Tkachenko Waves in Rapidly Rotating Bose-Einstein Condensates

We present a mean-field theory numerical study of Tkachenko waves of a vortex lattice in trapped atomic Bose-Einstein condenstates. Our results show remarkable qualitative and quantitative agreement with recent experiments at the Joint Institute for Laboratory Astrophysics. We extend our calculations beyond the conditions of the experiment, probing deeper into the incompressible regime where we find excellent agreement with analytical results. In addition, bulk excitations observed in the experiment are discussed.

Figure 1

(a),(b) $(1,0)$ mode in action: (a) theory and (b) experimental results. The curve is a fitted sine wave with wavelength $1.33R_{\mathrm{TF}}$. (c),(d) $(2,0)$ mode in action: (c) theory and (d) experimental results. The curves are drawn to guide the eye.

Figure 2

Calculated distortions of arrays with (a) ${\xi }^2/{ϵ}^2=0.007$ and (b) ${\xi }^2/{ϵ}^2=0.032$ for a $(1,0)$ mode. Solid circles: equilibrium vortex positions. Arrows: relative vortex displacement.

Figure 3

Energy of the $(1,0)$ mode versus ${\xi }^2/{ϵ}^2$. In Ref. 11 the energy of this mode is given as $1.43ϵ\Omega$. The height of the plot symbols represents the error bars for both experiment and theory. The estimate of computational error is described in the text following Ref. 15.

Figure 4

(a) Bulk $(2,0)$ mode in action. Overlaid curve is a sine wave of wavelength $1.33R_{\mathrm{TF}}$. (b) Phase plot of $\delta \rho$. Notice the $\pi$ phase change as a node is radially crossed. (c) Experimental atom removal region [4]. (d) Close-up of (b) on a vortex location. In all phase plots, phase changes from 0 to $2\pi$ as one goes from dark to light regions.