Introduction. Systems of ultracold atoms.

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Strongly correlated many-body systems from
electronic materials to ultracold atoms to
photons

• Introduction. Systems of ultracold atoms.
• Bogoliubov theory. Spinor condensates.
• Cold atoms in optical lattices. Band structure
  and semiclasical dynamics.
• Bose Hubbard model and its extensions
• Bose mixtures in optical lattices
• Quantum magnetism of ultracold atoms.
• Current experiments observation of
  superexchange
• Detection of many-body phases using noise
  correlations
• Fermions in optical lattices
• Magnetism and pairing in systems with
  repulsive interactions.
• Current experiments Mott state
• Experiments with low dimensional systems
• Interference experiments. Analysis of high
  order correlations
• Non-equilibrium dynamics

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Atoms in optical lattices. Bose Hubbard model
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Bose Hubbard model
In the presence of confining potential we also
need to include
Typically
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Bose Hubbard model. Phase diagram

M.P.A. Fisher et al., PRB (1989)
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Bose Hubbard model
Hamiltonian eigenstates are Fock states
Away from level crossings Mott states have a
gap. Hence they should be stable to small
tunneling.
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Bose Hubbard Model. Phase diagram

Mott insulator phase
Particle-hole excitation
Tips of the Mott lobes
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Gutzwiller variational wavefunction
Kinetic energy
z number of nearest neighbors
Interaction energy favors a fixed number of atoms
per well. Kinetic energy favors a superposition
of the number states.
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Gutzwiller variational wavefunction
Example stability of the Mott state with n atoms
per site
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Bose Hubbard Model. Phase diagram

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Bose Hubbard model Experiments with atoms in
optical lattices
Theory Jaksch et al. PRL (1998)
Experiment Kasevich et al., Science (2001)
Greiner et al., Nature (2001)
Phillips et al., J. Physics B
(2002)
Esslinger et al., PRL (2004)
many more
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Nature 41539 (2002)
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Shell structure in optical lattice
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Optical lattice and parabolic potential
Parabolic potential acts as a cut through the
phase diagram. Hence in a parabolic potential we
find a wedding cake structure.
Jaksch et al., PRL 813108 (1998)
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Shell structure in optical lattice
S. Foelling et al., PRL 97060403 (2006)
Observation of spatial distribution of lattice
sites using spatially selective microwave
transitions and spin changing collisions
n1
n2
Related work Campbell, Ketterle, et al. Science
313649 (2006)
superfluid regime
Mott regime
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arXive0904.1532
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Extended Hubbard modelCharge Density Wave and
Supersolid phases
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Extended Hubbard Model
Checkerboard phase
Crystal phase of bosons. Breaks translational
symmetry
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van Otterlo et al., PRB 5216176 (1995)
Variational approach Extension of the Gutzwiller
wavefunction
Supersolid superfluid phase with broken
translational symmetry
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Quantum Monte-Carlo analysis
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arXiv0906.2009
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Difficulty of identifying supersolid phases in
systems with parabolic potential
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Bose Hubbard model away from equilibrium. Dynamica
l Instability of strongly interacting bosons in
optical lattices
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Moving condensate in an optical lattice.
Dynamical instability
Theory Niu et al. PRA (01), Smerzi et al. PRL
(02) Experiment Fallani et al. PRL (04)
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Dynamical instability
Classical limit of the Hubbard model.
Discreet Gross-Pitaevskii equation
Current carrying states
Linear stability analysis States with pgtp/2 are
unstable
Amplification of density fluctuations
unstable
unstable
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Dynamical instability. Gutzwiller approximation
Wavefunction
Time evolution
We look for stability against small fluctuations
Phase diagram. Integer filling
Altman et al., PRL 9520402 (2005)
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Optical lattice and parabolic trap.
Gutzwiller approximation
The first instability develops near the edges,
where N1
U0.01 t J1/4
Gutzwiller ansatz simulations (2D)
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PRL (2007)
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Beyond semiclassical equations. Current decay by
tunneling
Current carrying states are metastable. They can
decay by thermal or quantum tunneling
Thermal activation
Quantum tunneling
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Current decay by thermal phase slips
Theory Polkovnikov et al., PRA (2005)
Experiments De Marco et al., Nature (2008)
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Current decay by quantum phase slips
Theory Polkovnikov et al., Phys. Rev. A (2005)
Experiment Ketterle et al., PRL (2007)
Dramatic enhancement of quantum fluctuations in
interacting 1d systems
d1 dynamical instability. GP regime
d1 dynamical instability. Strongly
interacting regime
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Engineering magnetic systems using cold atoms in
an optical lattice
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Two component Bose mixture in optical lattice
Two component Bose Hubbard model
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Two component Bose Hubbard model
In the regime of deep optical lattice we can
treat tunneling as perturbation. We consider
processes of the second order in t
We can combine these processes into anisotropic
Heisenberg model
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Two component Bose Hubbard model
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Quantum magnetism of bosons in optical lattices
Duan et al., PRL (2003)

