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:2112.11883 · Physical Review A

Determination of the scattering length of erbium atoms

A. Patscheider, Lauriane Chomaz, G. Natale, Daniel Petter, Manfred J. Mark, Simon Baier, Bing Yang, Reuben R. W. Wang, John L. Bohn, Francesca Ferlaino

Published Jun 8, 2022• 27 citations
arXivHTMLPDF

Abstract

An accurate knowledge of the scattering length is fundamental in ultracold quantum gas experiments and essential for the characterization of the system as well as for a meaningful comparison to theoretical models. Here, we perform a careful characterization of the $s$-wave scattering length ${a}_{\mathrm{s}}$ for the four highest-abundance isotopes of erbium, in the magnetic field range from 0 to 5 G. We report on cross-dimensional thermalization measurements and apply the Enskog equations of change to numerically simulate the thermalization process and to analytically extract an expression for the so-called number of collisions per rethermalization (NCPR) to obtain ${a}_{\mathrm{s}}$ from our experimental data. We benchmark the applied cross-dimensional thermalization technique with the experimentally more demanding lattice modulation spectroscopy and find good agreement for our parameter regime. Our experiments are compatible with a dependence of the NCPR with ${a}_{\mathrm{s}}$, as theoretically expected in the case of strongly dipolar gases. Surprisingly, we experimentally observe a dependency of the NCPR on the density, which might arise due to deviations from an ideal harmonic trapping configuration. Finally, we apply a model for the dependency of the background scattering length with the isotope mass, allowing us to estimate the number of bound states of erbium.

DOI
10.1103/physreva.105.063307

Related papers

Determination of the s -Wave Scattering Length of Chromium

Piet O. Schmidt, S. Hensler, J. Werner +4 more

We have measured the deca-triplet $s$-wave scattering length of the bosonic chromium isotopes $^{52}\mathrm{C}\mathrm{r}$ and $^{50}\mathrm{C}\mathrm{r}$. From the time constants for cross-dimensional thermalization in ultracold atomic samples, we have determined the magnitudes $|a(^{52}\mathrm{C}\mathrm{r})|=(170\ifmmode\pm\else\textpm\fi{}39){a}_{0}$ and $|a(^{50}\mathrm{C}\mathrm{r})|=(40\ifmmode\pm\else\textpm\fi{}15){a}_{0}$, where ${a}_{0}=0.053\text{ }\mathrm{n}\mathrm{m}$. By measuring the rethermalization rate of $^{52}\mathrm{C}\mathrm{r}$ over a wide temperature range and comparing the temperature dependence with the effective-range theory and numerical single-channel calculations, we have obtained strong evidence that the sign of $a(^{52}\mathrm{C}\mathrm{r})$ is positive. Rescaling our $^{52}\mathrm{C}\mathrm{r}$ model potential to $^{50}\mathrm{C}\mathrm{r}$ strongly suggests that $a(^{50}\mathrm{C}\mathrm{r})$ is positive also.

arXiv:quant-ph/0303069•Physical Review Letters•Nov 3, 2003•49 citations

s -wave scattering lengths of the strongly dipolar bosons Dy162 and Dy164

Yijun Tang, Andrew Sykes, Nathaniel Burdick +2 more

We report the measurement of the deca-heptuplet $s$-partial-wave scattering length $a$ of two bosonic isotopes of the highly magnetic element dysprosium: $a=112(10){a}_{0}$ for $^{162}\mathrm{Dy}$ and $a=92(8){a}_{0}$ for $^{164}\mathrm{Dy}$, where ${a}_{0}$ is the Bohr radius. The scattering lengths are determined by the cross-dimensional relaxation of ultracold gases of these Dy isotopes at temperatures above quantum degeneracy. In this temperature regime, the measured rethermalization dynamics can be compared to simulations of the Boltzmann equation using a direct-simulation Monte Carlo method employing the anisotropic differential scattering cross section of dipolar particles.

