1. Resonance Scattering in optical lattices and Molecules 崔晓玲 (IOP, CASTU) Collaborators: 王玉鹏 (IOP), Fei Zhou (UBC) 2010.08.02 大连

2. Outline Motivation/Problem: effective scattering in optical lattice –Confinement induced resonance –Validity of Hubbard model –Collision property of Bloch waves (compare with plane waves) Basic concept/Method –Renormalization in crystal momentum space Results –Scattering resonance purely driven by lattice potential –Criterion for validity of single-band Hubbard model –Low-energy scattering property of Bloch waves E-dependence, effective range Induced molecules, detection, symmetry

3. see for example: Nature 424, 4 (2003), JILA molecule B B0B0 asas EbEb Feature 1: Feshbach resonance driven by magnetic field Feature 2: Feshbach molecule only at positive a_s Motivation I: biatomic collision under confinements: induced resonance and molecules 3D Free space: s-wave scattering length

4. Motivation I: z biatomic collision under confinements: induced resonance and molecules 3D Free space: s-wave scattering length Confinement Induced Resonance and Molecules see for example: CIR in quasi-1D PRL 81,938 (98); 91,163201(03), M. Olshanii et al

5. biatomic collision under confinements: induced resonance and molecules 3D Free space: s-wave scattering length Confinement Induced Resonance and Molecules Motivation I: Feature 1: resonance induced by confinement Feature 2: induced molecule at all values of a_s see for example: CIR in quasi-1D expe: PRL 94, 210401 (05), ETH

6. biatomic collision under confinements: induced resonance and molecules 3D Free space: s-wave scattering length Confinement Induced Resonance and Molecules Motivation I: Q: whether there is CIR or induced molecule in 3D optical lattice? see for example: CIR in quasi-1D

7. Validity of single-band Hubbard model to optical lattice Motivation II: under tight-binding approximation: Q: how to identify the criterion quantitatively? break down in two limits:  shallow lattice potential  strong interaction strength

8. Scattered Bloch waves near the bottom of lowest band Motivation III: near E=0, quadratic dispersion defined by band mass free space Q: low-energy effective scattering (2 body, near E=0) free space ? explicitly, energy-dependence of scattering matrix, effective interaction range, property of bound state/molecule……

9. Solution to all Qs: two-body scattering problem in optical lattice for all values of lattice potential and interaction strength ! major difficulty: however,  state-dependent U  Unseparable: center of mass(R) and relative motion(r) Previous works are mostly based on single-band Hubbard model, except few exact numerical works (see, G. Orso et al, PRL 95, 060402, 2005; H. P. Buechler, PRL 104, 090402, 2010: both exact but quite time- consuming with heavy numerics, also lack of physical interpretation such as individual inter/intra-band contributions, construction of Bloch- wave molecule…

10. ----from basic concept of low-energy effective scattering First, based on standard scattering theory, Lippmann-Schwinger equation : = + E=0 Our method: momentum-shell renormalization implication of renormalization procedure to obtain low-energy physics!

11. RG eq: with boundary conditions: ----from basic concept of low-energy effective scattering Our method: momentum-shell renormalization Then, an explicit RG approach:

12. RG approach to optical lattice and results Simplification of U: XL Cui, YP Wang and F Zhou, Phys. Rev. Lett. 104, 153201 (2010) inter-band, to renormalize short-range contribution intra-band, specialty of OL

13. two-step renormalization Step I : renormalize all virtual scattering to higher-band states (inter-band) Step II : further integrate over lowest-band states (intra-band) Characteristic parameter: C 1 --- interband; C 2 --- intraband

14. 1. resonance scattering at E=0: Results resonance scattering of Rb-K mixture resonance at

15. For previous study in this limit see P.O. Fedichev et al, PRL 92, 080401 (2004). 2. Validity of Hubbard model: To safely neglect inter-band scattering, Condition I: : deep lattice potential Under these conditions, Hubbard limit Condition II: : weak interaction

16. cross section, phase shift set C 1 =0 In the opposite limit, Both intra- and inter-band contribute to low-E effective scattering, where C 1 can NOT be neglected!

17. as/aLas/aL E s-band 0 3. Symmetry between repulsive and attractive bound state: simply solvable: K conserved (semi-separated) state-independent U Zero-energy resonance scattering attractive and repulsive bound state

18. bound state for a general K: scattering continuum 0 12t B T-matrix and bound state:

19. repulsive a s >0 attractive -a s <0 Winkler et al, Nature 441, 853 (06) From particle-hole symmetry, scattering continuum 0 12t Resonance scattering and bound states near the bottom of lowest band for a negative a_s therefore imply resonance scattering and bound states near the top of the band for a positive a_s.

20. 4. E-dependence, effective range : compare with free space (all E): In Hubbard model regime, when Effective interaction range of atoms in optical lattice is set by lattice constant (finite, >> range in free space), even for two atoms near the band bottom! This leads to much exotic E-dependence of T-matrix in optical lattice.

21.  Effective scattering using renormalization approach  Optical lattice induced resonance scattering (zero-energy)  Large a_s, shallow v: interband + intraband  Small a_s, deep v: intraband (dominate) ------- validity criterion for single-band Hubbard model  Bound state induced above resonance –Binding energy, momentum distribution (for detection) –Mapping between attractive (ground state) and repulsive bound state via particle-hole symmetry  Exotic E-dependence of T-matrix / effective potential ------- due to finite-range set by lattice constant Conclusion Phys. Rev. Lett. 104, 153201 (2010)

23. Bound state/molecule above resonance (v>v c ): a two-body bound state/molecule : Real momentum distribution : no interband, C 1 =0 Smeared peak at discrete Q as v increases!!

24. Bound state: T(E B )=infty (Bethe-Salpeter eq) 0 12t repulsive a s >0 attractive -a s <0 Repulsive metastable excited above band top n q peaked at q=±pi Attractive ground state below band bottom n q peaked at q=0 K=0 bound state assume a 2-body wf: