The observed baryon asymmetry in the Universe is one of the important unsolved problems in cosmology. Possible models that can lead to this asymmetry are discussed in Refs. [1, 2]. One such model is due to Sakharov, where the asymmetry can arise under conditions of baryon number violation, C and CP-violation, and thermal non-equilibrium [3]. The Standard Model cannot account for the observed asymmetry [4], motivating theories beyond the Standard Model to introduce new sources of CP-violation [5]. Equivalently, since these theories respect CPT symmetry [6], they introduce new sources of T-violation. An elementary particle with a permanent electric dipole moment (EDM) violates both P and T symmetries [7, 8], and measurements of these EDMs have been exceptionally fruitful in testing physics beyond the Standard Model. Paramagnetic molecules such as ¹⁷⁴YbF [9], ²³²ThO [10] and ¹⁸⁰HfF⁺ [11] are excellent probes of P,T-violating physics. These molecules are primarily sensitive to the electron EDM, d _e , and the scalar P,T-violating electron-nucleon interaction, C _S . The most precise measurement thus far, in ThO, constrains new CP-violating interactions to mass scales above 30 TeV in some models and above 3 TeV in most [10].

These electron EDM measurements use paramagnetic molecules with a heavy atom of zero nuclear spin, I = 0. As a result, they are not sensitive to P,T-violating nuclear moments ¹ . Instead, one can use isotopes with nuclear spin to probe CP-violating effects within the nucleus. The lowest-order P,T-violating nuclear moment is the EDM (which requires I ≥ 1/2), but that is effectively screened by the outer electrons [13]. The screening is incomplete due to the finite size of the nucleus [12], but in paramagnetic atoms and molecules the remaining effect (the nuclear Schiff moment) is negligible compared to the contribution from the electron EDM. Conversely, in diamagnetic atoms and molecules the contribution from the electron EDM is small and the Schiff moment can be probed. At present, the most stringent limits on the size of nucleon EDMs and P,T-violating nucleon-nucleon interactions come from a measurement of the nuclear Schiff moment of ¹⁹⁹Hg [14], a diamagnetic atom with I = 1/2. An experiment is being constructed that aims to improve on these constraints by measuring the Schiff moment of ²⁰⁵Tl in a beam of TlF molecules, taking advantage of the strong polarisation in an electric field due to the small splitting between opposite parity rotational states [15]. Since the Schiff moment is a screened effect, it is interesting to consider the effect of higher-order moments. The next-order nuclear moment of interest is the magnetic quadrupole moment (MQM) (requires I ≥ 1), which is not screened and can contribute significantly to the P,T-violating energy shifts of atoms and molecules.

A spherical nucleus can acquire an MQM from the EDM of its valence proton or neutron [16], or from P,T-odd nucleon-nucleon interactions involving the valence nucleon [17]. If we denote the total nuclear MQM as M and the contribution to the MQM from a single proton or neutron as M 0 p / n , then for these nuclei we have M ∼ M 0 p / n . In nuclei with quadrupole deformation, there are many nucleons in open shells which collectively contribute to the nuclear MQM, which can increase M by an order of magnitude [18, 19]. Furthermore, the interaction of the nuclear MQM with the magnetic moment of the valence electron in the molecule is enhanced by relativistic effects due to the heavy atom [17]. Table 1 summarizes these effects for isotopologues of molecules currently used for electron EDM experiments.

In this paper, we consider how to make a measurement of nuclear MQMs using heavy, polar molecules. We first calculate the size of energy shifts in various states of a molecule due to the nuclear MQM. We then describe a procedure for preparing the molecules in the states most sensitive to the MQM. In particular, we introduce a generic state preparation and readout method which utilises the large tensor Stark shifts in molecules to induce a high-order coupling between the two states used for the measurement. Finally, we assess the sensitivity of such an experiment to CP-violating parameters in the hadronic sector, and compare this to other experiments such as Schiff moment or neutron EDM measurements. Our analysis applies to a wide range of molecules, though we will often use ¹⁷³YbF as a specific example. The nuclear spin of ¹⁷³Yb is 5/2, while that of ¹⁹F is 1/2. We note that the advantages of polyatomic molecules for measuring MQMs and other CP-violating effects are outlined in Ref. [26].

We consider a molecule in a ²Σ state with electron spin S and rotational angular momentum N. The nucleus of interest has spin I. The molecule is in an electric field E which is parallel to z. A suitable effective Hamiltonian is H = H rot + H Stark + H hyp + H M , (1) where H _rot = B N ² is the rotational energy, H Stark = − μ ⋅ E describes the Stark shift, H _hyp describes the hyperfine interactions, and H _M describes the nuclear MQM interaction. Here, B is the rotational constant and μ is the electric dipole moment operator. The nuclear MQM interaction can be written as H M = − W M M 2 I 2 I − 1 S T ̂ n , (2) where n is a unit vector along the internuclear axis and T ̂ is a second-rank tensor whose components are T i , j = I i I j + I j I i − 2 3 δ i , j I ( I + 1 ) [17, 18]. With the help of appendices 8.1 and 8.2 of Ref. [27], we can re-write this in spherical tensor notation as H M = W M M 2 I 2 I − 1 20 3 T ( 1 ) ( S , I 2 ) ⋅ n ≡ Q ⋅ n = Q z ′ , (3) where T⁽¹⁾(S, I ⁽²⁾) is a rank-1 tensor constructed from S and I ⁽²⁾; the latter is a rank-2 tensor constructed from the nuclear spin I. We see that the MQM interaction has been simplified to a vector Q that acts along the symmetry axis of the molecule, z′.

