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:2011.01978 · Entropy

Random Matrix Theory Analysis of a Temperature-Related Transformation in Statistics of Fano–Feshbach Resonances in Thulium Atoms

Emil T. Davletov, V. V. Tsyganok, Vladimir Khlebnikov, Daniil A. Pershin, Alexey V. Akimov

Published Dec 10, 2020• 0 citations
arXivHTMLPDF

Abstract

Recently, the transformation from random to chaotic behavior in the statistics of Fano-Feshbach resonances was observed in thulium atoms with rising ensemble temperature. We performed random matrix theory simulations of such spectra and analyzed the resulting statistics in an attempt to understand the mechanism of the transformation. Our simulations show that, when evaluated in terms of the Brody parameter, resonance statistics do not change or change insignificantly when higher temperature resonances are appended to the statistics. In the experiments evaluated, temperature was changed simultaneously with optical dipole trap depth. Thus, simulations included the Stark shift based on the known polarizability of the free atoms and assuming their polarizability remains the same in the bound state. Somewhat surprisingly, we found that, while including the Stark shift does lead to minor statistical changes, it does not change the resonance statistics and, therefore, is not responsible for the experimentally observed statistic transformation. This observation suggests that either our assumption regarding the polarizability of Feshbach molecules is poor or that an additional mechanism changes the statistics and leads to more chaotic statistical behavior.

DOI
10.3390/e22121394

Related papers

Random to Chaotic Statistic Transformation in Low-Field Fano-Feshbach Resonances of Cold Thulium Atoms

Vladimir Khlebnikov, D. A. Pershin, V. V. Tsyganok +5 more

Here, we report on the observation of a random to chaotic temperature transformation in the statistics of nearest-neighbor spacings of Fano-Feshbach resonances in the ultracold polarized gas of thulium-169 atoms. We associate this transformation to the appearance of so-called d resonances as well as the shift of other resonances with the temperature. In addition to this statistical change, it has been observed that the characters of s- and d-resonance temperature shifts are quite different: s resonances experience almost no shift or even negative shift with the temperature, while d resonances experience an obvious positive shift. The sign change was attributed to the difference in polarizability of Feshbach molecules and free thulium atoms. In addition, careful analysis of the broad Fano-Feshbach resonances enabled the determination of the sign of thulium's background scattering length. A rethermalization experiment made it possible to estimate a length value of a_{bg}=+144±38 a.u. This indicates that thulium atoms are suitable for achieving Bose-Einstein condensation.

arXiv:1902.00201•Physical Review Letters•Nov 19, 2019•37 citations

Characterizing the temperature dependence of Fano-Feshbach resonances of ultracold polarized thulium

Vladimir Khlebnikov, V. V. Tsyganok, D. A. Pershin +3 more

Recent studies demonstrated anomalous temperature shifts for some Fano-Feshbach resonances of thulium atoms. These anomalies were explained by the variation in light intensity in the optical dipole trap, which accompanied changes in temperature. In addition, a temperature-related transformation of the statistics of the inter-resonance spacing was demonstrated [Aikawa et al., Phys. Rev. Lett. 108, 210401 (2012)]. Here, we analyze the shifts of isolated $s$- and $d$-type Fano-Feshbach resonances of ultracold thulium atoms with temperature for a fixed depth of an optical dipole trap. The measurements are consistent with the three-body recombination-based theory of the temperature-related resonance shift and enable the extraction of the resonance parameters, particularly the magnetic moments of closed channel states. This parameter and the known polarizability of the open channel enable us to separate the contributions of the temperature and Stark shift to the overall shift of the resonances and show the dominant role of the Stark effect in the overall shift.

arXiv:2012.09223•Physical Review A•Feb 4, 2021•10 citations

Spectral statistics of molecular resonances in erbium isotopes: How chaotic are they?

Jordi Mur-Petit, Rafael A. Molina

We perform a comprehensive analysis of the spectral statistics of the molecular resonances in (166)Er and (168)Er observed in recent ultracold collision experiments [Frisch et al., Nature (London) 507, 475 (2014)] with the aim of determining the chaoticity of this system. We calculate different independent statistical properties to check their degree of agreement with random matrix theory (RMT), and analyze if they are consistent with the possibility of having missing resonances. The analysis of the short-range fluctuations as a function of the magnetic field points to a steady increase of chaoticity until B∼30 G. The repulsion parameter decreases for higher magnetic fields, an effect that can be interpreted as due to missing resonances. The analysis of long-range fluctuations allows us to be more quantitative and estimate a 20%-25% fraction of missing levels. Finally, a study of the distribution of resonance widths provides additional evidence supporting missing resonances of small width compared with the experimental magnetic field resolution. We conclude that further measurements with increased resolution will be necessary to give a final answer to the problem of missing resonances and the agreement with RMT.

