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 greater than or equal to2Omega. 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.10.1103/PhysRevLett.91.100402 91, 100402 (2003).