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arXiv:0812.3403 · Physical Review B

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

Ryan Barnett, Daniel K. Podolsky, Gil Refael

Published Jul 21, 2009• 57 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.80.024420

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Abstract In this paper, we develop an appropriate set of hydrodynamic equations for a U(N) invariant superfluid that couple the dynamics of superflow and magnetization. In the special case when both the superfluid and normal velocities are zero, the hydrodynamic equations reduce to a generalized version of Landau–Lifshitz equation for ferromagnetism with U(N) symmetry. When both velocities are non-zero, there appears couplings between the superflow and magnetization dynamics, and the superfluid velocity no longer satisfies the irrotational condition. On the other hand, the magnitude of magnetization is no longer a constant of motion as was the case for the standard Landau–Lifshitz theory. In comparison with the simple superfluid, the first and second sounds are modified by a non-zero magnetization through various thermodynamic functions. For U(2) invariant superfluid, we get both (zero-) sound wave and a spin wave at zero temperature. It is found that the dispersion of spin wave is always quadratic, which is consistent with microscopic analysis. In the appendix, we show that the hydrodynamic equation for a U(N) invariant superfluid can be obtained from the general hydrodynamic equation with arbitrary internal symmetries.
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We obtain the equations of motion for a ferromagnetic Bose condensate of arbitrary spin in the long-wavelength limit. We find that the magnetization of the condensate is described by a nontrivial modification of the Landau-Lifshitz equation, in which the magnetization is advected by the superfluid velocity. This hydrodynamic description, valid when the condensate wave function varies on scales much longer than either the density or spin healing lengths, is physically more transparent than the corresponding time-dependent Gross-Pitaevskii equation. We discuss the conservation laws of the theory and its application to the analysis of the stability of magnetic helices and Larmor precession. Precessional instabilities, in particular, provide a novel physical signature of dipolar forces.