Chiral edge states of vortex matter

We study the evolution of the excitation spectrum of a trapped rotating Bose-Einstein condensate as a function of increasing rotational frequency and discuss chiral edge states supported on the condensate surface. We describe these modes in the context of a mean-field Bogoliubov–de Gennes treatment. We find that they dominate the spectrum at fast rotation, and the dispersion relation for these excitations becomes linear in the limit of high angular momentum as in the quantum Hall effect. We also identify the edge states as primary agents in the nucleation and decay of the lattice.

Figure 1

Azimuthally symmetric (3,0) [6] (a) Tkachenko mode and (b) inertial mode. (c) Edge state wave. The overlayed lines have been extracted from the associated density-fluctuation plot.

Figure 2

(a) ${\rho }^{\prime }$ for a typical phonon mode; (b) ${\rho }^{\prime }$ for a the same phonon mode shown in (a) but with the short-distance structure around the cores smoothed out by interpolation

Figure 3

$m$ vs $\omega$. Solid line: diffused vorticity approximation of Eq. (15). Dotted line: full numerical solution of Eqs. (8, 9). Tm, Sm, and Im represent the Tkachenko modes, the surface modes, and the inertial modes, respectively.

Figure 4

(a) ${\rho }^{\prime }$ for a typical Tkachenko wave. (b) Amplitude and (c) phase of ${\rho }^{\prime }$ for long-wavelength $\lambda \sim R$ phonon and inertial mode which is well described by the diffused vorticity ansatz ${\rho }^{\prime }\left(\mathbf{r}\right)=f\left(r\right)e^{im\varphi }$.

Figure 5

(a) Cross section (along the $z$ axis) of the effective mean-field potential $\frac{1}{2}m{\omega }_{\perp }^2{\mathbf{r}}^2+2g{\mid {\Phi }_0\mid }^2-\mu )$ experienced by a particle in the trap; (b) the amplitude and (c) phase of the Bogoliubov quasiparticle state $\left(u\right)$ for this excitation. $R_L=⟨r⟩$ (dashed circle) for a state in the Landau state $h_{0,q}(r,\varphi )$ where $q$ is chosen to match the winding number of $u$ in region $I$.

Figure 6

(Color online) We plot the $N_p$, $l_z$, and $\delta$ defined in the text for different ground states. The labels Tm, Im, Sm approximately mark the locations of Tkachenko, inertial, and edge states, respectively. The depletion is multiplied by a factor of $10000∕N$.

Figure 7

(a) and (b) represent $\mid {\psi }_{HF}\mid$ for degenerate core-bound Hartree-Fock states [Eq. (30)], where ${\psi }_{HF}$ is the wave function. (c) $\mid {\psi }_{HF}\mid$ of an extended Hartree-Fock state. In all plots the overlayed white dots are the equilibrium vortex positions.