TFVS Two atoms in a 1D Optical Lattice
close to a Feshbach Resonance M. Wouters NQS2005, Camerino, july 6th, 2005 In collaboration with
G. Orso, L.P. Pitaevskii, S. Stringari
TFVS Two atoms in a 1D Optical Lattice
close to a Feshbach Resonance M. Wouters NQS2005, Camerino, july 6th, 2005 In collaboration with
G. Orso, L.P. Pitaevskii, S. Stringari
Motivation
Introduction Tunable interactions External periodic potentials Experiments with ultra-cold atoms: two-body physics in optical lattices ? M. Greiner, C. A. Regal, D.S. Jin, Proceedings of ICAP-2004 (Rio de Janeiro) condmat/0502539
Periodic Potential
Introduction Center of mass and relative motion don’t decouple
No ‘nice’ analytical wave functions d z1 r Recoil energy: sER z2 G. Orso, L.P. Pitaevskii, S. Stringari, M.W., cond-mat/0503096, accepted to PRL
Harmonic confinement
Introduction Harmonic confinement in z-direction:
separation of relative and c.om. motion lz 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
Integral equation
No separation of center of mass
and relative motion where is periodic Discrete translational symmetry Q is quasi-momentum of the molecule Method takes regular part Bethe-Peierls
Green’s function
Method Handle the singularity: Independent of energy and external potential Numerically
Tight Binding Analytically
Qualitative picture
Quasi-2D molecule
For large s h.o. Tightly bound molecule
Lattice = perturbation Results 0 Effective 3D situation
with anisotropic mass
Binding energy
Results Q = 0 S=20 10 5 0
Critical scattering length
Results Q = 0 Q = qB
Binding energy at resonance
Results Q = 0
Binding energy dispersion
Results Center of mass
motion and relative
motion are coupled Binding energy depends on the quasi-momentum d/|acr| increases with
quasi-momentum S=2.5
Band width dispersion
Results The bandwidth depends strongly on the scattering length (binding energy). Possible to extract experimentally from Bloch oscillations
Effective mass dispersion
Results Depends also strongly on the scattering length (binding energy). Possible to extract experimentally from Bloch/dipole oscilations
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)
Outline
Introduction
Method
Results
Conclusions/Perspectives Two atoms in a 1D Optical lattice
close to a Feshbach resonance
‘Resonant’ molecules
Introduction Scattering length For Schrödinger Equation : Bound State Matching the two expressions : if
Resonant molecules
Introduction with Green Function more formal Replace real interatomic potential with zero range
Pseudo-potential (Refs.) where if we choose
Tight Binding
Method Ansatz : Large s, small E d/acr Only lowest band contributes Width of Wannier function
Qualitative picture
Tightly bound molecule
Lattice = perturbation Quasi-2D molecule
For large s h.o. Results 0 Effective 3D situation
with anisotropic mass
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