Methods, analysis, and the treatment of systematic errors for the electron electric dipole moment search in thorium monoxide

We performed a spin precession measurement, resembling previous beam-based eEDM experiments [20, 41, 42], on ${}^{232}{\mathrm{Th}}^{16}{\rm{O}}$ molecules in a pulsed molecular beam generated by a cryogenic buffer gas beam source. Figure 4 shows a simplified schematic of the measurement. The molecules fly at velocity $v\approx 200$ m s⁻¹ into a magnetically shielded region with nominally uniform and parallel electric $\vec{{ \mathcal E }}$ and magnetic $\vec{{ \mathcal B }}$ fields. Molecule population is transfered from $| {X}^{1}{{\rm{\Sigma }}}^{+},J=1,M=\pm 1\rangle$ in the electronic ground state to the metastable $| H,J=1,M=\pm 1,{\rm{\Omega }}=\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}M\rangle \equiv | \pm ,\,\tilde{{ \mathcal N }}\rangle$ state manifold (in the $| \pm ,\,\tilde{{ \mathcal N }}\rangle$ nomenclature we use ± to refer to $M=\pm 1$) by optical pumping through the short-lived $| {A}^{3}{{\rm{\Pi }}}_{{0}^{+}},J=0,M=0\rangle$ state with a 943 nm laser. This results in an even distribution of population in an incoherent mixture of the four $| \pm ,\,\tilde{{ \mathcal N }}\rangle$ states in H¹² . Figure 3 shows the electronic states of ThO relevant to the eEDM measurement.

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In the absence of any experimental imperfections, we describe our system in terms of coordinate axes $+\hat{z}$ along $+\vec{{ \mathcal E }}$ (for a specified sign of applied field that we denote as positive, pointing approximately east to west in the lab) and $+\hat{x}$ along the direction of the molecular beam (which travels approximately south to north) such that $+\hat{y}$ is approximately aligned with gravity (see figure 4). Note that when we reverse the direction of the electric field, by construction the laboratory coordinate system does not change and the orientation of the electric field can be described by $\tilde{{ \mathcal E }}\equiv \mathrm{sgn}(\hat{z}\cdot \vec{{ \mathcal E }})=\pm 1$. Analogously, we reverse the direction of the magnetic field between two $\tilde{{ \mathcal B }}\equiv \mathrm{sgn}(\hat{z}\cdot \vec{{ \mathcal B }})=\pm 1$ states. Since the directions of the fields are encoded by $\tilde{{ \mathcal E }}$ and $\tilde{{ \mathcal B }}$, we define the magnitudes of the fields simply as ${{ \mathcal B }}_{z}\equiv | {{ \mathcal B }}_{z}|$ and ${ \mathcal E }\equiv | \vec{{ \mathcal E }}|$.

Preparation of a spin state occurs in an electric field such that the molecule is polarised and the molecule orientation, specified by ${ \mathcal N }$, can be spectroscopically chosen. A superposition of the $M=\pm 1$ sublevels is prepared by optically pumping on the transition at 1090 nm between states $| \pm ,\,\tilde{{ \mathcal N }}\rangle$ and $| {C}^{1}{{\rm{\Pi }}}_{1},J=1,M=0\rangle (| {\rm{\Omega }}=+1\rangle -\tilde{{ \mathcal P }}| {\rm{\Omega }}=-1\rangle )/\sqrt{2}\equiv | C,\tilde{{ \mathcal P }}\rangle$, where $\tilde{{ \mathcal P }}=\pm 1$ is the excited state parity¹³ , with laser light linearly polarised in the xy plane. The resulting state corresponds to having the total angular momentum of the molecule aligned in the xy plane. Because the σ electron's spin is aligned with $\vec{J}$, by the Wigner–Eckart theorem this is equivalent to aligning the spin [77], and we use this shorthand from here on. The state preparation laser frequency is tuned to spectroscopically select the molecule alignment $\tilde{{ \mathcal N }}$, while the nearly degenerate $M=\pm 1$ states remain unresolved. The excited state C, which decays at a rate ${\gamma }_{C}\approx 2\pi \times 0.3\,\mathrm{MHz}$, decays primarily ($\approx 75 \%$ [67]) to the ground state so that one superposition of the two $| \pm ,\,\tilde{{ \mathcal N }}\rangle$ states is optically pumped out of H and the remaining orthogonal superposition, which is 'dark' to the preparation laser beam, is the prepared state. The linear polarisation of the state preparation laser beam, ${\hat{\epsilon }}_{\mathrm{prep}}$, sets the relative coupling of each of the two $| \pm ,\,\tilde{{ \mathcal N }}\rangle$ states to $| C,\tilde{{ \mathcal P }}\rangle$ and determines the spin alignment angle of the remaining state in the laboratory frame. The bright superposition $| B({\hat{\epsilon }}_{\mathrm{prep}})\rangle$ is pumped away, and the orthogonal dark superposition $| D({\hat{\epsilon }}_{\mathrm{prep}})\rangle$ remains.

For the moment, we consider the specific case $\tilde{{ \mathcal P }}=+1$ and ${\hat{\epsilon }}_{\mathrm{prep}}=\hat{x}$, (the general case will be discussed in section 3.1.2). In this case, the prepared state

$$\begin{eqnarray}&&| \psi (t=0),\tilde{{ \mathcal N }}\rangle =\displaystyle \frac{1}{\sqrt{2}}(| +,\,\tilde{{ \mathcal N }}\rangle -| -,\,\tilde{{ \mathcal N }}\rangle )\end{eqnarray} \tag{ 8 }$$

has the electron spin aligned along the $\hat{y}$ axis. As the molecules traverse the spin precession region of length L = 22 cm (which takes a time $\tau \approx 1$ ms), the electric and magnetic fields exert torques on the electric and magnetic dipole moments, causing the spin to precess in the xy plane by angle $2\phi ;$ this corresponds to the state

$$\begin{eqnarray}&&| \psi (t=\tau ),\tilde{{ \mathcal N }}\rangle =\displaystyle \frac{1}{\sqrt{2}}({{\rm{e}}}^{-{\rm{i}}\phi }| +,\,\tilde{{ \mathcal N }}\rangle -{{\rm{e}}}^{+{\rm{i}}\phi }| -,\,\tilde{{ \mathcal N }}\rangle ),\end{eqnarray} \tag{ 9 }$$

where ϕ is given approximately by the sum of the Zeeman and eEDM contributions to the spin precession angles,

$$\begin{eqnarray}&&\phi =-(\tilde{{ \mathcal B }}{g}_{1}{\mu }_{{\rm{B}}}{{ \mathcal B }}_{z}+\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}{d}_{e}{{ \mathcal E }}_{\mathrm{eff}})\tau .\end{eqnarray} \tag{ 10 }$$

The sign of the eEDM term, $\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}$, arises from the relative orientation between ${\vec{{ \mathcal E }}}_{\mathrm{eff}}$ and the electron spin as illustrated in figure 2.

At the end of the spin precession region, we measure ϕ by optically pumping on the same $H\to C$ transition with the linearly polarised state readout laser beam. The polarisation alternates rapidly between two orthogonal linear polarisations $\hat{X}$ and $\hat{Y}$, such that each molecule is subject to excitation by both polarisations as it flies through the detection region, and we record the modulated fluorescence signals ${F}_{X}$ and ${F}_{Y}$ from the decay of C to the ground state at 690 nm. This procedure amounts to a projective measurement of the spin onto $\hat{X}$ and $\hat{Y}$, which are defined such that $\hat{X}$ is at an angle θ with respect to $\hat{x}$ in the xy plane. To determine ϕ we compute the asymmetry,

$$\begin{eqnarray}&&{ \mathcal A }\equiv \displaystyle \frac{{F}_{X}-{F}_{Y}}{{F}_{X}+{F}_{Y}}\propto \cos [2(\phi -\theta )].\end{eqnarray} \tag{ 11 }$$

We set ${{ \mathcal B }}_{z}$ and θ such that $\phi -\theta \approx (\pi /4)(2n+1)$ for integer n, so that the asymmetry is linearly proportional to small changes in ϕ and maximally sensitive to the eEDM. A simplified schematic of the experimental procedure just described is shown in figure 4.

By repeating the measurement of ϕ after having reversed any one of the signs $\tilde{{ \mathcal N }}$, $\tilde{{ \mathcal E }}$ or $\tilde{{ \mathcal B }}$, we may isolate the eEDM phase from the Zeeman phase. In practice, we repeat the phase measurement under all 2³ $(\tilde{{ \mathcal N }},\tilde{{ \mathcal E }},\tilde{{ \mathcal B }})$ experiment states to reduce the sensitivity of the eEDM measurement to other spurious phases, and we extract the phase ${\phi }^{{ \mathcal N }{ \mathcal E }}=-{d}_{e}{{ \mathcal E }}_{\mathrm{eff}}\tau ={\phi }_{\mathrm{EDM}}$. Here, we have introduced the notation ${\phi }^{u}$, discussed in detail in the next section, which we use throughout this document to refer to the component of ϕ that is odd under the set of switches listed in the superscript u, and implicitly even under those which are not listed (see section 3.1.2 and equation (23) for a rigorous definition). A component which is even under all switches is considered to be 'non-reversing' and is given an 'nr' superscript.

To fully describe the method by which we extracted d_e from the data in section 4, and to describe the systematic error models in section 5, we must introduce some additional formalism to describe the spin precession measurement to generalise the simple case described in the previous section.

We work in the regime in which the Stark shift in H is approximately linear, ${E}_{\mathrm{Stark}}\approx -\tilde{{ \mathcal N }}{D}_{1}{ \mathcal E }$, which holds when the Stark interaction energy is large compared to the Ω-doublet energy splitting ${{\rm{\Delta }}}_{{\rm{\Omega }},1}$ but small compared to the rotational energy scale, described by the H-state rotational constant ${B}_{H}\approx 2\pi \times$ 9.8 GHz, i.e. ${{\rm{\Delta }}}_{{\rm{\Omega }},1}\ll {D}_{1}{ \mathcal E }\ll {B}_{H}$. In this regime, the molecular alignment is approximately related to Ω by $\tilde{{ \mathcal N }}=\tilde{{ \mathcal E }}M{\rm{\Omega }};$ this relation is assumed throughout this document. This is a good approximation, but it is notable that due to the Stark interaction at first order in perturbation theory, each $| M,\tilde{{ \mathcal N }}\rangle$ state is a superposition of all four $| H,J,M,{\rm{\Omega }}\rangle$ states with J = 1, 2 and ${\rm{\Omega }}=\pm 1$. This effect is discussed further in sections 5.2.6 and 5.6.2.

Let us consider the preparation of a spin-aligned state again. Starting from an incoherent mixture of the four $| \pm ,\,\tilde{{ \mathcal N }}\rangle$ states, we perform optical pumping on the electric dipole transition between $| \pm ,\,\tilde{{ \mathcal N }}\rangle$ and $| C,\tilde{{ \mathcal P }}\rangle$, for a specific $\tilde{{ \mathcal N }}$, with laser light of polarisation ${\hat{\epsilon }}_{\mathrm{prep}}$ that is nominally linear in the xy plane. This step depletes the bright superposition state (see e.g. [78])

$$\begin{eqnarray}&&| B({\hat{\epsilon }}_{\mathrm{prep}},\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle =({\hat{\epsilon }}_{+1}^{* }\cdot {\hat{\epsilon }}_{\mathrm{prep}}^{* })| +,\,\tilde{{ \mathcal N }}\rangle -\tilde{{ \mathcal P }}({\hat{\epsilon }}_{-1}^{* }\cdot {\hat{\epsilon }}_{\mathrm{prep}}^{* })| -,\,\tilde{{ \mathcal N }}\rangle ,\end{eqnarray} \tag{ 12 }$$

where ${\hat{\epsilon }}_{\pm 1}=\mp (\hat{x}\pm {\rm{i}}\hat{y})/\sqrt{2}$ are unit vectors for circular polarisation. The corresponding dark state (with which the laser does not interact) is the orthogonal superposition

$$\begin{eqnarray}&&| D({\hat{\epsilon }}_{\mathrm{prep}},\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle =({\hat{\epsilon }}_{+1}^{* }\cdot {\hat{\epsilon }}_{\mathrm{prep}})| +,\,\tilde{{ \mathcal N }}\rangle +\tilde{{ \mathcal P }}({\hat{\epsilon }}_{-1}^{* }\cdot {\hat{\epsilon }}_{\mathrm{prep}})| -,\,\tilde{{ \mathcal N }}\rangle .\end{eqnarray} \tag{ 13 }$$

This dark state serves as the initial state, $| \psi (0),\tilde{{ \mathcal N }}\rangle =| D({\hat{\epsilon }}_{\mathrm{prep}},\tilde{{ \mathcal N }},\tilde{{ \mathcal P }}=+1)\rangle$, for the spin-precession experiment, where we fixed the state preparation laser frequency to address the excited state with parity $\tilde{{ \mathcal P }}=+1$. The state preparation laser polarisation can be parameterised as

$$\begin{eqnarray}&&{\hat{\epsilon }}_{\mathrm{prep}}=-{{\rm{e}}}^{-{\rm{i}}{\theta }_{\mathrm{prep}}}\cos {{\rm{\Theta }}}_{\mathrm{prep}}{\hat{\epsilon }}_{+1}+{{\rm{e}}}^{+{\rm{i}}{\theta }_{\mathrm{prep}}}\sin {{\rm{\Theta }}}_{\mathrm{prep}}{\hat{\epsilon }}_{-1},\end{eqnarray} \tag{ 14 }$$

where ${{\rm{\Theta }}}_{\mathrm{prep}}\approx \pi /4$ defines the ellipticity Stokes parameter ${({S}_{3}/I)}_{\mathrm{prep}}=\cos 2{{\rm{\Theta }}}_{\mathrm{prep}}\approx 0$, and ${\theta }_{\mathrm{prep}}$ defines the linear polarisation angle with respect to $\hat{x}$ in the xy plane. From here on, we refer to the ellipticity Stokes parameter as $S\equiv {S}_{3}/I$. There is a one-to-one correspondence between the dark state superposition and the projection of the laser polarisation ${\hat{\epsilon }}_{\mathrm{prep}}$ onto the xy plane. If the laser polarisation does not lie entirely in the xy plane, equations (12) and (13) are still appropriate, but require normalisation. Note that if the laser is linearly polarised, switching the excited state parity $\tilde{{ \mathcal P }}$ has the same effect on the dark state as rotating the laser polarisation angle by $\pi /2$.

Following the initial state preparation, the molecules traverse the spin-precession region with their forward velocity nominally along $\hat{x}$. In this region there are nominally uniform and parallel electric ($\vec{{ \mathcal E }}$) and magnetic ($\vec{{ \mathcal B }}$) fields, which produce energy shifts given by

$$\begin{eqnarray}&&E(M,\tilde{{ \mathcal N }})=-| M| {D}_{1}{ \mathcal E }\tilde{{ \mathcal N }}-{{Mg}}_{1}{\mu }_{{\rm{B}}}{{ \mathcal B }}_{z}\tilde{{ \mathcal B }}-M\eta {\mu }_{{\rm{B}}}{ \mathcal E }{{ \mathcal B }}_{z}\tilde{{ \mathcal N }}\tilde{{ \mathcal B }}-{{Md}}_{e}{{ \mathcal E }}_{\mathrm{eff}}\tilde{{ \mathcal N }}\tilde{{ \mathcal E }},\end{eqnarray} \tag{ 15 }$$

where D₁ is the electric dipole moment of $| H,J=1\rangle$. Here $\eta =0.79(1)$ nm V⁻¹ accounts for the ${ \mathcal E }$-dependent magnetic moment difference between the two sets of $\tilde{{ \mathcal N }}$ levels in $| H,J=1\rangle$ [59], as described in section 4.2.1. The energy shift terms that depend on the sign of M contribute to the spin precession angle ϕ, which is given by:

$$\begin{eqnarray}&&\phi =\displaystyle \frac{1}{2}{\int }_{0}^{L}(E(M=+1,\tilde{{ \mathcal N }})-E(M=-1,\tilde{{ \mathcal N }}))\displaystyle \frac{{\rm{d}}x}{v}.\end{eqnarray} \tag{ 16 }$$

This phase is dominated by the magnetic (Zeeman) interaction. The Stark shift, proportional to $| M|$, does not contribute. The state then evolves to:

$$\begin{eqnarray}&&| \psi (\tau ),\tilde{{ \mathcal N }}\rangle =({{\rm{e}}}^{-{\rm{i}}\phi }| +,\,\tilde{{ \mathcal N }}\rangle \langle +,\,\tilde{{ \mathcal N }}| \,+\,{{\rm{e}}}^{+{\rm{i}}\phi }| -,\,\tilde{{ \mathcal N }}\rangle \langle -,\,\tilde{{ \mathcal N }}| )| \psi (0),\tilde{{ \mathcal N }}\rangle ,\end{eqnarray} \tag{ 17 }$$

(recall $| \psi (0),\tilde{{ \mathcal N }}\rangle =| D({\hat{\epsilon }}_{\mathrm{prep}},\tilde{{ \mathcal N }},\tilde{{ \mathcal P }}=+1)\rangle$ per equation (13)) and molecules enter a detection region where the state is read out by optically pumping again between the $| H,J=1\rangle$ and $| C,J=1\rangle$ manifolds. This optical pumping is performed alternately by two laser beams with nominally orthogonal linear polarisations ${\hat{\epsilon }}_{X}$ and ${\hat{\epsilon }}_{Y}$ ¹⁴ . These beams excite the projection of $| \psi (\tau ),\tilde{{ \mathcal N }}\rangle$ onto the bright states

$$\begin{eqnarray}&&| B({\hat{\epsilon }}_{X},\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle \quad \mathrm{and}\quad | B({\hat{\epsilon }}_{Y},\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle ,\end{eqnarray} \tag{ 18 }$$

(with the same $\tilde{{ \mathcal N }}$ that was addressed in the state preparation optical pumping step, but with an independent choice of $\tilde{{ \mathcal P }}$) with probability ${P}_{X,Y}$ respectively. In the ideal case in which all laser polarisations are exactly linear, this probability is given by

$$\begin{eqnarray}&&{P}_{X,Y}(\phi ,{\theta }_{\mathrm{prep}},{\theta }_{X,Y},\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})={| \langle B({\hat{\epsilon }}_{X,Y},\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})| \psi (\tau ),\tilde{{ \mathcal N }}\rangle | }^{2}=[1-\tilde{{ \mathcal P }}\cos (2({\theta }_{\mathrm{prep}}-{\theta }_{X,Y}+\phi ))]/2,\end{eqnarray} \tag{ 19 }$$

where ${\theta }_{X,Y}$ are the linear polarisation angles of the state readout beams, with respect to $\hat{x}$. The result is a signal that varies sinusoidally with the precession angle ϕ. To measure these probabilities, we observe the associated modulated fluorescence signals, ${F}_{X,Y}={{fN}}_{0}{P}_{X,Y}$, where N₀ is the number of molecules in the addressed $\tilde{{ \mathcal N }}$ level at the state readout region, and f is the fraction of total fluorescence photons that are detected.

To distinguish between molecule number fluctuations and phase variations, we normalise with respect to the former by rapidly switching the state readout laser between the two orthogonal polarisations, ${\hat{\epsilon }}_{X,Y}$, every 5 μs. This is significantly quicker than fluctuations in the molecule number and is sufficiently quick that every molecule is interrogated by both polarisations (see section 4 or [64] for more details). We then form an asymmetry ${ \mathcal A }$, which is immune to molecule number fluctuations, given by

$$\begin{eqnarray}&&{ \mathcal A }=\displaystyle \frac{{F}_{X}-{F}_{Y}}{{F}_{X}+{F}_{Y}}=\tilde{{ \mathcal P }}\cos [2(\phi -\theta )],\end{eqnarray} \tag{ 20 }$$

where we have assumed that the readout polarisations are exactly orthogonal, given by ${\theta }_{X}={\theta }_{\mathrm{read}}$ and ${\theta }_{Y}={\theta }_{\mathrm{read}}+\pi /2$, and where we have defined $\theta \equiv {\theta }_{\mathrm{read}}-{\theta }_{\mathrm{prep}}$ ¹⁵ . In this equation and from now on unless otherwise noted, $\tilde{{ \mathcal P }}$ refers to the excited state parity that is addressed by the state readout laser, not to be confused with the excited state parity addressed by the state preparation laser, which is kept fixed.

The value of ${{ \mathcal B }}_{z}$ and the state preparation and readout laser beam polarisations are chosen so that $| \phi -\theta | \approx \pi /4$. This corresponds to the linear part of the asymmetry fringe in equation (20), where ${ \mathcal A }$ is most sensitive to, and linearly proportional to, small changes in ϕ (see figure 23). A variety of effects including imperfect optical pumping, decay from C back to H, elliptical laser polarisation and forward velocity dispersion, reduce the measurement sensitivity by a 'contrast' factor

$$\begin{eqnarray}&&{ \mathcal C }\equiv -\displaystyle \frac{1}{2}\displaystyle \frac{\partial { \mathcal A }}{\partial \theta }\approx \displaystyle \frac{1}{2}\displaystyle \frac{\partial { \mathcal A }}{\partial \phi },\end{eqnarray} \tag{ 21 }$$

with $| { \mathcal C }| \leqslant 1$. We measure this parameter by dithering $\theta ={\theta }^{\mathrm{nr}}+{\rm{\Delta }}\theta \tilde{\theta }$ (where ${\theta }^{\mathrm{nr}}$ is the average or 'non-reversing' polarisation angle) between states of $\tilde{\theta }=\pm 1$, with amplitude ${\rm{\Delta }}\theta =0.05\,\mathrm{rad}$. We found that typically $| { \mathcal C }| \approx 0.94$. We then extract the measured phase, ${\rm{\Phi }}={ \mathcal A }/(2{ \mathcal C })+q\pi /4$, by normalising the asymmetry measurements according to the measured contrast—see section 4 for more details on the data analysis methods used to evaluate this quantity. In the ideal case, the measured phase matches closely with the precession phase, ${\rm{\Phi }}\approx \phi$. However, a variety effects that are investigated closely in section 5 lead to slight deviations between these two quantities, which can contribute to systematic errors in the measurement.

