Nuclear-spin dependent parity violation in optically trapped polyatomic molecules

Abstract

Improved nuclear spin-dependent parity violation measurements will enable experimental determination of poorly known electroweak coupling parameters. Here, we investigate the suitability of optically trapped linear polyatomic molecules as probes of nuclear spin-dependent parity violation. The presence of closely spaced, opposite-parity $\ell$-doublets is a general feature of such molecules, allowing parity-violation-sensitive pairs of levels to be brought to degeneracy in magnetic fields typically 100 times smaller than in diatomics. Assuming laser cooling and trapping of polyatomics at the current state-of-the-art for diatomics, we expect to measure nuclear spin-dependent parity-violating matrix elements iW with 70 times better sensitivity than the current best measurements. Our scheme should allow for 10% measurements of iW in nuclei as light as Be or as heavy as Yb, with averaging times on the order of 10 days and 1 s, respectively.

Introduction

Measurements of nuclear spin-independent (NSI) and nuclear spin-dependent (NSD) parity violation (PV) are a means to probe standard model (SM) electroweak interactions on a tabletop scale¹. NSI-PV has been measured in protons and a number of heavy atoms^{2,3,4,5,6,7,8,9} and found to be in good agreement with SM PV predictions due to the weak charge Q_W. However, the only non-zero measurement (14% relative uncertainty) of NSD-PV in an atomic system comes from Cs¹⁰, and this result implies constraints on SM meson-nucleon couplings which are in disagreement with other atomic PV measurements^11,12. NSD-PV arises primarily from three interactions: vector electron-axial nucleon electroweak current coupling (V_eA_n), the nuclear anapole moment, and the combined effects of nuclear weak charge and normal hyperfine structure. The V_eA_n effect is described by two parameters C_2u and C_2d relating to spin-dependent Z₀ boson exchange between an electron and an up or down quark, respectively. These parameters are among the most poorly measured in the SM, with relative uncertainties 300 and 70%, respectively¹³. PV measurements may also probe beyond standard model physics^14,15. Searches for oscillating PV signals have been proposed as a means to detect axion-like particles, a leading dark matter candidate¹⁶.

A beam of cold diatomic molecules has been demonstrated^{17,18,19,20,21} to be a highly sensitive system for measuring NSD-PV effects. Mixing of opposite-parity quantum states from PV effects is amplified when the states have nearly the same energy⁶. The lowest two rotational states of diatomic molecules have opposite parity and may be brought to near degeneracy using a large magnetic field B. While this method is quite general, current systematic uncertainties are roughly 100 times too large to measure NSD-PV in the lightest nuclei where nuclear structure calculations are tractable²².

Recent advances in laser cooling have lead to optically trapped diatomic molecules with sub-Doppler temperature and single-molecule detection efficiency²³, while similar strides with polyatomic molecules have followed closely behind^{24,25,26,27,28}. Moreover, polyatomic molecules have been proposed as exquisite systems for precision measurements of fundamental symmetries^29,30,31 and time variation of fundamental constants^32,33.

Here, we show that linear asymmetric polyatomic molecules in an optical trap are well-suited for measurement of NSD-PV. Polyatomic molecules possess opposite-parity states 10–1000 times closer in energy than diatomics, requiring similarly smaller B-fields. Systematic uncertainties are reduced compared to beam experiments due to the lower magnetic field and a smaller interaction volume. Furthermore, these smaller fields may be produced without superconducting magnets, allowing trivial B-field reversal for detection and mitigation of systematic effects. We show that these molecules may be optically trapped using “magic” conditions where differential light shifts are small enough for a precise PV measurement. The obvious advantage of performing a precision measurement on trapped species compared to a beam is the increased interaction time τ. The sensitivity to any PV matrix element iW is $\delta W/W = 1/\tau \sqrt {N_{\mathrm{m}}}$, where N_m is the total number of measurements (iW is purely imaginary due to conservation of time-reversal symmetry). Assuming optical trapping of polyatomic molecules at the current state-of-the-art for diatomics^23,34,35, we expect at least a factor of 70 increase in PV sensitivity over the state-of-the-art BaF measurement^20,36. Our method is applicable to all laser-coolable polyatomic molecules with ²Σ ground states, and possibly others.

