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 Eugene Demler Harvard University $$ NSF, AFOSR, MURI, DARPA,

2. Probing many-body states Beyond analogies with condensed matter systems: Far from equilibrium quantum many-body dynamics 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 Optical conductivity STM Lattice modulation experiments

3. Outline Lattice modulation experiments in the Mott state. Linear response theory. Low temperatures (magnetically ordered) “High” temperatures (no magnetic order) Comparison to experiments Nonequilibrium spin dynamics Lattice modulation experiments: phase sensitive detection of d-wave pairing

4. 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

5. 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

6. 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

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

8. 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

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

10. 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.

11. 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

12. 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.

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

14. “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)

15. 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

16. 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

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

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

19. tt Two component Fermi/Bose mixture in optical lattice Example:. Mandel et al., Nature 425:937 (2003) Quantum magnetism in optical lattices Spin interactions can be tuned: AF vs ferro, easy plane vs axis. Duan et al., PRL 91:94514 (2003) Bosons

20. J J Use magnetic field gradient to prepare a stateObserve oscillations between and Observation of superexchange in a double well potential Theory: A.M. Rey et al., PRL (2007) Expts: S. Trotzky et al., Science 319:295 (2008)

21. 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 Contrast to incoherent dynamics: T.L. Ho arXiv:0808.2677

22. 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

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

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

25. 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?

26. 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

27. 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

28. 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

29. Scanning tunneling spectroscopy of high Tc cuprates

30. Conclusions Far from equilibrium quantum dynamics of many-body systems: new chapter in Hubbard model is being opened up by experiments with cold atoms Thanks to: Harvard-MIT