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arXiv:physics/0701123 · Physical Review A

Grid methods for cold molecules: Determination of photoassociation line shapes and rate constants

Shimshon Kallush, Ronnie Kosloff, Françoise Masnou-Seeuws

Published Apr 17, 2007• 14 citations
arXivHTMLPDF

Abstract

A description of photoassociation by cw laser is formulated in the framework of grid methods. The Hamiltonian describing one or several bound states coupled to a multiple of continuum manifolds via a radiative field is written in the energy representation and diagonalized. The generality of the treatment allows us to compute accurately and efficiently physical properties such as nonlinear high-intensity energy shifts, line shapes, and photoassociation rates both for isolated and nonisolated resonances. Application is given to sodium photoassociation in the experimental conditions of McKenzie et al. [Phys. Rev. Lett. 88, 090403 (2002)]. Inverted regions for the dependency of the rate vs the intensity and nonsymmetric line shapes are predicted to occur above the saturation limit. Comparison with the model of Bohn and Julienne [Phys. Rev. A 60, 414 (1999)] is discussed.

DOI
10.1103/physreva.75.043404

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A description of photoassociation by cw laser is formulated in the framework of grid methods. The Hamiltonian describing one or several bound states coupled to a multiple of continuum manifolds via a radiative field is written in the energy representation and diagonalized. The generality of the treatment allows us to compute accurately and efficiently physical properties such as nonlinear high-intensity energy shifts, line shapes, and photoassociation rates both for isolated and nonisolated resonances. Application is given to sodium photoassociation in the experimental conditions of McKenzie et al. [Phys. Rev. Lett. 88, 090403 (2002)]. Inverted regions for the dependency of the rate vs the intensity and nonsymmetric line shapes are predicted to occur above the saturation limit. Comparison with the model of Bohn and Julienne [Phys. Rev. A 60, 414 (1999)] is discussed.
We derive ultracold atom-atom photoassociation line shapes valid for intense light fields and investigate laser power effects in sodium and rubidium photoassociation to the purely long-range ${0}_{g}^{\ensuremath{-}}$ symmetry state. We consider intensities up to a few hundreds W/cm2,{\mathrm{W}/\mathrm{c}\mathrm{m}}^{2},W/cm2, a strongly saturating intensity for typical transitions of experimental interest. For these intensities the ${0}_{g}^{\ensuremath{-}}$ rotational spectrum is still well resolved; however, it is essential to couple the photoassociation resonance to both s- and d-wave ground-state channels. A low-energy d-wave shape resonance can have a profound effect on the line shape. Understanding the line shape is essential for precision spectroscopic analysis and could improve the extraction of ground-state scattering properties such as scattering lengths.
A theoretical analysis of photoassociation in ultracold metastable helium has been undertaken for the 2(^3)S1 + 2(^3)S1 to 2(^3)S1 + 2(^3)P transition. A full multichannel calculation is presented for two aspects of the system, a) the accurate calculation of energies for the bound vibrational levels of the dimer; and b) the non-perturbative calculation of the photoassociation line shapes allowing for abitrary laser intensity and detuning. A strong emphasis is placed on the accuracy of the calculations, and the elimination of approximations that have been used in previous investigations. The results indicate that full multichannel calculations are required rather than approximate single-channel calculations in some important situations, and are especially useful for making assignments between experiment and theory. Bound levels have been previously measured from the photoassociation of cold metastable thermal gases to energies close to the three asymptotes j = 0, 1, 2 of the 2(^3)S1 → 2(^3)Pj atomic transition. Previous analysis of these measurements was made using either approximate accumulated phase techniques based on long-range single-channel potentials, or using multichannel calculations and long-range potentials. Because accurate short-range electronic potentials have been calculated only recently, this thesis presents the first multichannel calculation for the levels near the j = 1, 2 asymptotes. The new short-range potentials had previously been used in an approximate single-channel calculation, but this type of calculation ignores the effects of non-adiabatic and Coriolis couplings. Presented here are the binding energies calculated from a fully multichannel technique which includes these couplings, and some differences of up to 10% from the single-channel calculations are observed. The detailed knowledge of the bound wave functions is then exploited to determine a set of observability criteria that produce a near unique assignment of theoretical levels to experimental observations. From these unique assignments, a 1% correction to the short-range electronic potentials has been implemented that significantly improves the agreement between theory and experiment. Experimental measurements have previously been made of line shifts in the photoassociation spectra of vibrational levels in the 0+u long-range potential that asymptotes to 2(^3)S1 + 2(^3)P0 and their variation with laser intensity. These results were previously analysed by performing an approximate second-order perturbative calculation of the line shift that is valid only for low laser intensities. In this thesis a completely non-perturbative calculation of the spectra and line shapes for abitrary laser intensity is presented for photoassociation to the levels probed in experiment. The techniques of dressed state formalism and a modified radiative coupling are employed to properly treat the laser coupling for large interatomic separations. These calculations produce resonance profiles superimposed on a very significant background, a feature that is not seen in photoassociation calculations of other atomic species. From the calculated line shapes, the line shifts and line widths were determined for varying intensity and a small nonlinear intensity-dependence of up to 2.5% was found for the line shift. The effects of invoking the dipole approximation for the laser coupling for the long-range molecular system are analysed by solving the equations using the complete laser coupling that does not assume the dipole approximation. Important differences are obtained between the profiles with and without the dipole approximation, and the photoassociation resonances are found to be more discernible at larger intensities when the complete laser coupling is used.
Since the first observation of ultracold molecules in Orsay, investigation of new ways of creating ultracold molecules inside cold atom traps is a challenging task for both theoretical and experimental groups. In this paper, we review some of our recent results on formation processes using cw laser photoassociation of cold atoms, involving population of long‐range molecular potential wells, or resonant coupling. We implemented accurate numerical methods based on a mapped Fourier grid approach, to solve time‐independent Schrödinger equation, in order to obtain quantitative predictions for ultracold molecule formation rates and detection efficiency.
In this thesis I present the results of photoassociation measurements in ensembles of atomic calcium over an intensity range of below 1 W/cm2 up to 600 W/cm2. The temperature of the atoms was reduced to 1 μK by laser and evaporative cooling. Afterwards the two most-weakly bound states in the excited molecular potentials that dissociate to the asymtote 3P1+1S0 have been investigated regarding the shape and absolute rate of the resonances. The investigation showed that the measured rates differ by more than an order of magnitude compared to theoretical calculations. The predictions are based on a scattering formalism by Bohn & Julienne and wave functions derived from coupled-channel calculations. The shape of the resonances, the width in particular, agrees well with the numerical calculations. Especially the power broadening at high intensity is reproduced by the experiment. Further the light shift introduced by the photoassociation laser has been measured which also differs from the predictions. In the further course of the thesis the numerical simulations were improved by considering additional influences like the exact dependence of the Franck-Condon density and the light shift of the photoassociation laser as a function of the collision energy. Additionally the modeling of the experimental setup with a special emphasis on the volume of the optical trap and thus the density of the atomic ensemble was improved and the influence of varying trap depth and photoassociation duration has been studied. Investigating possible explanations for the discrepancy of the measured rates compared to theory, I show that by a modification of the coupling parameter and the spontaneous decay rate of the molecular state the rates measured in the experiment can be reproduced theoretically, with the restriction that at high intensity the calculated line shape no longer agrees with the measurement. In the last chapter an improved experimental apparatus is presented that by using an optical lattice suppresses thermal broadening and thus simplifies the analysis substantially.
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.