# Electrostatic focusing of cold and heavy molecules for the ACME electron EDM
search

X. Wu1,2, P. Hu1, Z. Han1, D. G. Ang2, C. Meisenhelder2, G. Gabrielse3, J. M.
Doyle2, and D. DeMille1 1 Department of Physics, University of Chicago,
Chicago, IL 60637, USA  2 Department of Physics, Harvard University,
Cambridge, MA 02138, USA  3 Center for Fundamental Physics, Northwestern
University, Evanston, IL 60208, USA<EMAIL_ADDRESS>
###### Abstract

The current best upper limit for electron electric dipole moment (EDM),
$|d_{e}|<1.1\times 10^{-29}\,$e$\cdot$cm ($90$% confidence), was set by the
ACME collaboration in 2018. The ACME experiment uses a spin-precession
measurement in a cold beam of ThO molecules to detect $d_{e}$. An improvement
in statistical uncertainty would be possible with more efficient use of
molecules from the cryogenic buffer gas beam source. Here, we demonstrate
electrostatic focusing of the ThO beam with a hexapole lens. This results in a
factor of $16$ enhancement in the molecular flux detectable downstream, in a
beamline similar to that built for the next generation of ACME. We also
demonstrate an upgraded rotational cooling scheme that increases the ground
state population by $3.5$ times compared to no cooling, consistent with
expectations and a factor of $1.4$ larger than previously in ACME. When
combined with other demonstrated improvements, we project over an order of
magnitude improvement in statistical sensitivity for the next generation ACME
electron EDM search.

##  1 Introduction

The electron electric dipole moment (EDM), $d_{e}$, is an asymmetric charge
distribution along the electron’s spin. Searching for $d_{e}$ provides a
powerful probe for new physics beyond the Standard Model (SM) [1, 2]. The best
upper limit on its value, $|d_{e}|<1.1\times 10^{-29}\,$e$\cdot$cm ($90$%
c.l.), was set by the recent ACME measurement [3], which we refer to as ACME
II. This result severely constrains the parameter space for any new
$CP$-violating interactions mediated by particles in the energy range of
$3\sim 30\,$TeV [4]. Further improvement in the electron EDM search would test
many theories with new physics at even higher energies.

The ACME experiment is based on measuring spin precession of electrons in
thorium monoxide (ThO) molecules, which are present in the form of a
cryogenically cooled beam. The EDM search uses the metastable
$H\,^{3}\Delta_{1}$ state of ThO. This state possesses an effective intra-
molecular electric field of $\mathcal{E}\approx 78\,$GV/cm [5, 6], which
interacts with the electron EDM to produce the sought-for spin precession. To
further improve the ACME experiment, several upgrades are underway.

Envisioned already at the outset of the ACME program[7] but not previously
demonstrated, is the more efficient use of the molecules produced at the
source. Up to now, the ThO beams in ACME have been formed by simply trimming
the highly divergent beam with a series of aligned apertures. As a result,
only $\sim 0.04\%$ of the ThO produced in the initial beam could go through
the measurement region, dictated by the solid angle subtended by the detection
volume as seen from the source. One solution to increasing the number of
useful molecules is to focus the ThO beam with an electromagnetic lens, which
can exert a linear restoring force transverse to the beam and direct the
originally diverging molecular trajectories into the downstream detection
volume. Moreover, the upcoming version of ACME (referred to as ACME III) plans
to operate with a longer spin-precession region, designed for an increased
coherence time, based on the recently-measured longer lifetime of the $H$
state [8]. This upgrade of a longer beamline makes collimation from a
molecular lens even more important, since the detection volume is further
downstream. Recent spectroscopic studies on ThO [9] indicate that an
electrostatic lens could enhance molecular signal by focusing molecules in
their long-lived, metastable, and highly polarizable $Q\,^{3}\Delta_{2}$
electronic state. To perform the EDM search, molecules would be transferred
into the EDM-sensitive $H$ state after the lens, and the subsequent
experimental protocol would be the same as in ACME II.

Taming molecular beams via electrostatic interactions has been realized in
various forms, including the deceleration and trapping of polar molecules [10,
11, 12, 13]. Use of electrostatic lenses has also been applied previously, for
the first ammonia maser [14], in molecular scattering studies [15, 16], and
for precision measurements with molecules [17, 18, 19]. In this work, we
report the electrostatic focusing of a cryogenic buffer gas cooled beam (CBGB)
of ThO molecules, which gives rise to a factor of $16$ enhancement in the
detectable molecular flux in a test beamline similar to ACME III geometry. The
slow mean forward velocity of the beam, $\bar{v}_{f}$, and low temperatures of
the translational and rotational degrees of freedom (all standard properties
of CBGBs, as shown in various applications [20, 21, 22]), make our lens
unusually effective(see Section 3.3).

We also implement new and improved protocols for rotational cooling and single
quantum state transfer, which maximize the useful signal from beam focusing
and the preparation of molecules in the EDM-sensitive $H$ state. The
rotational cooling, which occurs upstream from the lens, accumulates molecular
population into the absolute ground state
$X\,^{1}\Sigma^{+}\,|JM\Omega=0,0,0\rangle$ 111$J$ and $M$ follow the usual
definition of angular momentum quantum numbers, and $\Omega=\Lambda+\Sigma$,
where $\Lambda$ and $\Sigma$ are the projection of total electronic orbital
angular momentum $L$ and total electronic spin $S$ on the internuclear axis,
respectively.. We develop a physically compact system that allows a relatively
short distance from the molecular source to the lens, hence providing a large
effective solid angle of molecules incident on the lens. To prepare molecules
for beam focusing, Stimulated Raman adiabatic passage (STIRAP) [23, 24]
transfers molecules into $|JM\Omega=2,2,-2\rangle$ in the $Q\,^{3}\Delta_{2}$
state, chosen for optimal lens operation. The STIRAP transfer from $X$ to $Q$
in ThO was recently demonstrated [9] to have $90\%$ efficiency for molecules
within the entire spatial and velocity capture range of the lens. The
combination of electrostatic focusing and these efficient quantum state
manipulations makes this molecular lens system a substantial upgrade for ACME
III.

The content of this paper is as follows. In Section 2 we briefly review the
properties of the $Q$ state, which enable efficient electrostatic focusing of
the ThO beam. Section 3 introduces an analytical model of the molecular lens,
which captures the principal concepts of lens design and performance. Section
4 describes the experimental setup. Section 5 discusses rotational cooling and
the STIRAP transfer. Finally, in Section 6 we present results of electrostatic
lensing of ThO in a realistic ACME beamline geometry, and compare with
expectations from Monte Carlo trajectory simulations.