arXiv:1506.03393•Physical Review A•Aug 10, 2015•89 citations

Scattering properties of ultra-cold chromium atoms

Piet O. Schmidt

In this work a gas of ultra-cold chromium atoms in a magnetic trap has been prepared and its elastic and inelastic scattering properties have been investigated with regard to Bose-Einstein condensation. Bose-Einstein condensation of dilute atomic gases is achieved using different cooling and trapping techniques. Deviating from the standard way we were able to devise a continuous loading mechanism for a magnetic trap as a result of the spectroscopic properties and the large magnetic dipole moment of chromium. It not only allows us to trap more atoms but also facilitates the subsequent preparation steps. After loading, the magnetic trap is compressed to increase the atomic density. In doing so we also increases the temperature of the atomic cloud. Doppler cooling of the atoms in the compressed trap reduces the temperature and increases the density. Subsequent evaporative cooling further reduces the temperature. The efficiency of the cooling mechanism is determined by the elastic and inelastic scattering properties of atomic species. Ultra-cold collisions between ground state atoms are dominated by s-wave collisions. They are characterized by a single parameter, the scattering length a. We were able to determine the temperature dependence of the elastic collision rate for the two bosonic isotopes 52Cr and 50Cr in a relaxation experiment. Comparing our results with the effective range theory allowed us not only to extract the magnitude of the scattering length, but also its sign. The sign of the scattering length is important since only condensates with a positive scattering length are stable. Our efforts to achieve Bose-Einstein condensation in chromium by evaporative cooling in a magnetic trap resulted in a maximum phase space density of 0.04. Further cooling reduced the phase space density due to increased loss of atoms from dipolar relaxation. We were able to obtain preliminary results on the magnetic field dependence of dipolar relaxation. Our experimental data is in excellent agreement with theory. These experiments confirm the theoretical prediction that the dipolar relaxation rate is independent from the details of the interaction potential but rather scales with the magnetic offset field and the magnetic dipole moment of the atoms. The findings of this work mark an important step towards the realization of a Bose-Einstein condensate with chromium atoms. Especially the knowledge of the elastic and inelastic scattering properties allow now to devise a successful strategy.

•Jan 1, 2003•5 citations

Measurement of elastic cross section for cold cesium collisions

S. A. Hopkins, S. C. Webster, J. Arlt +4 more

We have measured the time taken for a magnetically trapped cloud of cold cesium atoms in the ${(F=3,m}_{F}=\ensuremath{-}3)$ ground state to rethermalize from a nonequilibrium spatial and velocity distribution. From these measurements we infer the dependences of the elastic scattering cross section on temperature and magnetic field in the ranges $1\ensuremath{-}30\ensuremath{\mu}\mathrm{K}$ and 0.05--2.0 mT, respectively. We determine a lower bound on the magnitude of the $(3,\ensuremath{-}3)+(3,\ensuremath{-}3)$ s-wave scattering length of ${940a}_{0}.$