To isolate the eEDM term from other components of the energy shift in equation (15), the experiment is repeated under different conditions that are characterised by parameters whose sign is switched regularly during the experiment. The spin precession measurement is repeated for all 2⁴ experiment states defined by the four primary binary switch parameters: $\tilde{{ \mathcal N }}$, the molecular orientation relative to the applied electric field (changed every 0.5 s); $\tilde{{ \mathcal E }}$, the direction of the applied electric field in the laboratory (2 s); $\tilde{\theta }$, the sign of the readout polarisation dither (10 s); and $\tilde{{ \mathcal B }}$, the direction of the applied magnetic field in the laboratory (40 s). For each ($\tilde{{ \mathcal N }},\tilde{{ \mathcal E }},\tilde{{ \mathcal B }}$) state, the asymmetry ${ \mathcal A }(\tilde{{ \mathcal N }},\tilde{{ \mathcal E }},\tilde{{ \mathcal B }})$, contrast ${ \mathcal C }(\tilde{{ \mathcal N }},\tilde{{ \mathcal E }},\tilde{{ \mathcal B }})$, and measured phase ${\rm{\Phi }}(\tilde{{ \mathcal N }},\tilde{{ \mathcal E }},\tilde{{ \mathcal B }})$ are determined as described earlier. The data taken under all ${2}^{4}=16$ experimental states derived from these four binary switches constitutes a 'block' of data.

We can write the phase ${\rm{\Phi }}(\tilde{{ \mathcal N }},\tilde{{ \mathcal E }},\tilde{{ \mathcal B }})$ in terms of components with particular parity with respect to the experimental switches:

$$\begin{eqnarray}&&{\rm{\Phi }}(\tilde{{ \mathcal N }},\tilde{{ \mathcal E }},\tilde{{ \mathcal B }})={{\rm{\Phi }}}^{\mathrm{nr}}+{{\rm{\Phi }}}^{{ \mathcal N }}\tilde{{ \mathcal N }}+{{\rm{\Phi }}}^{{ \mathcal E }}\tilde{{ \mathcal E }}+{{\rm{\Phi }}}^{{ \mathcal B }}\tilde{{ \mathcal B }}+{{\rm{\Phi }}}^{{ \mathcal N }{ \mathcal E }}\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}+{{\rm{\Phi }}}^{{ \mathcal N }{ \mathcal B }}\tilde{{ \mathcal N }}\tilde{{ \mathcal B }}+{{\rm{\Phi }}}^{{ \mathcal E }{ \mathcal B }}\tilde{{ \mathcal E }}\tilde{{ \mathcal B }}+{{\rm{\Phi }}}^{{ \mathcal N }{ \mathcal E }{ \mathcal B }}\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}\tilde{{ \mathcal B }}.\end{eqnarray} \tag{ 22 }$$

We refer to these components as 'switch-parity channels'. A channel is said to be odd with respect to some subset of switches (labelled as superscripts) if it changes sign when any of those switches is performed. Thus it will also change sign if an odd number of those switches is performed. It is implicitly even under all other switches. We use this general notation throughout this document to refer to correlations of various measured quantities and experimental parameters with experiment switches. To generalise, if we have k binary experiment switches $({\tilde{{ \mathcal S }}}_{1},{\tilde{{ \mathcal S }}}_{2},\ldots ,{\tilde{{ \mathcal S }}}_{k})$ such that ${\tilde{{ \mathcal S }}}_{i}=\pm 1$, and we perform a measurement of the parameter $X({\tilde{{ \mathcal S }}}_{1},{\tilde{{ \mathcal S }}}_{2},\ldots ,{\tilde{{ \mathcal S }}}_{k})$ for a complete set of the 2^k switch states, then the component of X that is odd under the product of switches $[{\tilde{{ \mathcal S }}}_{a}{\tilde{{ \mathcal S }}}_{b}...]$ is given by

$$\begin{eqnarray}&&{X}^{{{ \mathcal S }}_{a}{{ \mathcal S }}_{b}...}\equiv \displaystyle \frac{1}{{2}^{k}}\displaystyle \sum _{{\tilde{{ \mathcal S }}}_{1}...{\tilde{{ \mathcal S }}}_{k}=\pm 1}[{\tilde{{ \mathcal S }}}_{a}{\tilde{{ \mathcal S }}}_{b}...]X({\tilde{{ \mathcal S }}}_{1},{\tilde{{ \mathcal S }}}_{2},\ldots ,{\tilde{{ \mathcal S }}}_{k}).\end{eqnarray} \tag{ 23 }$$

The switch parity behaviour of a given component is expressed in the superscript which lists the experimental switches with respect to which the component is odd. We order the switch labels in the superscripts such that the fastest switches are listed first and the slowest switches are listed last. Some components give particularly important physical quantities. Most notably, the eEDM precession phase is extracted from the $\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}$-correlated component of the measured phase: that is, in the ideal case ${{\rm{\Phi }}}^{{ \mathcal N }{ \mathcal E }}=-{d}_{e}{{ \mathcal E }}_{\mathrm{eff}}\tau$. Additionally, the Zeeman precession phase is nominally given by ${{\rm{\Phi }}}^{{ \mathcal B }}=-{\mu }_{{\rm{B}}}{g}_{1}{{ \mathcal B }}_{z}\tau$. Recall we label 'non-reversing' components with an 'nr' superscript. In a few cases, we drop the superscript parity because it is redundant. For example, we drop the superscript on the dominant components of the applied electric and magnetic fields, ${ \mathcal E }\equiv {{ \mathcal E }}^{{ \mathcal E }}$ and ${{ \mathcal B }}_{z}\equiv {{ \mathcal B }}_{z}^{{ \mathcal B }}$.

Many other experimental parameters are also varied between blocks of data to suppress and monitor systematic errors (figure 5). These 'superblock' switches include: excited-state parity addressed by the state readout laser beams, $\tilde{{ \mathcal P }}$ (chosen randomly after every block, with equal numbers of $\tilde{{ \mathcal P }}=\pm 1$); simultaneous change of the power supply polarity and interchange of leads connecting the electric field plates to their voltage supply, $\tilde{{ \mathcal L }}$ (4 blocks); a rotation of the state readout polarisation basis by ${\theta }_{\mathrm{read}}\to {\theta }_{\mathrm{read}}+\pi /2$ to interchange the roles of the X and Y beams, $\tilde{{ \mathcal R }}$ (8 blocks); and a global polarisation rotation of both state preparation and readout lasers by ${\theta }_{\mathrm{read}}\to {\theta }_{\mathrm{read}}+\pi /2$ and ${\theta }_{\mathrm{prep}}\to {\theta }_{\mathrm{prep}}+\pi /2$, $\tilde{{ \mathcal G }}$ (16 blocks).

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Additionally, the magnitude of the magnetic field, ${{ \mathcal B }}_{z}$, was switched on the timescale of 64–128 blocks ($\sim 1$ h), and the magnitude of the applied electric field, ${ \mathcal E }$, and the laser propagation direction, $\hat{k}\cdot \hat{z}$, were changed on timescales of $\sim 1$ day and $\sim 1$ week, respectively.

On these longer timescales, we also alternated between taking eEDM data under normal conditions, for which all experiment parameters were set to their nominally ideal values, and taking data with intentional parameter variations (IPVs), during which some experimental parameter was set to deviate from ideal so that we could monitor the size of the known systematic errors described in section 5.2.6. We took IPV data in which we varied (a) the non-reversing electric field ${{ \mathcal E }}^{\mathrm{nr}}$ and (b) the $\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}$-correlated Rabi frequency, ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$, to measure the sensitivity of the eEDM measurement to these parameters and we varied (c) the state preparation laser detuning ${{\rm{\Delta }}}_{\mathrm{prep}}$ to monitor the size of the residual ${{ \mathcal E }}^{\mathrm{nr}}$. These systematic errors are discussed in more detail in sections 3.2.5 and 5.2.6.

The details of the data analysis required to extract the eEDM-correlated phase ${{\rm{\Phi }}}^{{ \mathcal N }{ \mathcal E }}$ are described in section 4. A lower bound on the statistical uncertainty $\delta {{\rm{\Phi }}}^{{ \mathcal N }{ \mathcal E }}$ of the eEDM-correlated phase is given by photoelectron shot noise to be $\delta {{\rm{\Phi }}}^{{ \mathcal N }{ \mathcal E }}=1/(2| { \mathcal C }| \sqrt{N})$ for N detected photoelectrons [17, 79]. In the case where shot noise is the sole contribution, we can express the statistical uncertainty $\delta {d}_{e}$ in our measurement of the eEDM as

$$\begin{eqnarray}&&\delta {d}_{e}=\delta {{\rm{\Phi }}}^{{ \mathcal N }{ \mathcal E }}\displaystyle \frac{1}{{{ \mathcal E }}_{\mathrm{eff}}\tau }=\displaystyle \frac{1}{2| { \mathcal C }| \tau {{ \mathcal E }}_{\mathrm{eff}}\sqrt{\dot{N}T}},\end{eqnarray} \tag{ 24 }$$

where $\dot{N}\approx f\dot{{N}_{0}}$ is the measurement rate (equivalent to the photoelectron detection rate) and T is the integration time (recall f is the fraction of fluorescence photons detected and N₀ is the number of molecules in the addressed $\tilde{{ \mathcal N }}$ level). Further discussion of the achieved statistical uncertainty is presented in section 4.

In this section we provide an overview of our experimental procedure and the important components of our apparatus. The reader should consult subsequent subsections for further details. A schematic of the experimental apparatus is shown in figure 6.

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ThO molecules were produced via pulsed laser ablation of a ThO₂ ceramic target. This took place in a cryogenic neon buffer gas cell, held at a temperature of $\approx 16$ K, at a repetition rate of 50 Hz. The resulting molecular beam was collimated and had a forward velocity ${v}_{\parallel }\approx 200$ m s⁻¹. In the state readout region the molecular pulses had a temporal (spatial) length of around 2 ms (40 cm). The buffer gas beam source is described in detail in section 3.2.2.

After leaving the buffer gas source, the molecules had a velocity distribution and rotational level populations consistent with a Maxwell–Boltzmann distribution at a temperature of $\approx 4$ K. This was lower than the cell temperature due to expansion cooling, which enhanced the number of usable ThO molecules in the relevant rotational state. Concentration of population into a particular rotational level was performed via optical pumping and microwave mixing (see section 3.2.3). The molecules then passed through adjustable horizontal and vertical collimators consisting of a double layer of razor blades affixed to linear translation vacuum feedthroughs. Under normal running conditions, these collimators were withdrawn so that they did not affect the profile of the molecule beam in the spin-precession region; however, they were used to modify the spatial profile of the molecule beam during systematic checks to investigate the effect of molecule beam position and pointing. Just before the field plates, 126 cm from the beam source, the molecules passed through a 1 cm square collimating aperture, which determined the beam profile in the spin-precession region and prevented particles in the beam from being deposited on the field plates.

As described in section 3.1, a spin precession measurement was performed where the precession angle provided a measure of the interaction energy of an eEDM with the effective electric field, ${{ \mathcal E }}_{\mathrm{eff}}$, in the molecule. A pair of transparent, ITO-coated glass plates provided an electric field that polarised and aligned the molecules. Laser beams passed through these plates to perform state preparation and readout. Around the vacuum chamber were coils that provided a uniform magnetic field in the $+\hat{z}$ direction, and five layers of magnetic shielding which shielded against environmental magnetic fields. The electric and magnetic fields are discussed in detail in sections 3.2.5 and 3.2.6. The fluorescence induced by the state readout laser beam was collected by a set of eight lenses and transferred out of the spin-precession region using fibre bundles and light pipes (see section 3.2.7), where it was detected by photo-multiplier tubes¹⁶ .

The basic operation of our beam source [66, 71, 73, 74, 80–91] is depicted in figure 7. Neon buffer gas was flowed at a rate of $\approx 30$ SCCM (standard cubic centimetres per minute) through a copper cell held at a $T\approx 16$ K. The inside of the cell was cylindrical with a diameter of 13 mm and a length of 75 mm. Within the cell ThO was introduced at high temperature via laser ablation: overlapped beams of light with wavelengths 532 and 1064 nm emitted by a pulsed Nd:YAG laser¹⁷ were focussed onto a 1.9 cm diameter ${\mathrm{ThO}}_{2}$ target fabricated from pressed and sintered powder [92, 93]. The laser pulses had a duration of a few ns, a pulse energy up to approximately 100 mJ and a repetition rate of 50 Hz. The resulting hot plume of ejected particles, which contained ThO along with various other ablation byproducts, was cooled by collisions with the neon buffer gas, became entrained, and then exited the cell. The cell temperature was maintained by a combination of a pulse tube refrigerator¹⁸ and a resistive heater.

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The cell was surrounded by a 4 K copper shield that protected the cell from black-body radiation and cryopumped most of the neon emerging from the cell. This shield was also partially covered with activated charcoal that acted as a cryopump for residual helium in the neon buffer gas. We observed a background pressure of 10⁻⁷ Torr without any mechanical pumping of the beam source when cold and with no buffer gas flow. The 4 K shield had a stainless steel conical collimator with a circular aperture of diameter 6 mm, located 25 mm from the cell aperture, by which distance the expanding beam was sufficiently diffuse that intra-beam collisions were negligible and most trajectories were ballistic. This collimator thus functioned as a differential pumping aperture without affecting the beam's cooling, acceleration or expansion [74]. The collimator had a thermal standoff relative to the 4 K shield to which it was mounted so that it could be kept at a temperature above the freezing point of neon by a resistive heater to prevent ice buildup on the collimator adversely affecting the beam dynamics. Another layer of shielding surrounded the 4 K copper shield, constructed from aluminium and held at a temperature of 60 K. Both the 4 and 60 K radiation shields were thermally connected to the pulse tube by heat links made of flexible copper rope.

The aluminium vacuum chamber that housed the buffer gas beam source¹⁹ had windows on each side, providing optical access for both the ablation laser and spectroscopy lasers, the latter allowing characterisation and monitoring of beam properties. The ThO beam's forward velocity distribution was roughly Gaussian with mean ${v}_{\parallel }\approx 200$ m s⁻¹ and standard deviation ${\sigma }_{{v}_{\parallel }}\approx 13$ m s⁻¹, corresponding to a temperature of $\approx 5$ K. The rotational temperature was ${T}_{\mathrm{rot}}\approx 4$ K (rotational constant ${B}_{X}\approx 0.33$ cm⁻¹), meaning that $\approx 90$% of the population was contained in the levels J = 0–3. Upon exiting the cell, the beam had a FWHM angular spread of $\approx 45^\circ$. Several stages of collimation were applied before reaching the spin-precession region. The final collimator subtended a solid angle of $\approx 6\times {10}^{-5}\,\mathrm{sr}$, meaning 1 in $\sim 20\,000$ molecules exiting the cell reached the spin-precession region, where the precession measurement was performed (see figure 6).

ThO yields from a given ablation spot decreased significantly after $\sim {10}^{4}$–10⁵ YAG pulses ($\sim 10$ min), at which time the laser spot was moved to an un-depleted region via a motorised mirror to re-optimise the beam flux. Each target was found to provide acceptable levels of molecule flux for around 300 h of continuous running ($\approx 5\times {10}^{7}$ shots) before requiring replacement.

We observed that $\approx 2\,\mathrm{cm}$ downstream of (further from) the buffer gas beam source cell aperture, J-changing collisions were 'frozen out' [74], and the distribution of rotational state populations was fairly well described by a Boltzmann distribution with temperature ${T}_{\mathrm{rot}}\approx 4$ K. At this temperature the resulting fractions of molecules in the J = 0–3 levels were estimated to be 0.1, 0.3, 0.3 and 0.2 respectively.

As described in section 3.2.4, we transferred the initial ground state population into $| H,J=1\rangle$ via optical pumping. To enhance the population which was transferred, we accumulated population in a single rotational level of the ground state before state preparation. The scheme used to achieve this is illustrated in figure 8 and discussed in detail in [94]. The first stage of the process was the optical pumping of molecules out of $| X,J=2\rangle$ ($| X,J=3\rangle$), via $| C,{J}^{\prime }=1\rangle$ ($| C,{J}^{\prime }=2\rangle$) into $| X,J=0\rangle$ ($| X,J=1\rangle$) using laser light at 690 nm. The natural linewidth of the $X\to C$ transition is $\approx 2\pi \times 0.3\,\mathrm{MHz}$, however the usable molecules had a $\approx \pm 0.7$ m s⁻¹ transverse velocity spread, corresponding to a $1\sigma$ Doppler width of $\approx 2\pi \times 1.5\,\mathrm{MHz}$ at 690 nm. Because the lasers used had linewidths of $\lesssim 1\,\mathrm{MHz}$, to completely optically pump these molecules we relied on a combination of power broadening and extended interaction time. Optical pumping occured in a magnetically unshielded region where a background field ${ \mathcal B }\approx 500$ mG was present; however, the magnetic moment of X (C) is $\sim {\mu }_{{\rm{N}}}$ ($\approx {\mu }_{{\rm{B}}}/J(J+1)$), the nuclear magneton, which led to a Zeeman shift of $\sim 2\pi \times 400\,\mathrm{Hz}$ ($\lesssim 2\pi \times 400$ kHz) such that the M sublevels were not resolved by our lasers. The $| C,J=1\rangle$ state has an Ω-doublet splitting of ${{\rm{\Delta }}}_{{\rm{\Omega }},C,J=1}\approx 2\pi \times 51\,\mathrm{MHz}$ [95]. This splitting scales as ${{\rm{\Delta }}}_{{\rm{\Omega }},C,J}\propto J(J+1)$, meaning we could spectroscopically resolve the Ω-doublets for all $| C,J\rangle$. In addition, having no ${ \mathcal E }$-field present meant that the M sublevels of C and X remained unresolved and the energy eigenstates remained parity eigenstates. The X state is also insensitive to ${ \mathcal E }$-fields due to the lack of Ω-doublet substructure; opposite parity states are separated by $\sim 10\,\mathrm{GHz}$ and were hence unmixed. Laser beams with linear polarisation alternating between $\hat{x}$ and $\hat{y}$ were used to ensure that all population in $| X,J=2,3\rangle$ was addressed. This was achieved by directing around 10 passes of the beam, offset in x, through the vacuum chamber, passing through a quarter-wave plate twice in each pass, over a distance of around 2 cm.

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The laser light for rotational cooling was derived from home-built extended cavity diode lasers (ECDLs). The lasers were frequency-stabilised using a scanning transfer cavity with a computer-controlled servo [96]. Frequency-doubled light at 1064 nm from a frequency-stabilised Nd:YAG laser, locked to a molecular iodine line via modulation transfer spectroscopy [97], provided the reference for the transfer cavity.

After this first stage of rotational cooling, there was significantly greater population in the $| X,J=0\rangle$ state than in any of the $| X,J=1,M\rangle$ sublevels. We obtained a $\approx 25 \%$ increase in the J = 1 population by applying a continuous microwave field, resonant with the $J=0\to J=1$ transition; a sufficiently high microwave power combined with the inherent velocity dispersion of the molecule beam led to an equilibration of population between the coupled levels [94]. In this second stage of rotational cooling it was empirically observed that applying an electric field to lift the M_y sublevel degeneracy was necessary to obtain the increased population in $| X,J=1\rangle$. A pair of copper electric field plates (spacing $\approx 4$ cm) provided a field of $\approx 40\,{\rm{V}}$ cm⁻¹ in the $\hat{y}$ (vertical) direction. We applied microwaves resonant with the Stark-shifted $| J=0\rangle \to | J=1,{M}_{y}=0\rangle$ transition at a frequency of $2\pi \times 19.904\,521\,\mathrm{GHz}$ from an ex vacuo horn. Between the rotational cooling and spin-precession regions of the experiment (see figure 6) there was not a well-defined quantisation axis, and we observe that the populations of the $| J=1,M\rangle$ magnetic sublevels were equalised by the time the molecules reached the state preparation region.

Overall, we find that rotational cooling provided a factor of between 1.5 and 2.0 increase in the molecule fluorescence signal F in the state readout region. This gain factor was observed to vary slowly over time, possibly due to variations in the rotational temperature of the molecule beam, with significant changes sometimes observed when the ablation target was changed.

Following rotational cooling, the molecular beam passed into the spin-precession region, where the molecules experienced a nominally uniform electric field, $\vec{{ \mathcal E }}$, which was nominally collinear with a magnetic field, $\vec{{ \mathcal B }}$. Note that since neither of the states ${X}^{1}{{\rm{\Sigma }}}^{+}$ nor ${A}^{3}{{\rm{\Pi }}}_{{0}^{+}}$ have Ω-doublet structure, parity remained a good quantum number for these levels for the small ($\sim 100\,{\rm{V}}$ cm⁻¹) electric fields we applied.

We transferred the molecules into the H electronic state via optical pumping, as illustrated in figure 9. A 943 nm laser beam nominally propagating along $\hat{z}$ excited molecules from the $| X,J=1\rangle$ to $| A,J=0\rangle$. The laser beam passed through a quarter-wave plate, was retroflected and offset in x, then passed again through the quarter-wave plate, such that the molecules were pumped by two spatially separated laser beams of orthogonal polarisations, allowing all population in both the $| X,J=1,M=\pm 1\rangle$ levels to be excited. After excitation to A, the molecules could spontaneously decay into the $| H,J=1\rangle$ manifold of states. We observed a transfer efficiency from X to H of $\approx 0.3$ [94]. In this decay, five out of the six sublevels were populated; 1/6 of the population decayed to each of $| H,M=\pm 1,\tilde{{ \mathcal N }}=\pm 1\rangle$ and 1/3 to $| H,\tilde{{ \mathcal P }}=-1,M=0\rangle$ (see sections 2.2 and 3.1 for definitions of $\tilde{{ \mathcal N }}$ and $\tilde{{ \mathcal P }}$); decay to $| H,\tilde{{ \mathcal P }}=+1,M=0\rangle$ is forbidden. Of these five populated states, only one corresponded to the desired initial state described by equation (13), and only 1/6 of the population in the H state was in this desired state. We estimated a total transfer efficiency from $| X,J=1,M=\pm 1\rangle$ to the state in equation (13) of $30 \% \times 1/6=5 \%$.

