Nuclear-Spin Dependent Parity Violation in Optically Trapped Polyatomic Molecules

We investigate using 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 order the of 10 days and 1 second, respectively.

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 scale [1]. NSI-PV has been measured in protons and a number of heavy atoms [2][3][4][5][6][7] and found to be in good agreement with SM PV predictions due to the weak charge Q W . However, the only nonzero measurement (14 % relative uncertainty) of NSD-PV in an atomic system comes from Cs [8], and this result implies constraints on SM meson-nucleon couplings which are in disagreement with other atomic PV measurements [9,10]. NSD-PV arises primarily from two interactions: vector electron-axial nucleon electroweak current coupling (V e A n ), and the nuclear anapole moment. The V e A n effect is described by two parameters C 2u and C 2d relating to spin-dependent Z 0 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 %, respectively [11]. PV measurements may also probe beyond standard model physics [12,13]. Searches for oscillating PV signals have been proposed as a means to detect axion-like particles, a leading dark matter candidate [14].
A beam of cold diatomic molecules has been demonstrated [15][16][17][18][19] 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 energy [6]. 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 tractable [20].
Here, we show that linear asymmetric polyatomic molecules in an optical trap are well-suited for measurement of NSD-PV. Polyatomic molecules possess oppositeparity states 10 to 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 molecule 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 δW/W = 1/τ √ N 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 [21,31,32], we expect at least a factor of 70 increase in PV sensitivity over the state-of-the-art BaF measurement [18,33]. Our method is applicable to all laser-coolable polyatomic molecules with 2 Σ ground states, and possibly others.
Consider the properties of linear asymmetric molecules with a 2 Σ + 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 ℓ = ±v b , ±(v b − 2), . . . , ±1 or 0. Within this vibrational manifold, the effective Hamiltonian is where B is the rotational constant, γ is the spin-rotation (SR) constant, b and c are hyperfine (HF) constants, e is the electron charge, T 2 (∇E) · T 2 (Q) is a scalar product of rank-2 spherical tensors describing the elec-tric 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 [34,35]. The upper (lower) sign corresponds to the parity P of the closely spaced "ℓ-doublet" eigenstates N, v b ℓ , P = ± = 1 √ 2 (|N, +ℓ ± (−1) N −ℓ |N, −ℓ ). The ℓ-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/2π ∼ 1 GHz to 100 GHz. In an excited bending mode, opposite-parity states are spaced by only q b ≈ −2B 2 /ω b , which is on the order of q b /2π ∼ 10 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).
For a given electronic state of a molecule, the effective NSD-PV Hamiltonian is [36] Here, W p encodes the overlap of unpaired electrons with the nucleus and can be calculated from molecular spectroscopy with high precision for 2 Σ + states [15],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 = κ 2 + κ a + κ Q . κ 2 is proportional to the strength of the V e A n coupling, and is independent of nuclear mass A (for a typical nucleus, |κ 2 | ≃ 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 κ 2 and κ a [37]. 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-functional [38], quasirelativistic zero-order regular approximation [26], and (with an estimated 1.5 % accuracy) relativistic coupled-cluster [39] 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 approximately 10 %, assuming the SR/HF constants in Eq. 1 are known [15]. The SR/HF constants relating to a typical metal atom M differ by a approximately 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. In Table I [38,39], except RaOH for which calculation exists [26]. Values for κ2 assume C2N = −0.05; values for κa assume weak neutronneutron coupling g = 1 [40]. W m is the value of W with maximal (S ×Î) ·n .
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 (i) p ≈ 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) [41]. Laser cooling schemes have been proposed for molecules with an electron centered on atoms with a wide range of mass (as light as Be [42] and B [43], and as heavy as Yb [44] and Tl [45,46]), and extension to polyatomic species, while technically more complicated, is straightforward [24].
H eff p 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 |η; m F , + , |η ′ ; m F , − a "PV pair", withη denoting all other nominal quantum numbers when B = 0. So long as the ℓ-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 B B ∼ 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. 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 ℓdoublet splitting is smaller than the SR/HF splitting, states |η; m F , ± remain split by ∼ q b for any applied Bfield. In this case, the Zeeman interaction repels this ℓ-doublet from a neighboring |η ′ ; m F , ± doublet, preventing the crossing of PV pairs |η; m F , ± , |η ′ ; m F , ∓ . The first PV crossing actually occurs at µ B B ≈ B, as in a diatomic molecule [17], and requires an experimentally more challenging B ∼ 100 mT to 1000 mT [17].
