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Quantum Simulation MURI Review
Theoretical work by groups lead by
Luming Duan (Michigan)
Mikhail Lukin (Harvard)
Subir Sachdev (Harvard)
Peter Zoller (Innsbruck)
Hans-Peter Buchler (Stuttgart)
Eugene Demler (Harvard)
new Hamiltonianmanipulationcooling
Fermionic superfluid
in opt. lattice
(Ketterle, Bloch, Demler, Lukin,
Duan)
Spinor gases in optical lattices
(Ketterle, Demler)
Bose Hubbard QS
validation(Bloch)
Bose Hubbard
precision QS(Ketterle)
Single atom single site detection
(Greiner)
Quantum gas
microscope
Polaronphysics(Zwierlein)
QS of BCS-BEC
crossover, imb. Spin mix.(Ketterle, Zwierlein)
Extended Hubbardlong range int.(Molecules: Doyle, cavity:
Kasevich)
Itinerant ferromagnetism
(Ketterle, Demler)
superlattice
d-wave in plaquettes, noise corr. Detection(Rey, Demler, Lukin)
Fermi Hubbard
model, Mott(Bloch, Ketterle,
Demler, Lukin, Duan)
d-wave superfluidity Quantum magnetism
Bose-Hubbard modelFermionic superfluidity
lattice spin control
(Greiner, Thywissen)
Super-exchange interaction(Bloch, Demler,
Lukin, Duan)
MURI quantum simulation – map of achievements
Quantum magnetism
with ultracold atoms
Dynamics of magnetic domain
formation near Stoner transition
Theory:
David Pekker, Rajdeep Sensarma,
Mehrtash Babadi, Eugene Demler,
arXiv:0909.3483
Experiments:
G. B. Jo et al.,
Science 325:1521 (2009)
Stoner model of ferromagnetism
Mean-field criterion
U N(0) = 1
U – interaction strengthN(0) – density of states at Fermi level
Signatures of ferromagnetic correlations inparticles losses, molecule formation, cloud radius
Observation of Stoner transition by G.B. Jo et al., Science (2009)
Magnetic domains could not be resolved. Why?
Stoner Instability
New feature of cold atoms systems: non-adiabatic
crossing of Uc
Screening of U (Kanamori) occurs on times 1/EF
Two timescales in the system: screening and magnetic
domain formation
Magnetic domain formation takes place on much longer time scales: critical slowing down
Quench dynamics across Stoner instability
Find collective modes
Unstable modes determinecharacteristic lengthscale
of magnetic domains
For U<Uc damped collective modes ωq =ω’- i ω”
For U>Uc unstable collective modes ωq = + i ω”
For MIT experiments domain
sizes of the order of a few λF
Quench dynamics in D=3
Dynamics of magnetic domain formation near Stoner transition
Moving across transition at a finite rate
0 uu*
domains
coarsen
slow
growth
domains
freeze
Growth rate of magnetic domains
Domain size
Domains freeze when
Domain size at “freezing” point
Superexchange interaction
in experiments with double wells
Theory: A.M. Rey et al., PRL 2008Experiments: S. Trotzky et al., Science 2008
Quantum magnetism of bosons in optical lattices
Duan, Demler, Lukin, PRL (2003)
• Ferromagnetic
• Antiferromagnetic
For spin independent lattice and interactionswe find ferromagnetic exchange interaction.
