1. Conference on quantum fluids and strongly correlated systemsParis 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. Harvard-MIT Lattice modulation experiments with fermions in optical lattice Ehud Altman Weizmann Institute David Pekker Harvard University Rajdeep Sensarma Harvard University Eugene Demler Harvard University Dynamics of Hubbard model Thanks to I. Bloch, T. Esslinger, M. Lukin, A.M. Rey

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

5. U t t Fermions in optical lattice 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, w0 • X-Ray and neutron • scattering • Bragg spectroscopy, • TOF noise correlations • ARPES ??? • Lattice • modulation • experiments • Optical conductivity • STM

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 Comparison with DMFT+LDA models suggests that temperature is above the Neel transition Next step: observation of antiferromagnetic order 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. 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

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

13. 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

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

15. “Low” Temperature d h = + Incoherent part: dispersion 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 J, weight by J/t

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

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

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

19. 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 hole Doublon Production Rate Experiments

20. Doublon decay and relaxation

21. Relaxation of doublon hole pairs in the Mott state Energy Released ~ U • Relaxation requires • creation of ~U2/t2 • spin excitations • Energy carried by spin excitations ~ J =4t2/U Relaxation rate Very slow Relaxation Large U/t :

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

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

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

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

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

27. Lattice modulation as a probe of d-wave superfluids number of quasi-particles density-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)

28. Scanning tunneling spectroscopy of high Tc cuprates

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

30. Happy birthday Gora!