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Nonequilibrium dynamics of ultracold atoms in optical lattices. Lattice modulation experiments and more Ehud Altman Peter Barmettler Vladmir Gritsev David Pekker Matthias Punk Rajdeep Sensarma Mikhail Lukin Eugene Demler Weizmann Institute University of Fribourg Harvard, Fribourg Harvard University Technical University Munich Harvard University Harvard University Harvard University $$ NSF, AFOSR, MURI, DARPA, Fermionic Hubbard model From high temperature superconductors to ultracold atoms Atoms in optical lattice Antiferromagnetic and superconducting Tc of the order of 100 K Antiferromagnetism and pairing at sub-micro Kelvin temperatures 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 Mott state of fermions in optical lattice 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 Lattice modulation experiments with fermions in optical lattice. Probing the Mott state of fermions Related theory work: Kollath et al., PRA 74:416049R (2006) Huber, Ruegg, arXiv:0808:2350 Lattice modulation experiments Probing dynamics of the Hubbard model Modulate lattice potential Measure number of doubly occupied sites Main effect of shaking: modulation of tunneling Doubly occupied sites created when frequency w matches Hubbard U Lattice modulation experiments Probing dynamics of the Hubbard model R. Joerdens et al., arXiv:0804.4009 Mott state Regime of strong interactions U>>t. Mott gap for the charge forms at Antiferromagnetic ordering at “High” temperature regime All spin configurations are equally likely. Can neglect spin dynamics. “Low” temperature regime Spins are antiferromagnetically ordered or have strong correlations Schwinger bosons and Slave Fermions Bosons Fermions Constraint : Singlet Creation Boson Hopping 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 “Low” Temperature Schwinger bosons Bose condensed d Propagation of holes and doublons strongly affected by interaction with spin waves 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 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 “Low” Temperature Rate of doublon production • Low energy peak due to sharp quasiparticles • Broad continuum due to incoherent part • Spin wave shake-off peaks “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) 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 Lattice modulation experiments. Sum rule Ad(h) is the spectral function of a single doublon (holon) Sum Rule : Experiments: Possible origin of sum rule violation • Nonlinearity • Doublon decay The total weight does not scale quadratically with t Doublon decay and relaxation Relaxation of doublon hole pairs in the Mott state Energy Released ~ U Energy carried by creation of ~U2/t2 spin excitations ~J Relaxation requires spin excitations =4t2/U Relaxation rate Large U/t : Very slow Relaxation Alternative mechanism of relaxation UHB • Thermal escape to edges LHB m • Relaxation in compressible edges Thermal escape time Relaxation in compressible edges Doublon decay in a compressible state How to get rid of the excess energy U? Compressible state: Fermi liquid description p -h p -h Doublon can decay into a pair of quasiparticles with many particle-hole pairs U p -h p -p Doublon decay in a compressible state Perturbation theory to order n=U/t Decay probability To find the exponent: consider processes which maximize the number of particle-hole excitations Expt T. Esslinger et al. Doublon decay in a compressible state Fermi liquid description Single particle states Doublons Interaction Decay Scattering 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) Two component Bose mixture in optical lattice Example: . Mandel et al., Nature 425:937 (2003) t t Two component Bose Hubbard model Quantum magnetism of bosons in optical lattices Duan, Demler, Lukin, PRL 91:94514 (2003) Altman et al., NJP 5:113 (2003) • Ferromagnetic • Antiferromagnetic Observation of superexchange in a double well potential Theory: A.M. Rey et al., PRL (2007) J J Use magnetic field gradient to prepare a state Observe oscillations between and states Experiment: Trotzky et al., Science (2008) Preparation and detection of Mott states of atoms in a double well potential Comparison to the Hubbard model Beyond the basic Hubbard model Basic Hubbard model includes only local interaction Extended Hubbard model takes into account non-local interaction Beyond the basic Hubbard model Nonequilibrium spin dynamics in optical lattices Dynamics beyond linear response 1D: XXZ dynamics starting from the classical Neel state Coherent time evolution starting with Y(t=0) = Equilibrium phase diagram QLRO D • DMRG • XZ model: exact solution • D>1: sine-Gordon Bethe ansatz solution Time, Jt 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 XXZ dynamics starting from the classical Neel state Expected: critical slowdown near quantum critical point at D=1 Observed: fast decay at D=1 Lattice modulation experiments with fermions in optical lattice. Detecting d-wave superfluid state 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? Lattice modulation experiments Modulating hopping via modulation of the optical lattice intensity where 3 • Equal energy contours Resonantly exciting quasiparticles with 2 1 0 1 Enhancement close to the banana tips due to coherence factors 2 3 3 2 1 0 1 2 3 Lattice modulation as a probe of d-wave superfluids 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 pairing 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) Scanning tunneling spectroscopy of high Tc cuprates Conclusions Experiments with fermions in optical lattice open many interesting questions about dynamics of the Hubbard model Thanks to: Harvard-MIT Fermions in optical lattice U t Hubbard model plus parabolic potential t Probing many-body states Electrons in solids Fermions in optical lattice • Thermodynamic probes i.e. specific heat • X-Ray and neutron scattering • ARPES • System size, number of doublons as a function of entropy, U/t, w0 • Bragg spectroscopy, TOF noise correlations ??? • Optical conductivity • STM • Lattice modulation experiments