1. Nonequilibrium dynamics of ultracold atoms in optical lattices. Lattice modulation experiments and more Ehud Altman Weizmann Institute Peter Barmettler University of Fribourg Vladmir Gritsev Harvard, Fribourg David Pekker Harvard University Matthias Punk Technical University Munich Rajdeep Sensarma Harvard University Mikhail Lukin Harvard University Eugene Demler Harvard University $$ NSF, AFOSR, MURI, DARPA,

2. Antiferromagnetic and superconducting Tc of the order of 100 K Atoms in optical lattice Antiferromagnetism and pairing at sub-micro Kelvin temperatures Fermionic Hubbard model From high temperature superconductors to ultracold atoms

3. Outline Introduction. Recent experiments with fermions in optical lattice Lattice modulation experiments in the Mott state. Linear response theory Comparison to experiments Superexchange interactions in optical lattice Lattice modulation experiments with d-wave superfluids

4. Mott state of fermions in optical lattice

5. Signatures of incompressible Mott state of fermions in optical lattice Suppression of double occupancies T. Esslinger et al. arXiv:0804.4009 Compressibility measurements I. Bloch et al. arXiv:0809.1464

6. Lattice modulation experiments with fermions in optical lattice. Related theory work: Kollath et al., PRA 74:416049R (2006) Huber, Ruegg, arXiv:0808:2350 Probing the Mott state of fermions

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

8. Lattice modulation experiments Probing dynamics of the Hubbard model R. Joerdens et al., arXiv:0804.4009

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

10. Schwinger bosons and Slave Fermions BosonsFermions Constraint : Singlet Creation Boson Hopping

11. Schwinger bosons and slave fermions Fermion hopping Doublon production due to lattice modulation perturbation Second order perturbation theory. Number of doublons Propagation of holes and doublons is coupled to spin excitations. Neglect spontaneous doublon production and relaxation.

12. d h 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 J, weight by J/t Incoherent part: dispersion Propagation of holes and doublons strongly affected by interaction with spin waves Schwinger bosons Bose condensed “Low” Temperature

13. Propogation of doublons and holes Spectral function: Oscillations reflect shake-off processes of spin waves Hopping creates string of altered spins: bound states Comparison of Born approximation and exact diagonalization: Dagotto et al.

14. “Low” Temperature Rate of doublon production Low energy peak due to sharp quasiparticles Broad continuum due to incoherent part Spin wave shake-off peaks

15. “High” Temperature Atomic limit. Neglect spin dynamics. All spin configurations are equally likely. A ij (t ’ ) replaced by probability of having a singlet Assume independent propagation of doublons and holes. Rate of doublon production A d(h) is the spectral function of a single doublon (holon)

16. Propogation of doublons and holes Hopping creates string of altered spins Retraceable Path Approximation Brinkmann & Rice, 1970 Consider the paths with no closed loops Spectral Fn. of single holeDoublon Production Rate Experiments

17. A d(h) is the spectral function of a single doublon (holon) Sum Rule : Experiments: Possible origin of sum rule violation The total weight does not scale quadratically with t Nonlinearity Doublon decay Lattice modulation experiments. Sum rule

18. Doublon decay and relaxation

19. Energy Released ~ U  Energy carried by spin excitations ~ J =4t 2 /U  Relaxation requires creation of ~U 2 /t 2 spin excitations Relaxation of doublon hole pairs in the Mott state Relaxation rate Large U/t : Very slow Relaxation

20. Alternative mechanism of relaxation LHB UHB m Thermal escape to edges Relaxation in compressible edges Thermal escape time Relaxation in compressible edges

21. Doublon decay in a compressible state How to get rid of the excess energy U? Compressible state: Fermi liquid description Doublon can decay into a pair of quasiparticles with many particle-hole pairs U p-pp-p p-hp-h p-hp-h p-hp-h

22. Doublon decay in a compressible state To find the exponent: consider processes which maximize the number of particle-hole excitations Perturbation theory to order n=U/t Decay probability Expt T. Esslinger et al.

23. Doublon decay in a compressible state Fermi liquid description Single particle states Doublons Interaction Decay Scattering

24. Superexchange interaction in experiments with double wells Refs: Theory: A.M. Rey et al., Phys. Rev. Lett. 99:140601 (2007) Experiment: S. Trotzky et al., Science 319:295 (2008)

25. t t Two component Bose mixture in optical lattice Example:. Mandel et al., Nature 425:937 (2003) Two component Bose Hubbard model

26. Quantum magnetism of bosons in optical lattices Duan, Demler, Lukin, PRL 91:94514 (2003) Altman et al., NJP 5:113 (2003) Ferromagnetic Antiferromagnetic

27. J J Use magnetic field gradient to prepare a stateObserve oscillations between and states Observation of superexchange in a double well potential Theory: A.M. Rey et al., PRL (2007) Experiment: Trotzky et al., Science (2008)

28. Preparation and detection of Mott states of atoms in a double well potential

29. Comparison to the Hubbard model

30. Basic Hubbard model includes only local interaction Extended Hubbard model takes into account non-local interaction Beyond the basic Hubbard model

32. Nonequilibrium spin dynamics in optical lattices Dynamics beyond linear response

33. 1D: XXZ dynamics starting from the classical Neel state DMRG XZ model: exact solution D >1: sine-Gordon Bethe ansatz solution Time, Jt D Equilibrium phase diagram Y (t=0) = Coherent time evolution starting with QLRO

34. XXZ dynamics starting from the classical Neel state D<1, XY easy plane anisotropy Surprise: oscillations Physics beyond Luttinger liquid model. Fermion representation: dynamics is determined not only states near the Fermi energy but also by sates near band edges (singularities in DOS) D>1, Z axis anisotropy Exponential decay starting from the classical ground state

35. XXZ dynamics starting from the classical Neel state Expected: critical slowdown near quantum critical point at D =1 Observed: fast decay at D =1

36. Lattice modulation experiments with fermions in optical lattice. Detecting d-wave superfluid state

37. consider a mean-field description of the superfluid s-wave: d-wave: anisotropic s-wave: Setting: BCS superfluid Can we learn about paired states from lattice modulation experiments? Can we distinguish pairing symmetries?

38. Modulating hopping via modulation of the optical lattice intensity Lattice modulation experiments where Equal energy contours Resonantly exciting quasiparticles with Enhancement close to the banana tips due to coherence factors

39. Distribution of quasi-particles after lattice modulation experiments (1/4 of zone) Momentum distribution of fermions after lattice modulation (1/4 of zone) Can be observed in TOF experiments Lattice modulation as a probe of d-wave superfluids

40. number of quasi-particlesdensity-density correlations Peaks at wave-vectors connecting tips of bananas Similar to point contact spectroscopy Sign of peak and order-parameter (red=up, blue=down) Lattice modulation as a probe of d-wave pairing

41. Scanning tunneling spectroscopy of high Tc cuprates

42. Conclusions Experiments with fermions in optical lattice open many interesting questions about dynamics of the Hubbard model Thanks to: Harvard-MIT

43. Fermions in optical lattice t U t Hubbard model plus parabolic potential Probing many-body states Electrons in solids Fermions in optical lattice Thermodynamic probes i.e. specific heat System size, number of doublons as a function of entropy, U/t, w 0 X-Ray and neutron scattering Bragg spectroscopy, TOF noise correlations ARPES??? Optical conductivity STM Lattice modulation experiments