##  2 Molecular state for lensing

Figure 1: (a) The lowest-energy electronic states of ThO ($v=0$ vibrational
states shown). Energy levels (transitions) relevant in this work are shown in
blue (with arrows). Estimated leading contributions from Hund’s case (a) terms
[25] are shown where relevant. (b) Stark shift vs. applied electric field
strength for selected rotational levels in the $Q$ and $X$ state. All $M\neq
0$ levels are doubly degenerate in an external $\mathcal{E}$-field. Levels
with $M\Omega<0$ (red solid curves) are low-field seeking. These levels,
present in the $Q$ state but not in $X$, exhibit a strong linear Stark shift
for small fields; deviation from linearity is substantial at larger fields
(when the Stark shift is comparable to the splitting to the nearest rotational
level). All levels with $M\Omega=0$ (hence all $X$ state levels) have weaker,
quadratic Stark shifts for small fields. All low-field seeking sublevels of
the $J=2$ states are shown for both $X$ and $Q$ states. The zero of energy is
set at the field-free $J=2$ state energy for both electronic states. The $J=3$
states (dashed lines) have higher zero-field energy due to rotational
splitting.

Among the known and predicted electronic states of ThO (Fig. 1(a)), we
identified the $Q\,^{3}\Delta_{2}$ state to be an ideal candidate for
molecular beam focusing. Its relevant properties include [9]:

  * •

A long lifetime, $\tau_{Q}>62\,$ms ($90\%$ c.l.), such that negligible loss
from decay occurs along the entire ACME beamline, where the total fly-through
time is $\approx 11\,$ms.

  * •

A strong optical transition to the $C\,^{1}\Pi_{1}$ state of ThO, which
enables STIRAP transfer of the population between $X$ and $Q$ via $C$ with
about $90\%$ efficiency in both directions.

  * •

A strong linear Stark shift (Fig. 1 (b)), with a measured molecule-frame
electric dipole moment, $d_{Q}\approx 4.1\,$D. At the maximum usable field
strength (due to onset of nonlinearity in the Stark shift),
$\mathcal{E}_{nl}\approx 30\,$kV/cm, the Stark interaction gives rise to a
trap depth of $T_{trap}\approx 1.6\,$K. This corresponds to a lens capture
velocity $v_{cap}=\sqrt{2k_{B}T_{trap}/m}\approx 10\,$m/s for ThO molecules
prepared in the low-field-seeking $Q\,|JM\Omega=2,2,-2\rangle$ level (where
$k_{B}$ is the Boltzmann constant and $m$ is the molecular mass).

We note briefly the disadvantages that would be encountered in using other
long-lived states of ThO for focusing. For example, the
$X\,|JM\Omega=2,0,0\rangle$ ground state has a nearly 10-fold weaker Stark
shift than the $Q\,|JM\Omega=2,2,-2\rangle$ state (see Fig. 1 (b)), making it
far less effective for electrostatic focusing. The $H\,^{3}\Delta_{1}$ state
has similar Stark shifts to the Q state. However, its much shorter lifetime of
$\sim 4\,$ms [8] would lead to significant loss from decay over the focusing
beam path. In addition, dramatically higher laser power is needed to populate
the H state via STIRAP [26], for the full distribution of molecules that can
be captured by the lens.

##  3 Analytic model of electrostatic hexapole lens

###  3.1 General principle

Electrostatic focusing of molecular beams is closely analogous to focusing
beams of light. Hence, widely-known properties of geometric ray optics with
thick lenses offer an intuitive guide to designing an electrostatic lens and
anticipating its performance [27]. The essential goal is to image the
molecular beam source at a location near where the molecules will be detected
for the EDM search, while achieving the highest possible numerical aperture.
This analytical model provides an accurate estimate of the basic lens design
parameters, which can then be optimized further using a detailed simulation of
molecular trajectories. Here we review the essentials for electrostatic
hexapole focusing, appropriate for molecular states with linear Stark shifts.

Figure 2: (a) Cross-section of the hexapole lens and its $\mathcal{E}$-field
strength contour lines. In our design, the lens radius $R=d_{0}\approx 2\,$cm,
where $d_{0}$ is the electrode diameter, and a typical voltage of $\pm
V\lesssim\pm 20\,$kV is applied on the electrodes. (b) $\mathcal{E}$-field
strength distribution along the two cut-lines in (a), between two opposite
electrodes and between two opposite gaps. Also shown is the analytical model
for an ideal hexapole field distribution. The deviations from the ideal
distribution here are analogous to astigmatism in optical lenses.

An electrostatic hexapole lens consists of $6$ electrodes arranged at the
vertices of a regular hexagon (Fig. 2 (a)). For technical simplicity we use
cylindrical electrodes. A DC voltage $\pm V$ is applied, with alternating
polarities on adjacent electrodes. This results in a quadratic field strength
distribution near the central axis of the hexapole (Fig. 2 (b)), given by
$\mathcal{E}(r)=(3V/R^{3})r^{2}$, where $r$ is the radial coordinate and $R$
is the radius of the molecular lens (i.e. distance from the center to the
nearest electrode edge).

ThO molecules in the $Q\,|JM\Omega=2,2,-2\rangle$ state that pass through the
hexapole field will experience a quadratically varying potential, $W_{st}(r)$,
due to the positive linear Stark shift:
$W_{st}(r)=-\mathcal{D}\mathcal{E}(r)=-(3\mathcal{D}V/R^{3})r^{2}$, where the
state-dependent dipole $\mathcal{D}=d_{Q}\frac{\Omega M}{J(J+1)}$. This gives
rise to a linear restoring force
$\mathbf{F}(\mathbf{r})=-\textrm{d}W_{st}/\textrm{d}r\,\hat{r}=-m\omega^{2}r\,\hat{r}$,
which drives molecules into sinusoidal motion in the radial direction, with
angular frequency

$\omega=\left(\frac{d_{Q}}{m}\frac{-\Omega
M}{J(J+1)}\frac{6V}{R^{3}}\right)^{1/2}.$ (1)

We can then write the molecule’s radial displacement $r(x)$ at any axial
displacement $x$ in the hexapole field, knowing its initial radial
displacement $r_{i}$, radial velocity $\dot{r_{i}}$, and axial velocity
$v_{x}$:

$r(x)=r_{i}\cos(px)+\left(\frac{\dot{r_{i}}}{v_{x}p}\right)\sin(px),$ (2)

where the lens constant $p\equiv\omega/v_{x}$ [27]. The evolution of molecular
trajectories from the source to the detector can now be described in terms of
optical parameters, analogous to the description of optical rays deflected by
a thick lens:

$\left(\begin{array}[]{cc}r\\\
\dot{r}/v_{x}\end{array}\right)=\left(\begin{array}[]{cc}1&b^{\prime}\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}1&0\\\
-f^{-1}&1\end{array}\right)\left(\begin{array}[]{cc}1&a^{\prime}\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}r_{i}\\\
\dot{r}_{i}/v_{x}\end{array}\right),$ (3)

where $f=[p\sin(pL)]^{-1}$ is the effective focal length of the hexapole lens,
$L$ is the electrode length, and $a^{\prime}$ ($b^{\prime}$) is the distance
from the object (image) plane to the principal plane near the hexapole lens
entrance (exit). The principal planes are at a depth $\delta a=\delta
b=[1-\cos(pL)]/[p\sin(pL)]$ into the lens, such that $a^{\prime}\equiv
a+\delta a$ and $b^{\prime}\equiv b+\delta b$ where $a$ ($b$) is the distance
from object (image) to the lens entrance (exit).

