Strongly Interacting Atoms in
Optical Lattices
Javier von Stecher
JILA and Department of Physics , University of Colorado
In collaboration with
Victor Gurarie,
Leo Radzihovsky,
Ana Maria Rey
INT 2011
“Fermions from Cold Atoms to Neutron Stars:…
arXiv:1102.4593
to appear in PRL
Support
Strongly interacting Fermions:
…Benchmarking the Many-Body Problem.”
BCS-BEC crossover
a0>0
Molecular
BEC
a0=±
a0<0
Degenerate
Fermi Gas
(BCS)
Strongly interacting Fermions + Lattice:
…Understanding the Many-Body Problem?”
More challenging:
Not unique:
- different lattice structure
and strengths.
-Band structure, nontrivial
dispersion relations, …
-Single particle?, twoparticle physics??
Fermi-Hubbard model
Minimal model of interacting fermions in the tight-binding regime
Hopping Energy
Interaction Energy
J
U
i
i+1
Fermi-Hubbard model
Schematic phase diagram for the Fermi Hubbard model
• half-filling
• simple cubic
lattice
•3D
Experiments:
R. Jordens et al., Nature (2008)
U. Schneider et al., Science (2008).
Esslinger, Annual
Rev. of Cond. Mat.
2010
Open questions:
- d-wave superfluid phase?
- Itinerant ferromagnetism?
Beyond the single band Hubbard
Model
Many-Body Hamiltonian (bosons):
Hamiltonian parameters:
Extension of the Fermi
Hubbard Model:
Very complicated…
But, what is the new physics?
Zhai and Ho, PRL (2007)
Iskin and Sa´ de Melo, PRL(2007)
Moon, Nikolic, and Sachdev, PRL (2007)
…
New Physics: Orbital physics
Experiments:
“Orbital superfluidity”:
Populating Higher bands:
Raman
pulse
Long lifetimes ~100 ms (10-100 J)
T. Muller,…, I. Bloch PRL 2007
Scattering in Mixed Dimensions with Ultracold Gases
G. Lamporesi et al. PRL (2010)
JILA KRb Experiment
G. Wirth, M. Olschlager, Hemmerich
New Physics: Resonance Physics
Experiments:
Tuning interactions in lattices:
tune
interaction
Molecules of Fermionic Atoms in
an Optical Lattice
T. Stöferle , …,T. Esslinger PRL 2005
Two-body spectrum in a single site:
Theory and Experiment
Lattice induced resonances
Tight Binding + Short range interactions:
Good understanding of the onsite fewbody physics.
Resonance
• weak nonlocal
coupling
• Strong onsite
interactions.
Separation of energy scales
New degree of freedom:
internal and orbital structure of atoms and
molecules
Independent control of onsite and
nonlocal interactions
Lattice induced resonances
1D Feschbach resonance
scattering
continuum
Energy
Feshbach resonance in free space
Two-body level: weakly bound molecules
Many-body level: BCS-BEC crossover…
Interaction λ
bound
state
0
Lattice induced resonances
1D Feschbach resonance
Energy
Feshbach resonance + Lattice
bands
What is the many-body behavior?
What is the two-body behavior?
Resonances
Interaction λ
Two-body physics:
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
…
Many-body physics (tight –binding):
L. M. Duan PRL 2005, EPL 2008
Dickerscheid , …, Stoof PRA, PRL 2005
K. R. A. Hazzard & E. J. Mueller PRA(R) 2010
…
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.
Two 1D particles in a lattice
One Dimension:
y
z
x
Hamiltonian:
+ a weak lattice in the z-direction
Vx=Vy=200-500 Er, Vz=4-20 Er
1D interaction:
Confinement induced resonance
Two 1D particles in a lattice
One Dimension:
Bound States in 1D:
Form at any weak attraction.