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The 943 nm laser light was derived from a commercial ECDL and then amplified by a commercial tapered amplifier²⁰ , generating $\approx 400\,\mathrm{mW}$. As with the rotational cooling lasers, we verified that the power was sufficient to drive optical pumping to completion across the entire transverse velocity distribution of the molecular beam. This laser was also stabilised via the previously described (section 3.2.3) transfer cavity. The frequency of the laser light was monitored every 30–60 min by scanning across the molecular resonances, allowing for independent fine-tuning and compensation of long-term frequency changes ($\lesssim 2\pi \times 100\,\mathrm{kHz}$ per half hour) due to e.g. temperature drifts in the cavity.

Around 1 cm downstream of the optical pumping laser beam that transferred population to H, we prepared the initial state of H (equation (13)) by driving the transition between $| H,M=\pm 1,\tilde{{ \mathcal N }}\rangle$ and $| C,\tilde{{ \mathcal P }}=+1\rangle$ (see section 3.1 for more details) using laser light at 1090 nm. A distance L = 22 cm downstream of the preparation laser, a second 1090 nm laser beam was used to read out the molecule state via the same transition (but with the option to excite to either $\tilde{{ \mathcal P }}$ state). This laser light was also derived from a commercial ECDL. It was then amplified using a fibre amplifier²¹ , increasing the power to $\approx 250\,\mathrm{mW}$. AOMs were then used to split and frequency shift the light to address both $\tilde{{ \mathcal N }}$ states in the H state, allowing spectroscopic selection of molecular alignment, and of both $\tilde{{ \mathcal P }}$ levels in the C state. Switching between these frequencies was achieved with either RF switches²² or a DDS synthesiser²³ . Given the linear Stark shifts ${D}_{1}{ \mathcal E }\approx 2\pi \times 146\,\mathrm{MHz}$ ($2\pi \times 37$ MHz) in H with an applied electric field strength $| { \mathcal E }| =141\,{\rm{V}}$ cm⁻¹ (36 V cm⁻¹), and the excited state Ω-doublet splitting ${{\rm{\Delta }}}_{{\rm{\Omega }},C,J=1}\approx 50\,\mathrm{MHz}$ in C, these transitions were spectroscopically well-resolved. We fixed the nominal frequency of the state preparation laser to only address $\tilde{{ \mathcal P }}=+1$, but periodically switched the state readout laser frequency to address $\tilde{{ \mathcal P }}=\pm 1$ ($\sim 1\,\min$ period). The transition frequencies of the state preparation and state readout laser beams were changed synchronously to always address the same $\tilde{{ \mathcal N }}$ level, with a switch between $\tilde{{ \mathcal N }}$ levels every 0.5 s. The state preparation and readout laser beams were then independently amplified with a pair of fibre amplifiers²⁴ , providing $\sim 3$–4 W of power. Immediately before interrogating the molecules, the polarisation of the state readout laser beam was rapidly (100 kHz) switched between two orthogonal linear polarisations. The scheme for producing the $\tilde{{ \mathcal N }}$ and $\tilde{{ \mathcal P }}$ switches, and this fast polarisation switch, together with the corresponding laser transitions, is shown in figure 10. We now describe in detail how the appropriate frequency laser light was produced.

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Light from the ECDL was amplified and split equally, passing to two AOMs which produced shifts $\pm {\omega }_{{\rm{L}}}^{{ \mathcal N }}$ where ${\omega }_{{\rm{L}}}^{{ \mathcal N }}$ is half the splitting between the two $\tilde{{ \mathcal N }}$ states; these AOMs were switched on and off to perform the $\tilde{{ \mathcal N }}$ switch. The two frequency-shifted beams were combined and overlapped. For state preparation (lower branch of diagram), another AOM shifted the light by $+{\omega }_{{\rm{L}},1}$, into resonance with the lower Ω-doublet in C ($\tilde{{ \mathcal P }}=+1$). This light was then amplifed again and passed through an AOM to vary the power (used as a systematic check). For the state readout (upper branch of diagram), a single AOM switched frequency to produce shifts $+{\omega }_{{\rm{L}},\mathrm{2,3}}$ for the two $\tilde{{ \mathcal P }}$ states. A relative detuning between state preparation and readout laser beams (not shown) was also implemented with this AOM. (Shifts common to both beams were made by changing ${\omega }_{0}$.) The light was then amplified again and passed through an AOM to vary the power. Finally, polarisation switching was achieved with two AOMs switched on and off at 100 kHz, π out of phase with each other; light not diffracted (and frequency shifted by $-{\omega }_{{\rm{L}},\mathrm{PS}}$) by the first AOM was diffracted (and also frequency shifted by $-{\omega }_{{\rm{L}},\mathrm{PS}}$) by the second AOM. The diffracted light from each path was combined on a polarising beam splitter such that the linear polarisation of the final output beam alternated.

Based on the notation above we can now write the components of the frequencies of the state preparation and readout laser beams which do not reverse with any experimental switch as ${\omega }_{{\rm{L}},\mathrm{prep}}^{\mathrm{nr}}={\omega }_{{\rm{L}},0}+{\omega }_{{\rm{L}},1}$ and ${\omega }_{{\rm{L}},\mathrm{read}}^{\mathrm{nr}}={\omega }_{{\rm{L}},0}+({\omega }_{{\rm{L}},2}+{\omega }_{{\rm{L}},3})/2-{\omega }_{{\rm{L}},\mathrm{PS}}$, respectively. We can also write the $\tilde{{ \mathcal P }}$-correlated frequency component of the state readout laser as ${\omega }_{{\rm{L}},\mathrm{read}}^{{ \mathcal P }}=({\omega }_{{\rm{L}},2}-{\omega }_{{\rm{L}},3})/2$. We then write the detuning components as ${{\rm{\Delta }}}_{i}={\omega }_{{\rm{L}},i}-{\omega }_{{HC}}$ where $i\in \{\mathrm{prep},X,Y\}$ indexes the laser and ${\omega }_{{HC}}$ is the transition frequency between the line centres of the $| H,J=1\rangle$ and $| C,J=1\rangle$ manifolds²⁵ . We can rewrite this overall detuning in terms of various switch parity components:

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{i}={\omega }_{{\rm{L}},i}-{\omega }_{{HC},i}\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&=\,({\omega }_{{\rm{L}},i}^{\mathrm{nr}}+\tilde{{ \mathcal N }}{\omega }_{{\rm{L}}}^{{ \mathcal N }}+\tilde{{ \mathcal P }}{\omega }_{{\rm{L}}}^{{ \mathcal P }}{\delta }_{i,\{X,Y\}})-\left({\omega }_{{HC}}^{\mathrm{nr}}+\tilde{{ \mathcal N }}{D}_{1}| { \mathcal E }({x}_{i})\tilde{{ \mathcal E }}+{{ \mathcal E }}^{\mathrm{nr}}({x}_{i})| -\displaystyle \frac{1}{2}{{\rm{\Delta }}}_{{\rm{\Omega }},C,J=1}\tilde{{ \mathcal P }}{\delta }_{i,\{X,Y\}}\right)\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&=\,{{\rm{\Delta }}}_{i}^{\mathrm{nr}}+\tilde{{ \mathcal N }}{{\rm{\Delta }}}_{i}^{{ \mathcal N }}+\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}{{\rm{\Delta }}}_{i}^{{ \mathcal N }{ \mathcal E }}+\tilde{{ \mathcal P }}{{\rm{\Delta }}}_{i}^{{ \mathcal P }}{\delta }_{i,\{X,Y\}}.\end{eqnarray} \tag{ 27 }$$

In the above equations we have defined detuning components of given switch parities—we shall now explain each component in turn. ${{\rm{\Delta }}}_{i}^{{ \mathcal N }}=({\omega }_{{\rm{L}}}^{{ \mathcal N }}-{D}_{1}{ \mathcal E }({x}_{i}))$ is the mismatch between the Stark shift ${D}_{1}{ \mathcal E }({x}_{i})$ and the AOM frequency ${\omega }_{{\rm{L}}}^{{ \mathcal N }}$ used to switch between resonantly addressing the two $\tilde{{ \mathcal N }}$ states, where x_i is the x position of laser beam i. ${{\rm{\Delta }}}_{i}^{{ \mathcal N }{ \mathcal E }}={D}_{1}{{ \mathcal E }}^{\mathrm{nr}}({x}_{i})$ is a detuning component correlated like an eEDM signal which is due to a non-reversing component of the applied electric field. To understand this relation, consider figure 11. Recall that ${{\rm{\Delta }}}_{{\rm{\Omega }},C,J=1}$ is the Ω-doublet splitting of the C state. For a ${{ \mathcal E }}^{\mathrm{nr}}\ne 0$, $| { \mathcal E }|$, and hence the splitting between the $\tilde{{ \mathcal N }}$ levels in H, depends on $\tilde{{ \mathcal E }}$. If the laser frequency for each $\tilde{{ \mathcal N }}$ is set assuming ${{ \mathcal E }}^{\mathrm{nr}}=0$, a nonzero ${{ \mathcal E }}^{\mathrm{nr}}$ leads to blue or red detuning from resonance, correlated with $\tilde{{ \mathcal E }}$. Because the sign of the Stark shift is correlated with $\tilde{{ \mathcal N }}$, the resulting detuning is also correlated with $\tilde{{ \mathcal N }}$.

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${{\rm{\Delta }}}_{i}^{{ \mathcal P }}={\omega }_{{\rm{L}}}^{{ \mathcal P }}+{{\rm{\Delta }}}_{{\rm{\Omega }},C,J=1}/2$ is the mismatch between the excited state parity splitting and the AOM frequency, ${\omega }_{{\rm{L}}}^{{ \mathcal P }}=({\omega }_{{\rm{L}},3}-{\omega }_{{\rm{L}},2})/2$, used to switch between the two states (${\delta }_{i,\{X,Y\}}$ is the Kronecker delta, 1 if i = X or i = Y, zero else). We observed that ${{\rm{\Delta }}}^{{ \mathcal N }}$ (${{\rm{\Delta }}}^{{ \mathcal P }}$) was typically less than $2\pi \times 20\,\mathrm{kHz}$ ($2\pi \times 50$ kHz). Although we could measure ${{\rm{\Delta }}}^{{ \mathcal N }}$ with $\sim 2\pi \times 1\,\mathrm{kHz}$ precision, fluctuations in the Stark splitting, likely caused by thermally-induced fluctuations of the field plate spacing, limited our ability to zero out this correlated detuning.

We define ${{\rm{\Delta }}}^{\mathrm{nr}}=({{\rm{\Delta }}}_{\mathrm{prep}}^{\mathrm{nr}}+(1/2)({{\rm{\Delta }}}_{X}^{\mathrm{nr}}+{{\rm{\Delta }}}_{Y}^{\mathrm{nr}}))/2$ as the average non-reversing detuning of the state preparation and readout laser beams; its value typically fluctuated by $\sim 2\pi \times 0.1\,\mathrm{MHz}$ over several hours. Every 30–60 min the value of ${{\rm{\Delta }}}^{\mathrm{nr}}$ was scanned across the molecular resonance in the readout region using the Δ-tuning AOM (see figure 10), as an auxiliary optimisation. ${{\rm{\Delta }}}^{\mathrm{nr}}$ was set to the value where the fluorescence signal was maximum. This ensured that the average detuning of the state readout laser beams, $({{\rm{\Delta }}}_{X}^{\mathrm{nr}}+{{\rm{\Delta }}}_{Y}^{\mathrm{nr}})/2$, was zero, however, if the state preparation and readout laser beams were not exactly parallel, there could be a difference between ${{\rm{\Delta }}}_{i}^{\mathrm{nr}}$ due to the resulting difference in Doppler shifts. The effect of a detuning difference between the two state readout polarisations ${{\rm{\Delta }}}^{{XY}}=({{\rm{\Delta }}}_{X}^{\mathrm{nr}}-{{\rm{\Delta }}}_{Y}^{\mathrm{nr}})/2$ is discussed in section 5.3. Additionally, each day we scanned the frequency of the preparation laser across the molecule resonance while monitoring the contrast of our fluorescence signal to ensure ${{\rm{\Delta }}}_{\mathrm{prep}}^{\mathrm{nr}}$ was kept below $2\pi \times 0.2\,\mathrm{MHz}$ (an example scan is shown in figure 24). The ways in which detuning components can contribute to systematic errors are discussed in detail in sections 5.2.3 and 5.2.6.

Other polarisation switches of the state preparation and readout laser beams ($\tilde{{ \mathcal R }}$ and $\tilde{{ \mathcal G }}$) were controlled independently via half-wave plates mounted in high resolution rotation stages²⁶ . These switches and their use in the experiment are described in detail in section 4. Both beams were shaped using cylindrical lenses to be extended in y so all molecules in the beam were addressed. The Gaussian standard deviations of the beam intensities were 1.1 mm and 7.5 mm in the x and y directions, respectively [94]. The preparation laser beam was temporally modulated at 50 Hz with a chopper wheel, synchronous with the molecule beam pulses, to minimise the incident power on the field plates so as to reduce an important systematic error, described in sections 5.2.3 and 5.2.4.

The applied ${ \mathcal E }$-field was generated with a pair of 43 cm × 23 cm parallel conducting plates composed of $\approx 1.25\,\mathrm{cm}$ thick Borofloat glass, coated with a $\sim 200\,\mathrm{nm}$ layer of indium tin oxide on the inner faces²⁷ . The plates were transparent to the $X\to A$ optical pumping laser (943 nm), the $H\to C$ state preparation and readout lasers (1090 nm), and the $C\to X$ molecule fluorescence (690 nm). The outside faces of the electric field plates were prepared with a broadband anti-reflection (AR) coating with a specified <1% reflectivity at normal incidence from 600 to 1000 nm. The plates were made much larger than the precession region in order to minimise inhomogeneity of the field through which the molecules passed, and to enable large solid angle collection of fluorescence through the plates. One of the field plates was mounted in an aluminium frame fixed to the base of the vacuum chamber. The other field plate was secured a distance of 2.5 cm away in a kinematic aluminium frame. On the inward-facing surfaces, a frame of gold-plated copper clamped each field plate to the aluminium mounts and also functioned as a 'guard ring' electrode, suppressing the effect of fringing fields near the edges of the plate. The field plates were protected from impinging molecular beam particles by a $1\,\mathrm{cm}\times 1\,\mathrm{cm}$ square collimator fixed to the entrance of the assembly.

The applied electric field was controlled by a 20 bit DAC, amplified to produce up to ±200 V²⁸ . The field plate assembly was referenced to the vacuum chamber ground. Equal and opposite voltages, $\pm V$, were applied to each side of the assembly. The direction of the field (the $\tilde{{ \mathcal E }}$ switch) was reversed every 1–2 s by reprogramming the output of the DAC channels to reverse their polarity. The configuration of the electrical connections between the amplified voltage and the field plates, denoted by $\tilde{{ \mathcal L }}$, was reversed via a pair of mercury-wetted relays every 2.6 min²⁹ . Data were also taken with two different values of ${ \mathcal E }=36$ and 141 V cm⁻¹, varied on a $\sim 1$ day time scale.

We measured the homogoneity of the electric field in a number of ways which we shall describe in turn now. Firstly, an indirect measure was obtained by determining the spatial variation of the field plate separation d using a 'white light' Michelson interferometer [98]. A schematic of the setup is shown in figure 12. We directed a light beam at normal incidence through the electric field plates. This resulted in multiple reflected beams, but we restrict discussion to the reflections from the conducting surfaces as these are of primary interest and were efficiently experimentally isolated from all others. The reflected beams passed into a Michelson interferometer with one arm of fixed length (L₂) and one with length adjustable via a micrometre (L₁). Constructive (destructive) interference occured whenever the lengths of two reflected beam paths differed by an integer (odd half-integer) multiple of the wavelength of the light. This condition was restricted further by the use of a broadband superluminescent diode³⁰ with a short coherence length ${L}_{{\rm{c}}}$ (nominally ${L}_{{\rm{c}}}\approx 15\,\mu {\rm{m}}$). Thus the interference was only substantial when the two beams differed in length by $\lesssim {L}_{{\rm{c}}}$. This occurred when ${L}_{1}={L}_{2}$ (for reflections off the same surface) or when ${L}_{1}={L}_{2}\pm d$ (for reflections off surfaces spaced by d). The case where both beams reflected off the same surface was used as a reference to determine the position ${L}_{1}={L}_{2}$. A measure of this interference was achieved by producing a spatial interference pattern (inset figure 12) through a slight tilting of the arms of the interferometer. Analysis of the spatial Fourier components of the resulting interference pattern provided a quantitative measure of the interference fringe contrast; a plot of contrast versus arm position L₁ yielded a peak with width $\delta {L}_{1}\approx {L}_{{\rm{c}}}$. By performing this analysis while varying the path length L₁, the plate separation was deduced. This entire procedure was then performed over a range of transverse ($x,y$) positions on the field plates. The resulting data are shown in figure 13.

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This measurement clearly showed a bowing of the electric field plates; the plate separation varied approximately quadratically with the position in x. This is shown in the left-hand plot of figure 13. In the $\hat{x}$ direction we observed a maximum variation in the plate separation of around 20 μm. We saw a roughly 80 $\mu {\rm{m}}$ variation in the $\hat{y}$ (vertical) direction but note that the collimated molecular beam extended only over ±5 mm in y so the biggest plate spacing variation at a given x was $\approx 10\,\mu {\rm{m}}$. From these measurements and a typical applied voltage of $V=\pm 177\,{\rm{V}}$, we expected ${ \mathcal E }$ to vary by around 100 mV cm⁻¹ in the $\hat{x}$ direction and $\lesssim 15$ mV cm⁻¹ in the y direction in the region sampled by the molecules.

The indirect measurements of the spatial variation of the applied electric field provided by interferometric mapping of the field plate separation were later corroborated by direct measurements of $\vec{{ \mathcal E }}(x)$. Spatial variation of $\vec{{ \mathcal E }}$ could lead to the accumulation of geometric phases during the spin precession measurement [99]. There are known mechanisms by which such phases can contribute to eEDM-like systematic errors, as described in section 5.4, though simple estimates show that these effects are several orders of magnitude below the sensitivity of this measurement. However, additional ${ \mathcal E }$-field imperfections such as non-reversing fields, due to e.g. variations in the ITO coating, which could produce patch potentials, are known to contribute to eEDM-like systematic errors and are only revealed by more direct measurements of the electric field, which we will now describe.

We can write the electric field present in the precession region in the following manner:

$$\begin{eqnarray}&&\vec{{ \mathcal E }}\cdot \hat{z}={ \mathcal E }\tilde{{ \mathcal E }}+{{ \mathcal E }}^{\mathrm{nr}}+{{ \mathcal E }}^{{ \mathcal L }}\tilde{{ \mathcal L }}+{{ \mathcal E }}^{{ \mathcal E }{ \mathcal L }}\tilde{{ \mathcal E }}\tilde{{ \mathcal L }},\end{eqnarray} \tag{ 28 }$$

where, as usual, $\tilde{{ \mathcal E }}=\mathrm{sgn}(\hat{z}\cdot \vec{{ \mathcal E }})$ is the direction of the field in the spin-precession region and $\tilde{{ \mathcal L }}$ represents the binary state of the physical leads connecting the voltage supply to the field plates. The terms on the right-hand side are: ${ \mathcal E }\tilde{{ \mathcal E }}$, the intentionally applied electric field; ${{ \mathcal E }}^{\mathrm{nr}}$, a non-reversing electric field; ${{ \mathcal E }}^{{ \mathcal L }}$, a non-reversing electric field component from the power supply that can be reversed by switching $\tilde{{ \mathcal L }};$ and ${{ \mathcal E }}^{{ \mathcal E }{ \mathcal L }}\tilde{{ \mathcal E }}\tilde{{ \mathcal L }}$, a component of the applied field that is reversed by switching $\tilde{{ \mathcal E }}$ or $\tilde{{ \mathcal L }}$.

We directly measured the components of ${ \mathcal E }$ using the molecules themselves, in three different ways. The first method used Raman spectroscopy, driving a two-photon Lambda-type transition between $\tilde{{ \mathcal N }}$ levels in $| H,J=1\rangle$ as shown in figure 14. The Raman transfer was performed at positions between, but close to, the state preparation and readout laser beams, where there was sufficient optical access. The procedure was as follows: first, an $\hat{x}$-polarised state preparation laser beam depleted a superposition $| B(\hat{x},\tilde{{ \mathcal N }}=+1,\tilde{{ \mathcal P }}=+1)\rangle$ (recall $| B\rangle$ is the bright state as defined in section 3.1) by exciting it to the C state. Next, at a point downstream, two co-propagating, $\hat{x}$-polarised Raman beams were used to repopulate this depleted superposition by driving population from the other $\tilde{{ \mathcal N }}$ state, via the transition $| B(\hat{x},\tilde{{ \mathcal N }}=-1,\tilde{{ \mathcal P }}=+1)\rangle \to | C,\tilde{{ \mathcal P }}=+1\rangle \to | B(\hat{x},\tilde{{ \mathcal N }}=+1,\tilde{{ \mathcal P }}=+1)\rangle$. The frequencies of the two Raman beams were tuned with a pair of AOMs. The state readout laser then addressed the same transition as the preparation laser and excited the repopulated superposition to the C-state from which it spontaneously decayed back to X and fluoresced at 690 nm.

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Efficient transfer of population between the two $\tilde{{ \mathcal N }}$ states occurred for zero two-photon detuning ($\delta /2$ in figure 14). This condition was indicated by a peak in fluorescence, giving a measure of the Stark shifted energy, and hence the absolute size of the applied field, $| { \mathcal E }|$. This procedure was repeated for different positions of the Raman laser beams along the $\hat{x}$ direction. The non-reversing component of the electric field was found by repeating the measurement after reversing the applied voltages. An example of such a pair of scans is shown on the right of figure 14.

