Geometrical approach to hydrodynamics and low-energy excitations of spinor condensates

In this work, we derive the equations of motion governing the dynamics of spin-$F$ spinor condensates. We pursue a description based on standard physical variables (total density and superfluid velocity), alongside $2F$ “spin nodes:” unit vectors that describe the spin-$F$ state and also exhibit the point-group symmetry of a spinor condensate’s mean-field ground state. In the first part of our analysis, we derive the hydrodynamic equations of motion, which consist of a mass continuity equation, $2F$ Landau-Lifshitz equations for the spin nodes, and a modified Euler equation. In particular, we provide a generalization of the Mermin-Ho relation to spin one and find an analytic solution for the skyrmion texture in the incompressible regime of a spin-half condensate. In the second part, we study the linearized dynamics of spinor condensates. We provide a general method to linearize the equations of motion based on the symmetry of the mean-field ground state using the local stereographic projection of the spin nodes. We also provide a simple construction to extract the collective modes from symmetry considerations alone akin to the analysis of vibrational excitations of polyatomic molecules. Finally, we present a mapping between the spin-wave modes, and the wave functions of electrons in atoms, where the spherical symmetry is degraded by a crystal field. These results demonstrate the beautiful geometrical structure that underlies the dynamics of spinor condensates.

Figure 1

(Color online) Possible phases for the spin-one and spin-two condensates. The red dots on the unit sphere correspond to the unit vectors ${\mathbf{n}}_{\mathbf{i}}$ (reciprocal spinors) defined in Eq. (12). Spin-one condensates have ferromagnetic (1a) and nematic phases (1b) while spin-two condensates have ferromagnetic (2a), uniaxial nematic (2b), square biaxial nematic (2c), and a tetrahedral (2d) phases.

Figure 2

(Color online) A skyrmion configuration corresponding to Eq. (66).

Figure 3

(Color online) Normal modes of the cyclic state. Mode (a) is the optical mode corresponding to pure displacements in $w_4$. Modes (b), (c), and (d) are gapless modes corresponding rotating about the $x$, $y$, or $z$ axes, respectively. The axes of rotation for these modes are shown.

Figure 4

Normal modes for the hexagonal configuration of the spin-three condensate. Vectors moving into and out of the plane are denoted with “$-$” and “$+$,” respectively. By multiplying the set of parameters $\left\{z_i\right\}$ corresponding to these displacements by a factor of $i$, one can identify ${\nu }_1={\nu }_2$, ${\nu }_3={\nu }_4$, ${\nu }_5={\nu }_7$, ${\nu }_6={\nu }_8$, ${\nu }_9={\nu }_{11}$, and ${\nu }_{10}={\nu }_{12}$. The modes ${\nu }_1={\nu }_2$, ${\nu }_5={\nu }_7$, and ${\nu }_6={\nu }_8$ correspond to Goldstone excitations due to the broken-spin symmetry while all other modes are optical and gapped. The mode ${\nu }_{10}={\nu }_{12}$ has a set of displacement vectors with lengths differing by a factor of 2.