Strongly Interacting Atoms in Optical Lattices

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

2. a0=± a0>0 a0<0 Molecular BEC Degenerate Fermi Gas (BCS) Strongly interacting Fermions:…Benchmarking the Many-Body Problem.” BCS-BEC crossover

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

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

8. New Physics: Resonance Physics Experiments: Two-body spectrum in a single site: Theory and Experiment Tuning interactions in lattices: tune interaction Molecules of Fermionic Atoms in an Optical Lattice T. Stöferle , …,T. Esslinger PRL 2005

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

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

11. Lattice induced resonances Energy 1D Feschbach resonance Feshbach resonance + Lattice bands What is the many-body behavior? Interaction λ What is the two-body behavior? Resonances Two-body physics: Many-body physics (tight –binding): 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 …

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 One Dimension: y z x Hamiltonian: + a weak lattice in the z-direction Vx=Vy=200-500 Er, Vz=4-20 Er 1D interaction: Confinement induced resonance

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

15. Energy Energy K=(k1+k2) k Non interacting lattice spectrum Single particle Two particles K=0 k=0 2 + (1,0) 1 Tight-binding limit: 0 + (0,0) k1=K/2+k, k2=K/2-k

16. Non interacting lattice spectrum Two-body scattering continuum bands (1,1) (0,2) (0,2) (1,1) V0=4 Er V0=20 Er (0,1) (0,1) (0,0) (0,0) K a/(2 π) K a/(2 π)

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

18. Exact two-body solution Calculations in a finite lattice with periodic boundary conditions Bloch Theorem: Plane wave expansion: Single particle basis functions: Two particles: Very large basis set to reach convergence ~ 104-105

19. Two-atom spectrum (0,1) (0,0) (0,0) Two body spectrum as a function of the interaction strength for a lattice with V0=4 Er

20. Two-atom spectrum (0,1) (0,0) (0,0) Two body spectrum as a function of the interaction strength for a lattice with V0=4 Er

21. Two-atom spectrum Tight-binding regime

22. Two-atom spectrum Tight-binding regime

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

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

25. Two-atom spectrum Tight-binding regime

26. Energy K Effective Hamiltonian wa,i(r) Wm,i(R,r) ΔE • 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

27. Energy K Effective Hamiltonian d† a† ΔE • Ja, Jd, gex, g and εd are input parameters

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

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

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

31. Comparison model and exact solution (2,0) molecule: 2nd excited (1,0) molecule: 1st excited Molecules above and below! 21 sites and V0=20Er

32. Dimer Wannier Function • Jd, g and εd fitting parameters to match spectrum? Effective Hamiltonian matrix elements: • How to calculate gex? • is a three-body term ai† di† • Wannier function for dimers: wa,i(r) Neglected terms: Wm,i(R,r) Prescription to calculate all eff. Ham. Matrix elements

33. Energy 0 K Dimer Wannier Function bound state Extraction of the bare dimer: bare dimer (0,1) dimer Wannier Function • Extraction of Jd, g and εd : 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. Parity effects Positive parity Atoms in different bands or species: P=pd+p1+p2 Negative parity Rectangular lattice _ More dimensions: + + + extra degeneracies… more than one dimer

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

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?