1. Cold Atomic and Molecular Collisions
ICAP Summer School 2006
University of Innsbruck
Cold Atomic and Molecular Collisions
1. Basics
2. Feshbach resonances
3. Photoassociation
Paul S. Julienne
Quantum Processes and Metrology Group
Atomic Physics Division
NIST
And many others, especially
Eite Tiesinga, Carl Williams, Pascal Naidon (NIST),
Thorsten Köhler (Oxford), Bo Gao (Toledo), Roman Ciurylo (Torun)

2. Cold Collisions
“Good” -- Essential interactions for control and measurement
“Bad” -- Source of trapped atom loss, heating, and decoherence
Atom-atom collisions can be quantitatively understood
and controlled--essential for quantum gas studies.
What is a scattering length,
and why is it significant?
Scattering resonances are a key to measurement and control.
Photoassociation
Magnetically tunable “Feshbach” resonances
Collisions in tightly confining atom traps.
Basic concepts, illustrated by examples.

3. Two kinds of collision
Elastic--do not change internal state a + b --> a + b
Thermalization
Evaporation
Mean field of BEC
BEC-BCS crossover in Fermi gases
Phase change--quantum logic gates
Inelastic--change internal state a + b --> a’ + b’ + E
Spin relaxation
Photoassociation
Loss of trapped atoms
Decoherence
Spinor condensates (E=0)

4. Cold atomic and molecular collision basics
Potential energy curves
Properties of the long range potential
Bound and scattering states
Collision cross sections and rates
Partial waves and cross sections
Elastic and inelastic collisions
Threshold properties of collisions and bound states
Boson and fermion differences
The effects of trap confinement on collisions
How are molecular collisions different

5. Cold neutral atomic gases
maybe not exhaustive ...

6. Ca 4s2 1S0 atom 1S0 + 1S0 Van der Waals 1g -C6/R6 Chemical bonding
Spin
degeneracy
Ca 4s2 1S0 atom
Total angular
momentum
Born-Oppenheimer approximation = potential energy curve
1S0 + 1S0
1g
Van der Waals
-C6/R6
Chemical bonding

7. Li 1s22s 2S1/2 atom
2S1/2 + 2S1/2
Similar for Na K Rb Cs
Potentials like
H2 molecule

8. Photoassociated atoms
1 pK
1 nK
1 K
1 mK
1 K
1000 K
106 K
109 K
Interior of sun
Molecules
Buffer gas cooling
Decellerated beams
Photoassociated atoms
Molecular BEC
Surface of sun
Room temperature
Outer space (3K)
Cold He
Laser cooled atoms
Atomic clock atoms
Fermionic quantum gases
A block of dry ice has a surface temperature of degrees F (-78.5 degrees C).
liquid nitrogen, which boils at -320 degrees F (-196 degrees C)
0 kelvin = degree Fahrenheit
0 kelvin = degree Celsius
Bose-Einstein condensates

9. E/kB 1 pK 1 nK 1 K 1 mK 1 K 1000 K 106 K 109 K = h/p 0.76 a0
momentum
0.76 a0
= 0.04 nm
6Li
x 0.2 for Cs
E/h
21 kHz =
21 MHz =
21 Hz =
24 a0
= 1.3 nm
760 a0
24000 a0
7.6x106 a0
= 40 nm
= 1.3 m
= 40 m

10. Characteristic Lengths
De Broglie wavelength
~ a0 (1 m)
Van der Waals length
a0 ( nm)
Chemical bond
< 20 a0 (< 1nm)
Scattering length
Trap size
> a0 (10 m)
Lattice: 1000 a0 (50 nm)
Interparticle spacing
2000 a0 (100nm) at 1015cm-3
20000 a0 (1000nm) at 1012cm-3

11. Collision of two atoms
Separate center of mass RCM and relative R motion with
reduced mass .
Expand (R,E) in relative angular momentum basis lm.
l = 0, 1, 2… s-, p-, d-waves, …
Potential energy:
--> phase shift
Neutral atoms (S-state): V(R) --> -C6/R6 van der Waals
Solve Schrödinger equation for bound and scattering (R,E)
--> bound states En
--> scattering phases, amplitudes

