Lahs and Comparat (New J. Phys. 26, 125001 (2024)) systematically classified how parity (P), time-reversal (T), and combined PT violation modify the response of atoms and molecules to static, inhomogeneous, and oscillating electric and magnetic fields, expressing the responses through generalized polarizability coefficients for induced electric-dipole, magnetic-dipole, and electric-quadrupole moments. Their framework recovers known observables (the electron electric dipole moment (EDM) spin–electric-field coupling, P-odd optical rotation, and the CP-odd polarizability computed for rare gases in 2005) and predicts several new observables, including P-odd and T-odd terms that only appear with field gradients or time-varying fields. However, no numerical values of the newly identified coefficients exist for any candidate system, so none of the proposed new observables has a feasibility or sensitivity forecast. The open problem is to perform the dedicated atomic- and molecular-structure calculations for candidate systems — coefficient values with uncertainty estimates, signal-to-noise projections, and assessment of backgrounds such as linewidth-induced mixing — required to decide which of these observables are experimentally reachable.

Background

Tests of fundamental symmetries in atoms and molecules are among the most sensitive low-energy probes of physics beyond the Standard Model. Parity P is the symmetry of a system under spatial inversion, and time reversal T is the symmetry under reversal of the time coordinate; a permanent electric dipole moment (EDM) of a nondegenerate state, for example, is forbidden unless both P and T are violated, which is why EDM searches in atoms and molecules constrain CP-violating new physics. In the standard treatment, a symmetry-violating interaction (written generically as potentials VPV^P, VPTV^{PT}, VTV^T for P-odd, PT-odd, and T-odd couplings, such as electron–nucleon contact terms or the nuclear Schiff moment interaction, the effective coupling of atomic electrons to the symmetry-violating nuclear moment that replaces the point-nucleus contact term at finite nuclear size) mixes states of opposite symmetry into the state of interest, so that applied electromagnetic fields then induce responses that are classically forbidden. Ordinary polarizabilities are the second-order response coefficients describing the electric dipole, magnetic dipole, or electric quadrupole moment induced in a state by external electric field E\mathcal{E} and magnetic field H\mathcal{H}. Lahs and Comparat (2024) generalized this: they derived, by second-order nondegenerate perturbation theory, the complete set of induced electric-dipole ⟨d⟩\langle\boldsymbol{d}\rangle, magnetic-dipole ⟨μ⟩\langle\boldsymbol{\mu}\rangle, and electric-quadrupole ⟨Qij⟩\langle Q_{ij}\rangle responses of an atom or molecule in static, spatially inhomogeneous, and time-harmonic fields, with each response coefficient (a generalized polarizability) labeled by the symmetry violation (P, T, or PT) and the multipole combination (dipole–dipole, dipole–quadrupole, etc.) that produces it. The framework recovers well-known observables, such as the P-odd T-odd spin–electric-field coupling used in electron-EDM searches, P-odd T-even optical rotation, and the CP-odd polarizability βCP\beta^{\mathrm{CP}} (a magnetic moment induced by a static electric field, μCP=βCPE0\boldsymbol{\mu}^{\mathrm{CP}}=\beta^{\mathrm{CP}}\mathcal{E}_0), for which Ravaine, Kozlov, and Derevianko (Phys. Rev. A 72, 012101 (2005)) performed numerical third-order perturbation-theory calculations in the Dirac–Hartree–Fock formalism (the relativistic mean-field approximation to electronic structure) for the rare-gas atoms He through Rn, finding a steep Z5Z^5 scaling. Beyond these recovered cases, the 2024 work predicts genuinely new observables: for example, a P-odd dipole–quadrupole polarizability αPdQ\alpha^{dQ}_{P} that induces an electric dipole moment proportional to the field gradient ∂jE\partial_j\mathcal{E}, a T-odd dipole–quadrupole polarizability αTμQ\alpha^{\mu Q}_{T} that induces a magnetic moment from a field gradient, and oscillating-field responses αPμd\alpha^{\mu d}_{P} and αPTμd\alpha^{\mu d}_{PT} in which an oscillating electric field applied to a polarized sample induces an orthogonal magnetization either in phase (P-odd) or out of phase (PT-odd) with the field — effects invisible in static homogeneous fields. The authors also identified a practical background: the finite natural linewidth of near-lying excited states produces a linewidth-induced mixing factor GℓkG^{\ell k} that can mimic T-odd signals. The size of every new coefficient depends on the specific level structure (transition frequencies and dipole, quadrupole, and magnetic-dipole matrix elements between close-lying states) and on the assumed symmetry-violating potentials, which typically scale steeply with nuclear charge for contact interactions, so heavy atoms and molecules with near-degenerate opposite-parity levels are expected to be the most favorable systems. Without numerical values, however, no statement about experimental feasibility, optimal system choice, or expected signal-to-noise is possible for any of the new observables.

