# Cross-calibration of atomic pressure sensors and deviation from quantum
diffractive collision universality for light particles

Pinrui Shen1, Erik Frieling1, Katherine R. Herperger1, Denis Uhland1, Riley A.
Stewart1, Avinash Deshmukh1, Roman V. Krems2, James L. Booth3, and Kirk W.
Madison1 1Department of Physics and Astronomy, University of British Columbia,
6224 Agricultural Road, Vancouver, B.C., V6T 1Z1, Canada  2Department of
Chemistry, University of British Columbia, 6224 Agricultural Road, Vancouver,
B.C., V6T 1Z1, Canada  3Department of Physics, British Columbia Institute of
Technology, 3700 Willingdon Avenue, Burnaby, B.C. V5G 3H2, Canada.

###### Abstract

The total room-temperature, velocity-averaged cross section for atom-atom and
atom-molecule collisions is well approximated by a universal function
depending only on the magnitude of the leading order dispersion coefficient,
$C_{6}$. This feature of the total cross section together with the universal
function for the energy distribution transferred by glancing angle collisions
($p_{\rm{QDU6}}$) can be used to empirically determine the total collision
cross section and realize a self-calibrating, vacuum pressure standard. This
was previously validated for Rb+N2 and Rb+Rb collisions. However, the post-
collision energy distribution is expected to deviate from $p_{\rm{QDU6}}$ in
the limit of small $C_{6}$ and small reduced mass. Here we observe this
deviation experimentally by performing a direct cross-species loss rate
comparison between Rb+H2 and Li+H2 and using the ab initio value of
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}$. We find a velocity
averaged total collision cross section ratio,
$R=\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}:\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}=0.83(5)$.
Based on an ab initio computation of
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}=3.13(6)\times 10^{-15}$
m3/s, we deduce
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}=3.8(2)\times 10^{-15}$
m3/s, in agreement with a Rb+H2 ab initio value of
$\langle\sigma_{\mathrm{tot}}v\rangle_{\mathrm{Rb+H_{2}}}=3.57\times
10^{-15}\mathrm{m}^{3}/\mathrm{s}$. By contrast, fitting the Rb+H2 loss rate
as a function of trap depth to the universal function we find
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}=5.52(9)\times 10^{-15}$
m3/s. Finally, this work demonstrates how to perform a cross-calibration of
sensor atoms to extend and enhance the cold atom based pressure sensor.

###### pacs:

##  I Introduction

Laser cooled atoms confined in a magnetic or magneto-optical trap can be used
as sensors of absolute pressure and particle flux measurements in vacuum Booth
_et al._ (2011); Madison (2012); Arpornthip  _et al._ (2012); Yuan  _et al._
(2013); Moore  _et al._ (2015); Makhalov  _et al._ (2016); Makhalov and
Turlapov (2017); Scherschligt  _et al._ (2017, 2018); Xiang  _et al._ (2018);
Booth  _et al._ (2019); Shen  _et al._ (2020, 2021); Barker  _et al._ (2021);
Zhang  _et al._ (2022); Ehinger  _et al._ (2022). For example, the background
gas density or pressure can be determined by observing loss of trapped atoms
induced by collisions with ambient atoms or molecules. In general, the
quantitative determination of the background gas density or pressure requires
the knowledge of the thermally averaged total collision cross section between
the trapped atoms and all atoms or molecules in the background gas.

The total collision cross sections can be obtained from quantum scattering
calculations Makrides  _et al._ (2019, 2020, 2022a, 2022b). However, such
calculations are affected by the uncertainty in the underlying potential
energy surfaces (PESs) and are limited to molecular species with a small
number of active degrees of freedom. As an alternative, the total collision
cross sections can be obtained from measurements with a background gas of
known density. However, this requires an independent calibration of gas
density.

We have recently demonstrated a different approach to determine the collision
cross section by measuring the dependence of loss of trapped atoms on the trap
depth. Such measurements rely on the universal function ($p_{\rm{QDU6}}$) for
the energy distribution imparted to the sensor atom by quantum diffractive
collisions mediated by van der Waals interactions Booth  _et al._ (2019). This
was used to demonstrate a cold-atom based, self-calibrating, primary vacuum
pressure standard Booth  _et al._ (2019); Madison  _et al._ (2018); Shen  _et
al._ (2020, 2021). The validity of this pressure standard was proven for Rb+N2
collisions by comparison with an orifice flow pressure standard using an ion
gauge calibrated for N2 using both standards. The underlying principle, the
universality of quantum diffractive collisions as manifested in the
universality of the $p_{\rm{QDU6}}$ function, has also been independently
validated for Rb+Rb collisions Stewart  _et al._ (2022).

This collision universality relies on the collision cross section being
dominated by scattering determined by the long range part of PES,
parameterized by $C_{6}$, and on the velocity averaging of the cross-section
inherent in the trap loss measurement process. The velocity averaging removes
the oscillatory variation of the cross section with collision energy. These
“glory” oscillations arise from interference between the scattering amplitudes
due to the short- and long-range parts of the interaction potential Child
(1974). Therefore, deviations from the universal prediction are expected in
the limit of small reduced mass and small $C_{6}$ Booth  _et al._ (2019). For
collisions with a small reduced mass, the glory oscillation amplitude and
period can become so large that the total cross section does not average to
the underlying trend set by the long-range interaction. This effect is
compounded by the relatively small atomic polarizabilities of low mass
collision partners yielding relatively low $C_{6}$ values, enhancing the
effects of the core repulsion and compromising the universal description.

The main goal of this work is to obtain experimental evidence for the
deviation of the total collision cross section for light collision partners
from the values derived from the universal $p_{\rm{QDU6}}$ function. In order
to do this, we require an independent measurement of the total collision cross
section. We consider Rb+H2 collisions and develop an experiment to obtain the
collision cross section independently of $p_{\rm{QDU6}}$ by cross-calibration
with measurements for another trapped sensor atom. Specifically, we present
measurements utilizing Rb and Li sensor atoms, both exposed to the same H2
gas. We then use known information on Li+H2 collision cross sections to obtain
the total collision cross section for Rb+H2 collisions. The results obtained
experimentally are confirmed by rigorous quantum scattering calculations. We
show that the results thus obtained are inconsistent with the value of the
cross section obtained from the $p_{\rm{QDU6}}$ function, thus demonstrating
the departure of Rb+H2 diffractive collisions from universality.

##  II Review of QDU

The physical system considered here involves trapped atoms (Rb and Li) as
probe (sensor) particles of an ambient atomic or molecular gas at room
temperature. We define the total collision cross section $\sigma_{\rm tot}$ as
the sum of cross sections for elastic and all possible inelastic collisions
between trapped sensor atoms and atoms or molecules in the background gas.
Under certain conditions, as described in Refs. Booth  _et al._ (2019); Shen
_et al._ (2020), the total collision rate is (universal) determined by the
leading term in the long-range interaction between neutral collision partners
$C_{6}$ (where $C_{6}$ is an orientation angle averaged value in the case of
molecules whose interactions are anisotropic) as follows:

$\displaystyle\Gamma_{\rm{tot}}=n_{\mathrm{bg}}\langle\sigma_{\rm{tot}}\,v\rangle$
(1)

with

$\displaystyle\langle\sigma_{\rm{tot}}\,v\rangle$ $\displaystyle\approx$
$\displaystyle
10.45812\left(\frac{C_{6}}{\hbar}\right)^{\frac{2}{5}}\left(\frac{k_{\rm{B}}T}{m_{\mathrm{bg}}}\right)^{\frac{3}{10}}$
(2)
$\displaystyle+6.71531\left(\frac{C_{6}}{\hbar}\right)^{\frac{1}{5}}\left(\frac{m_{\mathrm{bg}}}{k_{\rm{B}}T}\right)^{\frac{1}{10}}\left(\frac{\hbar}{\mu}\right),$

where $n_{\mathrm{bg}}$ is the density of the background gas, $\mu$ is the
reduced mass of the collision pair, $m_{\mathrm{bg}}$ is the mass of the
background particle, and the angle brackets denote the average over the
collision velocities with the Maxwell-Boltzmann distribution for room
temperature (300 K).

The measurable depletion of sensor atoms from a trap is quantified by the loss
rate $\Gamma_{\rm{loss}}$ and can be written as

$\displaystyle\Gamma_{\rm{loss}}(U)=n_{\mathrm{bg}}\langle\sigma_{\rm{loss}}(U)\
v\rangle,$ (3)

where $\sigma_{\rm loss}$ is the total cross section for collisions that eject
sensor atoms from the trap. The loss rate $\Gamma_{\rm{loss}}$ depends on the
trap depth $U$. As $U\rightarrow 0$,
$\left<\sigma_{\rm{loss}}\,v\right>\rightarrow\left<\sigma_{\rm{tot}}\,v\right>$
and $\Gamma_{\rm{loss}}\rightarrow\Gamma_{\rm{tot}}$. In this limit, every
collision liberates a trapped sensor atom.