Results

Relevant properties of polyatomic molecules

Consider the properties of linear asymmetric molecules with a ²Σ⁺ electronic ground state. If a bending vibrational mode (with vibrational constant ω_b and quantum number v_b) is excited and all other vibrational modes are in their ground state, the molecule’s rotational angular momentum N has a projection along the molecular axis $\ell = \pm v_b, \pm (v_b - 2), \ldots , \pm 1$ or 0. Within this vibrational manifold, the effective Hamiltonian is

$$\begin{array}{*{20}{l}} H \hfill & = \hfill & {B_{\mathrm{e}}({\mathbf{N}}^2 - \ell ^2) \pm ( - 1)^N\frac{{q_b}}{2}{\mathbf{N}}^2 + \gamma {\mathbf{N}} \cdot {\mathbf{S}} + b{\mathbf{I}} \cdot {\mathbf{S}}} \hfill \\ {} \hfill & {} \hfill & { + cI_zS_z - e{\mathrm{T}}^2(\nabla {\mathbf{E}}) \cdot {\mathrm{T}}^2({\mathbf{Q}})} \hfill \\ {} \hfill & {} \hfill & { + (\mu _{\mathrm{B}}g_{\mathrm{S}}{\mathbf{B}} \cdot {\mathbf{S}} + \mu _{\mathrm{B}}g_{\mathrm{L}}{\mathbf{B}} \cdot {\mathbf{L}} + \mu _{\mathrm{N}}g_{\mathrm{I}}{\mathbf{B}} \cdot {\mathbf{I}}),} \hfill \end{array}$$

(1)

where B_e is the rotational constant, γ is the spin-rotation (SR) constant, b and c are hyperfine (HF) constants, e is the electron charge, T²(∇E)⋅T²(Q) is a scalar product of rank-2 spherical tensors describing the electric field gradient ▽E at the nucleus with quadrupole moment Q, S is the electron spin, L is the electron orbital angular momentum, and I is the nuclear spin^37,38. The upper (lower) sign corresponds to the parity P of the closely spaced “$\ell$-doublet” eigenstates $\left| {N,v_b^\ell ,P = \pm } \right\rangle = \frac{1}{{\sqrt 2 }}\left( {\left| {N, + \ell } \right\rangle \pm ( - 1)^{N - \ell }\left| {N, - \ell } \right\rangle } \right)$. The $\ell$-doublet is the key property of polyatomic molecules absent from diatomics which we wish to exploit for a PV measurement. In linear modes, as well as diatomics, opposite-parity levels are spaced by roughly the rotational constant B_e/2π ~1 GHz to 100 GHz. In an excited bending mode, opposite-parity states are spaced by only q_b ≈ $- 2B_{\mathrm{e}}^2/\omega _b$, which is on the order of $q_b/2\pi \sim 10\,{\mathrm{MHz}}$ to 100 MHz. The relative spacing between levels may be tuned to degeneracy via the Zeeman interaction (last three terms of Eq. (1), with μ_B, μ_N the Bohr and nuclear magneton, respectively; and g_S, g_L, and g_I the g-factors corresponding to S, L, and I, respectively). Examples are provided in Supplementary Note 1.