• Ferromagnetic
• Antiferromagnetic

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Two component Bose mixture in optical
lattice.Mean field theory Quantum fluctuations
Altman et al., NJP (2003)
Hysteresis
1st order
2nd order line
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Two component Bose Hubbard model
infinitely large Uaa and Ubb
New feature coexistence of checkerboard
phase and superfluidity
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Exchange Interactions in Solids
antibonding
bonding
Kinetic energy dominates antiferromagnetic state
Coulomb energy dominates ferromagnetic state
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Realization of spin liquid
using cold atoms in an optical lattice

Theory Duan, Demler, Lukin PRL (03)
Kitaev model Annals of Physics (2006)
H - Jx S six sjx - Jy S siy sjy - Jz S siz
sjz
Questions Detection of topological
order Creation and manipulation of spin liquid
states Detection of fractionalization, Abelian
and non-Abelian anyons Melting spin liquids.
Nature of the superfluid state
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Superexchange interaction in experiments with
double wells
Theory A.M. Rey et al., PRL 2008 Experiments S.
Trotzky et al., Science 2008
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Observation of superexchange in a double well
potential
Theory A.M. Rey et al., PRL 2008
Experiments S. Trotzky et al. Science 2008
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Preparation and detection of Mott states of atoms
in a double well potential
Reversing the sign of exchange interaction
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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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From two spins to a spin chain
Data courtesy of S. Trotzky (group of I. Bloch)
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1D XXZ dynamics starting from the classical Neel
state
P. Barmettler et al, PRL 2009
Ising-Order
Quasi-LRO
Equilibrium phase diagram
D
1

• DMRG
• Bethe ansatz
• XZ model exact solution

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XXZ dynamics starting from the classical Neel
state
Dlt1, XY easy plane anisotropy
Oscillations of staggered moment, Exponential
decay of envelope
Except at solvable xx point where
Dgt1, Z axis anisotropy
Exponential decay of staggered moment
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Behavior of the relaxation time with anisotropy
See also Sengupta, Powell Sachdev (2004)

• Moment always decays to zero. Even for high easy
  axis anisotropy
• Minimum of relaxation time at the QCP. Opposite
  of classical critical slowing.