Using this method we measured the electric field at x positions where there was sufficient optical access, i.e. near the state preparation and readout laser beams. The $\tilde{{ \mathcal E }}$-correlated two-photon detuning ${\delta }^{{ \mathcal E }}=2\pi \times 13\,\mathrm{kHz}$ ($2\pi \times 11$ kHz) allowed us to extract a value of the non-reversing electric field component, ${{ \mathcal E }}^{\mathrm{nr}}={\delta }^{{ \mathcal E }}/2{D}_{1}=-6.5\pm 0.3$ mV cm⁻¹ (−5.5 ± 0.3 mV cm⁻¹), in the state preparation (readout) region. We did not observe any significant variation within the individual regions. We also observed that this non-reversing component did not vary with the size of the reversing electric field.

The second method used to measure the electric field had the greatest utility because it allowed for spatially resolved measurements along x in the spin precession region with comparable precision to the Raman method without perturbing the experimental apparatus. This was achieved via microwave spectroscopy. A schematic of the experimental setup is shown in figure 16.

The measurement procedure began with optical pumping of molecules into the H-state. The molecules travelled through the spin-precession region until it was entirely occupied by the molecule pulse. At this time, a π-pulse of microwaves at $2\pi \times 39\,\mathrm{GHz}$ with nominal $\hat{y}$ polarisation was applied counter-propagating to the molecule beam. When on resonance, this transferred population from $| B(\hat{y},\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle$ to $| H,J=2,M=0,\tilde{{ \mathcal P }}\rangle$ (excitation to (from) either $\tilde{{ \mathcal P }}$ ($\tilde{{ \mathcal N }}$) state was permitted) as shown in figure 15. State readout was performed as usual (see section 3.1) by optically pumping with alternating polarisations $\hat{x}$ and $\hat{y}$. The measured asymmetry (as defined in equation (11)) served as a measure of the microwave transfer efficiency. The x position of the molecules at the time of the microwave pulse was mapped onto their arrival time in the detection region and, with knowledge of the longitudinal molecular beam velocity, v_∥, could be extracted. Thus, the spatial dependence of the resonant frequency, ${\omega }_{\mathrm{MW}}(x)$, was provided by the time-dependence of the asymmetry, ${ \mathcal A }(t)$. Due to the DC Stark shift, ${\omega }_{\mathrm{MW}}$ was linearly proportional to the electric field magnitude and $| { \mathcal E }(x)|$ could be directly extracted.

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We observed a resonance linewidth of $\approx 2\pi \times 25\,\mathrm{kHz}\approx 2\pi /T$ which was limited by the microwave π-pulse duration of $T=40\,\mu {\rm{s}}$. With our signal-to-noise, we were able to fit the resonance centre to a precision of $\sim 2\pi \times 1\,\mathrm{kHz}$, typically using $\sim 50$ detuning values and averaging over $\sim 50$ molecule pulses per detuning value. Example data obtained via this method are shown in figure 17.

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In these data, it is evident that the resonant frequency of the microwaves varied across the molecule pulse by around $2\pi \times 60\,\mathrm{kHz}$. The position x of the molecules at the time of the microwave pulse was assumed to be linearly related to the molecule arrival time in the state readout region. The observed spatial variation of ${ \mathcal E }$ was roughly consistent with expectations based on the measured variation of the plate spacing described above.

By switching $\tilde{{ \mathcal N }}$ and $\tilde{{ \mathcal E }}$ between measurements of the ${ \mathcal E }$-field we were able to extract ${{ \mathcal E }}^{\mathrm{nr}}$ from the $\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}$-correlated component of ${\omega }_{\mathrm{MW}}$. These measurements, shown in figure 18, were used to evaluate the corresponding systematic error in equation (86).

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We clearly saw a non-uniform ${{ \mathcal E }}^{\mathrm{nr}}$ across the spin precession region. The spatial variation shown in figure 18 was reproducible for the period of several weeks over which these measurements of the electric field were taken. We are unsure as to the origin of the ${{ \mathcal E }}^{\mathrm{nr}}$ but believe it may have been caused by patch potentials [100] present on the electric field plates. We observed unexplained disagreement between the two measurement methods (Raman spectroscopy versus microwave spectroscopy), but note that both report non-reversing fields of a few mV cm⁻¹ with the same sign.

The mapping between arrival time in the detection region and x position during the microwave pulse was approximate, suffering from spatial averaging due to a variety of effects. For example, velocity dispersion led to averaging of ${dx}\times {\sigma }_{{v}_{\parallel }}/{v}_{\parallel }$, where ${\sigma }_{{v}_{\parallel }}$ is the longitudinal velocity spread of the molecular beam and dx is the distance between microwave interrogation and state readout. This averaging distance was largest, $\approx 1.6\,\mathrm{cm}$, at the state preparation region. Spatial averaging also occurred across the $\approx 0.7\,\mathrm{cm}$ distance traversed during the $T=40\,\mu {\rm{s}}$ microwave pulse. Finally, there was averaging of the spatial position of the molecules due to the finite size of the state readout laser beam and the polarisation switching; molecules were optically pumped (with varying probability) throughout the $\approx 0.5\,\mathrm{cm}$ wide laser beam.

In addition to spatial averaging, uncertainty in the mean longitudinal velocity also contributed an uncertainty in position. Changes of $\approx 10$ m s⁻¹ between molecule pulses were quite typical over the course of the ${ \mathcal E }$-field measurement, giving an estimated position uncertainty of $\lesssim 1\,\mathrm{cm}$.

By adding the above contributions in quadrature we concluded that the range of positions from which the microwave-induced signals could have originated increased from around $\approx 1.3\,\mathrm{cm}$ at the state readout beam to $\approx 2.1\,\mathrm{cm}$ at the optical pumping beam. These ranges are shown as horizontal error bars at the extrema of position in figure 18.

We used a third method to measure ${ \mathcal E }$ and ${{ \mathcal E }}^{\mathrm{nr}}$ in situ throughout the eEDM dataset by performing 'IPV' tests with large ${{\rm{\Delta }}}_{\mathrm{prep}}$ (denoted by 'c' in figure 5). Detuning the state preparation laser resulted in a reduction in the measured contrast $| { \mathcal C }|$ as shown in figure 24(B). Setting ${{\rm{\Delta }}}_{\mathrm{prep}}\approx \pm {2}^{}\mathrm{MHz}$ gives $| { \mathcal C }| \approx 0.5$, and the contrast was then approximately linearly proportional to ${{\rm{\Delta }}}_{\mathrm{prep}}$ with a sensitivity of about $1/{\gamma }_{C}\approx 1/(2\pi \times 2\,\mathrm{MHz})$. Any variation in the electric field would change the Stark shift, and thus also ${{\rm{\Delta }}}_{\mathrm{prep}}$, resulting in a change in contrast. Thus, using the previously described spin precession scheme, we indirectly measured parity components of the electric field from the appropriate parity components of the contrast:

$$\begin{eqnarray}&&{D}_{1}{{ \mathcal E }}^{\mathrm{nr}}({x}_{\mathrm{prep}})\approx \displaystyle \frac{\partial {{\rm{\Delta }}}_{\mathrm{prep}}}{\partial | { \mathcal C }| }| { \mathcal C }{| }^{{ \mathcal N }{ \mathcal E }},\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&{D}_{1}{ \mathcal E }({x}_{\mathrm{prep}})\approx {\omega }_{{\rm{L}}}^{{ \mathcal N }}+\displaystyle \frac{\partial {{\rm{\Delta }}}_{\mathrm{prep}}}{\partial | { \mathcal C }| }| { \mathcal C }{| }^{{ \mathcal N }}.\end{eqnarray} \tag{ 30 }$$

We looked for variation of ${ \mathcal E }$ or ${{ \mathcal E }}^{\mathrm{nr}}$ every 3–4 h. Measurements of ${{ \mathcal E }}^{\mathrm{nr}}$ were consistent with the microwave measurements, with a constant value ${{ \mathcal E }}^{\mathrm{nr}}({x}_{\mathrm{prep}})=-4.8\pm {0.9}^{}\mathrm{mV}\,{\mathrm{cm}}^{-1}$. However, the mismatch ${{\rm{\Delta }}}^{{ \mathcal N }}={D}_{1}{ \mathcal E }-{\omega }_{{\rm{L}}}^{{ \mathcal N }}$ between the Stark shift ${D}_{1}{ \mathcal E }$ and the $\tilde{{ \mathcal N }}$-correlated laser frequency shift, ${\omega }_{{\rm{L}}}^{{ \mathcal N }}$, was found to drift significantly on the scale of around $2\pi \times {20}^{}\mathrm{kHz}\,{{\rm{d}}}^{-1}$. This drift of ${{\rm{\Delta }}}^{{ \mathcal N }}$ was servoed by tuning ${\omega }_{{\rm{L}}}^{{ \mathcal N }}$ after each measurement, ensuring $| {{\rm{\Delta }}}^{{ \mathcal N }}| \lt 2\pi \times 30\,\mathrm{kHz}$ at all times [94], see sections 5.2.6 and 5.6 for more details.

Our experimental scheme did not require the application of a magnetic field. This was not the case with some previous eEDM experiments, where the magnetic field was used to define a quantisation axis [41, 42], or to cause the precession of spin to a direction associated with maximum sensitivity [20, 101]. Instead we used the electric field to define a quantisation axis, and we used the relative polarisations of the state preparation and readout lasers to define the basis in which we read out the electron's spin precession with maximal sensitivity.

However, we regularly applied a magnetic field ${ \mathcal B }$ in order to perform searches for systematic errors. The phase accumulation induced by an eEDM $\delta {d}_{e}\approx 5\times {10}^{-29}$ $e\,\mathrm{cm}$ would have the same size as a Zeeman phase produced by a magnetic field of ${ \mathcal B }\approx 0.2\,\mu {\rm{G}}$, which is small compared to some of the magnetic field imperfections in the experiment. However, phases associated with magnetic-field-induced precession were distinguished from eEDM-induced precession by the use of the switches at our disposal (e.g. electric field reversal). Nevertheless, it was important to investigate, quantify and minimise the effects of such magnetic fields, as they could have coupled with other experimental imperfections to give eEDM-like phases.

Under normal operating conditions we ran the experiment at three different magnetic field magnitudes, corresponding to a relative precession phase of ${\phi }^{{ \mathcal B }}\approx q\tfrac{\pi }{4}$ for $q=0,1,2$. The required z-component of the field was then ${{ \mathcal B }}_{z}=q{{ \mathcal B }}_{0}\tilde{{ \mathcal B }}$, where ${{ \mathcal B }}_{0}=\tfrac{\pi }{4}\tfrac{1}{{g}_{1}{\mu }_{{\rm{B}}}\tau }\approx 20\,\mathrm{mG}$. We also had the ability to apply transverse magnetic field components along $\hat{x}$ and $\hat{y}$, and all five linearly independent first-order gradients. The various coils that we used are illustrated in figure 19.

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The primary magnetic field, ${{ \mathcal B }}_{z}$, was produced by two sets of rectangular coils, shown in orange in figure 19. These were wound on the surface of two hemicylindrical plastic shells, on the $\pm z$ sides of the spin-precession region. The coils were designed to maximise field uniformity and minimise distortion due to the boundary conditions imposed by the magnetic shielding. It was also possible to apply a $\partial {{ \mathcal B }}_{z}/\partial z$ gradient with these coils. Two end coils (red in figure 19), located on the $\pm x$ ends of the spin-precession region, enhanced the uniformity of the ${ \mathcal B }$-field along x and enabled application of a $\partial {{ \mathcal B }}_{z}/\partial x$. The main coils were powered by two separate commercial power supplies³¹ , and the end coils were powered by custom power supplies. The current flowing through these coils was continuously monitored throughout the course of the experiment by measuring with a digital multimeter the voltage dropped across precision resistors.

We used three sets of auxiliary magnetic field coils in systematic error searches. A pair of circular Helmholtz coils (yellow in figure 19) were wrapped around the same frame used for the main coils and were formed from ribbon cable. They provided a magnetic field in the $\pm \hat{x}$ directions and could also provide a $\partial {{ \mathcal B }}_{x}/\partial x$. Above and below the spin-precession region chamber ($\pm y$) there were four sets of rectangular coils (blue and green in figure 19). These allowed us to produce a field in the $\pm \hat{y}$ directions as well as all three associated first-order gradients. Note that the three first-order magnetic field gradients that we could not apply could be inferred from Maxwell's equations. A summary of the fields that we could apply is given in table 1.

Table 1.  A summary of the magnetic fields and magnetic field gradients that we could produce. The coil colours refer to figure 19.

Coil colour	Fields produced	Field gradients produced
Orange	${{ \mathcal B }}_{z}$	$\partial {{ \mathcal B }}_{z}/\partial z$
Red	${{ \mathcal B }}_{z}$	$\partial {{ \mathcal B }}_{z}/\partial x$, $\partial {{ \mathcal B }}_{z}/\partial z$
Yellow	${{ \mathcal B }}_{x}$	$\partial {{ \mathcal B }}_{x}/\partial x$
Blue	${{ \mathcal B }}_{y}$	$\partial {{ \mathcal B }}_{y}/\partial y$, $\partial {{ \mathcal B }}_{y}/\partial z$
Green	${{ \mathcal B }}_{y}$	$\partial {{ \mathcal B }}_{y}/\partial x$, $\partial {{ \mathcal B }}_{y}/\partial y$

Several measures were taken to minimise stray magnetic fields affecting the molecules. The simplest was to ensure no magnetised objects were placed within the spin-precession region. To ensure this, all components were fabricated from non-magnetic materials (e.g. no stainless steel). The magnetisation of all objects was also checked before installation by passing them across an AC-coupled magnetometer sensitive to 0.1 mG field variations.

The ambient ${ \mathcal B }$-field in the laboratory was dominated by that from the Earth's core ($\sim 500\,{\rm{m}}{\rm{G}}$ approximately along $\hat{x}+\hat{y}$). To suppress this and other DC/low-frequency fields, the spin-precession region was surrounded by a set of five concentric cylindrical magnetic shields constructed from $\approx 1.6\,\mathrm{mm}$ thick mu-metal³² . Each layer of shielding should have provided around a factor of 10 reduction in the DC magnetic field [69]; however, residual magnetisation of the mu-metal was found to limit the field components to $\gtrsim 20\,\mu {\rm{G}}$ for ${{ \mathcal B }}_{x}$ and ${{ \mathcal B }}_{y}$, and $\gtrsim 500\,\mu {\rm{G}}$ for ${{ \mathcal B }}_{y}$ ³³ . Each shielding layer was divided into two half-cylinders and two end caps. The outermost (innermost) shield was 132 cm (86 cm) long and had a diameter of 107 cm (76 cm). These shields had holes to allow lasers to pass through in the z direction, and to accommodate the molecule beam. There were also holes for the light pipes to extract molecule fluorescence, and some electric connections, in the x direction. Measurements and simulations showed that these holes had a negligible impact on the shielding efficiency. The shielding factor remained approximately constant up to an AC frequency $\sim 2\pi \times 3\,\mathrm{GHz}$ for which the wavelength becomes comparable to the size of any apertures in the shields, $\sim 10\,\mathrm{cm}$, and the magnetic field noise starts to penetrate the shields. However, our measurement was only sensitive to magnetic field noise at frequencies up to roughly the inverse of the spin precession time $1/\tau \approx 2\pi \times 1\,\mathrm{kHz}$ [79]. The aluminium vacuum chamber also shielded AC magnetic noise above a frequency $\sim 1/\pi \sigma {t}^{2}\mu \approx 2\pi \times 100\,\mathrm{Hz}$, where $\sigma \approx 3.5\times {10}^{7}$ S m⁻¹ is the electrical conductivity, $t\approx 1\,\mathrm{cm}$ is the thickness and μ is the permeability $\approx {\mu }_{0}$, the vacuum permeability [102, 103].

The relatively large (${ \mathcal B }\sim 10$ mG) fields applied by the ${{ \mathcal B }}_{z}$ coils caused the inner magnetic shields to become slightly magnetised, inducing a non-reversing magnetic field, ${{ \mathcal B }}^{\mathrm{nr}}\approx 30\,\mu {\rm{G}}$. In order to suppress this remanent field we performed a degaussing procedure on the magnetic shields by passing a 200 Hz sinusoidal current through sets of loosely wound ribbon cable coils which wrapped axially (in the xy plane at z = 0) between the shield layers. The maximum current amplitude was 1 A, sufficient to drive the mu-metal to saturation, and the amplitude was decreased with an exponential envelope over a period of 1 s. To fully degauss all layers of the magnetic shielding takes around 4 s. There was also a 1 s period of 'dead time' during which the main magnetic field was turned back on and allowed to settle. This degaussing procedure was repeated every time the applied magnetic field was changed, which occured approximately every 40 s.

Variations in the magnetic fields present were continuously measured throughout the experimental procedure. This was achieved using a set of four three-axis fluxgate magnetometers³⁴ , which were mounted in a tetrahedral configuration outside the spin-precession region vacuum chamber (but inside the magnetic shielding). We also used an additional fluxgate magnetometer which was positioned at a distance of around 1 m from the apparatus and outside of the magnetic shielding. By continuously recording the measurements provided by these magnetometers we were able to search for correlations of our data with the magnetic field present. In particular, we checked for the presence of a magnetic field correlated with the electric field, ${{ \mathcal B }}^{{ \mathcal E }}$, which would have been characteristic of a leakage current flowing between the electric field plates—an effect known to contribute a significant systematic error in previous eEDM experiments [70, 101].

Additional measurement of the magnetic fields was carried out by opening the vacuum system and passing a rotatable flux-gate magnetometer into the chamber. This allowed for measurement of the fields directly along the beam line. The freedom to rotate the magnetometers was crucial to distinguish between electronic offsets and ${\vec{{ \mathcal B }}}^{\mathrm{nr}}$ for fields $\lesssim 1$ mG. From these measurements we were able to directly characterise most of the magnetic fields and first-order field gradients, including non-reversing components. Example data obtained from these measurements are shown in figure 20. We saw that the applied fields were all flat to within 1 mG, and the non-reversing components, with the exception of ${{ \mathcal B }}_{y}^{\mathrm{nr}}$, were less than 50 μG. Systematic uncertainty due to these fields is discussed in section 5.7.

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As previously described, our experimental data consisted of laser-induced molecule fluorescence, emitted in all directions (with a well-defined angular distribution [64]) when the molecules were interrogated by the state readout laser beam. The apparatus for collecting this light is illustrated in figure 21. The fluorescence light passed through the transparent electric field plates, whose inner (outer) faces are ITO (AR) coated. Behind each field plate was a set of four AR-coated lens doublets, which collimated and then focussed the light. The optical axes of the doublets intersected a ray path from the centre of the fluorescing molecule region, accounting for refraction through the electric field plates. The first (second) lens of each doublet was a 75 mm³⁵ (50 mm³⁶ ) diameter spherical lens of focal length 50 mm (35 mm). On each side ($\pm z$), each of the four lens doublets focussed light onto one of four sections of a 'quadfurcated' fibre bundle³⁷ whose input ends were 9 mm in diameter and fastened in lens tubes. The output of the fibre bundle was connected to a 19 mm diameter fused quartz light pipe with optical couplant gel³⁸ in between. The light pipe passed out of the spin-precession region vacuum chamber and magnetic shields and directed the light onto a PMT³⁹ . Bandpass filters⁴⁰ were used to suppress backgrounds from e.g. scattered light. Detailed tests of the light collection were carried out [94] which estimated that $\approx 14$% of the fluorescence photons were collected. The major contributions to this efficiency were the finite solid angle subtended by the collection lenses ($\approx 50 \%$), finite coupling efficiency into fibre bundles ($\approx 60 \%$) and finite coupling efficiency between the fibre bundles and the light pipes ($\approx 50 \%$). In addition, the quantum efficiency of the PMT's was specified to be $\approx 10 \%$ , which further reduced the signal obtained.

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The data acqusition system performed the following three functions:

• (i)  
  Digital modulation of the experimental parameters necessary for acquiring the complete set of phase and contrast measurements required to extract the eEDM, as described in section 3.1.2.
• (ii)  
  Rapid (5 MSa s⁻¹) acquisition and storage of high-bandwidth fluorescence waveforms for the spin precession measurement.
• (iii)  
  Monitoring and logging of experimental parameters useful for checking the experimental state and for searching for systematic errors (e.g. magnetic fields, beam source temperatures).

All functions were coordinated with a LabVIEW-based software system.

Data acquisition timing was controlled by a digital delay generator⁴¹ . Every 20 ms, a TTL signal was produced which triggered the ablation laser Q-switch, in turn creating a pulse of molecules. Molecule fluorescence signals, measured as a PMT photocurrent, were captured on a 20 bit digital oscilloscope⁴² . The oscilloscope was triggered 6–7 ms after the ablation pulse, depending on the current molecule beam forward velocity, and recorded a 9 ms window of signal containing the entire molecule signal (1–2 ms) and several ms of background. The 100 kHz square wave that drove the fast polarisation switching of the state readout laser was synchronised with the 50 Hz Q-switch trigger so that the relative phase was fixed. The 5 MSa s⁻¹ data rate of the oscilloscope enabled resolution of the time-dependent structure within each 5 $\mu {\rm{s}}$ polarisation bin; this structure could vary on timescales as short as the C-state lifetime $1/{\gamma }_{C}\approx 500$ ns [67].

Signal waveforms, S(t), were captured from two PMTs—note that we were not counting individual photoelectrons, but instead amplified and read out a voltage proportional to the count rate. These waveforms were then transferred to the control PC where they were digitally averaged over 25 pulses to form one 'trace'. The traces were then written to a hard drive. A file containing auxiliary measurements was recorded synchronously with each fluorescence trace. This file included the states of the experimental switches and other auxiliary measurements such as ${ \mathcal E }$-field voltages, ${ \mathcal B }$-field currents, laser power and polarisation, magnetic field measurements, molecular beam buffer gas flow rate, buffer gas cell temperature, and the temperature, pressure and humidity in our lab. This data proved useful in searching for systematic errors as described in section 5.

