![Page 1: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/1.jpg)
• 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) collisionsOlivier Dulieu
Predoc’ school, Les Houches,september 2004
![Page 2: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/2.jpg)
How to create ultracold molecules using laser cooling?
Laser cooling of molecules:NO closed level-scheme
Laser cooling of atoms: closed level-scheme
![Page 3: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/3.jpg)
One proposal
• Based on the development of a Multiple Single Frequency Laser• Sequential cooling on electronic transitions: R,T,V
• Simulation on Cs2 B1uX, with chirped frequencies
![Page 4: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/4.jpg)
One proposal
• Based on the development of a Multiple Single Frequency Laser• Sequential cooling on electronic transitions: R,T,V
• Simulation on Cs2 B1uX, with chirped frequencies
![Page 5: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/5.jpg)
One proposal
• Based on the development of a Multiple Single Frequency Laser• Sequential cooling on electronic transitions: R,T,V
• Simulation on Cs2 B1uX, with chirped frequencies
![Page 6: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/6.jpg)
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
![Page 7: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/7.jpg)
One Solution: cold atom photoassociation
Ultracold molecule!!
First discussion
![Page 8: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/8.jpg)
First steps
![Page 9: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/9.jpg)
First observations
Ultracold molecule!!
![Page 10: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/10.jpg)
First reviews
![Page 11: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/11.jpg)
PA well-known at thermal energies:diffuse bands
From Stwalley&Wang, J. Mol. Spectrosc. 195, 194 (1999)
*2AAA
![Page 12: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/12.jpg)
![Page 13: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/13.jpg)
![Page 14: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/14.jpg)
PA at ultracold energies
Free-bound transition = quasibound-bound transition
)),;(()()( *2 JvnpnsAhnsAnsA jL
LA detuning
Energy balance
recoilDopplerLgPA EEEhEJvE 2),(
200 cm-1 @300K
10-4 cm-1 @100K
recoilDopplerbL EEEJvEh ),(
UltracoldExcited
Short-livedmolecules
![Page 15: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/15.jpg)
Stwalley&Wang, J. Mol. Spectrosc. 195, 194 (1999)
![Page 16: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/16.jpg)
PAS of cold Cs
Trap loss
REMPI
![Page 17: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/17.jpg)
Detection of PA
REMPI
TRAP LOSS
Ultracold molecules
Ex: Cs Ex:Na
![Page 18: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/18.jpg)
11 years of PA observations (1993-2004)
• 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
![Page 19: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/19.jpg)
PA: Probe of the long-range part of molecular potentials
![Page 20: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/20.jpg)
Long-range interactions between neutral atoms
Multipolar expansion (in 1/R) of electrostatic interaction:
32121 ).)(.(3.
)(R
ndndddRV dd
Stwalley&Wang, J. Mol. Spectrosc. 195, 194 (1999)
![Page 21: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/21.jpg)
Le Roy-Bernstein approach
How to make the link between observed transitions and long-range behavior of the potential?
