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The Journal of Chemical Physics

Mapped Fourier methods for long-range molecules: Application to perturbations in the Rb2(0u+) photoassociation spectrum

Viatcheslav Kokoouline, Olivier Dulieu, Ronnie Kosloff, Françoise Masnou-Seeuws

Published May 22, 1999• 237 citations
Publisher

Abstract

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.

DOI
10.1063/1.478860

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Theoretical treatment of channel mixing in excited Rb2 and Cs2 ultracold molecules: Determination of predissociation lifetimes with coordinate mapping

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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 ${\mathrm{Rb}}_{2}$ and ${\mathrm{Cs}}_{2}{0}_{u}^{+}{(ns+np}^{2}{P}_{3/2})$ spectra, where $n=5,6.$ 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 ${}^{87}{\mathrm{Rb}}_{2}$ is reproduced. A strong isotopic effect is found for the rubidium dimer, the lifetimes of ${}^{85}{\mathrm{Rb}}_{2}$ and ${}^{87}{\mathrm{Rb}}_{2}$ 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 ${P}_{1/2}$ dissociation limit. Excellent agreement is obtained with the numerical results, suggesting the possibility of fitting to experimental spectra.

Physical Review A•Aug 18, 2000•38 citations
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 mapped Fourier grid method, using a coordinate adapted to the local de Broglie wavelength associated to the vibrational motion in a molecule, and recently developed by Kokoouline et al. [J. Chem. Phys. 110, 9865 (1999)], is used to compute the perturbed binding energies and rotational constants of the Cs20u+(6s+6p2P1/2,3/2){\mathrm{Cs}}_{2}{0}_{u}^{+}(6s+6p{}^{2}{P}_{1/2,3/2})Cs2​0u+​(6s+6p2P1/2,3/2​) photoassociation spectrum below the lower dissociation limit. The results are shown to be very sensitive to the molecular potential curves and spin-orbit coupling. In the case of Rb20u+(5s+5p2P1/2,3/2),{\mathrm{Rb}}_{2}{0}_{u}^{+}(5s+5p{}^{2}{P}_{1/2,3/2}),Rb2​0u+​(5s+5p2P1/2,3/2​), a strong isotopic effect is also predicted. The results can be interpreted in the framework of generalized two-channel quantum defect theory, and generalized Lu-Fano plots are presented. This work suggests that parameters could easily be fitted to experimental spectra, eliminating the difficulties caused by limitations in the accuracy of the short-range molecular potentials.
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.
Recently, the method of Lu-Fano plots was extended to potentials with general ${R}^{\ensuremath{-}n}$ asymptotic behavior. The aim of this paper is to discuss the efficiency of this method for the interpretation of experimental vibrational spectra of long-range diatomic molecules when two channels are coupled. A Lu-Fano analysis is implemented on experimental data for the Cs20u+(6S+6P1/2,3/2){\mathrm{Cs}}_{2}{0}_{u}^{+}{(6S+6P}_{1/2,3/2})Cs2​0u+​(6S+6P1/2,3/2​) photoassociation spectra below the 6S+P1/2{6S+P}_{1/2}6S+P1/2​ and 6S+P3/2{6S+P}_{3/2}6S+P3/2​ dissociation limits. The parameters are fitted on level energies and predissociation widths, giving information about the asymptotic coefficients C3{C}_{3}C3​ for both channels and about a coupling parameter which determines the variation of predissociation widths as a function of detuning. This parameter can be extrapolated above the 6S+P3/2{6S+P}_{3/2}6S+P3/2​ threshold to yield the fine-structure transition cross section between the two channels. The probability for such a transition in a model where hyperfine structure effects are neglected is found to be 0.54.
Strong perturbation effects in the infrared fluorescence spectra recorded after the resonant excitation process ${X}^{1}{\ensuremath{\Sigma}}_{g}^{+}\ensuremath{\rightarrow}{A}^{1}{\ensuremath{\Sigma}}_{u}^{+}({0}_{u}^{+})\ensuremath{\rightarrow}(2{)}^{1}{\ensuremath{\Pi}}_{g}$ in Rb2{\mathrm{Rb}}_{2}Rb2​, have been observed by high resolution Fourier transform spectroscopy. They arise from complete coupling of the two states over the whole range of binding energies, yielding unusual molecular spectra with oscillating vibrational and rotational structures. A quantitative interpretation, based on Fourier grid Hamiltonian method calculations, of these newly observed phenomena is given. This opens new opportunities to study the triplet manifold in Rb2{\mathrm{Rb}}_{2}Rb2​, RbCs, and Cs2{\mathrm{Cs}}_{2}Cs2​, particularly the lowest state relevant for precise scattering length determination.
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.