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• 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 latticesQuantum magnetism of ultracold atoms.
Current experiments: observation of superexchange
• Detection of many-body phases using noise correlations
• Fermions in optical latticesMagnetism 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
Strongly correlated many-body systems: from electronic
materials to ultracold atoms to photons
Atoms in optical lattices.
Bose Hubbard model
Bose Hubbard model
tunneling of atoms between neighboring wells
repulsion of atoms sitting in the same well
U
t
In the presence of confining potential we also need to include
Typically
Bose Hubbard model. Phase diagram
M.P.A. Fisher et al.,
PRB (1989)21+n
Uµ
1
0
Mottn=1
n=2
n=3
Superfluid
Mott
Mott
Weak lattice Superfluid phase
Strong lattice Mott insulator phase
Bose Hubbard model
Hamiltonian eigenstates are Fock states
Uµ
0 1
Set .
Away from level crossings Mott states have a gap. Hence they should be stable to small tunneling.
Bose Hubbard Model. Phase diagram
Particle-hole excitation
Mott insulator phase
21+n
Uµ
1
0
Mottn=1
n=2
n=3
Superfluid
Mott
Mott
Tips of the Mott lobes
Gutzwiller variational wavefunction
Normalization
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.
Gutzwiller variational wavefunction
Transition takes place when coefficient before becomes negative. For large n this corresponds to
Take the middle of the Mott plateau
Expand to order
Example: stability of the Mott state with n atoms per site
Bose Hubbard Model. Phase diagram
21+n
Uµ
1
0
Mottn=1
n=2
n=3
Superfluid
Mott
Mott
Note that the Mott state only exists for integer filling factors.For even when atoms are localized,
make a superfluid state.
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 …
Nature 415:39 (2002)
Shell structure in optical lattice
Optical lattice and parabolic potential
Jaksch et al.,
PRL 81:3108 (1998)
Parabolic potential acts as a “cut” through the phase diagram. Hence in a parabolic
potential we find a “wedding cake” structure.
21+n
Uµ
1
0
Mottn=1
n=2
n=3
Superfluid
Mott
Mott
Shell structure in optical latticeS. Foelling et al., PRL 97:060403 (2006)
Observation of spatial distribution of lattice sites using spatially selective microwave transitions and spin changing collisions
superfluid regime Mott regime
n=1
n=2
Related work
Campbell, Ketterle, et al.
Science 313:649 (2006)
arXive:0904.1532
Extended Hubbard model
Charge Density Wave
and Supersolid phases
Extended Hubbard Model
- on site repulsion - nearest neighbor repulsion
Checkerboard phase:
Crystal phase of bosons.Breaks translational symmetry
van Otterlo et al., PRB 52:16176 (1995)
Variational approach
Extension of the Gutzwiller wavefunction
Supersolid – superfluid phase with broken translational symmetry
Quantum Monte-Carloanalysis
arXiv:0906.2009
Difficulty of identifying supersolid phases
in systems with parabolic potential
Bose Hubbard model away from equilibrium.Dynamical Instability of strongly interacting bosons in optical lattices
Moving condensate in an optical lattice. Dynamical instability
v
Theory: Niu et al. PRA (01), Smerzi et al. PRL (02)Experiment: Fallani et al. PRL (04)
Linear stability analysis: States with p>p/2 are unstable
Classical limit of the Hubbard model. Discreet Gross-Pitaevskii equation
Current carrying states
r
Dynamical instability
Amplification ofdensity fluctuations
unstableunstable
Dynamical instability. Gutzwiller approximation
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.2 0.4 0.6 0.8 1.0
d=3
d=2
d=1
unstable
stable
U/Uc
p/π
Wavefunction
Time evolution
Phase diagram. Integer filling
We look for stability against small fluctuations
Altman et al., PRL 95:20402 (2005)
The first instability
develops near the edges,
where N=1
0 100 200 300 400 500
-0.2
-0.1
0.0
0.1
0.2
0.00 0.17 0.34 0.52 0.69 0.86
Ce
nte
r o
f M
ass M
om
en
tum
Time
N=1.5
N=3
U=0.01 t
J=1/4
Gutzwiller ansatz simulations (2D)
Optical lattice and parabolic trap.
