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Lattice modulation experiments with fermions in optical lattice
Dynamics of Hubbard model
Ehud Altman Weizmann InstituteDavid Pekker Harvard UniversityRajdeep Sensarma Harvard UniversityEugene Demler Harvard University
Thanks to I. Bloch, T. Esslinger, M. Lukin, A.M. Rey
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
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
Outline
• Introduction. Recent experiments with fermions in optical lattice
Signatures of Mott state Observation of Superexchange
• Lattice modulation experiments in the Mott state. Linear response theory
• Comparison to experiments• Lattice modulation experiments with d-wave
superfluids
Mott state of fermions
in optical lattice
Signatures of incompressible Mott state
Suppression in the number of double occupancies Esslinger et al. arXiv:0804.4009
Signatures of incompressible Mott stateResponse to external potential I. Bloch et al., unpublished
Radius of the cloud as a functionof the confining potential
Next step: observation of antiferromagnetic order
Comparison with DMFT+LDA models suggests that temperature is above the Neel transition
However superexchange interactions have already been observed
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
Observation of superexchange in a double well potential.Reversing the sign of exchange interactions
Lattice modulation experiments with fermions in optical lattice.
Mott state
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
T. Esslinget 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
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.
“Low” Temperature
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
“Low” Temperature
Rate of doublon productionSpectral function
• Low energy peak due to sharp quasiparticles
• Broad continuum due to incoherent part
• Oscillations reflect shake-off processes of spin waves
“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
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
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 superfluids
Conclusions
Experiments with fermions in optical lattice openmany interesting questions about dynamics of the Hubbard model
Thanks to:
Harvard-MIT