INTRODUCTION TO PHYSICS OF ULTRACOLD COLLISIONS ZBIGNIEW IDZIASZEK Institute for Quantum Information, University of Ulm, 14 February 2008 Institute for Theoretical Physics, University of Warsaw and Center for Theoretical Physics, Polish Academy of Science
Institute for Quantum Information, University of Ulm, 14 February 2008. INTRODUCTION TO PHYSICS OF ULTRACOLD COLLISIONS. ZBIGNIEW IDZIASZEK. Institute for Theoretical Physics, University of Warsaw and Center for Theoretical Physics, Polish Academy of Science. Outline. - PowerPoint PPT Presentation
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INTRODUCTION TO PHYSICS OF ULTRACOLD COLLISIONS
ZBIGNIEW IDZIASZEK
Institute for Quantum Information,University of Ulm, 14 February 2008
Institute for Theoretical Physics, University of Warsaw
andCenter for Theoretical Physics, Polish Academy of Science
Outline
1. Characteristic scales associated with ultracold collisions
2. Wigner threshold laws
3. Scattering lengths and pseudopotentials
4. Quantum defect theory
5. Resonance phenomena:
- shape resonances
- Feshbach resonances
J. Weiner, V.S. Bagnato, S. Zilio, and P.S. Julienne, Rev. Mod. Phys. 71, 1 (1999)
(Ultra)cold atomic collisions
cold collisions
ultracold collisions
Typical interaction potential
long-range part: dispersion forces n
n
rCrV ~)(
- neutral atoms, both in S state: van der Waals interaction, n = 6
- atom in S state-charged particle (ion): polarization forces, n = 4
- neutral atoms with dipole moments dipole-dipole interaction, n = 3
V(r)
r
short-range part: chemical binding forces
centrifugal barrier: )1(2 2
2
llr
Long-range dispersion forces
At E0 (close to the threshold) scattering properties are determined by the part of the potential with the slowest decay at r
nn
rCrV ~)(
Characteristic scales
)( r
Length scale:
Energy scale:
Typical range of the potential
Height of the centrifugal barrier, determines contribution of higher partial waves
For EE* only s-wave (l = 0) collisions
Characteristic scales
Example values of R* and E* for different kinds of interactions
R*(a0) E*(mK) 6Li 31 29 40K 65 1.0 85Rb 83 0.35
Neutral atoms in S states (alkali) R* (a0) E* (K) 40Ca++ 87Rb 3989 0.198 9Be++ 87Rb 2179 2.23 40Ca++ 23Na 2081 1.37
Atom(S)-ion (alkali atom-alkali earth ion)
consequences for collisions in traps
• R* for atom-atom << size of the typical trapping potentials
• E* for atom-ion is 103 lower than for atom-atom
higher partial waves (l > 0) not negligible for ultracold atom-ion collisions (~K), whereas negligible for atom-atom collisions
• R* for atom-ion ~ size of the trapping potentials (rf + optical traps)
Knowledge of the scattering phases at a single value of energy allows to determine the scattering properties + position of bound states at different energies
3) Quantum-defect functions
Can be found analytically for inverse power-law potentials
Deep potential, wave function weakly depends on E
Shallow potential, wave function strongly depends on E
r>>R*
Quantum-defect theory of ultracold collisions
),(ˆ),(ˆ
ErgErf
),(),(),(
ErErgErf
R*
Rmin
Solutions with WKB-like normalization at small distances
Solutions with energy-like normalization at r
Analytic across threshold!
Non-analytic across threshold!
Linearly independent solutions of the radial Schrödinger equation
02
)1()(2 2
2
2
22
rEr
llrVr l
For large energies when semiclassical description becomes
applicable at all distances, two sets of solutions are the same
Quantum-defect theory of ultracold collisions
QDT functions connect f,ĝ with f,g,
Physical interpretation of C(E), tan (E) and tan (E):
In WKB approximation, small distances (r~Rmin)
For E, semiclassical description is valid at all distances
C(E) - rescaling
(E) and (E) – shift of the WKB phase
For E0, analytic behavior requires
Quantum-defect theory of ultracold collisions
Expressing the wave function in terms of f,ĝ functions
very weakly depends on energy: constE )(
QDT functions relates to observable quantities, e.g. scattering matrices
The same parameter predicts positions of the bound states
- QDT parameter (short-range phase)
Example: energies of the atom-ion molecular complex
Solid lines:quantum-defect theory for independent of E i l
Points:numerical calculations for ab-initio potentials for 40Ca+ - 23Na
Ab-initio potentials:O.P. Makarov, R. Côté, H. Michels, and W.W. Smith, Phys.Rev.A 67, 042705 (2005).