As was shown in Refs. Booth  _et al._ (2019); Shen  _et al._ (2020), the trap
depth dependence of $\Gamma_{\rm{loss}}$ (for small values of $U$) is a
universal function

$\displaystyle\langle\sigma_{\rm{loss}}(U)\ v\rangle$ $\displaystyle\approx$
$\displaystyle\langle\sigma_{\rm{tot}}\,v\rangle\left[1-\sum_{j=1}^{\infty}\beta_{j}\left(\frac{U}{U_{\rm{d}}}\right)^{j}\right],$
(4) $\displaystyle\approx$
$\displaystyle\langle\sigma_{\rm{tot}}\,v\rangle\left[1-p_{\rm{QDU6}}\left(\frac{U}{U_{\rm{d}}}\right)\right]$

where the coefficients $\beta_{j}$ are given in Tab. 1 and $U_{\rm{d}}$ is the
characteristic energy scale for quantum diffractive collisions,

$\displaystyle U_{\rm{d}}$ $\displaystyle=$
$\displaystyle\frac{4\pi\hbar^{2}}{m_{\mathrm{t}}\bar{\sigma}}\ =\
\frac{4\pi\hbar^{2}v_{p}}{m_{\mathrm{t}}\langle\sigma_{\rm{tot}}\,v\rangle}.$
(5)

Here $m_{\mathrm{t}}$ is the mass of the trapped atom and
$v_{p}=\sqrt{2k_{\rm{B}}T/m_{\mathrm{bg}}}$ is the most probable velocity of
the background particles (of mass $m_{\mathrm{bg}}$ at temperature $T$). The
expansion in Eq. 4 is valid for $U\leq U_{\rm{d}}$. The universal function
$p_{\rm{QDU6}}$ is the cumulative energy distribution function of the trapped
sensor atom after collision, representing the probability that the sensor atom
remains in a trap of depth $U$ after the collision. The trap loss probability
is $(1-p_{\rm{QDU6}})$.

Ref. Booth  _et al._ (2019) showed that the coefficients $\beta_{j}$ are the
same for different atomic and molecular species, provided the strength of the
van der Waals interaction is significant.

Term | $\beta_{j}$
---|---
1 | 0.673 (7)
2 | -0.477 (3)
3 | 0.228 (6)
4 | -0.0703 (42)
5 | 0.0123 (14)
6 | -0.0009 (2)
Table 1: The universal coefficients appearing in Eq. 4 were derived from
fitting the quantum scattering (QS) computations for
$\langle\sigma_{\rm{loss}}(U)\ v\rangle$ to a sixth order polynomial. The
derivation of $\beta_{j}$ can be found in Booth  _et al._ (2019).

This feature of the velocity averaged cross sections allows one to measure
$\langle\sigma_{\rm{tot}}\,v\rangle$ by recording the trap loss rate
$\Gamma_{\rm{loss}}=n_{\mathrm{bg}}\langle\sigma_{\rm{loss}}(U)\ v\rangle$ for
different trap depths, normalizing these measurements by the total collision
rate (i.e. the loss rate at zero trap depth, $\Gamma_{\rm{tot}}$), and fitting
the ratio $\Gamma_{\rm{loss}}/\Gamma_{\rm{tot}}=\langle\sigma_{\rm{loss}}(U)\
v\rangle/\langle\sigma_{\rm{tot}}\,v\rangle$ to the universal shape, Eq. 4
Booth  _et al._ (2019). The power of this approach is that any test gas
species can be calibrated following this prescription with no need for ab
initio computations requiring a priori knowledge of the interaction
potentials. Once the $\langle\sigma_{\rm{tot}}\,v\rangle$ are determined
experimentally, the loss of atoms from a magnetic trap with a fixed depth $U$
can be used to determine the gas pressure via Eq. 6,

$\displaystyle P$ $\displaystyle=$ $\displaystyle
nk_{\rm{B}}T=\left(\frac{\Gamma_{\rm{loss}}(U)}{\langle\sigma_{\rm{loss}}(U)\
v\rangle}\right)k_{\rm{B}}T.$ (6)

###  II.1 Origin of the Quantum Diffractive Collision Universal Law

The insensitivity of the velocity-averaged total and loss cross sections to
the short range details of the PES arises from velocity averaging in a regime
where the global variation of the cross section follows a trend set by the
long range PES. To illustrate this, consider Fig. 1, which shows a log-log
plot of a single-channel quantum scattering (QS) calculation of the total
collision cross section as a function of relative collision velocity for Rb-Ar
(the solid blue traces) and for Rb-He (the solid red traces) collisions. These
QS computations are performed using the Lennard-Jones shape PES for Rb-Ar and
Rb-He with the long-range coefficient, $C_{6}$, and the potential depth
information from Ref. Medvedev  _et al._ (2018). These computed cross sections
reveal the glory oscillations superimposed on the Jeffries-Born (JB)
prediction given in Eq. 7 and shown by the dashed red and blue lines in Fig.
1(a).

$\displaystyle\sigma_{\mathrm{JB}}$ $\displaystyle\sim$ $\displaystyle
8.0828\left[\frac{C_{6}}{\hbar v}\right]^{2/5}.$ (7)

Note that at much higher velocities, the variation of the cross section with
velocity follows a different trend set by the short range details of the PES
111At relative velocities high enough that the de Broglie wavelength of the
collision complex is on the order of the bound state minimum distance, the
cross section follows the trend $\sigma\sim\left[\frac{C_{12}}{\hbar
v}\right]^{2/11}$ for a Leonard-Jones potential.

Figure 1:  The total collision cross section as a function of relative
velocity is shown in (a) for Rb-Ar (blue) and Rb-He (red) collisions on a log-
log plot. The cross sections exhibit oscillations around a trend line modeled
by the Jeffries Born approximation
$\sigma_{\mathrm{JB}}\sim\left[\frac{C_{6}}{\hbar v}\right]^{2/5}$ (thin
dashed lines). Also shown are the Maxwell-Boltzmann velocity distributions for
the relative collision velocity given a stationary Rb sensor atom and room
temperature gas of Ar (thick black dashed line) and for He (thick grey dashed
line) atoms. The cumulative ratio $F(v)$ given by Eq.8 is shown in (b). Its
value at $v=\infty$ is the ratio of the true total velocity averaged cross
section $\langle\sigma_{\rm{tot}}\,v\rangle$ to that computed with the
Jeffries Born approximation. When the velocity averaged quantity
$\langle\sigma_{\rm{tot}}\,v\rangle$ is computed, for Rb-Ar collisions (blue),
we see that $F(\infty)\approx 1$ meaning that it is independent of the
oscillations and only depends on the average velocity and the magnitude of
$C_{6}$. For Rb-He, a residual dependence remains and $F(\infty)\neq 1$
because the larger oscillation amplitude and period is no longer averaged away
by convolution with the MB distribution.

It is clear that the character of the glory oscillations is very different for
these two species. The oscillation amplitudes and phases are determined by the
relative magnitude and phase of the long and short range scattering
contributions, and the number of oscillations is related to the number of
rotation-less bound states supported by the potential Child (1974). For Rb-Ar,
where the $C_{6}$ coefficient is close to a factor of 10 larger than that for
Rb-He collisions 222The theoretically predicted values for C6/$E_{H}a_{0}^{6}$
are 334(2) and 44.07(11) for Rb-Ar and Rb-He respectively. the total cross-
section is dominated by the long range contribution. Coupled with the the
larger reduced mass of the Rb-Ar collision system one observes a larger number
of lower amplitude glory oscillations, than those associated with Rb-He
collisions.

The trap loss rate measurements are inherently averaged over the Maxwell-
Boltzmann (MB) speed distribution of the room temperature background gas (the
dotted traces in Figure 1). When this average encompasses many oscillation
periods and the oscillation amplitudes are small, their effect is minimized
and the information about the short range PES carried by the glory
oscillations is removed. In these cases, the quantity
$\langle\sigma_{\rm{tot}}\,v\rangle$ is universal in that it depends primarily
on the average relative collision velocity and long-range character of the PES
set by the magnitude of $C_{6}$. By contrast, a small $C_{6}$ yields in larger
amplitude glory oscillations, and a small reduced mass results in fewer
oscillations within the MB velocity distribution. Thus, the averaged quantity
$\langle\sigma_{\rm{tot}}\,v\rangle$ retains information about the short range
portion of the PES and will deviate from the universal prediction.

To illustrate this point, consider Fig. 1(b). It shows the cumulative integral
ratio

$\displaystyle F(v)$ $\displaystyle=$ $\displaystyle\frac{\int_{0}^{v}4\pi
v_{\mathrm{bg}}^{3}\sigma(v_{\mathrm{bg}})e^{-m_{\mathrm{bg}}v_{\mathrm{bg}}^{2}/2k_{\rm{B}}T}dv_{\mathrm{bg}}}{\int_{0}^{v}4\pi
v_{\mathrm{bg}}^{3}\sigma_{\mathrm{JB}}(v_{\mathrm{bg}})e^{-m_{\mathrm{bg}}v_{\mathrm{bg}}^{2}/2k_{\rm{B}}T}dv_{\mathrm{bg}}}$
(8)

where $\sigma(v)$ is the QS computed value (shown in (a) by the solid lines)
and $\sigma_{\mathrm{JB}}(v)$ is the Jeffries Born approximation (shown in (a)
as the dashed lines) to the velocity dependent cross section. The value of
this function $F_{\infty}=F(\infty)$ denotes the ratio of the total velocity
averaged cross section $\langle\sigma_{\rm{tot}}\,v\rangle$ computed from the
full PES using the QS calculations to that found using the approximation. For
Rb-Ar collisions, we find that $F_{\infty}\approx 1$ indicating that the total
velocity averaged cross section is independent of the oscillations and thus
independent of the short range PES. For Rb-He collisions, because the
oscillation amplitude and period is larger, $F(\infty)\neq 1$. This indicates
that a residual dependence on the short range PES remains.

