Download Document

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
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
(Ultra)cold atomic collisions
cold collisions
ultracold collisions
J. Weiner, V.S. Bagnato, S. Zilio, and P.S. Julienne,
Rev. Mod. Phys. 71, 1 (1999)
Typical interaction potential
V(r)
2
centrifugal barrier:
l (l  1)
2r 2
r
long-range part:
dispersion forces
V (r ) ~ 
Cn
rn
- neutral atoms, both in S state:
van der Waals interaction, n = 6
- atom in S state-charged particle (ion):
polarization forces, n = 4
short-range part:
chemical binding forces
- neutral atoms with dipole moments
dipole-dipole interaction, n = 3
Long-range dispersion forces
At E0 (close to the threshold) scattering properties are determined by the part of the
potential with the slowest decay at r
V (r ) ~ 
Cn
rn
( r  )
Characteristic scales
Length scale:
Typical range of the potential
Energy scale:
Height of the centrifugal barrier, determines contribution of higher partial waves
For EE* only s-wave (l = 0) collisions
Characteristic scales
Example values of R* and E* for different kinds of interactions
Neutral atoms in S states (alkali)
6
Li
40
K
85
Rb
R*(a0)
31
65
83
E*(mK)
29
1.0
0.35
Atom(S)-ion (alkali atom-alkali earth ion)
40
Ca++ 87Rb
9
Be++ 87Rb
40
Ca++ 23Na
R* (a0)
3989
2179
2081
E* (K)
0.198
2.23
1.37
• R* for atom-atom << size of the typical trapping potentials
• R* for atom-ion ~ size of the trapping potentials (rf + optical traps)
 consequences for collisions in traps
• 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
Partial-wave expansion and phase shifts
 2




V
(
r
)

E

 r   0
 2

 2k 2
E
2
V(r)
Partial wave expansion r    Rl (r )Ylm  ,  
lm
 (r )
 2 2

 2l (l  1)

V
(
r
)


E

 Rl r   0
2
2
2


r
2

r


r
At large distances
Rl (r )  A jl (kr)  tan  l nl (kr) ~ sin( kr  l 2   l )
without potential: l=0
attractive potential: l > 0
repulsive potential: l < 0
Threshold laws for elastic collisions
Behavior of cross-sections at E0
Wigner threshold laws for short-range potentials
E. Wigner Phys. Rev. 73, 1002 (1948)
 l 1
decays faster than 1/rn
E 0
l (r )  A1r  A2 r
E0
l (r )  jl (kr)  tan  l nl (kr) (r  )
l
( r  )
Example: Yukawa potential
Smooth and continuous matching
For k  0
tan  l 
l (r )  kr  tan  l kr
l
A2 2l 1
k
~ k 2l 1
A1
l (r , k )  k l l (r , k  0)
Cross section for partial wave l σ l ~ k 4l
l 1
(r  )
Threshold laws for elastic collisions
Long-range dispersion potentials
First-order Born approximation
f l (k )  
mCn  n2  12  l  n2  32 
2
2

n
2

   l 
n
2
1
2
k n 3
(Landau-Lifshitz, QM)
For 2l < n-3 Wigner threshold law is preserved
tan  l ~ k 2l 1
For 2l >n-3 long-range contribution dominates
tan  l ~ k n 2
V (r ) ~ 
Cn
rn
e i 2 l  1
fl 
2ik
Exact treatment
 d 2 Cn l (l  1)
2
  2  n 
l (r , k )  0

k
2
r
r
 dr

Analytical solution at E=0
l(1) r,0  r J ( x)
l( 2) r,0  r N ( x)
  2l  1 /( n  2)
x
Cn
2
n  2 r n2
tan  l ~ c1k n  2  c 2 k 2l 1
Special case n=3 tan  l ~ k ln k
Scattering length
For l=0 Wigner threshold law: tan  0 ~ k
Scattering length
Physical interpretation:
 tan  0 (k ) 
a  lim  

k 0
k


R0 (r )  r  a
repulsive
attractive
Potential without bound states
Scattering length
Each time new bound state enters the potential a diverges and changes sign
a(V0)
10
V(r)
7.5
5
R0
2.5
r
20
40
60
80
V0
-2.5
V0
-5
-7.5
Higher partial waves
In the Wigner threshold regime tan  l ~ k 2l 1 ( 2l  n  3)
l-wave Scattering length
For p-wave
al 1 3  V
 tan  l (k ) 
al 2l 1  lim


