Statistics of electromagnetic transitions as a signature of chaos in many-electron atoms

Abstract

Using a configuration interaction approach we study statistics of the dipole matrix elements (E1 amplitudes) between the 14 lower states with J π = 4 − and 21st to 100th even states with J = 4 in the Ce atom (1120 lines). We show that the distribution of the matrix elements is close to Gaussian, although the width of the Gaussian distribution, i.e. the root-mean-square matrix element, changes with the excitation energy. The corresponding line strengths are distributed according to the Porter-Thomas law which describes statistics of transition strengths between chaotic states in compound nuclei. We also show how to use a statistical theory to calculate mean squared values of the matrix elements or transition amplitudes between chaotic many-body states. We draw some support for our conclusions from the analysis of the 228 experimental line strengths in Ce [J. Opt. Soc. Am. 8, 1545(1991], although direct comparison with the calculations is impeded by incompleteness of the experimental data. Nevertheless, the statistics observed evidence that highly excited many-electron states in atoms are indeed chaotic.

Quantum chaos in many-body systems: what can we learn from the Ce atom?

“…He studied configuration compositions of the eigenstates of the Ce atom using data from the tables [29], and came to the conclusion that the "eigenfunctions are random superpositions of some few basic states". Inspired by that work we conducted an extensive numerical study of the spectra and eigenstates of complex open-shell atoms, using the rare-earth atom of Ce as an example [8,16,[30][31][32]. The present paper summarizes our earlier findings, as well as gives new insights into the problem of quantum chaos in a real many-body system.…”

“…A statistical approach to calculation of meansquared values of the matrix elements between chaotic many-body states has been developed and described in [8,17,18,32], and we will only present the results here. SupposeM is a one-body operator 8M = α,β α|m|β a † α a β ≡ α,β m αβραβ , which causes transitions between the chaotic states a and b. Eq.…”

“…The expression for the mean-squared reduced matrix element retains the structure of Eqs. (15) and (16), but must be modified (see [8,32]). The result will depend on the rank k of the irreducible spherical tensor operatorM.…”

Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles

“…These formed the main motivation for the present study. Let us add that in the past, besides deriving the dilute limit formulas for the bivariate moments of the transition strength densities in some situations [24,25], there are suggestions of using a polynomial expansion theory [26] (later in [24] it was shown that the polynomial expansion starts with the GOE result and hence in general inappropriate) for transition strengths, a specialized theory for one-body transition operators [18,[27][28][29] and using the bivariate t-distribution form for transition strength densities [30].…”

Prediction of quantum many-body chaos in the protactinium atom

“…Chaos allows one to develop statistical theory and calculate matrix elements of different operators between extremely complex many-body states including electromagnetic transition probabilities and probabilities of other processes -see e.g. [11][12][13][14][15][16][17].…”