1- introduction, overview 2- hamiltonian of a diatomic molecule 3- molecular symmetries; hund’s...
TRANSCRIPT
• 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
How to create ultracold molecules using laser cooling?
Laser cooling of molecules:NO closed level-scheme
Laser cooling of atoms: 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 B1uX, 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 B1uX, 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 B1uX, 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
Ultracold molecule!!
First discussion
First steps
First observations
Ultracold molecule!!
First reviews
PA well-known at thermal energies:diffuse bands
From Stwalley&Wang, J. Mol. Spectrosc. 195, 194 (1999)
*2AAA
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
Stwalley&Wang, J. Mol. Spectrosc. 195, 194 (1999)
PAS of cold Cs
Trap loss
REMPI
Detection of PA
REMPI
TRAP LOSS
Ultracold molecules
Ex: Cs Ex:Na
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
PA: Probe of the long-range part of molecular potentials
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)
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
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
Pure long-range molecules (1)
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
)()(
)()()(
RVR
RRRVV
The 0g- pure long-range state (1)
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
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
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
• 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
PAS of the 0g- pure long-range state in Cs2
(2)
RKR
asymptotic
Quantum chemistry
Atomic radiative lifetime from PASAmiot et al, PRA 66, 052506(2002)
3
66
2
33 24
3
3
66
2
sp
prsCC
Non-relativistic
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
Double-well process
in Cs2
PA
SE
REMPI
PA and cold molecule formation in Cs2
REMPI spectra
Varying the PA laser frequency
Varying the REMPI laser frequency
Dion et al, EPJD 18, 365 (2002)
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
Resonant coupling process (1)C. M. Dion et al, PRL 86, 2253 (2001)
Resonant coupling process (2)
Next resonance
Resonant coupling process (3)
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)
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
• « 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)