Physical Review A•Feb 11, 2000•39 citations

Scattering lengths in isotopologues of the RbYb system

Mateusz Borkowski, Piotr Szymon Żuchowski, R. Ciuryło +7 more

We model the binding energies of rovibrational levels of the RbYb molecule using experimental data from two-color photoassociation spectroscopy in mixtures of ultracold ${}^{87}$Rb with various Yb isotopes. The model uses a theoretical potential based on state-of-the-art ab initio potentials, further improved by least-squares fitting to the experimental data. We have fixed the number of bound states supported by the potential curve, so that the model is mass scaled, that is, it accurately describes the bound-state energies for all measured isotopic combinations. Such a model enables an accurate prediction of the $s$-wave scattering lengths of all isotopic combinations of the RbYb system. The reduced mass range is broad enough to cover the full scattering lengths range from $\ensuremath{-}\ensuremath{\infty}$ to $+\ensuremath{\infty}$. For example, the ${}^{87}$Rb${}^{174}$Yb system is characterized by a large positive scattering length of $+880(120)$ a.u., while ${}^{87}$Rb${}^{173}$Yb has $a=\ensuremath{-}626(88)$ a.u. On the other hand ${}^{87}$Rb${}^{170}$Yb has a very small scattering length of $\ensuremath{-}11.5(2.5)$ a.u. confirmed by the pair's extremely low thermalization rate. For isotopic combinations including ${}^{85}$Rb the variation of the interspecies scattering lengths is much smoother ranging from $+39.0(1.6)$ a.u. for ${}^{85}$Rb${}^{176}$Yb to $+230(12)$ a.u. in the case of ${}^{85}$Rb${}^{168}$Yb. Hyperfine corrections to these scattering lengths are also given. We further complement the fitted potential with interaction parameters calculated from alternative methods. The recommended value of the van der Waals coefficient is ${C}_{6}=2837(13)$ a.u. agrees with but is more precise than the current state-of-the-art theoretical predictions [M. S. Safronova, S. G. Porsev, and C. W. Clark, Phys. Rev. Lett. 109, 230802 (2012)].

arXiv:1309.3131•Physical Review A•Nov 20, 2013•64 citations

Feshbach resonances in cold atomic gases

E. G. M. van Kempen

The research described in this thesis is part of an international effort with the aim of studying ultracold quantum degenerate gas samples like Bose-Einstein condensates and Fermi degenerate systems, consisting of atoms confined in magnetic or optical traps. The behavior of such samples is governed by the inter-atomic interactions and the resulting properties of the atom-atom scattering. At the relevant temperatures of 1nK to 10µK a key property is the atom-atom scattering length a. In chapter 2 a theoretical method is presented which enables one to describe the interaction and scattering of (ultra)cold atoms to unprecedented precision. It is also unparalleled in comprehensiveness: it allows the prediction of a large and varied set of experimental data for all isotopes of the same element. The method relies on the extraction from experiments of a few (phase) parameters which completely summarize the behavior of the atoms in the (ultra)cold regime. In chapter 3 the method is applied to the "workhorses" of cold-atom physics: the atomic species 85Rb and 87Rb. We extract the foregoing parameters to a very high precision from several recent high precision experiments, allowing us to predict e.g. the 87Rb spinor condensate to be ferromagnetic: a prediction for which the scattering length has to be calculated with a precision better than 1%. We also predict Feshbach resonances at experimentally accessible magnetic field strengths; resonances searched for and found by the experimental group of Rempe. In close collaboration with his group we "fine-tune" the interaction parameters found previously, by making use of only one of the observed resonances. We then obtain agreement with 42 out of the observed 43 resonance field strengths and are able to identify bound states inducing the Feshbach resonances at these locations. Chapter 4 describes the results of this research. With a thorough understanding of the rubidium interactions, we then switch to lithium which has a fermionic (6Li) and a bosonic (7Li) isotope. Both are being used in cold-atom experiments. In chapter 5 we evaluate the interaction parameters for lithium allowing us to predict magnetic field strengths for which a sample of fermionic 6Li atoms can be regarded as strongly interacting. Furthermore, a three-level method for measuring mean-field shifts, based on radio-frequency techniques, is introduced. For weak interactions we find proportionality of resonance shifts to interaction strengths. In the strongly interacting regime, however, these shifts become very small reflecting the quantum unitarity limit and many-body effects. Most interesting is the fact that in this regime the shifts are small both for large positive a and for large negative a, likely reflecting the universality of the interaction energy. In chapters 6 and 7 the interactions between lithium atoms are reinvestigated, making use of newly available experimental data and with the updated interaction parameters special attention is paid to locating field strengths at which magnetically tunable Feshbach resonances occur in the scattering of lithium atoms. In chapter 6 scattering events in a gas of (fermionic) 6Li atoms are studied. In chapter 7 we show that a consistent description of the 6Li+6Li system can be given. We discuss theoretical uncertainties for the position of the wide 6Li Feshbach resonance and present an analytic scattering model for this resonance, based on the inclusion of a field-dependent virtual open-channel state. We predict new Feshbach resonances for the 6Li-7Li system, and their importance for different types of crossover superfluidity models is discussed. Molecules created by magnetically sweeping over these resonances will have a fermionic character. One magnetic field strength is predicted at which two different fermionic molecules can be created simultaneously.