###  3.2 Boundary conditions

Besides the equations for the molecular trajectories, geometric and practical
constraints for the ACME III beamline need to be considered when designing the
lens.

  * •

We would like to set the distance from the source to the lens entrance to be
as small as possible, $a=a_{min}$, to obtain a maximum numerical aperture. In
practice, this is determined by how compact we can make the vacuum system at
the source (Sec. 4.1), the rotational cooling (Sec. 4.2), and the STIRAP
region (Sec. 4.3). For ACME III, we find $a_{min}=0.3\,$m is technically
feasible.

  * •

We would like the lens to capture as many trajectories as possible, hence the
maximal field strength, $\mathcal{E}_{max}$, should be roughly
$\mathcal{E}_{max}\approx\mathcal{E}_{nl}\approx 30\,$kV/cm where the Stark
shift starts to become nonlinear.

  * •

Everything within the transverse capture velocity of the lens, $v_{cap}$,
needs to actually make it inside the lens entrance. This sets the minimum lens
radius $R>R_{min}$, where $R_{min}\approx(v_{cap}/v_{x})\cdot
a_{min}+\phi_{ap}/2=1.7\,$cm, where $v_{x}\approx\bar{v}_{f}\approx 220\,$m/s
is the typical mean forward velocity of the ACME molecular beam, and
$\phi_{ap}=0.6\,$cm is the diameter of the first aperture after the source.

  * •

The maximum lens radius $R<R_{max}$ is constrained by the operational voltage
we apply while still keeping the maximum field strength near
$\mathcal{E}_{max}$. To ensure a robust system, we have found it convenient to
keep the long-term operational voltage below $\pm V^{*}\approx\pm 20\,$kV.
Thus, $R_{max}=3V^{*}/\mathcal{E}_{max}\approx 2\,$cm.

  * •

To maximize the molecular flux arriving at the detection volume, we would like
the lens to make an image somewhere between the entrance aperture and the
detection region of the spin precession measurement (at a distance $b_{1}$ and
$b_{2}$ away from the lens exit, respectively). Thus, the image-to-lens
distance, $b$, is constrained to $b_{1}\approx 0.6\,\textrm{m}<b<b_{2}\approx
1.6\,$m for the ACME III design.

These boundary conditions determine basic design parameters of the lens, or
constrain the ranges where the optimizations of these parameters can be
performed. As an initial estimate, we set an object distance $a=0.3\,$m, image
distance $b\approx 1\,$m, lens radius $R\approx 2\,$cm, and electrode voltage
$V=20\,$kV. From Eq. 1, we get an oscillation frequency $\omega=2\pi\times
91\,$Hz and the corresponding lens constant is $p=2.6\,\textrm{m}^{-1}$. Using

$\frac{1}{a^{\prime}}+\frac{1}{b^{\prime}}=\frac{1}{f},$ (4)

we can solve for the electrode length,

$L=\frac{1}{p}\,\textrm{cot}^{-1}\left(\frac{abp-1/p}{a+b}\right)=0.46\,\textrm{m},$
(5)

and the lens focal length,

$f=\frac{1}{p\sin(pL)}=\frac{\bar{v}_{f}}{\omega}\frac{1}{\sin((\omega/\bar{v}_{f})L)}=0.41\,\textrm{m}.$
(6)

For comparison, the result from Eqn. 5 is close to the optimized electrode
length, $L=0.53\,$m, obtained from a full numerical optimization taking into
account realistic spatial and velocity distributions at the molecular source,
the beamline geometry, and the nonlinearity in the Stark shift of the lensing
state (see Fig. 1 (b)). The forward velocity spread of the molecular beam
(nonlinear Stark shift) leads to the analogue of chromatic (spherical)
aberration in optical lenses.

###  3.3 Advantages of CBGB sources for molecular beam focusing

Equations 4 to 6 show that the lens performance depends critically on the
molecular source:

  * •

For a supersonic beam source, which has typically several times higher
$\bar{v}_{f}$, the electrode length and the focal length will increase almost
proportionally when the lens voltage is kept constant. To keep the same image-
to-lens distance, the lens entrance must be placed further from the source,
which reduces the solid angle subtended by the lens quadratically with
$\bar{v}_{f}$.

  * •

For a thermal source, which has a wide forward velocity spread $\Delta
v_{f}\sim\bar{v}_{f}$, beam focusing will suffer severely from chromatic
aberration, as $f\propto v_{f}$ approximately. In addition, since the lens
frequency $\omega$ depends on the internal state, only a tiny fraction of
rotational states populated in a thermal beam will be focused correctly.

Consequently, a CBGB molecular source operated in the hydrodynamic regime [28,
20] is ideal for electrostatic focusing, due to the small values of
$\bar{v}_{f}$ and $\Delta v_{f}/\bar{v}_{f}$, and its low rotational
temperature.

##  4 Experimental setup

###  4.1 Molecular source

Characterization of the electrostatic lens is performed using the ACME II
beamline [3], extended so that the source-to-detection length matches
approximately that planned for ACME III. ThO molecules are produced by laser
ablation of ThO2 in a cell containing Ne buffer gas at $\sim 17K$, and from
there extracted in a beam with mean forward velocity $\bar{v}_{f}\approx
220\,$m/s, $1\,\sigma$ velocity spread of $\Delta v_{f}/\bar{v}_{f}\approx
7\%$, and rotational temperature $T_{rot}\sim 4\,$K [29]. A $690\,$nm laser
tuned to the $X\to C$, $P(2)$ line ($P(J)$ denotes the $J\to J’=J-1$
transition) of ThO intersects the molecular beam just outside the buffer-gas
cell. The signal from absorption of this laser beam is used to normalize the
laser-induced fluorescence (LIF) signal downstream to cancel out slow drifts
in ablation yield. The same signal is used as the start time in a time-of-
flight (TOF) measurement from which the mean forward velocity of the molecular
beam $\bar{v}_{f}$ is extracted.

Figure 3: Schematic of the electrostatic hexapole lens test. The coordinate
system used throughout this manuscript is in the upper right. The location of
each component along the beamline axis is labelled at the bottom. The drawing
is not to scale, and emphasis is given to the critical components for the lens
test. A $6\,$mm diameter aperture at $x=2.5\,$cm acts as a $4\,$K cryogenic
pump for buffer-gas atoms. A $13\,$mm diameter aperture at $x=23\,$cm prevents
the direct line of sight for particles from hitting either the rotational
cooling optics or the lens electrodes. The distance from the source to the
entrance plane of the lens is $a=0.39\,$m, the shortest length possible with
the ACME II beam source. Another $2.5\,\textrm{cm}\times 2.5\,\textrm{cm}$
aperture at $x=200\,$cm prevents the molecular beam from hitting the electric
field plates.

