Tkachenko Modes of Vortex Lattices in Rapidly Rotating Bose-Einstein Condensates

We calculate the in-plane modes of the vortex lattice in a rotating Bose condensate, from the slowly rotating to mean-field quantum Hall limits. The Tkachenko mode frequency, linear in wave vector $k$ for lattice rotational velocities $\Omega$ much smaller than the lowest sound wave frequency in a finite system, becomes quadratic in $k$ in the opposite limit. The system also supports an inertial mode of frequency $\ge 2\Omega$. The calculated frequencies are in excellent agreement with recent observations of Tkachenko modes by I. Coddington, P. Engels, V. Schweikhard, and E. A. Cornell, Phys. Rev. Lett. 91, 100402 (2003).