1- introduction, overview 2- hamiltonian of a diatomic molecule
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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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) collisionsOlivier Dulieu
Predoc’ school, Les Houches,september 2004
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How to create ultracold molecules using laser cooling?
Laser cooling of molecules:NO closed level-scheme
Laser cooling of atoms: closed level-scheme
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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
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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
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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
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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
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One Solution: cold atom photoassociation
Ultracold molecule!!
First discussion
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First steps
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First observations
Ultracold molecule!!
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First reviews
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PA well-known at thermal energies:diffuse bands
From Stwalley&Wang, J. Mol. Spectrosc. 195, 194 (1999)
*2AAA
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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
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Stwalley&Wang, J. Mol. Spectrosc. 195, 194 (1999)
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PAS of cold Cs
Trap loss
REMPI
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Detection of PA
REMPI
TRAP LOSS
Ultracold molecules
Ex: Cs Ex:Na
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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
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PA: Probe of the long-range part of molecular potentials
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Long-range interactions between neutral atoms
Multipolar expansion (in 1/R) of electrostatic interaction:
32121 ).)(.(3.)(
RndndddRV dd
Stwalley&Wang, J. Mol. Spectrosc. 195, 194 (1999)
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Le Roy-Bernstein approachHow 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
21 vR
vR v RVEdRv
nn
RCDRV )(
222
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
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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
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Pure long-range molecules (1)
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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
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)()()()()(
RVRRRRV
V
The 0g- pure long-range state (1)
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Hund’s case (a) representationThe 0g
- pure long-range state (2)
)()()()()(
RVRRRRV
V
At large distances:-Atomic spin-orbitX
XX
)(...)( 88
66
33 RV
RC
RC
RCRV exch
)(...2)( 88
66
33 RV
RC
RC
RCRV exch
-Asymptotic expansion of V
2A
2A
32 fsA
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The 0g- pure long-range state (3)
Hund’s case (c) representation
)(31)(
32)()(
32
)()(32)(
32)(
31
2
RVRVARVRV
RVRVRVRVA
V
32 fsA
Diagonalization of the spin-orbit matrix
02
22A
AA
Flat potential1/R6
Attractive potential1/R3
interaction1/R3
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attractive
The 0g- pure long-range state (4)
)(8)(
21])([])[0( 6
232
33
2/32/3 RRCR
RCpsEpsV g
33
23)(
RCAR
23)(: ARR
6
23
33
2/32/3 34])([])[0(ARC
RCpsEpsV g
,..., 8
866
RC
RC
repulsive
0)(when R
33
2/32/3 )12(])([])[0(RCpsEpsV g
Potential well
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• PAS spectrum: 75 vibrational levels, J=2• Direct Potential Fit approach:
PAS of the 0g- pure long-range state in Cs2
(1)
)();()();()()()(
3
3
RVRCRRCRRRV
Vrel
rel
Amiot et al, PRA 66, 052506(2002)
• 9 Fitting parameters
• minimization
)(),(),(),(,,, //8
/63 RVRRRCCC exchrel
2/12
1 )()()(1
N
i
obscalc
iuiyiy
MN
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PAS of the 0g- pure long-range state in Cs2
(2)
RKR
asymptotic
Quantum chemistry
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Atomic radiative lifetime from PASAmiot et al, PRA 66, 052506(2002)
3
66
2
33 24
3366
2
sp
prsCC Non-relativistic
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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
Double-well caseObserved in:
Cs2, Rb2
Resonant couplingObserved in:
Cs2, RbCs,KRb
« not efficient » caseObserved in:Na2, K2, KRb, NaCs
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Double-well process
in Cs2
PASE
REMPI
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PA and cold molecule formation in Cs2
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REMPI spectra
Varying the PA laser frequency
Varying the REMPI laser frequency
Dion et al, EPJD 18, 365 (2002)
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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
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Resonant coupling process (1)C. M. Dion et al, PRL 86, 2253 (2001)
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Resonant coupling process (2)
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Next resonance
Resonant coupling process (3)
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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)
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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
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Tuning the scattering length Moerdjik et al,PRA 51, 4852 (1995)
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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)