Since the velocity-averaged loss cross section asymptotically approaches the
velocity-averaged total cross section as the trap depth goes to zero,
$\langle\sigma_{\rm{loss}}\ v\rangle$ is expected to also be insensitive to
the short range details of the PES at small trap depths. The trap-depth scale
over which this insensitivity persists is set by $U_{\rm{d}}$. In brief,
collisions that impart energies of $U_{\rm{d}}$ or less cannot carry any
information about variations of the potential on a length scale smaller than
the total cross section. This can be understood as a consequence of the Born
approximation.

Because universality results from the velocity averaging, the parameter regime
in which this universality is expected to break down is small $C_{6}$, small
reduced mass, and narrow velocity distributions (i.e. very cold background
gases). Although Ref. Booth  _et al._ (2019) explored the robustness of this
universality for the case of Rb-Ar collisions by varying the PES at short
range (thus changing the relative magnitudes of the short and long range
scattering contributions) and by varying the test gas temperature, the regime
of low reduced mass was only considered analytically and only briefly.
Therefore, this work focuses on the low reduced mass complexes Rb-H2 and
Li-H2. A full study of the parameter regimes where universality holds is
needed; however, such a study is beyond the scope of this paper.

##  III Experimental Procedure

Two experimental apparatuses were used in this work. Device (A), described in
Refs. Booth  _et al._ (2019); Shen  _et al._ (2020), is only capable of
trapping Rb atoms. It was used to perform an extensive study of the trap depth
dependence of the loss rate of Rb sensor atoms exposed to a H2 background gas
in order to extract the value of $\langle\sigma_{\rm{tot}}\,v\rangle$ based on
a QDU universality description. Device (B), described in Refs. Bowden  _et
al._ (2016), was used to perform loss rate measurements of both Rb and Li
atoms exposed to the same H2 background gas, thus providing a direct
measurement of
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}:\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}$.
Here we give a brief description of the two devices and the experimental
protocols used in this study.

###  III.1 Device (A)

Device (A) consists of an atom source (a 2D MOT) that is separated from and
provides a cold flux of atoms to the pressure measurement section composed of
a magneto-optic trap (a 3D MOT), a co-located magnetic trap (MT), and a suite
of ionization gauges (IGs), including a residual gas analyzer (RGA),
illustrated in Fig. 3(a). The H2 background gas was introduced by applying a
voltage to the non-evaporate getter pump (NEG, C400-2-DSK from SAES Getters)
that heats up the NEG and releases the absorbed gas species from the getter.
We used the RGA to detect which gases were present in the system when the NEG
is heated. We verified that H2 is the dominated gas species. For applied
voltages between 2.5 V and 4.5 V, the total background pressure contributions
from other gas species is less than 1 %.

Figure 2: Measured RGA signal as a function of the voltage applied across the
NEG. At each applied voltage, an RGA measurement is performed to determine the
partial pressure of the background gas species. At the same time, the ion
gauge (IG) is used to record the total background pressure. Red circles
represent the H2 gas, green squares represent the O2 gas, blue triangles
represent the H2O gas, and the black diamonds represent the CO gas. For
applied voltages in the range of [2.5 V, 4.5 V] we observe that the background
pressure contribution from other gas species is less than 1 %. We observed
that when the NEG voltage is larger than 4.5 V, the partial pressure of CO
exceeds 1 % of the total.

As discussed in Refs. Booth  _et al._ (2019); Shen  _et al._ (2020), to create
a cold ensemble of sensor atoms, we load a 3D MOT from a cold atomic beam
emerging from a 2D MOT. The MOT trapping (or “pump”) laser beams for both the
2D and 3D MOT are locked $\delta=12$ MHz below the D2 (52S1/2 $\rightarrow$
52P3/2) $F=2\rightarrow F^{\prime}=3$ transition of the 87Rb sensor atoms. The
“repump” laser light is tuned to resonance with the $F=1\rightarrow
F^{\prime}=2$ transition. The magnetic field configuration is a spherical
quadrupole for the 3D MOT and is operated with a gradient of 13.6 G/cm along
the axial direction. During the measurement sequence, approximately $10^{7}$
atoms are loaded into the 3D MOT, producing a fluorescence signal from an
amplified photo-detector of $V_{\rm{MOT}}$. Using a technique described in
Ref. Jooya  _et al._ (2013), we verified that our photodetector signal is
linear in the atom number. This is an essential requirement to correctly
deduce the atom loss rate from the decay of the fluorescence signal.

After loading the 3D MOT, the atoms are cooled by turning down the pump laser
power and changing the detuning of the pump laser frequency to $\delta=60$ MHz
below the $F=2\rightarrow F^{\prime}=3$ transition transition for 20 ms. The
pump light is then extinguished while leaving the repump light on. At the same
time, we apply a right-hand circularly polarized optical pumping (OP) beam
which is tuned into resonance with the $F=2\rightarrow F^{\prime}=2$
transition. An additional magnetic field, generated by a Helmholtz coil pair
is also introduced along the propagation direction of the OP beam in order to
drive $\sigma_{+}$ transitions. After 2 ms, the repump light, the optical
pumping light and the Helmholtz coils are extinguished and the quadrupole
magnetic field gradient is increased. Atoms that fall into the low-field
seeking sublevels ($m_{F}=1$ and $m_{F}=2$) of the $F=2$ manifold are retained
in the MT. Gravitational filtering is used to remove the $|F=2,m_{F}=1\rangle$
atoms to insure the loss rate measurements correspond to a single state and
trap depth. We set the axial magnetic field to 27 G/cm and hold the atoms for
200 ms so that only those atoms in the $|F=2,m_{F}=2\rangle$ state are
retained in the MT. Atoms in the $|F=2,m_{F}=1\rangle$ state fall out of the
trap as this gradient is insufficient to compensate the gravitational force.
After this quantum state purification step, we then increase the axial
magnetic field gradient from 27 G/cm to 272 G/cm in 10 ms and the atoms are
held for a period $t$. For trap depths smaller than 1.0 mK, we only increase
the axial magnetic field gradient to 81.6 G/cm so that the trap volume
explored by the sensor atoms is larger and thus the intra-trap two-body
collision rate – which scales with the sensor atom density – is minimized. For
this study of the trap depth dependent loss rate of Rb sensor atoms exposed to
a H2 background gas, we specifically chose to trap the $|F=2,m_{F}=2\rangle$
state to access the largest trap depths possible given our setup.

As in Refs. Booth  _et al._ (2019); Shen  _et al._ (2020), at the end of this
hold time, a radio frequency (RF) field, whose frequency is swept from a high
frequency to a lower frequency cutoff $\nu_{\mathrm{min}}$, is turned on to
eject all atoms above a certain energy from the trap. This step sets the trap
depth for each loss rate measurement. The remaining atoms in the MT below this
cutoff energy are recaptured in the 3D MOT using the same settings as used for
the initial MOT loading, and the signal, $V_{\rm{MT}}$, is recorded. We
compute the recaptured fraction,
$f_{\mathrm{recap}}=V_{\rm{MT}}(t)/V_{\rm{MOT}}$, where $V_{\rm{MOT}}$ is the
photodiode voltage just before the MOT to MT transfer. The quantity
$V_{\rm{MOT}}$ is proportional to the atom number transferred into the MT, and
this procedure reduces the effects of shot-to-shot variations in the number of
atoms loaded into the MOT Shen  _et al._ (2020). Following the methods
described in Ref. Stewart  _et al._ (2022), we eliminate the contributions
from the intra-trap two body collisions and verify that the recaptured
fraction versus hold time follows a one-body exponential decay law. Then, we
use the method of two-point measurements described in Ref. Jones  _et al._
(1996) to optimize the determination of the decay constant by placing half of
the sample points at the shortest hold time and the remaining half at 1.28
$\tau$, where $\tau$ is the lifetime of the cold sensor atoms. Using this two-
point measurement scheme, we obtain a smaller statistical uncertainty in the
decay rate than when using a sample set spaced linearly in time. An example
data set showing the decay rate measurements using this two point measurement
scheme is shown in the inset of Fig. 4. In this scheme, we have the following
expressions for the recaptured fractions at each hold time,

$\displaystyle\begin{cases}f_{j}(t_{j})=f_{0}\exp(-\Gamma_{\rm{exp}}t_{j})\\\
f_{k}(t_{k})=f_{0}\exp(-\Gamma_{\rm{exp}}t_{k}),\end{cases}$ (9)

where $t_{j}$ is the first hold time and $t_{f}$ is the second hold time. The
recaptured fraction at zero time is $f_{0}$, and $f_{j}$ and $f_{k}$ are the
recaptured fractions at the times $t_{j}$ and $t_{f}$, respectively. The
experimentally measured loss rate is a sum of the collision-induced loss due
to H2 gas ($\Gamma_{\rm{loss}}$) introduced by heating the NEG, and the
constant baseline loss rate, $\Gamma_{0}$. Namely,
$\Gamma_{\rm{exp}}=\Gamma_{\rm{loss}}+\Gamma_{0}$. Here $\Gamma_{0}$ includes
all other sources of sensor atom loss including those due to the residual
background vapor present before the NEG is heated and to other loss processes
not related to background gas collisions Booth  _et al._ (2019); Shen  _et
al._ (2020).