•Jan 1, 2006•1 citation
An accurate knowledge of the scattering length is fundamental in ultracold quantum gas experiments and essential for the characterization of the system as well as for a meaningful comparison to theoretical models. Here, we perform a careful characterization of the sss-wave scattering length as{a}_{\mathrm{s}}as​ for the four highest-abundance isotopes of erbium, in the magnetic field range from 0 to 5 G. We report on cross-dimensional thermalization measurements and apply the Enskog equations of change to numerically simulate the thermalization process and to analytically extract an expression for the so-called number of collisions per rethermalization (NCPR) to obtain as{a}_{\mathrm{s}}as​ from our experimental data. We benchmark the applied cross-dimensional thermalization technique with the experimentally more demanding lattice modulation spectroscopy and find good agreement for our parameter regime. Our experiments are compatible with a dependence of the NCPR with as{a}_{\mathrm{s}}as​, as theoretically expected in the case of strongly dipolar gases. Surprisingly, we experimentally observe a dependency of the NCPR on the density, which might arise due to deviations from an ideal harmonic trapping configuration. Finally, we apply a model for the dependency of the background scattering length with the isotope mass, allowing us to estimate the number of bound states of erbium.
We have measured the deca-triplet sss-wave scattering length of the bosonic chromium isotopes 52Cr^{52}\mathrm{C}\mathrm{r}52Cr and 50Cr^{50}\mathrm{C}\mathrm{r}50Cr. From the time constants for cross-dimensional thermalization in ultracold atomic samples, we have determined the magnitudes $|a(^{52}\mathrm{C}\mathrm{r})|=(170\ifmmode\pm\else\textpm\fi{}39){a}_{0}$ and $|a(^{50}\mathrm{C}\mathrm{r})|=(40\ifmmode\pm\else\textpm\fi{}15){a}_{0}$, where a0=0.053 nm{a}_{0}=0.053\text{ }\mathrm{n}\mathrm{m}a0​=0.053 nm. By measuring the rethermalization rate of 52Cr^{52}\mathrm{C}\mathrm{r}52Cr over a wide temperature range and comparing the temperature dependence with the effective-range theory and numerical single-channel calculations, we have obtained strong evidence that the sign of a(52Cr)a(^{52}\mathrm{C}\mathrm{r})a(52Cr) is positive. Rescaling our 52Cr^{52}\mathrm{C}\mathrm{r}52Cr model potential to 50Cr^{50}\mathrm{C}\mathrm{r}50Cr strongly suggests that a(50Cr)a(^{50}\mathrm{C}\mathrm{r})a(50Cr) is positive also.
We report the measurement of the deca-heptuplet sss-partial-wave scattering length aaa of two bosonic isotopes of the highly magnetic element dysprosium: a=112(10)a0a=112(10){a}_{0}a=112(10)a0​ for 162Dy^{162}\mathrm{Dy}162Dy and a=92(8)a0a=92(8){a}_{0}a=92(8)a0​ for 164Dy^{164}\mathrm{Dy}164Dy, where a0{a}_{0}a0​ is the Bohr radius. The scattering lengths are determined by the cross-dimensional relaxation of ultracold gases of these Dy isotopes at temperatures above quantum degeneracy. In this temperature regime, the measured rethermalization dynamics can be compared to simulations of the Boltzmann equation using a direct-simulation Monte Carlo method employing the anisotropic differential scattering cross section of dipolar particles.
In this work a gas of ultra-cold chromium atoms in a magnetic trap has been prepared and its elastic and inelastic scattering properties have been investigated with regard to Bose-Einstein condensation. Bose-Einstein condensation of dilute atomic gases is achieved using different cooling and trapping techniques. Deviating from the standard way we were able to devise a continuous loading mechanism for a magnetic trap as a result of the spectroscopic properties and the large magnetic dipole moment of chromium. It not only allows us to trap more atoms but also facilitates the subsequent preparation steps. After loading, the magnetic trap is compressed to increase the atomic density. In doing so we also increases the temperature of the atomic cloud. Doppler cooling of the atoms in the compressed trap reduces the temperature and increases