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Three-component wave function: Subjects of our research are Spin-1 BEC gases 23 Na, 39 K, 87 Rb: I=3/2, F=1 Density Global phase Spinor Spin-1 BEC: Three components wave fucntion

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Spin-1 BEC classified as Hamiltonian of the spin-1 condensate: Classification of spin-1 BEC: AFM and FM manifolds (Ho, PRL 1998) antiferromagnetic (AFM) for g 2 > 0 ferromagnetic (FM) for g 2 <0

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Hydrodynamics only confined in one of manifold Lamacraft PRA 77, 063622 (2012) : Initial state Kudo & Kawaguchi PRA 82, 053614(2010) Kuda & Kawaguchi PRA 82, 053614 (2012)

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Topological structure only confined in one of manifold Borgh & Ruostekoski PRA 87, 033617 (2013) Lamacraft PRA 77, 063622 (2012) : Initial state OPM: Mizushima, Machida, & Kita PRA 89, 030401 (2002) Khawaja & Stoof PRA 64, 043612 (2001) Stoof, Vliegen, & Khawaja PRL 87, 120407 (2001) Kawaguchi, Nitta, & Ueda PRL 100, 180403 (2008)

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Spin texture with a topological number d vector is order parameter for anti-ferromagnetic spinor

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Skyrmion is supposed to be topologically stable; Experimentally, it is not stable! Motivated us to re-exam the existing hydrodynamics of spin-1 BEC Successful creation of Skyrmion in AFM BEC.

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Same conclusion was reached by Yukawa & Ueda [PRA 86, 063614 (2012)] ψ AFM FM δ/2

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Strategy: project onto three orthogonal spinors to get three hydrodynamic equations (Refael, PRB 2009)

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Projection with η F : Dynamics of spin-1 BEC in FM manifold Projection with η A : <-Continuity Equation <-Euler Equation for spin-1 BEC Where <-Landau-Lifshitz equation

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However, projection with η F : Dynamics of spin-1 BEC in FM manifold Structure or dynamics of d-vector are forbidden! -> Hydrodynamics only confined into FM manifold become impossible! d vector is order parameter for ferromagnetic spinor Lamacraft PRA 77, 063622 (2012) Khawaja & Stoof PRA 64, 043612 (2001) Kawaguchi, Nitta, & Ueda PRL 100, 180403 (2008) Kuda & Kawaguchi PRA 82, 053614 (2012)

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Projection with η F : Dynamics of spin-1 BEC in AFM manifold Projection with η A : Same procedures were done in AFM manifold <-Landau-Lifshitz equation <-Continuity Equation <-Euler Equation for spin-1 BEC

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Dynamics of spin-1 BEC in AFM manifold Again… ! Not as strict as in FM manifold. But most structures do not fulfill these equations, including the Skyrmion structure We found one example allowed by these equations. “Uniform spiral spin structure” in 1 dimension projection with η F :

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“All” dynamics involve evolution into a mixed state (δ ≠ 0)

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Initially tried to understand unstable Skyrmion dynamics Instead found neither AFM nor FM sub-manifold supports a well-define d dynamics (FM; t=0) (FM+AFM, t>0) (AFM; t=0) (AFM+FM, t>0) Numerical solution of the Gross-Pitaevskii equation proves our claim

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Spiral AFM state can maintain dynamics entirely within the AFM manifold 17 n=1:

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Collapse of AFM Skyrmion - Mixing of FM manifold AFM Skyrmion collapse by FM manifold mixing Points are local magnetization; The fact that points have size means FM manifold is mixing in

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