Dissipative hydrodynamic equation of a ferromagnetic Bose-Einstein condensate: Analogy to magnetization dynamics in conducting ferromagnets

The hydrodynamic equation of a spinor Bose-Einstein condensate (BEC) gives a simple description of spin dynamics in the condensate. We introduce the hydrodynamic equation of a ferromagnetic BEC with dissipation originating from the energy dissipation of the condensate. The dissipative hydrodynamic equation has the same form as an extended Landau-Lifshitz-Gilbert (LLG) equation, which describes the magnetization dynamics of conducting ferromagnets in which localized magnetization interacts with spin-polarized currents. Employing the dissipative hydrodynamic equation, we demonstrate the magnetic domain pattern dynamics of a ferromagnetic BEC in the presence and absence of a current of particles, and discuss the effects of the current on domain pattern formation. We also discuss the characteristic lengths of domain patterns that have domain walls with and without finite magnetization.

Figure 1

(a) Time dependence of the averaged longitudinal magnetization, kinetic and MDDI energies in the presence (solid curves), and absence (dashed curves) of ${\mathit{v}}_{\mathrm{mass}}$. The dot-dashed curve in the graph of $E_{\mathrm{kin}}/h$ (middle panel) corresponds to the contribution from ${\mathit{v}}_{\mathrm{mass}}$. (b) Snapshots of longitudinal magnetization in which white and black correspond to positive and negative values of ${\widehat{f}}_z$, respectively. The size of each snapshot is $256\times 256$ $\mu$m. Here, we take $q/h=-50$ Hz, and $n_{\mathrm{tot}}=6n_{\mathrm{tot}}^{\left(0\right)}$, where $n_{\mathrm{tot}}^{\left(0\right)}=\sqrt{2\pi d^2}{n}_{3\mathrm{D}}^{\left(0\right)}$ with $n_{3\mathrm{D}}^{\left(0\right)}=2.3\times {10}^{14}$ cm${}^{-3}$ and $d=1.0$ $\mu$m. The damping rate is given by a typical value $\Gamma =0.03$. The other parameters are given by the typical values for a spin-1 ${}^{87}$Rb atom: $M=1.44\times {10}^{-25}$ Kg, $F=1$, and $g_F=-1/2$.

Figure 2

Comparison of domain formation dynamics between the hydrodynamic description and GP numerics. (a) Time dependence of the averaged longitudinal magnetization, kinetic and MDDI energies simulated by the hydrodynamic equation (left column), and the GP equation (right column). (b) Snapshots at $t=5$ s simulated by the hydrodynamic and GP equations. Solid, dashed, and dot-dashed curves in (a) are for $q/h=-50$, $-40$, and $-30$, respectively.

Figure 3

Domain pattern formation in the absence of a magnetic field. The direction of the stripes depends on the initial configuration of spins. The parameters are the same as those given in Fig. 1

Figure 4

Dependence of theoretical characteristic lengths on $-q/h$ for the hydrodynamic description. (a) Theoretically estimated characteristic lengths of a stripe pattern (solid curve) [Eq. (58)], a pattern at its emergence (dashed curve) [Eq. (49) with Eq. (46)], and the domain wall width (dot-dashed curve) [Eq. (54)] for $n_{\mathrm{tot}}=6n_{\mathrm{tot}}^{\left(0\right)}$. (b) Theoretical characteristic length of a stripe pattern [Eq. (58)] for $n_{\mathrm{tot}}=3n_{\mathrm{tot}}^{\left(0\right)}$ (dashed line), $6n_{\mathrm{tot}}^{\left(0\right)}$ (solid line), $10n_{\mathrm{tot}}^{\left(0\right)}$ (dot-dashed line), and $20n_{\mathrm{tot}}^{\left(0\right)}$ (dotted line).

Figure 5

Dependence of theoretical characteristic lengths on $-q/h$ for the GP equation. (a) Theoretically estimated characteristic lengths of the stripe pattern with fully magnetized domain walls (solid curve) [Eq. (58)], that of the stripe pattern without transverse magnetization (dot-dashed line) [Eq. (65)], and that of a pattern at its emergence (dashed curve) [Eq. (49) with Eq. (47)] for $n_{\mathrm{tot}}=6n_{\mathrm{tot}}^{\left(0\right)}$. (b) Theoretically estimated domain wall width of the stripe pattern with fully magnetized domain walls (solid curve) [Eq. (54)] and that of the stripe pattern without transverse magnetization (dot-dashed line) [Eq. (64)] for $n_{\mathrm{tot}}=6n_{\mathrm{tot}}^{\left(0\right)}$.