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

arXiv:1704.04134 · Physical Review A

Anisotropic optical trapping as a manifestation of the complex electronic structure of ultracold lanthanide atoms: The example of holmium

Hui Li, Jean-François Wyart, Olivier Dulieu, Maxence Lepers

Published Jun 26, 2017• 29 citations
arXivHTMLPDF

Abstract

The efficiency of optical trapping is determined by the atomic dynamic dipole polarizability, whose real and imaginary parts are associated with the potential energy and photon-scattering rate, respectively. In this article we develop a formalism to calculate analytically the real and imaginary parts of the scalar, vector, and tensor polarizabilities of lanthanide atoms. We assume that the sum-over-state formula comprises only transitions involving electrons in the valence orbitals like $6s,\phantom{\rule{0.16em}{0ex}}5d,\phantom{\rule{0.16em}{0ex}}6p$, and $7s$, while transitions involving $4f$ core electrons are neglected. Applying this formalism to the ground level of configuration $4{f}^{q}6{s}^{2}$, we restrict the sum to transitions implying the $4{f}^{q}6s6p$ configuration, which yields polarizabilities depending on two parameters: an effective transition energy and an effective transition dipole moment. Then, by introducing configuration-interaction mixing between $4{f}^{q}6s6p$ and other configurations, we demonstrate that the imaginary part of the scalar, vector, and tensor polarizabilities is very sensitive to configuration-interaction coefficients, whereas the real part is not. The magnitude and anisotropy of the photon-scattering rate are thus strongly related to the details of the atomic electronic structure. Those analytical results agree with our detailed electronic-structure calculations of the energy levels, Land\'e $g$ factors, transition probabilities, polarizabilities, and van der Waals ${C}_{6}$ coefficients, previously performed on erbium and dysprosium and presently performed on holmium. Our results show that, although the density of states decreases with increasing $q$, the configuration interaction between $4{f}^{q}6s6p,\phantom{\rule{0.16em}{0ex}}4{f}^{q\ensuremath{-}1}5d6{s}^{2}$, and $4{f}^{q\ensuremath{-}1}5{d}^{2}6s$ is surprisingly stronger in erbium ($q=12$) than in holmium ($q=11$), itself stronger than in dysprosium ($q=10$).

DOI
10.1103/physreva.95.062508

Related papers

Optical trapping of ultracold dysprosium atoms: transition probabilities, dynamic dipole polarizabilities and van der Waals C 6 coefficients

H Li, Jean-François Wyart, Olivier Dulieu +2 more

Abstract The efficiency of the optical trapping of ultracold atoms depends on the atomic dynamic dipole polarizability governing the atom-field interaction. In this article, we have calculated the real and imaginary parts of the dynamic dipole polarizability of dysprosium in the ground and first excited levels. Due to the high electronic angular momentum of those two states, the polarizabilities possess scalar, vector and tensor contributions that we have computed, on a wide range of trapping wavelengths, using the sum-over-state formula. Using the same formalism, we have also calculated the C 6 coefficients characterizing the van der Waals interaction between two dysprosium atoms in the two lowest levels. We have computed the energies of excited states and the transition probabilities appearing in the sums, using a combination of ab initio and least-square-fitting techniques provided by the Cowan codes and extended in our group. Regarding the real part of the polarizability, for field frequencies far from atomic resonances, the vector and tensor contributions are two-orders-of-magnitude smaller than the scalar contribution, whereas for the imaginary part, the vector and tensor contributions represent a noticeable fraction of the scalar contribution. Finally, our anisotropic C 6 coefficients are much smaller than those published in the literature.

arXiv:1607.05628•Journal of Physics B Atomic Molecular and Optical Physics•Dec 15, 2016•46 citations

Anisotropic optical trapping of ultracold erbium atoms

Maxence Lepers, Jean-François Wyart, Olivier Dulieu

Ultracold atoms confined in a dipole trap are submitted to a potential whose depth is proportional to the real part of their dynamic dipole polarizability. The atoms also experience photon scattering whose rate is proportional to the imaginary part of their dynamic dipole polarizability. In this article we calculate the complex dynamic dipole polarizability of ground-state erbium, a rare-earth atom that was recently Bose condensed. The polarizability is calculated with the sum-over-state formula inherent to second-order perturbation theory. The summation is performed on transition energies and transition dipole moments from ground-state erbium, which are computed using the Racah-Slater least-squares fitting procedure provided by the cowan codes. This allows us to predict nine unobserved odd-parity energy levels of total angular momentum $J=5$, 6, and 7, in the range 25 000--31 000 cm${}^{\ensuremath{-}1}$ above the ground state. Regarding the trapping potential, we find that ground-state erbium essentially behaves like a spherically symmetric atom, in spite of its large electronic angular momentum. We also find a mostly isotropic van der Waals interaction between two ground-state erbium atoms, characterized by a coefficient ${C}_{6}^{\mathrm{iso}}=1760$ a.u. To the contrary, the photon-scattering rate shows a pronounced anisotropy since it strongly depends on the polarization of the trapping light.

