1. 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

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. 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

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

5. Mott state of fermions in optical lattice

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

7. Signatures of incompressible Mott state Response to external potential I. Bloch et al., unpublished Radius of the cloud as a function of the confining potential Next step: observation of antiferromagnetic order Comparison with DMFT+LDA models suggests that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

8. 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)

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

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

11. 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)

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

13. Comparison to the Hubbard model

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

16. Observation of superexchange in a double well potential. Reversing the sign of exchange interactions

17. Lattice modulation experiments with fermions in optical lattice. Mott state

18. 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

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

20. 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

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

22. 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.

23. “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

24. “Low” Temperature Rate of doublon production Spectral function Low energy peak due to sharp quasiparticles Broad continuum due to incoherent part Oscillations reflect shake-off processes of spin waves

25. “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)

26. 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

27. 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

28. Doublon decay and relaxation

29. 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

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

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

32. 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?

33. 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

34. 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

35. 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

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