Outline%20of%20these%20lectures

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Outline of these lectures

• Introduction. Systems of ultracold atoms.
• Cold atoms in optical lattices.
• Bose Hubbard model
• Bose mixtures in optical lattices
• Detection of many-body phases using noise
  correlations
• Experiments with low dimensional systems
• Fermions in optical lattices
• Magnetism
• Pairing in systems with repulsive
  interactions
• Current experiments Paramagnetic Mott state
• Experiments on nonequilibrium fermion dynamics
• Lattice modulation experiments
• Doublon decay
• Stoner instability

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Ultracold fermions in optical lattices
3
Fermionic atoms in optical lattices
Experiments with fermions in optical lattice,
Kohl et al., PRL 2005
4
Quantum simulations with ultracold atoms
Atoms in optical lattice
Antiferromagnetism and pairing at sub-micro
Kelvin temperatures
Same microscopic model
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Positive U Hubbard model
Possible phase diagram. Scalapino, Phys. Rep.
250329 (1995)
Antiferromagnetic insulator
D-wave superconductor
-
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Fermionic Hubbard modelPhenomena predicted
Superexchange and antiferromagnetism (P.W.
Anderson) Itinerant ferromagnetism. Stoner
instability (J. Hubbard) Incommensurate spin
order. Stripes (Schulz, Zaannen, Emery,
Kivelson, Fradkin, White, Scalapino, Sachdev,
) Mott state without spin order. Dynamical Mean
Field Theory (Kotliar, Georges, Metzner,
Vollhadt, Rozenberg, ) d-wave
pairing (Scalapino, Pines,) d-density wave
(Affleck, Marston, Chakravarty, Laughlin,)
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Superexchange and antiferromagnetismin the
Hubbard model. Large U limit
Singlet state allows virtual tunneling and
regains some kinetic energy
Triplet state virtual tunneling forbidden by
Pauli principle
Effective Hamiltonian Heisenberg model
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Hubbard model for small U. Antiferromagnetic
instability at half filling
Analysis of spin instabilities. Random Phase
Approximation
Fermi surface for n1
Nesting of the Fermi surface leads to singularity
BCS-type instability for weak interaction
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Hubbard model at half filling
TN
Paramagnetic Mott phase one fermion per
site charge fluctuations suppressed no spin order
U
BCS-type theory applies
Heisenberg model applies
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SU(N) Magnetism with Ultracold Alkaline-Earth
Atoms
A. Gorshkov, et al., Nature Physics 2010
Alkaline-Earth atoms in optical lattice
Nuclear spin decoupled from electrons
SU(N2I1) symmetry ? SU(N) spin models ?
valence-bond-solid spin-liquid phases
orbital degree of freedom ? spin-orbital
physics ? Kugel-Khomskii model transition metal
oxides with perovskite structure ? SU(N) Kondo
lattice model for N2, colossal
magnetoresistance in manganese oxides and heavy
fermion materials
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Doped Hubbard model
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Attraction between holes in the Hubbard model
Loss of superexchange energy from 8 bonds
Loss of superexchange energy from 7 bonds
Single plaquette binding energy
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Pairing of holes in the Hubbard model
Non-local pairing of holes
Leading istability d-wave Scalapino et al, PRB
(1986)
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Pairing of holes in the Hubbard model
BCS equation for pairing amplitude
Q
-


-
Systems close to AF instability c(Q) is large
and positive Dk should change sign for kkQ
dx2-y2
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Stripe phases in the Hubbard model
Stripes Antiferromagnetic domains separated by
hole rich regions
Antiphase AF domains stabilized by stripe
fluctuations
First evidence Hartree-Fock calculations.
Schulz, Zaannen (1989)
Numerical evidence for ladders Scalapino, White
(2003) For a recent review see Fradkin
arXiv1004.1104
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Possible Phase Diagram
AF antiferromagnetic SDW- Spin Density
Wave (Incommens. Spin Order, Stripes) D-SC
d-wave paired
After several decades we do not yet know the
phase diagram
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Quantum simulations with ultracold atoms
Atoms in optical lattice
Antiferromagnetism and pairing at sub-micro
Kelvin temperatures
Same microscopic model
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How to detect fermion pairing
Quantum noise analysis of TOF images is more
than HBT interference
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Second order interference from the BCS superfluid
Theory Altman et al., PRA 7013603 (2004)
n(k)
k
BCS
BEC
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Momentum correlations in paired fermions
Greiner et al., PRL 94110401 (2005)
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Fermion pairing in an optical lattice
Second Order Interference In the TOF images
Normal State
Superfluid State
measures the Cooper pair wavefunction
One can identify unconventional pairing
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Current experiments
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Fermions in optical lattice. Next challenge
antiferromagnetic state
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Signatures of incompressible Mott state of
fermions in optical lattice
Suppression of double occupancies R. Joerdens et
al., Nature (2008)
Compressibility measurements U. Schneider et al.,
Science (2008)
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Fermions in optical lattice. Next challenge
antiferromagnetic state
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Negative U Hubbard model
U
t
t
Phase diagram Micnas et al., Rev. Mod.
Phys.,1990
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Hubbard model with attraction anomalous
expansion
Hackermuller et al., Science 2010

• Constant Entropy
• Constant Particle Number
• Constant Confinement

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Anomalous expansion Competition of energy and
entropy
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Lattice modulation experiments with fermions in
optical lattice.
Probing the Mott state of fermions
Sensarma, Pekker, Lukin, Demler, PRL (2009)
Related theory work Kollath et al., PRL (2006)
Huber, Ruegg, PRB
(2009) Orso,
Iucci, et al., PRA (2009)
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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., Nature 455204 (2008)
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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 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 t2/U,
weight by t/U
Oscillations reflect shake-off processes of spin
waves
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Rate of doublon production