the density. Subsequent evaporative cooling further reduces the temperature. The efficiency of the cooling mechanism is determined by the elastic and inelastic scattering properties of atomic species. Ultra-cold collisions between ground state atoms are dominated by s-wave collisions. They are characterized by a single parameter, the scattering length a. We were able to determine the temperature dependence of the elastic collision rate for the two bosonic isotopes 52Cr and 50Cr in a relaxation experiment. Comparing our results with the effective range theory allowed us not only to extract the magnitude of the scattering length, but also its sign. The sign of the scattering length is important since only condensates with a positive scattering length are stable. Our efforts to achieve Bose-Einstein condensation in chromium by evaporative cooling in a magnetic trap resulted in a maximum phase space density of 0.04. Further cooling reduced the phase space density due to increased loss of atoms from dipolar relaxation. We were able to obtain preliminary results on the magnetic field dependence of dipolar relaxation. Our experimental data is in excellent agreement with theory. These experiments confirm the theoretical prediction that the dipolar relaxation rate is independent from the details of the interaction potential but rather scales with the magnetic offset field and the magnetic dipole moment of the atoms. The findings of this work mark an important step towards the realization of a Bose-Einstein condensate with chromium atoms. Especially the knowledge of the elastic and inelastic scattering properties allow now to devise a successful strategy.
We have measured the time taken for a magnetically trapped cloud of cold cesium atoms in the ${(F=3,m}_{F}=\ensuremath{-}3)$ ground state to rethermalize from a nonequilibrium spatial and velocity distribution. From these measurements we infer the dependences of the elastic scattering cross section on temperature and magnetic field in the ranges $1\ensuremath{-}30\ensuremath{\mu}\mathrm{K}$ and 0.05--2.0 mT, respectively. We determine a lower bound on the magnitude of the $(3,\ensuremath{-}3)+(3,\ensuremath{-}3)$ s-wave scattering length of 940a0.{940a}_{0}.940a0​.
We model the binding energies of rovibrational levels of the RbYb molecule using experimental data from two-color photoassociation spectroscopy in mixtures of ultracold 87{}^{87}87Rb with various Yb isotopes. The model uses a theoretical potential based on state-of-the-art ab initio potentials, further improved by least-squares fitting to the experimental data. We have fixed the number of bound states supported by the potential curve, so that the model is mass scaled, that is, it accurately describes the bound-state energies for all measured isotopic combinations. Such a model enables an accurate prediction of the sss-wave scattering lengths of all isotopic combinations of the RbYb system. The reduced mass range is broad enough to cover the full scattering lengths range from $\ensuremath{-}\ensuremath{\infty}$ to $+\ensuremath{\infty}$. For example, the 87{}^{87}87Rb174{}^{174}174Yb system is characterized by a large positive scattering length of +880(120)+880(120)+880(120) a.u., while 87{}^{87}87Rb173{}^{173}173Yb has $a=\ensuremath{-}626(88)$ a.u. On the other hand 87{}^{87}87Rb170{}^{170}170Yb has a very small scattering length of $\ensuremath{-}11.5(2.5)$ a.u. confirmed by the pair's extremely low thermalization rate. For isotopic combinations including 85{}^{85}85Rb the variation of the interspecies scattering lengths is much smoother ranging from +39.0(1.6)+39.0(1.6)+39.0(1.6) a.u. for 85{}^{85}85Rb176{}^{176}176Yb to +230(12)+230(12)+230(12) a.u. in the case of 85{}^{85}85Rb168{}^{168}168Yb. Hyperfine corrections to these scattering lengths are also given. We further complement the fitted potential with interaction parameters calculated from alternative methods. The recommended value of the van der Waals coefficient is C6=2837(13){C}_{6}=2837(13)C6​=2837(13) a.u. agrees with but is more precise than the current state-of-the-art theoretical predictions [M. S. Safronova, S. G. Porsev, and C. W. Clark, Phys. Rev. Lett. 109, 230802 (2012)].
