# Trapped particle evolution driven by residual gas collisions

Avinash Deshmukh1, Riley A. Stewart1, Pinrui Shen1, James L. Booth2, and Kirk
W. Madison1,∗ 1Department of Physics & Astronomy, University of British
Columbia,
6224 Agricultural Road, Vancouver, B.C., V6T 1Z1, Canada  2 Physics
Department, British Columbia Institute of Technology, 3700 Willingdon Avenue,
Burnaby, B.C. V5G 3H2, Canada  ∗Corresponding author<EMAIL_ADDRESS>
###### Abstract

We present a comprehensive mathematical model and experimental measurements
for the evolution of a trapped particle ensemble driven by collisions with a
room-temperature background vapor. The model accommodates any trap geometry,
confining potential, initial trapped distribution, and other experimental
details; it only depends on the the probability distribution function
$P_{t}(E)$ for the collision-induced energy transfer to the trapped ensemble.
We describe how to find $P_{t}(E)$ using quantum scattering calculations and
how it can be approximated using quantum diffractive universality Booth  _et
al._ (2019). We then compare our model to experimental measurements of a 87Rb
ensemble energy evolution exposed to a room temperature background gas of Ar
by means of a single parameter fit for the total collision rate $\Gamma$. We
extracted a collision rate of $\Gamma=0.646(1)\ \text{s}^{-1}$. This is
compared to a value of 0.664(4) $\text{s}^{-1}$ found by the commonly used
method of zero-trap depth extrapolation, a 2.8% correction that is a result of
our model fully taking ensemble loss and heating into account. Finally, we
report a five-fold increase in the precision of our collision rate extraction
from the experimental data.

###### pacs:

34.50.-s, 34.50.Cx, 34.80.Bm, 34.00.00, 30.00.00, 67.85.-d, 37.10.Gh

††preprint: APS/??

##  I Introduction

In a vacuum, single particles or ensembles of single particles are constantly
perturbed by collisions with atoms and molecules in the residual background
vapor. Energetic elastic and inelastic collisions can result in neutral
particle loss from traps Migdall  _et al._ (1985); Bjorkholm (1988);
Hemmerling  _et al._ (2014); Rennick  _et al._ (2014) and ion replacement in
ion traps Obšil  _et al._ (2019). Collisions that do not impart enough energy
to free a trapped particle also constitute an important decoherence mechanism.
These collisions can induce heating in neutral ensembles Beijerinck (2000a,
b); Cornell  _et al._ (1999); Bali  _et al._ (1999), melting of ion Coulomb
crystals Kranzl  _et al._ (2022); Pagano  _et al._ (2018); Kiesenhofer  _et
al._ (2023), reordering of ion chains Aikyo  _et al._ (2020); Spivey  _et al._
(2022), decoherence in trapped-atom Gibble (2013) and trapped-ion optical
atomic clocks Hankin  _et al._ (2019), and localization-induced decoherence of
single levitated nanoparticles Magrini  _et al._ (2021); Aspelmeyer (2023).
Understanding the nature of these collisions is important for both producing
cold particle ensembles and using them in quantum sensing and quantum
information applications.

While understanding the qualitative role of neutral atom loss and ensemble
heating due to collisions with the residual background vapor was important for
achieving Bose-Einstein condensation in alkali metal gases Monroe  _et al._
(1993); Cornell  _et al._ (1999), a rigorous and quantitative understanding of
the two effects is essential for the emerging field of quantum vacuum
metrology. Here, precise measurements of a sensor particle’s collision rate
$\Gamma=n\langle\sigma(v)v\rangle$ can be used with knowledge of the
thermally-averaged, velocity-weighted collision cross section
$\langle\sigma(v)v\rangle$ to determine the background particle density $n$
Booth  _et al._ (2011); Madison (2012). Previous measurements of sensor atom
collision rates were conducted by observing the sensor-atom loss rate from
shallow magnetic Fagnan  _et al._ (2009); Van Dongen  _et al._ (2011);
Scherschligt  _et al._ (2017, 2018); Eckel  _et al._ (2018); Booth  _et al._
(2019); Shen  _et al._ (2020, 2021); Stewart  _et al._ (2022); Ehinger  _et
al._ (2022); Shen  _et al._ (2023); Barker  _et al._ (2023) and optical dipole
traps Makhalov  _et al._ (2016); Makhalov and Turlapov (2017). As illustrated
in Figure 1, the challenge with this approach lies in the fact that even for
exceedingly small trap depths, not every collision results in loss Fagnan  _et
al._ (2009); Van Dongen  _et al._ (2011). This complicates extracting the
particle’s collision rate from the observed trap loss rate as the
instantaneous loss rate varies in time due to its energy dependence. Indeed,
both the instantaneous loss rate and heating rate depend on the maximum trap
depth (defined for an atom with zero energy) and on the energy distribution of
the sensor atoms in the trap. Collisions that do not eject sensor atoms change
(heat) the energy distribution of the sensor ensemble, causing the effective
trap depth and corresponding loss rate to vary with time. This means that a
simple exponential loss model does not adequately capture the behavior of trap
loss. Furthermore, as shown here, extrapolation of this loss rate to zero trap
depth to obtain the total collision rate relies on data with very low signal-
to-noise ratio (SNR), because low trap depth measurements requires culling
high energy atoms. These measurements subsequently have a large uncertainty on
the total collision rate. This problem cannot be corrected by using higher
trap depth measurements, because the variation in the loss rate sharply
decreases at small trap depths. This flattening of the loss rate curve cannot
be observed at larger trap depths, which makes interpolation from large trap
depth inaccurate. Our recent work has accounted for the role of the sensor-
atom energy distribution and has added heating corrections to loss rate
measurements Stewart  _et al._ (2022); Shen  _et al._ (2021, 2023); however, a
more complete description requires an analysis of how the trapped particle
energy distribution evolves in time due to these collisions.

