Figure 1

Real and imaginary parts of the lowest Bogoliubov energies ${\epsilon }^{(\ell )}$ for $\ell =0,\pm 1$, where the $m=\pm 1$ components of the eigenfunction are proportional to $e^{\pm i\ell \varphi }$. The two energies ${\epsilon }^{(\pm 1)}$ are degenerate due to the axisymmetry of the system. We have taken the parameters of spin-1 ${}^{87}\mathrm{Rb}$ atoms, where the spin-independent interaction strength $g_0^{2\mathrm{D}}$ is related to the spin-dependent strength $g_1^{2\mathrm{D}}$ by $g_0^{2\mathrm{D}}=-216.1g_1^{2\mathrm{D}}$.Reuse & Permissions

Figure 2

Degree of Bogoliubov excitation $P_{\pm 1}$ in Eq. (5). The interaction strengths are $g_0^{2\mathrm{D}}=2200$ and $g_1^{2\mathrm{D}}=-10.18$. The initial state is the ground state ${\Psi }_0$ of Eq. (2) for the $m=0$ component plus a small amount of random noise in the $m=-1$ component. The dashed line is proportional to $e^{0.141\omega t}$. The insets show the density-phase profiles at $\omega t=130$, where the size of the frame is $16\times 16$ in units of $a_{\mathrm{ho}}=[\hslash /(M\omega ){]}^{1/2}$. The heights of the vertical lines in the insets indicate $|{\psi }_m{|}^2{a}_{\mathrm{ho}}^2/N$, which is 0.001 for $m=\pm 1$ and 0.0025 for $m=0$.Reuse & Permissions

Figure 3

(a) Time evolution of the fraction $n_{-1}$ (dashed curves) and the orbital angular momentum per particle $L_{-1}$ (solid curves) in the $m=-1$ component with ($\gamma =0.03$, red curves) and without ($\gamma =0$, blue curves) dissipation. The interaction strengths are the same as in Fig. 2. The initial state is given by ${\psi }_0={\Psi }_0$, ${\psi }_{-1}={10}^{-4}r(e^{i\varphi }+1.0001e^{-i\varphi }){\Psi }_0$, and ${\psi }_1=0$, where ${\Psi }_0$ is the ground state solution of Eq. (2). (b)–(d) Snapshots of the density-phase profiles and spin textures. The heights of the vertical lines in the density-phase profiles show $|{\psi }_m{|}^2{a}_{\mathrm{ho}}/N$, which is 0.001 for $m=\pm 1$ and 0.0025 for $m=0$. The length of the vector in the bottom panels is proportional to $(F_x^2+F_y^2{)}^{1/2}$ and the color represents $(F_x^2+F_y^2{)}^{1/2}/(|{\psi }_{-1}{|}^2+|{\psi }_0{|}^2+|{\psi }_1{|}^2)$. The size of the frame is $16\times 16$ for the density-phase profiles and $12\times 12$ for the spin textures in units of $a_{\mathrm{ho}}$.Reuse & Permissions

Figure 4

Energy and density-phase profiles obtained by the variational method with Eq. (6). The interaction strengths are the same as in Fig. 2. The size of the frame of the insets is $16\times 16$ in units of $a_{\mathrm{ho}}$ and the heights of the vertical lines show $|{\psi }_m{|}^2{a}_{\mathrm{ho}}^2/N$, which is 0.0005 for $m=\pm 1$ and 0.0025 for $m=0$.Reuse & Permissions