ULTRACOLD COLLISIONS IN THE PRESENCE OF TRAPPING POTENTIALS ZBIGNIEW IDZIASZEK Institute for Quantum Information, University of Ulm, 18 February 2008 Institute for Theoretical Physics, University of Warsaw and Center for Theoretical Physics, Polish Academy of Science
31
Embed
ULTRACOLD COLLISIONS IN THE PRESENCE OF TRAPPING POTENTIALS ZBIGNIEW IDZIASZEK Institute for Quantum Information, University of Ulm, 18 February 2008 Institute.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
ULTRACOLD COLLISIONS IN THE
PRESENCE OF TRAPPING POTENTIALS
ZBIGNIEW IDZIASZEK
Institute for Quantum Information,University of Ulm, 18 February 2008
Institute for Theoretical Physics, University of Warsaw
and
Center for Theoretical Physics, Polish Academy of Science
OutlineOutline
1. Binary collisions in harmonic traps
- collisions in s-wave
- collisions in higher partial waves
3. Scattering in quasi-1D and and quasi-2D traps
- confinement –induced resonances
2. Energy dependent scattering length
4. Feshbach resonances
SystemSystem
1. Ultracold atoms in the trapping potential
Typical trapping potentials are harmonic close to the center
magnetic traps, optical dipole traps, electro-magnetic traps for charged particles, ...
Interactions can be modeled via contact pseudopotential
2. Characteristic range of interaction R* << length scale of the trapping potential
- Very accurate for neutral atoms
- Not applicable for charged particles, e.g. for atom-ion collisions
222222
2
1)( zyxmV zyxT r
R*
trap size
CM and relative motions can be separated in harmonic potential
Axially symmetric trap:
Contact pseudopotential for s-wave scattering (low energies):
Hamiltonian (harmonic-oscillator units)
Two ultracold atoms in harmonic trapTwo ultracold atoms in harmonic trap
length unit: energy unit:
Relative motion
We expand into basis of harmonic oscillator wave functions
Contact pseudopotential affects only states with mz=0 and k even
(non vanishing at r=0 )
radial:
axial:
For mz0 or k odd trivial solution:
Two ultracold atoms in harmonic trapTwo ultracold atoms in harmonic trap
Integral representation can be obtained from:
Eigenenergies:
Eigenfunctions:
Substituting expansion into Schrödinger equation and projecting on
Two ultracold atoms in harmonic trapTwo ultracold atoms in harmonic trap
Energy spectrum for = 5 Energy spectrum for = 1/5
Two ultracold atoms in harmonic trapTwo ultracold atoms in harmonic trap
Energy spectrum in cigar-shape traps ( > 1)
Energy spectrum in pancake-shape traps ( < 1)
For
Z.I., T. Calarco, PRA 71, 050701 (2005)
For
T. Stöferle et al., Phys. Rev. Lett. 96, 030401 (2006)
Bound state for positive and negative energies due to the trap
Comparison of theory vs. experiment: atoms in optical lattice
T. Bush et al., Found. Phys. 28, 549 (1998)
• solid line – theory (spherically symmetric trap)
• points – experimental data
-10 -5 0 5 10-2
0
2
4
6
8
En
erg
y [E
/]
a/aHO
Two ultracold atoms in harmonic trapTwo ultracold atoms in harmonic trap
)(
)(
23
21
E
E
a
dmd
Energy spectrum and wave functions for
very elongated cigar-shape trap
Energy spectrum for = 100
exact energies
1D model + g1D
First excited state
Elongated in the direction of weak trapping
Size determined by the strong confinement
Wave function is nearly isotropic
Trap-induced bound state (a < 0)
Two ultracold atoms in harmonic trapTwo ultracold atoms in harmonic trap
Dip in the center due to the strong interaction
Identical fermions can only interact in odd partial waves (l = 2n+1)
Hamitonian of the relative motion:
Two ultracold atoms in harmonic trapTwo ultracold atoms in harmonic trap
Energy spectrum for = 1/10Energy spectrum for = 1/10
Two ultracold fermions in harmonic trap
No interactions in higher partial waves at E0 (Wigner threshold law)12~tan l
l k
Scattering for l > 0 can be enhanced in the presence of resonances Feshbach resonances
Fermi pseudopotential - applicable for: k R* 1, k a 1/ k R*
s-wave scattering lenght:
In the tight traps (large k) or close to resonances (large a)