arXiv:1310.6149•Physical Review A•Feb 5, 2014•48 citations

van der Waals coefficients for interactions of dysprosium and erbium atoms with alkali-metal and alkaline-earth-metal atoms

Klaudia Zaremba-Kopczyk, Michał Tomza, Maxence Lepers

The long-range part of the interatomic interactions plays a substantial role in the collisional dynamics of ultracold gases. Here, we report on the calculation of the isotropic and anisotropic ${C}_{6}$ coefficients characterizing the van der Waals interaction between dysprosium or erbium atoms in the two lowest energy levels and the ground-state alkali-metal (Li, Na, K, Rb, Cs, Fr) or alkaline-earth-metal (Be, Mg, Ca, Sr, Ba) atoms. The calculations are done using the integral of dynamic dipole polarizabilities at so-called imaginary frequencies of the two interacting atoms. For all atom pairs, we find that the isotropic ${C}_{6}$ coefficients are two or three orders of magnitude larger than the anisotropic ones. Those coefficients are essential for modeling collisional properties of heteronuclear quantum mixtures containing highly magnetic dysprosium or erbium atoms and alkali-metal or alkaline-earth-metal atoms.

arXiv:2311.17745•Physical Review A•Jan 18, 2024•0 citations

Anisotropic polarizability of erbium atoms

Jan Hendrik Becher, Simon Baier, K. Aikawa +4 more

We report on the determination of the dynamical polarizability of ultracold erbium atoms in the ground and in one excited state at three different wavelengths, which are particularly relevant for optical trapping. Our study combines experimental measurements of the light shift and theoretical calculations. In particular, our experimental approach allows us to isolate the different contributions to the polarizability, namely, the isotropic scalar and anisotropic tensor part. For the latter contribution, we observe a clear dependence of the atomic polarizability on the angle between the laser-field-polarization axis and the quantization axis, set by the external magnetic field. Such an angle dependence is particularly pronounced in the excited-state polarizability. We compare our experimental findings with the theoretical values, based on semiempirical electronic structure calculations, and we observe a very good overall agreement. Our results pave the way to exploit the anisotropy of the tensor polarizability for spin-selective preparation and manipulation.

arXiv:1710.07162•Physical Review A•Jan 19, 2018•46 citations

Dynamic polarizabilities of rare-earth-metal atoms and dispersion coefficients for their interaction with helium atoms

Xi Chu, A. Dalgarno, Gerrit C. Groenenboom

The dynamic scalar and tensor polarizabilities of the rare-earth-metal atoms are calculated with time-dependent density functional theory. The frequency-dependent polarizabilities at imaginary frequencies are used to determine the isotropic and orientation-dependent van der Waals coefficients for the interactions of the rare-earth-metal atoms with helium atoms. The static polarizabilities are compared with other theoretical values and with experimental results. The agreement is satisfactory in most cases but there are some exceptions where the discrepancy between theory and experiment is significant. The derived isotropic van der Waals coefficients range between 37 and 50 ${E}_{H}{a}_{0}^{6}$ and the orientation-dependent coefficients between 2 and $\ensuremath{-}1$ ${E}_{H}{a}_{0}^{6}$. Thus the ratio of elastic to inelastic scattering cross sections is expected to be substantial and any one of the rare-earth-metal atoms is an excellent candidate for trapping and cooling in a He gas.

Physical Review A•Mar 29, 2007•50 citations

Dynamic dipole polarizabilities of heteronuclear alkali dimers: optical response, trapping and control of ultracold molecules