To keep the calculation simple, we first neglect H _hyp. Its effects are considered later. We find it advantageous to use a basis where the rotational and spin wavefunctions are separated, so we choose the basis set denoted by |G, M _G ; N, M _N ⟩, where G = S + I is the total spin and M _G and M _N are the projections of G and N on the laboratory z-axis. H _rot + H _Stark is diagonal in G, M _G and M _N , but the electric field mixes states of different N. After diagonalizing H _rot + H _Stark we obtain the eigenfunctions | G , M G ; N ̃ , M N 〉 = ∑ N a N | G , M G ; N , M N 〉 and corresponding eigenvalues E N ̃ , M N . Here, N ̃ is a label which connects a state adiabatically to its field-free state.

The energy shift due to H _M is Δ E M ′ = 〈 G , M G ; N ̃ , M N | Q z ′ | G , M G ; N ̃ , M N 〉 = W M M ζ η . (4) In this equation, ζ depends only on the electron and nuclear spins and is evaluated in Appendix A, whereas η is independent of the spins and expresses the degree of alignment between the internuclear axis of the molecule and the laboratory z-axis, often known as the polarization factor. It is given by η = − 1 μ mol d E N ̃ , M N d E , (5) where μ _mol is the dipole moment along the internuclear axis. Figure 1 shows the evaluation of ζ for a few selected values of S and I. In all the examples given in Figure 1, the largest MQM energy shift is found for the states |G, M _G = ±G⟩, where G does not take its maximum possible value. It is interesting that the configuration where the electron and nuclear spins are parallel is not the one most sensitive to the nuclear MQM. We find this to be true for larger values of S and I as well.

Now we consider the effect of H _hyp, which includes the spin-rotation interaction, the magnetic dipole hyperfine interaction and the electric quadrupole hyperfine interaction [28]. To handle this, we introduce the spin of the second nucleus I ₂, an intermediate angular momentum F ₁ = G + N and the total angular momentum F = F ₁ + I ₂. Using the field-free coupled basis, |N, G, F ₁, F, M _F ⟩, we diagonalize H _rot + H _Stark + H _hyp to find the new eigenstates | N ̃ , G ̃ , F ̃ 1 , F ̃ , M F 〉 which are adiabatically linked to the field-free states. Then, we calculate the MQM energy shift Δ E M = 〈 N ̃ , G ̃ , F ̃ 1 , F ̃ , M F | H M | N ̃ , G ̃ , F ̃ 1 , F ̃ , M F 〉 . (6) The matrix elements needed to calculate ΔE _M are given in Appendix A.

Although Eq. 6 will be more accurate than Eq. 4, we expect the latter to work well for most ²Σ molecules since they have no spin-orbit coupling and the largest spin-rotation coupling is typically much smaller than the hyperfine and rotational energies. As a concrete example, for the ²Σ ground state of ¹⁷³YbF, the electron-spin-rotation coupling strength is γ = −13 MHz, the hyperfine Fermi contact strength between the electron and Yb nuclear spins is b _F (Yb) = −1.98 GHz, and the rotational constant is B = 7.24 GHz [28, 29]. The fluorine nucleus has spin I ₂ = 1/2 but this gives rise to a much smaller hyperfine splitting than that of the Yb nucleus (b _F (F) = 0.17 GHz). Consequently, the decoupled basis | N , M N 〉 | G , M G 〉 | I 2 , M I˙ 2 〉 is a good first approximation to the exact eigenstates.

Table 2 lists the MQM energy shifts of all states that correlate to the lowest rotational level of ground state ¹⁷³YbF, in a static electric field of 18 kV/cm. At this field, the polarization factor is η = 0.676. We have given both ΔE _M calculated using Eq. 6 and the approximate result Δ E M ′ calculated using Eq. 4. The two sets of results are similar, as we would expect from the arguments above. These states, together with the MQM energy shifts, are shown pictorially in Figure 2. The large splitting between the G ̃ = 2 and G ̃ = 3 states comes from the hyperfine interaction between the electron spin and the Yb nuclear spin. The much smaller splitting into states with F ̃ = G ̃ ± I 2 is due to the interaction with the F nuclear spin. States of the same F ̃ and G ̃ but different |M _F | are split by the tensor part of the Stark interaction. This is small for the state with N ̃ = 0 , but much larger for all other states. In the absence of the MQM or other P, T-violating interactions, states of the same { G ̃ , F ̃ , | M F | } are degenerate. The MQM interaction lifts this degeneracy by the amounts indicated in the figure.