• Sharp absorption edge due to coherent
  quasiparticles

• Broad continuum due to incoherent part

• Spin wave shake-off peaks

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High Temperature
Retraceable Path Approximation Brinkmann Rice,
1970
Consider the paths with no closed loops
Original Experiment R. Joerdens et al., Nature
455204 (2008)
Theory Sensarma et al., PRL 103, 035303 (2009)
Spectral Fn. of single hole
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Temperature dependence

• Reduced probability to find a singlet on
  neighboring sites

Density
Psinglet
Radius
Radius
D. Pekker et al., upublished
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Fermions in optical lattice.Decay of repulsively
bound pairs
Ref N. Strohmaier et al., PRL 2010
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Fermions in optical lattice.Decay of repulsively
bound pairs Doublons repulsively bound
pairs What is their lifetime?
Direct decay is not allowed by energy conservation
Excess energy U should be converted to kinetic
energy of single atoms
Decay of doublon into a pair of quasiparticles
requires creation of many particle-hole pairs
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Fermions in optical lattice.Decay of repulsively
bound pairs
Experiments N. Strohmaier et. al.
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Relaxation of doublon- hole pairs in the Mott
state
Energy U needs to be absorbed by spin
excitations

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

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

Relaxation rate
Very slow, not relevant for ETH experiments
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Doublon decay in a compressible state
Excess energy U is converted to kinetic energy of
single atoms
Compressible state Fermi liquid description
Doublon can decay into a pair of quasiparticles
with many particle-hole pairs
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Doublon decay in a compressible state
Perturbation theory to order nU/6t Decay
probability
Doublon Propagator
Interacting Single Particles
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Doublon decay in a compressible state
N. Strohmaier et al., PRL 2010
To calculate the rate consider processes which
maximize the number of particle-hole excitations
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Why understanding doublon decay rate is important
Prototype of decay processes with emission of
many interacting particles. Example resonance
in nuclear physics (i.e. delta-isobar) Analogy
to pump and probe experiments in condensed matter
systems Response functions of strongly
correlated systems at high frequencies.
Important for numerical analysis. Important for
adiabatic preparation of strongly correlated
systems in optical lattices
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Surprises of dynamics in the Hubbard model
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Expansion of interacting fermions in optical
lattice
U. Schneider et al., arXiv1005.3545
New dynamical symmetry identical slowdown of
expansion for attractive and repulsive interaction
s
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Competition between pairing and ferromagnetic
instabilities in ultracold Fermi gases near
Feshbach resonances
arXiv1005.2366
D. Pekker, M. Babadi, R. Sensarma, N. Zinner, L.
Pollet, M. Zwierlein, E. Demler
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Stoner model of ferromagnetism
Spontaneous spin polarization decreases
interaction energy but increases kinetic energy
of electrons
Mean-field criterion
U interaction strength N(0) density of
states at Fermi level
Existence of Stoner type ferromagnetism in a
single band model is still a subject of debate
Theoretical proposals for observing Stoner
instability with ultracold Fermi gases Salasnich
et. al. (2000) Sogo, Yabu (2002) Duine,
MacDonald (2005) Conduit, Simons (2009)
LeBlanck et al. (2009)
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Experiments were done dynamically. What are
implications of dynamics? Why spin domains could
not be observed?
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• Is it sufficient to consider effective model
  with repulsive interactions when analyzing
  experiments?
• Feshbach physics beyond effective repulsive
  interaction

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Feshbach resonance
Interactions between atoms are intrinsically
attractive Effective repulsion appears due to low
energy bound states
Example
scattering length
V0 tunable by the magnetic field Can tune through
bound state
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Feshbach resonance
Two particle bound state formed in vacuum
Stoner instability
BCS instability
Molecule formation and condensation
This talk Prepare Fermi state of weakly
interacting atoms. Quench to the
BEC side of Feshbach resonance.
System unstable to both molecule formation
and Stoner ferromagnetism. Which
instability dominates ?
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Many-body instabilities
Imaginary frequencies of collective modes
Magnetic Stoner instability
Pairing instability
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Many body instabilities near Feshbach resonance
naïve picture
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Pairing instability regularized
bubble is UV divergent
Change from bare interaction to the scattering
length
Instability to pairing even on the BEC side
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Pairing instability
Intuition two body collisions do not lead to
molecule formation on the BEC side of Feshbach
resonance. Energy and momentum conservation laws
can not be satisfied.
This argument applies in vacuum. Fermi sea
prevents formation of real Feshbach molecules by
Pauli blocking.
Molecule
Fermi sea
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Stoner instability
Stoner instability is determined by two
particle scattering amplitude
Divergence in the scattering amplitude arises
from bound state formation. Bound state is
strongly affected by the Fermi sea.
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Stoner instability
RPA spin susceptibility
Interaction Cooperon
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Stoner instability
Pairing instability always dominates over
pairing If ferromagnetic domains form, they form
at large q
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Pairing instability vs experiments
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Summary

• Introduction. Magnetic and optical trapping of
  ultracold atoms.
• Cold atoms in optical lattices.
• Bose Hubbard model. Equilibrium and dynamics
• Bose mixtures in optical lattices
• Quantum magnetism of ultracold atoms.
• Detection of many-body phases using noise
  correlations
• Experiments with low dimensional systems
• Interference experiments. Analysis of high
  order correlations
• Fermions in optical lattices
• Magnetism and pairing in systems with
  repulsive interactions. Current experiments
  paramgnetic Mott state, nonequilibrium dynamics.
• Dynamics near Fesbach resonance. Competition of
  Stoner instability and pairing

• Detection of many-body phases
• Nonequilibrium dynamics

Emphasis of these lectures