In some cases, it may be advantageous to measure iW in an excited rotational or vibrational state. For example, consider 171 YbOH (I( 171 Yb) = 1/2). We may estimate the relevant parameters in Eq. 1 by reduced-mass-scaling (where appropriate) the constants B, ω b=2 , γ, b(H), and c(H) from the 174 YbOH isotopologue (I( 174 Yb) = 0) [47]; by assuming constants b(Yb) and c(Yb) to be the same as in chemically similar 171 YbF; and by taking q b = −2B 2 /ω 2 . With these parameters, the ℓ-doublet in v b ℓ = 1 1 , N = 1 is smaller than the SR and Yb HF interactions. For all possible PV pairs, the value of (S ×Î(Yb)) ·n is only nonzero due to the small state mixing from the HF interaction with the H nucleus. However, in v b ℓ = 1 1 , N = 2, 3 , the ℓ-doublet is larger than the Yb HF splitting and multiple PV pairs with (S ×Î(Yb)) ·n ∼ 0.1 exist. We now propose a procedure to trap polyatomic molecules and measure matrix elements iW of H eff p . A cyrogenic buffer gas beam source creates a slow, cold beam of the desired molecule species [48,49]. Molecules are sequentially laser-slowed [50], trapped using a magneto-optical trap (MOT) [51], and loaded into a red-detuned optical dipole trap (ODT) while performing Λ-enhanced cooling [32]. 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 v b ℓ = 1 1 , N, P = (−1) N state [52,53]. The PV signal is measured by the Stark interference method, which has been examined in detail elsewhere [6,7,54]. We summarize the main points closely following the notation of Ref. [33]. We apply a static magnetic field B = Bẑ 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 E = E 0 cos(ω E t)ẑ is applied to drive the transition between the near degenerate levels. The effective Hamiltonian H eff ± for the two level system can be written [33] H (3) Here ∆ 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 ≪ d E 0 , ∆ ≪ ω E , and assuming for now that α ′ = 0, the PV signal S = |c From Eq. 4, we see that the PV transition amplitude (first term in square brackets) interferes with the E1 transition amplitude (second term in square brackets). The interference term changes sign under a reversal of either E, B, or ∆. The PV matrix element W may be extracted through an asymmetry measurement [17]  where the ellipse denotes higher order terms in W/∆. Detection using optical cycling should provide shot noiselimited readout.
An inhomogeneous B-field will also produce broadening; typically δ∆ ≈ µ B δB [19]. We require δB ≪ 1/δU µ B 1 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 [18,19]. We expect that the ability to easily reverse a smaller field will further aid in de-tecting and eliminating B-field inhomogeneities. Moreover, the field must only be homogenous over the interaction volume, which is approximately 100 times smaller in an ODT compared a beam. Systematic uncertainties involving field gradients should similarly by reduced a factor of 100 from the smaller interaction volume.
From Eq. 2, we see that Stark and NSD-PV 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 a homogeneous 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 r [19], or by microwave depletion spectroscopy [60]. Finally, investigating multiple PV pairs in the same molecule provides strong systematic error rejection by varying the ratio (S ×Î) ·n /d by a calculable, possibly large, amount [54]. For example, in PV pairs with different signs of (S ×Î) ·n /d, contributions to 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, δ∆ is typically limited by interaction time τ , with τ ∼ 100 µs [18].
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 [21,52,61,62]. 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 cm 3 /s. Comparing with the trapping conditions of Ref. [32] with N = 1300 molecules at density n = 6 × 10 8 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 10 5 , or with additional cooling.
We now estimate the sensitivity of our method to NSD-PV matrix elements, δW = 1/τ √ RN T , 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 peak sensitivity is achieved with τ = 1/δ∆ ≈ 1/(U δθ) ≈ 1.6 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 ODT [32]. Under these conditions, our expected experimental sensitivity is δW ≈ 2π × 1 Hz/ √ Hz. This represents a factor of 70 improvement over the best to-date NSD-PV measurement in BaF [18,19,33]. More ambitiously, one could plausibly expect N = 10 5 to 10 6 could be achieved with improved loading and cooling efficiency [28,63].
With the proposed sensitivity, it should be possible to separate contributions to κ from nuclear anapole (κ a ∝ A 2/3 ) and V e A n (κ 2 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 133 Cs [8], a precise measurement of PV in a light system would be especially illuminating. In light nuclei, κ ≈ κ 2 . As stated, κ 2 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 scale [12]. 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 ( 13 C, 14 N, and either 9 Be, 10 Be, or 25 Mg) with T 100 hours per nucleus. The nuclear structure of 14 N is of special interest and well studied due to the anomalously long 14 C→ 14 N half-life and their role in radiocarbon dating [20].
Because iW is enhanced by ≈ Z 2 A 2/3 in heavy species [38], RaOH appears especially promising [26]. However, the high mass is a hinderance to effective laserslowing by the standard methods for molecules [49,50]. Moreover, the longest-lived Ra isotope possessing nuclear spin ( 225 Ra) has a half-life of only 15 days. Nevertheless, MOTs of atomic 225 Ra with typical N = 1000 have been produced for atomic electric dipole measurements [64]; 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 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 ℓ-doublets) even in their vibrational ground state. For most 2 A 1 states (analogue of 2 Σ), 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.