Nuclear spin-dependent parity violation

For a given electronic state of a molecule, the effective NSD-PV Hamiltonian is³⁹

$$H_{\mathrm{p}}^{{\mathrm{eff}}} = \kappa W_{\mathrm{p}}({\mathbf{S}} \times \widehat {\mathbf{I}}) \cdot \widehat {\mathbf{n}}.$$

(2)

Here, W_p encodes the overlap of unpaired electrons with the nucleus and can be calculated with high accuracy by ab initio or semiempirical methods, $\widehat {\mathbf{n}}$ is a unit vector along the molecular axis, and κ is the measurable parameter of interest. In a given nucleus, various NSD-PV effects contribute to κ = κ₂ + κ_a + κ_Q. κ₂ is proportional to the strength of the V_eA_n coupling, and is independent of nuclear mass A (for a typical nucleus, $|\kappa _2| \simeq 0.05$). κ_a is proportional to the nuclear anapole moment and is proportional to A^2/3. κ_Q is due to the combined effects of nuclear weak charge and normal hyperfine structure, and is negligible compared to κ₂ and κ_a⁴⁰. Measurements in several nuclei are required to distinguish among the different NSD-PV effects.

Ultimately, the ability to precisely determine κ and its underlying contributions will be limited to the accuracy of theoretical values of W_p. Calculations of W_p have been performed on several diatomic molecules via Dirac-Hartree-Fock and relativistic density-functional⁴¹, quasirelativistic zero-order regular approximation^29,42, and (with an estimated 1.5% accuracy) relativistic coupled-cluster⁴³ methods. While such calculations are beyond the scope of this proposal, a semiempirical method may be used to calculate W_p for any species to ~10%, assuming the SR/HF constants in Eq. (1) are known¹⁷. The SR/HF constants relating to a typical metal atom M differ by a ~10% between laser-coolable monofluorides MF and MX molecules, where X is a suitable ligand with charge state −1 (e.g., OH, NC, or CCH). Thus, we expect similar 10% accuracy when estimating W_p for MX, using either MX SR/HF constants for a semiemprical calculation or a more detailed method described above for the corresponding MF. This approximation is in line with available theoretical values W_p(Ra) in RaF^41,42 and RaOH²⁹. In Table 1, we give a list of laser-coolable polyatomic molecules MX, with W_p from calculations on MX where available or MF otherwise (see also Supplementary Note 2).

Table 1 Parameters for parity violation measurement in polyatomic molecules

In general, Eq. (2) should be summed over all nuclei i with spin I_i  ≥ 1/2 in a molecule. The PV signal is easiest to interpret when the unpaired electron is centered on one atom in the molecule, i.e. $W_{\mathrm{p}}^{(i)} \approx 0$ for all but one atom. A single-atom-centered unpaired electron is also a defining characteristic of laser-coolable molecules: this electron interacts negligibly with the nuclear vibration of the molecule, leading to electronic transitions which are highly diagonal in vibrational quantum number (and thus requires a small number of vibrational repump lasers for cooling)⁴⁴. Laser cooling schemes have been proposed for molecules with an electron centered on atoms with a wide range of mass (as light as Be⁴⁵ and B⁴⁶, and as heavy as Yb⁴⁷ and Tl^48,49), and extension to polyatomic species, while technically more complicated, is straightforward²⁷.

$H_{\mathrm{p}}^{{\mathrm{eff}}}$ is a pseudoscalar interaction, which connects states with different parity P and the same lab frame angular momentum projection m_F. We dub such states $\left| {\tilde \eta ;m_F, + } \right\rangle ,\left| {\tilde \eta \prime ;m_F, - } \right\rangle$ a “PV pair”, with $\tilde \eta$ denoting all other nominal quantum numbers when B = 0. So long as the $\ell$-doublet splitting is not smaller than all SR/HF splittings, PV pairs of a given rotational manifold N cross in an applied B-field when μ_BB  ~ q_b. This situation is common for light molecules due to their typically smaller HF interactions and larger q_b arising from their larger rotational constant B_e. Typical values of q_b imply a modest field of B ~1 mT to 10 mT will bring a PV pair to degeneracy. For instances where the $\ell$-doublet splitting is smaller than all SR and HF intervals, states $\left| {\tilde \eta ;m_F, \pm } \right\rangle$ remain split by ~q_b for any applied B-field. In this case, the Zeeman interaction repels this $\ell$-doublet from a neighboring $\left| {\tilde \eta \prime ;m_F, \pm } \right\rangle$ doublet, preventing the crossing of PV pairs $\left| {\tilde \eta ;m_F, \pm } \right\rangle ,\left| {\tilde \eta \prime ;m_F, \mp } \right\rangle$. Examples are provided in Supplementary Note 1. Regardless of the hierarchy of $\ell$-doubling, SR, and HF interactions, PV crossings always occur between neighboring rotational levels at μ_BB ≈ B_e, as in a diatomic molecule¹⁹, but require an experimentally more challenging B ~100 mT to 1000 mT¹⁹.

In some cases, it may be advantageous to measure iW in a more highly-excited rotational or vibrational state. For example, consider ¹⁷¹YbOH (I(¹⁷¹Yb) = 1/2). We may estimate the relevant parameters in Eq. (1) by reduced-mass-scaling (where appropriate) the constants B_e, ω_b=2, γ, b(H), and c(H) from the ¹⁷⁴YbOH isotopologue (I(¹⁷⁴Yb) = 0)⁵⁰; by assuming constants b(Yb) and c(Yb) to be the same as in chemically similar ¹⁷¹YbF; and by taking $q_b = - 2B_{\mathrm{e}}^2/\omega _2$. With these parameters, the $\ell$-doublet in $\left| {v_b^\ell = 1^1,N = 1} \right\rangle$ is smaller than the SR and Yb HF interactions. For all possible PV pairs, the value of $\left\langle {({\mathbf{S}} \times \widehat {\mathbf{I}}({\mathrm{Yb}})) \cdot \widehat {\mathbf{n}}} \right\rangle$ is only nonzero due to the small state mixing from the HF interaction with the H nucleus. However, in $\left| {v_b^\ell = 1^1,N = 2,3} \right\rangle$, the $\ell$-doublet is larger than the Yb HF splitting and multiple PV pairs with $\left\langle {({\mathbf{S}} \times \widehat {\mathbf{I}}({\mathrm{Yb}})) \cdot \widehat {\mathbf{n}}} \right\rangle$ ~0.1 exist.

Measurement procedure

We now propose a procedure to trap polyatomic molecules and measure matrix elements iW of $H_{\mathrm{p}}^{{\mathrm{eff}}}$. A cryogenic buffer gas beam source creates a slow, cold beam of the desired molecule species^51,52. Molecules are sequentially laser-slowed⁵³, trapped using a magneto-optical trap (MOT)⁵⁴, and loaded into a red-detuned optical dipole trap (ODT) while performing Λ-enhanced cooling³⁵. Then, one of the cooling laser frequencies is turned off in order to optically pump molecules into a single, optically dark SR/HF state. Stimulated Raman adiabatic passage completes state preparation by efficiently transferring to the $\left| {v_b^\ell = 1^1,N,\eta ,m_F,P = ( - 1)^N} \right\rangle$ state^55,56. Performing the measurement in an ODT allows for long interaction times while only requiring precise field control over a small volume.

The PV signal is measured by the Stark interference method, which has been examined in detail elsewhere^6,7,19. We summarize the main points closely following the notation of Cahn et al.³⁶. We apply a static magnetic field ${\mathbf{B}}\, = \,B\widehat {\mathbf{z}}$ to shift a particular PV pair to near degeneracy. We denote the time-dependent probability amplitudes of these states c_±(t), and assume an initial state c₋(0) = 1, c₊ (0) = 0. An oscillating electric field ${\mathbf{E}} = E_0{\mathrm{cos}}(\omega _Et)\widehat {\mathbf{z}}$ is applied to drive the transition between the near-degenerate levels. The effective Hamiltonian $H_ \pm ^{{\mathrm{eff}}}$ for the two-level system can be written³⁶

$$H_ \pm ^{{\mathrm{eff}}} = \left( {\begin{array}{*{20}{c}} \Delta & {d\,E_0\,{\mathrm{cos}}(\omega _E\,t) + i\,W} \\ {d\,E_0\,{\mathrm{cos}}(\omega _E\,t) - i\,W} & { - \alpha \prime \,E_0^2{\mathrm{cos}}^2(\omega _Et)/2} \end{array}} \right).$$

(3)

Here Δ is the small detuning from degeneracy, d is the transition dipole moment, and α′ is the differential polarizability of the two states. In the limit where $W \ll d{\kern 1pt} E_0,{\mathrm{\Delta }} \ll \omega _E$, and assuming for now that α′ = 0, the PV signal S = |c₊(t)|² is

$$S \approx 4\left[ {2\frac{W}{{\mathrm{\Delta }}}\frac{{d\,E_0}}{{\omega _E}} + \left( {\frac{{d\,E_0}}{{\omega _E}}} \right)^2} \right]{\mathrm{sin}}^2\left( {\frac{{{\mathrm{\Delta }}t}}{2}} \right).$$

(4)

From Eq. (4), we see that the combined PV and electric dipole transition probability (first term in square brackets) interferes with the standard electric dipole transition probability (second term in square brackets). The interference term changes sign under a reversal of either E, B, or Δ. The PV matrix element iW may be extracted through an asymmetry measurement¹⁹

$${\cal{A}} = \frac{{S( + E_0) - S( - E_0)}}{{S( + E_0) + S( - E_0)}} = 2\frac{W}{{\mathrm{\Delta }}}\frac{{\omega _E}}{{d\,E_0}} + \ldots ,$$

(5)

where the ellipse denotes higher order terms in W/Δ. Detection using optical cycling should provide shot noise-limited readout⁵⁷.

Estimates of broadening and systematic uncertainties

Sensitivity to iW is expected to be limited by inhomogeneous broadening of Δ. For optically trapped molecules, the detuning uncertainty δΔ is dominated by the differential Stark shift due to the optical trap. Employing certain “magic” polarization conditions in an ODT^{58,59,60,61,62} will set α′ = 0, U′ = 0, and $\delta {\mathrm{\Delta }} = \delta U\prime = \frac{{\partial U\prime }}{{\partial \theta }}\delta \theta$, where U is the trap depth, U′ is the differential trap depth, and θ is an angle related to the polarization. We investigated three such magic conditions for PV pairs in MgNC and YbOH and find $\frac{{\partial U\prime }}{{\partial \theta }} = U/(\theta - \theta _{{\mathrm{magic}}})$ is typical. Figure 1 demonstrates magic conditions for one PV pair in MgNC. In producing magic conditions with a general elliptical polarization, we estimate $\delta \theta /\theta \sim 10^{ - 3}$⁶¹, while for linear polarized light, δθ/θ < 10⁻⁴ is possible using high-quality Glan-type polarizers. We, therefore, expect δU′/U ~ 10⁻⁴ using a magic angle trap^58,59, where the ODT linear polarization is rotated by $\theta _{{\mathrm{magic}}} = {\mathrm{cos}}^{ - 1}\left( {\frac{1}{{\sqrt 3 }}} \right) \approx 54.7^\circ$ with respect to the quantization axis. For U = 2π × 1 MHz, δΔ = 2π × 100 Hz.

Fig. 1

Polarization sensitivity for magic optical trapping. The Differential ac Stark shift U′ is plotted as a function of polarization angle θ as defined in the figures for the nominal |N = 1, J = 1/2, F = 3, m_F = 3, P  = +〉, |N = 1, J = 3/2, F = 3, m_F = 3, P = −〉 parity violation (PV) pair in ²⁵MgNC. For each configuration U = 1.2 MHz for the PV pair when U′ = 0 at θ_magic. The calculation is performed with B ≈ 8.072 mT such that the PV pair is degenerate in the absence of the trapping field. (Red) linear polarization in the x, z-plane, (blue) elliptical polarization in x, z-plane, (green) elliptical polarization in x, y-plane

An inhomogeneous B-field will also produce broadening; typically δΔ ≈ μ_BδB²¹. We require $\delta B \ll 1/\delta U\mu _B \, \lesssim \, 1\, {\mathrm{nT}}$ for B-field inhomogeneity to not limit sensitivity. This implies δB/B = 10⁻⁷ for the largest fields we may require, B = 10 mT. For comparison, the recent BaF NSD-PV measurement demonstrated δB/B = 10⁻⁸, even with the much more experimentally challenging B ≈ 460 mT^20,21. We expect that the ability to easily reverse a smaller field will further aid in detecting and eliminating B-field inhomogeneities. Moreover, the field must only be homogenous over a small, fixed interaction volume for an ODT. For measurements in a molecular beam, the interaction volume grows proportional to the interaction time; in a fountain or free fall configuration, the interaction volume grows proportional to the square of the interaction time. The smaller interaction volume in an ODT implies that systematic uncertainties involving field gradients should also be reduced. In order to prevent temporal fluctuations from limiting sensitivity, we will require fluctuations δB ≲ 1 nT for all timescales longer than 1/ω_E, the shortest relevant timescale in the experiment.

From Eq. (3), we see that electric dipole and NSD-PV transition amplitudes are π/2 out of phase, and there is no interference in a static E-field. However, the presence of a non-reversing E-field E_nr still poses an issue. In the molecule frame, a static E_nr has significant frequency components at axial and radial ODT frequencies ω_z, ω_r, and multiples, sums, and differences thereof. Assuming uncorrelated trap oscillations, this effect will lead to an inhomogeneous broadening much smaller than that of the differential ac Stark shift. Accurate measurement of E_nr is possible by Stark interference with a reversible pulsed dc field E_r²¹, or by microwave depletion spectroscopy⁶³. Finally, investigating multiple PV pairs in the same molecule provides strong systematic error rejection by varying the ratio $\left\langle {({\mathbf{S}} \times \widehat {\mathbf{I}}) \cdot \widehat {\mathbf{n}}} \right\rangle /d$ by a calculable, possibly large, amount¹⁹ (see Supplementary Note 3). For example, in PV pairs with different signs of $\left\langle {({\mathbf{S}} \times \widehat {\mathbf{I}}) \cdot \widehat {\mathbf{n}}} \right\rangle /d$, contributions to ${\cal{A}}$ from actual NSD-PV will switch sign, but contributions from E_nr will not.

Other relaxation mechanisms are expected to lead to negligible broadening compared to differential ac Stark shifts. For example, in a beam experiment, δΔ is typically limited by interaction time τ, with τ ~ 100 μs²⁰. In an ODT, τ may easily exceed 1 s. Trapped molecule lifetimes τ_trap = 0.5 s to 25 s have been reported in a variety of traps^23,55,64,65. With near-ideal vacuum conditions, we expect trap lifetimes τ_trap ~ 10 s, limited by vibrational decay. The loss rate due to off-resonant scattering from the trapping laser can typically be made R_sc ≲ 1 s⁻¹ by using standard mid-infrared wavelength fiber lasers. Typical inelastic collision cross sections are expected to be σ_in ≲ 10⁻⁹ cm³/s. Comparing with the trapping conditions of Cheuk et al.³⁵, with N = 1300 molecules at density n = 6 × 10⁸ cm⁻³, we estimate an inelastic molecule-molecule collision rate of R_in ≈ 0.5 s⁻¹. Therefore, collisions will become important when trapped molecule number N ≳ 10⁵, or with additional cooling.