Gutzwiller approximation
PRL (2007)
Beyond semiclassical equations. Current decay by tunneling
phase
jphase
jphase
j
Current carrying states are metastable. They can decay by thermal or quantum tunneling
Thermal activation Quantum tunneling
Current decay by thermal phase slips
Theory: Polkovnikov et al., PRA (2005) Experiments: De Marco et al., Nature (2008)
Current decay by quantum phase slips
Theory: Polkovnikov et al., Phys. Rev. A (2005) Experiment: Ketterle et al., PRL (2007)
Dramatic enhancementof quantum fluctuations
in interacting 1d systems
d=1 dynamicalinstability.GP regime
d=1 dynamicalinstability.Strongly interactingregime
Engineering magnetic systems
using cold atoms in an optical lattice
tt
Two component Bose mixture in optical lattice
Two component Bose Hubbard model
Example: . Mandel et al., Nature (2003)
We consider two component Bose mixture in the n=1 Mott state with equal number of and atoms.
We need to find spin arrangement in the ground state.
In the regime of deep optical lattice we can treat tunnelingas perturbation. We consider processes of the second order in t
We can combine these processes into anisotropic Heisenberg model
Two component Bose Hubbard model
Two component Bose Hubbard model
Quantum magnetism of bosons in optical lattices
Duan et al., PRL (2003)
• Ferromagnetic
• Antiferromagnetic
Two component Bose mixture in optical lattice.
Mean field theory + Quantum fluctuations
2 ndorder line
Hysteresis
1st order
Altman et al., NJP (2003)
Two component Bose Hubbard model
+ infinitely large Uaa and Ubb
New feature:coexistence of
checkerboard phase
and superfluidity
Exchange Interactions in Solids
antibonding
bonding
Kinetic energy dominates: antiferromagnetic state
Coulomb energy dominates: ferromagnetic state
Questions:Detection of topological orderCreation and manipulation of spin liquid statesDetection of fractionalization, Abelian and non-Abelian anyonsMelting spin liquids. Nature of the superfluid state
Realization of spin liquid
using cold atoms in an optical latticeTheory: Duan, Demler, Lukin PRL (03)
H = - Jx Σ σix σj
x - Jy Σ σiy σj
y - Jz Σ σiz σj
z
Kitaev model Annals of Physics (2006)
Superexchange interaction
in experiments with double wells
Theory: A.M. Rey et al., PRL 2008Experiments: S. Trotzky et al., Science 2008
J
J
Use magnetic field gradient to prepare a state
Observe oscillations between and states
Observation of superexchange in a double well potentialTheory: A.M. Rey et al., PRL 2008
Experiments:S. Trotzky et al.Science 2008
Reversing the sign of exchange interaction
Preparation and detection of Mott statesof atoms in a double well potential
Comparison to the Hubbard model
Basic Hubbard model includesonly local interaction
Extended Hubbard modeltakes into account non-localinteraction
Beyond the basic Hubbard model
Beyond the basic Hubbard model
From two spins to a spin chain
Spin oscillations ?
Data courtesy of Data courtesy of
S. S. TrotzkyTrotzky
(group of I. Bloch)(group of I. Bloch)
1D: XXZ dynamics starting from the classical Neel state
• DMRG• Bethe ansatz
• XZ model: exact solution
Time, Jt
∆Equilibrium phase diagram:
Ψ(t=0) =
Quasi-LRO
1
Ising-Order
P. Barmettler et al, PRL 2009
XXZ dynamics starting from the classical Neel state
∆<1, XY easy plane anisotropy
Oscillations of staggered moment, Exponential decay of envelope
∆>1, Z axis anisotropy
Exponential decay of staggered moment
Except at solvable xx point where:
Behavior of the relaxation time with anisotropy
- Moment always decays to zero. Even for high easy axis anisotropy
- Minimum of relaxation time at the QCP. Opposite of classical critical slowing.
See also: Sengupta,
Powell & Sachdev (2004)
Magnetism in optical lattices
Higher spins and higher symmetries
F=1 spinor condensates
Spin symmetric interaction of F=1 atoms
Antiferromagnetic Interactions for
Ferromagnetic Interactions for
Antiferromagnetic spin F=1 atoms in optical lattices
Hubbard Hamiltonian
Symmetry constraints
Demler, Zhou, PRL (2003)
Nematic Mott Insulator
Spin Singlet Mott Insulator
Nematic insulating phase for N=1
Effective S=1 spin model Imambekov et al., PRA (2003)
When the ground state is nematic in d=2,3.
One dimensional systems are dimerized: Rizzi et al., PRL (2005)
Nematic insulating phase for N=1.
Two site problem
12
0 -2 4
1
Singlet state is favored when
One can not have singlets on neighboring bonds.
