Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Ground-state potential of the Ca dimer from Fourier-transform spectroscopy

Olivier Allard, A. Pashov, H. Knöckel, E. Tiemann

Published Oct 10, 2002• 73 citations
Publisher

Abstract

The $B{}^{1}{\ensuremath{\Sigma}}_{\mathrm{u}}^{+}\ensuremath{\rightarrow}X{}^{1}{\ensuremath{\Sigma}}_{\mathrm{g}}^{+}$ system in ${}^{40}{\mathrm{Ca}}_{2}$ was studied by Fourier-transform spectroscopy. Transitions to 730 ground-state levels were induced by the blue-green lines of an ${\mathrm{Ar}}^{+}$ radiation and the single-mode radiation of a frequency-doubled continuous-wave Nd:YAG yttrium aluminum garnet laser. An accurate potential-energy curve, which covers 99.8% of the well depth, was derived directly from the experimentally observed differences between ground-state levels. Long-range analysis led to a revised value of the ground-state dissociation energy, $1102.08(9){\mathrm{cm}}^{\ensuremath{-}1},$ which differs by more than $7{\mathrm{cm}}^{\ensuremath{-}1}$ from the previous experimental determination [C. R. Vidal, J. Chem. Phys. 72, 1864 (1980)]. A first estimate of the s-wave scattering length is reported.

DOI
10.1103/physreva.66.042503

Related papers

Cold collisions of ground-state calcium atoms in a laser field: A theoretical study

Béatrice Bussery‐Honvault, Jean–Michel Launay, Robert Moszyński

State-of-the-art ab initio techniques have been applied to compute the potential-energy curves for the ground $X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ and excited ${}^{1}{\ensuremath{\Pi}}_{g}(4s3d)$ states of the calcium dimer in the Born-Oppenheimer approximation. The weakly bound ground state was calculated by symmetry-adapted perturbation theory, while the strongly bound excited state was computed using a combination of the linear-response theory within the coupled-cluster singles and doubles framework for the core-valence electronic correlation and of the full configuration interaction for the valence-valence correlation. The ground-state potential has been corrected by considering the relativistic terms resulting from the first-order many-electron Breit theory, and the retardation corrections. The magnetic electronic transition dipole moment governing the ${}^{1}{\ensuremath{\Pi}}_{g}{\ensuremath{\leftarrow}}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ transitions has been obtained as the first residue of the polarization propagator computed with the coupled-cluster method restricted to single and double excitations. The computed energies and transition moments have been analytically fitted and used in the dynamical calculations of the rovibrational energy levels, ground-state scattering length, photoassociation intensities at ultralow temperatures, and spontaneous emission coefficients from the ${}^{1}{\ensuremath{\Pi}}_{g}(4s3d)$ to the $X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ state. The spectroscopic constants of the theoretical ground-state potential are in a good agreement with the experimental values derived from the Fourier-transform spectra [O. Allard et al., Eur. Phys. J. D (to be published)]. The theoretical s-wave scattering length for the ground state is $a=44\mathrm{bohrs},$ suggesting that it should be possible to obtain a stable Bose-Einstein condensate of calcium atoms. Finally, the computed photoassociation intensities and spontaneous emission coefficients suggest that it should be possible to obtain cold calcium molecules by photoassociation of cold atoms to the first ${}^{1}{\ensuremath{\Pi}}_{g}$ state followed by a spontaneous emission to the ground state.

