Precision atomic clocks now reach fractional frequency uncertainties near 10−1810^{-18}, where the shift of atomic energy levels by the blackbody radiation (BBR) of the ambient environment is a leading systematic. A 2025 work derived the complete multipole decomposition of the thermal (finite-temperature) one-loop self-energy of a bound electron and showed that, beyond the familiar electric-dipole thermal Stark shift, the thermal QED (TQED) framework necessarily produces a magnetic-dipole thermal Zeeman shift (BBRZ), an electric-quadrupole term, a diamagnetic contribution from the Dirac negative continuum, and relativistic corrections to the self-energy operator and to the wave functions — but presented numerical results only for hydrogen and a few light hydrogen-like ions (He+^+, Li2+^{2+}, Ne9+^{9+}), with the many-electron clock species in its tables covered through published quantum-mechanical data. Follow-up works by the same group delivered a fully relativistic evaluation for hydrogen-like ions of arbitrary nuclear charge and a dipole-level BBR Stark computation for alkali-metal atoms with core polarization. What no work has done is evaluate the full set of TQED one-loop thermal corrections for many-electron atoms and ions using correlated many-body wavefunctions, where the required matrix elements involve configuration mixing and core-valence correlation. The open problem is to carry out this extension and to determine which of the beyond-dipole and relativistic thermal terms enter the uncertainty budgets of realistic clock and precision-spectroscopy systems, including the high-nn Rydberg regime in which the diamagnetic and relativistic terms grow rapidly with principal quantum number.