###  4.2 State preparation region 1: rotational cooling

It takes two steps to transfer molecules from the $\sim 4\,$K Boltzmann
distribution over rotational states in the CBGB into the single, desired
lensing state $Q\,|JM\Omega=2,2,-2\rangle$. First, a rotational cooling stage
optically pumps molecules into the absolute ground state,
$X\,|JM\Omega=0,0,0\rangle$. Second, a STIRAP process transfers the population
from $X$ into $Q$.

The rotational cooling stage consists of a set of multi-pass optics which
allows the rotational cooling laser beams to intersect the molecule cloud $13$
times, within a $2.5\,$cm total interaction length. All optics are mounted
inside the vacuum chamber to keep the system compact and hence to minimize the
undesired expansion of the molecular beam before it reaches the lens entrance.
The in-vacuum multi-pass system consists of a pair of anti-reflection coated
fused silica (FS) right-angle prisms of $3.6\,$cm hypotenuse-length, offset by
$3$mm (see Fig. 3), bonded to a FS base plate using UHV compatible and UV-
curable epoxy 222Model: OPTOCAST 3553-LVUTF-HM. We measured the outgassing
rate to be $\lesssim 10^{-12}\,\textrm{torr}\cdot$L/s/cm2 with a residual gas
analyzer.. The $13$-pass optical path is first aligned outside the vacuum,
with uncured epoxy in place, then exposed to a UV lamp to fix the relative
position of each component. The assembly is subsequently heat treated to cure
sections that are possibly underexposed to the UV light, and vacuum-baked to
$150\,$ degree C before the final installation inside the beamline. The
exiting laser beam is directed by a pair of FS mirrors (epoxied on the same
base plate) though a vacuum window where the transmitted power can be
monitored. A pair of copper field plates is used to apply an
$\mathcal{E}$-field of between $0$ and $\approx 150\,$V/cm along the
$\hat{y}$-axis (perpendicular to both the laser propagation axis, $\hat{z}$,
and the beamline axis, $\hat{x}$).

###  4.3 State preparation region 2: STIRAP transfer to the lensing state

The STIRAP transfer from $X\,|JM\Omega=0,0,0\rangle$ to
$Q\,|JM\Omega=2,2,-2\rangle$ needs to avoid populating other $M$ and $\Omega$
sublevels of the $Q\,|J=2\rangle$ manifold. The degeneracy of $\Omega~{}(M)$
is lifted by applying homogenous fields of $|\mathcal{E}|\approx 50\,$V/cm
($|\mathcal{B}|\approx 5\,$G) along the $\hat{z}$-axis, generated by in-vacuum
field plates (coils). The $\sigma$-polarized STIRAP laser beams go through
slit openings on the $\mathcal{E}$-field plates before intersecting the
molecular beam. The level diagram of the STIRAP transfer is detailed in Sec.
5.2.

###  4.4 Electrostatic lens

The electrostatic hexapole lens system consists of $6$ stainless steel
cylindrical electrodes $53\,$cm in length, $19\,$mm in diameter, and with
$9.5\,$mm edge-to-edge spacing between adjacent electrodes (see Fig. 2(a)). Up
to $\pm 30\,$kV high voltage (HV) is applied, corresponding to an
$\mathcal{E}$-field as large as $63\,$kV/cm between adjacent
electrodes.333Multiple Geiger-Müller counters are installed around the lens
system to monitor possible X-ray generation during HV operation [30]. X-ray
count rates above the background level trigger a safety interlock to turn off
the HV system. The exact dimensions of the lens system are optimized for
maximal molecular flux at the detection volume downstream, as determined by
trajectory simulations that include non-ideal behaviors (e.g. molecular beam
velocity and spatial spreads, nonlinearity in the Stark shift of the
$Q\,|JM\Omega=2,2,-2\rangle$ state, etc.). The entire hexapole electrode
assembly sits on a home-built $5$-axis kinematic mount, which allows in situ
in-vacuum alignment.

###  4.5 State preparation region 3: STIRAP transfer back to the ground state

To perform the same spin precession measurement protocol as in ACME II,
molecules must enter the EDM measurement region in the
$X\,|JM\Omega=0,0,0\rangle$ state. Hence, molecules exiting the lens in
$Q\,|JM\Omega=2,2,-2\rangle$ should be transferred back to the ground state.
This can be accomplished via $Q\textendash C\textendash X$ STIRAP, performed
just after the end of the lens electrodes (similar in design to the pre-lens
STIRAP region). We note that $90\%$ efficient $Q\textendash C\textendash X$
STIRAP transfer was demonstrated in [9]. This paper focuses on the performance
of the state preparation and electrostatic focusing, so for all of the results
described here, this post-lens STIRAP region is not used and is marked as
optional in Fig. 3. Instead, we monitor only signals from molecules exiting
the lens in the $Q$ state.

###  4.6 Detection region

The molecules travel $\approx 150\,$cm from the end of the lens to the
downstream detection region, where molecular population in the $Q$ state is
measured by exciting the $Q\to I$ line at $746\,$nm (see Fig. 1 (a)) and
detecting the $512\,$nm $I\rightsquigarrow X$ LIF signal. Up to $500\,$mW of
$746\,$nm laser light is used in detection, expanded vertically (along the
$\hat{y}$-axis) to $\approx 3\,$cm $1/e^{2}$-diameter, and collimated along
the $\hat{x}$-axis to about $2\,$mm $1/e^{2}$-diameter.

##  5 State preparation before the lens: principle and results

###  5.1 Rotational Cooling

The rotational cooling scheme is designed to occupy minimal length along the
molecular beam and to be capable of saturating molecules in the wide
transverse spatial and velocity capture range of the lens. The former
minimizes the transverse size of the beam that contains molecules within the
lens capture range. The latter ensures the entire population from the lens
capture range can be accumulated in the ground state.

Figure 4: Rotational cooling scheme and result. (a) Level diagrams for
rotational cooling on the $X\to C$, $P(2)$ and $Q(1)$ lines of ThO. The
straight arrows depict the rotational pumping lines. The squiggle arrows
depict spontaneous decays, into the desired final state (red), and out of the
optical pumping system (blue). (b) Schematic of the optical components to
combine, amplify, and modulate the two laser beams driving the rotational
cooling transitions. (c) Modulation timing diagram of the polarizations,
frequencies, and $\mathcal{E}$-field applied during rotational cooling. (d)
Power dependence of the ground state population gain resulting from rotational
cooling. Measurements (red circles) are compared with the expected rotational
cooling gain (blue band) and the result obtained in ACME II (black dashed
line) [3]. The blue area indicates the $\pm 1\,\sigma$ uncertainty range in
the expected gain (see A). Error bars of the data points represent the
standard error in the mean over $10$ consecutive measurements, each consisting
of $64$ molecule pulses with rotational cooling on and then another $64$ with
rotational cooling off.

