1. L. PruvostLab. A. Cotton, Orsay, F Molecular spectroscopy 2008 1 Molecular spectroscopy 2008 Weakly bound molecules. Analysis by the Lu-Fano method coupled to the LeRoy-Bernstein model Laboratoire Aimé Cotton CNRS, bat 505 Orsay Laurence PRUVOST

2. L. PruvostLab. A. Cotton, Orsay, F Molecular spectroscopy 2008 2 Introduction Context of cold molecule formation How to find efficient schemes ? Good knowledge of the molecular spectroscopy : location and properties of the molecular levels. Adapt the Lu-Fano method to analyze the weakly-bound molecules ► A method which allows us to exhibit the coupling between molecular states and to find quasi-resonant coupled levels. Weakly bound molecules. Analysis by the Lu-Fano method coupled to the LeRoy-Bernstein model

3. L. PruvostLab. A. Cotton, Orsay, F Molecular spectroscopy 2008 3 Photo-association of cold atoms - molecule formation Two neighbouring cold atoms (5s+5s), submitted to a resonant laser light, are photo-associated to a weakly-bound excited molecule M*. (1) Photo-association Rb(5s) + Rb(5s) + h  Rb 2 * The lifetime of the molecule M* is very short. The molecule either spontaneously decays to atoms or to a more stable molecule. (2) : spontaneous emission Rb 2 *  h ’ + Rb + Rb (3) : molecule formation Rb 2 *  h ’’ + Rb 2 Cold molecule formation ► To favour the process (1) : excitation of a weakly bound molecule. Efficient free-bound transition. ► To favour the process (3) : large Franck-Condon factor at small R value.

4. L. PruvostLab. A. Cotton, Orsay, F Molecular spectroscopy 2008 4 Cold molecule formation To favour the process (1) : (excitation of a weakly bound molecule) and to favour the process (3) (large Franck-Condon factor at small R value) A solution : coupled potentials ► The wavefunction of the intermediate excited molecule has two regions of probability Dion et al. PRL, 86, 2253, 2001

5. L. PruvostLab. A. Cotton, Orsay, F Molecular spectroscopy 2008 5 0u+ molecular states Couplings between molecular potentials (s 1/2 -p 1/2 )0u+ and (s 1/2 -p 3/2 )0u+ due to Spin-orbit Spin-other orbit spin-spin interaction... ► Search quasi-resonant levels of the two potentials ► How to find the best candidate ? + R

6. L. PruvostLab. A. Cotton, Orsay, F Molecular spectroscopy 2008 6 Experimental data: 87 Rb 2, trap-loss spectra below the 5s 1/2 -sp 1/2 limit Among 0u+ resonances, how to find the best 0u+ level ? 3 molecular series : 0g-, 0u+ and 1g Close to the dissociation limit = weakly bound molecules Each line position gives the binding energy of the molecule. ► Adapt the Lu-Fano method to analyze the data. Jelassi, Viaris, Pruvost, LAC, Orsay

7. L. PruvostLab. A. Cotton, Orsay, F Molecular spectroscopy 2008 7 Weakly bound molecules Dipole-dipole interaction For large internuclear distances, R, the binding energy of the molecule is due to the dipole- dipole interaction V(R) = -c 3 /R 3 Or 1/R 6 depending on molecular symmetry. BKW Solution of -c n /R n for a 1/R n potential, the BKW approach (LeRoy-Bernstein and Stwalley 1970) gives the power law v D -v= (  E n ) (n-2)/2n v vibrational quantum number, E n parameter defined from mass and dipole element (c n ) v D constant, whose integer part is the number of levels lying above the dissociation. NOT INCLUDED in the model : Short range effects, Coupling with another molecular potential r1 r1 R r2r2 R>> r 1, r 2 bottom Core LRB

8. L. PruvostLab. A. Cotton, Orsay, F Molecular Spectroscopy 2008 8 Data analysis The LeRoy-Bernstein power law, v D -v = (  E 3 ) 1/6, is extensively applied to analyze weakly bound molecules. One of the method to determine c 3. How to improve the analysis ? The Lu-Fano method adapted/applied to weakly bound molecules. Definitions v D -v = (  E 3 ) 1/6 v* = v D -v effective v ibrational quantum number  = v* - Int(v*) molecular vibrational quantum defect ► Lu-Fano graph:  versus the binding energy  Remarks  varies from 0 to 1. If the LRB law is satisfied,  =  D is a constant. Lu-Fano for Rydberg atomic states (1970) V(r)=-1/r n*= (  R yd ) ½  = n* - Int(n*) See also Kokooline et al. PRA 62, 022504, 2000; PRA 65, 62710, 2002.

