1. Strongly Interacting Atoms in
Optical Lattices
Javier von Stecher
JILA and Department of Physics , University of Colorado
Support
INT 2011
“Fermions from Cold Atoms to Neutron Stars:…
arXiv:1102.4593
to appear in PRL
In collaboration with
Victor Gurarie,
Leo Radzihovsky,
Ana Maria Rey
5. Fermi-Hubbard model
Schematic 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?
6. 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?
7. 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 physics
Experiments:
Raman
pulse
Long lifetimes ~100 ms (10-100 J)
G. Wirth, M. Olschlager, Hemmerich
“Orbital superfluidity”:
8. New Physics: Resonance Physics
Experiments:
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
9. 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
10. 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
Energy
0
bound
state
scattering
continuum
Lattice induced resonances
11. Feshbach resonance + LatticeFeshbach resonance + Lattice
What is the many-body behavior?
Interaction λ
Energy
1D Feschbach resonance
bands
P.O. Fedichev, M. J. Bijlsma, and P. Zoller PRL 2004
G. Orso et al, PRL 2005
X. Cui, Y. Wang, & F. Zhou, PRL 2010
H. P. Buchler, PRL 2010
N. Nygaard, R. Piil, and K. Molmer PRA 2008
…
L. M. Duan PRL 2005, EPL 2008
Dickerscheid , …, Stoof PRA, PRL 2005
K. 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
12. 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.
13. 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
14. Two 1D particles in a lattice
Hamiltonian:Hamiltonian:
1D interaction:
Confinement induced resonance
One Dimension:
1D dimers with 40
K
H. Moritz, …,T. Esslinger PRL 2005
Bound States in 1D:
Form at any weak attraction.
15. Non interacting lattice spectrum
Energy
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
16. 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
17. 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 pairsU<0, attractive bound pairs
18. 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:
19. 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)
20. 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)
23. Avoided crossing between a molecular
band and the two-atom continuum
Interaction
Energy
dimer
continuum
K=0
Interaction
Energy
K=π/a
First excited dimer crossing
24. How can we understand this qualitatively change in the atom-dimer coupling?
Interaction
Energy
Interaction
Energy
K=0
K=π/a
Second excited dimer crossing
26. Effective Hamiltonian
Energy
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)
28. 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
29. 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
30. 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
31. Comparison model and exact solution
(1,0) molecule: 1st
excited (2,0) molecule: 2nd
excited
21 sites and V =20E
Molecules above and below!Molecules above and below!
32. 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
33. 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.
(g≈1.7 J for (0,1) dimer)
34. Effective Hamiltonian parameters
•Construct dimer Wannier
function
•Extract eff. Hamiltonian
parameters
Single band Hubbard model:
… and symmetric coupling
Enhanced assisted tunneling!
35. P=pd+p1+p2
Atoms in different bands or species:
More dimensions:
extra degeneracies…
more than one dimer
Parity effects
Positive
parityNegative
parity+ +
+
_
Rectangular latticeRectangular lattice
gb
ga
36. Experimental observation:
Initialize system in dimer state.
Change interactions with time.
Measure molecule fraction as a
function of quasimomentum.
Ramp Experiment:
Time
Energy
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):
37. 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?