We performed the recapture fraction measurements at the two hold times
sequentially in order to minimize the influence of fluctuations in the MOT to
MT transfer that affect $f_{0}$. This procedure allows us to make the
assumption that the zero time recapture fraction is the same for each two-
point measurement pair. Using this assumption and the log of the ratio of the
expressions in Eq. 9, we find,

$\displaystyle\ln\Big{[}\frac{f_{j}(t_{j})}{f_{k}(t_{k})}\Big{]}=\Gamma_{\rm{loss}}(t_{k}-t_{j})+\Gamma_{0}(t_{k}-t_{j})$
$\displaystyle=\frac{\langle\sigma_{\rm{loss}}(U)\
v\rangle\Big{[}(P_{k}-P_{0})t_{k}-(P_{j}-P_{0})t_{j}\Big{]}}{\rm{i_{\rm{g}}}k_{\rm{B}}T}+\Gamma_{0}(t_{k}-t_{j}).$
(10)

We have used Eq. 6 to rewrite the loss rate due to the added gas on the RHS as
$\Gamma_{\rm{loss}}^{(j)}=n_{\rm{H}_{2}}^{(j)}\langle\sigma_{\rm{loss}}(U)\
v\rangle=\frac{(P_{j}-P_{0})}{\rm{i_{\rm{g}}}k_{\rm{B}}T}\langle\sigma_{\rm{loss}}(U)\
v\rangle$. Here $P_{j}$ and $P_{k}$ are the pressure readings from the
(uncalibrated) IG for the measurements corresponding to the holding times
$t_{j}$ and $t_{k}$, respectively. The IG response to the added H2 gas is
obtained by subtracting the IG pressure readings at the background level,
$P_{0}$, from the measured pressures $P_{j}$ and $P_{k}$. (Note that these
pressure readings need not be accurate. The only requirement of the IG is that
its response is linear in the gas pressure.) The H2 calibration factor of the
ion gauge is $\rm{i_{\rm{g}}}$, and $T$ is the ambient temperature recorded by
a thermal couple gauge, which is the same for different hold times. We repeat
this two point measurement procedure for many different trap depths and thus
map out the function of $\langle\sigma_{\rm{loss}}(U)\
v\rangle/\rm{i_{\rm{g}}}$ versus the trap depth.

The last term in Eq. 10 quantifies the contributions from loss mechanisms that
do not depend on the density of the H2 gas species introduced and can be
obtained from the pre-heating (baseline) loss rate measurements. That is, the
pressure of the ambient gas $P_{0}$ and the baseline loss rate for a variety
of trap depths $\Gamma_{0}(U)$ are determined before any H2 gas is introduced
into the system. For this work, $\Gamma_{\rm{loss}}$ is two orders of
magnitude higher than $\Gamma_{0}$.

Figure 3: Experiment A and B setup. Panel (a) shows a schematic of the
experiment apparatus A including the Rb source stage, the 2D MOT section, the
3D MOT section (sensing section) and the pumping section. 2D, 3D MOT light and
the push light are indicated by red arrows. H2 gas is introduced into the
system by heating the NEG, located on the bottom side of the apparatus. The
RGA installed on the top side of the apparatus is used to monitor the partial
pressures of all gases introduced into the system by heating the NEG. The
pressure change of the H2 gas is recorded by the ionization gauge (IG). Panel
(b) shows the apparatus of device B. Rb and Li atoms leave the ovens on the
right, passing through a differential pumping tube and through the Zeeman
slower until being captured by the MOT. The NEG (shown) on the left was heated
to increase the H2 partial pressure in the MOT region.

###  III.2 Device (B)

This apparatus was used to perform loss rate measurements of both Rb and Li
atoms exposed to the same H2 background gas. The aim was to determine the
ratio of the total cross sections,

${R=\frac{\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}}{\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}}.}$
(11)

For these measurements, the Li and Rb atoms were confined in very shallow
traps to ensure that the measured loss rate is approximately equal to the
total collision rate (i.e. $\Gamma_{\rm{loss}}\approx\Gamma_{\rm{tot}}$). In
this limit, the cross section ratio can be obtained by inspecting the ratio of
the Li and Rb loss rates. In addition, the exact value of the H2 gas density
need not be known. All that is required is that the H2 gas density be the same
for the Rb and Li loss rate measurements. To extract the cross section for
collisions with only H2 molecules, loss rate measurements of Rb and Li were
made in pairs where the H2 gas density was the same for each pair. Then, as
described below, these pair measurements were made at a variety of different
H2 gas densities.

In device (B), effusive ovens emit thermal, parallel atomic beams of the two
species – Rb and Li – which are subsequently decelerated and cooled in a
Zeeman slower region by a counter-propagating bi-chromatic optical beam. The
atoms are then captured in a 3D MOT and transferred into a co-located MT. The
3D MOT region vacuum is maintained by an ion pump and a NEG and is separated
from the oven region by a differential pumping tube. The NEG is the same model
as the one used in device (A). In order to increase the H2 pressure above its
baseline pressure while keeping the other gasses relatively constant, we
applied a voltage across the NEG within the same range as in device (A). Since
we are interested in the loss rate ratio of the Li and Rb sensor ensembles
when exposed to the same H2 gas, we could have co-trapped both sensor
ensembles in the same MT and simultaneously measured the Rb and Li loss rates.
However, this was not done because the optimal operating parameters for
slowing, trapping and transferring atoms from the MOT to the MT are
significantly different for Rb and Li. Instead, we interlaced the loss rate
measurements for the two species as described below, in order to to minimize
systematic errors associated with pressure changes during the measurements. As
in device A, we measured the atom numbers by observing the atomic fluorescence
signal of the atoms in the MOT using a photo-detector. We verified the
linearity of our photo-detector signal with atom number by direct comparison
with the atom number determined by absorption images of our atoms (using a
different detector) over a large range of atom numbers. The atomic
fluorescence signal was used for this study because it has a much higher
sensitivity than the absorption image method.

For the Lithium measurements, the MOT pump laser was detuned by
$\delta_{\mathrm{pump}}=50$ MHz below the D2 (22S1/2 $\rightarrow$ 22P3/2)
$F=3/2\rightarrow F^{\prime}=5/2$ transition of the 6Li sensor atoms. After a
MOT loading sequence of 10 s, $(1-2)\times 10^{7}$ cooled Li atoms reside in
the MOT, corresponding to a fluorescence signal of
$V_{\mathrm{MOT,Li}}\sim(1.4-2.8)$ V. After the MOT is loaded, we extinguish
the Zeeman slowing optical beam and block the thermal atomic beams using a
mechanical beam block. We then initiate a 100 ms ramp of the quadrupole
magnetic field gradient of the MOT from 10 G/cm to 30 G/cm, followed by a
cooling sequence for which we change the detuning of the pump and repump laser
to $\delta_{\mathrm{pump}}=-13.6$ MHz and $\delta_{\mathrm{repump}}=-9$ MHz.
After this 4 ms of cooling, the Li atoms are optically pumped to the $F=1/2$
state in 10 $\mu$s by extinguishing the repump light before the pump light.
Atoms in the $|F=1/2,m_{F}=-1/2\rangle$ state are low field seeking and are
trapped in the MT given their total energy is less than the maximum potential
energy, $U_{\mathrm{max}}/k_{\rm{B}}=300\;\mu$K. This maximum potential energy
is a fundamental limit resulting from the magnetic moment sign change of the
$|F=1/2,m_{F}=-1/2\rangle$ state at a field of approximately 30 G Shen  _et
al._ (2020). Because of the temperature of atoms in the MOT (on the order of
1mK) and the limited trap depth, we typically transfer only around 2%
([2-4]$\times 10^{5}$) of atoms from the MOT to the MT. After transfer, we
hold the atoms for a time $t$. After this hold time, we recapture those atoms
remaining in the MT into the 3D MOT using the same settings as used for the
initial MOT loading. As described for device (A), we record the MOT
fluorescence after recapture and compute the recaptured fraction,
$f_{\mathrm{recap}}=V_{\rm{MT}}(t)/V_{\rm{MOT}}$, where $V_{\rm{MOT}}$ is the
photodiode voltage just before the MOT to MT transfer. No radio-frequency
radiation is applied to set the trap depth for the Li measurement. Rather, we
rely on the fundamental trap depth limit described above for the
$|F=1/2,m_{F}=-1/2\rangle$ state.

For the Rb measurements, the 3D MOT light operates on the same transition as
device (A), but the experimentally optimal values differ slightly. During
loading, the trapping laser is detuned to $\delta_{\mathrm{pump}}=-20$ MHz,
below the pump transition. After loading the MOT for 2 s, we observed a
fluorescence signal of $V_{\mathrm{MOT,Rb}}=(1.6-3.2)$ V, corresponding to
$(0.8-1.6)\times 10^{7}$ Rb atoms. After the MOT was loaded, we further cooled
the trapped ensemble for 15 ms by detuning the pump light to
$\delta_{\mathrm{pump}}=35$ MHz below the $F=2\rightarrow F^{\prime}=3$
transition. The repump light is extinguished for the last 10 ms of this
procedure to optically pump the atoms into the $F=1$ hyperfine manifold. We
then extinguished all light and increased the quadruple magnetic field axial
gradient from 20 G/cm to 60 G/cm and held the atoms in the
$|F=1,m_{F}=-1\rangle$ state for a time $t$. This MOT to MT transfer procedure
is similar to that for device (A). The principal difference is that the atoms
in device (B) are trapped in the $|F=1,m_{F}=-1\rangle$ state and thus no spin
polarization beam or gravitational filtering was required since this is the
only trapped state for the $F=1$ manifold. Immediately after the atoms are
transferred into the MT and then immediately before recapture of the atoms
into the MOT (after the hold time of $t$ seconds), we apply a radio frequency
(RF) field for a duration of 100 ms and whose frequency is swept from a high
value to a lower value to eject all atoms above a certain energy from the
trap. We chose the RF field parameters to set the magnetic trap depth to
$U\leq 200~{}\mathrm{\mu K}$. As with device (A), atoms remaining in the MT
below this cutoff energy are recaptured in the 3D MOT. The fluorescence is
then recorded, and the recaptured fraction is found.