Physical Review A•Sep 23, 2003•51 citations

Ground-state properties of Ca2 from narrow-line two-color photoassociation

Evgenij Pachomow, Veit Peter Dahlke, E. Tiemann +2 more

By two-color photoassociation of $^{40}\mathrm{Ca}$ four weakly bound vibrational levels in the ${\mathrm{Ca}}_{2} X{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Sigma}}}_{g}^{+}$ ground-state potential were measured, using highly spin-forbidden transitions to intermediate states of the coupled system ${^{3}\mathrm{\ensuremath{\Pi}}}_{u}$ and ${^{3}\mathrm{\ensuremath{\Sigma}}}_{u}^{+}$ near the ${^{3}P}_{1}+{^{1}S}_{0}$ asymptote. From the observed binding energies, including the least bound state, the long-range dispersion coefficients ${\mathrm{C}}_{6},{\mathrm{C}}_{8},{\mathrm{C}}_{10}$ and a precise value for the $s$-wave scattering length of 308.5(50)${a}_{0}$ were derived. From mass scaling we also calculated the corresponding scattering length for other natural isotopes. From the Autler-Townes splitting of the spectra, the molecular Rabi frequency has been determined as a function of the laser intensity for one bound-bound transition. The observed value for the Rabi frequency is in good agreement with calculated transition moments based on the derived potentials, assuming a dipole moment being independent of internuclear separation for the atomic pair model.

arXiv:1702.00710•Physical Review A•Apr 25, 2017•14 citations

Re-examination of the Cs2 ground singlet X1Σg+ and triplet a3Σu+ states

V. B. Sovkov, Feng Xie, A. M. Lyyra +3 more

This paper clarifies the disagreement in the depth of the potential energy curve of the cesium dimer singlet ground state which has lasted for nearly a decade. We point out that the origin of this disagreement must be a technical misprint in the values of the three binding energies reported by Danzl et al. [Science 321, 1062 (2008)], while the X1Σg+ state potential reported by Coxon and Hajigeorgiou [J. Chem. Phys. 132, 094105 (2010)], based on experimental data by Amiot and Dulieu [J. Chem. Phys. 117, 5155 (2002)], is quite correct. We have recalculated the potential energy function of the triplet ground state a3Σu+ by using the available experimental data spanning both the attractive and the repulsive branches so that the potential energy function complies asymptotically with the singlet ground state X1Σg+ potential energy function by Coxon and Hajigeorgiou. This is important for the simulation of the near dissociation properties such as Feshbach resonances, which are typically observed in modern experiments with ultracold atoms and molecules.

The Journal of Chemical Physics•Sep 12, 2017•18 citations

Improved Models for the Potential Energy Functions of the Ground Singlet and Lowest-Lying Triplet States of the Cesium Dimer

Jesse Baldwin

The Morse/Long Range (MLR) potential has become one of the most reliable and highly \nused potential energy functions for diatomic molecules. It includes the theoretical long range \nbehaviour that diatomic molecules are known to exhibit as they approach the dissociation \nlimit. Heavy alkali metals with adjacent electronic states often exhibit strong coupling \nbetween the spin and orbital angular momentum. The ground state X¹Σg⁺ and the lowest \nlying triplet state aᶟΣᵤ⁺ of Cs₂ exhibit such coupling effects and as a result, modeling the \nhighest vibrational states of these states is a non-trivial problem. Utilizing scattering length \nvalues obtained from published analysis of 60 Feshbach resonances, the correct form of the \npotential energy function was determined. Moreover, the scattering length values were used \nto determine the correct leading dispersion coefficient that describes the true form of the \nlong-range potential energy functions. All previous attempts to determine global potential \nenergy functions for these states have considered only the optical spectroscopic data. This is \nthe first ever effort attempting to use scattering lengths determined from cold atom collision \nexperiments in a combined analysis with conventional spectroscopic data.

•Aug 23, 2012•1 citation

Coupling of the XΣ+1 and aΣ+3 states of KRb

A. Pashov, O. Docenko, M. Tamanis +3 more

A comprehensive study of the electronic states at the $4s+5s$ asymptote in $\mathrm{KRb}$ is presented. Abundant spectroscopic data on the $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ state were collected by Fourier-transform spectroscopy, which allows one to determine an accurate experimental potential energy curve up to $14.8\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$. The existing data set [C. Amiot et al., J. Chem. Phys. 112, 7068 (2000)] on the ground state $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ was extended by several additional levels lying close to the atomic asymptote. In a coupled channels fitting routine complete molecular potentials for both electronic states were fitted. Along with the line frequencies of the molecular transitions, recently published positions of Feshbach resonances in $^{40}\mathrm{K}$ and $^{87}\mathrm{Rb}$ mixtures [F. Ferlaino et al., Phys. Rev. A 74, 039903 (2006)] were included in the fit. This makes the derived potential curves capable for an accurate description of observed cold collision features so far. Predictions of scattering lengths and Feshbach resonances in other isotopic combinations are reported.