Problem Statement

For the generalized P-, T-, and PT-odd polarizability framework of Lahs and Comparat (New J. Phys. 26, 125001 (2024)), perform the dedicated atomic- and molecular-structure calculations needed to assess experimental feasibility: identify promising candidate systems (atoms and molecules whose level structures provide the large transition moments and near-lying states that enhance the responses), compute the numerical values — with uncertainty estimates — of the generalized polarizability coefficients for each proposed observable, in particular the newly identified channels that have no existing values (the P-odd static-field and field-gradient-induced electric-dipole terms such as αPdQ\alpha^{dQ}_{P}, the T-odd field-gradient-induced magnetic-dipole term αTμQ\alpha^{\mu Q}_{T}, and the oscillating-field response coefficients αPμd\alpha^{\mu d}_{P} and αPTμd\alpha^{\mu d}_{PT} governing in-phase and out-of-phase induced magnetization), and derive from them the expected signal sizes and achievable signal-to-noise ratios in realistic experimental configurations, including an assessment of whether the linewidth-induced mixing background GℓkG^{\ell k} can be suppressed or corrected below the projected signal for the affected T- and PT-odd observables. The objective is a quantitative feasibility verdict for each proposed observable in the framework's leading candidate systems, sufficient to decide which of the new symmetry-violation observables merit experimental searches. Completion requires a per-observable reachable/not-reachable verdict that rests on that observable's computed coefficients with their uncertainty estimates together with the corresponding signal-to-noise and, where applicable, background projections; the newly identified channels named above are examples rather than an exhaustive list of the required coefficients, and any observable left untreated is reported as left open.

References

  1. Sebastian Lahs and Daniel Comparat, "Polarizabilities as probes for P, T, and PT violation", New Journal of Physics 26, 125001 (2024). DOI: 10.1088/1367-2630/ad985a, https://doi.org/10.1088/1367-2630/ad985a; arXiv:2407.14694, https://arxiv.org/abs/2407.14694
  2. Sebastian Lahs, Daniel Comparat, Fiona Kirk, and Benjamin M. Roberts, "Atomic observables induced by cosmic fields", Physical Review D 113, 015041 (2026). DOI: 10.1103/j7s3-wjg6, https://doi.org/10.1103/j7s3-wjg6; arXiv:2510.08007, https://arxiv.org/abs/2510.08007
  3. Lei Cong, Wei Ji, Pavel Fadeev, Filip Ficek, Min Jiang, Victor V. Flambaum, Haosen Guan, Derek F. Jackson Kimball, Mikhail G. Kozlov, Yevgeny V. Stadnik, and Dmitry Budker, "Spin-dependent exotic interactions", Reviews of Modern Physics 97, 025005 (2025). DOI: 10.1103/RevModPhys.97.025005, https://doi.org/10.1103/RevModPhys.97.025005; arXiv:2408.15691, https://arxiv.org/abs/2408.15691
  4. Boris Ravaine, Mikhail G. Kozlov, and Andrei Derevianko, "Atomic CP-violating polarizability", Physical Review A 72, 012101 (2005). DOI: 10.1103/PhysRevA.72.012101, https://doi.org/10.1103/PhysRevA.72.012101; arXiv:hep-ex/0504017, https://arxiv.org/abs/hep-ex/0504017
  5. A. Derevianko and M. G. Kozlov, "CP-violating magnetic moments of atoms and molecules", Advances in Atomic, Molecular, and Optical Physics 58, 77–112 (2010). DOI: 10.1016/S1049-250X(10)05807-6, https://doi.org/10.1016/S1049-250X(10)05807-6
  6. V. V. Flambaum, J. S. M. Ginges, and G. Mititelu, "Parity and time reversal violating nuclear polarizability" (2000). arXiv:nucl-th/0010100, https://arxiv.org/abs/nucl-th/0010100