After extracting the measured phase ${{\rm{\Phi }}}_{j}(\tilde{{ \mathcal N }},\tilde{{ \mathcal E }},\tilde{{ \mathcal B }})$, we performed the basis change described in equation (23), from this experiment switch state basis to the experiment switch parity basis, denoted by ${{\rm{\Phi }}}_{j}^{p}$, where p is a placeholder for a given experiment switch parity.

We observed that the molecule beam forward velocity, and hence the spin precession time ${\tau }_{j}$, fluctuated by up to 10% over a 10 min time period. Since ${{ \mathcal B }}_{z}$ and g₁ are known from auxiliary measurements to a precision of around 1%, we were able to extract ${\tau }_{j}$ from each block from the Zeeman precession phase measurement, ${{\rm{\Phi }}}_{j}^{{ \mathcal B }}=-{\mu }_{{\rm{B}}}{g}_{1}{{ \mathcal B }}_{z}{\tau }_{j}$ (see section 3.1.2). Velocity dispersion caused ${\tau }_{j}$ to vary across the molecule pulse with a nominally linear dependence on time after ablation, t, however we observed significant deviations from linearity. Thus, we fit ${\tau }_{j}$ to a 3rd order polynomial in t in order to evaluate ${\bar{\tau }}_{j}$. Then, we evaluated the measured spin precession frequencies defined as

$$\begin{eqnarray}&&{\omega }_{j}^{p}={{\rm{\Phi }}}_{j}^{p}/{\bar{\tau }}_{j},\end{eqnarray} \tag{ 34 }$$

for all phase channels p (see equation (22) for definition). We extracted the eEDM from ${\omega }_{j}^{{ \mathcal N }{ \mathcal E }}$, which in the absence of systematic errors would be given by ${\omega }^{{ \mathcal N }{ \mathcal E }}=-{d}_{e}{{ \mathcal E }}_{\mathrm{eff}}$ independent of j.

From here on we will drop the j subscript that denotes a grouping of n adjacent asymmetry points about a particular time after ablation t_j; it is implicit that independent phase measurements were computed from many separate groups of data, each with different values of t_j across the duration of the molecule pulse. At the end of the analysis, and whenever it was convenient to do so, we implicitly performed weighted averaging across the j subscript.

Other phase channels could be used to search for and monitor systematic errors, discussed in detail in section 5, or to measure properties of ThO, as is the case with ${\omega }^{{ \mathcal N }{ \mathcal B }}$. We discuss the latter case here. This channel provided a measure of ${\rm{\Delta }}g$, the magnetic moment difference between upper and lower $\tilde{{ \mathcal N }}$-levels, arising from perturbations due to other electronic and rotational states [59, 78]. Because this difference limits the extent to which the $\tilde{{ \mathcal N }}$ reversal can suppress certain systematic errors [69], it is an important quantity both in our experiment and in other experiments measuring eEDMs in molecules with Ω-doublet structure [105]. Figure 25 illustrates an observed linear dependence ${\rm{\Delta }}g/2=\eta { \mathcal E }$, as predicted [59, 78]. Since ${ \mathcal E }$ and ${{ \mathcal B }}_{z}$ are precisely known from auxiliary measurements, the constant η can be directly calculated from our angular frequency measurements:

$$\begin{eqnarray}&&\eta =-\displaystyle \frac{{\omega }^{{ \mathcal N }{ \mathcal B }}}{{\mu }_{{\rm{B}}}{ \mathcal E }{{ \mathcal B }}_{z}}.\end{eqnarray} \tag{ 35 }$$

Our measured value of $\eta =-0.79\pm 0.01\,\mathrm{nm}\,{{\rm{V}}}^{-1}$ was approximately half of what one would compute using the methods developed to understand the effect in the PbO molecule [78, 106]. This discrepancy was subsequently understood as being primarily due to coupling to other fine-structure components in the ${}^{3}{\rm{\Delta }}$ manifold [59, 66]. The ${\omega }^{{ \mathcal N }{ \mathcal B }}$ channel illustrates the importance of understanding phase channels besides that corresponding to the eEDM.

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As described in section 3.1, the molecules initially enter the state preparation laser beam in an incoherent mixture of the two states $| \pm ,\,\tilde{{ \mathcal N }}\rangle$. The bright state $| B({\hat{\epsilon }}_{\mathrm{prep}},\tilde{{ \mathcal N }},{\tilde{{ \mathcal P }}}_{\mathrm{prep}})\rangle$ is then optically pumped away through $| C,{\tilde{{ \mathcal P }}}_{\mathrm{prep}}\rangle$ leaving behind the dark state $| D({\hat{\epsilon }}_{\mathrm{prep}},\tilde{{ \mathcal N }},{\tilde{{ \mathcal P }}}_{\mathrm{prep}})\rangle$ as the initial state for the spin precession. The molecules then undergo spin precession by angle ϕ evolving to a final state $| {\psi }_{f}\rangle =U(\phi )| D({\hat{\epsilon }}_{\mathrm{prep}},\tilde{{ \mathcal N }},{\tilde{{ \mathcal P }}}_{\mathrm{prep}})\rangle$ where $U(\phi )={\sum }_{\pm }{{\rm{e}}}^{\mp {\rm{i}}\phi }| \pm ,\,\tilde{{ \mathcal N }}\rangle \langle \pm ,\,\tilde{{ \mathcal N }}|$ is the spin precession operator. The molecules then enter the state readout laser that optically pumps the molecules with alternating polarisations ${\hat{\epsilon }}_{X}$ and ${\hat{\epsilon }}_{Y}$ (which are nominally linearly polarised and orthogonal) between $| \pm ,\,\tilde{{ \mathcal N }}\rangle$ and $| C,{\tilde{{ \mathcal P }}}_{\mathrm{read}}\rangle$. For each polarisation, the optical pumping results in a fluorescence count rate proportional to the projection of the state onto the bright state, ${F}_{X,Y}={{fN}}_{0}| \langle B({\hat{\epsilon }}_{X,Y},\tilde{{ \mathcal N }},{\tilde{{ \mathcal P }}}_{\mathrm{read}})| {\psi }_{f}\rangle {| }^{2}$ where f is the photon detection efficiency, and N₀ is the number of molecules in the addressed $\tilde{{ \mathcal N }}$ level. We then compute the asymmetry, ${ \mathcal A }=({F}_{X}-{F}_{Y})/({F}_{X}+{F}_{Y})$, dither the linear polarisation angles in the state readout laser beams to evaluate the fringe contrast, ${ \mathcal C }=(\partial { \mathcal A }/\partial \phi )/2\approx -(\partial { \mathcal A }/\partial {\theta }_{\mathrm{read}})/2$, and extract the measured phase, ${\rm{\Phi }}={ \mathcal A }/(2{ \mathcal C })+q\pi /4$ ⁴⁵ . We then report the result of the measurement in terms of an equivalent phase precession frequency $\omega ={\rm{\Phi }}/\tau$ where $\tau \approx {1}^{}\mathrm{ms}$ is the spin precession time, which was measured for each block as described in section 4.2.1.

Let us first consider the idealised case in which all laser polarisations are exactly linear, ${{\rm{\Theta }}}_{i}=\pi /4$ for each laser $i\in \{\mathrm{prep},X,Y\}$, the angle between the state preparation laser polarisation ($\mathrm{prep}$) and state readout basis ($X,Y$) is $\pi /4$, ${\theta }_{\mathrm{read}}-{\theta }_{\mathrm{prep}}=-\pi /4$, and the accumulated phase is small, $| \phi | \ll 1$ (i.e. no magnetic field is applied). Under these conditions, the measured phase Φ is equal to the accumulated phase ϕ. Now consider the effect of adding polarisation offsets $d{\vec{\epsilon }}_{i}$ to each of the three laser beams such that ${\hat{\epsilon }}_{i}\to {\hat{\epsilon }}_{i}+\kappa d{\vec{\epsilon }}_{i}$, where $\kappa =1$ is a perturbation parameter. It is useful to cast the polarisation imperfections in terms of linear angle imperfections, ${\theta }_{i}\to {\theta }_{i}+\kappa d{\theta }_{i}$ and ellipticity imperfections, ${{\rm{\Theta }}}_{i}\to {{\rm{\Theta }}}_{i}+\kappa d{{\rm{\Theta }}}_{i}$ where ${S}_{i}=-2d{{\rm{\Theta }}}_{i}$ is the laser ellipticity Stokes parameter; these are related by

$$\begin{eqnarray}&&\displaystyle \frac{\hat{z}\cdot ({\hat{\epsilon }}_{i}\times d{\vec{\epsilon }}_{i})}{{\hat{\epsilon }}_{i}\cdot {\hat{\epsilon }}_{i}}=d{\theta }_{i}-{\rm{i}}{d}{{\rm{\Theta }}}_{i}.\end{eqnarray} \tag{ 37 }$$

Note that laser polarisations can have a nonzero projection in the $\hat{z}$ direction, but we assume in the discussion above that ${\hat{\epsilon }}_{i}$ represents a normalised projection of the laser polarisation onto the xy plane⁴⁶ . With these polarisation imperfections in place, the measured phase Φ gains additional terms:

$$\begin{eqnarray}&&{\rm{\Phi }}=\phi +\kappa (d{\theta }_{\mathrm{prep}}-\displaystyle \frac{1}{2}(d{\theta }_{X}+d{\theta }_{Y}))-{\kappa }^{2}{\tilde{{ \mathcal P }}}_{\mathrm{prep}}{\tilde{{ \mathcal P }}}_{\mathrm{read}}d{{\rm{\Theta }}}_{\mathrm{prep}}(d{{\rm{\Theta }}}_{X}-d{{\rm{\Theta }}}_{Y})+O({\kappa }^{3}),\end{eqnarray} \tag{ 38 }$$

up to second order in κ. In the eEDM measurement, we switch between two values of $\tilde{{ \mathcal P }}\equiv {\tilde{{ \mathcal P }}}_{\mathrm{read}}$, the parity of the excited state addressed during state readout, and we set ${\tilde{{ \mathcal P }}}_{\mathrm{prep}}=+1$, the parity of the excited state addressed during state preparation. It is worth dwelling on equation (38) for a moment. A rotation of all polarisations by the same angle leaves the measured phase unchanged: $d{\theta }_{i}\to d{\theta }_{i}+d\theta \Longrightarrow {\rm{\Phi }}\to {\rm{\Phi }}$, as expected. A deviation in the relative angle between the state preparation and readout beams, $d{\theta }_{\mathrm{prep}}\to d{\theta }_{\mathrm{prep}}+d\theta$ and $d{\theta }_{X,Y}\to d{\theta }_{X,Y}-d\theta$, enters into the phase measurement as ${\rm{\Phi }}\to {\rm{\Phi }}+2d\theta$, but is benign so long as $d\theta$ is uncorrelated with the expected eEDM signal. The laser ellipticities affect the phase measurement only when the state readout beams differ in ellipticity, and this contribution to the phase can be distinguished from the others by switching the excited state parity, $\tilde{{ \mathcal P }}$. This last term is particularly interesting because it allows for multiplicative couplings between polarisation imperfections in the state preparation and state readout beams to contribute to the measured phase.

Although the polarisation imperfection terms in equation (38) are uncorrelated with the $\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}$ and hence do not contribute to the systematic error, we will see in later sections that additional imperfections can lead to changes in the molecule state that is prepared or read out that are equivalent to correlations $d{\theta }_{i}^{{ \mathcal N }{ \mathcal E }}$ and $d{{\rm{\Theta }}}_{i}^{{ \mathcal N }{ \mathcal E }}$. The framework of equation (38) is useful for understanding how these correlations result in systematic errors in the eEDM measurement extracted from ${{\rm{\Phi }}}^{{ \mathcal N }{ \mathcal E }}$.

In this section we describe in detail how interference between multipole transition amplitudes can lead to a measured phase that mimicks an eEDM spin precession phase. We develop a general framework illustrating how such phases depend on laser polarisation and pointing.

In an applied electric field, opposite parity levels are mixed, allowing both odd parity (E1, M2,...) and even parity (M1, E2,...) electromagnetic multipole amplitudes to contribute when driving an optical transition. These amplitudes depend on the orientation of the electric field relative to the light polarisation $\hat{\epsilon }$ and the laser pointing direction $\hat{k}$. This S.I. effect forms the basis of precise measurements of weak interactions through parity non-conserving amplitudes in atoms and molecules [109–111]. However, it can also generate a systematic error in searches for permanent electric dipole moments which look for spin precession correlated with the orientation of an applied electric field. These S.I. amplitudes have been calculated and measured for optical transitions in Rb [112, 113] and Hg [114, 115], and have been included in the systematic error analysis in the Hg EDM experiment [108, 116].

In this section, we consider S.I. as a source of systematic errors in the ACME experiment. There are two important differences between molecular and atomic systems. First, molecular states such as the ${H}^{3}{{\rm{\Delta }}}_{1}$ state in ThO can be highly polarisable and opposite parity states can be completely mixed by the application of a modest laboratory electric field. Second, molecular selection rules can be much weaker than atomic selection rules: the ${H}^{3}{{\rm{\Delta }}}_{1}\to {C}^{3}{{\rm{\Pi }}}_{1}$ transition that we drive is nominally an E1 forbidden spin-flip transition (${\rm{\Delta }}{\rm{\Sigma }}=1$, where Σ is the projection of the total electron spin S = 1 onto the intermolecular axis), but these states have significant subdominant contributions from other spin-orbit terms [76], between some of which the E1 transition is allowed. Both of these effects significantly amplify the effect of S.I. in molecules relative to atoms. In this section we will derive the effect of S.I. on the measured phase Φ.

Consider a plane wave vector potential $\vec{A}$ with real amplitude A₀, oscillating at frequency ω, that is resonant with a molecular optical transition $| g\rangle \to | e\rangle$,with wave vector $\vec{k}=(\omega /c)\hat{k}$, and complex polarisation $\hat{\epsilon }$:

$$\begin{eqnarray}&&\vec{A}(\vec{r},t)={A}_{0}\hat{\epsilon }{{\rm{e}}}^{{\rm{i}}\vec{k}\cdot \vec{r}-{\rm{i}}\omega t}+{\rm{c}}.{\rm{c}}.\end{eqnarray} \tag{ 39 }$$

The interaction Hamiltonian ${H}_{\mathrm{int}}$ between this classical light field and the molecular system is given by:

$$\begin{eqnarray}&&{H}_{\mathrm{int}}(t)=-\displaystyle \sum _{a}\displaystyle \frac{{e}^{a}}{{m}^{a}}\vec{A}({\vec{r}}^{a},t)\cdot {\vec{p}}^{a}\end{eqnarray} \tag{ 40 }$$

where a indexes a sum over all of the particles in the system with charge e^a, mass m^a, position ${\vec{r}}^{a}$ and momentum ${\vec{p}}^{a}$. Typically we apply the multipole expansion on the transition matrix element between states $| g\rangle$ and $| e\rangle ;$ the matrix element can then be written as

$$\begin{eqnarray}&&{ \mathcal M }\equiv \langle e| {H}_{\mathrm{int}}| g\rangle ={{\rm{i}}{A}}_{0}{\omega }_{{eg}}\displaystyle \sum _{\lambda =1}^{\infty }\langle e| \hat{\epsilon }\cdot {\vec{E}}_{\lambda }+(\hat{k}\times \hat{\epsilon })\cdot {\vec{M}}_{\lambda }| g\rangle ,\end{eqnarray} \tag{ 41 }$$

where ${\vec{E}}_{\lambda }$ describes the electric interaction of order $O({(\vec{k}\cdot \vec{r})}^{\lambda -1})$ and ${\vec{M}}_{\lambda }$ describes the magnetic interaction of order $O(\alpha {(\vec{k}\cdot \vec{r})}^{\lambda -1})$ (where α is the fine structure constant) such that

$$\begin{eqnarray}{\vec{E}}_{\lambda } & = & \displaystyle \frac{{(i)}^{\lambda -1}}{\lambda !}\displaystyle \sum _{a}{e}^{a}{\vec{r}}^{a}{(\vec{k}\cdot {\vec{r}}^{a})}^{\lambda -1},\\ {\vec{M}}_{\lambda } & = & \displaystyle \frac{{(i)}^{\lambda -1}}{(\lambda -1)!}\displaystyle \sum _{a}\left(\displaystyle \frac{{e}^{a}}{2{m}^{a}}\right)\left[{(\vec{k}\cdot {\vec{r}}^{a})}^{\lambda -1}\left(\displaystyle \frac{1}{\lambda +1}{\vec{L}}^{a}+\displaystyle \frac{1}{2}{g}^{a}{\vec{S}}^{a}\right)+\left(\displaystyle \frac{1}{\lambda +1}{\vec{L}}^{a}+\displaystyle \frac{1}{2}{g}^{a}{\vec{S}}^{a}\right){(\vec{k}\cdot {\vec{r}}^{a})}^{\lambda -1}\right],\end{eqnarray} \tag{ 42 }$$

where L^a is the orbital angular momentum, S^a is the spin angular momentum, and g^a is the spin g-factor for particle of index a (see e.g. [117]). For typical atomic or molecular optical transitions, if all moments are allowed, we expect the dominant corrections to the leading order E1 transition moment to be on the order of M1/E1 $\sim \,\alpha \sim {10}^{-2}$–10⁻³ and E2/E1$\sim {{ka}}_{0}\sim {10}^{-3}$–10⁻⁴, where a₀ is the Bohr radius. We note, however, that it is possible that these corrections could be larger in our particular case, since the $H\to C$ E1 transition is nominally forbidden and has a very weak amplitude. In this work we neglect the higher order contributions beyond E2, though the effects may by evaluated by using the expansion above.

During the state preparation and readout of the molecule state, transitions are driven between the state $| g\rangle ={\sum }_{\pm }{d}_{\pm }| \pm ,\,\tilde{{ \mathcal N }}\rangle$ and $| e\rangle =| C,\tilde{{ \mathcal P }}\rangle$, where d_± are state amplitudes that denote the particular superposition in $| H\rangle$ that is being interrogated. The particular d_± combination that results in ${ \mathcal M }=0$ describes the state that is dark, and the orthogonal state is bright and is optically pumped away.

It is convenient to expand the Hamiltonian ${H}_{\mathrm{int}}$ in terms of spherical tensor operators. Furthermore, the laser is only resonant with ${\rm{\Delta }}M=\pm 1$ transitions, so the spherical tensor operators with angular momentum projections other than ±1 can be reasonably omitted. In table 3, we factor the first 4 multipole operators into products of molecule and light field operators and express the molecular operators in terms of spherical tensors ${T}_{\pm 1}^{\lambda }$ of rank $\lambda =1,2$. The E1 and M1 terms consist of vector operators with $\lambda =1$. The E2 and M2 operators are rank 2 cartesian operators which can have spherical tensor operator contributions for $\lambda =0,1,2$. The rank $\lambda =0$ components of the E2 and M2 operators, and the $\lambda =1$ component of the E2 operator, vanish. The rank $\lambda =1$ component of the M2 operator does not vanish, but the light field angular dependence of this operator is equivalent to E1, so we may treat it as such.

Table 3.  Only spherical tensor operators ${T}_{q}^{\lambda }$ with projection $q=\pm 1$ contribute to the $| H\rangle \to | C\rangle$ transition amplitude. With this simplifying assumption, we can write the matrix element for each multipole operator in the form shown at the top of this table, which factors the molecule properties and the light properties (where ${\hat{\epsilon }}_{\pm }=\mp (\hat{x}\pm {\rm{i}}\hat{y})/\sqrt{2}$ are the spherical basis vectors and $\hat{z}$ is the direction of the applied static electric field). Here, the molecular operators ${O}^{\lambda }$ and the corresponding light vectors $\vec{V}({O}^{\lambda })$ are listed for the E1, M1, E2, and M2 operators. To evaluate a transition matrix element one should substitute the operators of interest from the 3rd and 4th columns into the expression at the top of the table. The subscripts i and j index cartesian components.