arXiv:0706.2849•Physical Review A•Aug 30, 2007•111 citations

High precision spectroscopy on beams of atoms and dimers

Stephan Falke

This thesis is concerned with long range behavior of the interaction potential of two neutral atoms.Two aspects are of special interest: First, how the long range behavior changes if one of the atoms is replaced by another isotope.Second, how to model the influence of a laser field on long range interaction.Several high precision spectroscopy experiments on particle beams are carried out.The transition frequencies of the D lines of 39,40,41 K are measured with frequency comb generated by a self-referenced femtosecond laser, which is locked to a Cs-clock.Discrepancies in the literature are removed and new values for the hyperfine structures of the 4p 1/2 and 4p 3/2 state and the isotope shift of both D lines are derive.On a beam of potassium dimers the A 1 Σ + u state of K 2 is studied up to the dissociation limit complementing and extending photoassociation spectroscopy of ultracold ensembles.Local perturbation due to the spin-orbit coupling with the b 3 Π u state are identified.Molecular hyperfine structure of levels close to the asymptote is observed and modeled by atomic parameter only.Improved values of the atomic radiative lifetime of the states 4p 1/2 and 4p 3/2 are derived from the long range parameter C 3 of the molecule.Also, an improved value for the dissociation energy of the X 1 Σ + g ground state is obtained and recipes for the investigation of cold collision via spectroscopy of the molecular structure at the ground state asymptotes are derived.For the first time asymptotic levels of two isotopomers of K 2 are studied and a comparison of asymptotic spectra of two isotopomers 39 K 2 and 39 K 41 K shows that corrections to the Born-Oppenheimer potential are needed.Optical coupling between asymptotic ground state levels and long range levels of an excited state has been suggested as tool for influencing cold collisions.Here, we model lineshapes for an experiment which investigates such a coupling systematically in Na 2 by laser spectroscopy in a beam.A fit program for the lineshape is used to compare experimental and theoretical values and to check the validity of the applied model from multichannel quantum defect theory.The dependency on the detuning of the coupling laser is reproduced but other details failed.To allow similar studies are reported for alkali metals on the simpler quantum system from alkali earth elements the development of an oven for a continuous beam of Ca 2 for Doppler-free spectroscopy is discussed including a major change of design and first experiences are reported.The heating of the oven is, unlike those for the beams applied above, outside the vacuum tank because of the high temperatures needed for the oven.

