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

Fermionic Hubbard model

From high temperature superconductors to ultracold atoms Antiferromagnetic and superconducting Tc of the order of 100 K Atoms in optical lattice 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

d

“Low” Temperature Schwinger bosons Bose condensed 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.

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)

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 A d(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 spin excitations ~ J =4t 2 /U Relaxation rate Large U/t :  Relaxation requires creation of Very slow Relaxation ~U 2 spin excitations /t 2

Alternative mechanism of relaxation

UHB LHB m • Thermal escape to edges • 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 Doublon can decay into a pair of quasiparticles with many particle-hole pairs

U p h p h 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) =

QLRO

Equilibrium phase diagram 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 • Equal energy contours 2 1 3 0 1 2 3 3 2 1 0 1 2 3 Resonantly exciting quasiparticles with Enhancement close to the banana tips due to coherence factors

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

t Fermions in optical lattice U Hubbard model plus parabolic potential t Probing many-body states Electrons in solids • Thermodynamic probes i.e. specific heat • X-Ray and neutron scattering • ARPES • Optical conductivity • STM Fermions in optical lattice • System size, number of doublons as a function of entropy

, U/t

, w 0 • Bragg spectroscopy, TOF noise correlations ???

• Lattice modulation experiments