$\langle e| {H}_{\mathrm{int}}({O}^{\lambda })| g\rangle ={{\rm{i}}{A}}_{0}{\omega }_{{eg}}[{\hat{\epsilon }}_{+1}^{* }\langle e| {T}_{+1}^{\lambda }({O}^{\lambda })| g\rangle +{\hat{\epsilon }}_{-1}^{* }\langle e| {T}_{-1}^{\lambda }({O}^{\lambda })| g\rangle ]\cdot \vec{V}({O}^{\lambda })+...$
Term	Tensor	Molecular Operator, ${O}^{\lambda }$	Light Vector, $\vec{V}({O}^{\lambda })$
	rank, λ
E1	1	${{\rm{\Sigma }}}_{a}{e}^{a}{r}_{i}^{a}$	$\hat{\epsilon }$
M1	1	${{\rm{\Sigma }}}_{a}\tfrac{{e}^{a}}{2{m}^{a}}({L}_{i}^{a}+{g}^{a}{S}_{i}^{a})$	$\hat{k}\times \hat{\epsilon }$
E2	2	$\tfrac{\omega }{2c}{\sum }_{a}{e}^{a}{r}_{i}^{a}{r}_{j}^{a}$	$\tfrac{{\rm{i}}}{\sqrt{2}}[\hat{\epsilon }(\hat{k}\cdot \hat{z})+\hat{k}(\hat{\epsilon }\cdot \hat{z})]$
M2	2	$\tfrac{\omega }{c}{{\rm{\Sigma }}}_{a}\tfrac{{e}^{a}}{2{m}^{a}}\left\{{r}_{i}^{a},\tfrac{1}{3}{L}_{j}^{a}+\tfrac{1}{2}{g}^{a}{S}_{j}^{a}\right\}$	$\tfrac{{\rm{i}}}{\sqrt{2}}[\hat{k}((\hat{k}\times \hat{\epsilon })\cdot \hat{z})+(\hat{k}\times \hat{\epsilon })(\hat{k}\cdot \hat{z})]$

Using well-known properties of angular momentum matrix elements [62], we may write the transition matrix element in the following form,

$$\begin{eqnarray}&&{ \mathcal M }={{\rm{i}}{A}}_{0}{\omega }_{{eg}}{c}_{{\rm{E}}1}\displaystyle \frac{1}{\sqrt{2}}{[{(-1)}^{J+1}\tilde{{ \mathcal P }}]}^{(1-\tilde{{ \mathcal N }}\tilde{{ \mathcal E }})/2}({\hat{\epsilon }}_{-1}^{* }{d}_{+}+\tilde{{ \mathcal P }}{(-1)}^{J^{\prime} }{\hat{\epsilon }}_{+1}^{* }{d}_{-})\cdot {\vec{\varepsilon }}_{\mathrm{eff}},\end{eqnarray} \tag{ 43 }$$

such that ${\vec{\varepsilon }}_{\mathrm{eff}}$ is the 'effective E1 polarisation' (i.e. including the effects of interference between multipole transition matrix elements is equivalent to an E1 transition with this polarisation) with the form

$$\begin{eqnarray}&&{\vec{\varepsilon }}_{\mathrm{eff}}=\hat{\epsilon }-{a}_{{\rm{M}}1}{\rm{i}}\hat{n}\times (\hat{k}\times \hat{\epsilon })+{a}_{{\rm{E}}2}(\tilde{{ \mathcal P }}){\rm{i}}(\hat{k}(\hat{\epsilon }\cdot \hat{n})+\hat{\epsilon }(\hat{k}\cdot \hat{n}))+...\end{eqnarray} \tag{ 44 }$$

where $\hat{n}=\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}\hat{z}$ is the orientation of the internuclear axis in the laboratory frame, ${a}_{{\rm{E}}2}(\tilde{{ \mathcal P }})={c}_{{\rm{E}}2}(\tilde{{ \mathcal P }})/(\sqrt{2}{c}_{{\rm{E}}1})$ and ${a}_{{\rm{M}}1}={c}_{{\rm{M}}1}/{c}_{{\rm{E}}1}$ are real dimensionless ratios describing the strength of the M1 and E2 matrix elements relative to E1, and the c coefficients are matrix elements,

$$\begin{eqnarray}&&{c}_{{\rm{E}}1}=\langle C,J,0,1| {\rm{E}}1| H,J^{\prime} ,1,1\rangle ,\end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}&&{c}_{{\rm{M}}1}=\langle C,J,0,1| {\rm{M}}1| H,J^{\prime} ,1,1\rangle ,\end{eqnarray} \tag{ 46 }$$

$$\begin{eqnarray}&&{c}_{{\rm{E}}2}(\tilde{{ \mathcal P }})=\langle C,J,0,1| {\rm{E}}2| H,J^{\prime} ,1,1\rangle +\tilde{{ \mathcal P }}{(-1)}^{J}\langle C,J,0,1| {\rm{E}}2| H,J^{\prime} ,1,-1\rangle ,\end{eqnarray} \tag{ 47 }$$

which are defined using the state notation $| A,J,M,{\rm{\Omega }}\rangle$ for electronic state A, and 'E1, M1, E2' refer to the corresponding molecular operators in table 3. It is useful to define the Rabi frequency ${{\rm{\Omega }}}_{{\rm{r}}}=| { \mathcal M }|$ as the magnitude of the amplitude connecting to the bright state, and the unit vector ${\hat{\varepsilon }}_{\mathrm{eff}}$ corresponding to the projection of ${\vec{\varepsilon }}_{\mathrm{eff}}$ onto the xy plane,

$$\begin{eqnarray}&&{\hat{\varepsilon }}_{\mathrm{eff}}=\displaystyle \frac{{\vec{\varepsilon }}_{\mathrm{eff}}-({\vec{\varepsilon }}_{\mathrm{eff}}\cdot \hat{z})\hat{z}}{\sqrt{| {\vec{\varepsilon }}_{\mathrm{eff}}{| }^{2}-| {\vec{\varepsilon }}_{\mathrm{eff}}\cdot \hat{z}{| }^{2}}}.\end{eqnarray} \tag{ 48 }$$

This completely determines the bright and dark states, which have been previously defined in equations (12) and (13) for solely E1 transition matrix elements.

The odd parity E1 and even parity M1 and E2 contributions to the effective polarisation differ by a factor of $\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}$, which is correlated with the expected eEDM signal. Expanding the effective E1 polarisation in terms of switch parity components, ${\hat{\varepsilon }}_{\mathrm{eff}}={\hat{\varepsilon }}_{\mathrm{eff}}^{\mathrm{nr}}+\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}d{\vec{\varepsilon }}_{\mathrm{eff}}^{{ \mathcal N }{ \mathcal E }}$, and evaluating the effective $\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}$ correlated polarisation imperfections using equation (37), we find that the bright and dark states have effective polarisation correlations given by:

$$\begin{eqnarray}&&\displaystyle \frac{\hat{z}\cdot ({\hat{\varepsilon }}_{\mathrm{eff}}^{\mathrm{nr}}\times d{\vec{\varepsilon }}_{\mathrm{eff}}^{{ \mathcal N }{ \mathcal E }})}{{\hat{\varepsilon }}_{\mathrm{eff}}^{\mathrm{nr}}\cdot {\hat{\varepsilon }}_{\mathrm{eff}}^{\mathrm{nr}}}\approx \,d{\theta }_{\mathrm{eff}}^{{ \mathcal N }{ \mathcal E }}-{\rm{i}}{d}{{\rm{\Theta }}}_{\mathrm{eff}}^{{ \mathcal N }{ \mathcal E }}\end{eqnarray} \tag{ 49 }$$

$$\begin{eqnarray}&&\approx {\rm{i}}({a}_{{\rm{M}}1}-{a}_{{\rm{E}}2}(\tilde{{ \mathcal P }}))(\hat{\epsilon }\cdot \hat{z})((\hat{k}\times \hat{\epsilon })\cdot \hat{z}).\end{eqnarray} \tag{ 50 }$$

It is useful to use a particular parameterisation of the laser pointing $\hat{k}$ and polarisation $\hat{\epsilon }$ to expand the expression in equation (50) in terms of pointing and polarisation imperfections. The state preparation laser $\hat{k}$-vector is aligned along (or against) the $\hat{z}$ direction in the laboratory, so it is convenient to parameterise the pointing deviation from normal by spherical angle ${\vartheta }_{k}$, and the direction of this pointing imperfection by polar angle ${\varphi }_{k}$ in the xy plane, such that:

$$\begin{eqnarray}&&\hat{k}=\cos {\varphi }_{k}\sin {\vartheta }_{k}\hat{x}+\sin {\varphi }_{k}\sin {\vartheta }_{k}\hat{y}+\cos {\vartheta }_{k}\hat{z}.\end{eqnarray} \tag{ 51 }$$

We may use a parameterisation for the polarisation $\hat{\epsilon }$ that is similar to that in equation (14), but a slight modification is required to ensure that $\hat{k}\cdot \hat{\epsilon }=0$:

$$\begin{eqnarray}&&\hat{\epsilon }={N}_{\epsilon }(-{{\rm{e}}}^{-{\rm{i}}\theta }\cos {\rm{\Theta }}{\hat{\epsilon }}_{+1}+{{\rm{e}}}^{{\rm{i}}\theta }\sin {\rm{\Theta }}{\hat{\epsilon }}_{-1}+{\epsilon }_{z}\hat{z}),\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}&&{\epsilon }_{z}=-\displaystyle \frac{1}{\sqrt{2}}\tan {\theta }_{k}({{\rm{e}}}^{-{\rm{i}}(\theta -{\varphi }_{k})}\cos {\rm{\Theta }}+{{\rm{e}}}^{{\rm{i}}(\theta -{\varphi }_{k})}\sin {\rm{\Theta }}),\end{eqnarray} \tag{ 53 }$$

where ${N}_{\epsilon }$ is a normalisation constant that ensures that ${\hat{\epsilon }}^{* }\cdot \hat{\epsilon }=1$. With these parameterisations in place, and expanding about small ellipticities $d{\rm{\Theta }}$ such that ${\rm{\Theta }}=\pi /4+d{\rm{\Theta }}$, and small laser pointing deviation, ${\vartheta }_{k}\ll 1$, we find that the $\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}$-correlated effective laser polarisation imperfections are given by:

$$\begin{eqnarray}&&d{\theta }_{\mathrm{eff}}^{{ \mathcal N }{ \mathcal E }}\approx -\displaystyle \frac{1}{2}({a}_{{\rm{M}}1}-{a}_{{\rm{E}}2}(\tilde{{ \mathcal P }})){\vartheta }_{k}^{2\ }{S}_{i}^{\ }\cos (2(\theta -{\varphi }_{k})),\end{eqnarray} \tag{ 54 }$$

$$\begin{eqnarray}&&d{{\rm{\Theta }}}_{\mathrm{eff}}^{{ \mathcal N }{ \mathcal E }}\approx -\displaystyle \frac{1}{2}({a}_{{\rm{M}}1}-{a}_{{\rm{E}}2}(\tilde{{ \mathcal P }})){\vartheta }_{k}^{2\ }\sin (2(\theta -{\varphi }_{k})),\end{eqnarray} \tag{ 55 }$$

where ${S}_{i}=-2d{{\rm{\Theta }}}_{i}$ describe the laser ellipticities. Hence, following equation (38), there is a systematic error in ${\omega }^{{ \mathcal N }{ \mathcal E }}$:

$$\begin{eqnarray}&&{\omega }_{{\rm{S}}.{\rm{I}}.}^{{ \mathcal N }{ \mathcal E }}=\displaystyle \frac{1}{\tau }\displaystyle \frac{1}{4}({a}_{{\rm{M}}1}-{a}_{{\rm{E}}2}(\tilde{{ \mathcal P }}))\ \end{eqnarray} \tag{ 56 }$$

$$\begin{eqnarray}&&\times \,[{\vartheta }_{k,\mathrm{prep}}^{2}(-2{S}_{\mathrm{prep}}{c}_{\mathrm{prep}}+\tilde{{ \mathcal P }}{s}_{\mathrm{prep}}({S}_{X}-{S}_{Y}))\end{eqnarray} \tag{ 57 }$$

$$\begin{eqnarray}&&+\,{\vartheta }_{k,X}^{2}({S}_{X}{c}_{X}+\tilde{{ \mathcal P }}{S}_{\mathrm{prep}}{s}_{X})+{\vartheta }_{k,Y}^{2}({S}_{Y}{c}_{Y}-\tilde{{ \mathcal P }}{S}_{\mathrm{prep}}{s}_{Y})],\end{eqnarray} \tag{ 58 }$$

where ${c}_{i}\equiv \cos (2({\theta }_{i}-{\varphi }_{i,k}))$ and ${s}_{i}\equiv \sin (2({\theta }_{i}-{\varphi }_{i,k}))$ describe the dependence of the systematic error on the difference between the linear polarisation angle ${\theta }_{i}$ and the pointing angle ${\varphi }_{i,k}$ in the xy plane.

There is another contribution to this systematic error that arises when the coupling to the off-resonant opposite parity excited state $| C,-\tilde{{ \mathcal P }}\rangle$ is also taken into account. This additional contribution becomes significant when the ellipticities are comparable to or smaller than ${\gamma }_{C}/{{\rm{\Delta }}}_{{\rm{\Omega }},C,J=1}\approx 0.5 \%$.

The eEDM channel, ${\omega }^{{ \mathcal N }{ \mathcal E }}$, was defined to be even under the superblock switches (including $\tilde{{ \mathcal P }}$), hence those terms proportional to $\tilde{{ \mathcal P }}$ in the equation above do not contribute to our reported result. Additionally, the $\tilde{{ \mathcal G }}$ and $\tilde{{ \mathcal R }}$ switches rotate the polarisation angles for each laser by roughly ${\theta }_{i}\to {\theta }_{i}+\pi /2$ periodically and the resulting ${\omega }^{{ \mathcal N }{ \mathcal E }}$ signal is averaged over these states. Provided that the pointing drift is much slower than the timescale of these switches, and to the extent that the laser polarisations constituting the $\tilde{{ \mathcal R }}$ and $\tilde{{ \mathcal G }}$ states are orthogonal, then these systematic errors should dominantly contribute to the ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal G }}$ and ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal R }}$ channels which were found to be consistent with zero (see figure 35).

An indirect limit on the size of the systematic error due to S.I., ${\omega }_{{\rm{S}}.{\rm{I}}.}^{{ \mathcal N }{ \mathcal E }}$, may be estimated by assuming a reasonable suppression factor by which the effects in ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal R }}$ and ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal G }}$ may 'leak' into ${\omega }^{{ \mathcal N }{ \mathcal E }}$. We monitored the pointing drift on a beam profiler and observed pointing drifts up to $d{\vartheta }_{k}\sim 50\,\mu \mathrm{rad}$ throughout a full set of superblock states. The absolute pointing misalignment angle was not well known but was estimated to be larger than ${\vartheta }_{k}\gtrsim 0.5\,\mathrm{mrad}$. Hence we may estimate a conservative suppression factor $d{\vartheta }_{k}/{\vartheta }_{k}\lesssim 1/10$ by which pointing drift may contaminate ${\omega }^{{ \mathcal N }{ \mathcal E }}$ from ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal R }}$ and ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal G }}$. The two $\tilde{{ \mathcal R }}$ states are very nearly orthogonal, but the $\tilde{{ \mathcal G }}$ states deviate sufficiently from orthogonal (see section 5.2.5) such that the leakage from ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal G }}\to {\omega }^{{ \mathcal N }{ \mathcal E }}$ will dominate the systematic error; we estimate a suppression factor of about ${c}_{p}^{\mathrm{nr}}/{c}_{p}^{{ \mathcal G }}\sim {s}_{p}^{\mathrm{nr}}/{s}_{p}^{{ \mathcal G }}\sim 1/5$. Based on the upper limits on the measured values for ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal R }}$ and ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal G }}$ combined with leakage from ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal R }}$ and ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal G }}$ into ${\omega }^{{ \mathcal N }{ \mathcal E }}$ due to pointing drift, and leakage from ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal G }}$ into ${\omega }^{{ \mathcal N }{ \mathcal E }}$ due to non-orthogonality of the two $\tilde{{ \mathcal G }}$ states, we estimate the possible size of the systematic error to be ${\omega }_{{\rm{S}}.{\rm{I}}.}^{{ \mathcal N }{ \mathcal E }}\lesssim {1}^{}\mathrm{mrad}\,{{\rm{s}}}^{-1}$.

Note that the mechanism for this systematic error was not discovered until after the publication of our result [19] and hence was not included in our systematic error analysis there. Furthermore, since we did not observe this effect, this systematic error does not match any of the inclusion criteria outlined in section 5.10 and hence is not included in the systematic error budget in this paper. Since we did not understand the mechanism for this systematic error while running the apparatus, we were not able to place direct limits on the size of this systematic error. We estimate that the absolute pointing deviation from ideal was at most ${5}^{}\mathrm{mrad}$ and the ellipticity of each laser was no more than ${S}_{i}\approx 5 \%$. The E1/M1 interference coefficient is ${a}_{{\rm{M}}1}\approx 0.1$ for the $H\to C$ transition. This gives an estimate of ${\omega }_{{\rm{S}}.{\rm{I}}.}^{{ \mathcal N }{ \mathcal E }}\sim {0.1}^{}\mathrm{mrad}\,{{\rm{s}}}^{-1}$ before suppression due to the $\tilde{{ \mathcal R }}$ and $\tilde{{ \mathcal G }}$ switches. Hence, we do not believe that this systematic error significantly shifted the result of our measurement.

In this section we describe contributions to the measured phase Φ that depend on the AC Stark shifts induced by the state preparation and readout lasers. We describe mechanisms by which such phase contributions may arise, and we describe mechanisms by which $\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}$ correlated experimental imperfections may couple to these phases to result in eEDM-mimicking phases. Concise descriptions of some of the effects described here can be found in [66, 67, 94].

During our search for systematic errors as described in section 5.1, we empirically found that there was a contribution to the measured phase $d{\rm{\Phi }}({\rm{\Delta }},{{\rm{\Omega }}}_{{\rm{r}}})$ that had an unexpected linear dependence on the laser detuning Δ, a quadratic dependence on laser detuning Δ in the presence of a nonzero magnetic field, and a linear dependence on small changes to the magnitude of the Rabi frequency, $d{{\rm{\Omega }}}_{{\rm{r}}}/{{\rm{\Omega }}}_{{\rm{r}}}$, in the presence of a nonzero magnetic field,

$$\begin{eqnarray}&&d{\rm{\Phi }}({\rm{\Delta }},{{\rm{\Omega }}}_{{\rm{r}}})=\displaystyle \sum _{i}[{\alpha }_{{\rm{\Delta }},i}{{\rm{\Delta }}}_{i}+{\alpha }_{{{\rm{\Delta }}}^{2},i}{{\rm{\Delta }}}_{i}^{2}+{\beta }_{d{{\rm{\Omega }}}_{{\rm{r}},i}}(d{{\rm{\Omega }}}_{{\rm{r}},i}/{{\rm{\Omega }}}_{{\rm{r}},i})+...].\end{eqnarray} \tag{ 59 }$$

where $i\in \{\mathrm{prep},X,Y\}$ indexes the state preparation and readout lasers. The coefficients we measured were ${\alpha }_{{\rm{\Delta }}}\sim {1}^{}\mathrm{mrad}/(2\pi \times \,\mathrm{MHz})$, ${\alpha }_{{{\rm{\Delta }}}^{2}}\sim {1}^{}\mathrm{mrad}/{(2\pi \times \mathrm{MHz})}^{2}$ and ${\beta }_{d{{\rm{\Omega }}}_{{\rm{r}}}}\sim {10}^{-3}$. We performed these measurements by independently varying the laser detunings ${{\rm{\Delta }}}_{i}$ across resonance using AOMs or modulating the laser power using AOMs with the set-up depicted in figure 10 and extracting the measured phase Φ. Examples of such measurements are given in figure 31.

We determined that this behaviour can be caused by mixing between bright and dark states, due to a small non-adiabatic laser polarisation rotation or Zeeman interaction present during the optical pumping used to prepare and read out the spin state. The mixed bright and dark states differ in energy by the AC Stark shift, which leads to a relative phase accumulation between the bright and dark state components that depends on the laser parameters Δ and ${{\rm{\Omega }}}_{{\rm{r}}}$. We shall now derive the AC Stark shift phase that results in equation (59), under simplifying assumptions amenable to analytic calculations.

Consider a three level system consisting of the bright $| B(\hat{\varepsilon },\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle$ and dark $| D(\hat{\varepsilon },\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle$ states and the lossy excited state $| C,\tilde{{ \mathcal P }}\rangle$ with decay rate ${\gamma }_{C}$. For simplicity, assume that there is no applied magnetic field for the time being. In this system, the instantaneous eigenvectors (depicted in figure 29(C)) are

$$\begin{eqnarray}&&| {B}_{\pm }\rangle \equiv \pm {\kappa }_{\pm }| C,\tilde{{ \mathcal P }}\rangle +{\kappa }_{\mp }| B(\hat{\varepsilon },\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle {,}^{}| D\rangle \equiv | D(\hat{\varepsilon },\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle ,\end{eqnarray} \tag{ 60 }$$

and the instantaneous eigenvalues are

$$\begin{eqnarray}&&{E}_{B\pm }=\displaystyle \frac{1}{2}({\rm{\Delta }}\pm \sqrt{{{\rm{\Delta }}}^{2}+{{\rm{\Omega }}}_{{\rm{r}}}^{2}}){,}^{}{E}_{D}=0,\end{eqnarray} \tag{ 61 }$$

such that the mixing amplitudes ${\kappa }_{\pm }$ are given by

$$\begin{eqnarray}&&{\kappa }_{\pm }=\displaystyle \frac{1}{\sqrt{2}}\sqrt{1\pm \displaystyle \frac{{\rm{\Delta }}}{\sqrt{{{\rm{\Delta }}}^{2}+{{\rm{\Omega }}}_{{\rm{r}}}^{2}}}}.\end{eqnarray} \tag{ 62 }$$

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The effect of the decay of the excited state (which occurs almost entirely to states outside of the three level system) may be taken into account by adding an anti-Hermitian operator term in the Schrodinger equation, $| \dot{\psi }\rangle =-{\rm{i}}(H-{\rm{i}}\tfrac{1}{2}{\rm{\Gamma }})| \psi \rangle$, where ${\rm{\Gamma }}={\gamma }_{C}| C,\tilde{{ \mathcal P }}\rangle \langle C,\tilde{{ \mathcal P }}|$ is the decay operator. This formulation is equivalent to the Lindblad master equation,

$$\begin{eqnarray}&&\dot{\rho }=-{\rm{i}}[H,\rho ]-\displaystyle \frac{1}{2}\{{\rm{\Gamma }},\rho \},\end{eqnarray} \tag{ 63 }$$

that governs the time evolution of the density matrix $\rho =| \psi \rangle \langle \psi |$. In practice, we implement this decay term by calculating the time evolution of the system according to H, and then making the substitution ${\rm{\Delta }}\to {\rm{\Delta }}-{\rm{i}}{\gamma }_{C}/2$ before calculating squares of amplitudes.

It is useful to work in the dressed state basis, $| D\rangle$, $| {B}_{\pm }\rangle$, (basis C in figure 29) because these are nearly stationary states and have simple time evolution in the case that laser polarisation and Rabi frequency are stationary. If we allow the laser polarisation to vary in time, then the dressed state basis varies in time, and the system evolves according to the Hamiltonian,

$$\begin{eqnarray}&&\tilde{H}={{UHU}}^{\dagger }-{\rm{i}}{U}{\dot{U}}^{\dagger },\end{eqnarray} \tag{ 64 }$$

where U is the transformation from time independent basis A to time dependent basis C (from figure 29), ${{UHU}}^{\dagger }$ is diagonal, and $-{\rm{i}}{U}{\dot{U}}^{\dagger }$ is a fictitious force term that arises because we are working in a non-inertial frame when the laser polarisation is time dependent [77].