Estimated sensitivity to parity violating effects

We now estimate the sensitivity of our method to NSD-PV matrix elements, $\delta W = 1/\tau \sqrt {RNT}$, where R is the repetition rate, N the number of trapped molecules per measurement, and T the total measurement time. We assume molecules are trapped in a U = 2π × 1 MHz deep magic angle trap, with δθ/θ = 10⁻⁴. The combined effects of all relaxation times considered should allow for interaction times of τ ≲ 1 s, but optimum sensitivity is achieved by setting the interaction time equal to the coherence time of the system: $\tau = \frac{1}{{\delta {\mathrm{\Delta }}}} \approx \frac{1}{U}\frac{\theta }{{\delta \theta }} \approx 1.6\,{\mathrm{ms}}$. Allowing t_MOT = 50 ms to load the MOT and t_trans = 40 ms for state transfer, repetition rates R = 10 s⁻¹ should be possible. We expect that with molecules produced from an isotopically enriched source, N ≈ 1000 for all species; this would be equivalent to the best to-date sample of directly cooled molecules in an ODT³⁵. Under these conditions, our expected experimental sensitivity is δW ≈ 2π × 1 Hz/$\sqrt {{\mathrm{Hz}}}$. This represents a factor of 70 improvement over the best to-date NSD-PV measurement in BaF^20,21,36. More ambitiously, one could plausibly expect N = 10⁵ to 10⁶ could be achieved with improved loading and cooling efficiency^31,66.

Discussion

With the proposed sensitivity, it should be possible to separate contributions to κ from nuclear anapole (κ_a ∝ A^2/3) and V_eA_n (κ₂ A-independent) effects by measuring NSD-PV in a variety of nuclei. With the only non-zero NSD-PV measurement to-date in the heavy ¹³³Cs¹⁰, a precise measurement of PV in a light system would be especially illuminating. In light nuclei, κ ≈ κ₂. As stated, κ₂ depends upon C_2u,d which are among the most poorly know SM parameters and are suppressed at tree level. Thus, a precise measurement of NSD-PV in light systems could potentially be sensitive to beyond SM physics above the 1 TeV scale¹⁴. In molecules such as BeNC and MgNC, the nuclear and molecular calculations are highly tractable. Furthermore, W_p(N) ~ W_p(C) ~ 2π × 10 mHz in these systems; a single species could provide a 10% measurement of κ for three nuclei (¹³C, ¹⁴N, and either ⁹Be, ¹⁰Be, or ²⁵Mg) with T ≲ 100 hours per nucleus. The nuclear structure of ¹⁴N is of special interest and well studied due to the anomalously long ¹⁴C → ¹⁴N half-life and their role in radiocarbon dating²².

Because iW is enhanced by ≈ Z²A^2/3 in heavy species^41,42, RaOH appears especially promising²⁹. However, the high mass is a hinderance to effective laser-slowing by the standard methods for molecules^52,53. Moreover, the longest-lived Ra isotope possessing nuclear spin (²²⁵Ra) has a half-life of only 15 days. New techniques beyond those proposed here may be required to produce trapped RaOH in sufficient quantity for precision measurement. Nevertheless, MOTs of atomic ²²⁵Ra with typical N = 1000 have been produced for atomic electric dipole measurements⁶⁷; with our expected sensitivity, only N = 300 total molecules would need to be detected for δW/W = 0.1.

We have shown that optically trapped polyatomic molecules offer a dramatically enhanced sensitivity to parity-violating effects and additional checks of systematic errors. Restriction to laser-coolable species still allows for measurement of NSD-PV in nuclei with a wide range of masses, necessary for the determination of key SM parameters and tests of beyond SM physics. The improved sensitivity should enable measurements of NSD-PV even in light nuclei where calculations are highly accurate.

Here, we have only considered linear asymmetric molecules. Symmetric top molecules possess k-doublets of opposite parity (similar to $\ell$-doublets) even in their vibrational ground state. For most ²A₁ states (analogue of ²Σ), the k-doublet splitting (~10 kHz) is smaller than SR/HF and thus not suitable for Zeeman tuning PV pairs to degeneracy for a NSD-PV measurement. However, the close spacing and miniscule differential Zeeman and ac Stark shifts of k-doublets may make symmetric top molecules in a magnetic or optical trap ideal for measuring NSI-PV.