##  IV Discussion of Results

###  IV.1 Trap loss measurements of Rb exposed to H2

Using apparatus A, we performed an extensive study of the trap loss rate
versus trap depth of 87Rb atoms exposed to a H2 background gas over a much
wider range of depths than previously studied Shen  _et al._ (2020). In
particular, we made loss rate measurements from 0.5 mK to 6 mK, in an effort
to define the shape of the loss rate versus trap depth curve more precisely
and facilitate a clearer comparison with the universal prediction. However, as
the trap depth increases, the collision-induced heating of the trapped
ensemble becomes more pronounced, confounding the loss rate measurements. Here
we performed additional measurements and data analysis to properly account for
this effect.

Collisions that fail to eject sensor atoms from the trap change the sensor
ensemble energy distribution over time Beijerinck (2000a, b). Because the
effective trap depth depends on the sensor-atom energy distribution in the
trap, this results in a time-dependent and monotonically decreasing effective
trap depth. If not accounted for, this leads to a systematic overestimate of
the measured loss rate with trap depth and, consequently, a systematic
underestimate in the deduced $\langle\sigma_{\rm{tot}}\,v\rangle$ value.

Although the fitting uncertainty for $\langle\sigma_{\rm{tot}}\,v\rangle$
decreases as the range of trap depths increases, the systematic error caused
by unaccounted sensor-ensemble heating will increase with trap depth. We have
found that the ensemble heating error becomes dominant once the scaled trap
depth, $U/U_{\rm{d}}$, exceeds 0.3 Shen  _et al._ (2020). Therefore, the
uncertainty in determining $\langle\sigma_{\rm{tot}}\,v\rangle$ is limited by
ensemble heating. To resolve this problem and eliminate the resulting
systematic uncertainty, we apply a correction to the trap depth by measuring
the energy distribution of the ensemble as a function of the hold time. This
allows us to infer the correct effective trap depth as a function of the hold
time (details are provided in Appendix A). After properly accounting for
ensemble heating, we obtained corrected values of
$\langle\sigma_{\rm{loss}}(U)\ v\rangle/\rm{i_{\rm{g}}}$ as a function of the
trap depth. For each trap depth, we repeated the measurements for three times
and we repeated the whole measurement sequence on three different calendar
dates. The data are shown in Fig. 4(a). The overlap of the data on the three
different dates indicates the reproducibility of the measurements.

One goal of this measurement is to determine the precise shape of
$\langle\sigma_{\rm{loss}}(U)\ v\rangle$ as a function of the trap depth
without any assumption that it follows a particular curve. For this purpose,
we fit the combined data on different dates to a second order polynomial. This
fit is also used to extrapolate and find the value of
$\langle\sigma_{\rm{loss}}(U)\ v\rangle/\rm{i_{\rm{g}}}$ at zero trap depth,
which is needed to normalize the loss rate data as shown in Fig. 4(b). The
best fit to the data is

$\displaystyle\frac{\langle\sigma_{\rm{loss}}(U)\
v\rangle/\rm{i_{\rm{g}}}}{1\times 10^{-15}\rm{m}^{3}/\rm{s}}$ $\displaystyle=$
$\displaystyle 8.82(6)-0.38(4)\left(\frac{U}{\rm{mK}}\right)$ (12)
$\displaystyle\hskip 28.0pt+\ 0.022(6)\left(\frac{U}{\rm{mK}}\right)^{2}.$

Another goal of this measurement is to determine the value of
$\langle\sigma_{\rm{tot}}\,v\rangle$ and $\rm{i_{\rm{g}}}$ simultaneously by
fitting these data to the universal function, Eq. 4. As expected, the fitted
values on different dates agree with each other. We find
$\langle\sigma_{\rm{tot}}\,v\rangle=[5.49(6),5.54(7),5.55(8)]\times 10^{-15}$
m3/s, and $\rm{i_{\rm{g}}}=[0.632(7),0.625(7),0.645(10)]$. The universal
function fit to the combined data set is shown in Fig. 4(b). The (heating-
corrected) values obtained by fitting the combined data set are
$\langle\sigma_{\rm{tot}}\,v\rangle=5.52(9)\times\ 10^{-15}$ m3/s and
$\rm{i_{\rm{g}}}=0.633(10)$.

Figure 4:  Measured trap loss rate coefficients as a function of trap depth
for 87Rb atoms held in a magnetic trap and exposed to a H2 gas. The data in
(a) shows the value of $\langle\sigma_{\rm{loss}}(U)\
v\rangle/\rm{i_{\rm{g}}}$ as a function of the trap depth after accounting for
sensor ensemble heating (see appendix). Different colors represent data taken
on three different calendar dates. Each datapoint is the average of three two-
point decay rate measurements. Examples of two-point decay measurements, each
at a different trap depth, are shown in the inset. The black curve in (a) is
the second order polynomial fit to the data provided in Eq. 12 and used to
extrapolate the value at $U=0$. By instead fitting the same data to the
universal function, Eq. 4, we determine the value of
$\langle\sigma_{\rm{tot}}\,v\rangle$ and $\rm{i_{\rm{g}}}$ for the data taken
on three different dates. Fitting the red circles we obtain
$\langle\sigma_{\rm{tot}}\,v\rangle=5.49(6)\times
10^{-15}\;\mathrm{m}^{3}/\mathrm{s}$ and $\rm{i_{\rm{g}}}=0.632(7)$ . For the
blue triangles, we have $\langle\sigma_{\rm{tot}}\,v\rangle=5.54(7)\times
10^{-15}\;\mathrm{m}^{3}/\mathrm{s}$ and $\rm{i_{\rm{g}}}=0.625(7)$. For the
green squares, we have $\langle\sigma_{\rm{tot}}\,v\rangle=5.58(6)\times
10^{-15}\;\mathrm{m}^{3}/\mathrm{s}$ and $\rm{i_{\rm{g}}}=0.645(10)$. The
universal fit to the combined set of data yields
$\langle\sigma_{\rm{tot}}\,v\rangle=5.52(9)\times
10^{-15}\;\mathrm{m}^{3}/\mathrm{s}$ and $\rm{i_{\rm{g}}}=0.633(10)$. The
universal plot of all the data (the loss rate coefficient is normalized by the
value at zero trap depth and the trap depth is re-scaled by $U_{\rm{d}}$) is
shown in (b), and the grey dashed line represents the universal fit function.

###  IV.2 Cross species sensor atom comparison

In order to determine the Rb+H2 cross section independently from that
determined by the universality fit, we used device (B) to perform loss rate
measurements of trapped Rb and of trapped Li atoms exposed to the same H2
background gas pressure. By confining the atoms in very shallow magnetic traps
where $\Gamma_{\rm{loss}}\approx\Gamma_{\rm{tot}}$, we were able to determine
the ratio of the Li+H2 and Rb+H2 cross sections,
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}:\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}$.
Using this measured ratio and the ab initio computed value for the Li+H2 cross
section, we were able to determine the Rb+H2 cross section and compare this to
that found using the universality fit described in Sec. IV.1

For these measurements, the aim was simply to determine the ratio of the total
cross sections. Therefore, the exact value of the H2 gas density need not be
known. In fact, all that is required is that the _change_ of the H2 gas
density be the same for each pair of measurements of the Rb and Li trap loss
rates, $\Gamma_{\rm{Rb+H}_{2}}$ and $\Gamma_{\rm{Li+H}_{2}}$. For traps of
zero depth with respect to collisions with H2 molecules
($\Gamma_{\rm{loss}}\approx\Gamma_{\rm{tot}}$), we have

$\displaystyle\Gamma_{\rm{Rb+H}_{2}}$ $\displaystyle=$ $\displaystyle
n_{\rm{H}_{2}}\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}+\Gamma_{\rm{(Rb,0)}}$
(13) $\displaystyle\Gamma_{\rm{Li+H}_{2}}$ $\displaystyle=$ $\displaystyle
n_{\rm{H}_{2}}\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}+\Gamma_{\rm{(Li,0)}}$
(14)

where $\Gamma_{\rm{(Rb,0)}}$ and $\Gamma_{\rm{(Li,0)}}$ are the Rb and Li
baseline loss rates measured before the NEG is heated. As for device (A),
$n_{\rm{H}_{2}}$ is the additional number density of H2 gas resulting from
heating the NEG. As the NEG was heated, $n_{\rm{H}_{2}}$ increased, and
measurements of $\Gamma_{\rm{Rb+H}_{2}}$ and $\Gamma_{\rm{Li+H}_{2}}$ are made
in pairs with minimum time between the two measurements so that the value of
$n_{\rm{H}_{2}}$ is the same for each data pair. In Fig. 5 we plot the ordered
pairs $(\Gamma_{\rm{Rb+H}_{2}},\Gamma_{\rm{Li+H}_{2}})$ for different values
of $n_{\rm{H}_{2}}$ in a 2D scatter plot where the abscissa is the Rb loss
rate and the ordinate is the Li loss rate. We then perform an orthogonal
distance regression (ODR) fit of the data to a line. The slope of this line is
the ratio:

$\displaystyle
R=\frac{\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}}{\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}}.$
(15)

From the data, we found $R=0.83\pm 0.05$. Based on the theoretically predicted
value from Ref. Makrides  _et al._ (2019, 2022a) of
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}=3.13(6)\times 10^{-15}$
m3/s, we can infer that
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}=3.8(2)\times 10^{-15}$
m3/s.

