Lattice modulation experiments with fermions in optical lattice

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Lattice modulation experiments with fermions in
optical lattice

• Dynamics of Hubbard model

Ehud Altman Weizmann
Institute David Pekker
Harvard University Rajdeep Sensarma
Harvard University Eugene Demler
Harvard University
Thanks to I. Bloch, T. Esslinger, M. Lukin, A.M.
Rey
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Fermionic Hubbard model
From high temperature superconductors to
ultracold atoms
Atoms in optical lattice
Antiferromagnetism and pairing at sub-micro
Kelvin temperatures
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Fermions in optical lattice
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, w0

• X-Ray and neutron
• scattering

• Bragg spectroscopy,
• TOF noise correlations

• ARPES

???

• Lattice
• modulation
• experiments

• Optical conductivity
• STM

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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

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• Mott state of fermions
• in optical lattice

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Signatures of incompressible Mott state
Suppression in the number of double occupancies
Esslinger et al. arXiv0804.4009
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Signatures of incompressible Mott state
Response to external potential I. Bloch et al.,
unpublished
Radius of the cloud as a function of the
confining potential
Comparison with DMFTLDA models suggests that
temperature is above the Neel transition
Next step observation of antiferromagnetic order
However superexchange interactions have already
been observed
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Superexchange interaction in experiments with
double wells
Refs Theory A.M. Rey et al., Phys. Rev. Lett.
99140601 (2007) Experiment S. Trotzky et al.,
Science 319295 (2008)
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Two component Bose mixture in optical lattice
Example . Mandel et al., Nature
425937 (2003)
Two component Bose Hubbard model
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Quantum magnetism of bosons in optical lattices
Duan, Demler, Lukin, PRL 9194514 (2003) Altman
et al., NJP 5113 (2003)

• Ferromagnetic
• Antiferromagnetic

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Observation of superexchange in a double well
potential
Theory A.M. Rey et al., PRL (2007)
Experiment Trotzky et al., Science (2008)
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Preparation and detection of Mott states of atoms
in a double well potential
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Comparison to the Hubbard model
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Beyond the basic Hubbard model
Basic Hubbard model includes only local
interaction
Extended Hubbard model takes into account
non-local interaction
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Beyond the basic Hubbard model
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Observation of superexchange in a double well
potential. Reversing the sign of exchange
interactions
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Lattice modulation experiments with fermions in
optical lattice.Mott state
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Lattice modulation experiments Probing
dynamics of the Hubbard model
Measure number of doubly occupied sites
Main effect of shaking modulation of tunneling
Doubly occupied sites created when frequency w
matches Hubbard U
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Lattice modulation experiments Probing
dynamics of the Hubbard model
T. Esslinget et al., arXiv0804.4009
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Mott state
Regime of strong interactions Ugtgtt.
High temperature regime
All spin configurations are equally likely. Can
neglect spin dynamics.
Spins are antiferromagnetically ordered or have
strong correlations
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Schwinger bosons and Slave Fermions
Constraint
Singlet Creation
Boson Hopping
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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
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Propagation of holes and doublons strongly
affected by interaction with spin waves
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
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Spectral function
Rate of doublon production

• Low energy peak due to sharp quasiparticles

• Broad continuum due to incoherent part

• Oscillations reflect shake-off processes
• of spin waves

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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)
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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
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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
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Doublon decay and relaxation
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Relaxation of doublon hole pairs in the Mott state
Energy Released U

• Relaxation requires
• creation of U2/t2
• spin excitations

• Energy carried by
• spin excitations
• J 4t2/U

Relaxation rate
Very slow Relaxation
Large U/t
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Alternative mechanism of relaxation

• Thermal escape to edges

• Relaxation in compressible edges

Thermal escape time
Relaxation in compressible edges
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Lattice modulation experiments with fermions in
optical lattice.Detecting d-wave superfluid state
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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?
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Lattice modulation experiments
Modulating hopping via modulation of the optical
lattice intensity
where

• Equal energy
• contours

Resonantly exciting quasiparticles with
Enhancement close to the banana tips due to
coherence factors
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Lattice modulation as a probe of d-wave
superfluids
Momentum distribution of fermions after lattice
modulation (1/4 of zone)
Distribution of quasi-particles after lattice
modulation experiments (1/4 of zone)
Can be observed in TOF experiments
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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 (redup,
  bluedown)

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Conclusions
Experiments with fermions in optical lattice
open many interesting questions about dynamics of
the Hubbard model
Thanks to
Harvard-MIT