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Strongly Interacting Atoms in Optical Lattices
Javier von Stecher
JILA and Department of Physics , University of Colorado
SupportINT 2011
“Fermions from Cold Atoms to Neutron Stars:…
arXiv:1102.4593to appear in PRL
In collaboration withVictor Gurarie, Leo Radzihovsky,Ana Maria Rey
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Strongly interacting Fermions:…Benchmarking the Many-Body Problem.”
BCS-BEC crossover
a0<0a0>0 a0=±
Degenerate Fermi Gas
(BCS)
Molecular BEC
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Strongly interacting Fermions + Lattice:…Understanding the Many-Body Problem?”
?More challenging:
-Band structure, nontrivial dispersion relations, …
-Single particle?, two-particle physics??
Not unique:
- different lattice structure and strengths.
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Interaction EnergyHopping Energy
J
U
i+1
i
Fermi-Hubbard modelMinimal model of interacting fermions in the tight-binding regime
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Fermi-Hubbard modelSchematic phase diagram for the Fermi Hubbard model
Esslinger, Annual Rev. of Cond. Mat. 2010
• half-filling • simple cubic lattice •3D
Experiments:R. Jordens et al., Nature (2008)U. Schneider et al., Science (2008).
Open questions:- d-wave superfluid phase?- Itinerant ferromagnetism?
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Many-Body Hamiltonian (bosons):
Hamiltonian parameters:
Beyond the single band Hubbard Model
Extension of the Fermi Hubbard Model:
Zhai and Ho, PRL (2007)Iskin and Sa´ de Melo, PRL(2007)Moon, Nikolic, and Sachdev, PRL (2007)…
Very complicated…But, what is the new physics?
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T. Muller,…, I. Bloch PRL 2007
Populating Higher bands:
Scattering in Mixed Dimensions with Ultracold Gases G. Lamporesi et al. PRL (2010)JILA KRb Experiment
New Physics: Orbital physicsExperiments:
Ramanpulse
Long lifetimes ~100 ms (10-100 J)
G. Wirth, M. Olschlager, Hemmerich
“Orbital superfluidity”:
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New Physics: Resonance PhysicsExperiments:
T. Stöferle , …,T. Esslinger PRL 2005
Tuning interactions in lattices:
Molecules of Fermionic Atoms in an Optical Lattice
tune interaction
Two-body spectrum in a single site: Theory and Experiment
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Lattice induced resonances
Resonance
Tight Binding + Short range interactions:
• Strong onsite interactions.
Good understanding of the onsite few-body physics.
• weak nonlocal coupling
New degree of freedom: internal and orbital structure of atoms and molecules
Separation of energy scales
Independent control of onsite and nonlocal interactions
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Feshbach resonance in free spaceFeshbach resonance in free space
Two-body level: weakly bound molecules
Many-body level: BCS-BEC crossover…
1D Feschbach resonance
Interaction λ
Ener
gy
0bound state
scatteringcontinuum
Lattice induced resonances
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Feshbach resonance + LatticeFeshbach resonance + Lattice
What is the many-body behavior?
Interaction λ
Ener
gy
1D Feschbach resonance
bands
P.O. Fedichev, M. J. Bijlsma, and P. Zoller PRL 2004G. Orso et al, PRL 2005X. Cui, Y. Wang, & F. Zhou, PRL 2010H. P. Buchler, PRL 2010N. Nygaard, R. Piil, and K. Molmer PRA 2008…
L. M. Duan PRL 2005, EPL 2008Dickerscheid , …, Stoof PRA, PRL 2005K. R. A. Hazzard & E. J. Mueller PRA(R) 2010…
Two-body physics: Many-body physics (tight –binding):
What is the two-body behavior? Resonances
Lattice induced resonances
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Our strategy
• Start with the simplest case– Two particles in 1D + lattice.
• Benchmark the problem:– Exact two-particle solution
• Gain qualitative understanding– Effective Hamiltonian description
Two-body calculations are valid for two-component Fermi systems and bosonic systems .Below, we use notation assuming bosonic statistics.
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Two 1D particles in a lattice
y
xz
+ a weak lattice in the z-direction+ a weak lattice in the z-direction
Hamiltonian:Hamiltonian:
1D interaction:
Confinement induced resonance
One Dimension:
Vx=Vy=200-500 Er, Vz=4-20 Er
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Two 1D particles in a lattice
Hamiltonian:Hamiltonian:
1D interaction:
Confinement induced resonance
One Dimension:
1D dimers with 40K
H. Moritz, …,T. Esslinger PRL 2005
Bound States in 1D:Form at any weak attraction.
