Stability of coreless vortices in ferromagnetic spinor Bose-Einstein condensates

We study the energetic and dynamic stability of coreless vortices in nonrotated spin-1 Bose-Einstein condensates, trapped with a three-dimensional optical potential and a Ioffe-Pritchard field. The stability of stationary vortex states is investigated by solving the corresponding Bogoliubov equations. We show that the quasiparticle excitations corresponding to axisymmetric stationary states can be taken to be eigenstates of angular momentum in the axial direction. Our results show that coreless vortex states can occur as local or global minima of the condensate energy or become energetically or dynamically unstable depending on the parameters of the Ioffe-Pritchard field. The experimentally most relevant coreless vortex state containing a doubly quantized vortex in one of the hyperfine spin components turned out to have very nontrivial stability regions, and especially a quasiperiodic dynamic instability region which corresponds to splitting of the doubly quantized vortex.

Figure 1

(Color online) One possible spin texture of a coreless vortex in a pancake-shaped condensate in the presence of the Ioffe-Pritchard field. The spin of the condensate tends to align with the external field.

Figure 2

Stability phase diagram for the axially symmetric ⟨2,1,0⟩ vortex state. Top, $\lambda =0.2$; middle, $\lambda =1.0$; and bottom, $\lambda =5.0$. The left panels show the different stability phases and the right panels the maximum value of $\mid \mathrm{Im}\{{\omega }_q∕{\omega }_r\}\mid$ for each point in the $(B_{\perp },B_z)$ plane as a gray-scale plot: bright regions correspond to dynamically stable regions, and dark ones dynamically unstable. In the region denoted as “local minimum” ⟨2,1,0⟩ is a local minimum of the mean-field energy and the polar state $⟨1,0,-1⟩$ is the ground state of the system. The dynamic instability region indicates the section of the parameter space where dynamic instability modes can appear.

Figure 3

Stability phase diagram of the polar vortex $⟨1,0,-1⟩$ (right panel) and the lowest-energy excitations as a function of $B_z$ for fixed ${\mu }_{\mathrm{B}}{B}_{\perp }∕2\hslash {\omega }_r=0.5$ (left panel). The phase diagram is symmetric with respect to reversing the sign of $B_z$.

Figure 4

Quasiperiodic structure of the dynamic instability as a function of $B_z$ for the axially symmetric ⟨2,1,0⟩ vortex and ${\mu }_{\mathrm{B}}{B}_{\perp }∕2\hslash {\omega }_r=0.7$. Left, $\lambda =5.0$; middle, $\lambda =1.0$; and right, $\lambda =0.2$.

Figure 5

Negative-energy excitations and corresponding positive-energy excitations with ${\kappa }_q=\pm 2$ for the ferromagnetic vortex state ⟨2,1,0⟩. Here ${\mu }_{\mathrm{B}}{B}_{\perp }∕2\hslash {\omega }_r=0.5$ and $\lambda =1.0$ are fixed. The dynamic instabilities start to occur for ${\mu }_{\mathrm{B}}{B}_z∕2\hslash {\omega }_r\gtrsim 1.16$, which is denoted by the solid line. The intervals denoted by the dashed lines and the roman numerals I, II, III, and IV correspond to regions of values of $B_z$ for which the ⟨2, 1, 0⟩ vortex state is dynamically stable (cf. Fig. 2).

Figure 6

(Color online) Isosurfaces of particle densities in different spin components corresponding to vortex state ⟨2,1,0⟩ and a slight excitation of a dynamic instability mode. The trap asymmetry parameter is $\lambda =0.2$. Population of the dynamic instability mode is 1% of the total number of particles. The isosurfaces correspond to values ${\mid {\Psi }_1\left(\mathbf{r}\right)\mid }^2=2.6\times {10}^{-5}N∕a_r^3$, ${\mid {\Psi }_0\left(\mathbf{r}\right)\mid }^2=1.0\times {10}^{-5}N∕a_r^3$, and ${\mid {\Psi }_{-1}\left(\mathbf{r}\right)\mid }^2=1.0\times {10}^{-5}N∕a_r^3$.