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Outline of these lectures

Outline of these lectures. Introduction. Systems of ultracold atoms. Cold atoms in optical lattices. Bose Hubbard model Bose mixtures in optical lattices Detection of many-body phases using noise correlations Experiments with low dimensional systems Fermions in optical lattices

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Outline of these lectures

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  1. Outline of these lectures • Introduction. Systems of ultracold atoms. • Cold atoms in optical lattices. • Bose Hubbard model • Bose mixtures in optical lattices • Detection of many-body phases using noise correlations • Experiments with low dimensional systems • Fermions in optical lattices Magnetism Pairing in systems with repulsive interactions Current experiments: Paramagnetic Mott state • Experiments on nonequilibriumfermion dynamics Lattice modulation experiments Doublon decay • Stoner instability

  2. Ultracold fermions in optical lattices

  3. U t t Fermionic atoms in optical lattices Experiments with fermions in optical lattice, Kohl et al., PRL 2005

  4. Antiferromagnetic and superconducting Tc of the order of 100 K Quantum simulations with ultracold atoms Atoms in optical lattice Antiferromagnetism and pairing at sub-micro Kelvin temperatures Same microscopic model

  5. Positive U Hubbard model Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995) Antiferromagnetic insulator D-wave superconductor -

  6. Fermionic Hubbard modelPhenomena predicted Superexchange and antiferromagnetism (P.W. Anderson) Itinerant ferromagnetism. Stoner instability (J. Hubbard) Incommensurate spin order. Stripes(Schulz, Zaannen, Emery, Kivelson, Fradkin, White, Scalapino, Sachdev, …) Mott state without spin order. Dynamical Mean Field Theory (Kotliar, Georges, Metzner, Vollhadt, Rozenberg, …) d-wave pairing (Scalapino, Pines,…) d-density wave (Affleck, Marston, Chakravarty, Laughlin,…)

  7. Superexchange and antiferromagnetismin the Hubbard model. Large U limit Singlet state allows virtual tunneling and regains some kinetic energy Triplet state: virtual tunneling forbidden by Pauli principle Effective Hamiltonian: Heisenberg model

  8. Q=(p,p) Hubbard model for small U. Antiferromagnetic instability at half filling Analysis of spin instabilities. Random Phase Approximation Fermi surface for n=1 Nesting of the Fermi surface leads to singularity BCS-type instability for weak interaction

  9. paramagnetic Mott phase Hubbard model at half filling TN Paramagnetic Mott phase: one fermion per site charge fluctuations suppressed no spin order U BCS-type theory applies Heisenberg model applies

  10. SU(N) Magnetism with Ultracold Alkaline-Earth Atoms |e> = 3P0 698 nm 150 s ~ 1 mHz Ex: 87Sr(I = 9/2) |g> = 1S0 A. Gorshkov, et al., Nature Physics 2010 Alkaline-Earth atoms in optical lattice: Nuclear spin decoupled from electrons SU(N=2I+1) symmetry → SU(N) spin models ⇒ valence-bond-solid & spin-liquid phases • orbital degree of freedom ⇒ spin-orbital physics → Kugel-Khomskii model [transition metal oxides with perovskite structure] → SU(N) Kondo lattice model [for N=2, colossal magnetoresistance in manganese oxides and heavy fermion materials]

  11. Doped Hubbard model

  12. Attraction between holes in the Hubbard model Loss of superexchange energy from 8 bonds Loss of superexchange energy from 7 bonds Single plaquette: binding energy

  13. -k’ k’ spin fluctuation k -k Pairing of holes in the Hubbard model Non-local pairing of holes Leading istability: d-wave Scalapino et al, PRB (1986)

  14. -k’ k’ spin fluctuation k -k Pairing of holes in the Hubbard model BCS equation for pairing amplitude Q - + + - Systems close to AF instability: c(Q) is large and positive Dk should change sign for k’=k+Q dx2-y2

  15. Stripe phases in the Hubbard model Stripes: Antiferromagnetic domains separated by hole rich regions Antiphase AF domains stabilized by stripe fluctuations First evidence: Hartree-Fock calculations. Schulz, Zaannen (1989) Numerical evidence for ladders: Scalapino, White (2003); For a recent review see Fradkin arXiv:1004.1104

  16. T AF pseudogap D-SC SDW n=1 doping Possible Phase Diagram AF – antiferromagnetic SDW- Spin Density Wave (Incommens. Spin Order, Stripes) D-SC – d-wave paired After several decades we do not yet know the phase diagram

  17. Antiferromagnetic and superconducting Tc of the order of 100 K Quantum simulations with ultracold atoms Atoms in optical lattice Antiferromagnetism and pairing at sub-micro Kelvin temperatures Same microscopic model

  18. How to detect fermion pairing Quantum noise analysis of TOF images is more than HBT interference

  19. n(r’) k F n(r) Second order interference from the BCS superfluid Theory: Altman et al., PRA 70:13603 (2004) n(k) k BCS BEC

  20. Momentum correlations in paired fermions Greiner et al., PRL 94:110401 (2005)

  21. Fermion pairing in an optical lattice Second Order Interference In the TOF images Normal State Superfluid State measures the Cooper pair wavefunction One can identify unconventional pairing

  22. Current experiments

  23. TN current experiments Mott U Fermions in optical lattice. Next challenge: antiferromagnetic state

  24. Signatures of incompressible Mott state of fermions in optical lattice Suppression of double occupancies R. Joerdens et al., Nature (2008) Compressibility measurements U. Schneider et al., Science (2008)

