Download (Ultra-)Cold Molecules (Ultra

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1- Introduction, overview
2- Hamiltonian of a diatomic molecule
3- Molecular symmetries; Hund’s cases
4- Molecular spectroscopy
5- Photoassociation of cold atoms
6- Ultracold (elastic) collisions
Olivier Dulieu
Predoc’ school, Les Houches,september 2004
How to create ultracold molecules
using laser cooling?
Laser cooling of atoms:
closed level-scheme
Laser cooling of molecules:
NO closed level-scheme
One proposal
• Based on the development of a Multiple Single Frequency Laser
• Sequential cooling on electronic transitions: R,T,V
• Simulation on Cs2 B1PuX, with chirped frequencies
One proposal
• Based on the development of a Multiple Single Frequency Laser
• Sequential cooling on electronic transitions: R,T,V
• Simulation on Cs2 B1PuX, with chirped frequencies
One proposal
• Based on the development of a Multiple Single Frequency Laser
• Sequential cooling on electronic transitions: R,T,V
• Simulation on Cs2 B1PuX, with chirped frequencies
One exception?
• Direct laser cooling of BeH, CaH, at Los
Alamos
• Alkaline-earth
hydrides
have
Rydberg
transitions similar to the D1, D2 lines in alkali
atoms (good spectral isolation), with almost
diagonal FC factors matrix (99%)
• BeH: theoretical benchmark for open-shell
molecules
• CaH/CaD: degenerate quantum gases
One Solution: cold atom photoassociation
First
discussion
Ultracold
molecule!!
First
steps
First
observations
Ultracold
molecule!!
First
reviews
PA well-known at thermal energies:
diffuse bands
A  A    A2*
From Stwalley&Wang, J. Mol. Spectrosc. 195, 194 (1999)
PA at ultracold energies
A(ns)  A(ns)  h L  A2* ((ns  np j ; v, J ))
Ultracold
Excited
Short-lived
molecules
Energy balance
EPA (v, J )  2 E g  h L  E  EDoppler  Erecoil
detuning
   A  L
h L  Eb (v, J )  E  EDoppler  Erecoil
200 cm-1
@300K
10-4 cm-1
@100mK
Free-bound transition
= quasibound-bound transition
Stwalley&Wang, J. Mol. Spectrosc. 195, 194 (1999)
PAS of
cold Cs
Trap
loss
REMPI
Detection of PA
Ex: Cs
REMPI
TRAP
LOSS
Ultracold molecules
Ex:Na
11 years of PA observations (1993-2004)
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Li2: Hulet (Rice,US), Zimmerman (Tübingen, D)
Na2: Lett (NIST, US), VanderStraten (Utrecht, NL)
K2: Gould, Stwalley (Storrs, US)
Rb2: Heinzen (Austin, US), Gabbanini (Pisa, I)
Cs2: Pillet (Orsay, F), Stwalley (Storrs, US)
H2: Walraven (Amsterdam, NL)
He2: Leduc, Cohen-Tannoudji (Paris, F)
Ca2: Tiemann, Riehle (Hannover/Braunschweig, D)
Yb2: (Tokyo, JP)
RbCs: DeMille (Yale, US)
KRb: Marcassa, Bagnato (São Carlos, BR), Stwalley (Storrs, US)
NaCs: Bigelow (Rochester, US)
Sr2: (Boulder, US)
In progress: LiCs (Freiburg, D)….
Also: PA in condensates
PA: Probe of the long-range part
of molecular potentials
Long-range interactions between neutral atoms
Multipolar expansion (in 1/R) of electrostatic interaction:
 
   
d .d  3(d1.n )( d 2 .n )
Vd  d ( R)  1 2
R3
Stwalley&Wang, J. Mol. Spectrosc. 195, 194 (1999)
Le Roy-Bernstein approach
LeRoy&Bernstein, J. Chem.Phys. 52, 3869 (1970)
How to make the link between observed transitions and long-range
behavior of the potential?
2m
1
v 
2


R2 ( v )
R1 ( v )
dREv  V ( R)


h2n

Ev  D  K n 
n 2 
 (2m) Cn 
n  6 : Ev  vD  v 
1/ 2
1
n2
vD  v 
2n
n2
Cn
V ( R)  D  n
R
 (n  2)(1  1 n) 

K n  
 2(1 2  1 n) 
(fractional) vibrational quantum number
at the dissociation limit
3
n  3 : Ev  vD  v 
6
-No solution for n=2
-Limited to a single potential
-Rotation ( 1/R2) not included
Accumulated phase method:
Numerical approach for higher flexibility
Moerdjik et al, PRA 51, 4852 (1995)
Almost constant phase F(R0) at
this point R0 for all upper lying
vibrational levels
If:
-A single level is known
-The asymptotic potential is known
Inward integration of the Schrödinger
equation down to R0, with limit condition
on the logarithmic derivative of F(R0)
Fitting strategy:
F( R0 )  F 0  F E ( D  E )  F J J ( J  1)  ...
Parameters: F( R0 ), D, Cn
Scattering length
Crubellier etal, Eur. Phys. J. D, 6, 211 (1999)
Pure long-range molecules (1)
Pure long-range molecules (2)
0g (ns  np3/ 2 )
R-3
R-3R-6, R-8
Quantum chemistry
Spies, 1989
R-3R-6, R-8+exchange
0g (6s  6 p3 / 2 )
The 0g- pure long-range state (1)
V P ( R)   PP ( R)  P ( R) 