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Magnetism in optical lattices Higher spins and
higher symmetries
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F1 spinor condensates
Spin symmetric interaction of F1 atoms
Ferromagnetic Interactions for
Antiferromagnetic Interactions for
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Antiferromagnetic spin F1 atoms in optical
lattices
Hubbard Hamiltonian
Demler, Zhou, PRL (2003)
Symmetry constraints
Nematic Mott Insulator
Spin Singlet Mott Insulator
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Nematic insulating phase for N1
Effective S1 spin model
Imambekov et al., PRA (2003)
the ground state is nematic in d2,3.
When
One dimensional systems are dimerized Rizzi et
al., PRL (2005)
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Nematic insulating phase for N1. Two site
problem
Singlet state is favored when
One can not have singlets on neighboring
bonds. Nematic state is a compromise. It
corresponds to a superposition of
and
on each bond
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SU(N) Magnetism with Ultracold Alkaline-Earth
Atoms
A. Gorshkov et al., Nature Physics (2010)
Example 87Sr (I 9/2)
nuclear spin decoupled from electrons
SU(N2I1) symmetry
SU(N) spin models
Example Mott state with nA atoms in sublattice A
and nB atoms in sublattice B
Phase diagram for nA nB N
There are also extensions to models with
additional orbital degeneracy
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Learning about order from noiseQuantum noise
studies of ultracold atoms
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Quantum noise
Classical measurement collapse of
the wavefunction into eigenstates of x
Histogram of measurements of x
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Probabilistic nature of quantum mechanics
Bohr-Einstein debate on spooky action at a
distance
Einstein-Podolsky-Rosen experiment
Measuring spin of a particle in the left
detector instantaneously determines its value in
the right detector
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Aspects experimentstests of Bells inequalities
S
Correlation function
Classical theories with hidden variable require
Quantum mechanics predicts B2.7 for the
appropriate choice of qs and the state
Experimentally measured value B2.697. Phys. Rev.
Let. 4992 (1982)
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Hanburry-Brown-Twiss experiments
Classical theory of the second order coherence
Hanbury Brown and Twiss, Proc. Roy. Soc.
(London), A, 242, pp. 300-324
Measurements of the angular diameter of
Sirius Proc. Roy. Soc. (London), A, 248, pp.
222-237
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Quantum theory of HBT experiments
Glauber, Quantum Optics and Electronics (1965)
HBT experiments with matter
Experiments with neutrons Ianuzzi et al., Phys
Rev Lett (2006)
For bosons
Experiments with electrons Kiesel et al., Nature
(2002)
Experiments with 4He, 3He Westbrook et al.,
Nature (2007)
For fermions
Experiments with ultracold atoms Bloch et al.,
Nature (2005,2006)
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Shot noise in electron transport
Proposed by Schottky to measure the electron
charge in 1918
Spectral density of the current noise
Related to variance of transmitted charge
When shot noise dominates over thermal noise
Poisson process of independent transmission of
electrons
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Shot noise in electron transport
Current noise for tunneling across a Hall bar on
the 1/3 plateau of FQE
Etien et al. PRL 792526 (1997) see also Heiblum
et al. Nature (1997)
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Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices Hanburry-Brown-
Twiss experiments and beyond
Theory Altman et al., PRA (2004)
Experiment Folling et al., Nature (2005)
Spielman et al., PRL (2007)
Tom et al. Nature (2006)
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Time of flight experiments
Quantum noise interferometry of atoms in an
optical lattice
Second order coherence
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Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices
Experiment Folling et al., Nature (2005)
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Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices
Second order correlation function
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Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices
Relate operators after the expansion to operators
before the expansion. For long expansion times
use steepest descent method of integration
TOF experiments map momentum distributions to
real space images
Second order real-space correlations after TOF
expansion can be related to second order momentum
correlations inside the trapped system
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Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices
Example Mott state of spinless bosons
Only local correlations present in the Mott state
G - reciprocal vectors of the optical lattice
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Quantum noise in TOF experiments in optical
lattices
Boson bunching arises from the Bose enhancement
factors. A single particle state with
quasimomentum q is a supersposition of states
with physical momentum qnG. When we detect a
boson at momentum q we increase the probability
to find another boson at momentum qnG.
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Quantum noise in TOF experiments in optical
lattices
Another way of understanding noise correlations
comes from considering interference of two
independent condensates
After free expansion
Oscillations in the second order correlation
function
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Quantum noise in TOF experiments in optical
lattices Mott state of
spinless bosons
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Interference of an array of independent
condensates
Hadzibabic et al., PRL 93180403 (2004)
Smooth structure is a result of finite
experimental resolution (filtering)
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Second order correlations. Experimental issues.
Autocorrelation function

• Complications we need to consider
• finite resolution of detectors
• projection from 3D to 2D plane

s detector resolution
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Second order coherence in the insulating state of
bosons.
Experiment Folling et al., Nature (2005)
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Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices
Example Band insulating state of spinless
fermions
Only local correlations present in the band
insulator state
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Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices
Example Band insulating state of spinless
fermions
We get fermionic antibunching. This can be
understood as Pauli principle. A single particle
state with quasimomentum q is a supersposition of
states with physical momentum qnG. When we
detect a fermion at momentum q we decrease
the probability to find another fermion at
momentum qnG.
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Second order coherence in the insulating state of
fermions.
Experiment Tom et al. Nature (2006)
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Second order correlations asHanburry-Brown-Twiss
effect
Bosons/Fermions
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Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices
Second order correlation function
Additional contribution to second order
correlation function
We expect to get new peaks in the correlation
function when
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Probing spin order in optical lattices
Correlation function measurements after TOF
expansion.
Extra Bragg peaks appear in the second order
correlation function in the AF phase.
This reflects
doubling of
the
unit cell by
magnetic order.