2 l 1
k 0

k
- scattering volume

Pseudopotentials
At very low energies only s-wave scattering is present
Total cross-section:
de Broglie wavelength
 (k )  4a 2 - depends on a single parameter

2
 R0 range of the potential
k
particles do not resolve details of the potential
shape independent approximation
J. Weiner et al. RMP 71 (1999)
Pseudopotentials
Fermi pseudopotential
regularization operator
V (r) 
(removes divergences of the 3D wave function at r0)
4 a

 (r) r
m
r
2
E. Fermi, La Ricerca Scientifica, Serie II 7, 13 (1936)
V(r)
Asymptotic solution
as (r )
 (r )
R0
as (r )
r
Pseudopotential
Pseudopotential supports single bound state for a>0
Ebind
2

2ma 2
(r) ~ e r a
Correct for a weakly bound state with E<<E*
Pseudopotentials
Generalized pseudopotential for all partial waves
K. Huang & C. N. Yang, Phys. Rev. 105, 767 (1957)
Correct version of Huang & Yang potential:
l-wave scattering length
R. Stock et al, PRL 94, 023202 (2005)
A. Derevianko, PRA 72, 044701 (2005)
ZI & TC, PRL 96, 013201 (2006)
For particular partial waves it can be simplified ...
Pseudopotential for p-wave scattering
Pseudopotential for d-wave scattering
Pseudopotentials
Test: square-well potential + harmonic confinement
V(r)
R0
r
Pseudopotential method valid for
V0
Scattering volume
Energy spectrum for R0=0.01d
Energy spectrum for R0=0.2 d
Quantum-defect theory of ultracold collisions
Seaton, Proc. Phys. Soc. London 88, 801 (1966)
Green, Rau and Fano, PRA 26, 2441 (1986)
Mies, J. Chem. Phys. 80, 2514 (1984).
Asymptotic behavior, the same as for the real physical potential
1) Reference potential(s)
V (r )  
Cn
 E
rn
( r  )
Arbitrary at small r (model potential)
2) Quantum-defect parameters
Characterize the behavior of the wave function
at small distances (~Rmin)
R*
Shallow potential, wave function
strongly depends on E
Independent of energy for a wide range of
kinetic energies
3) Quantum-defect functions
r>>R*
Rmin Deep potential, wave function
weakly depends on E
Scattering phases (r~)  quantum defect parameters (r~Rmin)
Can be found analytically for inverse power-law potentials
Knowledge of the scattering phases at a single value of energy allows to determine the
scattering properties + position of bound states at different energies
Quantum-defect theory of ultracold collisions
Linearly independent solutions of the radial Schrödinger equation
 2 2

 2l (l  1)


V
(
r
)


E

 l r   0
2
2
2r
 2 r

fˆ (r , E )
gˆ (r , E )
Solutions with WKB-like normalization
at small distances
f (r , E )
g (r , E )
 (r , E )
Analytic across threshold!
R*
Solutions with energy-like
normalization at r
Rmin
For large energies when semiclassical description becomes
applicable at all distances, two sets of solutions are the same
Non-analytic across threshold!
Quantum-defect theory of ultracold collisions

QDT functions connect f,ĝ with f,g,
In WKB approximation, small distances (r~Rmin)
Physical interpretation of C(E), tan (E) and tan (E):
C(E) - rescaling
(E) and (E) – shift of the WKB phase
For E, semiclassical description is
valid at all distances
For E0, analytic behavior requires
Quantum-defect theory of ultracold collisions

Expressing the wave function in terms of f,ĝ functions
 - QDT parameter (short-range phase)
 very weakly depends on energy:
 ( E )  const
QDT functions relates  to observable quantities, e.g. scattering matrices
The same parameter predicts positions of the bound states
Quantum-defect theory of ultracold collisions
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).