Tkachenko Oscillations and the Compressibility of a Rotating Bose-Einstein Condensate

The elastic oscillations of the vortex lattice of a cold Bose gas (Tkachenko modes) are shown to play a crucial role in the saturation of the compressibility sum rule, as a consequence of the hybridization with the longitudinal degrees of freedom. The presence of the vortex lattice is responsible for a $q^2$ behavior of the static structure factor at small wave vectors $q$, which implies the absence of long range order in 2D configurations at zero temperature. Sum rules are used to calculate the Tkachenko frequency in the presence of harmonic trapping. Results are derived in the Thomas-Fermi regime and compared with experiments as well as with previous theoretical estimates.

Figure 1

Lowest Tkachenko frequency of a trapped Bose gas in units of ${\omega }_0=\sqrt{\hslash \Omega /4m{R}_{\perp }^2}$ as a function of the angular velocity $\Omega$. The solid line is the sum rule result, while the dashed line is the prediction of Eq. (5) with $q=5.45/R_{\perp }$ (see text). Experimental points are taken from Refs. 9, 20 using the 3D Thomas-Fermi value for $R_{\perp }$. The prediction of Ref. 6 corresponds to ${\omega }_T/{\omega }_0=5.45$.

Figure 2

Density change of the lowest Tkachenko mode at $\Omega /{\omega }_{\perp }=0.9$. The inset shows the corresponding displacement $ϵ$ of the vortex lattice at the angles $\pi /6$, $\pi /2$, and $5\pi /6$. The amplitude of the oscillation corresponds to $\delta \Omega /\Omega =0.05$.