Two optical transitions at $690\,$nm are driven for rotational cooling: the
$P(2)$ and $Q(1)$ lines ($Q(J)$ denotes the $J\to J’=J$ transition) of the
$X\to C$ transition (Fig. 4 (a)). These serve to pump population initially in
$J=2$ and $1$ levels, respectively, into $J=0$ in $X$. The maximal pumping
efficiency on the $P(2)$ line is obtained at zero ambient $\mathcal{E}$-field,
under which the parity ($\mathcal{P}$) of the Lambda-doublet in the excited
$C\,|J^{\prime}=1\rangle$ state remains a good quantum number. Population
excited to $C\,|J^{\prime\mathcal{P}^{\prime}}=1^{-}\rangle$ (the upper level
of the doublet) can only decay to the target state
$X\,|J^{\mathcal{P}}=0^{+}\rangle$ or back to the initial state
$X\,|J^{\mathcal{P}}=2^{+}\rangle$. Pumping on the $Q(1)$ line, on the other
hand, requires an external $\mathcal{E}$-field to mix the parity eigenstates
in $C$. Otherwise, direct optical pumping from
$X\,|J^{\mathcal{P}}=1^{-}\rangle$ to $X\,|J^{\mathcal{P}}=0^{+}\rangle$ is
parity-forbidden.

When the $J=2$ and $J=1$ populations are fully depleted by this scheme, the
ground state population should increase by a factor of $3.7^{+0.4}_{-0.3}$,
based on the previously measured vibronic branching ratios from $C$ to all
other states [9] and the Hönl-London factors for the $X\to C$ transition (see
A). By adding another laser to pump the $X\,|J=3\rangle$ population (using the
$P(3)$ line), in principle another factor of $1.17$ increase in population
could be gained, but we do not include this extra step here.

To minimize the length needed for rotational cooling, the two laser beams
driving the $P(2)$ and $Q(1)$ lines are overlapped in space, then delivered
via a common optical fiber (Fig. 4 (b)) to the multi-pass optics (see Sec.
4.2) inside the beamline. For molecules to interact with both laser
frequencies, we switch rapidly between the two beams (Fig. 4 (c)). The $P(2)$
and $Q(1)$ beams are prepared in orthogonal linear polarizations, then
combined on a polarization beam splitter (PBS), before passing through a
polarization electro-optical modulator (EOM) and an output PBS. We switch the
EOM to allow either the $P(2)$ or the $Q(1)$ beam to pass through the output
PBS (at typically $15\,$dB extinction ratio) at any given time. For optimal
operation, the full frequency-switching cycle is performed in $16\,\mu$s. The
optical power of this overlapped laser beam is amplified from $10\,$mW to
$250\,$mW by a $690\,$nm semiconductor tapered amplifier. Synchronized with
the laser beam switching, a homogeneous $\mathcal{E}$-field is also switched
between $0$ and $\approx 150\,$V/cm in the region where the laser intersects
the molecular beam. To cover the $20\,$MHz FWHM (full width at half maximum)
transverse Doppler width of the molecule cloud, the optical beam is spectrally
broadened by a phase EOM, adding up to $90$ sidebands of $330\,$kHz spacing.
(This sideband spacing corresponds to the natural linewidth of $X\to C$
transition, where the $C$ state lifetime is $\tau_{C}\approx 480\,$ns [31,
32].) To access all $M$ sublevels in the $J=2$ and $J=1$ levels and to
eliminate Zeeman dark states, the polarization of the laser beam is switched
between $\hat{s}$ and $\hat{p}$ by a Pockels cell before it is delivered to
the beamline. Each polarization is on for $2\,\mu$s, corresponding to about
$4\times\tau_{C}$; this is found empirically to give optimal performance. The
rise and fall times of both the frequency and polarization switching pulses
are $<20\,$ns, ensuring a sudden (non-adiabatic) change between excitation
conditions, as needed for efficient optical pumping in this system. The
temporal modulation scheme for rotational cooling is summarized in Fig. 4 (c).
With these, we reduce the length of the rotational cooling region from
$\approx 20\,$cm in ACME II to $\approx 4\,$cm now.

Results when using this compact rotational cooling setup are shown in Fig. 4
(d). We define the gain factor as the ratio of signals from the
$X\,|JM\Omega=0,0,0\rangle$ population with the rotational cooling turned on
versus off. The data demonstrates that the rotational cooling gain saturates
at $\approx 3.5$, consistent with expectations based on known branching ratios
for decay from the $C$ state [9]. This is a factor of $1.4$ improvement from
the ACME II rotational cooling gain [3], likely due to the fast polarization
and frequency switching, better $\mathcal{E}$-field control, and higher laser
intensity available.

###  5.2 State transfer via STIRAP

Figure 5: Level diagrams for the STIRAP transfer from $X\,|J=0\rangle$ to
$Q\,|JM\Omega=2,2,-2\rangle$ via the intermediate $C\,|JM\Omega=1,1,-1\rangle$
level. Black line segments indicate the Stark-shifted energy levels, and green
line segments indicate the Zeeman shifts of each level. The splitting $W_{Z}$
is the Zeeman shift of $Q\,|JM\Omega=2,2,-2\rangle$, while $\Delta W_{S}$ is
the differential Stark-shift relative to the adjacent $|M|=1$ levels. $\delta$
is the one-photon detuning of the STIRAP laser beams.

The transfer of ThO population from its ground state to the
$Q\,|JM\Omega=2,2,-2\rangle$ state used for lensing is performed with near
unit efficiency using a stimulated Raman adiabatic passage (STIRAP) process.
In our previous studies [9], a similar STIRAP transfer between
$X\,|JM=0,0\rangle$ and $Q\,|JM=2,0\rangle$, via an intermediate state
$C\,|JM=1,0\rangle$, was fully characterized, with $90\%$ transfer efficiency
in either direction (i.e. $X\textendash C\textendash Q$ and $Q\textendash
C\textendash X$). Here, we need to transfer into the stretched state $M=2$
instead of $M=0$. This is realized by applying $\sigma$-polarization in both
the Pump ($690\,$nm) and Stokes ($1196\,$nm) beams for the $X\textendash
C\textendash Q$ STIRAP (Fig. 5). Since $\pm M$ states are degenerate with an
offset $\mathcal{E}$-field alone, to lift the degeneracy between $M=\pm 2$
states, and hence prevent STIRAP-transferring to the opposite stretched state
due to imperfection in laser polarizations, an offset $\mathcal{B}$-field of
$5\,$Gauss is applied in addition to the offset $\mathcal{E}$-field of
$50.1\,$V/cm. Both $\mathcal{B}$\- and $\mathcal{E}$-fields are along the
$\hat{z}$-axis. The former induces a Zeeman shift of $W_{Z}=9.6\,$MHz between
$M=2$ and $M=0$ levels, while the latter induces a differential Stark-shift of
$\Delta W_{S}=34.2\,$MHz between$|M|=2$ and $|M|=1$ levels. To avoid losses
due to spontaneous emission from the $C$ state, the one-photon detuning of the
STIRAP laser is blue-shifted by $\delta\approx 18\,$MHz from resonance.
Simulations based on prior results show that this STIRAP transfer should
saturate over $97\%$ of the population within the lens capture range.