12. S-wave scattering phase shift
Wavelength 2/k
Noninteracting
atoms
Phase
Interacting
atoms

14. x0
k-independent
node

15. d v d Cross section  Classical balls with distance of
closest approach d (diameter) d v
d
define an area with
(10-12 cm2)
and a collision rate
= nv
(typical: 1 s-1, MOT
104 s-1, BEC)
Rate constant Kv 
Time between collisions = 1/= 1/(Kn) ≈ 1s (MOT), 100s (BEC)

16. Classical picture b No potential With potential b R E E R E=0
Interaction
region b R E
Centrifugal potential
Centrifugal barrier E R
E=0

17. q R Quantum scattering theory target source scattered wave
drops off as 1/R
source
target R q
detector
dA=R2d dW
cross section
atoms/s scattered flux into d
plane wave flux (atoms/cm2/s)

18. radial Schrödinger equation
Partial wave expansion
Expansion of a plane wave (Messiah, Quantum Mechanics, Vol.1, Appendix B.III):
Geometric
Dynamic
where
and is a solution to the
radial Schrödinger equation
Centrifugal
potential

19. radial Schrödinger equation
Add an interaction potential V(R)
where represents a plane + scattered wave:
and is a solution to the
radial Schrödinger equation

20. Centrifugal barrier collision energy E Expectation:
For small fixed E the cross
section has contributions from
a few partial waves or
atoms only collide via the
lowest few partial waves

21. Elastic cross section  for like Na atoms
Many (even) partial waves

22. Identical particle collisions
Identical atoms in same internal state a:
Bosons: even only
Fermions: odd only
Identical atoms in different internal states a,b:
Boson, Fermions: even and odd
+1, boson
-1, fermion
See Stoof, Koelman and Verhaar, Phys. Rev. B 38, 4688 (1988).
Jim Burke’s thesis

23. Collision cross section
Solve Schrödinger equation for each
Get phase shift
Identical bosons: even
Identical fermions: odd
Nonidentical species: all
van der Waals potential:

24. Threshold properties
Upper bound = 1 (unitarity limit)

25. Interpretation of scattering length
Na+Na 1 K
-100
100
200
Internuclear separation R (a )
A = -100 a
A = 0 a
A= +100 a Y
(R) x
3 different short-range potentials
with
3 different scattering lengths
Huang and Yang, Phys. Rev. 105, 767 (1957)
Also E. Fermi (1936), Breit (1947),
Blatt and Weisskopf (1952)

26. Particle (atom pair) in a box
 = reduced mass
Higher Energy
Lower Energy

27. A word about normalization
Energy-normalized:
Thus
e.g., energy width from Fermi Golden Rule matrix element:
Also, “classical time” normalization:
Probability in dR proportional to time in dR:

28. More semiclassical considerations
WKB phase-amplitude form:
Validity criterion:

29. Semiclassical considerations continued
Rvdw
Julienne and Mies, J. Opt. Soc. Am. B 89, 2257 (1989)
For R << Rvdw

31. x0
k-independent
node

33. Classical
turning
point
(1 mK)
Peak of
d-wave
Barrier
(5 mK) x0

34. -1 -0.5 0.5 10 20 30 40 50 60 Short-range Y (R) compared E/k = 1.4 m K
0.5 10 20 30 40 50 60
Short-range Y
(R) compared
E/k
= 1.4 m K B Y
(R)
d-wave
s-wave
s-wave amplitude =
d-wave amplitude = 5.07x10 -5
R(a )

37. Scattering and last bound state near threshold (normalized to same value at small R)x0 x0

38. Van der Waals potential
Write the Schrödinger equation in length and energy units of
The potential becomes
This potential has exact analytic solutions and many
useful properties.
B. Gao, Phys. Rev. A 58, 1728, 4222 (1998) + series of papers.
See Lett, Jones, Tiesinga, Julienne, Rev. Mod. Phys. 78, 483 (2006).

39. “Size” of potential V(R)
Gribakin and Flambaum, Phys. Rev. A 48, 546 (1993)

40. Noninteracting atoms
Interacting atoms