FSRCC two-valence calculations of clock transition properties, dipole polarizability and isotope shifts in Fermionic and Bosonic Sr

TL;DR

This work addresses the Sr clock transition problem by applying an ab initio two-valence FSRCC framework for $^{87}$Sr and $^{88}$Sr, complemented by PRCC for the ground-state polarizability and corrected for Breit, QED, and perturbative triples. It reports excitation energies in good agreement with experimental data and E1, M1, and hyperfine-structure matrix elements within experimental uncertainties, with the clock-state lifetime for $^{87}$Sr aligning with measurements and the $^{88}$Sr lifetime showing notable differences from model-potential results. The PRCC polarizability agrees with the measured value and validates the treatment of core and valence correlations, while isotope-shift parameters differ from MCDF, illustrating the importance of all-order correlation and relativity. Overall, the work provides robust ab initio benchmarks for Sr clock transitions, demonstrating that nonlinear CC effects, Breit, QED, and perturbative triples are essential for accuracy and enabling precision metrology and tests of fundamental physics.

Abstract

We employ an all-particle multireference Fock-space relativistic coupled-cluster (FSRCC) theory to study the $5s^2{\;^1}S_0 - 5s5p{\;^3P^o_0}$ clock transition in both Fermionic and Bosonic isotopes of Sr. We compute the excitation energies, E1 and M1 transition amplitudes, hyperfine reduced matrix elements, and isotope shifts using FSRCC theory. Further, we calculate the lifetime of the metastable clock states for $^{87}$Sr and $^{88}$Sr. Furthermore, we employ perturbed relativistic coupled-cluster (PRCC) theory to compute the ground state electric dipole polarizability of Sr. To improve the accuracy of our results, we incorporate the corrections from the relativistic and quantum electrodynamical (QED) effects, and perturbative triples to all our calculations. Our computed excitation energies are in good agreement with the experimental data for low-lying excited states. Our results for E1, M1 and HFS reduced matrix elements are within the experimental error bars, however, with slight difference from the previous calculations due to more accurate treatment of electron correlations in our calculation. Our computed lifetime of the clock state for $^{87}$Sr is within the error bars of the available experimental results, whereas for $^{88}$Sr, it is an order of magnitude smaller than the only available calculation using model potential. Our PRCC result for the ground state polarizability is in good agreement with the experiment, and smaller than previous calculations. As can be expected, our FSRCC results on isotope shift parameters show differences from the MCDF calculations. From the detailed analysis of our results, we find that the corrections from the Breit interaction, QED effects, and perturbative triples are crucial to get accurate clock transition properties in Sr.

Figures (8)

• Figure 1: The diagrammatic representation of closed-shell, one-valence and two-valence single and double CC operators.
• Figure 2: The Golstone diagrams contributing to perturbative triples ${\widetilde{R}}_3$ (diagram (a)) and correlation energy (diagrams (b - g)) arising from perturbative triples. Diagram (h) represents a two-body effective operator using which valence-states are coupled to give atomic state functions, as shown in Fig. \ref{['coupling']}.
• Figure 3: Angular factors arising from the coupling of single-particle states of a two-body effective energy operator.
• Figure 4: Schematic energy level diagram showing the $5s^2{\;^1}S_0 - 5s5p{\;^3P^o_0}$ clock transition in $^{87}$Sr and $^{88}$Sr via a HFS induced and a two-photon E1+M1 channel, respectively.
• Figure 5: Panels (a - c) show the convergence trend for energy of $^3P_{0}$, $^1S_{0}$ and $^1P_{1}$ states with augmentation of orbitals in $s$, $p$, $d$, $f$ and $g$ symmetries. Panels (d - f) show the convergence trend for E1, HFS, and M1 reduced matrix elements with basis size.