Dynamics of vortex matter in rotating two-species Bose-Einstein condensates

In a rotating two-component Bose-Einstein condensate (BEC), the traditional triangular vortex lattice can be replaced by a rectangular vortex lattice or even a structure characterized in terms of vortex sheets, depending on the interspecies interactions. We study the dynamics of this system by analyzing the Bogoliubov excitation spectrum. Excitations familiar to BEC vortex systems are found such as Tkachenko modes, hydrodynamic modes, and surface waves, however, the complex two-component morphology also gives rise to phenomena including shear flow between vortex sheets.

Figure 1

(Color online) Dots and crosses represent in-phase and out-of phase modes, respectively. Out-of-phase modes gradually depart from in-phase modes decreasing energies as interspecies interaction increases. The spectra of in-phase modes are not seriously affected by the change of $g_{12}$. (a) $g_{12}=0.1g$, (b) $g_{12}=0.5g$, (c) $g_{12}=0.9g$.

Figure 2

(Color online) Three ground states depending on different rotational frequencies with $g_{12}=0.75g$. Bright clouds represent one species with dots the positions of vortices while crosses are vortices of the other component. (a) $\Omega =0.4{\omega }_{\mathrm{tr}}$, (b) $\Omega =0.6{\omega }_{\mathrm{tr}}$, (c) $\Omega =0.9{\omega }_{\mathrm{tr}}$.

Figure 3

(Color online) Energy vs angular momentum for the three ground states in Fig. 2. Each point represents a normal mode. Tkachenko modes are identified as negative angular momentum states in the low-energy regime. Surface modes are divided into in-phase (dots) and out-of-phase (crosses) modes. Note that the gap between spectral lines of in-phase and out-of-phase surface modes decreases at higher angular frequencies. (a) $\Omega =0.4{\omega }_{\mathrm{tr}}$, (b) $\Omega =0.6{\omega }_{\mathrm{tr}}$, (c) $\Omega =0.9{\omega }_{\mathrm{tr}}$.

Figure 4

(Color online) Energies of normal modes as a function of $g_{12}$ showing the transition from phase-overlapped to phase-separated state. The actual transition interaction strength $g_{12}^T$ is slightly larger than $g$ due to the kinetic energy contribution.

Figure 5

(Color online) Three ground states depending on different rotational frequencies with $g_{12}=1.1g$. Bright clouds represent one species with the dots the positions of vortices while crosses are vortices of the other component. (a) is close to the nonrotating PS condensate, except that it has phase singularities stabilized by the presence of the other component. $\Omega =0.2{\omega }_{\mathrm{tr}}$. (b) The first emergence of a vortex sheet (dots) that effectively divides the condensate into two pieces. $\Omega =0.3{\omega }_{\mathrm{tr}}$. (c) A typical vortex-sheet state for high rotational frequency. $\Omega =0.8{\omega }_{\mathrm{tr}}$.

Figure 6

(Color online) Spectrum of three different vortex-sheet states corresponding to Figs. 5a, 5b, 5c. (a) $\Omega =0.2{\omega }_{\mathrm{tr}}$, (b) $\Omega =0.3{\omega }_{\mathrm{tr}}$, (c) $\Omega =0.8{\omega }_{\mathrm{tr}}$.

Figure 7

(Color online) Amplitude of density fluctuation $n^{\prime }$ of Tkachenko modes with wavelength (a) longer and (b) shorter than intersheet spacing.

Figure 8

(Color online) A snapshot of the most highly excited Tkachenko mode. Arrows are added to show the motion of vortices flowing along each sheet.