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.