Angular Momentum of a Bose-Einstein Condensate in a Synthetic Rotational Field

By applying a position-dependent detuning to a spin-orbit-coupled Hamiltonian with equal Rashba and Dresselhaus coupling, we exploit the behavior of the angular momentum of a harmonically trapped Bose-Einstein condensed atomic gas and discuss the distinctive role of its canonical and spin components. By developing the formalism of spinor hydrodynamics, we predict the precession of the dipole oscillation caused by the synthetic rotational field, in analogy with the precession of the Foucault pendulum, the excitation of the scissors mode, following the sudden switching off of the detuning, and the occurrence of Hall-like effects. When the detuning exceeds a critical value, we observe a transition from a vortex free, rigidly rotating quantum gas to a gas containing vortices with negative circulation which results in a significant reduction of the total angular momentum.

Figure 1

(a) Total density and (b) velocity field of a spin-orbit-coupled BEC confined in an anisotropic harmonic trap obtained from GP calculations, corresponding to $\mathrm{\Omega }=8E_r$ ($E_r=k_0^2/2m=h\times 1775\mathrm{Hz}$), $\eta =0.0025E_r$ and trapping frequencies $({\omega }_x,{\omega }_y)=2\pi \times (50\sqrt{2},50)\mathrm{Hz}$, scattering length $a=100a_0$ where $a_0$ is the Bohr radius and the Thomas-Fermi radius $R_x^{\mathrm{TF}}\approx 13.8\mu \mathrm{m}$, $R_y^{\mathrm{TF}}\approx 19.6\mu \mathrm{m}$. For (a), the relative density is represented by colors with blue for zero density and yellow for maximum density. The size and color of the arrows in (b) reflects the magnitude of the velocity field. Red and long arrows correspond to large velocities while blue and short arrows correspond to small velocities.

Figure 2

Foucault precession of the spin-orbit-coupled BEC. (a) Time-dependence of the center-of-mass position $(\overline{x},\overline{y})$ of the spin-orbit-coupled Bose gas in the presence of a position dependent detuning. The dynamics is initiated by exciting a dipole mode along the $x$ direction, achieved by suddenly shifting the trap center to $x_0=3\mu \mathrm{m}$. The black lines are the hydrodynamic results and the red dots are the results obtained from the GP calculation. (b) Precession of the dipole oscillation. The starting point of the motion is (0,0) (blue dot) and the blue arrow indicates the precession direction. The system parameters are the same as in Fig. 1.

Figure 3

(a) Excitation of the scissors mode after turning off the position dependent detuning. The red dashed lines are the long symmetric axes of the BEC and $T=2\pi /{\omega }_{\mathrm{sc}}$ is the period of the scissors mode. The other system parameters are the same as in Fig. 1. The relative density is represented by colors with blue for zero density and yellow for maximum density. (b), (c) Excitation of the Hall-like effect. A periodic modulation of the trap center $x_0(t)=A(t)\sin (\omega t)$ along the $x$ direction causes a resonant response of the dipole oscillation along the $y$ direction if $\omega ={\omega }_y$. The amplitude $A(t)$ is, here, linearly ramped to the final value $A=0.18\mu \mathrm{m}$ at $t=500\mathrm{ms}$ and then held for another 250 ms. The dipole oscillations along $x$ and $y$ are shown in (b) and (c), respectively, where the black solid lines are the hydrodynamic prediction and the red dots are the results from the GP simulation. The trap frequencies are $({\omega }_x,{\omega }_y)=2\pi \times (120,30)\mathrm{Hz}$, the Raman coupling $\mathrm{\Omega }=8E_r$ and the detuning gradient $\eta =0.002E_r$.

Figure 4

Dependence of the total angular momentum $⟨{\widehat{L}}_z⟩$ (blue circles), the canonical $⟨{\widehat{L}}_z^c⟩$ (red triangles), and the spin-dependent $⟨{\widehat{L}}_z^s⟩$ (green squares) contributions on the detuning gradient $\eta$ of a spin-orbit-coupled atomic gas in an isotropic harmonic trap with $({\omega }_x,{\omega }_y)=2\pi \times (50,50)\mathrm{Hz}$ and the Thomas-Fermi radius $R^{\mathrm{TF}}\approx 17.9\mu \mathrm{m}$. The results are obtained by numerical solving of the GP equations. The Raman coupling strength is $\mathrm{\Omega }={\mathrm{\Omega }}_c$. Top panels show typical density profile and velocity field of the two phases blow and above ${\eta }_c$. The inset panel shows the velocity field overlapped with the density profile near the center of a vortex. For contour plots, the relative density is represented by colors with blue for zero density and red for maximum density.