1D dimers with 40K
Hamiltonian:
1D interaction:
H. Moritz, …,T. Esslinger PRL 2005
Confinement induced resonance
Non interacting lattice spectrum
Single particle
Two particles
k=0
K=0
1
0
k
Energy
Energy
2
+
(1,0)
(0,0)
K=(k1+k2)
Tight-binding limit:
k1=K/2+k, k2=K/2-k
+
Non interacting lattice spectrum
Two-body scattering continuum bands
V0=20 Er
V0=4 Er
(1,1)
(0,2)
(0,2)
(1,1)
(0,1)
(0,1)
(0,0)
(0,0)
K a/(2 π)
K a/(2 π)
Two particles in a lattice, single
band Hubbard model
Tight-binding approximation
Nature 2006
Grimm, Daley,
Zoller…
J
U
i
U<0, attractive bound pairs
i+1
U>0, repulsive bound pairs
Exact two-body solution
Calculations in a finite lattice with
periodic boundary conditions
Bloch Theorem:
Plane wave expansion:
Single particle basis functions:
Two particles:
Very large basis set to reach convergence ~ 104-105
Two-atom spectrum
(0,1)
(0,0)
(0,0)
Two body spectrum as a function of the interaction strength for a lattice with V0=4 Er
Two-atom spectrum
(0,1)
(0,0)
(0,0)
Two body spectrum as a function of the interaction strength for a lattice with V0=4 Er
Two-atom spectrum
Tight-binding regime
Two-atom spectrum
Tight-binding regime
First excited dimer crossing
Avoided crossing between a molecular
band and the two-atom continuum
dimer
Energy
K=0
continuum
Interaction
Energy
K=π/a
Interaction
Second excited dimer crossing
Energy
K=0
Interaction
Energy
K=π/a
Interaction
How can we understand this qualitatively change in the atom-dimer coupling?
Two-atom spectrum
Tight-binding regime
Energy
Effective Hamiltonian
ΔE
K
wa,i(r)
Wm,i(R,r)
 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
Effective Hamiltonian
Ja
Energy
Jd
d†
a†
ΔE
K
gex
 Ja, Jd, gex, g and εd are input parameters
g
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
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
Parity effects
Atom-dimer interaction in quasimomentum space:
Prefer to couple at :
K=π/a (max K)
K=0 (min K)
K = center of mass quasi momentum
atoms
molecules
k
Energy
Energy
Energy
molecules
k
k
Comparison model and exact solution
(1,0) molecule: 1st excited
(2,0) molecule: 2nd excited
Molecules above and below!
21 sites and V0=20Er
Dimer Wannier Function
 Jd, g and εd fitting parameters to match spectrum?
gex
Effective Hamiltonian matrix elements:
 How to calculate gex?
is a three-body term
i
i+1
 Wannier function for dimers:
ai †
di†
wa,i(r)
Wm,i(R,r)
Prescription to calculate all
eff. Ham. Matrix elements
Neglected terms:
Dimer Wannier Function
Extraction of the bare dimer:
bound state
(0,1) dimer Wannier Function
Energy
bare dimer
0
K
 Extraction of Jd, g and εd : excellent agreement with the fitting values.
(g1.7 J for (0,1) dimer)
Effective Hamiltonian parameters
•Construct dimer Wannier
function
•Extract eff. Hamiltonian
parameters
Single band Hubbard model:
Enhanced assisted tunneling!
… and symmetric coupling
Parity effects
ga
Atoms in different bands or species:
P=pd+p1+p2
Rectangular lattice
Negative
parity
More dimensions:
gb
extra degeneracies…
more than one dimer
_
+
+
+
Positive
parity
Experimental observation:
Observe quasimomentum dependence of atom-dimer coupling
Ramp Experiment:
 Initialize system in dimer state.
dimer state
 Measure molecule fraction as a
function of quasimomentum.
Dimer fraction (Landau-Zener):
Energy
 Change interactions with time.
Scattering continuum
dimer
fraction
Time
Also K-dependent quantum beats…
N. Nygaard, R. Piil, and K. Molmer PRA 2008
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:
What is the many-body physics of the effective Hamiltonian?