To avoid population loss from $Q\,|JM\Omega=2,2,-2\rangle$ due to spin flips
as molecules travel through the volume between the STIRAP region to the
hexapole lens, a sufficiently strong $\mathcal{E}$-field needs to be present
along the trajectories of molecules. This is achieved using an arrangement of
planar electrodes where each STIRAP field plate is aligned with two electrodes
in the hexapole lens that have the same polarity (Fig. 9). This ensures no
reversal of electric field direction along the molecules’ trajectories during
the transit. Details of the electrode arrangement and the resulting field
strength distribution in the transit volume are elaborated in B.

##  6 Results from molecular beam focusing

The electric hexapole lens is optimized for maximal molecular flux at the new
ACME detection region (approximately $1.5\,$m downstream from the lens exit).
With ThO population prepared in the $Q\,|JM\Omega=2,2,-2\rangle$ state, we
characterize its performance and compare it with trajectory simulations. Three
key figures are benchmarked: the enhancement factor in the molecular flux, the
transverse velocity distribution, and the transverse spatial profile of the
lens-focused molecular beam in the detection region. The first of these is an
essential factor in projecting the total EDM sensitivity improvement, while
the other two determine the laser power requirements for preparation and
readout of the spin precession in the ACME III experiment.

###  6.1 Molecular flux

Figure 6: (a) Measured and simulated flux gain from molecular beam focusing
versus the voltage applied on the lens electrodes. The lens gain factor is
defined as the ratio of signals detected downstream at a given lens voltage
versus that with the lens turned off. Error bars of the data points are
derived from the standard error in the mean over $\approx 6$ consecutive
measurements, each consisting of $64$ molecule pulses with lens voltages
applied, and then with lens electrodes grounded. The results from trajectory
simulations are shown for a range of beam mean forward velocities,
$\bar{v}_{f}$, from $210$ to $220\,$m/s. (b) to (e) Simulated molecular
trajectories at selected lens voltages, projected onto the $x$-$z$ plane.
Trajectories that make it into the detection region are depicted by green
traces, while those with transverse velocity within lens capture range but do
not make it to the detection are depicted in red. Black vertical line segments
indicate fixed apertures for the molecular beam.

The molecular flux enhancement from electrostatic focusing is given by the
ratio between the detected molecular signal in the downstream region, with
high voltage applied to the lens electrodes versus with the electrodes
grounded. This molecular signal is taken at the maximum probe laser intensity,
which saturates over $99\%$ of the Doppler spread of the $Q\to I$ transition,
and after integrating over the temporal profile of each molecular pulse. The
lens gain factor is plotted in Fig. 6 (a), as a function of electrode voltage.
As shown there, the largest molecular flux is obtained with $\pm 14\,$kV
applied on the hexapole electrodes, yielding an enhancement factor of
$16.2(6)$.

The dependence of the measured enhancement factor with respect to the voltage
applied on the lens is consistent with expectations based on our trajectory
simulations. The simulation takes into account the realistic spatial
distribution of the molecular beam, its transverse and forward velocity
spreads, and constraints in beamline geometry. The simulation result, which
includes no fitting parameters, is shown in Fig. 6 (a). The good agreement
with experimental data is obtained when the exact nonlinearity of the lensing
state’s Stark shift is included in the simulation [33]. The range of simulated
results is set by the uncertainty in the molecular beam mean forward velocity
$\bar{v}_{f}$, as determined from the in situ TOF measurement between the
absorption signal upstream and the LIF detection downstream. We observe a
small difference in the TOF signal between when the lens is operated at the
optimal signal (at $\pm 14\,$kV), corresponding to $\bar{v}_{f}=220\,$m/s, and
when the lens is turned off, corresponding to $\bar{v}_{f}=210\,$m/s. This
$10\,$m/s difference is taken as the uncertainty in $\bar{v}_{f}$.

To make sense of the dependence of the gain factor versus applied lens
voltage, we take a closer look at the simulated trajectories at selected
voltages on the lens, starting from the same initial sample distributions.
They are shown in Fig. 6 (b) to (e). The number of green traces shown in each
plot is proportional to the predicted integrated signal at the corresponding
voltage. For comparison, Fig. 6 (b) and (d) also plot trajectories (at reduced
line density for clarity) that are within the lens capture range at $\pm
14\,$kV, but do not make it into the detection region. At $0\,$kV (Fig. 6
(b)), the molecules follow ballistic trajectories, since no focusing force is
applied. The least number of molecules arrive at the detection region here,
due to the beam divergence. As the lens voltage is increased, more
trajectories are bent towards the forward direction, thanks to the increasing
focusing force. At $\pm 9\,$kV (Fig. 6 (c)), the trajectories become nearly
collimated after leaving the molecular lens, but the number of trajectories
arriving at the detection region is not yet optimized. At $\pm 14\,$kV (Fig. 6
(d)), the molecular lens is making an image of the beam source right in front
of the molecular beam aperture (at the entrance of the spin precession region,
see Fig. 3), according to Eqn. 4, but blurred by aberrations. This yields the
maximal detected signal, which corresponds to about $38\%$ of the transversely
slow trajectories (those within the velocity capture range of the lens) out of
the beam source, and about $95\%$ of those transversely slow trajectories
arriving at the molecular lens. Further increasing the lens voltage causes
over-focusing. At $\pm 19\,$kV (Fig. 6 (e)), the image plane shifts visibly
upstream, to about $60\,$cm before the molecular beam aperture according to
Eqn. 4. Due to clipping of the over-focused trajectories on this aperture, the
detected signal decreases to $\sim 70\%$ of the optimal value.

###  6.2 Transverse velocity distribution

Figure 7: Measured transverse Doppler spectra of the molecular beam in the
detection region, at selected lens voltages. Each spectrum is proportional to
the transverse ($\hat{z}$) velocity distribution of the beam, where $1\,$MHz
detuning corresponds to $0.75\,$m/s velocity. Error bars of the data points
represent the standard error in the mean over $\approx 4$ consecutive
measurements, each consisting of $64$ molecule pulses. The range of
predictions from simulated trajectories is set by the uncertainty in
$\bar{v}_{f}$ of the molecular beam, as in Fig. 6 (a).

To quantitatively confirm more details of the molecular trajectories, the
frequency of the probe laser is scanned to determine the distribution of
molecular transverse velocities, via the Doppler shift, at various lens
voltages. These Doppler spectra are then compared with expectations based on
the trajectory simulations. From the trajectory plots in Fig. 6 (b) and (c),
we anticipate that the velocity distribution should narrow when the lens
voltage increases from $0$ to $\pm 9\,$kV, as the molecular beam gets more
collimated and molecular trajectories becomes almost parallel. When the
voltage increases further to $\pm 14$ and $\pm 19\,$kV, we expect the
distributions to widen again, as the image plane moves upstream of the
detection region (Fig. 6 (d) to (e)).

