1. Symmetry Breaking in Scalar, Spinor, and Rotating Bose-Einstein condensates Hiroki Saito FB18 Aug. 24, 2006 Univ. of Electro-Communications, Japan Tokyo Institute of Technology, Japan Yuki Kawaguchi Masahito Ueda Univ. of Arizona, Tucson Rina Kanamoto

2. Contents Symmetry breaking in BEC 1. rotation symmetry breaking in attractive BEC 2. clockwise-counterclockwise symmetry breaking in ferromagnetic BEC 3. chiral symmetry breaking in dipolar BEC

3. 1D attractive BEC 1D infinite space bright soliton translation symmetry breaking 1D ring rotation symmetry breaking

4. Critical localization on a ring density excitation energy RK, HS, and MU, PRA 67 013608 (2003); PRA 68 043619 (2003); PRL 94 090904 (2005); PRA 73 033611 (2006) zero point energy attractive interaction energy > localization Goldstone mode gN

5. Single vortex ------ ring geometry axi-symmetric vortex state topological defect ringlike geometry attractive interaction rotation symmetry breaking

6. Symmetry breaking in rotating attractive BEC HS and MU, PRL 89 190402 (2002); PRA 69 013604 (2004) split-merge cycle excitation energy rotation symmetry breaking Goldstone mode

7. Magnetization of spinor BEC Ferromagnetic interaction (e.g. spin-1 87 Rb) The initial state is prepared in non-magnetic state. (e.g. m F =0) violation of spin conservation How does magnetization occur in isolated systems? spin texture e.g. domain structure HS and MU, PRA 72, 023610 (2005)

8. Topological texture formation polar-core vortex energy cost at domain walls ferromagnetic energy > spontaneous formation of topological texture HS, YK, and MU, PRL 96, 065302 (2006) degenerate

9. Polar-core vortex formation initial state m = 0 + noise m=1 component clockwise-counterclockwise symmetry breaking

10. Energy of broken symmetry state

11. Recent experiment in Berkeley Sadler et al., cond-mat/0605351 magnitude of spin direction of spin

12. Dipolar interaction dipolar interaction long range anisotropic 52 Cr dipolar BEC has been realized in Stuttgart. Griesmaier et al., PRL 94 10401 (2005) dipolar magnetic field YK, HS, and MU, PRL 96, 080405 (2006)

13. Ground state spin texture spin-1 87 Rb BEC ferromagnetic dipole energy kinetic energy << x-z plane x-y plane single domain dipole energy kinetic energy >> magnetic closure

14. Chiral symmetry breaking dipole energy kinetic energy ~ x-z plane x-y plane Chiral symmetry breaking YK, HS, and MU, to appear in PRL

15. Spontaneous circulation orbital angular momentum dipole interaction / kinetic energy Nonzero circulation by chiral symmetry breaking

16. Summary 1. Rotation symmetry breaking vortex split by attractive interaction 2. Clockwise-counterclockwise symmetry breaking topological spin texture formation 3. Chiral symmetry breaking ground state with dipolar interaction