V  

 P ( R)
V ( R) 

The 0g- pure long-range state (2)
Hund’s case (a) representation
A
2
A
2
V ( R)   PP (X
R)  P (X
R) 

V  



(
R
)
V
(
R
)
P X


P
C3P C6P C8P
P
V ( R)  3  6  8  ...  Vexch
( R)
R
R
R
P
At large distances:
-Atomic spin-orbit A  2 fs 3
-Asymptotic expansion of V
C3 C6 C8

V ( R)  2 3  6  8  ...  Vexch
( R)
R
R
R

The 0g- pure long-range state (3)
Diagonalization of the spin-orbit matrix
Hund’s case (c) representation
Attractive potential
1/R3
 A

 2
 A

 2
A 

2
0 

A  2 fs 3
A 1 P

2 
2 
P
  V ( R)  V ( R)
V ( R)  V ( R) 
3
3

V  2 3
 2 

2 P
1 
P
V ( R)  V ( R)
 A  V ( R)  V ( R) 

3
3
 3





interaction
1/R3
Flat potential
1/R6
The 0g- pure long-range state (4)
 C C 
C3 1  2
8C32
V (0 [ s  p3 / 2 ])  E ([ s  p3 / 2 ])  3 
 ( R)  6   ( R)   66 , 88 ,...
 R R 
R 2 
R


g
R   :  ( R) 
3 A C3
 ( R) 
 3
2 R
3A
2
2
C
4
C
3
V (0 g [ s  p3 / 2 ])  E ([ s  p3 / 2 ])  33 
R 3 AR 6
attractive
Potential
well
when  ( R)  0
V (0 g [ s  p3 / 2 ])  E ([ s  p3 / 2 ])  ( 2  1)
C3
R3
repulsive
PAS of the 0g- pure long-range state in Cs2 (1)
• PAS spectrum: 75 vibrational levels, J=2
• Direct Potential Fit approach:
Amiot et al, PRA 66, 052506(2002)
 V P ( R)   PP ( R)
 P ( R)   rel (C3 ; R) 

V  



(
R
)


(
C
;
R
)
V
(
R
)
rel
3
 P

/P
/P
/P
• 9 Fitting parameters C3 , C6 , C8 ,  PP ( R),  P ( R),  rel ( R), Vexch ( R)
• minimization
 1
 
 N  M
 ycalc (i )  yobs (i ) 



u (i )
i 1 

N
2 1/ 2



PAS of the 0g- pure long-range state in Cs2 (2)
asymptotic
RKR
Quantum
chemistry
Atomic radiative lifetime from PAS
Amiot et al, PRA 66, 052506(2002)
Non-relativistic

3
6s r 6 p
C
C 

2
3
P
3
2

3
4 6 p 6 s
  
 
 2 
3
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Cold molecule formation processes
Main requirement: stabilization of the excited population in a bound
state
Solution: « R »-transfer of the probability density
« not efficient » case
Observed in:
Na2, K2, KRb, NaCs
Double-well case
Observed in:
Cs2, Rb2
Resonant coupling
Observed in:
Cs2, RbCs,KRb
Double-well process
in Cs2
REMPI
PA
SE
PA and cold molecule formation in Cs2
REMPI spectra
Varying the REMPI
laser frequency
Dion et al, EPJD 18, 365 (2002)
Varying the PA laser frequency
Predicted vibrational population in the lowest
3 + state, after decay of 0 - PA levels in Cs
u
g
2
Vibrational level
Of the
a3u+ state
Detuning of the 0g- PA level
Resonant coupling process (1)
C. M. Dion et al, PRL 86, 2253 (2001)
Resonant coupling process (2)
Resonant coupling process (3)
Next resonance
PA rates, shifts, line shapes: references
(non exhaustive)
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Thorsheim et al, PRL 58, 2420 (1987)
Napolitano et al, PRA 73, 1352 (1994)
Julienne, J. Research NIST 101, 487 (1996)
Pillet et al, JPB 30, 2801 (1997)
Côté & Dalgarno, PRA 58, 498 (1998)
Javanainen & Mackie, PRA 58, R789 (1998)
Bohn& Julienne, PRA 60, 414 (1999)
Mackie & Javanainen, PRA 60, 3174 (1999)
Jones et al, PRA 61, 012501 (1999)
Drag et al, IEEE J. Quantum Electronics 36, 1378 (2001)
Montalvão & Napolitano, PRA 64, 011403(R) (2001)
C. M. Dion et al, PRL 86, 2253 (2001)
Dion et al, EPJD 18, 365 (2002)
Simoni et al, PRA 66, 063406 (2002)
A short tutorial on Feshbach resonances
• Resonance: a bound state embedded in a continuum
• Shape resonance, Feshbach resonance
Collision in channel i with a resonance
Tuning the scattering length
Moerdjik et al,PRA 51, 4852 (1995)
Bibliography
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« Interactions in ultracold gases: from atoms to molecules », ed. by M.
Weidemüller and C. Zimmermann, Wiley VCH (2003); nice collection of
tutorials and research papers from a workshop and training school held
in Heidelberg in 2002, in the framework of the EU Network « Cold
Molecules »
J.T. Bahns, P.L. Gould, W.C. Stwalley, Adv. At. Mol. Opt. Physics 42, 171
(2000)
F. Masnou-Seeuws, P. Pillet, Adv. At. Mol. Opt. Physics 47, 53 (2001)
O. Dulieu, F. Masnou-Seeuws, JOSA B, (2003)