1- introduction, overview 2- hamiltonian of a diatomic molecule 3- molecular symmetries; hund’s...

• 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)