An MQM measurement can be done in a similar way to measurements of the electron EDM or the nuclear Schiff moment [10, 15, 30]. Using the notation ² |F, M _F ⟩, let us define states |0⟩ = |F, F⟩ and | ± 〉 = 1 2 | F , + F 〉 ± | F , − F 〉 . Molecules are first prepared in |0⟩, typically by optical pumping, and then transferred to | + ⟩. This state evolves for time τ in the presence of parallel electric and magnetic fields, becoming | ψ 〉 = 1 2 e i ϕ | F , + F 〉 + e − i ϕ | F , − F 〉 = cos ⁡ ϕ | + 〉 + i ⁡ sin ⁡ ϕ | − 〉 . Here, ϕ = ϕ M + ϕ B = Δ E M + Δ E B τ / ℏ , where ΔE _B is the (absolute) Zeeman shift of the states. Finally, the populations in | + ⟩ and | − ⟩ are read out, typically by transferring the population in | + ⟩ to one state (e.g., back to |0⟩) and the population in | − ⟩ to a state of sufficiently different energy that the two are easily resolved (e.g., by laser-induced fluorescence detection).

A particular challenge in this measurement scheme, which does not typically arise in related experiments, is the preparation of a superposition of ±M _F states where M _F and −M _F differ by several units. For example, consider the levels of ¹⁷³YbF shown in Figure 2. The states most sensitive to the MQM have G ̃ = 2 , F ̃ = 5 / 2 , M F = ± 5 / 2 . In that case, the measurement scheme calls for a superposition of a pair of states that differ in M _F by 5 units of angular momentum. Previously, optical pumping light modulated at a harmonic of the Larmor frequency has been used to prepare and study coherences within states of high angular momentum [31]. In the following, we describe a general method to prepare the states we need.

Finally, we consider the sensitivity of a molecular MQM experiment to hadronic CP-violating parameters, and compare related experiments. Again, we use ¹⁷³YbF as a prototypical example. The parameters of the experiment are the coherence time τ and the number of molecules measured N. At the shot-noise limit, the statistical uncertainty of an ideal frequency measurement will be δ f = 1 / ( 2 π τ N ) . Recent experimental work and case studies [15, 33–35] suggest that measurements using the molecules considered here can reach a sensitivity of approximately 1 mHz/ H z . This could be achieved using a high flux focussed molecular beam from a cryogenic source, giving N ≈ 3 × 10⁸ molecules per second and τ ≈ 10 ms, or an ultracold molecular beam giving N ≈ 3 × 10⁶ molecules per second and τ ≈ 100 ms, or ultracold molecules in an optical lattice giving N ≈ 3 × 10⁴ molecules per second and τ ≈ 1 s. In this case, the uncertainty reaches δf = 1 μHz in about 300 h of measurement time.

If we assume that the CP-violating energy splitting in ¹⁷³YbF is due only to the MQM of the Yb nucleus, 2Δ_CP = 0.42W _M M, then the uncertainty is equivalent to a 2σ-sensitivity on the MQM of 4.5 × 10⁻¹³ e fm². From this, we can calculate the expected level to which we can measure various CP-violating parameters, using the parameter sensitivities compiled in Table 3. We compare these to the experimental constraints set by the Schiff moment measurement of ¹⁹⁹Hg [14] and the projected sensitivities of the proposed Schiff moment measurement of ²⁰⁵Tl from a TlF beam experiment [15]. The former is S (¹⁹⁹ H g < 3.1 × 10⁻¹³ e fm³, and the latter has an anticipated statistical sensitivity of δf = 45 nHz which corresponds to ³ δS (²⁰⁵ T l = 9.1 × 10⁻¹⁴ e fm³.

Table 4 shows the 95% confidence limit constraints on CP-violating parameters set by the ¹⁹⁹Hg experiment, as well as projected 2σ-sensitivities for experiments using TlF and YbF, given the uncertainties quoted above. We find that a measurement of the MQM of ¹⁷³Yb using YbF molecules can make large improvements relative to the current limits on CP-violating parameters obtained from the Hg experiment. The measurement also gives greater sensitivities than obtainable from the proposed TlF experiment, even though the projected frequency uncertainty of the TlF experiment is 20 times smaller.

Measurements of nuclear MQMs in isotopologues of heavy, paramagnetic molecules have the potential to probe hadronic CP-violating physics at a level well beyond current limits. We have calculated the molecular energy shifts to be expected, showing which states should be used. We find that measurements will require molecules prepared in a superposition of magnetic sub-levels that differ by many units of angular momentum, and we have described a general method for preparing these states. These experiments would benefit from the recent advances in laser cooling applied to molecules [38]. The nuclear spin of the heavy atom in the molecule leads to large hyperfine intervals and a more complex hyperfine structure which makes the laser cooling more difficult than for the isotopologues cooled so far. Nevertheless, a recent study [39] shows how cooling of the required molecules can be done with relatively small additions to existing experiments.