•Jan 1, 2007•1 citation
The B1\ensuremathΣu+\ensuremath→X1\ensuremathΣg+B{}^{1}{\ensuremath{\Sigma}}_{\mathrm{u}}^{+}\ensuremath{\rightarrow}X{}^{1}{\ensuremath{\Sigma}}_{\mathrm{g}}^{+}B1\ensuremathΣu+​\ensuremath→X1\ensuremathΣg+​ system in 40Ca2{}^{40}{\mathrm{Ca}}_{2}40Ca2​ was studied by Fourier-transform spectroscopy. Transitions to 730 ground-state levels were induced by the blue-green lines of an Ar+{\mathrm{Ar}}^{+}Ar+ radiation and the single-mode radiation of a frequency-doubled continuous-wave Nd:YAG yttrium aluminum garnet laser. An accurate potential-energy curve, which covers 99.8% of the well depth, was derived directly from the experimentally observed differences between ground-state levels. Long-range analysis led to a revised value of the ground-state dissociation energy, 1102.08(9)cm\ensuremath−1,1102.08(9){\mathrm{cm}}^{\ensuremath{-}1},1102.08(9)cm\ensuremath−1, which differs by more than 7cm\ensuremath−17{\mathrm{cm}}^{\ensuremath{-}1}7cm\ensuremath−1 from the previous experimental determination [C. R. Vidal, J. Chem. Phys. 72, 1864 (1980)]. A first estimate of the s-wave scattering length is reported.
State-of-the-art ab initio techniques have been applied to compute the potential-energy curves for the ground $X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ and excited ${}^{1}{\ensuremath{\Pi}}_{g}(4s3d)$ states of the calcium dimer in the Born-Oppenheimer approximation. The weakly bound ground state was calculated by symmetry-adapted perturbation theory, while the strongly bound excited state was computed using a combination of the linear-response theory within the coupled-cluster singles and doubles framework for the core-valence electronic correlation and of the full configuration interaction for the valence-valence correlation. The ground-state potential has been corrected by considering the relativistic terms resulting from the first-order many-electron Breit theory, and the retardation corrections. The magnetic electronic transition dipole moment governing the ${}^{1}{\ensuremath{\Pi}}_{g}{\ensuremath{\leftarrow}}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ transitions has been obtained as the first residue of the polarization propagator computed with the coupled-cluster method restricted to single and double excitations. The computed energies and transition moments have been analytically fitted and used in the dynamical calculations of the rovibrational energy levels, ground-state scattering length, photoassociation intensities at ultralow temperatures, and spontaneous emission coefficients from the ${}^{1}{\ensuremath{\Pi}}_{g}(4s3d)$ to the $X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ state. The spectroscopic constants of the theoretical ground-state potential are in a good agreement with the experimental values derived from the Fourier-transform spectra [O. Allard et al., Eur. Phys. J. D (to be published)]. The theoretical s-wave scattering length for the ground state is a=44bohrs,a=44\mathrm{bohrs},a=44bohrs, suggesting that it should be possible to obtain a stable Bose-Einstein condensate of calcium atoms. Finally, the computed photoassociation intensities and spontaneous emission coefficients suggest that it should be possible to obtain cold calcium molecules by photoassociation of cold atoms to the first ${}^{1}{\ensuremath{\Pi}}_{g}$ state followed by a spontaneous emission to the ground state.
By two-color photoassociation of 40Ca^{40}\mathrm{Ca}40Ca four weakly bound vibrational levels in the ${\mathrm{Ca}}_{2} X{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Sigma}}}_{g}^{+}$ ground-state potential were measured, using highly spin-forbidden transitions to intermediate states of the coupled system ${^{3}\mathrm{\ensuremath{\Pi}}}_{u}$ and ${^{3}\mathrm{\ensuremath{\Sigma}}}_{u}^{+}$ near the 3P1+1S0{^{3}P}_{1}+{^{1}S}_{0}3P1​+1S0​ asymptote. From the observed binding energies, including the least bound state, the long-range dispersion coefficients C6,C8,C10{\mathrm{C}}_{6},{\mathrm{C}}_{8},{\mathrm{C}}_{10}C6​,C8​,C10​ and a precise value for the sss-wave scattering length of 308.5(50)a0{a}_{0}a0​ were derived. From mass scaling we also calculated the corresponding scattering length for other natural isotopes. From the Autler-Townes splitting of the spectra, the molecular Rabi frequency has been determined as a function of the laser intensity for one bound-bound transition. The observed value for the Rabi frequency is in good agreement with calculated transition moments based on the derived potentials, assuming a dipole moment being independent of internuclear separation for the atomic pair model.
This paper clarifies the disagreement in the depth of the potential energy curve of the cesium dimer singlet ground state which has lasted for nearly a decade. We point out that the origin of this disagreement must be a technical misprint in the values of the three binding energies reported by Danzl et al. [Science 321, 1062 (2008)], while the X1Σg+ state potential reported by Coxon and Hajigeorgiou [J. Chem. Phys. 132, 094105 (2010)], based on experimental data by Amiot and Dulieu [J. Chem. Phys. 117, 5155 (2002)], is quite correct. We have recalculated the potential energy function of the triplet ground state a3Σu+ by using the available experimental data spanning both the attractive and the repulsive branches so that the potential energy function complies asymptotically with the singlet ground state X1Σg+ potential energy function by Coxon and Hajigeorgiou. This is important for the simulation of the near dissociation properties such as Feshbach resonances, which are typically observed in modern experiments with ultracold atoms and molecules.