  25. TN current experiments Mott U Fermions in optical lattice. Next challenge: antiferromagnetic state

  26. Negative U Hubbard model |U| t t Phase diagram: Micnas et al., Rev. Mod. Phys.,1990

  27. Hubbard model with attraction: anomalous expansion Hackermuller et al., Science 2010 • Constant Entropy • Constant Particle Number • Constant Confinement

  28. Anomalous expansion: Competition of energy and entropy

  29. Lattice modulation experiments with fermions in optical lattice. Probing the Mott state of fermions Sensarma, Pekker, Lukin, Demler, PRL (2009) Related theory work: Kollath et al., PRL (2006) Huber, Ruegg, PRB (2009) Orso, Iucci, et al., PRA (2009)

  30. Modulate lattice potential Lattice modulation experiments Probing dynamics of the Hubbard model Measure number of doubly occupied sites Main effect of shaking: modulation of tunneling Doubly occupied sites created when frequency w matches Hubbard U

  31. Lattice modulation experiments Probing dynamics of the Hubbard model R. Joerdens et al., Nature 455:204 (2008)

  32. Mott gap for the charge forms at Antiferromagnetic ordering at “Low” temperature regime Mott state Regime of strong interactionsU>>t. “High” temperature regime All spin configurations are equally likely. Can neglect spin dynamics. Spins are antiferromagnetically ordered or have strong correlations

  33. “Low” Temperature d h = + Incoherent part: dispersion Schwinger bosons Bose condensed Propagation of holes and doublons strongly affected by interaction with spin waves Assume independent propagation of hole and doublon (neglect vertex corrections) Self-consistent Born approximation Schmitt-Rink et al (1988), Kane et al. (1989) Spectral function for hole or doublon Sharp coherent part: dispersion set by t2/U, weight by t/U Oscillations reflect shake-off processes of spin waves

  34. “Low” Temperature Rate of doublon production • Sharp absorption edge due to coherent quasiparticles • Broad continuum due to incoherent part • Spin wave shake-off peaks

  35. d h “High” Temperature Retraceable Path Approximation Brinkmann & Rice, 1970 Consider the paths with no closed loops Original Experiment: R. Joerdens et al., Nature 455:204 (2008) Theory: Sensarma et al., PRL 103, 035303 (2009) Spectral Fn. of single hole

  36. Temperature dependence Density Psinglet Radius Radius D. Pekker et al., upublished Reduced probability to find a singlet on neighboring sites

  37. Fermions in optical lattice.Decay of repulsively bound pairs Ref: N. Strohmaier et al., PRL 2010

  38. Fermions in optical lattice.Decay of repulsively bound pairs Doublons – repulsively bound pairs What is their lifetime? Direct decay is not allowed by energy conservation Excess energy U should be converted to kinetic energy of single atoms Decay of doublon into a pair of quasiparticles requires creation of many particle-hole pairs

  39. Fermions in optical lattice.Decay of repulsively bound pairs Experiments: N. Strohmaier et. al.

  40. Relaxation of doublon- hole pairs in the Mott state Energy U needs to be absorbed by spin excitations • Relaxation requires • creation of ~U2/t2 • spin excitations • Energy carried by spin excitations ~ J =4t2/U Relaxation rate Very slow, not relevant for ETH experiments

  41. p-h p-h U p-h p-p Doublon decay in a compressible state Excess energy U is converted to kinetic energy of single atoms Compressible state: Fermi liquid description Doublon can decay into a pair of quasiparticles with many particle-hole pairs

  42. Doublon decay in a compressible state Perturbation theory to order n=U/6t Decay probability Doublon Propagator Interacting “Single” Particles

  43. Doublon decay in a compressible state N. Strohmaier et al., PRL 2010 Expt: ETHZ Theory: Harvard To calculate the rate: consider processes which maximize the number of particle-hole excitations

  44. Why understanding doublon decay rate is important Prototype of decay processes with emission of many interacting particles. Example: resonance in nuclear physics: (i.e. delta-isobar) Analogy to pump and probe experiments in condensed matter systems Response functions of strongly correlated systems at high frequencies. Important for numerical analysis. Important for adiabatic preparation of strongly correlated systems in optical lattices

  45. Surprises of dynamics in the Hubbard model

  46. Expansion of interacting fermions in optical lattice U. Schneider et al., arXiv:1005.3545 New dynamical symmetry: identical slowdown of expansion for attractive and repulsive interactions

  47. Competition between pairing and ferromagnetic instabilities in ultracold Fermi gases near Feshbach resonances arXiv:1005.2366 D. Pekker, M. Babadi, R. Sensarma, N. Zinner, L. Pollet, M. Zwierlein, E. Demler

  48. U N(0) = 1 Stoner model of ferromagnetism Spontaneous spin polarization decreases interaction energy but increases kinetic energy of electrons Mean-field criterion U – interaction strength N(0) – density of states at Fermi level Existence of Stoner type ferromagnetism in a single band model is still a subject of debate Theoretical proposals for observing Stoner instability with ultracold Fermi gases: Salasnich et. al. (2000); Sogo, Yabu (2002); Duine, MacDonald (2005); Conduit, Simons (2009); LeBlanck et al. (2009); …

  49. Experiments were done dynamically. What are implications of dynamics? Why spin domains could not be observed?

  50. Is it sufficient to consider effective model with repulsive interactions when analyzing experiments? Feshbach physics beyond effective repulsive interaction

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