Ferromagnetic ordering is favored by the boson
enhancement factor
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
Comparison to the Hubbard model
[Picture: Greiner (2002)]
Two-Orbital SU(N) Magnetism with
Ultracold Alkaline-Earth Atoms
Alkaline-Earth atoms in optical lattice:Ex: 87Sr (I = 9/2)
|g> = 1S0
|e> = 3P0
698 nm150 s ~ 1 mHz
• orbital degree of freedom � spin-orbital physics→ Kugel-Khomskii model [transition metal oxides with perovskite
structure]
→ SU(N) Kondo lattice model [for N=2, colossal magnetoresistance in
manganese oxides and heavy fermion materials]
Nuclear spin decoupled from electrons SU(N=2I+1) symmetry
→ SU(N) spin models � valence-bond-solid & spin-liquid phases
A. Gorshkov, et al., arXiv:0905.2963 (poster)
Mott state of the fermionic Hubbard model
Signatures of incompressible Mott state of fermions in optical lattice
Suppression of double occupanciesR. Joerdens et al., Nature (2008)
Compressibility measurementsU. Schneider, I. Bloch et al., Science (2008)
Fermions in optical lattice. Next challenge:
antiferromagnetic state
TN
U
Mott
currentexperiments
Lattice modulation experiments with fermions in optical lattice
Probing the Mott state of fermionsPekker, Sensarma, Lukin, Demler (2009)
Pekker, Pollet, unpublished
Measure number of doubly occupied sites
Modulate lattice potential
Doublon/hole production rate determined by
the spectral functions of excitations
Experiments by ETH Zurich group and others
Experiment:
R. Joerdens et al.,
Nature 455:204 (2008)
Spectral Function for
doublons/holes
Rate of doublon production
All spin configurations are equally likely.Can neglect spin dynamics
Temperature dependence comes from
probability of finding nearest neighbors
“High” Temperature
“Low” Temperature
Rate of doublon production
• Sharp absorption edge due to coherent quasiparticles
• Spin wave shake-off peaks
Spins are antiferromagnetically ordered
Decay probability
Doublon decay in a compressible stateHow to get rid of the excess energy U?
Doublon can decay into apair of quasiparticles with many particle-hole pairs
Perturbation theory to order n=U/6t
Consider processes which maximize the number of particle-hole excitations
N. Strohmaier, D. Pekker, et al., arXiv:0905.2963
d-wave pairing in the fermionic
Hubbard model
new Hamiltonianmanipulationcooling
Fermionic superfluid
in opt. lattice
(Ketterle, Bloch, Demler, Lukin,
Duan)
Spinor gases in optical lattices
(Ketterle, Demler)
Bose Hubbard QS
validation(Bloch)
Bose Hubbard
precision QS(Ketterle)
Single atom single site detection
(Greiner)
Quantum gas
microscope
Polaronphysics(Zwierlein)
QS of BCS-BEC
crossover, imb. Spin mix.(Ketterle, Zwierlein)
Extended Hubbardlong range int.(Molecules: Doyle, cavity:
Kasevich)
Itinerant ferromagnetism
(Ketterle, Demler)
superlattice
d-wave in plaquettes, noise corr. Detection(Rey, Demler, Lukin)
Fermi Hubbard
model, Mott(Bloch, Ketterle,
Demler, Lukin, Duan)
d-wave superfluidity Quantum magnetism
Bose-Hubbard modelFermionic superfluidity
lattice spin control
(Greiner, Thywissen)
Super-exchange interaction(Bloch, Demler,
Lukin, Duan)
MURI quantum simulation – map of achievements
A.M. Rey et al., EPL 87, 60001 (2009)
Using plaquettes to reach d-wave pairing
x
y
JJ’
Superlattice was a usefultool for observing magneticsuperexchange.
Can we use it to createand observe d-wavepairing?
Minimal system exhibitingd-wave pairing is a 4-siteplaquette
Ground State properties of a plaquetteGround State properties of a plaquette
• Unique singlet (S=0)
• d-wave symmetry
• Unique singlet (S=0)
• s-wave symmetry
d-wave symmetry
d-wave pair creation operator
Singlet creation operator
1
3
2
4
4
2
Scalapino,Trugman(1996)
Altman, Auerbach(2002)
Vojta, Sachdev(2004)
Kivelson et al.,(2007)
For weak coupling there are two possible competing configurations:
VS4 2 , 42 3 3
Which one is lower in energy is determined by the binding energy
1. Prepare 4 atoms in a single plaquette: A
plaquette is the minimum system that exhibits d-
wave symmetry.
3. Weakly connect the 2D array to realize and
study a superfluid exhibiting long range d-wave
correlations.
Melt the plaquettes into a 2D lattice,
experimentally explore the regime
unaccessible to theory
4.