Background

The most accurate atomic clocks (optical lattice and single-ion clocks based on, e.g., Sr, Yb, Al+^+, Hg+^+, Yb+^+) reach fractional frequency uncertainties around 10−1810^{-18}. At this level, the thermal electromagnetic radiation emitted by the apparatus at room temperature — blackbody radiation (BBR), described by Planck's distribution at temperature TT — shifts atomic transition frequencies at a measurable level and is one of the largest systematic corrections. The dominant piece is the dynamical Stark shift: in second-order perturbation theory the BBR electric field couples to the atom through the frequency-dependent (dynamical) electric-dipole polarizability, giving a shift of order (kBT)4(k_{\rm B}T)^4 times that polarizability, where kBk_{\rm B} is Boltzmann's constant. Farley and Wing (1981) computed such BBR-induced shifts and depopulation rates for Rydberg levels, and Porsev and Derevianko (2006) developed the multipolar theory of the BBR shift for clock transitions; high-accuracy many-electron calculations followed, e.g., Safronova and Safronova (2011) for Ca+^+. Because BBR is isotropic, the leading magnetic coupling appears as a thermal Zeeman shift of hyperfine and Zeeman-sublevel transitions (the BBRZ shift), known since Farley and Wing's work and recently reanalyzed in detail for optical clocks by Tang et al. (2024). A separate, more fundamental route to all of these effects is thermal QED: one computes the one-loop self-energy of the bound electron — the radiative correction due to emission and reabsorption of a virtual photon — using the thermal photon propagator (the finite-temperature field-theory technique introduced by Dolan and Jackiw and by Donoghue, Holstein, and Robinett), in which the photon occupation numbers follow the Planck distribution. Solovyev, Labzowsky, and Plunien (2015) gave the QED derivation of the BBR Stark shift and line broadening, and Solovyev (2020) formulated thermal QED for bound states generally. In this framework the thermal part of the self-energy can be decomposed into multipoles and expanded in the binding parameter αZ\alpha Z (fine-structure constant times nuclear charge), and each order reproduces a known quantum-mechanical effect plus additional, genuinely relativistic terms that the phenomenological quantum-mechanical treatment misses. Lopez-Rodriguez, Bobylev, Kvasov, Zalialiutdinov, and Solovyev (2025) completed this program for the hydrogen atom: they recovered the thermal Stark and Zeeman shifts, the thermal quadrupole interaction, relativistic corrections to the self-energy operator and to the wave function, the diamagnetic shift arising from the negative-energy (Dirac-sea) part of the spectrum, ΔEadia=π3e245 β4⟨a∣r2∣a⟩,β=(kBT)−1,\Delta E^{\rm dia}_{a}=\frac{\pi^{3}e^{2}}{45\,\beta^{4}}\langle a|r^{2}|a\rangle,\qquad \beta=(k_{\rm B}T)^{-1}, and the effect of finite level lifetimes (widths) that regularize the resonant denominators, with the induced level widths obtained from the imaginary part of the same one-loop correction. Their numerical tables cover hydrogen together with the light hydrogen-like ions He+^+, Li2+^{2+}, and Ne9+^{9+} computed within the one-electron formalism, while the BBRZ entries for the many-electron clock species Be, Mg, Al+^+, In+^+, Hg+^+ recycle quantum-mechanical values from Tang et al. (2024); for many-electron systems they assume the corrections are of the same order of magnitude as in hydrogen. The same group then published two follow-ups: Reiter et al. (2026) evaluated the thermal one-loop self-energy fully relativistically (no αZ\alpha Z expansion) for hydrogen and hydrogen-like ions of arbitrary ZZ, verifying the 1/Z41/Z^{4} scaling of the nonrelativistic dipole limit and finding 'magic' temperatures at which thermal frequency shifts vanish; and Bobylev et al. (2026) computed dipole polarizabilities and BBR Stark shifts for alkali-metal atoms with a Dirac-Fock (the relativistic Hartree–Fock method) plus core-polarization model, the first step toward many-electron systems but still restricted to the leading electric-dipole term with a simplified treatment of electron correlation (the polarization of the closed-shell core by the valence electron), rather than the configuration-interaction and coupled-cluster wavefunctions that carry the correlation (configuration mixing) needed for 10−1810^{-18}-level accuracy. Consequently, the relativistic and beyond-dipole terms of the TQED decomposition have never been evaluated for any many-electron atom or ion, even though the 2025 analysis found that in hydrogen they exceed the BBRZ shift and are comparable to the dynamical corrections to the dc-Stark shift, and that the diamagnetic term grows rapidly with principal quantum number nn (about 19 Hz for the n=100n=100, l=0l=0 hydrogen state at 300 K), making high-nn Rydberg states a regime where several neglected terms become large; a recent conventional quantum-mechanical analysis by Potvliege (2026) likewise finds the electric-quadrupole thermal shift comparable to the diamagnetic one in Rydberg states, though it does not employ the thermal-QED framework.

Problem Statement

Extend the finite-temperature QED one-loop self-energy program, established quantitatively only for hydrogen and hydrogen-like ions, to many-electron atoms and ions: derive and numerically evaluate, with correlated many-body wavefunctions (including core polarization and configuration mixing), the complete set of blackbody-radiation-induced level shifts and widths that the thermal one-loop multipole decomposition predicts — the dynamical electric-dipole Stark shift, the magnetic-dipole thermal Zeeman (BBRZ) shift with its relativistic corrections, the electric-quadrupole and higher multipole terms, the diamagnetic (negative-continuum) contribution, the relativistic corrections to the self-energy operator and to the bound-state wave functions, and the finite-lifetime (width) regularization of resonant denominators — for atomic states used in optical clocks and precision spectroscopy. The deliverable is a quantitative account of which of these thermal terms reach or exceed the 10−1810^{-18} fractional-frequency uncertainty level for specific clock systems and transitions (including any where quadrupole coupling is enhanced), together with the behavior of the rapidly growing diamagnetic and relativistic terms for high-nn Rydberg states and an assessment of their experimental detectability. The single overarching objective is a complete, correlation-aware TQED thermal-shift theory for realistic many-electron atomic systems; the individual terms are the components of that one decomposition and are to be evaluated within a common many-body framework rather than as disconnected estimates.

References

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