Nematic state is a compromise. It correspondsto a superposition of and
on each bond
SU(N) Magnetism with Ultracold Alkaline-Earth Atoms
Example: 87Sr (I = 9/2)
nuclear spin decoupled from electrons SU(N=2I+1) symmetry
SU(N) spin models
A. Gorshkov et al., Nature Physics (2010)
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
Learning about order from noise
Quantum noise studies of ultracold atoms
Quantum noiseClassical measurement:
collapse of the wavefunction into eigenstates of x
Histogram of measurements of x
Probabilistic nature of quantum mechanics
Bohr-Einstein debate on spooky action at a distance
Measuring spin of a particle in the left detectorinstantaneously determines its value in the right detector
Einstein-Podolsky-Rosen experiment
Aspect’s experiments:tests of Bell’s inequalities
SCorrelation function
Classical theories with hidden variable require
Quantum mechanics predicts B=2.7 for the appropriate choice of θ‘s and the state
Experimentally measured value B=2.697. Phys. Rev. Let. 49:92 (1982)
+
-
+
-1 2θ1 θ2
S
Hanburry-Brown-Twiss experimentsClassical theory of the second order coherence
Measurements of the angular diameter of Sirius
Proc. Roy. Soc. (London), A, 248, pp. 222-237
Hanbury Brown and Twiss,
Proc. Roy. Soc. (London),
A, 242, pp. 300-324
Quantum theory of HBT experiments
For bosons
For fermions
Glauber,Quantum Optics and Electronics (1965)
HBT experiments with matter
Experiments with 4He, 3He
Westbrook et al., Nature (2007)
Experiments with neutrons
Ianuzzi et al., Phys Rev Lett (2006)
Experiments with electrons
Kiesel et al., Nature (2002)
Experiments with ultracold atoms
Bloch et al., Nature (2005,2006)
Shot noise in electron transport
e- e-
When shot noise dominates over thermal noise
Spectral density of the current noise
Proposed by Schottky to measure the electron charge in 1918
Related to variance of transmitted charge
Poisson process of independent transmission of electrons
Shot noise in electron transport
Current noise for tunneling across a Hall bar on the 1/3
plateau of FQE
Etien et al. PRL 79:2526 (1997)see also Heiblum et al. Nature (1997)
Quantum noise analysis of time-of-flight
experiments 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)
Time of flight experiments
Quantum noise interferometry of atoms in an optical lattice
Second order coherence
Cloud after expansion
Cloud before expansion
Experiment: Folling et al., Nature (2005)
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices
Second order correlation function
Cloud after expansion
Cloud before expansion
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices
Here and are taken after the expansion time t.Two signs correspond to bosons and fermions.
Relate operators after the expansion to operators before the expansion. For long expansion times use steepest descent
method of integration
Second order real-space correlations after TOF expansioncan be related to second order momentum correlations
inside the trapped system
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices
TOF experiments map momentum distributions to real space images
Example: Mott state of spinless bosons
Only local correlations present in the Mott state
G - reciprocalvectors of the
optical lattice
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices
Quantum noise in TOF experiments in optical lattices
We get bunching when corresponds to oneof the reciprocal vectors of the original lattice.
Boson bunching arises from the Bose enhancement factors. A single particle state with quasimomentum q is a
supersposition of states with physical momentum q+nG.
When we detect a boson at momentum q we increase the probability to find another boson at momentum q+nG.
Quantum noise in TOF experiments in optical lattices
Another way of understanding noise correlations comes fromconsidering interference of two independent condensates
Oscillations in the second order correlation function
After free expansion
When condensates 1 and 2 are not correlated
We do not see interference in .
Quantum noise in TOF experiments in optical latticesMott state of spinless bosons
0 200 400 600 800 1000 1200-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Interference of an array of independent condensates
Hadzibabic et al., PRL 93:180403 (2004)
Smooth structure is a result of finite experimental resolution (filtering)
0 200 400 600 800 1000 1200-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Second order correlations. Experimental issues.
Complications we need to consider:- finite resolution of detectors
- projection from 3D to 2D plane
σ – detector resolution
Autocorrelation function
In Mainz experiments and
The signal in is
- period of the optical lattice
Second order coherence in the insulating state of bosons.
Experiment: Folling et al., Nature (2005)
Example: Band insulating state of spinless fermions
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices
Only local correlations present in the band insulator state
Example: Band insulating state of spinless fermions
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices
We get fermionic antibunching. This can be understoodas Pauli principle. A single particle state with quasimomentum
q is a supersposition of states with physical momentum q+nG.
When we detect a fermion at momentum q we decrease theprobability to find another fermion at momentum q+nG.
Second order coherence in the insulating state of fermions.
Experiment: Tom et al. Nature (2006)
Second order correlations as
Hanburry-Brown-Twiss effect
Bosons/Fermions
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices
Bosons with spin. Antiferromagnetic order
Second order correlation function
New local correlations
Additional contribution to second order correlation function
- antiferromagnetic wavevector
We expect to get new peaks in the correlation function when
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 theunit cell by
magnetic order.
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