The Morse/Long Range (MLR) potential has become one of the most reliable and highly \nused potential energy functions for diatomic molecules. It includes the theoretical long range \nbehaviour that diatomic molecules are known to exhibit as they approach the dissociation \nlimit. Heavy alkali metals with adjacent electronic states often exhibit strong coupling \nbetween the spin and orbital angular momentum. The ground state X¹Σg⁺ and the lowest \nlying triplet state aᶟΣᵤ⁺ of Cs₂ exhibit such coupling effects and as a result, modeling the \nhighest vibrational states of these states is a non-trivial problem. Utilizing scattering length \nvalues obtained from published analysis of 60 Feshbach resonances, the correct form of the \npotential energy function was determined. Moreover, the scattering length values were used \nto determine the correct leading dispersion coefficient that describes the true form of the \nlong-range potential energy functions. All previous attempts to determine global potential \nenergy functions for these states have considered only the optical spectroscopic data. This is \nthe first ever effort attempting to use scattering lengths determined from cold atom collision \nexperiments in a combined analysis with conventional spectroscopic data.
A comprehensive study of the electronic states at the 4s+5s4s+5s4s+5s asymptote in KRb\mathrm{KRb}KRb is presented. Abundant spectroscopic data on the $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ state were collected by Fourier-transform spectroscopy, which allows one to determine an accurate experimental potential energy curve up to 14.8A˚14.8\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}14.8A˚. The existing data set [C. Amiot et al., J. Chem. Phys. 112, 7068 (2000)] on the ground state $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ was extended by several additional levels lying close to the atomic asymptote. In a coupled channels fitting routine complete molecular potentials for both electronic states were fitted. Along with the line frequencies of the molecular transitions, recently published positions of Feshbach resonances in 40K^{40}\mathrm{K}40K and 87Rb^{87}\mathrm{Rb}87Rb mixtures [F. Ferlaino et al., Phys. Rev. A 74, 039903 (2006)] were included in the fit. This makes the derived potential curves capable for an accurate description of observed cold collision features so far. Predictions of scattering lengths and Feshbach resonances in other isotopic combinations are reported.
This thesis is concerned with long range behavior of the interaction potential of two neutral atoms.Two aspects are of special interest: First, how the long range behavior changes if one of the atoms is replaced by another isotope.Second, how to model the influence of a laser field on long range interaction.Several high precision spectroscopy experiments on particle beams are carried out.The transition frequencies of the D lines of 39,40,41 K are measured with frequency comb generated by a self-referenced femtosecond laser, which is locked to a Cs-clock.Discrepancies in the literature are removed and new values for the hyperfine structures of the 4p 1/2 and 4p 3/2 state and the isotope shift of both D lines are derive.On a beam of potassium dimers the A 1 Σ + u state of K 2 is studied up to the dissociation limit complementing and extending photoassociation spectroscopy of ultracold ensembles.Local perturbation due to the spin-orbit coupling with the b 3 Π u state are identified.Molecular hyperfine structure of levels close to the asymptote is observed and modeled by atomic parameter only.Improved values of the atomic radiative lifetime of the states 4p 1/2 and 4p 3/2 are derived from the long range parameter C 3 of the molecule.Also, an improved value for the dissociation energy of the X 1 Σ + g ground state is obtained and recipes for the investigation of cold collision via spectroscopy of the molecular structure at the ground state asymptotes are derived.For the first time asymptotic levels of two isotopomers of K 2 are studied and a comparison of asymptotic spectra of two isotopomers 39 K 2 and 39 K 41 K shows that corrections to the Born-Oppenheimer potential are needed.Optical coupling between asymptotic ground state levels and long range levels of an excited state has been suggested as tool for influencing cold collisions.Here, we model lineshapes for an experiment which investigates such a coupling systematically in Na 2 by laser spectroscopy in a beam.A fit program for the lineshape is used to compare experimental and theoretical values and to check the validity of the applied model from multichannel quantum defect theory.The dependency on the detuning of the coupling laser is reproduced but other details failed.To allow similar studies are reported for alkali metals on the simpler quantum system from alkali earth elements the development of an oven for a continuous beam of Ca 2 for Doppler-free spectroscopy is discussed including a major change of design and first experiences are reported.The heating of the oven is, unlike those for the beams applied above, outside the vacuum tank because of the high temperatures needed for the oven.