Due to its low mass and the small maximum trap depth
($U_{\mathrm{max}}/k_{\rm{B}}=300\;\mu$K) for the $|F=1/2,m_{F}=-1/2\rangle$
state, Li is always in the zero trap depth limit with $\frac{U}{U_{d}}\simeq
0.1$%. For Rb, we use radio-frequency radiation (as describe above) to limit
the magnetic trap depth to $U<200~{}\mathrm{\mu K}$, for which the deviation
of $\left<\sigma_{\rm{loss}}\,v\right>$ from
$\left<\sigma_{\rm{tot}}\,v\right>$ is less than 0.1% as per Eq. 12. Thus we
expect trap depth effects to be negligible compared to our statistical
uncertainty.

Figure 5: Trap loss rate measurements for an ensemble of 6Li and 87Rb atoms
exposed to the same H2 background gas produced by heating the NEG. As the H2
pressure increased, the loss rates increased for both ensembles. Here the loss
rates are plotted as the ordered pair
$(\Gamma_{\rm{Rb+H}_{2}},\Gamma_{\rm{Li+H}_{2}})$, and the data is fit to a
line whose slope provides the ratio of the loss rates and thus the ratio of
the cross sections,
$R=\frac{\Gamma_{\rm{Li+H}_{2}}}{\Gamma_{\rm{Rb+H}_{2}}}=\frac{\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}}{\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}}$.
For these data, we find $R=0.83\pm 0.05$.

###  IV.3 Ab Initio calculation predictions

In addition to experimentally investigating the rate coefficient ratio
$R={\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}}:{\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}}$,
we have also computed it. We find an _ab initio_ prediction of
$R\approx{0.869}$, which is within the range of experimentally measured value.
Specifically, we have performed coupled channel (CC) calculations to find the
Rb+H2 and Li+H2 cross sections. The details of these calculations is provided
in the appendix.

In brief, the Rb+H2 cross section was calculated using a PES kindly provided
to us from the authors of Ref. Upadhyay  _et al._ (2019). This PES was
obtained by _ab initio_ calculations with the unrestricted coupled cluster
method with single, double and perturbative triple excitations [UCCSD(T)]
Deegan and Knowles (1994). The Li+H2 cross section was found using a PES
published in Refs. Makrides  _et al._ (2019, 2022a). To validate the accuracy
of our coupled channel calculations, we first recomputed the rate coefficient
for Li + H2 collisions. We found a value of
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}=3.101(6)\times 10^{-15}$
m3/s, in good agreement with the value of
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}=3.13(6)\times 10^{-15}$
m3/s from Refs. Makrides  _et al._ (2019, 2022a). We then computed the cross
sections and rate coefficients for Rb + H2 collisions using the PES from
Upadhyay  _et al._ (2019), yielding
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}=3.60(7)\times 10^{-15}$
m3/s. Using these ab initio values, we predict a cross section ratio of
$R=0.869$ in good agreement with the experimental value of $R=0.83(5)$.

Although the accuracy of the Rb+H2 PES has not been independently validated,
the computed cross section based on this PES is in good agreement with the
value found experimentally by comparison with Li. Since both the experimental
value and theoretical estimate for the Rb+H2 cross section agree with each
other within their respective error bars and are inconsistent with the value
extracted from fitting the Rb loss rate curve given the universal function
(approximately 50% larger), this indicates that the cross sections for Rb+H2
and Rb+He collisions previously determined using the universality result are
systematically overestimated Booth  _et al._ (2019); Shen  _et al._ (2020,
2021).

##  V Cross-Species Calibration

The application of cold atom trap losses to pressure measurements is an
exciting realization of a fully quantum pressure sensor. The key ingredient
for this sensor is the determination of the velocity-averaged total collision
cross-section between the trapped species and the background gas,
$\langle\sigma_{\rm{tot}}\,v\rangle$. There are three methods for determining
these quantities: calibration of the loss rate against a known pressure, ab
initio quantum scattering computation of $\langle\sigma_{\rm{tot}}\,v\rangle$,
or relying on the universality description of the quantum diffractive
collisions. No single method can span the entire range of collision partners.
Calibration against a known background pressure can be implemented for a
limited number of inert gas species where the pressure can be determined
independently using an orifice flow standard. The ab initio QS computation
technique may be limited to collisions between light species by the
computation complexity associated with accurately finding the PES for many-
electron systems. The universality approach, as shown here, is inaccurate for
collisions between low reduced mass and low polarizability collision partners.
To overcome its limitations, as we have demonstrated in this study, we
leveraged a cross-species calibration for the loss rate coefficients. While we
have calibrated $\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}$ to the ab
initio value of $\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}$, the
inverse method can readily be applied. Namely, when low mass species such as
Li are used as sensor atoms, it is unlikely that the quantum diffractive
universality description will be applicable to any collision induced loss
measurements, regardless of the collision partner. The strength of using
trapped Li sensor atoms, the ability to perform precise QS computations of
$\langle\sigma_{\rm{tot}}\,v\rangle$, is realized for only a handful of
particles. However, the Li cross sections can be calibrated against those for
a heavy sensor atom such as Rb, where the QDU technique is available for a
wide range of background gas collisions, including other significant vacuum
constituents such as N2 and Ar. In short, cross-species calibration allows us
to exploit the strengths of each sensor atom to effect an optimum quantum
pressure sensor/standard.

##  VI Conclusions

The main goal of this work has been to demonstrate, experimentally, an
anticipated deviation of the total collision cross section for light collision
partners from the predictions of the universal $p_{\rm{QDU6}}$ function Booth
_et al._ (2019); Shen  _et al._ (2020, 2021). We have carefully characterized
the loss rate coefficient, $\left<\sigma_{\rm{loss}}\,v\right>$, as a function
of trap depth for Rb-H2 collisions and fit these data to the universal
prediction. This yielded a value of
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}=5.52(9)\times 10^{-15}$
m3/s. Next, we implemented a ratiometric technique which compared the loss
rates of Li atoms and Rb atoms, confined in very shallow magnetic traps,
simultaneously exposed to background H2. The choice of Li for cross-species
calibration is motivated by the simplicity of Li - H2 interactions, which
makes highly accurate predictions of Li + H2 scattering cross sections
feasible. The ab initio predictions of cross sections for Li + H2 collisions
at room temperature were previously published by other authors Makrides  _et
al._ (2019, 2022a). To ensure the accuracy of our results, in the present
work, we have recomputed the rate coefficients for Li + H2 collisions,
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}=3.101(6)\times 10^{-15}$
m3/s, in good agreement with the value of
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}=3.13(6)\times 10^{-15}$
m3/s from Refs. Makrides  _et al._ (2019, 2022a). Under our experimental
conditions, the loss rate ratio is
$R=\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}:\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}=0.83(5)$,
yielding $\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}=3.8(2)\times
10^{-15}$ m3/s when calibrated to the Li-H2 total collision cross section
value. We have also computed the cross sections and rate coefficients for Rb +
H2 collisions using the PES from Upadhyay  _et al._ (2019), yielding
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}=3.60(7)\times 10^{-15}$
m3/s. Thus, the ab initio value of $R=0.869$ agrees with the experimentally
determined value and verifies that the experimental value for
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}$ deviates from the value
derived from the universal function, which would predict $R=0.567(3)$.

The universality of quantum diffractive collisions was previously exploited to
demonstrate self-calibrating, primary vacuum pressure standard based on
trapped Rb atoms Booth  _et al._ (2019); Madison  _et al._ (2018); Shen  _et
al._ (2020, 2021). The energy distribution function $p_{\rm{QDU6}}$ was shown
to be universal for collisions of neutral species because diffractive
collisions are predominantly determined by the long-range part of PES and
because the dependence of the total scattering cross section on short-range
interactions is removed by Maxwell-Boltzmann averaging. However, both the
relative contribution of long-range glancing collisions and the effect of
Maxwell-Boltzmann averaging are expected to diminish for species with small
mass and, consequently, small polarizability.

The present work demonstrates this expected departure of Rb + H2 diffractive
collisions from universality. We have shown that this limitation can be
overcome by performing a cross-calibration of
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Rb+H}_{2}}$ to
$\langle\sigma_{\rm{tot}}\,v\rangle_{\rm{Li+H}_{2}}$. However, we recognize
that there is a limited number of collision partners which are amenable to
this ab initio approach. This has several important implications: First, we
believe that heavy polarizable atoms may be a preferred choice of pressure
sensors. These can be calibrated for a wide range of collision partners via
the universality method while light collision partners, such as H2 and He, can
readily be calibrated against other sensor species, such as Li, for which
$\langle\sigma_{\rm{tot}}\,v\rangle$ can be computed accurately. Second, the
inverse method, namely a lower mass, lower polarizability sensor atom, like
Li, can be calibrated to a heavier mass sensor atom (such as Rb) for a wide
range of collision partners accessible to the universal quantum diffractive
collision formalism. Finally, our work brings to light the necessity of
assessing the limitations of universality of quantum diffractive collisions
for light background species as well as light sensor atoms.