9. L. PruvostLab. A. Cotton, Orsay, F Molecular Spectroscopy 2008 9 Lu-Fano method applied to weakly bound molecules Method Extract the binding energy from the experimental data, . A good knowledge of the dissociation limit is required. Deduce v*, the effective vibrational quantum number v* = (  / E 3 ) 1/6 Deduce  the vibrational quantum defect  = v* - Int[v*] Plot the Lu-Fano graph (  versus  Jelassi, Viaris, Pruvost, PR A. 74, 12510,2006 The LF graph exhibits sharp variations, signatures of a coupling between two series (cf Rydberg states)

10. L. PruvostLab. A. Cotton, Orsay, F Molecular spectroscopy 2008 10 A model with 2 series of levels Assumption: V constant in the vicinity of E 2 Diagonalisation of V 1 E2E2 Demkov, Ostrovski, J Phys B, 28, 403,1995 Cohen-Tannoudji, Dupont-Roc, Grynberg, processus d’interaction entre photons et atomes, p 52. 11 E2E2

11. L. PruvostLab. A. Cotton, Orsay, F Molecular spectroscopy 2008 11 Characterisation of the coupling Previous model slightly modified because the non-perturbed quantum defect linearly depends on the energy Fit with Tan[  (  -  1 )]. Tan[  (  -  2 )/  2 ]=  2 K 2 and  1 =  1  -   Perturbing level  2 = 4.724 cm -1 Coupling constantK = 0.1221 Quantum defect at  =0  1  = 0.6932 Linear variation  = -0.0448  1  = 0.6932 ±0.0167  = -0.0448 ±0.0031  2 = 4.724 ±0.066  2 = 6.803 ±0.084 K = 0.1221 ±0.0086

12. L. PruvostLab. A. Cotton, Orsay, F DAMOP 2008 12 Consequences 1. Short range potential : the location of the barrier and of the minimum is deduced from the quantum defect at  =0 and the slope . R c =17.9 a.u. R’ c =19.3 a.u. 2. First predissociated level : an extrapolation of the Lu-Fano graph gives predictions for the first level of (5s - 5p 3/2 ) 0 u + located above the (5s-5p 1/2 ) 0 u + dissociation limit. @ +2.1 cm -1,  ~ 4 cm -1 Experimental confirmation Jelassi, Viaris, Pruvost PRA 74, 2006

13. L. PruvostLab. A. Cotton, Orsay, F DAMOP 2008 13 Consequences 3. Wavefonction mixing deduced from the coupling  = cos   1 (R) + sin   2 (R) @ 4.72 cm -1 ;  = 31.8° (72%, 28%)  1 (R) external, max at the turning point R 1 =82 ua  2 (R) internal, max at the turning point R 2 =24 ua ► Cold molecule formation in the ground state by increasing the probability near R=0, cold molecule formation is enhanced. ► Experimental confirmation: detection of cold Rb 2 for a laser detuning of 4.72 cm -1 Pisa group: Fioretti et al. JPB 40, 2007. 4.72 cm -1

14. L. PruvostLab. A. Cotton, Orsay, F Molecular Spectroscopy 2008 14 Application to other molecules: example of Cs 2 Case of 0 u + levels of Cs 2 (Pichler, Stwalley, 2004) Jelassi, Viaris, Pruvost, Pichler, Stwalley, accepted to PRA. ► 5 purburbing levels located at 5.91 cm ⁻ ¹, 16.92 cm ⁻ ¹, 28.13 cm ⁻ ¹, 39.27 cm ⁻ ¹, and 50.33 cm ⁻ ¹ ► Wave-function mixing In a 2-level model In a multi-level model

15. L. PruvostLab. A. Cotton, Orsay, F Molecular Spectroscopy 2008 15 Conclusion The Lu-Fano method adapted for weakly-bound molecules. LRB law used to convert the binding energy to a molecular quantum defect. The Lu-Fano graph allows us to - measure the coupling between the series. Then, predissociated levels are predicted and the wavefunction mixings are deduced. A semi-empirical method which only requires the asymptotic behaviour (c 3 value) of the potential and the location of the dissociation limit. Possible application to others molecules (homonuclear, heteronuclear) 0g- 1 channel 0u+ 2 channels

16. L. PruvostLab. A. Cotton, Orsay, F Molecular spectroscopy 2008 16 The group Photographe: Benoit Lantin LAC Haikel Jelassi, Fabienne Diry, LP Michael Mestre, Bruno Viaris de Lesegno