Assuming that the polarisation is nearly linear, ${\rm{\Theta }}\approx \pi /4$, but allowing the polarisation to rotate slightly, and allowing for a nonzero two photon detuning due to the Zeeman shift $\delta =-{g}_{1}{\mu }_{{\rm{B}}}{{ \mathcal B }}_{z}\tilde{{ \mathcal B }}$, the Hamiltonian in the dressed state picture is:

$$\begin{eqnarray}\tilde{H}=\left(\begin{array}{ccc}0 & -{\rm{i}}{\dot{\chi }}^{* }{\kappa }_{+} & -{\rm{i}}{\dot{\chi }}^{* }{\kappa }_{-}\\ {\rm{i}}\dot{\chi }{\kappa }_{+} & {E}_{B-} & -\tfrac{{\rm{i}}}{2}\tfrac{{\dot{{\rm{\Omega }}}}_{{\rm{r}}}{\rm{\Delta }}-{{\rm{\Omega }}}_{{\rm{r}}}\dot{{\rm{\Delta }}}}{{{\rm{\Delta }}}^{2}+{{\rm{\Omega }}}^{2}}\\ {\rm{i}}\dot{\chi }{\kappa }_{-} & \tfrac{{\rm{i}}}{2}\tfrac{{\dot{{\rm{\Omega }}}}_{{\rm{r}}}{\rm{\Delta }}-{{\rm{\Omega }}}_{{\rm{r}}}\dot{{\rm{\Delta }}}}{{{\rm{\Delta }}}^{2}+{{\rm{\Omega }}}_{{\rm{r}}}^{2}} & {E}_{B+}\end{array}\right)\quad \begin{array}{c}| D\rangle \\ | {B}_{-}\rangle \\ | {B}_{+}\rangle \end{array},\end{eqnarray} \tag{ 65 }$$

where $\dot{\chi }=\dot{{\rm{\Theta }}}-{\rm{i}}(\dot{\theta }+\delta )$ can be considered to be a complex polarisation rotation rate, ${\dot{{\rm{\Omega }}}}_{{\rm{r}}}$ is the rate of change of the Rabi frequency, and $\dot{{\rm{\Delta }}}$ is the rate of change of the detuning. Note that this Hamiltonian implies that the effect of a two photon detuning arising from the Zeeman shift is equivalent to that of a linear polarisation rotating at a constant rate.

We may then apply first order time-dependent perturbation theory in this picture to determine the extent of bright/dark state mixing due to $\dot{\chi }$ in the time evolution of the system. If we parameterise the time-dependent state as

$$\begin{eqnarray}&&| \psi (t)\rangle ={c}_{D}(t)| D\rangle +{c}_{B+}(t)| {B}_{+}\rangle +{c}_{B-}(t)| {B}_{-}\rangle ,\end{eqnarray} \tag{ 66 }$$

then in the case of a uniform laser field ${\dot{{\rm{\Omega }}}}_{{\rm{r}}}=0$, of duration t and with a constant detuning $\dot{{\rm{\Delta }}}=0$, the time evolution of the coefficients is given at first order by:

$$\begin{eqnarray}&&{c}_{D}(t)={c}_{D}(0)-\displaystyle \sum _{\pm }{\int }_{0}^{t}{\dot{\chi }}^{* }(t^{\prime} ){\kappa }_{\mp }(t^{\prime} ){{\rm{e}}}^{-{{\rm{i}}{E}}_{B\pm }t^{\prime} }{c}_{B\pm }(0)\,{\rm{d}}{t}^{\prime} ,\end{eqnarray} \tag{ 67 }$$

$$\begin{eqnarray}&&{c}_{B\pm }(t)={{\rm{e}}}^{-{{\rm{i}}{E}}_{B\pm }t}{c}_{B\pm }(0)+{{\rm{e}}}^{-{{\rm{i}}{E}}_{B\pm }t}{\int }_{0}^{t}\dot{\chi }(t^{\prime} ){\kappa }_{\mp }(t^{\prime} ){{\rm{e}}}^{{{\rm{i}}{E}}_{B\pm }t^{\prime} }{c}_{D}(0)\,{\rm{d}}{t}^{\prime} .\end{eqnarray} \tag{ 68 }$$

In the state preparation region, the molecules begin in an incoherent mixture of the states $| B({\hat{\varepsilon }}_{\mathrm{prep}},\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle$ and $| D({\hat{\varepsilon }}_{\mathrm{prep}},\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle$ and then enter the state preparation laser beam. In the ideal case of uniform laser polarisation, molecules that were in the bright state are optically pumped out of the three level system, and molecules that are in the dark state remain there; this results in a pure state, $| D({\hat{\varepsilon }}_{\mathrm{prep}},\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle$. However, if there is a small polarisation rotation by amount $d\chi \equiv {\int }_{0}^{t}\dot{\chi }(t^{\prime} ){\rm{d}}{t}^{\prime} \equiv d{\rm{\Theta }}-{\rm{i}}(d\theta -{g}_{1}{\mu }_{{\rm{B}}}{{ \mathcal B }}_{z}\tilde{{ \mathcal B }}t)$, such that $| d\chi | \ll 1$, then the dark state obtains a bright state admixture that may not be completely optically pumped away before leaving the laser beam⁴⁷ . In this case, the final state can be written as

$$\begin{eqnarray}&&| D({\hat{\varepsilon }}_{\mathrm{prep}}^{\prime },\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle =| D({\hat{\varepsilon }}_{\mathrm{prep}},\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle +d\chi {\rm{\Pi }}| B({\hat{\varepsilon }}_{\mathrm{prep}},\tilde{{ \mathcal N }},\tilde{{ \mathcal P }})\rangle \end{eqnarray} \tag{ 69 }$$

where ${\hat{\varepsilon }}_{\mathrm{prep}}^{\prime }$ is the effective polarisation that parameterises the initial state in the spin precession region

$$\begin{eqnarray}&&{\hat{\varepsilon }}_{\mathrm{prep}}^{\prime }={\hat{\varepsilon }}_{\mathrm{prep}}+d\chi {\rm{\Pi }}{\rm{i}}\hat{z}\times {\hat{\varepsilon }}_{\mathrm{prep}}^{* },\end{eqnarray} \tag{ 70 }$$

and Π is an amplitude that accounts for the AC Stark shift phase and the time dependent dynamics of the non-adiabatic mixing due to the polarisation rotation,

$$\begin{eqnarray}&&{\rm{\Pi }}=\displaystyle \sum _{\pm }{({\kappa }_{\mp })}^{2}{{\rm{e}}}^{-{{\rm{i}}{E}}_{B\pm }t}{\int }_{0}^{t}{\rm{d}}{t}{^{\prime} }^{}\displaystyle \frac{\dot{\chi }(t^{\prime} )}{d\chi }{{\rm{e}}}^{{{\rm{i}}{E}}_{B\pm }t^{\prime} }.\end{eqnarray} \tag{ 71 }$$

The deviations between the effective polarisation and the actual laser polarisation can be viewed as effective polarisation imperfections,

$$\begin{eqnarray}&&d{\theta }_{\mathrm{prep},\mathrm{eff}}=-d{{\rm{\Theta }}}_{\mathrm{prep}}\mathrm{Im}{\rm{\Pi }}+(d{\theta }_{\mathrm{prep}}-{g}_{1}{\mu }_{{\rm{B}}}{{ \mathcal B }}_{z}\tilde{{ \mathcal B }}t)\mathrm{Re}{\rm{\Pi }},\end{eqnarray} \tag{ 72 }$$

$$\begin{eqnarray}&&d{{\rm{\Theta }}}_{\mathrm{prep},\mathrm{eff}}=-d{{\rm{\Theta }}}_{\mathrm{prep}}\mathrm{Re}{\rm{\Pi }}-(d{\theta }_{\mathrm{prep}}-{g}_{1}{\mu }_{{\rm{B}}}{{ \mathcal B }}_{z}\tilde{{ \mathcal B }}t)\mathrm{Im}{\rm{\Pi }},\end{eqnarray} \tag{ 73 }$$

that lead to shifts in the measured phase Φ as described in equation (38). For definiteness, consider the case in which the polarisation rotation rate $\dot{\chi }(t^{\prime} )=d\chi /t$ is a constant for the duration of the optical pumping pulse t. In this case,

$$\begin{eqnarray}&&{\rm{\Pi }}=\displaystyle \sum _{\pm }{({\kappa }_{\mp })}^{2}{{\rm{e}}}^{-{{\rm{i}}{E}}_{B\pm }t/2}\mathrm{sinc}({E}_{B\pm }t/2).\end{eqnarray} \tag{ 74 }$$

This function has the property that $\mathrm{Im}{\rm{\Pi }}$ is an odd function in Δ that can take on values up to order unity across resonance (a frequency range on the order of ${\gamma }_{C}$) and is exactly zero on resonance. $\mathrm{Re}{\rm{\Pi }}$ is an even function quadratic in Δ about resonance, and depends on Rabi frequency on resonance. If the laser beam intensity reduces quickly as the molecule leaves it then most of the AC Stark shift phase arises from the last Rabi flopping period before the molecule exits the laser beam (provided $\dot{\chi }$ is nonzero during that time). If the intensity reduces slowly, the AC Stark shift phase can be exacerbated since the bright state amplitude is not as effectively optically pumped away while ${{\rm{\Omega }}}_{{\rm{r}}}\lt {\gamma }_{C}$. Nevertheless, beamshaping tests shown in figure 31 and numerical simulations indicate that Π is not very sensitive to the shape of the spatial intensity profile of the laser beam or the shape of the spatial variation of the polarisation.

If we consider only the first order contribution to the shift in the measured phase in equation (38), $d{\theta }_{\mathrm{prep},\mathrm{eff}}$, and neglect the second order shift that arises due to $d{{\rm{\Theta }}}_{\mathrm{prep},\mathrm{eff}}$, then we can relate the parameters in equation (59) to the amplitude Π accounting for the AC Stark shift phase and the complex polarisation rotation $d\chi$, by

$$\begin{eqnarray}&&{\alpha }_{{\rm{\Delta }},\mathrm{prep}}\approx -\displaystyle \frac{\partial \mathrm{Im}{\rm{\Pi }}}{\partial {{\rm{\Delta }}}_{\mathrm{prep}}}d{{\rm{\Theta }}}_{\mathrm{prep}},\end{eqnarray} \tag{ 75 }$$

$$\begin{eqnarray}&&{\alpha }_{{{\rm{\Delta }}}^{2},\mathrm{prep}}\approx \displaystyle \frac{{\partial }^{2}\mathrm{Re}{\rm{\Pi }}}{\partial {{\rm{\Delta }}}_{\mathrm{prep}}^{2}}(d{\theta }_{\mathrm{prep}}-g{\mu }_{{\rm{B}}}{{ \mathcal B }}_{z}\tilde{{ \mathcal B }}t),\end{eqnarray} \tag{ 76 }$$

$$\begin{eqnarray}&&{\beta }_{d{{\rm{\Omega }}}_{{\rm{r}}},\mathrm{prep}}\approx {{\rm{\Omega }}}_{{\rm{r}}}\displaystyle \frac{\partial \mathrm{Re}{\rm{\Pi }}}{\partial {{\rm{\Omega }}}_{{\rm{r}}}}(d{\theta }_{\mathrm{prep}}-g{\mu }_{{\rm{B}}}{{ \mathcal B }}_{z}\tilde{{ \mathcal B }}t).\end{eqnarray} \tag{ 77 }$$

We can interpret these results as follows. The linear dependence of the measured phase on detuning, ${\alpha }_{{\rm{\Delta }},\mathrm{prep}}$, comes from a spatially varying ellipticity in the x direction coupling to the AC Stark shift phase. Similarly, the quadratic dependence of Φ on Δ, ${\alpha }_{{{\rm{\Delta }}}^{2},\mathrm{prep}}$, and the dependence of Φ on a relative change in ${{\rm{\Omega }}}_{{\rm{r}}}$, ${\beta }_{d{{\rm{\Omega }}}_{{\rm{r}}},\mathrm{prep}}$, come from either a spatially varying linear polarisation in the x direction or a Zeeman shift, each coupling to the AC Stark shift phase. Here, we only analysed the phase shift that results from AC Stark shift effects in the state preparation laser beam, but there is an analogous phase shift in the state readout beam.

There are several other subdominant effects that also contribute to the AC Stark shift phase behaviour described in equation (38) in the presence of polarisation imperfections. The opposite parity excited state $| C,-\tilde{{ \mathcal P }}\rangle$ couples strongly to the dark state, but the mixing between these two states is weak because the transition frequency is off-resonant by a detuning ${{\rm{\Delta }}}_{{\rm{\Omega }},C,J=1}\approx 2\pi \times 51\,\mathrm{MHz}\gg {\gamma }_{C}$. In the case that an optical pumping laser has nonzero ellipticity, the bright state gains a weak coupling to the opposite-parity excited state proportional to this ellipticity. Then, two-photon bright-dark state mixing ensues in such a way that the mixing amplitude, and hence the measured phase, depends on the laser detuning.

The rapid polarisation switching of the state readout beam can also introduce AC Stark shift-induced phases in the absence of a polarisation gradient, if the average ellipticity between the two polarisations is nonzero. Suppose a particular molecule is first excited by the ${\hat{\epsilon }}_{X}$ polarised beam. The two bright eigenstates $| {B}_{\pm }\rangle$ are mostly optically pumped away, resulting in a fluorescence signal F_X. The population remaining in the bright eigenstates acquires a phase relative to the dark state, due to the AC Stark shift. Then the molecules are optically pumped by the ${\hat{\epsilon }}_{Y}$ polarised beam. If there is a nonzero average ellipticity, ${\hat{\epsilon }}_{Y}$ is not quite orthogonal to ${\hat{\epsilon }}_{X}$ and the new bright eigenstates that give rise to the fluorescence signal F_Y are superpositions of the former bright and dark states that acquired a relative AC Stark shift phase. This results in a fluorescence signal, and hence measured phase component, that depends linearly on laser detuning Δ.

The AC Stark shift phases described in the previous section can be induced by polarisation gradients in $\hat{x}$ across the state preparation and readout laser beams. In this section we describe a known mechanism by which these arose. Recall that these laser beams passed through transparent, ITO-coated electric field plates. For an absorbance α and laser intensity I, the rate of heat deposition into the plates is $\dot{Q}(x,y)=\alpha \,I(x,y)$. The laser beam profile is stretched in the y direction to ensure that all molecules are addressed. For simplicity we assume that the heating distribution, $\dot{Q}(x,y)=\dot{Q}(x)$, is completely uniform in the y direction. We also assume that there are no shear stresses, i.e. local expansion of the glass is isotropic. Under these assumptions, the relationship between the heating rate, $\dot{Q}$, and the internal stress tensor ${\sigma }_{{ij}}$ (where $i,j$ are Cartesian indices) is

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}{\sigma }_{{yy}}}{\partial {x}^{2}}=\displaystyle \frac{E{\alpha }_{V}}{\kappa }\dot{Q}(x),\end{eqnarray} \tag{ 78 }$$

where E, ${\alpha }_{V}$ and κ are the Young's modulus, coefficient of thermal expansion, and thermal conductivity, respectively [118]. Unit vectors $\hat{x}$ and $\hat{y}$ correspond to the principal axes of the stress tensor due to the symmetry of the heating function, hence the off-diagonal (shear) elements are zero, ${\sigma }_{{xy}}=0$. The other diagonal component, ${\sigma }_{{xx}}$, is uniform across the plates, and equal to ${\sigma }_{{yy}}$ far away from the laser. The stress-optical law states that the birefringence and stress are linearly proportional along the principal axes of the stress tensor [119]. The difference between the indices of refraction in the x and y directions is then ${\rm{\Delta }}n=K({\sigma }_{{xx}}-{\sigma }_{{yy}})$, where $K\approx 4\times {10}^{-6}\,{{\rm{MPa}}}^{-1}$ is the stress-optical coefficient for Borofloat glass [120]. The retardance of an incident laser beam of index i is ${{\rm{\Gamma }}}_{i}=2\pi {\rm{\Delta }}n(t/\lambda )$, where t is the thickness of the field plates (in the z direction), and λ is the wavelength of light. Hence, in this limit, the retardance due to thermal stress-induced birefringence is related to the laser intensity by:

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}{\rm{\Gamma }}}{\partial {x}^{2}}=\eta \displaystyle \frac{t}{\lambda }I(x),\end{eqnarray} \tag{ 79 }$$

where $\eta =2\pi {KE}{\alpha }_{V}\alpha /\kappa \approx 26\times {10}^{-6}$ W⁻¹ is a material constant of Borofloat glass [120]. The ellipticity imprinted on the nominally linearly polarised laser beam is given by

$$\begin{eqnarray}&&{S}_{i}={{\rm{\Gamma }}}_{i}(x)\sin (2({\theta }_{i}-{\phi }_{{\rm{\Gamma }},i})),\end{eqnarray} \tag{ 80 }$$

where ${\theta }_{i}$ is the linear polarisation angle and ${\phi }_{{\rm{\Gamma }},i}$ is the orientation of the fast axis of the birefringent material (nominally $\hat{x}$ in our case).

Assuming the laser has total power P, a Gaussian profile in x with standard deviation w_x, and a top-hat profile in y with half width w_y, the intensity is given by

$$\begin{eqnarray}&&I(x)=\displaystyle \frac{P}{\sqrt{8\pi }{w}_{x}{w}_{y}}{{\rm{e}}}^{-\tfrac{{x}^{2}}{2{w}_{x}^{2}}}\end{eqnarray} \tag{ 81 }$$

where $2{w}_{y}\gg {w}_{x}$. There is then an analytic solution to equation (79) from which we extracted a retardance gradient in the laser tail, $x={w}_{x}$, of

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\rm{\Gamma }}}{\partial x}\approx \displaystyle \frac{\mathrm{erf}(1/\sqrt{2})P\kappa t}{4{w}_{y}\lambda }\approx 0.03\,\mathrm{rad}\,{\mathrm{mm}}^{-1}\end{eqnarray} \tag{ 82 }$$

for a nominal laser power of $\approx 2$ W. Similar results were obtained from numerical finite element analysis. Thermal stress-induced birefringence has been observed in similar systems such as in UHV vacuum windows [121], laser output windows [122], and Nd:YAG rods [123].

The estimates of the ellipticity gradient agree well with measurements of the polarisation of the beam, as shown in figure 30. These polarimetry measurements were adapted from the procedure described in [124]; a polarimeter was constructed consisting of a rotating quarter-wave plate, fixed polariser, and fast photodiode. The use of a fast photodetector allows for polarimetry of the probe beam during the 100 kHz polarisation switching. The resolution of the system was such that we could quickly measure the normalised circular Stokes parameter, S, to a few percent, which is sufficient to measure typical birefringence gradients of $\sim 10 \%$ across the beam.

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We were able to suppress the magnitude of the AC Stark shift phases in several different ways that are illustrated in figure 31. The ellipticity gradient across the state preparation laser beam was suppressed by tuning the linear polarisation angle: as per equation (80), the ellipticity gradient is proportional to $\sin (2{\theta }_{\mathrm{prep}}-2{\phi }_{{\rm{\Gamma }},\mathrm{prep}})$, which vanishes when the polarisation is aligned along a birefringence axis, i.e. ${\theta }_{\mathrm{prep}}={\phi }_{{\rm{\Gamma }},\mathrm{prep}},{\phi }_{{\rm{\Gamma }},\mathrm{prep}}+\pi /2$. To determine ${\phi }_{{\rm{\Gamma }},\mathrm{prep}}$ we measured the total accumulated phase as a function of laser detuning for various ${\theta }_{\mathrm{prep}}$ and then extracted the slope ${\alpha }_{{\rm{\Delta }},\mathrm{prep}}^{\mathrm{nr}}=\partial {{\rm{\Phi }}}^{\mathrm{nr}}/\partial {{\rm{\Delta }}}_{\mathrm{prep}}$ for small detuning values. Note that when fitting the phase versus detuning data we found that cubic functions provided significantly better fits over the detuning ranges used (see figure 31(B)). We then selected ${\theta }_{\mathrm{prep}}$ to minimise ${\alpha }_{{\rm{\Delta }},\mathrm{prep}}^{\mathrm{nr}}$. This suppressed ${\alpha }_{{\rm{\Delta }},\mathrm{prep}}^{\mathrm{nr}}$ by about a factor of 50 relative to its original value, to ${\alpha }_{{\rm{\Delta }},\mathrm{prep}}^{\mathrm{nr}}\lesssim 0.1$ mrad/(2$\pi \,\times$ MHz).

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Another method implemented to suppress AC Stark shift phases was to reduce the time-averaged power of the state preparation laser incident on the field plates. We used a chopper wheel to modulate the laser at 50 Hz, synchronous with the molecular beam pulses, with a 50% duty cycle. We estimated the time scale for thermal changes to be on the order of $Q/\dot{Q}\sim 2\rho {{Cw}}_{x}^{2}/\kappa \sim {10}^{}{\rm{s}}$, where ρ and C are the density and heat capacity of Borofloat respectively, so did not anticipate any significant transient effects to be introduced. This modification reduced the retardance gradient, and hence the value of ${\alpha }_{{\rm{\Delta }},\mathrm{prep}}^{\mathrm{nr}}$, by about a factor of two, as shown in figure 31(C).

Finally, ${\alpha }_{{\rm{\Delta }},\mathrm{prep}}^{\mathrm{nr}}$ was suppressed by shaping of the laser beam intensity profile. AC Stark shift phases were most significant at the downstream edge of the state preparation laser beam. Here, the intensity is such that bright-dark state mixing is still occurring but the bright state is not efficiently optically pumped away. By making the spatial intensity profile drop off more rapidly, we reduced the time that molecules spent in this intermediate intensity regime. This was achieved by taking advantage of the aspherical distortion introduced by misaligning a telescope immediately before the laser beam entered the spin-precession region. This suppressed ${\alpha }_{{\rm{\Delta }},\mathrm{prep}}$ and ${\beta }_{{{\rm{\Delta }}}^{2},\mathrm{prep}}$ by $\approx 2$, as shown in figure 31(C). In addition to a phase suppression, we noticed that the optimal laser polarisation angle changed after implementing the steps described, as can be seen in figure 31(C). The reason for this change is not definitively known, but we suspect that as we suppressed the birefringent contribution to the AC Stark shift phase, the non-birefringent contributions (i.e. the phase due to nonzero ellipticity causing bright-dark state mixing via the off-resonant opposite parity excited state) became fractionally larger, and we needed to tune the polarisation angle to obtain cancellation between these two classes of effects.