Romain Véxiau, Dimitri Borsalino, Maxence Lepers +4 more

In this article we address the general approach for calculating dynamical dipole polarizabilities of small quantum systems, based on a sum-over-states formula involving in principle the entire energy spectrum of the system. We complement this method by a few-parameter model involving a limited number of effective transitions, allowing for a compact and accurate representation of both the isotropic and anisotropic components of the polarizability. We apply the method to the series of ten heteronuclear molecules composed of two of (Li,Na,K,Rb,Cs) alkali-metal atoms. We rely on both up-to-date spectroscopically-determined potential energy curves for the lowest electronic states, and on our systematic studies of these systems performed during the last decade for higher excited states and for permanent and transition dipole moments. Such a compilation is timely for the continuously growing researches on ultracold polar molecules. Indeed the knowledge of the dynamic dipole polarizabilities is crucial to model the optical response of molecules when trapped in optical lattices, and to determine optimal lattice frequencies ensuring optimal transfer to the absolute ground state of initially weakly-bound molecules. When they exist, we determine the so-called ‘magic frequencies’ where the ac-Stark shift and thus the viewed trap depth, is the same for both weakly-bound and ground-state molecules.

arXiv:1707.02168•International Reviews in Physical Chemistry•Aug 29, 2017•51 citations
The efficiency of optical trapping is determined by the atomic dynamic dipole polarizability, whose real and imaginary parts are associated with the potential energy and photon-scattering rate, respectively. In this article we develop a formalism to calculate analytically the real and imaginary parts of the scalar, vector, and tensor polarizabilities of lanthanide atoms. We assume that the sum-over-state formula comprises only transitions involving electrons in the valence orbitals like 6s,5d,6p6s,\phantom{\rule{0.16em}{0ex}}5d,\phantom{\rule{0.16em}{0ex}}6p6s,5d,6p, and 7s7s7s, while transitions involving 4f4f4f core electrons are neglected. Applying this formalism to the ground level of configuration 4fq6s24{f}^{q}6{s}^{2}4fq6s2, we restrict the sum to transitions implying the 4fq6s6p4{f}^{q}6s6p4fq6s6p configuration, which yields polarizabilities depending on two parameters: an effective transition energy and an effective transition dipole moment. Then, by introducing configuration-interaction mixing between 4fq6s6p4{f}^{q}6s6p4fq6s6p and other configurations, we demonstrate that the imaginary part of the scalar, vector, and tensor polarizabilities is very sensitive to configuration-interaction coefficients, whereas the real part is not. The magnitude and anisotropy of the photon-scattering rate are thus strongly related to the details of the atomic electronic structure. Those analytical results agree with our detailed electronic-structure calculations of the energy levels, Land'e ggg factors, transition probabilities, polarizabilities, and van der Waals C6{C}_{6}C6​ coefficients, previously performed on erbium and dysprosium and presently performed on holmium. Our results show that, although the density of states decreases with increasing qqq, the configuration interaction between 4fq6s6p,4fq\ensuremath−15d6s24{f}^{q}6s6p,\phantom{\rule{0.16em}{0ex}}4{f}^{q\ensuremath{-}1}5d6{s}^{2}4fq6s6p,4fq\ensuremath−15d6s2, and 4fq\ensuremath−15d26s4{f}^{q\ensuremath{-}1}5{d}^{2}6s4fq\ensuremath−15d26s is surprisingly stronger in erbium (q=12q=12q=12) than in holmium (q=11q=11q=11), itself stronger than in dysprosium (q=10q=10q=10).
Abstract The efficiency of the optical trapping of ultracold atoms depends on the atomic dynamic dipole polarizability governing the atom-field interaction. In this article, we have calculated the real and imaginary parts of the dynamic dipole polarizability of dysprosium in the ground and first excited levels. Due to the high electronic angular momentum of those two states, the polarizabilities possess scalar, vector and tensor contributions that we have computed, on a wide range of trapping wavelengths, using the sum-over-state formula. Using the same formalism, we have also calculated the C 6 coefficients characterizing the van der Waals interaction between two dysprosium atoms in the two lowest levels. We have computed the energies of excited states and the transition probabilities appearing in the sums, using a combination of ab initio and least-square-fitting techniques provided by the Cowan codes and extended in our group. Regarding the real part of the polarizability, for field frequencies far from atomic resonances, the vector and tensor contributions are two-orders-of-magnitude smaller than the scalar contribution, whereas for the imaginary part, the vector and tensor contributions represent a noticeable fraction of the scalar contribution. Finally, our anisotropic C 6 coefficients are much smaller than those published in the literature.