arXiv:1506.08086•Physical Review E•Oct 5, 2015•46 citations

Approach to chaos in ultracold atomic and molecular physics: Statistics of near-threshold bound states for Li+CaH and Li+CaF

Matthew D. Frye, Masato Morita, Christophe L. Vaillant +2 more

We calculate near-threshold bound states and Feshbach resonance positions for atom--rigid-rotor models of the highly anisotropic systems Li+CaH and Li+CaF. We perform statistical analysis on the resonance positions to compare with the predictions of random matrix theory. For Li+CaH with total angular momentum $J=0$ we find fully chaotic behavior in both the nearest-neighbor spacing distribution and the level number variance. However, for $J>0$ we find different behavior due to the presence of a nearly conserved quantum number. Li+CaF $(J=0)$ also shows apparently reduced levels of chaotic behavior despite its stronger effective coupling. This may indicate the development of another good quantum number relating to a bending motion of the complex. However, continuously varying the rotational constant over a wide range shows unexpected structure in the degree of chaotic behavior, including a dramatic reduction around the rotational constant of CaF. This demonstrates the complexity of the relationship between coupling and chaotic behavior.

arXiv:1511.08744•Physical Review A•May 23, 2016•29 citations

Ab Initio Random Matrix Theory of Molecular Electronic Structure

Zhen Tao, Victor Galitski

We use ab initio electronic-structure methods to investigate random-matrix theory (RMT) universality in molecular electronic structure. Using single-reference electronic structure methods, including Hartree-Fock, configuration-interaction singles (CIS), density functional theory, and linear-response time-dependent density-functional theory, we compute single-particle orbital energies and many-electron excitations of several representative molecules (benzene, alanine, 1-phenylethylamine, methyloxirane, and helicene chains). For generic low-symmetry geometries, the unfolded spectra of these ab initio Hamiltonians exhibit Wigner-Dyson level statistics of the Gaussian orthogonal ensemble (GOE). For extended helicene chains we explicitly restrict to bound valence excitations below the ionization threshold and still observe GOE statistics, indicating that the RMT universality is present for physical states of direct relevance to real molecules. We further explore the electric and magnetic field dependence of the molecular electronic spectra. The variance of electric polarizability (level curvature K) is predicted to be non-analytic in the magnetic field which serves as an infrared cutoff, <K^2> proportional to log(1/|B|). We observe a transition to the Gaussian unitary ensemble (GUE) by increasing the magnetic fields, although it occurs only at magnetic fields far beyond experimentally accessible scales. Our results indicate that random matrix universality provides a general framework for organizing ab initio predictions of interacting electron spectra in complex systems.

cond-mat.str-elCross-listedarXiv:2602.21299•Feb 26, 2026•0 citations

The loss spectrum of cold polarized thulium atoms in optical dipole trap at low magnetic fields

Vladimir Khlebnikov, D. A. Pershin, V. V. Tsyganok +5 more

Here we report on observing losses of thulium atoms from optical dipole trap at different magnetic fields. Loss spectra in range of 0 - 24 G magnetic fields were obtained at two temperatures of 2.2 and 12 µK with 14 mG resolution. The spectrum at higher temperature has richer structure showing the appearance of new Fano-Feschbach resonances.