LeRoy&Bernstein, J. Chem.Phys. 52, 3869 (1970)
)(
)(
2/12
1
)(2
2
1 vR
vR v RVEdRv
nn
R
CDRV )(
2
22
1
2
2
)2(
n
n
D
n
nn
n
nv vvC
hKDE
)121(2
)11()2(
n
nnKn
(fractional) vibrational quantum number at the dissociation limit
6
3
:3
:6
vvEn
vvEn
Dv
Dv
-No solution for n=2-Limited to a single potential-Rotation ( 1/R2) not included
![Page 22: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/22.jpg)
Accumulated phase method: Numerical approach for higher flexibility
Moerdjik et al, PRA 51, 4852 (1995)
Crubellier etal, Eur. Phys. J. D, 6, 211 (1999)
Almost constant phase (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 (R0) Fitting strategy:
...)1()()( 00 JJEDR JE
Parameters: nCDR ,),( 0
Scattering length
![Page 23: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/23.jpg)
Pure long-range molecules (1)
![Page 24: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/24.jpg)
Pure long-range molecules (2)
)(0 2/3npnsg
)66(0 2/3psg R-3
R-3R-6, R-8
Quantum chemistrySpies, 1989
R-3R-6, R-8+exchange
![Page 25: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/25.jpg)
)()(
)()()(
RVR
RRRVV
The 0g- pure long-range state (1)
![Page 26: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/26.jpg)
Hund’s case (a) representation
The 0g- pure long-range state (2)
)()(
)()()(
RVR
RRRVV
At large distances:
-Atomic spin-orbitXX
X
)(...)(8
86
63
3 RVR
C
R
C
R
CRV exch
)(...2)(88
66
33 RV
R
C
R
C
R
CRV exch
-Asymptotic expansion of V
2
A
2
A
32 fsA
![Page 27: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/27.jpg)
The 0g- pure long-range state (3)
Hund’s case (c) representation
)(3
1)(
3
2)()(
3
2
)()(3
2)(
3
2)(
3
1
2
RVRVARVRV
RVRVRVRVA
V
32 fsA
Diagonalization of the spin-orbit matrix
02
22A
AA
Flat potential1/R6
Attractive potential1/R3
interaction1/R3
![Page 28: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/28.jpg)
attractive
The 0g- pure long-range state (4)
)(
8)(
2
1])([])[0(
6
232
33
2/32/3 RR
CR
R
CpsEpsV g
33
2
3)(
R
CAR
2
3)(:
ARR
6
23
33
2/32/3 3
4])([])[0(
AR
C
R
CpsEpsV g
,...,88
66
R
C
R
C
repulsive
0)(when R
33
2/32/3 )12(])([])[0(R
CpsEpsV g
Potential well
![Page 29: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/29.jpg)
• PAS spectrum: 75 vibrational levels, J=2• Direct Potential Fit approach:
PAS of the 0g- pure long-range state in Cs2
(1)
)();()(
);()()()(
3
3
RVRCR
RCRRRVV
rel
rel
Amiot et al, PRA 66, 052506(2002)
• 9 Fitting parameters
• minimization
)(),(),(),(,,, //8
/63 RVRRRCCC exchrel
2/12
1 )(
)()(1
N
i
obscalc
iu
iyiy
MN
![Page 30: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/30.jpg)
PAS of the 0g- pure long-range state in Cs2
(2)
RKR
asymptotic
Quantum chemistry
![Page 31: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/31.jpg)
Atomic radiative lifetime from PASAmiot et al, PRA 66, 052506(2002)
3
66
2
33 24
3
3
66
2
sp
prsCC
Non-relativistic
![Page 32: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/32.jpg)
Cold molecule formation processes• Main requirement: stabilization of the excited population in a
bound state• Solution: « R »-transfer of the probability density
Double-well caseObserved in:
Cs2, Rb2
Resonant couplingObserved in:
Cs2, RbCs,KRb
« not efficient » caseObserved in:Na2, K2, KRb, NaCs
![Page 33: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/33.jpg)
Double-well process
in Cs2
PA
SE
REMPI
![Page 34: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/34.jpg)
PA and cold molecule formation in Cs2
![Page 35: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/35.jpg)
REMPI spectra
Varying the PA laser frequency
Varying the REMPI laser frequency
Dion et al, EPJD 18, 365 (2002)
![Page 36: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/36.jpg)
Predicted vibrational population in the lowest 3u
+ state, after decay of 0g- PA levels
in Cs2
Detuning of the 0g- PA level
Vibrational levelOf the
a3u+ state
![Page 37: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/37.jpg)
Resonant coupling process (1)C. M. Dion et al, PRL 86, 2253 (2001)
![Page 38: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/38.jpg)
Resonant coupling process (2)
![Page 39: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/39.jpg)
Next resonance
Resonant coupling process (3)
![Page 40: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/40.jpg)
PA rates, shifts, line shapes: references(non exhaustive)
• 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)
![Page 41: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/41.jpg)
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
![Page 42: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/42.jpg)
Tuning the scattering length Moerdjik et al,PRA 51, 4852 (1995)
![Page 43: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/43.jpg)
![Page 44: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of](https://reader035.vdocuments.mx/reader035/viewer/2022062320/56649d975503460f94a809af/html5/thumbnails/44.jpg)
Bibliography
• « 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)