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Resonance Scattering in optical lattices and Molecules

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. Outline. Motivation/Problem: effective scattering in optical lattice Confinement induced resonance Validity of Hubbard model

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Resonance Scattering in optical lattices and Molecules

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  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. as B B0 molecule Eb Motivation I: biatomic collision under confinements: induced resonance and molecules • 3D Free space: s-wave scattering length see for example: Nature 424, 4 (2003), JILA Feature 1: Feshbach resonance driven by magnetic field Feature 2: Feshbach molecule only at positive a_s

  4. Motivation I: 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 z PRL 81,938 (98); 91,163201(03), M. Olshanii et al

  5. Motivation I: 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 Feature 2: induced molecule at all values of a_s Feature 1: resonance induced by confinement expe: PRL 94, 210401 (05), ETH

  6. Motivation I: 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 Q: whether there is CIR or induced molecule in 3D optical lattice?

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

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

  9. Solution to all Qs: two-body scattering problem in optical lattice for all values of lattice potential and interaction strength ! however, • state-dependent U • Unseparable: center of mass(R) and relative motion(r) • major difficulty: • 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. E=0 + = Our method: momentum-shell renormalization ----from basic concept oflow-energy effective scattering First, based on standard scattering theory, Lippmann-Schwinger equation : implication of renormalization procedure to obtain low-energy physics!

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

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

  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:C1 --- interband; C2 --- intraband

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

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

  16. cross section , phase shift In the opposite limit, Both intra- and inter-bandcontribute to low-E effectivescattering, where C1 can NOT be neglected! set C1=0

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

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

  19. Winkler et al, Nature 441, 853 (06) 12t repulsive as>0 scattering continuum 0 attractive -as<0 From particle-hole symmetry, 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 positivea_s.

  20. 4. E-dependence, effective range : In Hubbard model regime , when compare with free space (all E): 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. Conclusion • 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 Phys. Rev. Lett. 104, 153201 (2010)

  22. Thanks for attention !

  23. no interband, C1=0 Smeared peak at discrete Q as v increases!! Bound state/molecule above resonance (v>vc): a two-body bound state/molecule : Real momentum distribution :

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

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