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Non interacting lattice spectrumEnergy
k
+
+
k=0
Single particleSingle particle Two particlesTwo particles
Tight-binding limit:Tight-binding limit:
k1=K/2+k, k2=K/2-k
Energy
K=(k1+k2)
K=0
(1,0)
(0,0)0
1
2
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Non interacting lattice spectrum
V0=4 Er
K a/(2 π)
(0,0)
(0,1)
(0,2)
(1,1)
V0=20 Er
K a/(2 π)
(0,0)
(0,1)
(0,2)
(1,1)
Two-body scattering continuum bands
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Two particles in a lattice, single band Hubbard model
Tight-binding approximation
Nature 2006 Grimm, Daley, Zoller…
J
U
i+1i
U>0, repulsive bound pairs U<0, attractive bound pairs
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Calculations in a finite lattice with periodic boundary conditions
Exact two-body solution
Plane wave expansion:Plane wave expansion:
Single particle basis functions:
Two particles:
Very large basis set to reach convergence ~ 104-105
Bloch Theorem:Bloch Theorem:
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Two-atom spectrum
Two body spectrum as a function of the interaction strength for a lattice with V0=4 Er
(0,0)
(0,1)
(0,0)
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Two-atom spectrum
Two body spectrum as a function of the interaction strength for a lattice with V0=4 Er
(0,0)
(0,1)
(0,0)
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Two-atom spectrum
Tight-binding regime
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Two-atom spectrum
Tight-binding regime
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Avoided crossing between a molecular band and the two-atom continuum
Interaction
Ener
gy
dimer
continuum
K=0
Interaction
Ener
gy
K=π/a
First excited dimer crossing
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How can we understand this qualitatively change in the atom-dimer coupling?
Interaction En
ergy
Interaction
Ener
gy
K=0
K=π/a
Second excited dimer crossing
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Two-atom spectrum
Tight-binding regime
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Effective HamiltonianEnergy
K
ΔE
Atoms and dimers are in the tight-binding regime.
They are hard core particles (both atoms and dimers).
Leading terms in the interaction are produced by hopping of one particle.
L. M. Duan PRL 2005, EPL 2008
wa,i(r) Wm,i(R,r)
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Effective HamiltonianEnergy
K
ΔE
Ja, Jd, gex, g and εd are input parameters
d†
a†
ggex
JaJd
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Parity effects The atomic and dimer wannier functions are symmetric or antisymmetric with respect to the center of the site.
Parity effects on the atom-dimer interaction:
S coupling
g-1 g+1
g+1= g-1
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Parity effects The atomic and dimer wannier functions are symmetric or antisymmetric with respect to the center of the site.
Parity effects on the atom-dimer interaction:
AS coupling
g-1 g+1
g+1= -g-1
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Parity effects
Atom-dimer interaction in quasimomentum space:
K = center of mass quasi momentum
Prefer to couple at :
K=π/a (max K)
K=0 (min K)
Energy
k
atoms
Energy
k
molecules
Energy
k
molecules
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Comparison model and exact solution
(1,0) molecule: 1st excited (2,0) molecule: 2nd excited
21 sites and V0=20Er
Molecules above and below!Molecules above and below!
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Dimer Wannier Function
Effective Hamiltonian matrix elements:
Jd, g and εd fitting parameters to match spectrum?
i+1i
gex
How to calculate gex?is a three-body term
Neglected terms:
wa,i(r)
Wannier function for dimers:
di†
ai†
Wm,i(R,r)
Prescription to calculate all eff. Ham. Matrix elements
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Dimer Wannier Function
(0,1) dimer Wannier Function
0
Energy
K
bound state
bare dimer
Extraction of the bare dimer:
Extraction of Jd, g and εd : excellent agreement with the fitting values. (g1.7 J for (0,1) dimer)
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Effective Hamiltonian parameters
•Construct dimer Wannier function•Extract eff. Hamiltonian parameters
Single band Hubbard model:
… and symmetric couplingEnhanced assisted tunneling!
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P=pd+p1+p2
Atoms in different bands or species:
More dimensions:
extra degeneracies…more than one dimer
Parity effects
Positive parity
Neg
ative
pa
rity
+ ++
_
Rectangular latticeRectangular lattice
gb
ga
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Experimental observation:
Initialize system in dimer state.
Change interactions with time.
Measure molecule fraction as a function of quasimomentum.
Ramp Experiment:
Time En
ergy
dimer state
Scattering continuum
dimer fraction
Observe quasimomentum dependence of atom-dimer coupling
N. Nygaard, R. Piil, and K. Molmer PRA 2008
Also K-dependent quantum beats…
Dimer fraction (Landau-Zener):
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Summary Lattice induced resonances (Lattice + Resonance + Orbital Physics)can be used to tuned lattice systems in new regimes.
The orbital structure of atoms and dimer plays a crucial role in the qualitative behavior of the atom-dimer coupling.
The momentum dependence of the molecule fraction after a magnetic ramp provides an experimental signature of the lattice induced resonances.
Outlook:Outlook: What is the many-body physics of the effective Hamiltonian?