The research described in this thesis is part of an international effort with the aim of studying ultracold quantum degenerate gas samples like Bose-Einstein condensates and Fermi degenerate systems, consisting of atoms confined in magnetic or optical traps. The behavior of such samples is governed by the inter-atomic interactions and the resulting properties of the atom-atom scattering. At the relevant temperatures of 1nK to 10µK a key property is the atom-atom scattering length a. In chapter 2 a theoretical method is presented which enables one to describe the interaction and scattering of (ultra)cold atoms to unprecedented precision. It is also unparalleled in comprehensiveness: it allows the prediction of a large and varied set of experimental data for all isotopes of the same element. The method relies on the extraction from experiments of a few (phase) parameters which completely summarize the behavior of the atoms in the (ultra)cold regime. In chapter 3 the method is applied to the "workhorses" of cold-atom physics: the atomic species 85Rb and 87Rb. We extract the foregoing parameters to a very high precision from several recent high precision experiments, allowing us to predict e.g. the 87Rb spinor condensate to be ferromagnetic: a prediction for which the scattering length has to be calculated with a precision better than 1%. We also predict Feshbach resonances at experimentally accessible magnetic field strengths; resonances searched for and found by the experimental group of Rempe. In close collaboration with his group we "fine-tune" the interaction parameters found previously, by making use of only one of the observed resonances. We then obtain agreement with 42 out of the observed 43 resonance field strengths and are able to identify bound states inducing the Feshbach resonances at these locations. Chapter 4 describes the results of this research. With a thorough understanding of the rubidium interactions, we then switch to lithium which has a fermionic (6Li) and a bosonic (7Li) isotope. Both are being used in cold-atom experiments. In chapter 5 we evaluate the interaction parameters for lithium allowing us to predict magnetic field strengths for which a sample of fermionic 6Li atoms can be regarded as strongly interacting. Furthermore, a three-level method for measuring mean-field shifts, based on radio-frequency techniques, is introduced. For weak interactions we find proportionality of resonance shifts to interaction strengths. In the strongly interacting regime, however, these shifts become very small reflecting the quantum unitarity limit and many-body effects. Most interesting is the fact that in this regime the shifts are small both for large positive a and for large negative a, likely reflecting the universality of the interaction energy. In chapters 6 and 7 the interactions between lithium atoms are reinvestigated, making use of newly available experimental data and with the updated interaction parameters special attention is paid to locating field strengths at which magnetically tunable Feshbach resonances occur in the scattering of lithium atoms. In chapter 6 scattering events in a gas of (fermionic) 6Li atoms are studied. In chapter 7 we show that a consistent description of the 6Li+6Li system can be given. We discuss theoretical uncertainties for the position of the wide 6Li Feshbach resonance and present an analytic scattering model for this resonance, based on the inclusion of a field-dependent virtual open-channel state. We predict new Feshbach resonances for the 6Li-7Li system, and their importance for different types of crossover superfluidity models is discussed. Molecules created by magnetically sweeping over these resonances will have a fermionic character. One magnetic field strength is predicted at which two different fermionic molecules can be created simultaneously.