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Physical Review Letters

Scissors-Mode Vibrations and the Emergence of SU(3) Symmetry from the Projected Deformed Mean Field

Yang Ren Sun, Cheng-Li Wu, K.H. Bhatt, Mike Guidry, D.H. Feng

Published Jan 26, 1998• 24 citations
Publisher

Abstract

Starting from a deformed potential we construct separate bases of collective neutron and proton rotational states by exact angular momentum projection. These rotational states are then coupled by diagonalizing a residual pairing plus quadrupole interaction. Many new bands emerge that are not found in the rotation of the usual BCS condensate, and may correspond to the geometrical scissors mode and its generalizations. These excitation modes can be understood as rotational bands built on spin- $1\ensuremath{\Elzxh}$ phonon excitations; they exhibit a nearly perfect dynamical $\mathrm{SU}(3)$ fermion spectrum, even though there is no explicit dynamical symmetry in our model.

DOI
10.1103/physrevlett.80.672

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Starting from a deformed potential we construct separate bases of collective neutron and proton rotational states by exact angular momentum projection. These rotational states are then coupled by diagonalizing a residual pairing plus quadrupole interaction. Many new bands emerge that are not found in the rotation of the usual BCS condensate, and may correspond to the geometrical scissors mode and its generalizations. These excitation modes can be understood as rotational bands built on spin- 1\ensuremath\Elzxh1\ensuremath{\Elzxh}1\ensuremath\Elzxh phonon excitations; they exhibit a nearly perfect dynamical SU(3)\mathrm{SU}(3)SU(3) fermion spectrum, even though there is no explicit dynamical symmetry in our model.
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.
As a direct generalization of our previous work, we construct a manifestly rotation invariant Hamiltonian describing a system composed of a tensor boson with spin I = 2 — such as, for example, phonon of nuclear quadrupole surface vibration — in which special cubic and quartic self-interactions are taken into account. All the ground state solutions are obtained. It is shown that these solutions can be classified into three classes: The first is the normal vacuum of boson, which is manifestly rotation symmetry. The second class of solutions can be viewed as generated from a single chosen state by rotating it into an arbitrary direction. The third one is Goldstone modes built on this chosen ground state. This chosen state — an exact ground state solution of our rotation invariant Hamiltonian — is not an eigenstate of angular momentum, but contains infinitely many different angular momentum states in it; thus, typifying “spontaneous breakdown” of “rotation symmetry”. It is further shown that this state can be naturally identified to an “intrinsic” deformed state in nuclear collective model of Bohr and Mottelson. The results are examined from the standpoint of Goldstone theorem and detailed discussions are given to explore how Goldstone theorem is modified in our finite dimensional system.
We present an exact diagonalization study of the breathing mode collective excitations for a rotating Bose-Einstein condensate of N=10N = 10N=10 spinless bosons interacting via repulsive finiterange Gaussian potential and harmonically confined in quasi-two-dimension. The yrast state and the low-lying excited states are variationally obtained in given subspaces of the quantized total angular momentum LLL employing the beyond lowest Landau level approximation in slowly rotating regime with 0≤L<2N0\le L< 2N0≤L<2N. For a given LLL, the low-energy eigenspectra (bands) are obtained in weakly to moderately interacting regime. Further, for a given interaction, the split in low-lying eigenenergies with increasing LLL is the precursor to spontaneous symmetry breaking of the axisymmetry associated with the entry of the first vortex. With increase in repulsive interaction, the value of the first breathing mode increases for stable total angular momentum states L=0L = 0L=0 and NNN, but decreases for intermediate 0<L<N0 < L < N0<L<N metastable states. The position of the observed first breathing modes in the eigenspectrum remains unchanged as the interaction is varied over several orders of magnitude.
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.
We study the collective oscillations of spin-orbit-coupled Bose-Einstein condensates in the presence of position-dependent detuning. Specifically, we explore the quadrupole modes of the system using both numerical and analytical approaches based on the Gross-Pitaevskii equation and hydrodynamic theory. Due to spin-orbit coupling and the synthetic magnetic field, the x y scissors mode couples with a superposition of the three diagonal quadrupole modes ( x 2 , y 2 , and z 2 ), resulting in the characteristic beating effect. The remaining two scissors modes, x z and y z , are coupled, giving rise to a Lissajous-like pattern that is highly sensitive to the excitation method and orientation of the synthetic magnetic field. Furthermore, we find that anisotropic interactions as well as the direction of the synthetic magnetic field, can significantly influence the oscillation amplitude and frequency of the quadrupole modes. These findings highlight the potential of Bose-Einstein condensates under synthetic magnetic fields for quantum sensing applications, such as magnetic field gradient measurements, and provide a promising foundation for future experimental research and technological development.
Quantum-deformation of the vibrational symmetry SU(3) supset SU(2) supset SO(2) of the two-dimensional SU(3) interacting boson model is studied from the point of view of the breaking of the dynamical symmetry. It is shown that, in agreement with the earlier work of Gupta et al. (1992) for the one-dimensional problem, the real and complex (more so the complex) q-deformations take this dynamical symmetry very close to the other (rotational) dynamical symmetry SU(3) supset SO(3) supset SO(2) of the two-dimensional SU(3) group. Also, the intermediate results of mixing these two symmetries are obtained, since q-deformations vary continuously. The purely imaginary q-deformation in SU q (3) supset SU q (2) supset SO q (2) restores the SU(3) symmetry for a parameter value larger than 1. Applications to real nuclei are also made.