Polarized states and domain walls in spinor Bose-Einstein condensates

We study spin-polarized states and their stability in the antiferromagnetic phase of spinor $\left(F=1\right)$ quasi-one-dimensional Bose-Einstein condensates. Using analytical approximations and numerical methods, we find various types of polarized states, including patterns of the Thomas-Fermi type, structures featuring a pulse in one component inducing a hole in the other components, states with holes in all three components, and domain walls (DWs). The stability analysis based on the Bogoliubov–de Gennes equations reveals intervals of weak oscillatory instability in families of these states, except for the DWs, which are always stable. The development of the instabilities is examined by means of direct simulations.

Figure 1

(Color online) Top panels: Examples of two stable spin-polarized states for $\Delta \theta =0$ (left panel) and $\Delta \theta =\pi$ (right panel), both obtained for $\Omega =0.1$ and $\mu =2$. Wave functions ${\psi }_{\pm 1}$ are identical and are depicted by the solid line, while the wave function ${\psi }_0$ is depicted by the dashed line. In the left panel, $q_{-1}=q_{+1}=0.5$ and $q_0=1$, while in the right panel $q_{-1}=q_{+1}=0.25$ and $q_0=0.5$. Bottom panel: The spatial distribution of the total mean-field spin $f$; note that in this case $f_y=f_z=0$ and $f_x=\pm f$ for $\Delta \theta =0$ or $\pi$, respectively.

Figure 2

(Color online) Spin-polarized state with a hole in the ${\psi }_{\pm 1}$ components, and a pulselike shape of the ${\psi }_0$ field, for $\Omega =0.1$ and $\Delta \theta =0$. The top left and right panels show unstable and stable states, with $\mu =2$ and $\mu =3$, respectively; solid and dashed lines depict components ${\psi }_{\pm 1}$ and ${\psi }_0$. The two panels in the second row display spectral planes $\left({\lambda }_r,{\lambda }_i\right)$ of the (in)stability eigenvalues for the same states. Note that the instability (of the state pertaining to $\mu =2$) is of the oscillatory type, being accounted for by a quartet of eigenvalues with nonzero real parts. The solid and dashed lines in the third-row panel show the normalized number of atoms (norm), $N$, in components ${\psi }_{\pm 1}$ and ${\psi }_0$, respectively, as a function of chemical potential $\mu$. The bottom panel shows the maximal growth rate, $\text{max}\left({\lambda }_r\right)$, as a function of $\mu$, which reveals the instability window for $1.81\le \mu \le 2.15$, with a maximum instability growth rate ${\left({\lambda }_r\right)}_{\text{max}}\approx 1.3\times {10}^{-3}$ at $\mu \approx 2$. The latter value corresponds to the unstable state shown in the top left panel.

Figure 3

(Color online) Top panels: Contour density plots display the evolution of the weakly unstable solution shown in Fig. 2 (the left and right panels represent, respectively, ${\psi }_{+1}\equiv {\psi }_{-1}$ and ${\psi }_0$ components). Because of the extremely small growth rate of the instability, it manifests itself only at $t>4000$. Middle panels: The respective wave-function profiles at $t=0$ (solid lines) and $t=10000$ (dashed lines); as above, the left and right panels show, respectively, wave functions ${\psi }_{\pm 1}$ and ${\psi }_0$. Bottom panel: Snapshots of the distribution of the total mean-field spin $f=f_x$ at $t=0$ (solid line) and $t=10000$ (dashed line).

Figure 4

(Color online) Same as Fig. 2 but for a state with one hole in each of the ${\psi }_{\pm 1}$ components and two holes in the ${\psi }_0$ component. In this case, the instability is driven by two quartets in the eigenvalue spectrum of small perturbations, which lead to instability in the interval $2.58\le \mu \le 3.22$. The maximum instability growth rate is $\text{max}\left({\lambda }_r\right)\approx 1.8\times {10}^{-3}$ at $\mu \approx 2.9$. The unstable state shown in the top right panel corresponds to $\mu =3$.

Figure 5

(Color online) Same as Fig. 3, for the unstable state shown in the top right panel of Fig. 4 (it pertains to $\mu =3$). The instability manifests itself at large times $\left(t>3500\right)$ and results in strong oscillatory deformation of the spinor condensate; this is clearly observed in the contour plots of the densities (top panels), the snapshots of the wave function profiles (middle panels), and the snapshots of the mean-field spin distribution (bottom panel).

Figure 6

(Color online) Initialization of the system when three spatially separated traps, $V_{-1}$, $V_0$ (shown by dashed parabolas), and $V_{+1}$ (the solid parabola), with equal strengths $\left(\Omega =0.1\right)$ and centers placed at $x=-10$, 0, and $+10$, hold the TF states of the ${\psi }_{-1}$, ${\psi }_0$, and ${\psi }_{+1}$ components, respectively. Then, two traps ($V_{-1}$ and $V_0$), centered at $x=-10$ and $x=0$, are turned off, and a stationary solution, supported solely by the trap $\left(V_{+1}\right)$ centered at $x=10$, is looked for by means of the fixed-point algorithm, using the configuration with the three mutually shifted components as an input.

Figure 7

(Color online) Top left panel: Wave functions of the components (${\psi }_{+1}$ and ${\psi }_{-1}$ are identical) in a stationary state found from the initial configuration prepared as shown in Fig. 6. The resulting spin-polarized state has the form of a domain-wall structure between the ${\psi }_0$ and ${\psi }_{\pm 1}$ components. The parameters are $\Omega =0.1$ and $\mu =2$. The top right panel shows the wave functions of the domain-wall state found at $\mu =1.43$. Middle panel: The norm of each component in the domain-wall structure vs the chemical potential (the dependences for $\Delta \theta =0$ and $\pi$ are identical). Bottom panels: The spatial distribution of the total mean-field spin (solid lines). For $\mu =2$ (bottom left) $f=f_x$, while for $\mu =1.43$ (bottom right) $f=\sqrt{f_x^2+f_z^2}\approx f_x$; in the latter case, the $f_x$ and $f_z$ components are depicted by dashed and dashed-dotted lines, respectively.

Figure 8

(Color online) Evolution of the domain-wall structure shown in the top left panel of Fig. 7, to which a random perturbation was added. Shown in the left and right panels are spatiotemporal contour plots of the densities in components ${\psi }_{\pm 1}$ (identical to each other) and ${\psi }_0$, respectively.

Figure 9

(Color online) Same as Fig. 8 but for the state shown in the top right panel of Fig. 7. The left, middle, and right panels show, respectively, the densities in the ${\psi }_{-1}$, ${\psi }_0$, and ${\psi }_{+1}$ components. Noteworthy is a stationary dark-soliton-like structure, located at $x\approx 13$ in the ${\psi }_{-1}$ component.