensuremath{-}{\ensuremath{\Omega}}^{2})1\ensuremath{-}3{\ensuremath{\Omega}}^{2},$ 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.