The experimental results confirm both the expected lineshapes and the relative
heights of the signal predicted by the simulations. The simulated Doppler
spectra at the four selected lens voltages are plotted, along with the
measurements, in Fig. 7. The simulation curves shown are a convolution between
the lineshapes predicted by trajectory simulation and that of the probe laser,
which is stabilized to typically $1\,$MHz. Only one free parameter — a global
vertical scaling on the simulation curves — is used for all four plots in Fig.
7 (a) to (d). The good agreement in both the lineshapes and relative signal
heights across the different lens voltage demonstrates that the numerical
model captures very well the dynamics of molecular beam focusing.

It is useful to compare the transverse Doppler width with that obtained in
ACME II, when the electric focusing was not available. This ratio of widths is
important to understand how much additional laser power will be needed,
relative to before, to saturate the probe signal across the entire velocity
distribution when the lens is in operation. In ACME II using a probe laser at
$703\,$nm, the $1\,\sigma$ Doppler width was $2.1\,$MHz [34]. The $1\,\sigma$
Doppler width here at the optimal beam focusing (Fig. 7 (c)) is $2.3\,$MHz.
After deconvolving the laser linewidth and accounting for the different probe
wavelength here, this is equivalent to a $1\,\sigma$ width of $2.4\,$MHz at
$703\,$nm. Thus, at optimized lens voltages, we will need to take into account
the $\approx 15\%$ wider transverse velocity distribution when preparing and
probing the spin precession in ACME III.

###  6.3 Transverse spatial distribution

Figure 8: Transverse spatial distributions of the molecular beam along the
transverse ($\hat{y}$) direction, at selected lens voltages. Error bars of the
data points represent the standard error in the mean over $\approx 4$
consecutive measurements, each consisting of $64$ molecule pulses. The range
of predictions from trajectory simulations is set by the uncertainty in
$\bar{v}_{f}$ of the molecular beam.

To characterize the molecular beam size, a spatial scan of the probe laser is
performed. Here, the probe laser beam is expanded vertically (along $\hat{y}$)
to about $3\,$cm $1/e^{2}$ diameter, then clipped to $5\,$mm size with a thin
slit. This thin slit is scanned along $\hat{y}$ in steps. The fluorescence
signal detected at each vertical step, corrected for the spatially dependent
photon collection efficiency444The inhomogeneity in photon collection
efficiency results from the fact that the fluorescence collection optics are
mounted only above the molecular beam. The functional form of the collection
efficiency versus $y$ position is from ray optics simulations, using the known
geometry of the collection optics., is plotted in Fig. 8 at lens voltages of
$0$, $\pm 9$, $\pm 14$, and $\pm 19\,$kV, alongside the predictions from
trajectory simulations. The predictions take into account the non-saturated
optical transition at the extreme values of the $y$-position (due to the
finite extent of the expanded laser beam), as well as the finite height of the
clipped laser beam. Only one global scaling parameter is applied for all $4$
plots of predictions in Fig. 8. Thus, the experimental data and numerical
predictions agree very well for both the spatial distribution and the overall
intensity of the molecular beam.

The narrowest spatial profile is obtained when the signal from lens focusing
is maximized (at $\pm 14\,$kV); here, the FWHM of the distribution is about
$1.3\,$cm (after deconvolving the finite height of the probe laser beam). This
agrees with the general observation from the sample trajectories in Fig. 6 (b)
to (e), where the beam’s spatial profile in the detection region is also the
narrowest at the condition for the optimal flux. This transverse spatial
profile is different from that in ACME II, performed without beam focusing,
which had a flat-top distribution (similar to the shape in Fig. 8 (a), taken
at lens off) with $2.3\,$cm FWHM. This narrower transverse spatial extent of
the molecular beam helps both to lower the laser power required to saturate
the detection in ACME III, and to in principle reduce the systematic error
associated with the transverse gradients of magnetic and electric fields
witnessed in ACME II [3].

##  7 Conclusions and outlook

We have demonstrated a molecular lens system that provides an enhancement of
over one order of magnitude in useful flux from a cold beam of heavy ThO
molecules, as needed for improving the ACME EDM search. We employ a new
quantum state preparation scheme that achieves a factor of $1.4$ improvement
relative to the rotational cooling scheme in ACME II. This enhanced ground
state population is then transferred into the quantum state used for lensing
via STIRAP. The molecular beam is focused with an electrostatic hexapole lens,
yielding a factor of $16$ enhancement in signal size detected from the $Q$
state, compared to when the lens is turned off, in a test beamline similar to
the ACME III geometry. 555It is useful to compare this with other
configurations, using our validated trajectory simulations. For example, in
the case of no lens but a realistic beamline of shorter length connecting the
beam source and the expanded EDM measurement region of ACME III, the
electrostatic focusing gives a factor of $12$ enhancement in flux. In the case
where the EDM measurement region was not expanded but instead remained the
same as in ACME II, the molecular lens would provide a factor of $11$ higher
flux. Under these conditions, the transverse velocity and spatial
distributions of the lens-focused beam are found to be closely comparable to
those used in ACME II. This implies that no significant increase of laser
power is required to saturate the preparation and readout of the spin
precession, compared to that used in ACME II.

This flux-enhancing system will be incorporated into the upcoming ACME III
measurement. Taking into account the $90\%$ STIRAP efficiency for both pre-
lens ($X\textendash C\textendash Q$) and post-lens ($Q\textendash C\textendash
X$) state transfer, the molecular beam focusing and improved rotational
cooling demonstrated here will enhance the signal size by a factor of $\approx
19$ in ACME III. If shot-noise limited detection can again be achieved such
that the sensitivity grows as the square root of the flux, this will
correspond to an improvement in the EDM statistical sensitivity by $4.4$
times. This improvement consists of two parts: a factor of $3.5$ from the
rotational cooling plus lens system if it was optimized for the shorter
beamline in ACME II; and a factor of $1.3$ from additional beam collimation by
the lens in the expanded spin-precession region of ACME III. When combined
with other demonstrated upgrades, including the longer coherence time to take
advantage of the recently-measured $H$ state lifetime [8], detectors with
higher efficiency [35], and a reduction of technical noise [36], we project an
improved statistical sensitivity to the electron EDM by potentially $30$ times
compared to the current best limit [3] from ACME II.

This work was supported by the National Science Foundation, the Gordon and
Betty Moore Foundation, and the Alfred P. Sloan Foundation. We thank Oskari
Timgren, Vincent Bernado, James Chow, Stan Cotreau, and Jesse Ward for their
tremendous technical support. We also thank Zack Lasner, Cristian Panda,
Nicholas Hutzler, and Takahiko Masuda for helpful discussions.

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##  Appendix A Data-based limit of rotational cooling gain

In this section, we derive a data-based upper bound for the population gain in
the ground state resulting from the rotational cooling. In an ensemble of ThO
molecules in the $X$ electronic state and whose rotational energy levels
follow a thermal distribution at temperature $T_{rot}$, the probability to
find a particle in the $J$-th rotational energy level is

$P_{J}=\frac{1}{Z}(2J+1)\exp\left(-\frac{J(J+1)B_{R}}{k_{B}T_{rot}}\right),$
(7)

where $Z=\sum_{J}{(2J+1)\exp\left(-J(J+1)B_{R}/(k_{B}T_{rot})\right)}$ is the
partition function, and $B_{R}=0.33264\,$cm-1 is the rotational constant. At
$T_{rot}=3.8(5)\,$K [29], the distribution in the lowest few rotational energy
levels are given in Table 1.

