1. Strongly Interacting Atoms in Optical Lattices Javier von Stecher JILA and Department of Physics, University of Colorado Support INT 2011 “Fermions from Cold Atoms to Neutron Stars:… arXiv:1102.4593 to appear in PRL In collaboration with Victor Gurarie, Leo Radzihovsky, Ana Maria Rey

2. Strongly interacting Fermions: …Benchmarking the Many-Body Problem.” BCS-BEC crossover a 0 <0 a 0 >0 a 0 =±  Degenerate Fermi Gas (BCS) Molecular BEC

3. Strongly interacting Fermions + Lattice: …Understanding the Many-Body Problem?” ? More challenging: -Band structure, nontrivial dispersion relations, … -Single particle?, two- particle physics?? Not unique: - different lattice structure and strengths.

4. Interaction Energy Hopping Energy J U i+1i Fermi-Hubbard model Minimal model of interacting fermions in the tight-binding regime

5. Fermi-Hubbard model Schematic phase diagram for the Fermi Hubbard model Esslinger, Annual Rev. of Cond. Mat. 2010 half-filling simple cubic lattice 3D Experiments: R. Jordens et al., Nature (2008) U. Schneider et al., Science (2008). Open questions: - d-wave superfluid phase? - Itinerant ferromagnetism?

6. Many-Body Hamiltonian (bosons): Hamiltonian parameters: Beyond the single band Hubbard Model Extension of the Fermi Hubbard Model: Zhai and Ho, PRL (2007) Iskin and Sa´ de Melo, PRL(2007) Moon, Nikolic, and Sachdev, PRL (2007) … Very complicated… But, what is the new physics?

7. T. Muller,…, I. Bloch PRL 2007 Populating Higher bands: Scattering in Mixed Dimensions with Ultracold Gases G. Lamporesi et al. PRL (2010) JILA KRb Experiment New Physics: Orbital physics Experiments: Raman pulse Long lifetimes ~100 ms (10-100 J) G. Wirth, M. Olschlager, Hemmerich “Orbital superfluidity”:

8. New Physics: Resonance Physics Experiments: Two-body spectrum in a single site: Theory and Experiment

9. Lattice induced resonances Resonance Tight Binding + Short range interactions: Strong onsite interactions. Good understanding of the onsite few- body physics. weak nonlocal coupling New degree of freedom: internal and orbital structure of atoms and molecules Separation of energy scales Independent control of onsite and nonlocal interactions

10. Feshbach resonance in free space Two-body level: weakly bound molecules Many-body level: BCS-BEC crossover… 1D Feschbach resonance Interaction λ Energy 0 bound state scattering continuum Lattice induced resonances

11. Feshbach resonance + Lattice What is the many-body behavior? Interaction λ Energy 1D Feschbach resonance bands P.O. Fedichev, M. J. Bijlsma, and P. Zoller PRL 2004 G. Orso et al, PRL 2005 X. Cui, Y. Wang, & F. Zhou, PRL 2010 H. P. Buchler, PRL 2010 N. Nygaard, R. Piil, and K. Molmer PRA 2008 … L. M. Duan PRL 2005, EPL 2008 Dickerscheid, …, Stoof PRA, PRL 2005 K. R. A. Hazzard & E. J. Mueller PRA(R) 2010 … Two-body physics: Many-body physics (tight –binding): What is the two-body behavior? Resonances Lattice induced resonances

12. Our strategy Start with the simplest case – Two particles in 1D + lattice. Benchmark the problem: – Exact two-particle solution Gain qualitative understanding – Effective Hamiltonian description Two-body calculations are valid for two-component Fermi systems and bosonic systems. Below, we use notation assuming bosonic statistics.

13. Two 1D particles in a lattice y xz + a weak lattice in the z-direction Hamiltonian: 1D interaction: Confinement induced resonance One Dimension: V x =V y =200-500 E r, V z =4-20 E r

14. Two 1D particles in a lattice Hamiltonian: 1D interaction: Confinement induced resonance One Dimension: 1D dimers with 40 K H. Moritz, …,T. Esslinger PRL 2005 Bound States in 1D: Form at any weak attraction.

