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The Journal of Chemical Physics

Quasibound states in long-range alkali dimers: Grid method calculations

Olivier Dulieu, Ronnie Kosloff, Françoise Masnou-Seeuws, Goran Pichler

Published Dec 22, 1997• 27 citations
Publisher

Abstract

A local minimum is found in the 0g+ long range potential curves of the K2 and Rb2 alkali dimers. This well-of magnitude 42 cm−1 for K2 and 93 cm−1 for Rb2—is located above the first ns+n2P3/2 dissociation limit and metastable states could be populated using laser light blue detuned compared to the resonance line. To compute the previously unknown energies and lifetimes of these quasibound states, two grid methods are employed. One method is based on diagonalizing a Fourier grid Hamiltonian, the other uses a propagation technique in imaginary time to filter out vibrational eigenfunctions. Equivalent results are given by both methods. Then the lifetimes are extracted from the correlation function obtained by propagation in real time of these numerical vibrational wave functions. The methods are employed both in adiabatic representation with one electronic potential curve and in diabatic representation with two potential curves. Two quasibound states are found for K2, and three for Rb2 above seven stable bound states. Their lifetimes vary from 20 ps to 3 ns.

DOI
10.1063/1.474179

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A local minimum is found in the 0g+ long range potential curves of the K2 and Rb2 alkali dimers. This well-of magnitude 42 cm−1 for K2 and 93 cm−1 for Rb2—is located above the first ns+n2P3/2 dissociation limit and metastable states could be populated using laser light blue detuned compared to the resonance line. To compute the previously unknown energies and lifetimes of these quasibound states, two grid methods are employed. One method is based on diagonalizing a Fourier grid Hamiltonian, the other uses a propagation technique in imaginary time to filter out vibrational eigenfunctions. Equivalent results are given by both methods. Then the lifetimes are extracted from the correlation function obtained by propagation in real time of these numerical vibrational wave functions. The methods are employed both in adiabatic representation with one electronic potential curve and in diabatic representation with two potential curves. Two quasibound states are found for K2, and three for Rb2 above seven stable bound states. Their lifetimes vary from 20 ps to 3 ns.
Numerical calculations of vibrational levels of alkali dimers close to the dissociation limit are developed in the framework of a Fourier Grid Hamiltonian method. The aim is to interpret photoassociation experiments in cold atom samples. In order to avoid the implementation of very large grids we propose a mapping procedure adapted to the asymptotic R−n behavior of the long-range potentials. On a single electronic potential, this allows us to determine vibrational wave functions extending up to 500a0 using a minimal number of grid points. Calculations with two electronic states, A 1Σu+ and b 3Πu states, both correlated to the Rb(5s)+Rb(5p) dissociation limit, coupled by fine structure are presented. We predict strong perturbation effects in the Rb2(0u+) spectrum, manifested under the 5s, 5p 2P1/2 dissociation limit by an oscillatory behavior of the rotational constants.
The treatment of the dynamics of ultracold molecules requires new theoretical tools. Previous work of the present authors [J. Chem. Phys. 110, 9865 (1999)] for calculation of vibrational levels by a Fourier grid representation with use of adaptative coordinates is generalized here to the treatment of the bound-continuum interaction in a two-channel problem. Two numerical methods are presented: a time-dependent method using a Chebyshev propagator to compute the correlation function and a time-independent method with diagonalization of a Hamiltonian that includes an absorbing optical potential. In both cases the adaptative coordinate is defined by a numerical rather than an analytical procedure. Lifetimes are reported for the predissociated levels of the Rb2{\mathrm{Rb}}_{2}Rb2​ and Cs20u+(ns+np2P3/2){\mathrm{Cs}}_{2}{0}_{u}^{+}{(ns+np}^{2}{P}_{3/2})Cs2​0u+​(ns+np2P3/2​) spectra, where n=5,6.n=5,6.n=5,6. The two numerical methods give similar results. The lifetimes increase with the vibrational quantum number proportionally to the classical vibration period estimated from the Le Roy--Bernstein law for an asymptotic ${R}^{\ensuremath{-}3}$ potential, and the energy variation can be fitted to an analytical formula. The results are shown to be very sensitive to the molecular parameters, potentials, and couplings. The measured width of 8.5 GHz reported by Cline et al. [Phys. Rev. Lett. 73, 632 (1994)] for one predissociated level of 87Rb2{}^{87}{\mathrm{Rb}}_{2}87Rb2​ is reproduced. A strong isotopic effect is found for the rubidium dimer, the lifetimes of 85Rb2{}^{85}{\mathrm{Rb}}_{2}85Rb2​ and 87Rb2{}^{87}{\mathrm{Rb}}_{2}87Rb2​ levels differing by a factor of 3. Finally, we present a third approach, in the framework of a generalized two-channel quantum-defect theory, where lifetimes are determined by extrapolation of parameters fitted to Lu-Fano plots of computed bound levels below the P1/2{P}_{1/2}P1/2​ dissociation limit. Excellent agreement is obtained with the numerical results, suggesting the possibility of fitting to experimental spectra.
