Data and Analysis of the Spin- Orbit Coupled Mixed A 1 Σ u + and b 3 Π u States of Cs 2 Andrey Stolyarov Moscow State University Tom Bergeman SUNY Stony Brook Supported by the Russian Foundation for Basic Research and by the US NSF.

Cs 2 ab initio Potentials from A. V. Stolyarov Subject of this talk: Empirical Potentials

One motivation: Production of cold ground state molecules Cs 2 : Innsbruck, Two STIRAP processes: Applied Phys. B 95, 219 (2009)

Its nice to know the energy level structure so as to know where to tune the lasers! For this purpose, we have analyzed experimental data on Rb 2, RbCs and Cs 2 A and b states. These states are highly perturbed by spin-orbit interactions. We have used DVR numerical methods to calculate eigenvalues of Hamiltonian matrices.

Experimental data

Back to Cs 2 A 1 Σ u + and b 3 Π u States Authors of the recent publication (PRA 83, (2011)) J.-M. Bai, E. H. Ahmed, B. Beser, Y. Guan, S. Kotochigova and A. M. Lyyra, Temple University S. Ashman, C. M. Wolfe, and John Huennekens, Lehigh University Feng Xie, D. Li and Li Li, Tsinghua University M. Tamanis and R. Ferber, University of Latvia, Riga A.Drozdova, E. Pazyuk and A. V. Stolyarov, Moscow State University J. G. Danzl and H.-C. Nägerl, University of Innsbruck N. Bouloufa, O. Dulieu, and C. Amiot, Université Paris-Sud, Orsay H. Salami and T. Bergeman, SUNY Stony Brook

Experimental Term Values Lower resolution monochromator data from Tsinghua University – valuable information on low levels of the b state. Higher resolution data from LIF FTS, from OODR polarization spectroscopy, and from cold molecule spectra

OODR Polarization Spectroscopy Data (Temple U.) See talk by Jianmei Bai RD03 on Thursday

Diatomic Molecule Hamiltonian: Lowest Excited 3 Π and 1 Σ + States V( 1 Σ + )+(x+2)B -Δ od 0 0 -Δ od V( 3 Π)-Δ d +(x+2)B -B√(2x) 0 0 -B√(2x) V( 3 Π)+B(x+2) -B√[2(x-2)] 0 0 -B√[2(x-2)] V( 3 Π)+Δ d +B(x-2) 1Σ+3Π01Σ+3Π0 3Π13Π1 H(R) = H K + H v (R) + H rot (R) 3Π23Π2 H V (R) + H rot (R) = B = ħ 2 /2μR 2 Hamiltonian Matrix x = J(J+1)

Fit Data Directly to Potentials and Spin-Orbit Functions Previous (10 years ago): fit to Dunham coefficients E(v,J) = Σ i,j Y i,j (v+1/2) i [J(J+1)] j RKR Potentials Current practice: Fit to potentials of form 1.“Hannover” (from E. Tiemann et al.) Supplemented by short range and long range expansions with typical forms: V 1 = a + b/R n V(exchange) = a R b exp(-cR) V(dispersion) =

Fit Data Directly to Potentials and Spin-Orbit Functions Current practice (cont’d): 2. Morse Long Range (from Le Roy) In principle, this potential eliminates the need to piecewise potentials at small and large R. Whether the exchange potential and all the terms in a dispersion potential are accurately represented remains an open question. 3. Expanded Morse Oscillator (AVS) V EMO (R) = D e [1-e -α(R-Re) ] α(R)= Σ i=0 a i [(R p – R p ref )/(R p +R p ref )] i

The Discrete Variable Representation (DVR) Method * DVR method computes perturbed wavefunctions to a high degree of accuracy, because all mesh points are used to obtain d 2 /dR 2 Kinetic Energy is represented by a full matrix over the mesh points, The potential energy elements are diagonal, V ii’ =  ii’  (R i ) Hamiltonian matrix, n × n n=number of R mesh points × number of channels (, etc). * Colbert and Miller, Journal of Chemical Physics 96, 1982 (1992)

“Reduced” Term values vs. J(J+1) Steepest + slopes: b 3 Π 1u Intermediate sIopes: b 3 Π 0u+ Negative slopes: A 1 Σ u + At higher energies, A 1 Σ u + and b 3 Π 0u+ levels are highly mixed: Term values near A(v=0)

Fitted Potentials for the A 1 Σ u + and b 3 Π u States of Cs 2 A1Σu+A1Σu+ b3Πub3Πu But b 3 Π 2u potential comes from ab initio calculations only – no data

Avoided crossings Information on b 3 Π 1u levels comes only from avoided crossing regions

Spin-Orbit Functions ECP1,2 Stolyarov at Moscow State MR-RAS-CI from Kotochigova at Temple U. The off-diagonal spin-orbit coupling function is more than 4 times as large as the typical vibrational spacing, hence requires extensive deperturbation analysis.

Second-order spin-orbit (SO2) effects e-f splitting in b 3 Π 0± comes from SO coupling with A 1 Σ u + and from SO2 coupling with other ungerade states SO coupling with a 3 Σ u + pushes the f parity components up, especially for R < 4.5 Å

Franck-Condon factors for Cs 2 A-X transitions Transitions to low v b 3 Π 0u+ states acquire strength by mixing with the A state. Only A state component contributes, so FC factors oscillate with v

A-X Franck-Condon factors (cont’d) A majority b 3 Π 0u+ state has a large FC factor to the X state when the A state component is in phase with an X state wavefunction

Summary and Conclusions We have compiled data on the Cs 2 A 1 Σ u + and b 3 Π u states from several sources and also added new data. These states are coupled by large spin-orbit function. The perturbative interactions have been fit to close to experimental accuracy by two numerical methods. Anomalies in Franck-Condon factors to the X state can be understood in view of the wavefunctions of the coupled states.