15. Non interacting lattice spectrum Energy k + + k=0 Single particle Two particles Tight-binding limit: k 1 =K/2+k, k 2 =K/2-k Energy K=(k 1 +k 2 ) K=0 (1,0) (0,0) 0 1 2

16. Non interacting lattice spectrum V 0 =4 E r K a/(2 π) (0,0) (0,1) (0,2) (1,1) V 0 =20 E r K a/(2 π) (0,0) (0,1) (0,2) (1,1) Two-body scattering continuum bands

17. Two particles in a lattice, single band Hubbard model Tight-binding approximation Nature 2006 Grimm, Daley, Zoller… U>0, repulsive bound pairsU<0, attractive bound pairs

18. Calculations in a finite lattice with periodic boundary conditions Exact two-body solution Plane wave expansion: Single particle basis functions: Two particles: Very large basis set to reach convergence ~ 10 4 -10 5 Bloch Theorem:

19. Two-atom spectrum Two body spectrum as a function of the interaction strength for a lattice with V 0 =4 E r (0,0) (0,1) (0,0)

20. Two-atom spectrum Two body spectrum as a function of the interaction strength for a lattice with V 0 =4 E r (0,0) (0,1) (0,0)

21. Two-atom spectrum Tight-binding regime

22. Two-atom spectrum Tight-binding regime

23. Avoided crossing between a molecular band and the two-atom continuum dimer continuum K=0 K=π/a First excited dimer crossing

24. How can we understand this qualitatively change in the atom-dimer coupling? K=0 K=π/a Second excited dimer crossing

25. Two-atom spectrum Tight-binding regime

26. Effective Hamiltonian Energy K ΔEΔE  Atoms and dimers are in the tight-binding regime.  They are hard core particles (both atoms and dimers).  Leading terms in the interaction are produced by hopping of one particle. L. M. Duan PRL 2005, EPL 2008 w a,i (r)W m,i (R,r)

27. Effective Hamiltonian Energy K ΔEΔE  J a, J d, g ex, g and ε d are input parameters d†d† a†a†

28. Parity effects  The atomic and dimer wannier functions are symmetric or antisymmetric with respect to the center of the site.  Parity effects on the atom-dimer interaction: S coupling g -1 g +1 g +1 = g -1

29. Parity effects  The atomic and dimer wannier functions are symmetric or antisymmetric with respect to the center of the site.  Parity effects on the atom-dimer interaction: AS coupling g -1 g +1 g +1 = -g -1

30. Parity effects Atom-dimer interaction in quasimomentum space: K = center of mass quasi momentum Prefer to couple at : K=π/a (max K) K=0 (min K) Energy k atoms Energy k molecules Energy k molecules

31. Comparison model and exact solution (1,0) molecule: 1 st excited (2,0) molecule: 2 nd excited 21 sites and V 0 =20E r Molecules above and below!

32. Dimer Wannier Function Effective Hamiltonian matrix elements:  J d, g and ε d fitting parameters to match spectrum?  How to calculate g ex ? is a three-body term Neglected terms: w a,i (r)  Wannier function for dimers: di†di† ai†ai† W m,i (R,r) Prescription to calculate all eff. Ham. Matrix elements

33. Dimer Wannier Function (0,1) dimer Wannier Function 0 Energy K bound state bare dimer Extraction of the bare dimer:  Extraction of J d, g and ε d : excellent agreement with the fitting values. (g  1.7 J for (0,1) dimer)

34. Effective Hamiltonian parameters Construct dimer Wannier function Extract eff. Hamiltonian parameters Single band Hubbard model: … and symmetric coupling Enhanced assisted tunneling!

35. P=p d +p 1 +p 2 Atoms in different bands or species: More dimensions: extra degeneracies… more than one dimer Parity effects Positive parity Negative parity ++ + _ Rectangular lattice

36. Experimental observation:  Initialize system in dimer state.  Change interactions with time.  Measure molecule fraction as a function of quasimomentum. Ramp Experiment: Time Energy dimer state Scattering continuum dimer fraction Observe quasimomentum dependence of atom-dimer coupling N. Nygaard, R. Piil, and K. Molmer PRA 2008 Also K-dependent quantum beats… Dimer fraction (Landau-Zener):

37. Summary  Lattice induced resonances (Lattice + Resonance + Orbital Physics)can be used to tuned lattice systems in new regimes.  The orbital structure of atoms and dimer plays a crucial role in the qualitative behavior of the atom-dimer coupling.  The momentum dependence of the molecule fraction after a magnetic ramp provides an experimental signature of the lattice induced resonances. Outlook:What is the many-body physics of the effective Hamiltonian?