Persistent currents in two-component condensates in a toroidal trap

The stability of persistent currents in a two-component Bose-Einstein condensate in a toroidal trap is studied in both the miscible and the immiscible regimes. In the miscible regime we show that superflow decay is related to linear instabilities of the spin-density Bogoliubov mode. We find a region of partial stability, where the flow is stable in the majority component while it decays in the minority component. We also characterize the dynamical instability appearing for a large relative velocity between the two components. In the immiscible regime the stability criterion is modified and depends on the specific density distribution of the two components. The effect of a coherent coupling between the two components is also discussed.

Figure 1

Spin, density, and single-component speeds of sound. For concreteness, the densities have been calculated in Thomas-Fermi approximation (see the Appendix).

Figure 2

Bogoliubov excitation spectrum for density (${\omega }_d$) and spin (${\omega }_s$) modes. The symbols correspond to the discretized values of $k$ (see text) arising from the ring geometry. For concreteness, the densities have been calculated in Thomas-Fermi approximation (see the Appendix).

Figure 3

Stability diagram in the miscible regime as obtained from imaginary-time simulations. The solid line represents the spin speed of sound $c_s$ [see Eq. (10)] computed at the density maxima by taking into account the corrective factor for 2D geometries. The upper dashed line represents the boundary for dynamical instability when $v_b=0$, as obtained from Bogoliubov analysis.

Figure 4

Lines of energetical instability for $v_b=\hslash {\kappa }_b/m{R}_0$ for different ${\kappa }_b$. The instability corresponds to the maximum velocity for the $a$ component that leads to a positive spin-mode frequency.

Figure 5

(Left panel) Time dependence of angular momentum of components $a$ (dashed line) and $b$ (solid line) in real-time dynamics in the partially stable region. (Right panels) Density snapshots of components $a$ (top row) and $b$ (bottom row) at times $t=260{\omega }_{\perp }^{-1}$ in panels (a) and (b), $t=305{\omega }_{\perp }^{-1}$ in panels (c) and (d), and $t=560{\omega }_{\perp }^{-1}$ in panels (e) and (f). The density (in units of $a_{\perp }^{-2}$) is represented in the color scale. For this case $P_z=0.8$ and $\kappa =7$.

Figure 6

Dynamical instability for ${\kappa }_a=20$, ${\kappa }_b=0$. (Top row) Real and imaginary parts of the dispersion relation for the spin mode. When the imaginary part vanishes, positive- (negative-) norm solutions are represented in solid (dashed) lines. (Bottom rows) Snapshots of majority component densities (in units of $a_{\perp }^{-2}$) during real-time dynamics, at times $t=31.5$ ms (a), $t=47.7$ ms (b), and $t=57.4$ ms (c).

Figure 7

Stability diagram in the phase-separated regime for the majority component, for different values of the polarization and as a function of the imaginary time, $\tau$. The color scale represents the value of the circulation. In the insets, we have plotted the final density of the majority component for $P_z=0.95$ (top) and $P_z=0.6$ (bottom).

Figure 8

Thomas-Fermi density profiles for a two-component mixture in a displaced harmonic trap. Dashed line, $N_a=N_b$; solid lines, $P_z=0.4$.