Ultracold atoms confined in a dipole trap are submitted to a potential whose depth is proportional to the real part of their dynamic dipole polarizability. The atoms also experience photon scattering whose rate is proportional to the imaginary part of their dynamic dipole polarizability. In this article we calculate the complex dynamic dipole polarizability of ground-state erbium, a rare-earth atom that was recently Bose condensed. The polarizability is calculated with the sum-over-state formula inherent to second-order perturbation theory. The summation is performed on transition energies and transition dipole moments from ground-state erbium, which are computed using the Racah-Slater least-squares fitting procedure provided by the cowan codes. This allows us to predict nine unobserved odd-parity energy levels of total angular momentum J=5J=5J=5, 6, and 7, in the range 25 000--31 000 cm${}^{\ensuremath{-}1}$ above the ground state. Regarding the trapping potential, we find that ground-state erbium essentially behaves like a spherically symmetric atom, in spite of its large electronic angular momentum. We also find a mostly isotropic van der Waals interaction between two ground-state erbium atoms, characterized by a coefficient C6iso=1760{C}_{6}^{\mathrm{iso}}=1760C6iso​=1760 a.u. To the contrary, the photon-scattering rate shows a pronounced anisotropy since it strongly depends on the polarization of the trapping light.
The long-range part of the interatomic interactions plays a substantial role in the collisional dynamics of ultracold gases. Here, we report on the calculation of the isotropic and anisotropic C6{C}_{6}C6​ coefficients characterizing the van der Waals interaction between dysprosium or erbium atoms in the two lowest energy levels and the ground-state alkali-metal (Li, Na, K, Rb, Cs, Fr) or alkaline-earth-metal (Be, Mg, Ca, Sr, Ba) atoms. The calculations are done using the integral of dynamic dipole polarizabilities at so-called imaginary frequencies of the two interacting atoms. For all atom pairs, we find that the isotropic C6{C}_{6}C6​ coefficients are two or three orders of magnitude larger than the anisotropic ones. Those coefficients are essential for modeling collisional properties of heteronuclear quantum mixtures containing highly magnetic dysprosium or erbium atoms and alkali-metal or alkaline-earth-metal atoms.
We report on the determination of the dynamical polarizability of ultracold erbium atoms in the ground and in one excited state at three different wavelengths, which are particularly relevant for optical trapping. Our study combines experimental measurements of the light shift and theoretical calculations. In particular, our experimental approach allows us to isolate the different contributions to the polarizability, namely, the isotropic scalar and anisotropic tensor part. For the latter contribution, we observe a clear dependence of the atomic polarizability on the angle between the laser-field-polarization axis and the quantization axis, set by the external magnetic field. Such an angle dependence is particularly pronounced in the excited-state polarizability. We compare our experimental findings with the theoretical values, based on semiempirical electronic structure calculations, and we observe a very good overall agreement. Our results pave the way to exploit the anisotropy of the tensor polarizability for spin-selective preparation and manipulation.
The dynamic scalar and tensor polarizabilities of the rare-earth-metal atoms are calculated with time-dependent density functional theory. The frequency-dependent polarizabilities at imaginary frequencies are used to determine the isotropic and orientation-dependent van der Waals coefficients for the interactions of the rare-earth-metal atoms with helium atoms. The static polarizabilities are compared with other theoretical values and with experimental results. The agreement is satisfactory in most cases but there are some exceptions where the discrepancy between theory and experiment is significant. The derived isotropic van der Waals coefficients range between 37 and 50 EHa06{E}_{H}{a}_{0}^{6}EH​a06​ and the orientation-dependent coefficients between 2 and $\ensuremath{-}1$ EHa06{E}_{H}{a}_{0}^{6}EH​a06​. Thus the ratio of elastic to inelastic scattering cross sections is expected to be substantial and any one of the rare-earth-metal atoms is an excellent candidate for trapping and cooling in a He gas.
In this article we address the general approach for calculating dynamical dipole polarizabilities of small quantum systems, based on a sum-over-states formula involving in principle the entire energy spectrum of the system. We complement this method by a few-parameter model involving a limited number of effective transitions, allowing for a compact and accurate representation of both the isotropic and anisotropic components of the polarizability. We apply the method to the series of ten heteronuclear molecules composed of two of (Li,Na,K,Rb,Cs) alkali-metal atoms. We rely on both up-to-date spectroscopically-determined potential energy curves for the lowest electronic states, and on our systematic studies of these systems performed during the last decade for higher excited states and for permanent and transition dipole moments. Such a compilation is timely for the continuously growing researches on ultracold polar molecules. Indeed the knowledge of the dynamic dipole polarizabilities is crucial to model the optical response of molecules when trapped in optical lattices, and to determine optimal lattice frequencies ensuring optimal transfer to the absolute ground state of initially weakly-bound molecules. When they exist, we determine the so-called ‘magic frequencies’ where the ac-Stark shift and thus the viewed trap depth, is the same for both weakly-bound and ground-state molecules.