In this work, we present a comprehensive theoretical treatment of trapped
particle loss and heating due to collisions with a room-temperature background
vapor Deshmukh (2023). We derive an analytic expression for the trapped
particle energy distribution evolution in time, based solely on a measured
ensemble energy distribution (as the initial condition), the total collision
rate, and the probability distribution function $P_{t}(E)$ for the energy
transferred in a collision. Our treatment differs from prior work on the trap
depth dependence of ensemble heating rates Beijerinck (2000a, b); Stewart  _et
al._ (2022); Shen  _et al._ (2021, 2023) in that it is valid for all time and
for any initial energy distribution. Moreover, it does not rely on a
particular form for the transferred energy distribution. We present a direct
comparison of the theory with experimental measurements of a magnetically
trapped rubidium-87 ensemble exposed to a background vapor of argon. By
analyzing the energy evolution of the particles in the trap, we determine the
total collision rate with a precision of better than 0.2%. This represents a
significant improvement (more than a factor of 5) in the precision for the
total collision rate determination. Our theory was recently applied in Ref.
Eckel (2023) to correct the total collision rate reported in a comparison
between two cold atom vacuum standards and a classical vacuum metrology
apparatus Barker  _et al._ (2023).

$E$(a)(b)$E$(c)(d) Figure 1: A schematic showing how background collisions
cause the trapped energy ensemble to evolve. The vertical axis denotes energy
and the horizontal axis is spatial position in a linear trap potential;
however, the form of the confining potential is irrelevant to the analysis
presented. The dashed horizontal line demarcates the maximum trap depth
$E_{m}$. (a) Upon loading in the trap, all of the particles (blue circles)
have a low energy and have not undergone any collisions with the background.
(b) At short times, some of the particles (light blue circles) have undergone
a single collision, which causes them to go to a new state (orange circles).
Particles with a new energy larger than $E_{m}$ are lost. Those with less
remain in the trap, but with a higher energy than before – the ensemble has
heated. (c) At longer times, particles continue to undergo collisions from a
zero to one collision state and also from a one to two collision state (green
circles). The one collision particles have a higher energy than they had at
the initial time, leading to a higher loss rate. (d) At long times, the energy
distribution bears very little similarity to the initial distribution. More
particles are in collided states than in initial states, and there are some
particles that have undergone three collisions (purple solid circles). These
particles can have an energy very close to $E_{m}$, and thus have a very small
probability of being retained after a fourth collision. This allows the
truncation of our analysis after a few collision orders.

##  II Evolution Model

We consider an ensemble of non-interacting particles with mass $m$ subject to
a spatially dependent trap potential $U$. A particle with initial energy
$E_{i}$ and momentum $\textbf{p}_{0}$, subject to $k$ collisions which
transfer momenta $\textbf{p}_{1},...,\textbf{p}_{k}$ to the particle
respectively, will have a final energy $E_{f}$ given by:

$E_{f}=E_{i}+\sum_{k=1}^{n}\left(E_{k}+\dfrac{\textbf{p}_{k}}{m}\cdot\sum_{i=0}^{k-1}\textbf{p}_{i}\right)$
(1)

where $E_{k}=p_{k}^{2}/2m$. Details are shown in Appendix A. Since the
probability of a collision event is independent of the number of previous
collisions undergone by the particle, we model the collisions as Poissonian
events that occur with a constant collision rate $\Gamma$. The fraction
$\rho_{k}$ of the ensemble that will have undergone $k$ collisions in a time
$t$ is then:

$\rho_{k}(\Gamma t)=\dfrac{(\Gamma t)^{k}e^{-\Gamma t}}{k!}$ (2)

The energy distribution of the ensemble $f(E,t)$ at time $t$ will be the sum
of the energy distributions $f_{k}(E)$ corresponding to the subset of
particles that underwent exactly $k$ collisions weighted by the Poissonian
probability that $k$ collision events occurred. We can write this as: Deshmukh
(2023)

$f(E,t)=\sum_{k=0}^{\infty}\dfrac{(\Gamma t)^{k}e^{-\Gamma t}}{k!}f_{k}(E)$
(3)

The distribution $f_{k}(E)$ can be computed using the initial energy
distribution $f_{0}(E)$ and the probability $P_{t}(E)$ that a collision
transfers an energy $E$. Since the incident velocities of the background
particles are isotropic, the momentum imparted by the collisions is also
isotropic. Then, if we average over the ensemble of particles at a certain
energy, the contribution of the cross-term $\textbf{p}\cdot\textbf{p}_{t}/m$
appearing in Eq. 1 will vanish. Having appropriate cross-terms such that
$E_{f}<E_{i}$ is the only way we can have a collision that cools the trapped
particle, so we do not need to consider such collisions here. Explicitly, this
means that $P_{t}(E)$ vanishes for $E<0$. We can therefore perform $k$
convolutions of $f_{0}(E)$ with $P_{t}(E)$ for all positive energies to find
the distribution $f_{k}(E)$:

$\displaystyle f_{k}(E)=\int_{0}^{\infty}$ $\displaystyle dE_{1}\
P_{t}(E-E_{1})$ (4)
$\displaystyle\times\prod_{j=2}^{k}\left(\int_{0}^{\infty}dE_{j}P_{t}(E_{j-1}-E_{j})\right)f_{0}(E_{k})$

where $P_{t}(E<0)=0$ and the product denotes iterated integrals. Substituting
this into our expression for the full distribution, we have:

$\displaystyle f(E,t)=\sum_{k=0}^{\infty}$ $\displaystyle\dfrac{(\Gamma
t)^{k}e^{-\Gamma t}}{k!}\int_{0}^{\infty}dE_{1}\ P_{t}(E-E_{1})$ (5)
$\displaystyle\times\prod_{j=2}^{k}\left(\int_{0}^{\infty}dE_{j}P_{t}(E_{j-1}-E_{j})\right)f_{0}(E_{k})$

Sample plots showing how the total energy distribution and the populations of
the $f_{k}(E)$ distributions change in time are shown in Fig. 2.

In traps of finite depth, particles above some energy $U$ are not trapped, and
all collisions that increase the energy above this trap depth result in loss.
In this case, we only convolve the distributions up to the energy $U$, so the
previous expression becomes:

$\displaystyle f(E,t)=$
$\displaystyle\Theta(U-E)\sum_{k=0}^{\infty}\dfrac{(\Gamma t)^{k}e^{-\Gamma
t}}{k!}\int_{0}^{U}dE_{1}\ P_{t}(E-E_{1})$ (6)
$\displaystyle\times\prod_{j=2}^{k}\left(\int_{0}^{U}dE_{j}P_{t}(E_{j-1}-E_{j})\right)f_{0}(E_{k})$

Because $P_{t}(E)$ does not necessarily vanish for $E>U$, the convolution up
to $U$ does not result in a normalized distribution. This reflects loss from
the trap; particles that are transferred an energy greater than the trap depth
will be lost from the trap and will not appear in experimentally measured
energy distributions. We therefore see that this expression provides the
trapped particle energy distribution accounting for both heating and loss.

