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