Transcript Document

Lattice modulation experiments
with fermions in optical lattice
Dynamics of Hubbard model
Ehud Altman
David Pekker
Rajdeep Sensarma
Eugene Demler
Weizmann Institute
Harvard University
Harvard University
Harvard University
Thanks to I. Bloch, T. Esslinger, M. Lukin, A.M. Rey
Fermionic Hubbard model
From high temperature superconductors to ultracold atoms
Atoms in optical lattice
Antiferromagnetic and
superconducting Tc
of the order of 100 K
Antiferromagnetism and
pairing at sub-micro Kelvin
temperatures
Fermions in optical lattice
U
t
Hubbard model plus parabolic potential
t
Probing many-body states
Electrons in solids
Fermions in optical lattice
• Thermodynamic probes
i.e. specific heat
• X-Ray and neutron
scattering
• ARPES
• System size, number of doublons
as a function of entropy, U/t, w0
• Bragg spectroscopy,
TOF noise correlations
???
• 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 state
Response to external potential
I. Bloch et al., unpublished
Radius of the cloud as a function
of the confining potential
Comparison with DMFT+LDA models suggests
that temperature is above the Neel transition
Next step: observation of antiferromagnetic order
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)
Two component Bose mixture in optical lattice
Example:
. Mandel et al., Nature 425:937 (2003)
t
t
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
Observation of superexchange in a double well potential
Theory: A.M. Rey et al., PRL (2007)
J
J
Use magnetic field gradient to prepare a state
Observe oscillations between
and
states
Experiment:
Trotzky et al.,
Science (2008)
Preparation and detection of Mott states
of atoms in a double well potential
Comparison to the Hubbard model
Beyond the basic Hubbard model
Basic Hubbard model includes
only local interaction
Extended Hubbard model
takes into account non-local
interaction
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 experiments
Probing dynamics of the Hubbard model
Modulate lattice potential
Measure number of doubly
occupied sites
Main effect of shaking: modulation of tunneling
Doubly occupied sites created when frequency w matches Hubbard U
Lattice modulation experiments
Probing 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
All spin configurations are equally likely.
Can neglect spin dynamics.
“Low” temperature regime
Spins are antiferromagnetically ordered
or have strong correlations
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
“Low” Temperature
d
Propagation of holes and doublons strongly
affected by interaction with spin waves
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
“Low” Temperature
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
“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 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
Doublon decay and relaxation
Relaxation of doublon hole pairs in the Mott state
Energy Released ~ U
Energy carried by
creation of ~U2/t2
spin excitations
~J
 Relaxation requires
spin excitations
=4t2/U
Relaxation rate
Large U/t :
Very slow Relaxation
Alternative mechanism of relaxation
UHB
• Thermal escape to edges
LHB
m
• Relaxation in compressible edges
Thermal escape time
Relaxation in compressible edges
Lattice modulation experiments
with fermions in optical lattice.
Detecting d-wave superfluid state
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?
Lattice modulation experiments
Modulating hopping via modulation
of the optical lattice intensity
where
3
• Equal energy
contours
Resonantly exciting
quasiparticles with
2
1
0
1
Enhancement close to the banana
tips due to coherence factors
2
3
3
2
1
0
1
2
3
Lattice modulation as a probe
of d-wave superfluids
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)
Conclusions
Experiments with fermions in optical lattice open
many interesting questions about dynamics of the
Hubbard model
Thanks to:
Harvard-MIT