1. Conference on quantum fluids and strongly correlated systems Paris 2008

2. Gora’s Lessons Theory is not done for theory’s sake. You can be a respectable theorist even if your predictions can be checked in experiments When you are senior and very important you should still enter new fields and learn about new areas of physics You do not hide behind the backs of younger collaborators. Do your own calculations! Forgetting something from Landau and Lifshitz can not be excused under any circmstances Entertainment is an important part of physics discussions

3. Lattice modulation experiments with fermions in optical lattice Dynamics of Hubbard model Ehud Altman Weizmann Institute David Pekker Harvard University Rajdeep Sensarma Harvard University Eugene Demler Harvard University Thanks to I. Bloch, T. Esslinger, M. Lukin, A.M. Rey Harvard-MIT

4. 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

5. 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

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

7. Mott state of fermions in optical lattice

8. Signatures of incompressible Mott state Suppression in the number of double occupancies Esslinger et al. arXiv:0804.4009

9. Signatures of incompressible Mott state Response to external potential I. Bloch, A. Rosch, et al., arXiv:0809.1464 Radius of the cloud as a function of the confining potential Next step: observation of antiferromagnetic order Comparison with DMFT+LDA models suggests that temperature is above the Neel transition However superexchange interactions have already been observed

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

11. 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

12. Lattice modulation experiments Probing dynamics of the Hubbard model T. Esslinget et al., arXiv:0804.4009

13. 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

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

15. 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.

16. “Low” Temperature 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

17. 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.

18. “Low” Temperature Rate of doublon production Low energy peak due to sharp quasiparticles Broad continuum due to incoherent part

19. “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)

20. 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

21. Lattice modulation experiments. Sum rule 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

22. Doublon decay and relaxation

23. 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

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

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

26. 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?

27. 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

28. 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

29. 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 superfluids

30. Scanning tunneling spectroscopy of high Tc cuprates

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

32. Happy birthday Gora!