Collective Oscillations of Vortex Lattices in Rotating Bose-Einstein Condensates

The complete low-energy collective-excitation spectrum of vortex lattices is discussed for rotating Bose-Einstein condensates by solving the Bogoliubov–de Gennes equation, yielding, e.g., the Tkachenko mode recently observed at JILA. The totally symmetric subset of these modes includes the transverse shear, common longitudinal, and differential longitudinal modes. We also solve the time-dependent Gross-Pitaevskii equation to simulate the actual JILA experiment, obtaining the Tkachenko mode and identifying a pair of breathing modes. Combining both approaches allows one to unambiguously identify every observed mode.

Figure 1

(a) Density profile of the condensate with 37 vortices at $\Omega =0.7{\omega }_r$. (b) Collective excitations up to $\omega =3.5{\omega }_r$. (c) Lowest excitations encircled by a dashed line in (b) as functions of $q_{\theta }$ are displayed as functions of the symmetry index $m$. Here, TK, BR, DP, and QP denote the Tkachenko, breathing, dipole, and quadrupole modes, respectively.

Figure 2

Oscillation patterns for the (a) lowest TK, (b) second-lowest TK, (c) lowest BR, and (d) second-lowest BR modes are indicated by the filled circles and solid lines. Empty circles and dotted lines correspond to the equilibrium state, ${\varphi }_g$.

Figure 3

(a) Transverse oscillations of vortices on three concentric circles, $j=1,2,3$, from the trap center. (b) Fourier analyses of these oscillation patterns, ${\theta }_j(t)$, and the longitudinal vortex motion, $r_j(t)$.

Figure 4

Frequencies of three modes obtained from the BdG and TDGP approaches are compared with the JILA data (filled squares, diamonds, and triangles) as functions of $\Omega$. In the main panel the TK (open squares), in the inset both the first BR (open diamonds) and the second BR (open triangles). The solid line represents the dispersion relation in Ref. 14.