Scissors modes of a Bose-Einstein condensate in a synthetic magnetic field

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.

Figure 1

(a) Phase diagram of the condensate in a synthetic magnetic field as a function of the atom number $N$ and the relevant parameter $\eta$ fixing the position dependence of the detuning (see text). The two spin components can be overlapped, exhibit vortices, or be separated, depending on the value of the relevant $\eta$ and $N$. (b) A typical distribution of the condensate obtained by numerically solving the GP equation for $\eta =0.005,0.01,0.013E_r$ at $N={10}^5$. Note that there are two vortices at $y=0$ for $\eta =0.01E_r$.

Figure 2

(a) Illustration of a spin-orbit-coupled BEC in the presence of a position-dependent detuning $\delta \left(y\right)=\eta k_{0}y$, which induces a synthetic magnetic field ${\mathbf{B}}_{\text{syn}}$ along the $z$ direction and a rotational velocity field in the horizontal plane. (b) Dependence of the quadrupole modes eigenfrequencies on the Raman coupling strength for $\eta =0$. The three red curves are the eigenfrequencies corresponding to the width oscillation of the condensate [Eq. (5)] and the blue curve corresponds to the oscillation frequency ${\omega }_{xy}/2\pi$ of the scissors mode in the horizontal plane. The trapping frequencies are $({\omega }_x,{\omega }_y,{\omega }_z)/2\pi =(50\sqrt{3},50,35)$Hz. At the two points indicated by the green star and cyan dot, the scissors mode frequency ${\omega }_{xy}$ and one of the three hybrid quadrupole modes are degenerate. In this work, we shall focus on the green star point at $\mathrm{\Omega }=6E_r$ where the degeneracy condition Eq. (6) is satisfied.

Figure 3

(a), (b) Time evolution of the scissors mode $xy$ in the horizontal plane and the quadrupole mode characterized by the operator $Q=({\omega }_x^2/{\omega }_y^2)x^2-y^2$ in the presence of a detuning gradient $\eta =0.001E_r$. The trapping frequencies are $(f_x,f_y,f_z)=(50\sqrt{3},50,35)$ Hz and the Raman coupling strength is $\mathrm{\Omega }=6E_r$ to meet the degeneracy condition Eq. (6). The dynamics is excited by a sudden rotation of the harmonic trap by a small angle ${\phi }_0=3^{\circ }$ in the $x\text{−}y$ plane.

Figure 4

Time evolution of the scissors modes in the vertical planes $\langle xz\rangle$ and $\langle yz\rangle$. The trapping frequencies are $(f_x,f_y,f_z)=(50\sqrt{3},50,35)\text{Hz}, \mathrm{\Omega }=6E_r$. Thus, $\sqrt{m/m^{*}}{\omega }_x={\omega }_y$, a beat appears for the two coupled modes. The detuning gradient $\eta =0.001E_r$. The dynamics is excited by a sudden rotation of the harmonic trap by a small angle ${\theta }_0=3^{\circ }$ in the $x\text{−}z$ plane.

Figure 5

Illustration of a BEC gyroscope realized via a synthetic magnetic field. The polar angle $\theta \left(t\right)$ corresponds to a fast scissors mode oscillation in the vertical plane and the azimuthal angle $\phi \left(t\right)$ corresponds to a slower precession around the symmetry axis $z$ due to the synthetic magnetic field.