Vortex-splitting and phase-separating instabilities of coreless vortices in $F=1$ spinor Bose-Einstein condensates

The low-lying excitations of coreless vortex states in $F=1$ spinor Bose-Einstein condensates (BECs) are theoretically investigated using the Gross-Pitaevskii and Bogoliubov–de Gennes equations. The spectra of the elementary excitations are calculated for different spin-spin interaction parameters and ratios of the number of particles in each sublevel. There exist dynamical instabilities of the vortex state which are suppressed by ferromagnetic interactions, and, conversely, enhanced by antiferromagnetic interactions. In both of the spin-spin interaction regimes, we find vortex-splitting instabilities in analogy with scalar BECs. In addition, a phase-separating instability is found in the antiferromagnetic regime.

Figure 1

(Color online) A set of complex-frequency modes in the complex plane. Four complex-frequency modes exist together.

Figure 2

(Color online) Spatial profile of the order parameter for $g_s^{\prime }=-0.001$ (a)–(d) and for $g_s^{\prime }=0.001$ (e)–(h). The magnetization $M$ is (a) $-0.90$, (b) $-0.30$, (c) 0.33, (d) 0.89, (e) $-0.90$, (f) $-0.30$, (g) 0.34, and (h) 0.90. The solid, dashed, and dash-dotted lines correspond to the $m_F=1$, 0, and $-1$ components, respectively. The order parameter corresponding to the $m_F=0$ component in the antiferromagnetic case, denoted by the dotted line, is purely imaginary.

Figure 3

(Color online) The ratio of the atoms in hyperfine spin states and total number of the atoms $N_i∕N$ as a function of magnetization for $g_s^{\prime }$ equals to $\left(\mathrm{a}\right)$ $-0.001$, (b) $-0.01$, (c) 0.001, and (d) 0.01. The solid, dashed, and dash-dotted lines correspond to $m_F=1$, 0, and $-1$ components, respectively. The total number of atoms in the two-dimensional (2D) plane is fixed to $N=1.5\times {10}^3{d}^{-1}$.

Figure 4

(Color online) Excitation spectrum for $M=-0.9$ and $g_s^{\prime }=-0.001$. The triangle at [$q_{\theta }=0$, $\mathrm{Re}\left(E_{\mathit{q}}\right)=0$] corresponds to the GP solution. The labels indicate the majority component in the quasiparticle amplitudes $u_i$ and $v_i$. This spectrum includes the complex-frequency modes denoted by the triangles at [$q_{\theta }=\pm 2$, $\mathrm{Re}\left(E_{\mathit{q}}\right)=\pm 1.82$].

Figure 5

(Color online) The absolute values of the imaginary parts of the complex-frequency eigenvalues are shown as a function of the magnetization $M$ for $g_s^{\prime }$ equals to $\left(\mathrm{a}\right)$ $-0.001$, (b) $-0.01$, (c) 0.001, and (d) 0.01. The $q_{\theta }=\pm 2$ and $\pm 1$ modes are indicated by dashed and solid lines, respectively. The insets in (c) and (d) show the detailed structure in the vicinity of $M=-1$ for $g_s^{\prime }=0.001$ and 0.01. There are no complex-frequency modes found for $M>-0.8$ in the ferromagnetic case.

Figure 6

(Color online) Magnetization dependence of the excitation energies of the $q_{\theta }=\pm 2$ modes for $g_s^{\prime }=-0.001$. The insets show the details of the spectrum where complex eigenenergies appear. The eigenenergy of the $q_{\theta }=-2$ mode is plotted as $-\mathrm{Re}\left(E_{\mathit{q}}\right)$. The majority component in the quasiparticle amplitudes $u_i$ and $v_i$ of the corresponding excitation energies is indicated by the solid, dashed, and dash-dotted lines for the $m_F=1$, 0, and $-1$ components in $q_{\theta }=2$ mode, respectively. The dots indicate $q_{\theta }=-2$ mode, which is dominated by the $m_F=-1$ component. The complex-frequency modes are labeled by triangles in both $q_{\theta }=\pm 2$ modes.

Figure 7

(Color online) Magnetization dependence of the lowest-energy excitations with $q_{\theta }=-2$ for different values of $g_s^{\prime }$. The solid, dashed, dash-dotted, and dotted lines indicate the $g_s^{\prime }=-0.01$, $-0.001$, 0.001, and 0.01 cases, respectively. The shift toward positive energy increases with $g_s^{\prime }$ decreasing from positive to negative values. The inset shows the absolute values of the quasiparticle amplitudes $\{u_i,v_i\}$ for $g_s=-0.001$ and $M=-0.95$. $u_0$ and $v_0$ are neglected since they are vanishingly small.

Figure 8

(Color online) Density profiles ${\mid {\Psi }_1\mid }^2$ (left column), ${\mid {\Psi }_0\mid }^2$ (center column), and ${\mid {\Psi }_{-1}\mid }^2$ (right column) perturbed by excitation modes. The upper row corresponds to the complex-frequency $q_{\theta }=\pm 2$ modes for $g_s^{\prime }=-0.001$ and $M=-0.9$, the middle row to the complex-frequency $q_{\theta }=\pm 1$ modes for $g_s^{\prime }=0.01$ and $M=0.2$, and the lower row to the lowest real-frequency $q_{\theta }=-1$ mode for $g_s^{\prime }=0.01$ and $M=0.3$. The density profile of the $m_F=0$ component ${\mid {\Psi }_0\mid }^2$ in the upper row is neglected since it is vanishingly small. The complex modes with both positive and negative $q_{\theta }$ are equally superposed since the modes appear as a result of the energy and angular momentum conservation. The field of view is $10d\times 10d$. We take $\lambda =100$ to show the essential qualitative features of the fluctuation.

Figure 9

(Color online) Detailed structures of the excitation spectra. We show $q_{\theta }=\pm 2$ modes for $g_s^{\prime }=-0.001$ (a) and $-0.01$ (b), and $q_{\theta }=1$ (c) and $-1$ (d) modes for $g_s^{\prime }=0.01$. The eigenenergies for $q_{\theta }=-2$ and $-1$ are plotted as $-\mathrm{Re}\left(E_{\mathit{q}}\right)$. For $q_{\theta }=2$ and 1 modes, the majority components of the spectrum are indicated by solid, dashed, and dash-dotted lines corresponding to $m_F=1$, 0, and $-1$, respectively. For the $q_{\theta }=-2$ and $-1$ modes, these are indicated by filled squares and dots corresponding to the $m_F=0$ and $-1$ components, respectively. The complex-frequency modes are labeled by triangles in all cases.