Nonequilibrium dynamics of ultracold atoms in optical lattices. Lattice modulation experiments and more Ehud Altman Weizmann Institute Peter Barmettler.

Nonequilibrium dynamics of ultracold atoms in optical lattices.

Lattice modulation experiments and more

Ehud Altman Weizmann InstitutePeter Barmettler University of FribourgVladmir Gritsev Harvard, FribourgDavid Pekker Harvard UniversityMatthias Punk Technical University MunichRajdeep Sensarma Harvard UniversityMikhail Lukin Harvard UniversityEugene Demler Harvard University

$$ NSF, AFOSR, MURI, DARPA,

Antiferromagnetic and superconducting Tc of the order of 100 K

Atoms in optical lattice

Antiferromagnetism and pairing at sub-micro Kelvin temperatures

Fermionic Hubbard modelFrom high temperature superconductors to ultracold atoms

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 measurementsI. Bloch et al. arXiv:0809.1464

Lattice modulation experiments with fermions in optical lattice.

Related theory work: Kollath et al., PRA 74:416049R (2006) Huber, Ruegg, arXiv:0808:2350

Probing the Mott state of fermions

Lattice modulation experimentsProbing 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 matches Hubbard U

Lattice modulation experimentsProbing 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

“Low” temperature regime

All spin configurations are equally likely.Can neglect spin dynamics.

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

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.

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

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.

“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 holesHopping 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

Ad(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

Doublon decay and relaxation

Energy Released ~ U

Energy carried by

spin excitations ~ J =4t2/U

Relaxation requires creation of ~U2/t2

spin excitations

Relaxation of doublon hole pairs in the Mott state

Relaxation rate

Large U/t : Very slow Relaxation

Alternative mechanism of relaxation

LHB

UHB

• 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 apair of quasiparticles with many particle-hole pairs

Up-p

p-h

p-h

p-h

Doublon decay in a compressible state

To find the exponent: consider processes which maximize the number of particle-hole excitations

Perturbation theory to order n=U/tDecay probability

ExptT. Esslingeret 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)

t

t

Two component Bose mixture in optical latticeExample: . Mandel et al., Nature 425:937 (2003)

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

J

J

Use magnetic field gradient to prepare a state

Observe oscillations between and states

Observation of superexchange in a double well potentialTheory: A.M. Rey et al., PRL (2007)

Experiment:Trotzky et al.,Science (2008)

Preparation and detection of Mott statesof atoms in a double well potential

Comparison to the Hubbard model

Basic Hubbard model includesonly local interaction

Extended Hubbard modeltakes into account non-localinteraction

Beyond the basic Hubbard model

Beyond the basic Hubbard model

Nonequilibrium spin dynamicsin optical lattices

Dynamics beyond linear response

1D: XXZ dynamics starting from the classical Neel state

• DMRG• XZ model: exact solution• >1: sine-Gordon Bethe ansatz solution

Time, Jt

Equilibrium phase diagram

(t=0) =Coherent time evolution starting with

QLRO

XXZ dynamics starting from the classical Neel state

<1, XY easy plane anisotropy

Surprise: oscillationsPhysics 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)

>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 =1

Observed: fast decay at =1

Lattice modulation experiments with fermions in optical lattice.

Detecting d-wave superfluid state

• 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?

Modulating hopping via modulation

of the optical lattice intensity

Lattice modulation experiments

where

3 2 1 0 1 2 3

3

2

1

0

1

2

3

• Equal energy contours

Resonantly exciting quasiparticles with

Enhancement close to the bananatips due to coherence factors

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

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)

Lattice modulation as a probe of d-wave pairing

Scanning tunneling spectroscopy of high Tc cuprates

Conclusions

Experiments with fermions in optical lattice openmany interesting questions about dynamics of the Hubbard model

Thanks to:

Harvard-MIT

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, 0

• X-Ray and neutron scattering

• Bragg spectroscopy, TOF noise correlations

• ARPES ???

• Optical conductivity• STM

• Lattice modulation experiments