Download Julienne lecture notes July 12th

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)
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.
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)
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
Cold neutral atomic gases
maybe not exhaustive ...
Ca 4s2 1S0 atom
Spin
degeneracy
Total angular
momentum
Born-Oppenheimer approximation = potential energy curve
1S
0
1
g
+ 1S0
Van der Waals
-C6/R6
Chemical bonding
Li 1s22s 2S1/2 atom
2S
1/2
Potentials like
H2 molecule
+ 2S1/2
Similar for
Na
K
Rb
Cs
109 K
106
K
1000 K
1K
1 mK
1 K
1 nK
1 pK
Interior of sun
Surface of sun
Room temperature
Outer space (3K)
Cold He
Molecules
Buffer gas cooling
Decellerated beams
Laser cooled atoms
Photoassociated atoms
Atomic clock atoms
Fermionic quantum gases
Bose-Einstein condensates Molecular BEC
E/kB
momentum
109 K
= h/p
106 K
6Li
x 0.2 for Cs
1000 K
0.76 a0
= 0.04 nm
1K
24 a0
= 1.3 nm
21 MHz = 1 mK
760 a0
= 40 nm
21 kHz = 1 K
24000 a0
= 1.3 m
E/h
21 Hz = 1 nK
1 pK
7.6x106 a0 = 40 m
Characteristic Lengths
De Broglie wavelength
Van der Waals length
Chemical bond
h

p
1/ 4


1 2C
x 0   2 6 

2 


Scattering length
~ 20000 a0 (1 m)
30 - 100 a0 (1.5 - 5 nm)
< 20 a0 (< 1nm)
  A  
A
> 200000 a0 (10 m)
Trap size

Interparticle spacing

Lattice: 1000 a0 (50 nm)
2000 a0 (100nm) at 1015cm-3
20000 a0 (1000nm) at 1012cm-3
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
S-wave scattering phase shift
Wavelength 2/k
Noninteracting
atoms
Interacting
atoms
Phase
x0
k-independent
node
Cross section 
Classical balls with distance of
closest approach d (diameter)
v
d
define an area with d (10-12
2
cm )
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)
Classical picture
No potential
With potential
b
Interaction
region
Centrifugal barrier
Centrifugal potential
E
E
E=0
b
R
R
Quantum scattering theory
source
R
target
scattered wave
drops off as 1/R
q
d
dA=R2d
detector
cross section
atoms/s scattered flux into d
plane wave flux (atoms/cm2/s)
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
Add an interaction potential V(R)
where
represents a plane + scattered wave:
and
is a solution to the
radial Schrödinger equation
Centrifugal barrier
collision energy E
barrier
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
Elastic cross section  for like Na atoms
Many (even) partial waves
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 http://jilawww.colorado.edu/pubs/thesis/burke/
Collision cross section
Solve Schrödinger equation for each
Get phase shift
Identical bosons: even
Identical fermions: odd
Nonidentical species: all
van der Waals potential:
Threshold properties
Upper bound = 1 (unitarity limit)
Interpretation of scattering length
(a)
Na+Na 1 K
3 different short-range potentials
with
3 different scattering lengths
(R)
1 m
See below
0
10000
Internuclear separation R (a 0)
2
4


Vˆ 
A (R) R
m
R
sin[ k(R  A)]
k1/ 2
as R  
(R) 
20000
x0
(R)
A = -100 a 0
A = 0 a0

Huang and Yang, Phys. Rev. 105, 767 (1957)
A= +100 a 0
Also E. Fermi (1936), Breit (1947),
Blatt and Weisskopf (1952)
-100
0
100
200
Internuclear separation R (a0)
Particle (atom pair) in a box
 = reduced mass
2



E 
2 L A 
2
2



E  
2 L
Higher Energy
2
0 +A
L
2



E   +A 
2 L
2
0
L
-A 0
2




E  2  A 
L+
2
Lower Energy
L
  2 
L 

E per particle for N pairs
+A
mL3
N +A
L3 m
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:
More semiclassical considerations
WKB phase-amplitude form:
Validity criterion:
Semiclassical considerations continued
Rvdw
For R << Rvdw
Julienne and Mies, J. Opt. Soc. Am. B 89, 2257 (1989)
x0
k-independent
node
x0
Peak of
d-wave
Barrier
(5 mK)
Classical
turning
point
(1 mK)
Short-range (R) compared
0.5
E/k = 1.4 K
B
0
(R)
d-wave
s-wave
-0.5
s-wave amplitude = 0.1418
d-wave amplitude = 5.07x10
-1
0
10
20
-5
30
R(a )
0
40
50
60
Scattering Length vs Control Parameter
400
200
As (Bohr) 0
-200
-400
9900
10000
10100 10200 10300
Control parameter
10400
10500
x
0
0.0
Energy of last bound state (GHz)
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
-1000
-500
0
500
Scattering Length A (a )
0
1000
2.0
Scattering and
last bound state
near threshold
(normalized to
same value at
small R)
A = +101 a 0
1.5
Continuum, 1.4 K
1.0
x0
0.50

0.0
-0.50
Bound
-1.0
-1.5
0
50
100
R
150
200
150
200
2.0

1.5
A s = -96 a0
1.0
x0
Continuum, 1.4 K
0.50
0.0
Bound
-0.50
-1.0
-1.5
0
50
100
R
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).
“Size” of potential V(R)
Gribakin and Flambaum, Phys. Rev. A 48, 546 (1993)
Rvdw(a 0 ) Evdw(mK) (MHz)
6
Li
40
K
85
Rb
133
Cs
31
65
83
101
29
1.0
0.35
0.13
614
21
7.3
2.7
Noninteracting atoms
Interacting atoms