A modified semi-empirical Tang-Toennies potential model is used to describe the a3Σu+ potentials of the alkali dimers. These potentials are currently of interest in connection with the laser manipulation of the ultracold alkali gases. The fully analytical model is based on three experimental parameters, the well depth De, well location Re, and the harmonic vibrational frequency ωe of which the latter is only slightly optimized within the range of the literature values. Comparison with the latest spectroscopic data shows good agreement for Na2, K2, Rb2, and Cs2, comparable to that found with published potential models with up to 55 parameters. The differences between the reduced potential of Li2 and the conformal reduced potentials of the heavier dimers are analyzed together with why the model describes Li2 less accurately. The new model potential provides a test of the principle of corresponding states and an excellent first order approximation for further optimization to improve the fits to the spectroscopic data and describe the scattering lengths and Feshbach resonances at ultra-low temperatures.
We calculate potential energy curves for heavy alkali-metal dimers, Rb2{\mathrm{Rb}}_{2}Rb2​ and Cs2{\mathrm{Cs}}_{2}Cs2​ in which one of the atoms is in a highly excited Rydberg states. The method combines numerical integration of coupled equations, describing interaction of electron with the ground-state atom in the field of the Coulomb core of the Rydberg atom, with subsequent matching of the obtained wave function with the Coulomb Green's function in the form of the Kirchhoff integral. The spin-orbit interaction for the Rydberg electron is also included. The results show the existence of several groups of states. Most interesting of them are dominated either by the 3S{}^{3}S3S symmetry near the ground-state atom or by the 3PJ{}^{3}{P}_{J}3PJ​ symmetry, J=0,1,2.J=0,1,2.J=0,1,2. All states, except the 3P1{}^{3}{P}_{1}3P1​ state, exhibit oscillatory dependence of energy on the internuclear distance that can support long-range molecular Rydberg states [e.g., ``trilobite states'' for the 3S{}^{3}S3S symmetry, Greene et al., Phys. Rev. Lett. 85, 2458 (2000)]. These states possess large diagonal and transition dipole moments which are expressed analytically in terms of the Coulomb wave functions and calculated in a broad range of internuclear separations.
The adiabatic potential energy curves of Rb2{\mathrm{Rb}}_{2}Rb2​ in the long-range Rydberg electronic states are calculated using the two-electron RRR-matrix method [M. Tarana and R. \ifmmode \check{C}\else \v{C}\fi{}ur\'{\i}k, Phys. Rev. A 95, 042515 (2017)] for the intermediate internuclear separations between 37 and 200 a.u. The results are compared with the zero-range models to find a region of the internuclear distances where Fermi's pseudopotential approach provides accurate energies. A finite-range potential model of the atomic perturber is used to calculate the wave functions of the Rydberg electron and their features specific for the studied range of internuclear distances are identified.
The photoassociation of alkali metal molecules (Rb2, Cs2, and RbCs) at short range is investigated in theory. Based on ab initio calculations to rationalize Franck–Condon filtering, the target states of photoassociation have been obtained. They correspond to vibrational transition levels from excited state (Rb2: v′{v}{\kern 1pt} 'v′ = 17–21, Cs2: v′{v}{\kern 1pt} 'v′ = 21–25, and RbCs: v′{v}{\kern 1pt} 'v′ = 18–22) to ground state. By using quantum wave-packet dynamic methods, the yields with time evaluation of the selected states are calculated interacting with a resonant laser pulse. Using gaussian pulse leads to the yield of Rb2 up to 82% at 860 fs. After a laser pulse, the positive chirp promotes the yield of vibrational states to increase, but the negative chirp inhibits it. That is to say, by changing the laser parameters and pulse shapes, it is much easier to control the photochemical process along desired direction. For photoassociation of alkali-metal dimers, homonuclear molecules have a better response to laser pulse than heteronuclear molecules. The efficiency of photochemical reaction of homonuclear molecules is much better and the photoassociative yield is much higher. Those conditions will provide an important reference and suggest a scheme for a feasible photoassociation for further experimental and theoretical research.