Nonequilibrium dynamics of ultracold atoms in optical lattices. Lattice modulation experiments and more

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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 Eugene Demler
Harvard University
NSF, AFOSR, MURI, DARPA,
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Probing many-body states
Beyond analogies with condensed matter
systems Far from equilibrium quantum many-body
dynamics
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Outline

• Lattice modulation experiments in the Mott state.
  
• Linear response theory.
• Low temperatures (magnetically
  ordered)
• High temperatures (no magnetic
  order)
• Comparison to experiments
• Nonequilibrium spin dynamics
• Lattice modulation experiments phase
• sensitive detection of d-wave pairing

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Signatures of incompressible Mott state of
fermions in optical lattice
Suppression of double occupancies T. Esslinger
et al. arXiv0804.4009
Compressibility measurements I. Bloch et al.
arXiv0809.1464
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Lattice modulation experiments with fermions in
optical lattice.Mott state
Related theory work Kollath et al., PRA
74416049R (2006)
Huber, Ruegg, arXiv08082350
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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
R. Joerdens 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
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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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Schwinger bosons Bose condensed
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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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
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Rate of doublon production

• Low energy peak due to sharp quasiparticles

• Broad continuum due to incoherent part

• Spin wave shake-off peaks

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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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Lattice modulation experiments Probing
dynamics of the Hubbard model
R. Joerdens et al., arXiv0804.4009
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Nonequilibrium spin dynamics in optical lattices
Dynamics beyond linear response
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Two component Fermi/Bose mixture in optical
lattice
Quantum magnetism in optical lattices
Spin interactions can be tuned AF vs ferro,
easy plane vs axis. Duan et al., PRL 9194514
(2003)
Bosons
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Observation of superexchange in a double well
potential
Theory A.M. Rey et al., PRL (2007) Expts S.
Trotzky et al., Science 319295 (2008)
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1D XXZ dynamics starting from the classical Neel
state
Coherent time evolution starting with
Equilibrium phase diagram
QLRO

• DMRG
• XZ model exact solution
• Dgt1 sine-Gordon
• Bethe ansatz solution

Contrast to incoherent dynamics T.L. Ho
arXiv0808.2677
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XXZ dynamics starting from the classical Neel
state
Dlt1, 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)
Dgt1, Z axis anisotropy
Exponential decay starting from the classical
ground state
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XXZ dynamics starting from the classical Neel
state
Expected critical slowdown near quantum
critical point at D1
Observed fast decay at D1
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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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Scanning tunneling spectroscopy of high Tc
cuprates
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Conclusions
Far from equilibrium quantum dynamics of
many-body systems new chapter in Hubbard model
is being opened up by experiments with cold atoms
Thanks to