Observation of Tkachenko Oscillations in Rapidly Rotating Bose-Einstein Condensates

We directly image Tkachenko waves in a vortex lattice in a dilute-gas Bose-Einstein condensate. The low (sub-Hz) resonant frequencies are a consequence of the small but nonvanishing elastic shear modulus of the vortex-filled superfluid. The frequencies are measured for rotation rates as high as 98% of the centrifugal limit for the harmonically confined gas. Agreement with a hydrodynamic theory worsens with increasing rotation rate, perhaps due to the increasing fraction of the volume displaced by the vortex cores. We also observe two low-lying $m=0$ longitudinal modes at about 20 times higher frequency.

Figure 1

(1,0) Tkachenko mode excited by atom removal (a) taken 500 ms after the end of the blasting pulse, (b) taken 1650 ms after the end of the blasting pulse. BEC rotation is counterclockwise. Lines are sine fits to the vortex lattice.

Figure 2

Measured oscillation amplitude for a typical excitation $\Omega =0.92$ and $2.2\times {10}^6$ atoms. Fit is to a sine wave times an exponential decay and yields a frequency of 0.85 Hz and a $Q$ of 3. The oscillation amplitude is expressed as the average amplitude of the sine wave fits to the vortex oscillation in units of the radial Thomas-Fermi radius (roughly the azimuthal displacement of a vortex a distance 0.33 $R_{\rho }$ from the condensate center). Both values are in expansion.

Figure 3

(a) shows the adjusted [15] ($n=1$, $m=0$) Tkachenko oscillation frequencies as a function of scaled rotation rate $\Omega$. (b) shows the (1,0) frequency as a function of the theory parameter $ϵ\Omega$. The dotted line is the theory line ${\nu }_{10}=1.43ϵ\Omega ({\omega }_{\rho }/2\pi )$ from Ref. 10. Note that the low number data shows much worse agreement with theory. (c) demonstrates the divergence from theory as the ratio of vortex core area to unit cell area increases. $A_{\mathrm{v}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x}}$ is $\pi {\xi }^2$ where the healing length $\xi =(8\pi na{)}^{-1/2}$ (here $n$ is density-weighted average density and $a$ is the $s$-wave scattering length). Lattice cell area $A_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}$ is $\sqrt{3}{b}^2/2$ (here $b$ is the nearest-neighbor vortex spacing). For all plots black squares and triangles refer to high and low atom number experiments, respectively.

Figure 4

On the left are the locations where atoms are removed from the cloud. For the (1,0) excitations the atoms are removed from the shaded region in the center. For the (2,0) mode atoms are removed from the shaded ring half a condensate radius out. The image on the right is the resulting (2,0) mode, where the black line has been added to guide the eye.