Manifold Mixing in Temporal Evolution of Spin-1 BEC

1. Manifold Mixing in Temporal Evolution of Spin-1 BEC Yun-Tak Oh Sungkyunkwan University Yun-Tak Oh, Panjin Kim, Jin-Hong Park, Jung Hoon Han, arXiv:1309.5683

2. CONTENTS 1. Meaning of AFM & FM manifolds in spin-1 BEC and hydrodynamics within them 2. Our (new) approach to spin-1 hydrodynamics 3.Conclusion

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

4. Classification of spin-1 BEC: AFM and FM manifolds (Ho, PRL 1998) Hamiltonian of the spin-1 condensate: Spin-1 BEC classified as ferromagnetic (FM) for g2<0 antiferromagnetic (AFM) for g2 > 0

5. Hydrodynamics only confined in one of manifold Spin-1 BEC FM AFM Kudo & Kawaguchi PRA 82, 053614(2010) Lamacraft PRA 77, 063622 (2012) Kuda & Kawaguchi PRA 82, 053614 (2012) : Initial state

6. Topological structure only confined in one of manifold OPM: OPM: Spin-1 BEC FM AFM Borgh & Ruostekoski PRA 87, 033617 (2013) Lamacraft PRA 77, 063622 (2012) Mizushima, Machida, & Kita PRA 89, 030401 (2002) Khawaja & Stoof PRA 64, 043612 (2001) : Initial state Stoof, Vliegen, & Khawaja PRL 87, 120407 (2001) Kawaguchi, Nitta, & Ueda PRL 100, 180403 (2008)

7. Example of topologically stable structure: Skyrmion d vector is order parameter for anti-ferromagnetic spinor Spin texture with a topological number

8. Unexpected collapse of Skyrmion in AFM manifold Shin group, PRL 108, 035301 (2012) Successful creation of Skyrmion in AFM BEC. Skyrmion is supposed to be topologically stable; Experimentally, it is not stable! Motivated us to re-exam the existing hydrodynamics of spin-1 BEC

9. New expression of three components wave function: AFM+FM FM ψ Same conclusion was reached by Yukawa & Ueda [PRA 86, 063614 (2012)] δ/2 AFM

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

11. Dynamics of spin-1 BEC in FM manifold Projection with ηF : <-Continuity Equation <-Euler Equation for spin-1 BEC Where Projection with ηA : <-Landau-Lifshitz equation

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

13. Dynamics of spin-1 BEC in AFM manifold Same procedures were done in AFM manifold Projection with ηF : <-Landau-Lifshitz equation Projection with ηA : <-Continuity Equation <-Euler Equation for spin-1 BEC

14. Dynamics of spin-1 BEC in AFM manifold Again…! projection with ηF : 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

15. “All” dynamics involve evolution into a mixed state (δ ≠ 0) Spin-1 BEC FM AFM

16. Conclusion • 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

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

18. 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