Hydrodynamic description of spin-1 Bose-Einstein condensates

We establish a complete set of hydrodynamic equations for a spin-1 Bose-Einstein condensate, which are equivalent to the multicomponent Gross-Pitaevskii equations and expressed in terms of only observable physical quantities: the spin and the nematic (or quadrupolar) tensor in addition to the density of particles and the mass current that appear in the hydrodynamic description of a scalar condensate. The obtained hydrodynamic equations involve a generalized Mermin-Ho relation that is valid regardless of the spatiotemporal dependence of the spin polarization. Low-lying collective modes for phonons and magnons are reproduced by linearizing the hydrodynamic equations. We also apply the single-mode approximation to the hydrodynamic equations and find a complete set of analytic solutions.

Figure 1

Euler angles $\alpha$, $\beta$, and $\gamma$ specifying an arbitrary rotation of the condensate wave function in real space. The Euler rotation of a spinor wave function at the coordinate ${\zeta }_{\mu }\left(\mathit{R}\right)$ is described by the matrix representation of the rotation operator $\widehat{\mathcal{U}}(\alpha \left(\mathit{R}\right),\beta \left(\mathit{R}\right),\gamma \left(\mathit{R}\right))$ given in Eq. (36). The vector $\mathit{f}$ represents the spin vector.

Figure 2

Cartesian representation of the condensate wave functions at position $\mathit{R}$ plotted in terms of $r(\theta ,\varphi )$ defined in Eq. (44) with its two axes ${\tilde{\mathit{e}}}_1$ and ${\tilde{\mathit{e}}}_2$ indicated by red arrows. These two axes are on the plane delimited by the closed contour perpendicular to the unit eigenvector of the nematic tensor ${\mathit{e}}_3$ at $\mathit{R}$. For the sake of simplicity, here we set the Euler angles $\alpha =0$ and $\beta =0$ so that ${\mathit{e}}_3$ is parallel to the $z$ axis and $r(\theta ,\varphi )=|{\zeta }_{\mu }(0,0,\gamma ;\vartheta )V_{\mu }(\theta ,\varphi )|$. The polarization parameter $\vartheta$ is taken to be $\vartheta =3\pi /16$. Note that ${\tilde{\mathit{e}}}_1$ and ${\tilde{\mathit{e}}}_2$ coincide with the major and minor axes of the condensate wave function, respectively.

Figure 3

Time and $\tilde{q}$ dependences of $\left|\mathit{f}\right(t\left)\right|$ (each leftmost panel) and snapshots of the condensate wave functions with the principal axis ${\tilde{\mathit{e}}}_1$ indicated by red arrows for the case of (a) $\tilde{q}=-2/5<u_0$, (b) $\tilde{q}=-u_0$, and (c) $\tilde{q}=-1/4>-u_0$, where $\tilde{q}$ satisfies $-2u_0<\tilde{q}<0$. Here, $\vartheta \left(0\right)$ is taken to be $\vartheta \left(0\right)=3\pi /16$ and hence $u_0=\cos 3\pi /8$ and $f_0=\sin 3\pi /8$. The period is given by $2T\left[k\left(\tilde{q}\right)\right]$, where $T\left[k\left(\tilde{q}\right)\right]$ is the first-kind elliptic integral with the elliptic modulus $k\left(\tilde{q}\right)$ given by Eq. (A20). We here use the following abbreviations: (a) $k_{<}\equiv k(-2/5)$, (b) $k_c\equiv k(-u_0)$, and (c) $k_{>}\equiv k(-1/4)$. The trajectory of ${\tilde{\mathit{e}}}_1$, which is indicated by solid closed curves with arrows, undergoes a transition at a branch point (b) with $\tilde{q}=-u_0$. The principal axes swing around their initial positions for the case of $\tilde{q}<-u_0$ as shown in (a), whereas they rotate about the spin vector (or the eigenvector of the nematic tensor ${\mathit{e}}_3$) for the case of $\tilde{q}>-u_0$ as shown in (c), because for $\tilde{q}=-u_0$ the major and minor axes exchange each other at $t=T\left(k_c\right)$, where the spin vector is fully polarized and the major and minor axes can take any mutually orthogonal directions.