Experimental study of universal few-body physics with ultracold atoms
INTRODUCTION TO PHYSICS OF ULTRACOLD COLLISIONS
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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 and Center for Theoretical Physics, Polish Academy of Science
description
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
Transcript of INTRODUCTION TO PHYSICS OF ULTRACOLD COLLISIONS
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)
Partial-wave expansion and phase shifts
02
)1()(2 2
2
2
22
rREr
llrVr l
0)(2
2
rErV
2
22kE
lm
lml YrR ,)(rPartial wave expansion
At large distances
r
V(r)
)(r
)sin(~)(tan)()( 2 lllll lkrkrnkrjArR
without potential: l=0
attractive potential: l > 0
repulsive potential: l < 0
Threshold laws for elastic collisions
decays faster than 1/rn
Smooth and continuous matching
Example: Yukawa potential
Wigner threshold laws for short-range potentialsE. Wigner Phys. Rev. 73, 1002 (1948)
)()(0 121 rrArArE ll
l
)()(tan)()(0 rkrnkrjrE llll
)(tan)(0For 1 rkrkrrk ll
ll
1212
1
2 ~tan lll kk
AA
)0,(),( krkkr ll
l
Cross section for partial wave l ll kσ 4~
Behavior of cross-sections at E0
Long-range dispersion potentials
First-order Born approximation
nn
rCrV ~)(
3
21
2
23
2
2
21
222
)(
nn
n
n
nn
l kl
lmCkf
(Landau-Lifshitz, QM)
For 2l < n-3 Wigner threshold law is preserved 12~tan ll k
2~tan nl k ik
efli
l 212
For 2l >n-3 long-range contribution dominates
Exact treatment
0),()1( 222
2
krk
rll
rC
drd
lnn
Analytical solution at E=0
)(0,)1( xJrrl
)(0,)2( xNrrl
)2/(12 nl
222
n
n
rC
nx
Threshold laws for elastic collisions
122
21 cc~tan ln
l kk
Special case n=3 kkl ln~tan
Scattering length
For l=0 Wigner threshold law: k~tan 0
Physical interpretation:
Potential without bound states
attractiverepulsive
Scattering length
kka
k
)(tanlim 0
0
arrR )(0
Scattering length
Higher partial waves
l-wave Scattering length
120
12 )(tanlim ll
k
ll k
ka
For p-wave - scattering volume Val 3
1
In the Wigner threshold regime 12~tan ll k )32( nl
Each time new bound state enters the potential a diverges and changes sign
V(r)
r
R0
V0
20 40 60 80
-7.5
-5
-2.5
2.5
5
7.5
10
V0
a(V0)
Pseudopotentials
At very low energies only s-wave scattering is present
Total cross-section: 24)( ak
de Broglie wavelength 02 Rk
range of the potential
particles do not resolve details of the potential shape independent approximation
- depends on a single parameter
J. Weiner et al. RMP 71 (1999)
Fermi pseudopotential
rrm
aV
)(4)(2
rr E. Fermi, La Ricerca Scientifica, Serie II 7, 13 (1936)
2
2
2maEbind
are ~(r)
regularization operator
(removes divergences of the 3D wave function at r0)
Pseudopotential supports single bound state for a>0
Correct for a weakly bound state with E<<E*
Pseudopotentials
r
V(r)
)(r )(ras
)(ras
R0 Pseudopotential
Asymptotic solution
Pseudopotentials
Generalized pseudopotential for all partial waves K. Huang & C. N. Yang, Phys. Rev. 105, 767 (1957)
Correct version of Huang & Yang potential:
R. Stock et al, PRL 94, 023202 (2005)A. Derevianko, PRA 72, 044701 (2005)
ZI & TC, PRL 96, 013201 (2006)
l-wave scattering length
For particular partial waves it can be simplified ...
Pseudopotential for d-wave scattering
Pseudopotential for p-wave scattering
Test: square-well potential + harmonic confinement
V(r)
r
R0
V0
Energy spectrum for R0=0.01d
Scattering volume
Energy spectrum for R0=0.2 d
Pseudopotential method valid for
Pseudopotentials
Quantum-defect theory of ultracold collisions
R*
Rmin
Seaton, Proc. Phys. Soc. London 88, 801 (1966)Green, Rau and Fano, PRA 26, 2441 (1986)
Mies, J. Chem. Phys. 80, 2514 (1984).
1) Reference potential(s)
Asymptotic behavior, the same as for the real physical potential
Arbitrary at small r (model potential)
)()( rErCrV n
n
2) Quantum-defect parameters
Characterize the behavior of the wave function at small distances (~Rmin)
Independent of energy for a wide range of kinetic energies
Scattering phases (r~) quantum defect parameters (r~Rmin)
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).
Quantum-defect theory of ultracold collisions
