TFVS

1
TFVS
Two atoms in a 1D Optical Lattice close to a
Feshbach Resonance
M. Wouters
In collaboration with G. Orso, L.P. Pitaevskii,
S. Stringari
NQS2005, Camerino, july 6th, 2005
2
Motivation
Introduction
Experiments with ultra-cold atoms

• Tunable interactions

M. Greiner, C. A. Regal, D.S. Jin, Proceedings of
ICAP-2004 (Rio de Janeiro) condmat/0502539
3
Periodic Potential
Introduction
d
Recoil energy
sER
r
z1
z2

• Center of mass and relative motion dont
  decouple
• No nice analytical wave functions

G. Orso, L.P. Pitaevskii, S. Stringari, M.W.,
cond-mat/0503096, accepted to PRL
4
Harmonic confinement
Introduction
Harmonic confinement in z-direction separation
of relative and c.om. motion
Bound state for any a!
1 D.S. Petrov, G.V. Shlyapnikov, Phys. Rev. A
64 (2000) 2 Z. Idziaszek and T. Calarco,
quant-ph/0410163
5
Integral equation
Method
No separation of center of mass and relative
motion
Bethe-Peierls
takes regular part
Discrete translational symmetry
Q is quasi-momentum of the molecule
6
Greens function
Method
Independent of energy and external potential
Handle the singularity

• Numerically
• Tight Binding

Analytically
7
Qualitative picture
Results
0
8
Binding energy
Results
Q 0
S20
10
5
0
9
Critical scattering length
Results
Q 0
Q qB
10
Binding energy at resonance
Results
Q 0
11
Binding energy dispersion
Results
Center of mass motion and relative motion are
coupled
Binding energy depends on the quasi-momentum
S2.5
d/acr increases with quasi-momentum
12
Band width dispersion
Results
The bandwidth depends strongly on the scattering
length (binding energy).
Possible to extract experimentally from Bloch
oscillations
13
Effective mass dispersion
Results
Depends also strongly on the scattering length
(binding energy).
Possible to extract experimentally from
Bloch/dipole oscilations
14
Conclusions/Perspectives

• Exact numerical method for any value of the laser
  intensity and scattering length
• Binding energy
• Tunneling properties
• 2D-3D optical lattices
• Scattering properties
• Analytical treatment
• Many body physics in 1D/2D/3D optical lattices
  (coupled layers/tubes/Hubbard model)

15
(No Transcript)
16
Outline
Two atoms in a 1D Optical lattice close to a
Feshbach resonance

• Introduction
• Method
• Results
• Conclusions/Perspectives

17
Resonant molecules
Introduction
Schrödinger Equation
Scattering length
For
Bound State
if
Matching the two expressions
18
Resonant molecules
Introduction
more formal
with Green Function
Replace real interatomic potential with zero
range Pseudo-potential (Refs.)
where
if we choose
19
Tight Binding
Method
Large s, small E
Ansatz
d/acr
Only lowest band contributes
Width of Wannier function
20
Qualitative picture
Results
0