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Nonequilibrium dynamics of
ultracold atoms in optical lattices.
Lattice modulation experiments and more
Ehud Altman
Peter Barmettler
Vladmir Gritsev
David Pekker
Matthias Punk
Rajdeep Sensarma
Eugene Demler
Weizmann Institute
University of Fribourg
Harvard, Fribourg
Harvard University
Technical University Munich
Harvard University
Harvard University
$$ NSF, AFOSR, MURI, DARPA,
Probing many-body states
Electrons in solids
Fermions in optical lattice
• Thermodynamic probes
i.e. specific heat
• X-Ray and neutron
scattering
• System size, number of doublons
as a function of entropy, U/t, w0
• Optical conductivity
• STM
• Bragg spectroscopy,
TOF noise correlations
• Lattice
modulation
experiments
Beyond analogies with condensed matter systems:
Far from equilibrium quantum many-body dynamics
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
Signatures of incompressible Mott state
of fermions in optical lattice
Suppression of double occupancies
T. Esslinger et al. arXiv:0804.4009
Compressibility measurements
I. Bloch et al. arXiv:0809.1464
Lattice modulation experiments
with fermions in optical lattice.
Mott state
Related theory work: Kollath et al., PRA 74:416049R (2006)
Huber, Ruegg, arXiv:0808:2350
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
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
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
Bosons
Fermions
Constraint :
Singlet Creation
Boson Hopping
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
Schwinger bosons Bose condensed
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
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
“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 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
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
Lattice modulation experiments
Probing dynamics of the Hubbard model
R. Joerdens et al., arXiv:0804.4009
Nonequilibrium spin dynamics
in optical lattices
Dynamics beyond linear response
Quantum magnetism in optical lattices
Two component Fermi/Bose mixture in optical lattice
Example:
t
. Mandel et al., Nature 425:937 (2003)
t
Spin interactions can be tuned:
AF vs ferro, easy plane vs axis.
Duan et al., PRL 91:94514 (2003)
Bosons
Observation of superexchange in a double well potential
J
Theory: A.M. Rey et al., PRL (2007)
Expts: S. Trotzky et al., Science 319:295 (2008)
J
Use magnetic field gradient to prepare a state
Observe oscillations between
and
1D: XXZ dynamics starting from the classical Neel state
Coherent time evolution
starting with
Y(t=0) =
Equilibrium phase diagram
QLRO
D
• DMRG
• XZ model: exact solution
• D>1: sine-Gordon
Bethe ansatz solution
Time, Jt
Contrast to incoherent dynamics:
T.L. Ho arXiv:0808.2677
XXZ dynamics starting from the classical Neel state
D<1, 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)
D>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 D=1
Observed:
fast decay
at D=1
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)
Scanning tunneling spectroscopy
of high Tc cuprates
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:
Harvard-MIT