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Physical Review A

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

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

Published Aug 18, 2000• 38 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.62.032716

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Theoretical treatment of channel mixing in excited Rb2 and Cs2 ultracold molecules: Perturbations in 0u+ photoassociation and fluorescence spectra

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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 ${\mathrm{Cs}}_{2}{0}_{u}^{+}(6s+6p{}^{2}{P}_{1/2,3/2})$ 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 ${\mathrm{Rb}}_{2}{0}_{u}^{+}(5s+5p{}^{2}{P}_{1/2,3/2}),$ 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.

Physical Review A•Jul 17, 2000•48 citations

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

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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.

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Formation of ultracold molecules by photoassociation: theoretical developments

Olivier Dulieu, Françoise Masnou-Seeuws

In recent years several groups of researchers have succeeded in making samples of cold molecules, at temperatures below 1 K, and of ultracold molecules, at temperatures below 1 mK. We concentrate on the formation of Cs2 dimers. Starting with laser-cooled atoms, photoassociation forms molecules in an excited electronic state. Various mechanisms are presented for stabilizing these molecules by spontaneous emission into a bound level of the ground electronic state. The design of new formation schemes requires precise knowledge of molecular potential curves that is well beyond the accuracy of ab initio calculations. We show that analytical long-range potentials can be fitted on photoassociation spectra and yield, through the determination of the C3 long-range coefficient, a value of the radiative lifetime of the 6P3/2 cesium level with an unprecedented accuracy. For short-range potentials it is possible to avoid explicit calculations by using asymptotic methods that generalize to long-range molecules the quantum-defect concepts developed for Rydberg atoms. Considering two coupled channels, we show how generalized Lu–Fano plots can extract precise information from experimental spectra.

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

Olivier Dulieu, Ronnie Kosloff, Françoise Masnou-Seeuws +1 more

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.

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Lu-Fano plot for interpretation of the photoassociation spectra

Viatcheslav Kokoouline, Cyril Drag, P. Pillet +1 more

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 ${\mathrm{Cs}}_{2}{0}_{u}^{+}{(6S+6P}_{1/2,3/2})$ photoassociation spectra below the ${6S+P}_{1/2}$ and ${6S+P}_{3/2}$ dissociation limits. The parameters are fitted on level energies and predissociation widths, giving information about the asymptotic coefficients ${C}_{3}$ 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 ${6S+P}_{3/2}$ 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.

Physical Review A•Jun 7, 2002•14 citations

Utlracold Rb2 Molecules: Formation, Detection and Trapping in an Optical Dipole Trap

Hyewon Pechkis

This thesis comprises two topics in the field of the ultracold atomic and molecular physics, both involving formation of ultracold 85Rb 2 by photoassociation (PA). Typically, these ground-state molecules are formed mainly in the very highest vibrational levels, rather than the low-v levels that will be required for most future investigations of ultracold molecular physics and chemistry. The first topic of my research is a method for circumventing this problem: resonant enhancement (RE) of photoassociative ground-state molecule production. The key idea of RE is to photoassociate pairs of colliding atoms into an excited state that has a bimodal vibrational probability distribution, with one peak at long range to enhance the photoassociation rate and another at shorter range to enhance radiative decay into deeply bound levels of the ground state. This can be arranged thanks to the strong spin-orbit perturbative interaction between the two 0+u states converging to two different asymptotes, (5s + 5p1/2) and (5s + 5 p3/2). This method enhances ground-state molecule formation in the region well below the dissociation limit. The experimental work described here was done at the University of Connecticut in collaboration with C. P. Koch at the Institut für Theoretische Physik, Freie Universität Berlin, Germany, who performed the theoretical analysis [1].^ The second part of my thesis is the investigation of ultracold molecules in an optical dipole trap. We have constructed a quasi-electrostratic optical trap (QUEST) for 85Rb using a focused CO2 laser [2]. This provides a very high density of ultracold atoms, up to 10 12 cm-3, providing much higher photoassociation rates. It also keeps the molecules trapped once they have formed. The QUEST provides an ample trapping lifetime for both trapped atoms and molecules, yielding about 4-5×106 atoms and about 9×105 high-v molecules. The long lifetime of the trapped molecules and atoms, combined with their high density, allows us to study interactions such as atom-molecule or molecule-molecule collisions in the ultracold regime. A preliminary experiment on Rb2 molecular decay is described. ^

•Jan 1, 2010•0 citations
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{R}^{\ensuremath{-}3}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.
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.
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.
In recent years several groups of researchers have succeeded in making samples of cold molecules, at temperatures below 1 K, and of ultracold molecules, at temperatures below 1 mK. We concentrate on the formation of Cs2 dimers. Starting with laser-cooled atoms, photoassociation forms molecules in an excited electronic state. Various mechanisms are presented for stabilizing these molecules by spontaneous emission into a bound level of the ground electronic state. The design of new formation schemes requires precise knowledge of molecular potential curves that is well beyond the accuracy of ab initio calculations. We show that analytical long-range potentials can be fitted on photoassociation spectra and yield, through the determination of the C3 long-range coefficient, a value of the radiative lifetime of the 6P3/2 cesium level with an unprecedented accuracy. For short-range potentials it is possible to avoid explicit calculations by using asymptotic methods that generalize to long-range molecules the quantum-defect concepts developed for Rydberg atoms. Considering two coupled channels, we show how generalized Lu–Fano plots can extract precise information from experimental spectra.
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.
This thesis comprises two topics in the field of the ultracold atomic and molecular physics, both involving formation of ultracold 85Rb 2 by photoassociation (PA). Typically, these ground-state molecules are formed mainly in the very highest vibrational levels, rather than the low-v levels that will be required for most future investigations of ultracold molecular physics and chemistry. The first topic of my research is a method for circumventing this problem: resonant enhancement (RE) of photoassociative ground-state molecule production. The key idea of RE is to photoassociate pairs of colliding atoms into an excited state that has a bimodal vibrational probability distribution, with one peak at long range to enhance the photoassociation rate and another at shorter range to enhance radiative decay into deeply bound levels of the ground state. This can be arranged thanks to the strong spin-orbit perturbative interaction between the two 0+u states converging to two different asymptotes, (5s + 5p1/2) and (5s + 5 p3/2). This method enhances ground-state molecule formation in the region well below the dissociation limit. The experimental work described here was done at the University of Connecticut in collaboration with C. P. Koch at the Institut für Theoretische Physik, Freie Universität Berlin, Germany, who performed the theoretical analysis [1].^ The second part of my thesis is the investigation of ultracold molecules in an optical dipole trap. We have constructed a quasi-electrostratic optical trap (QUEST) for 85Rb using a focused CO2 laser [2]. This provides a very high density of ultracold atoms, up to 10 12 cm-3, providing much higher photoassociation rates. It also keeps the molecules trapped once they have formed. The QUEST provides an ample trapping lifetime for both trapped atoms and molecules, yielding about 4-5×106 atoms and about 9×105 high-v molecules. The long lifetime of the trapped molecules and atoms, combined with their high density, allows us to study interactions such as atom-molecule or molecule-molecule collisions in the ultracold regime. A preliminary experiment on Rb2 molecular decay is described. ^