We observed much smaller AC Stark shift phases in the state readout laser beam than in the state preparation laser beam. This is not surprising since the effect is largely birefringent; the contributions to the effective polarisation imperfections for the ${\hat{\epsilon }}_{X}$ and ${\hat{\epsilon }}_{Y}$ polarised lasers should be opposite in sign, $d{\theta }_{X}\propto \sin (2({\theta }_{\mathrm{read}}-{\phi }_{{\rm{\Gamma }},\mathrm{read}}))$, $d{\theta }_{Y}\propto \sin (2({\theta }_{\mathrm{read}}-{\phi }_{{\rm{\Gamma }},\mathrm{read}}+\pi /2))$, such that they cancel each other in the measured phase (see equation (38)). The residual AC Stark shift phases measured in the state readout beam gave ${\alpha }_{{\rm{\Delta }},\mathrm{read}}^{\mathrm{nr}}\approx {0.5}^{}\mathrm{mrad}/(2\pi \times \,\mathrm{MHz})$. This was sufficiently small that the methods of suppression described above were only implemented in the state preparation region.

In the discussion above, we described how polarisation imperfections can lead to contributions to the measured phase that depend on the AC Stark shifts and hence on the laser detunings ${{\rm{\Delta }}}_{i}$ and Rabi frequencies ${{\rm{\Omega }}}_{{\rm{r}},i}$. However, these phases only produce a systematic error in ${\omega }^{{ \mathcal N }{ \mathcal E }}$ if there is a nonzero correlation ${{\rm{\Delta }}}_{i}^{{ \mathcal N }{ \mathcal E }}$ or ${{\rm{\Omega }}}_{{\rm{r}},i}^{{ \mathcal N }{ \mathcal E }}$ of the laser detuning or Rabi frequency. We observed such correlations and discuss them in this section. We will also describe how we evaluated the associated systematic errors.

In section 3.2.4 (see figure 11) we discussed how a non-reversing component of the applied electric field, ${{ \mathcal E }}^{\mathrm{nr}}$, could produce a ${{\rm{\Delta }}}^{{ \mathcal N }{ \mathcal E }}$. In an entirely analogous manner, the Rabi frequency magnitude ${{\rm{\Omega }}}_{{\rm{r}}}$ of the $H\to C$ transition can exhibit the following correlations:

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{r}},i}={{\rm{\Omega }}}_{{\rm{r}},i}^{\mathrm{nr}}+\tilde{{ \mathcal N }}{{\rm{\Omega }}}_{{\rm{r}},i}^{{ \mathcal N }}+\tilde{{ \mathcal N }}\tilde{{ \mathcal P }}{{\rm{\Omega }}}_{{\rm{r}},i}^{{ \mathcal N }{ \mathcal P }}+\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}{{\rm{\Omega }}}_{{\rm{r}},i}^{{ \mathcal N }{ \mathcal E }}+....\end{eqnarray} \tag{ 83 }$$

Here, ${{\rm{\Omega }}}_{{\rm{r}},i}^{\mathrm{nr}}$ is the dominant component of the Rabi frequency for laser $i\in \{\mathrm{prep},X,Y\}$, which could fluctuate in time on the order of 5% due to laser power instability. ${{\rm{\Omega }}}_{{\rm{r}},i}^{{ \mathcal N }}$ is generated by a laser power difference between the $\tilde{{ \mathcal N }}$ states. This arose because we routed the laser light along different paths through a series of AOMs for each state. We measured this effect with photodiodes and found that the largest fractional power correlation was ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }}/{{\rm{\Omega }}}_{{\rm{r}}}^{\mathrm{nr}}\approx 2.5\times {10}^{-3}$. An additional contribution to ${{\rm{\Omega }}}_{{\rm{r}},i}^{{ \mathcal N }}$ and a contribribution to ${{\rm{\Omega }}}_{{\rm{r}},i}^{{ \mathcal N }{ \mathcal P }}$ on the same order arises due to Stark mixing between rotational levels in H and C, leading to $\tilde{{ \mathcal N }}$- and $\tilde{{ \mathcal N }}\tilde{{ \mathcal P }}$-correlated transition amplitudes on the $H\to C$ transition.

Although we did not observe a laser power correlation with $\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}$ we did observe signals consistent with a Rabi frequency correlation, ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$. A nonzero $\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}$-correlated fluorescence signal (as defined in section 4.1) that also reversed with the laser propagation direction $\hat{k}\cdot \hat{z}$, ${F}^{{ \mathcal N }{ \mathcal E }}/{F}^{\mathrm{nr}}\approx -(2.4\times {10}^{-3})(\hat{k}\cdot \hat{z})$, together with a nonzero ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal B }}\approx (2.5{}^{}\mathrm{mrad}\,{{\rm{s}}}^{-1})({{ \mathcal B }}_{z}\,{\mathrm{mG}}^{-1})(\hat{k}\cdot \hat{z})$, provided the first evidence that a nonzero ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$ existed in our system. We believe that this fluorescence correlation arises from a linear dependence of the fluorescence signal size on Rabi frequency, ${F}^{{ \mathcal N }{ \mathcal E }}=(\partial F/\partial {{\rm{\Omega }}}_{{\rm{r}}}^{\mathrm{nr}}){{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$, which is nonzero since the state readout transitions were not fully saturated. We believe that the signal in ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal B }}$ was caused by a coupling between the Rabi-frequency correlation and the ${ \mathcal B }$-odd AC Stark shift phase, ${\omega }^{{ \mathcal N }{ \mathcal E }{ \mathcal B }}=\tfrac{1}{\tau }{\beta }_{d{{\rm{\Omega }}}_{{\rm{r}}}}^{{ \mathcal B }}{{ \mathcal B }}_{z}({{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}/{{\rm{\Omega }}}_{{\rm{r}}}^{\mathrm{nr}})$. We were able to verify a linear dependence of both of these channels on ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$ by intentionally correlating the laser intensity with $\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}$ using AOMs; this is shown for the ${{\rm{\Phi }}}^{{ \mathcal N }{ \mathcal E }{ \mathcal B }}$ channel in figure 32. Varying the size of this artificial ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$ allowed us to measure the value present in the experiment under normal operating conditions, ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}/{{\rm{\Omega }}}_{{\rm{r}}}^{\mathrm{nr}}\,=(-8.0\pm 0.8)\times {10}^{-3}(\hat{k}\cdot \hat{z})$. ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$ can couple to ${\beta }_{d{{\rm{\Omega }}}_{{\rm{r}}},i}^{\mathrm{nr}}$ as per equations (59) and (77) to result in a systematic error in ${\omega }^{{ \mathcal N }{ \mathcal E }}$. A nonzero ${\beta }_{d{{\rm{\Omega }}}_{{\rm{r}}},i}^{\mathrm{nr}}$ can be produced by a linear polarisation angle gradient (not observed in the experiment) or by a non-reversing Zeeman shift component ${g}_{1}{\mu }_{{\rm{B}}}{{ \mathcal B }}_{z}^{\mathrm{nr}}$.

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While searching for a model to explain the intrinsic ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$, we developed the S.I. model presented in section 5.2.2. For unnormalized effective polarisation ${\vec{\varepsilon }}_{\mathrm{eff}}={\vec{\varepsilon }}_{\mathrm{eff}}^{\mathrm{nr}}+\tilde{{ \mathcal N }}\tilde{{ \mathcal E }}d{\vec{\varepsilon }}_{\mathrm{eff}}^{{ \mathcal N }{ \mathcal E }}$, this model predicts ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}/{{\rm{\Omega }}}_{{\rm{r}}}^{\mathrm{nr}}\approx \mathrm{Re}({\vec{\varepsilon }}_{\mathrm{eff}}^{\mathrm{nr}* }\cdot d{\vec{\varepsilon }}_{\mathrm{eff}})\approx -\mathrm{Im}[({a}_{{\rm{M}}1}+{a}_{{\rm{E}}2})](\hat{k}\cdot \hat{z})$, which correctly predicts the dependence of ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$ on the laser propagation direction $\hat{k}\cdot \hat{z}$. However, the factors ${a}_{{\rm{M}}1}$ and ${a}_{{\rm{E}}2}$, which correspond to the ratio of M1 and E2 amplitudes to the E1 amplitude, must be real for a plane wave, so $\mathrm{Im}[({a}_{{\rm{M}}1}+{a}_{{\rm{E}}2})]=0$. Hence this model fails to explain this Rabi frequency correlation unless there is some additional effect that introduces a phase shift between the E1 and M1 amplitudes. For example, interference between the E1 amplitude due to the incident laser beam, and a phase shifted M1 amplitude due to a (low intensity) reflected beam can lead to a nonzero ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$ by this model. However, this phase factor oscillates spatially on the scale of the light wavelength, which is very small compared to the size of the molecule cloud and hence should average out over the entire molecular beam cloud. The origin of the intrinsic ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$ is still not fully understood, and we are continuing to explore models to understand this effect.

Given the empirical AC Stark shift phase model in equation (59), the resulting systematic errors in the frequency measurement are given by

$$\begin{eqnarray}&&{\omega }_{{{ \mathcal E }}^{\mathrm{nr}}}^{{ \mathcal N }{ \mathcal E }}=\displaystyle \frac{1}{\tau }\displaystyle \sum _{i\in \{\mathrm{prep},X,Y\}}{\alpha }_{{\rm{\Delta }},i}^{\mathrm{nr}}{D}_{1}{{ \mathcal E }}^{\mathrm{nr}}({x}_{i}),\end{eqnarray} \tag{ 84 }$$

$$\begin{eqnarray}&&{\omega }_{{{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}}^{{ \mathcal N }{ \mathcal E }}=\displaystyle \frac{1}{\tau }\displaystyle \sum _{i\in \{\mathrm{prep},X,Y\}}{\beta }_{d{{\rm{\Omega }}}_{{\rm{r}}},i}^{\mathrm{nr}}({{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}/{{\rm{\Omega }}}_{{\rm{r}}}^{\mathrm{nr}}).\end{eqnarray} \tag{ 85 }$$

Early in the experiment, we observed a nonzero systematic shift ${\omega }_{{{ \mathcal E }}^{\mathrm{nr}}}^{{ \mathcal N }{ \mathcal E }}$ and took the steps outlined in section 5.2.5 to suppress it. To verify that the steps taken were effective, we examined ${\omega }^{{ \mathcal N }{ \mathcal E }}$ as a function of an intentionally applied non-reversing electric field. The resulting data are shown in figure 33. The original slope, $\partial {\omega }^{{ \mathcal N }{ \mathcal E }}/\partial {{ \mathcal E }}^{\mathrm{nr}}=(6.7\pm 0.4)(\mathrm{rad}\,{{\rm{s}}}^{-1})\,{({\rm{V}}{{\rm{c}}{\rm{m}}}^{-1})}^{-1}$, corresponded to a systematic shift of ${\omega }_{{{ \mathcal E }}^{\mathrm{nr}}}^{{ \mathcal N }{ \mathcal E }}\approx -34\,\mathrm{mrad}\,{{\rm{s}}}^{-1}$ when combined with the measured ${{ \mathcal E }}^{\mathrm{nr}}\approx -{5}^{}\mathrm{mV}\,{\mathrm{cm}}^{-1}$. Following the modifications described above, the $\partial {\omega }^{{ \mathcal N }{ \mathcal E }}/\partial {{ \mathcal E }}^{\mathrm{nr}}$ slope was greatly suppressed, reducing the systematic error to ${\omega }_{{{ \mathcal E }}^{\mathrm{nr}}}^{{ \mathcal N }{ \mathcal E }}\lt 1\,\mathrm{mrad}\,{{\rm{s}}}^{-1}$, well below the statistical uncertainty in the measurement of ${\omega }^{{ \mathcal N }{ \mathcal E }}$.

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Because we observed that the parameters ${{ \mathcal E }}^{\mathrm{nr}}$ and ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$ caused systematic errors in ${\omega }^{{ \mathcal N }{ \mathcal E }}$, we intermittently measured the size of the associated systematic errors throughout the datasets that were used for our reported result. We measured $\partial {\omega }^{{ \mathcal N }{ \mathcal E }}/\partial {{ \mathcal E }}^{\mathrm{nr}}$ by applying a range of large non-reversing electric fields, up to around 70 times that present under normal running conditions. The value of $\partial {\omega }^{{ \mathcal N }{ \mathcal E }}/\partial {{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$ was measured by applying a correlated laser power ${P}^{{ \mathcal N }{ \mathcal E }}$ in the state preparation and state readout beams with a magnitude corresponding to an applied ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$ that was up to 20 times that measured under normal operating conditions. These parameters were measured for multiple values of the magnetic field magnitude, ${{ \mathcal B }}_{z}$, for which different state readout laser beam polarisations were required. Due to known birefringent behaviour of the AC stark shift phases, we allowed for this possibility for all AC stark shift phase systematic errors. We measured $\partial {\omega }^{{ \mathcal N }{ \mathcal E }}/\partial {{ \mathcal E }}^{\mathrm{nr}}$ for both $\hat{k}\cdot \hat{z}=\pm 1$, but the ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$ systematic error was only discovered after the $\hat{k}\cdot \hat{z}=+1$ dataset and hence $\partial {\omega }^{{ \mathcal N }{ \mathcal E }}/\partial {{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$ was only monitored during the $\hat{k}\cdot \hat{z}=-1$ dataset. The ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$ systematic error during the $\hat{k}\cdot \hat{z}=+1$ dataset was determined from auxiliary measurements of the AC Stark shift phase. As described in section 3.2.5, ${{ \mathcal E }}^{\mathrm{nr}}(x)$ exhibits significant spatial variation along the beam-line axis, x. However the ${{ \mathcal E }}^{\mathrm{nr}}$ that was intentionally applied to determine $\partial {\omega }^{{ \mathcal N }{ \mathcal E }}/\partial {{ \mathcal E }}^{\mathrm{nr}}$ was spatially uniform, and hence these measurements were insensitive to the difference $({{ \mathcal E }}^{\mathrm{nr}}({x}_{\mathrm{prep}})-{{ \mathcal E }}^{\mathrm{nr}}({x}_{\mathrm{read}}))$ between the state preparation laser beam at x_prep and the state readout beam at x_read. For this reason, we deduced the systematic error proportional to the difference $({{ \mathcal E }}^{\mathrm{nr}}({x}_{\mathrm{prep}})-{{ \mathcal E }}^{\mathrm{nr}}({x}_{\mathrm{read}}))$ from auxiliary measurements of the AC Stark shift phase parameters, ${\alpha }_{{\rm{\Delta }},i}^{\mathrm{nr}}$.

In summary, the systematic errors proportional to ${{ \mathcal E }}^{\mathrm{nr}}$ and ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$ that were evaluated and subtracted from ${\omega }^{{ \mathcal N }{ \mathcal E }}$ to report a measured value of ${\omega }_{T}^{{ \mathcal N }{ \mathcal E }}$ can be expressed as

$$\begin{eqnarray}&&{\omega }_{{{ \mathcal E }}^{\mathrm{nr}}}^{{ \mathcal N }{ \mathcal E }}=\left(\displaystyle \frac{\partial {\omega }^{{ \mathcal N }{ \mathcal E }}}{\partial {{ \mathcal E }}^{\mathrm{nr}}}\right)\displaystyle \frac{1}{2}({{ \mathcal E }}^{\mathrm{nr}}({x}_{\mathrm{prep}})+{{ \mathcal E }}^{\mathrm{nr}}({x}_{\mathrm{read}}))+\displaystyle \frac{1}{\tau }({\alpha }_{{\rm{\Delta }},\mathrm{prep}}^{\mathrm{nr}}-{\alpha }_{{\rm{\Delta }},X}^{\mathrm{nr}}-{\alpha }_{{\rm{\Delta }},Y}^{\mathrm{nr}})\displaystyle \frac{1}{2}({{ \mathcal E }}^{\mathrm{nr}}({x}_{\mathrm{prep}})-{{ \mathcal E }}^{\mathrm{nr}}({x}_{\mathrm{read}})),\end{eqnarray} \tag{ 86 }$$

$$\begin{eqnarray}{\omega }_{{{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}}^{{ \mathcal N }{ \mathcal E }}=\left\{\begin{array}{cc}\tfrac{1}{\tau }{\displaystyle \sum }_{i\in \{\mathrm{prep},X,Y\}}{\beta }_{d{{\rm{\Omega }}}_{{\rm{r}}},i}^{\mathrm{nr}}\left(\tfrac{{{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}}{{{\rm{\Omega }}}_{{\rm{r}}}^{\mathrm{nr}}}\right) & \,\,\,\,\,\,\,\,(\hat{k}\cdot \hat{z})=+1\\ \left(\tfrac{\partial {\omega }^{{ \mathcal N }{ \mathcal E }}}{\partial {{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}}\right){{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }} & (\hat{k}\cdot \hat{z})=-1\end{array}\right.,\end{eqnarray} \tag{ 87 }$$

where $(\partial {\omega }^{{ \mathcal N }{ \mathcal E }}/\partial {{ \mathcal E }}^{\mathrm{nr}})$ and $(\partial {\omega }^{{ \mathcal N }{ \mathcal E }}/\partial {{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }})$ were monitored by IPVs (see section 3.1.2) throughout the dataset used for our reported result, and ${{ \mathcal E }}^{\mathrm{nr}}({x}_{\mathrm{prep}})$, ${{ \mathcal E }}^{\mathrm{nr}}({x}_{\mathrm{read}})$, ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }{ \mathcal E }}$, ${\alpha }_{{\rm{\Delta }},i}^{\mathrm{nr}}$, and ${\beta }_{d{{\rm{\Omega }}}_{{\rm{r}}},i}^{\mathrm{nr}}$ were obtained from auxiliary measurements. These two systematic errors account for almost all of the systematic offset that was subtracted from ${\omega }^{{ \mathcal N }{ \mathcal E }}$ to obtain ${\omega }_{T}^{{ \mathcal N }{ \mathcal E }}$ as described in section 5.10.

The correlated components of the state preparation and readout laser beam detunings are described in detail in section 3.2.4. Each detuning component was separately exaggerated and in some cases multiple components were simultaneously exaggerated. Most of the detuning terms in equation (27) were exaggerated to $\pm 2\pi \,\times 1$–2 MHz. No detuning or detuning correlation produced a significant shift in ${\omega }^{{ \mathcal N }{ \mathcal E }}$ other than ${{\rm{\Delta }}}^{{ \mathcal N }{ \mathcal E }}$ caused by ${{ \mathcal E }}^{\mathrm{nr}}$, discussed in section 5.2.6. In some cases, shifts in other phase channels were induced, but all shifts were consistent with well-understood AC Stark shift and asymmetry models described in sections 5.2.3 and 5.3. For example, the combination of nonzero ${{\rm{\Delta }}}^{{ \mathcal N }}$ and ${{\rm{\Delta }}}^{\mathrm{nr}}$ coupled to the ${ \mathcal B }$-dependent component of the AC stark shift phase (equation (77)) induces a significant shift in ${\omega }^{{ \mathcal N }{ \mathcal B }}$ (see equation (59)). Asymmetry correlations also resulted from these detuning correlations, but these were only manifested in channels odd with respect to $\tilde{{ \mathcal P }}$ and $\tilde{{ \mathcal R }}$, and hence had no plausible effect on ${\omega }^{{ \mathcal N }{ \mathcal E }}$. Because the YbF eEDM experiment [101] observed unexplained dependence of the measured eEDM value on state preparation radio-frequency detuning, we included a systematic error contribution from all detuning imperfections in our systematic error budget.

Similar to detuning imperfections, the state preparation and readout lasers could have imperfect pointing and correlated intensities. Ideally the laser propagation direction, $\hat{k}$, would have been parallel to the laboratory electric field. This would have diminished the amount of $\hat{z}$ polarised light experienced by the molecules, which could drive unwanted off-resonant transitions, and prevented stray retroflection from the ITO field plate surfaces. Using this ITO retroflection as a guide, we aligned $\hat{k}$ perpendicular to the field plate surface, and therefore parallel to $\hat{{ \mathcal E }}$, to within $\sim 3$ mrad. To test for errors related to imperfect pointing, both the state preparation and readout pointing misalignments were exaggerated in the x-direction to ±10 mrad, as was the relative pointing of the $\hat{X}$ and $\hat{Y}$ state readout beams. The vacuum windows and ∼3.8 cm wide holes in the magnetic shields prevented us from further misaligning the beams. To decouple pointing imperfections from detuning imperfections, the state preparation and readout laser frequencies were tuned to resonance after each pointing adjustment. No shift in ${\omega }^{{ \mathcal N }{ \mathcal E }}$ was observed and no systematic error contribution from pointing imperfections was included. Pointing imperfections were only observed to affect the signal asymmetry, as previously discussed in section 5.3.

Unlike laser pointing and detuning, there was no 'ideal' value for laser intensity. The state preparation and readout laser intensities were chosen such that we were driving optical pumping to completion on the $H\to C$ transition without producing unnecessary thermal stress on the field plates. We decreased each laser intensity by a factor of four to check that there was no variation in ${\omega }^{{ \mathcal N }{ \mathcal E }}$. We observed a nonzero ${{\rm{\Omega }}}_{{\rm{r}}}^{{ \mathcal N }}$ caused by the $\tilde{{ \mathcal N }}$-correlated seed power into the high-power fibre amplifiers and by Stark mixing between rotational levels in H and C as discussed in section 5.2.6. We exaggerated this imperfection by a factor of 20. Only ${\omega }^{{ \mathcal N }{ \mathcal B }}$ was shifted, consistent with our understanding of the ${ \mathcal B }$-correlated AC Stark shift phase. These intensity systematic error checks were not included in the systematic error budget.