AIP conference proceedings•Jan 1, 2020•1 citation
Recently, the transformation from random to chaotic behavior in the statistics of Fano-Feshbach resonances was observed in thulium atoms with rising ensemble temperature. We performed random matrix theory simulations of such spectra and analyzed the resulting statistics in an attempt to understand the mechanism of the transformation. Our simulations show that, when evaluated in terms of the Brody parameter, resonance statistics do not change or change insignificantly when higher temperature resonances are appended to the statistics. In the experiments evaluated, temperature was changed simultaneously with optical dipole trap depth. Thus, simulations included the Stark shift based on the known polarizability of the free atoms and assuming their polarizability remains the same in the bound state. Somewhat surprisingly, we found that, while including the Stark shift does lead to minor statistical changes, it does not change the resonance statistics and, therefore, is not responsible for the experimentally observed statistic transformation. This observation suggests that either our assumption regarding the polarizability of Feshbach molecules is poor or that an additional mechanism changes the statistics and leads to more chaotic statistical behavior.
Here, we report on the observation of a random to chaotic temperature transformation in the statistics of nearest-neighbor spacings of Fano-Feshbach resonances in the ultracold polarized gas of thulium-169 atoms. We associate this transformation to the appearance of so-called d resonances as well as the shift of other resonances with the temperature. In addition to this statistical change, it has been observed that the characters of s- and d-resonance temperature shifts are quite different: s resonances experience almost no shift or even negative shift with the temperature, while d resonances experience an obvious positive shift. The sign change was attributed to the difference in polarizability of Feshbach molecules and free thulium atoms. In addition, careful analysis of the broad Fano-Feshbach resonances enabled the determination of the sign of thulium's background scattering length. A rethermalization experiment made it possible to estimate a length value of a_{bg}=+144±38 a.u. This indicates that thulium atoms are suitable for achieving Bose-Einstein condensation.
Recent studies demonstrated anomalous temperature shifts for some Fano-Feshbach resonances of thulium atoms. These anomalies were explained by the variation in light intensity in the optical dipole trap, which accompanied changes in temperature. In addition, a temperature-related transformation of the statistics of the inter-resonance spacing was demonstrated [Aikawa et al., Phys. Rev. Lett. 108, 210401 (2012)]. Here, we analyze the shifts of isolated sss- and ddd-type Fano-Feshbach resonances of ultracold thulium atoms with temperature for a fixed depth of an optical dipole trap. The measurements are consistent with the three-body recombination-based theory of the temperature-related resonance shift and enable the extraction of the resonance parameters, particularly the magnetic moments of closed channel states. This parameter and the known polarizability of the open channel enable us to separate the contributions of the temperature and Stark shift to the overall shift of the resonances and show the dominant role of the Stark effect in the overall shift.
We perform a comprehensive analysis of the spectral statistics of the molecular resonances in (166)Er and (168)Er observed in recent ultracold collision experiments [Frisch et al., Nature (London) 507, 475 (2014)] with the aim of determining the chaoticity of this system. We calculate different independent statistical properties to check their degree of agreement with random matrix theory (RMT), and analyze if they are consistent with the possibility of having missing resonances. The analysis of the short-range fluctuations as a function of the magnetic field points to a steady increase of chaoticity until B∼30 G. The repulsion parameter decreases for higher magnetic fields, an effect that can be interpreted as due to missing resonances. The analysis of long-range fluctuations allows us to be more quantitative and estimate a 20%-25% fraction of missing levels. Finally, a study of the distribution of resonance widths provides additional evidence supporting missing resonances of small width compared with the experimental magnetic field resolution. We conclude that further measurements with increased resolution will be necessary to give a final answer to the problem of missing resonances and the agreement with RMT.
We calculate near-threshold bound states and Feshbach resonance positions for atom--rigid-rotor models of the highly anisotropic systems Li+CaH and Li+CaF. We perform statistical analysis on the resonance positions to compare with the predictions of random matrix theory. For Li+CaH with total angular momentum J=0J=0J=0 we find fully chaotic behavior in both the nearest-neighbor spacing distribution and the level number variance. However, for J>0J>0J>0 we find different behavior due to the presence of a nearly conserved quantum number. Li+CaF (J=0)(J=0)(J=0) also shows apparently reduced levels of chaotic behavior despite its stronger effective coupling. This may indicate the development of another good quantum number relating to a bending motion of the complex. However, continuously varying the rotational constant over a wide range shows unexpected structure in the degree of chaotic behavior, including a dramatic reduction around the rotational constant of CaF. This demonstrates the complexity of the relationship between coupling and chaotic behavior.
We use ab initio electronic-structure methods to investigate random-matrix theory (RMT) universality in molecular electronic structure. Using single-reference electronic structure methods, including Hartree-Fock, configuration-interaction singles (CIS), density functional theory, and linear-response time-dependent density-functional theory, we compute single-particle orbital energies and many-electron excitations of several representative molecules (benzene, alanine, 1-phenylethylamine, methyloxirane, and helicene chains). For generic low-symmetry geometries, the unfolded spectra of these ab initio Hamiltonians exhibit Wigner-Dyson level statistics of the Gaussian orthogonal ensemble (GOE). For extended helicene chains we explicitly restrict to bound valence excitations below the ionization threshold and still observe GOE statistics, indicating that the RMT universality is present for physical states of direct relevance to real molecules. We further explore the electric and magnetic field dependence of the molecular electronic spectra. The variance of electric polarizability (level curvature K) is predicted to be non-analytic in the magnetic field which serves as an infrared cutoff, <K^2> proportional to log(1/|B|). We observe a transition to the Gaussian unitary ensemble (GUE) by increasing the magnetic fields, although it occurs only at magnetic fields far beyond experimentally accessible scales. Our results indicate that random matrix universality provides a general framework for organizing ab initio predictions of interacting electron spectra in complex systems.