Table 1: Thermal rotational distribution of the $X$ state of ThO at
$T_{rot}=3.8\,$K.

  * Rotational level $J$ | $0$ | $1$ | $2$ | $3$ | $4$ | $\geq 5$
---|---|---|---|---|---|---
Relative population $P_{J}$ [$\%$] | 12 | 28 | 28 | 19 | 8.8 | 4.0

We have previously determined the vibronic branching ratio for the decay
$C\,|\nu^{\prime}=0\rangle\rightsquigarrow X\,|\nu=0\rangle$ in ThO to be
$\xi=0.74(6)$ [9], where $\nu$ is the vibrational quantum number. Based on
Hönl-London factors, the rotational branching ratios, $\eta_{i}$, for
$C\,|J^{\prime\mathcal{P}^{\prime}}=1^{-}\rangle\rightsquigarrow
X\,|J^{\mathcal{P}}=i^{+}\rangle$ (where $i=0$ or $2$) are $\eta_{0}=2/3$ and
$\eta_{2}=1/3$. In addition, based on the parity selection rule, spontaneous
decay of $C\,|J^{\prime\mathcal{P}^{\prime}}=1^{-}\rangle\rightsquigarrow
X\,|J^{\mathcal{P}}=1^{-}\rangle$ is forbidden, while decay of
$C\,|J^{\prime\mathcal{P}^{\prime}}=1^{+}\rangle\rightsquigarrow
X\,|J^{\mathcal{P}}=1^{-}\rangle$ has $\eta_{1}=1$.

When optically pumping on the $P(2)$ line with no external $\mathcal{E}$-field
applied, the probability to decay to $X\,|J=i\rangle$ (where $i=0$ or $2$) is
$p_{i}=\xi\cdot\eta_{i}$. Thus, pumping on the $P(2)$ transition alone would
ideally increase the ground state $X\,|J=0\rangle$ population by

$\displaystyle P_{2,0}$ $\displaystyle=$ $\displaystyle
p_{0}P_{2}+p_{0}(P_{2}p_{2})+p_{0}(P_{2}p_{2}^{2})+p_{0}(P_{2}p_{2}^{3})+\cdots$
$\displaystyle=$ $\displaystyle p_{0}P_{2}\frac{1}{1-p_{2}}$ $\displaystyle=$
$\displaystyle 18\,\%~{}(\textrm{of the total rotational population}).$

The optical pumping on $Q(1)$ takes place with an external field
$\mathcal{E}=150\,$V/cm applied, which almost fully mixes the opposite parity
levels in $C\,|J^{\prime}=1\rangle$. This modifies the rotational branching
ratios in the decay $C\,|J^{\prime}=1\rangle\rightsquigarrow X\,|J\rangle$ to
be $\eta^{*}_{J}=1/2\cdot\eta_{J}$. The total branching ratio into
$X\,|J\rangle$ is then $p_{J}^{*}=\xi\cdot\eta_{J}^{*}$. The population that
lands in $X\,|J=2\rangle$ will be pumped by the $P(2)$ transition in
subsequent cycles. (This is possible because we switch between the $Q(1)$ and
$P(2)$ pumping configurations rapidly, and each molecule experiences each
laser frequency multiple times during rotational cooling.) Overall, optical
pumping on the $Q(1)$ transition (jointly with pumping on the $P(2)$ line)
would ideally increase the $X\,|J=0\rangle$ population by

$\displaystyle P_{1,0}$ $\displaystyle=$ $\displaystyle
p_{0}^{*}P_{1}+p_{0}^{*}(P_{1}p_{1}^{*})+p_{0}^{*}(P_{1}p_{1}^{*2})+p_{0}^{*}(P_{1}p_{1}^{*3})+\cdots$
$\displaystyle+p_{0}(P_{1}p_{2}^{*})+p_{0}(P_{1}p_{2}^{*})p_{2}+p_{0}(P_{1}p_{2}^{*})p_{2}^{2}+p_{0}(P_{1}p_{2}^{*})p_{2}^{3}+\cdots$
$\displaystyle+p_{0}(P_{1}p_{1}^{*}p_{2}^{*})+p_{0}(P_{1}p_{1}^{*}p_{2}^{*})p_{2}+p_{0}(P_{1}p_{1}^{*}p_{2}^{*})p_{2}^{2}+\cdots$
$\displaystyle+p_{0}(P_{1}p_{1}^{*2}p_{2}^{*})+p_{0}(P_{1}p_{1}^{*2}p_{2}^{*})p_{2}+p_{0}(P_{1}p_{1}^{*2}p_{2}^{*})p_{2}^{2}+\cdots$
$\displaystyle+\cdots$ $\displaystyle=$ $\displaystyle
p_{0}^{*}P_{1}\frac{1}{1-p_{1}^{*}}+p_{0}(P_{1}p_{2}^{*})\frac{1}{1-p_{2}}\frac{1}{1-p_{1}^{*}}$
$\displaystyle=$ $\displaystyle 14\,\%~{}(\textrm{of the total rotational
population}).$

Here, the first line takes into account the population falling into
$X\,|J=1\rangle$ in the first and subsequent spontaneous decays, and the
$n$-th line ($n>1$) accounts for the population landing in $X\,|J=2\rangle$
after the $(n-1)$-th optical pumping cycle, etc. They can all be added in
geometric series.

Thus, the expected result for the rotational cooling gain involving initial
population in the $J=1$ and $J=2$ levels is
$G\equiv(P_{0}+P_{1,0}+P_{2,0})/P_{0}=3.7^{+0.4}_{-0.3}$, where the error bars
are the quadrature sum of the uncertainties from the vibronic branching ratio,
$\Delta G_{\xi}=^{+0.4}_{-0.3}$, and the rotational temperature in the
Boltzmann distribution, $\Delta G_{T}=^{+0.2}_{-0.1}$.

##  Appendix B Electrode configuration between STIRAP region and lens
electrodes

To prevent population loss due to nonadiabatic transitions from
$Q\,|JM\Omega=2,2,-2\rangle$ into other $M$ sublevels, we need the
$\mathcal{E}$-field strength to be nonzero everywhere along the molecules’
pathway from the STIRAP region to the lens region. Our electrode configuration
to ensure this is shown in Fig. 9 (a), where the field plates are aligned with
the two hexapole electrodes of the same polarities. In this case, the
$\mathcal{E}_{z}$ component always stays in the same direction along the
molecules’ pathway. The resulting $\mathcal{E}$-field strength distribution in
the transit region is plotted in Fig. 9 (b).

Figure 9: Two arrangements of STIRAP field plates with respect to the hexapole
lens electrodes. (a) Each field plate is aligned with the two hexapole
electrodes that have the same polarity. In (b) is the corresponding field
strength contour in a cross-section plane in the transit region between field
plates and hexapole electrodes.