2. Connect two plaquettes into a
superplaquette and study the dynamics of a
d-wave pair. Use it to measure the pairing
gap.
?
Using plaquettes to reach d-wave pairing
Phase sensitive probe of d-wave pairing
From noise correlations to phase sensitive measurements in systems of ultra-cold atoms
T. Kitagawa, A. Aspect, M. Greiner, E. Demler
Second order interference from paired states
)'()()',( rrrr nnn −≡∆
n(r)
n(r’)
n(k)
k
0),( =Ψ−∆BCS
n rr
BCS
BEC
kF
Theory: Altman et al., PRA 70:13603 (2004)
Momentum correlations in paired fermions
Experiments: Greiner et al., PRL 94:110401 (2005)
How to measure the molecular
wavefunction?
How to measure the non-trivial symmetry of ψ(p)?
We want to measure the relative phase between components of the molecule at different wavevectors
Two particle interference
Coincidence count on detectorsmeasures two particle interference
χ – χphase controlled by beam
splitters and mirrors
Two particle interference
Implementation for atoms: Bragg pulse before expansion
Bragg pulse mixes statesk and –p = k-G
-k and p =-k+G
Coincidence count for states k and p depends on two particle interference and measures phase of the molecule wavefunction
� Description of s-wave Feshbach resonance in an optical lattice
�p-wave Feshbach resonance in an optical lattice and its phase diagram
Research by Luming Duan’s group: poster
� Low-dimensional effective Hamiltonian and 2D BEC-BCS crossover
� Anharmonicity induced resonance in a lattice
Confinement induced
(shift) resonance
Anharmonicity
induced resonance
Other theory projects by the Harvard group
Fermionic Mott states
with spin imbalance.
B. Wunsch (poster)
One dimensional
systems: nonequilibrium
dynamics and noise.
T. Kitagawa (poster)
Spin liquids.Detection of
fractional statistics and
interferometry of anions.
L. Jiang, A. Gorshkov,
Demler, Lukin, Zoller, et al. ,
Nature Physics (2009)
Fermions in spin dependent optical lattice. N. Zinner, B. Wunsch (poster).
Bosons in optical lattice with disoreder. D. Pekker (poster).
Dynamical preparation of AF state using bosons with ferromagnetic interactions.
M. Gullans, M. Rudner
new Hamiltonianmanipulationcooling
Fermionic superfluid
in opt. lattice
(Ketterle, Bloch, Demler, Lukin,
Duan)
Spinor gases in optical lattices
(Ketterle, Demler)
Bose Hubbard QS
validation(Bloch)
Bose Hubbard
precision QS(Ketterle)
Single atom single site detection
(Greiner)
Quantum gas
microscope
Polaronphysics(Zwierlein)
QS of BCS-BEC
crossover, imb. Spin mix.(Ketterle, Zwierlein)
Extended Hubbardlong range int.(Molecules: Doyle, cavity:
Kasevich)
Itinerant ferromagnetism
(Ketterle, Demler)
superlattice
d-wave in plaquettes, noise corr. Detection(Rey, Demler, Lukin)
Fermi Hubbard
model, Mott(Bloch, Ketterle,
Demler, Lukin, Duan)
d-wave superfluidity Quantum magnetism
Bose-Hubbard modelFermionic superfluidity
lattice spin control
(Greiner, Thywissen)
Super-exchange interaction(Bloch, Demler,
Lukin, Duan)
MURI quantum simulation – map of achievements
Summary
Quantum magnetism with ultracold atoms
• Dynamics of magnetic domain formation near Stoner transition
• Superexchange interaction in experiments with double wells
• Two-Orbital SU(N) Magnetism with Ultracold Alkaline-Earth Atoms
Mott state of the fermionic Hubbard model • Signatures of incompressible Mott state of fermions in optical lattice• Lattice modulation experiments with fermions in optical lattice• Doublon decay in a compressible state
Making and probing d-wave pairing in the fermionic Hubbard model• Using plaquettes to reach d-wave pairing• Phase sensitive probe of d-wave pairing