##  VII Contributions

Author contributions are as follows. P.S., A.D., and R.S. took and analyzed
the high trap depth loss rate measurements for Rb expoed to H2. E.F. and D.U.
took and analyzed the combined Rb+H2 and Li+H2 measurements. R.V.K. and K.R.H.
performed and analyzed the Rb+H2 and Li+H2 coupled-channel quantum scattering
calculations. P.S. and J.L.B. performed and analyzed preliminary, single-
channel, quantum scattering calculations. K.W.M. and J.L.B. obtained funding
for the research and conceived and planned the study. K.W.M. wrote the
manuscript with contributions from the other co-authors.

###### Acknowledgements.

We acknowledge financial support from the Natural Sciences and Engineering
Research Council of Canada (NSERC/CRSNG) and the Canadian Foundation for
Innovation (CFI). This work was done at the Center for Research on Ultra-Cold
Systems (CRUCS). P.S. acknowledges support from the DFG within the GRK 2079/1
program. K.R.H. acknowledges financial support from the Quantum Electronic
Science & Technology (QuEST) Award given by the Stewart Blusson Quantum Matter
Institute.

##  Appendix A Heating rate corrections

In this section, we describe how ensemble heating is accounted for in our
determination of $\langle\sigma_{\rm{tot}}\,v\rangle$. For clarity, we repeat
here the expression in Eq. 10 describing the fraction of sensor atoms
remaining in the MT at times $t_{j}$ and $t_{k}$.

$\displaystyle\ln\Big{[}\frac{f_{j}(t_{j})}{f_{k}(t_{k})}\Big{]}=\Gamma_{\rm{loss}}(t_{k}-t_{j})+\Gamma_{0}(t_{k}-t_{j})$
$\displaystyle=\frac{\overline{\langle\sigma_{\rm{loss}}(U)\
v\rangle}\Big{[}(P_{k}-P_{0})t_{k}-(P_{j}-P_{0})t_{j}\Big{]}}{\rm{i_{\rm{g}}}k_{\rm{B}}T}+\Gamma_{0}(t_{k}-t_{j}).$
(16)

Here, we have replaced $\langle\sigma_{\rm{loss}}(U)\ v\rangle$ with
$\overline{\langle\sigma_{\rm{loss}}(U)\ v\rangle}$, indicating that the rate
coefficient, modeled by the universal function in Eq. 4, is averaged over the
energy distribution of the ensemble, $\rho(E)$ Shen  _et al._ (2021). This
accounts for the trap depth of each atom in the ensemble with total energy
$E\neq 0$ being different than the maximum trap depth, $E_{\mathrm{max}}$.

$\displaystyle\overline{\langle\sigma_{\rm{loss}}(U)\
v\rangle}=\frac{\int_{0}^{E_{\rm{max}}}\left<\sigma_{\rm{loss}}(E_{\rm{max}}-E)\
v\right>\rho(E)dE}{\int_{0}^{E_{\rm{max}}}\rho(E)dE}.$ (17)

We can now include collision-induced ensemble heating by explicitly writing
the ensemble energy distribution as a time dependent function, $\rho(E,t)$,
and evaluating the integrated loss rate coefficient from zero time to the
measurement time $t$. In this case, the ratio of the remaining atoms satisfies

$\displaystyle\ln\Bigg{[}\frac{f_{j}(t_{j})}{f_{k}(t_{k})}\Bigg{]}-\Gamma_{0}(t_{k}-t_{j})$
$\displaystyle=\frac{(P_{k}-P_{0})}{\rm{i_{\rm{g}}}k_{\rm{B}}T}\int_{0}^{t_{k}}\frac{\int_{0}^{E_{\rm{max}}}\left<\sigma_{\rm{loss}}(E_{\rm{max}}-E)\
v\right>\rho(E,t)dE}{\int_{0}^{E_{\rm{max}}}\rho(E,t)dE}dt$
$\displaystyle-\frac{(P_{j}-P_{0})}{\rm{i_{\rm{g}}}k_{\rm{B}}T}\int_{0}^{t_{j}}\frac{\int_{0}^{E_{\rm{max}}}\left<\sigma_{\rm{loss}}(E_{\rm{max}}-E)\
v\right>\rho(E,t)dE}{\int_{0}^{E_{\rm{max}}}\rho(E,t)dE}dt.$

To simplify the expression, we can factor out the loss rate coefficient at
zero time, $\overline{\langle\sigma_{\rm{loss}}(U)\ v\rangle}_{t=0}$, which
gives the following final expression,

$\displaystyle\ln\Bigg{[}\frac{f_{j}(t_{j})}{f_{k}(t_{k})}\Bigg{]}-\Gamma_{0}(t_{k}-t_{j})$
$\displaystyle=\frac{\overline{\langle\sigma_{\rm{loss}}(U)\
v\rangle}_{t=0}}{\rm{i_{\rm{g}}}k_{\rm{B}}T}\Big{[}(P_{k}-P_{0})H(t_{k})-(P_{j}-P_{0})H(t_{j})\Big{]},$
(19)

where we define the heating factor as,

$\displaystyle H(t^{\prime})$ $\displaystyle\equiv$
$\displaystyle\frac{\int_{0}^{t^{\prime}}\frac{\int_{0}^{E_{\rm{max}}}\left<\sigma_{\rm{loss}}(E_{\rm{max}}-E)\
v\right>\rho(E,t)dE}{\int_{0}^{E_{\rm{max}}}\rho(E,t)dE}dt}{\frac{\int_{0}^{E_{\rm{max}}}\left<\sigma_{\rm{loss}}(E_{\rm{max}}-E)\
v\right>\rho(E,t=0)dE}{\int_{0}^{E_{\rm{max}}}\rho(E,t=0)dE}}$ (20)
$\displaystyle\equiv$
$\displaystyle\frac{\int_{0}^{t^{\prime}}\frac{\int_{0}^{E_{\rm{max}}}\left<\sigma_{\rm{loss}}(E_{\rm{max}}-E)\
v\right>\rho(E,t)dE}{\int_{0}^{E_{\rm{max}}}\rho(E,t)dE}dt}{\overline{\langle\sigma_{\rm{loss}}(U)\
v\rangle}_{t=0}}.$

In order to evaluate these expressions, we measured the sensor ensemble energy
distribution as a function of the hold time. Specifically, we determined
$\rho(E,t)$ from the measured cumulative energy distribution of the atoms in
the trap. This was done by measuring the number of atoms remaining in the trap
after ejecting atoms above an energy cutoff using a radio-frequency field as
described in Ref. Shen  _et al._ (2020). We found that the distribution is
well approximated by a Maxwell Boltzmann distribution with an energy offset,

$\displaystyle\rho(E,t)$ $\displaystyle=$
$\displaystyle\Theta(E-E_{\rm{min}})\cdot
2\left(\frac{E-E_{\rm{min}}}{\pi}\right)^{\frac{1}{2}}\left(\frac{1}{k_{B}T(t)}\right)^{\frac{3}{2}}e^{-\frac{E-E_{\rm{min}}}{k_{B}T(t)}}.$

We measured the evolution of $\rho(E,t)$, specifically the variation of $T$
and $E_{\mathrm{min}}$ with hold time, at the two different trapping currents
used in this study, $I=60$ A and $I=200$ A. We observed that
$E_{\mathrm{min}}$ was constant and that the temperature varied linearly in
time as shown in figure 6. The ensemble temperature is modeled by
$T(t)=T_{0}+mt$.

The experimentally determined ensemble energy distribution can then be used to
evaluate the integrals in Eq. 20 to determine the heating factor. Using this,
we can solve for the value of $\overline{\langle\sigma_{\rm{loss}}(U)\
v\rangle}_{t=0}/\rm{i_{\rm{g}}}$, by plugging the computed heating factors
into Eq. 19,

$\displaystyle\frac{\overline{\langle\sigma_{\rm{loss}}(U)\
v\rangle}_{t=0}}{\rm{i_{\rm{g}}}k_{\rm{B}}T}=\frac{\ln\Big{[}\frac{f_{j}(t_{j})}{f_{k}(t_{k})}\Big{]}-\Gamma_{0}(t_{k}-t_{j})}{\Big{[}(P_{k}-P_{0})H(t_{k})-(P_{j}-P_{0})H(t_{j})\Big{]}}.$
(22)

Figure 6: The measured temperature of the sensor ensemble versus the holding
time for atoms trapped in $F=2$ state at the magnetic trapping current, $I=60$
A (open blue squares) and $I=200$ A (blue filled squares). The ensemble
temperature is observed to increase linearly with the holding time. The dashed
lines are linear fits to the data, $T(t)=T_{0}+mt$, with
$T_{0}=377(26),944(69)\;\mu$K and $m=115(14),339(36)\;\mu$K/s respectively.

However, the right-hand side of Eq. 22 also depends on the unknown parameter,
$\langle\sigma_{\rm{loss}}(U)\ v\rangle$ (included in the heating factor).
This necessitates a self-consistent method to find the solution of Eq. 22. To
achieve this, we used the following iteration fitting method to determine the
final $\langle\sigma_{\rm{tot}}\,v\rangle$ and $\rm{i_{\rm{g}}}$ values:

  1. 1.

We pick an initial guess value for $\langle\sigma_{\rm{tot}}\,v\rangle$ to
determine a guess for the $\langle\sigma_{\rm{loss}}(U)\ v\rangle$ function
and feed this into the expression for $H$, Eq. 20, to compute the values of
$H(t_{j})$ and $H(t_{k})$ for the right-hand side of Eq. 22.

  2. 2.

Using the computed $H$ values, we solve for the left-hand side,
$\frac{\overline{\langle\sigma_{\rm{loss}}(U)\
v\rangle}_{t=0}}{\rm{i_{\rm{g}}}}$, of Eq. 22.

  3. 3.

We then fit the computed results,
$\frac{\overline{\langle\sigma_{\rm{loss}}(U)\
v\rangle}_{t=0}}{\rm{i_{\rm{g}}}}$, in step 2 to the universal fitting
function to obtain fitted values $\langle\sigma_{\rm{tot}}\,v\rangle$ and
$\rm{i_{\rm{g}}}$.

  4. 4.

We replace the initial guessed value for $\langle\sigma_{\rm{tot}}\,v\rangle$
with the newly fitted result and repeat the steps from 2 to 4 until the fitted
$\langle\sigma_{\rm{tot}}\,v\rangle$ converges.

We have found this iteration method works well and observed a convergence of
the $\langle\sigma_{\rm{tot}}\,v\rangle$ value after 3 steps even when the
initial guess was off by a factor of 6, as illustrated by the simulated
results shown in figure 7. Convergence happens because if the initial guess of
$\langle\sigma_{\rm{tot}}\,v\rangle$ is overestimated (underestimated), the
heating rate will also be overestimated (underestimated) which leads to a
smaller (larger) fitted $\langle\sigma_{\rm{tot}}\,v\rangle$ result.
Therefore, the guessed value at each iteration step will be closer to the true
value. Using this method we could determine the
$\langle\sigma_{\rm{tot}}\,v\rangle$ value with systematic errors from heating
removed.

Figure 7: Fitted $\langle\sigma_{\rm{tot}}\,v\rangle$ results as a function of
the iteration step for atoms trapped in $F=2$ state. The
$\langle\sigma_{\rm{tot}}\,v\rangle$ value at step zero represents the value
of the initial guess. Here the initial guesses are varied by more than a
factor of 6 and yet they all converge to the same
$\langle\sigma_{\rm{tot}}\,v\rangle$ value, $5.49\times
10^{-15}\rm{m}^{3}/\rm{s}$.

##  Appendix B Quantum Scattering Calculations

To describe the quantum scattering dynamics for Li+H2 and Rb+H2 collisions, we
solve the time-independent Schrödinger equation using the coupled channel
approach of Arthurs and Dalgarno Arthurs and Dalgarno (1960).

We represent the interaction potential energy of both Li+H2 and Rb+H2 as a
Legendre polynomial expansion,

$U(R,\theta)=\sum_{\lambda}U_{\lambda}(R)P_{\lambda}(\mathrm{cos}\theta),$
(23)

where $R$ is the magnitude of the centre of mass (COM) of H2 to the COM of the
atom, and $\theta$ is the angle between $\vec{R}$ and the molecular axis. For
Li+H2, the coefficients $U_{\lambda}(R)$ are provided by Makrides et al. for
$\lambda=0,2,4$ and $R<20$ a.u.Makrides  _et al._ (2019, 2022a) The Rb+H2
interaction potential ${U}(R,\theta)$ from Ref. Upadhyay  _et al._ (2019) is
used with permission from Upadhyay et al. For both systems, we employed a
Degli-Esposti-Werner function,

$\begin{split}{f(x)}&{=(g_{0}+g_{1}x+g_{2}x^{2}+g_{3}x^{3}+g_{4}x^{4}+g_{5}x^{5})}\\\
&{\times\mathrm{exp}(-a_{1}x-a_{2})-\frac{1}{2}(1+\mathrm{tanh}(1+tx))}\\\
&{\times(C_{6}x^{-6}+C_{8}x^{-8}+C_{10}x^{-10}),}\end{split}$ (24)

in order to interpolate and extrapolate the data Esposti and Werner (1990).

The collision cross sections are computed for energies from 0.1 cm-1 to 2000
cm-1. Calculations for a given initial rotational state $N_{i}$ were performed
seperately, for $N_{i}=0-6$. Additionally, to account for the two spin isomers
of molecular Hydrogen, calculations with orthohydrogen can only have odd
values of the $N$ (1,3,…) and parahydrogen can only have even values of $N$
(0,2,…). This is because protons are fermions which require an antisymmetric
wavefunction; since orthohydrogen has symmetric nuclear spin, it must have
antisymmetric rotational wavefunctions, and since parahydrogen has
antisymmetric nuclear spin, it must have symmetric rotational wavefunctions.

For Li+H2, a basis of $J=0-125$ is employed. A calculation with a given
initial state $N_{i}$, employs a basis with the maximum rotational quantum
number $N_{\mathrm{max}}$=12. For the $R$-propagaton procedure, we employ a
grid from 2 a.u. to 50 a.u. with 0.0025 a.u. spacing.

For Rb+H2, we use a basis with $J_{\mathrm{max}}=300$ and
$N_{\mathrm{max}}$=12. The $R$-propagation grid goes from 4 a.u. to 120 a.u.
for collision energies greater than 1 cm-1, and 4 a.u. to 250 a.u. for
energies less than 1 cm-1.

Figure 8: Total cross-section in log-log format plotted as a function of
velocity for Li+H2 (left) and Rb+H2 (right). The legend differentiates the
different initial rotational states $N_{i}$.

Given the total cross-section $\sigma_{N_{i}}(E_{\mathrm{col}})$ from the
scattering calculations (see Figure 8), our goal is to determine the averaged
thermalized rate coefficient

$K_{\mathrm{all}}(T)=\frac{\Gamma}{n_{\mathrm{BG}}},$ (25)

where $\Gamma$ is the trap loss rate and $n_{\mathrm{BG}}$ is the background
gas density. We first calculate the total collision rate coefficient
$\langle\sigma v\rangle_{N_{i}}$ for a given initial state by integrating the
product of the cross-section and the velocity $v$ of the molecule over the
Maxwell-Boltzmann distribution for collision energy $p(E_{\mathrm{col}})$,

$\langle\sigma
v\rangle_{N_{i}}\;=\int\sigma(E_{\mathrm{col}})v_{N_{i}}p(E_{\mathrm{col}})dE_{\mathrm{col}}$
(26)

where

$p(E_{\mathrm{col}})=2\sqrt{\frac{E_{\mathrm{col}}}{\pi}}\left(\frac{1}{k_{B}T}\right)^{3/2}\mathrm{exp}\left(-\frac{E_{\mathrm{col}}}{k_{B}T}\right).$
(27)

Next, we let $L_{N_{i}}(T)$ (see Figure 9) denote the thermally-averaged rate
coefficient for a given initial state,

$\begin{split}L_{N_{i}}(T)=\frac{\mathrm{exp}(-E_{N_{i}}/k_{B}T)}{Z}\langle\sigma
v\rangle_{N_{i}},\end{split}$ (28)

which includes the partition function

$Z=\sum_{N}(2N+1)(2-(-1)^{N})\mathrm{exp}(-E_{N}/k_{B}T)$ (29)

for all possible initial states $N=0\rightarrow 6$. In the above equation,
$(2-(-1)^{N})$ accounts for the ratio 3:1 of orthohydrogen to parahydrogen.
Then, the averaged thermalized rate coefficient is

$\langle\sigma_{\rm{tot}}\,v\rangle=\sum_{N_{i}}(2N_{i}+1)(2-(-1)^{N_{i}})\times
L_{N_{i}}(T).$ (30)

For room temperature $T=300$ K, we calculated
$\langle\sigma_{\rm{tot}}\,v\rangle\approx 3.570\times
10^{-15}\;\mathrm{m}^{3}/\mathrm{s}$ for Rb+H2 and
$\langle\sigma_{\rm{tot}}\,v\rangle\approx 3.101\times 10^{-15}$ m3/s for
Li+H2, converged to four significant figures. Then, the ratio

$\begin{split}R&=\frac{\langle\sigma_{\rm{tot}}\,v\rangle^{\mathrm{Li}+\ce{H2}}(T=300~{}\mathrm{K})}{\langle\sigma_{\rm{tot}}\,v\rangle^{\mathrm{Rb}+\ce{H2}}(T=300~{}\mathrm{K})}\\\
&=\frac{3.101\times 10^{-15}\;\mathrm{m}^{3}/\mathrm{s}}{3.570\times
10^{-15}\;\mathrm{m}^{3}/\mathrm{s}}\\\ &=0.8686\end{split}$ (31)

is within the error bars of the experiment.

Figure 9: Top row: the averaged thermalized rate coefficient is plotted as a
function of temperature for Li+H2 (top left) and Rb+H2 (top right). The final
theory values that we quote in the paper are taken from the thick black line
at 300 K in these subfigures. Bottom row: Equation 28 plotted for initial
state $N_{i}=0$ only, as the curve for the other initial states are almost
overlapping.

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