(a)(b)(c)(d) Figure 2: Cumulative energy distributions for the collision sub-
ensembles. We plot the Poisson probability weighted cumulative energy
distributions of the $k$-collision sub-ensembles $f_{k}(E)$ calculated using
Eqn. 4 at different times. Our initial distribution is a Maxwell-Boltzmann
distribution for an 87Rb gas at $T=300\ \mathrm{\mu K}$, and we use the
transfer energy distribution for the ${}^{87}\text{Rb}+\text{Ar}$ interaction
discussed in Section III. (a) At $\Gamma t=0$, the total distribution (dashed
black) is equal to the initial distribution $f_{0}(E)$ (blue) because the
ensemble has not yet undergone collisions. (b) At $\Gamma t=1$, the ensemble
is still dominated by $f_{0}(E)$, but there is also a small contribution from
$f_{1}(E)$ (orange) and a much smaller contribution from $f_{2}(E)$ (green).
We see the shape of the total distribution as a result of these new
contributions. (c) At $\Gamma t=3$, there are a similar number of particles in
the $f_{0}(E)$ and $f_{1}(E)$ sub-ensembles. The contribution of the
$f_{2}(E)$ has increased, and $f_{3}(E)$ (purple) is nonzero. (d) At $\Gamma
t=8$, the distribution is now dominated by the $f_{1}(E)$ sub-ensemble, and
$f_{0}(E)$ and $f_{2}(E)$ have approximately equal populations. The total
distribution has now changed significantly from the initial distribution, and
only reflects $f_{0}(E)$ in the small energy regime.

##  III The Energy Transfer Distribution

We now discuss the energy transfer probability distribution from a thermal
background to a trapped particle assuming elastic scattering. We will neglect
inelastic collisions in the transferred energy distribution, because we assume
that all inelastic collisions generate trap loss. We discuss two methods for
finding $P_{t}(E)$. The first method is to compute it from first principles
using interaction potentials and quantum scattering calculations. The second
method is to use a prediction based on quantum diffractive universality, under
which interactions between the trapped and background species that are
sufficiently dominated by a long range van-der Waals type interaction follow a
universal trap depth dependent loss function Booth  _et al._ (2019); Shen  _et
al._ (2020).

Figure 3: The transfer energy distribution, $P_{t}(E)$ for rubidium-87 atoms
subject to collisions by argon atoms at room temperature. The solid blue line
is $P_{t}(E)$ obtained from Eqn. 11 using the scattering amplitudes computed
by quantum scattering calculations as input. The orange dotted line is
obtained by differentiating the trap depth dependent loss rate based on
quantum diffractive universality (shown in Eqn. 13) with parameters found in
Ref Shen  _et al._ (2021). These distributions are not normalized over the
range of energies shown, reflecting the fact that collisions can cause loss.
The two curves are very similar; the largest deviation between the two is
3.6%, and the cumulate difference between the them is 2.28%. As we discuss in
Section IV, the total collision rate extracted using these two different
$P_{t}(E)$ curves differ by 0.61%. Since the quantum scattering calculations
require significant computational resources, it may be more convenient and
sufficiently accurate to simply use the universal polynomial.

###  III.1 Transfer probability from quantum scattering calculations

We consider a trapped particle of mass $m_{t}$ and velocity $\textbf{v}_{t}$
subject to an incident background particle of mass $m_{b}$ and velocity
$\textbf{v}_{b}$. We can reduce the analysis of the collision to the case of
single incident particle in the center of mass frame with reduced mass
$\mu=m_{t}m_{b}/(m_{t}+m_{b})$ and velocity
$\textbf{v}=\textbf{v}_{t}-\textbf{v}_{b}$ subject to the interaction
potential. Consider such a particle in the plane-wave state $\ket{\textbf{k}}$
with an initial momentum $\hbar\textbf{k}$. The probability amplitude of the
particle transitioning into a state $\ket{\textbf{k}^{\prime}}$ with momentum
$\hbar\textbf{k}^{\prime}$ is given by the scattering amplitude
$\mathcal{F}(\textbf{k},\textbf{k}^{\prime})$. Because we are in the center of
mass frame, this amplitude is only nonzero if
$|\mathbf{k}|=|\mathbf{k^{\prime}}|$.

We assume that the potential energy surface of the interaction potential is
spherically symmetric. In this case, the scattering amplitude must also be
rotationally invariant, and it can be written as:

$\mathcal{F}(\textbf{k},\textbf{k}^{\prime})=\sum_{\ell=0}^{\infty}(2\ell+1)\mathcal{F}_{\ell}(k)P_{\ell}(\cos\theta)$
(7)

where $\theta$ is the angle between k and $\textbf{k}^{\prime}$, $P_{\ell}$
are the Legendre polynomials, and the partial wave amplitudes
$\mathcal{F}_{\ell}$ are functions we must compute. We can re-write the
scattering amplitudes as functions of the incoming velocity of the background
particle and the scattering angle as:

$\mathcal{F}_{v}(\theta)=\sum_{\ell=0}^{\infty}(2\ell+1)\mathcal{F}_{\ell}(v/\mu\hbar)P_{\ell}(\cos\theta)$
(8)

The probability $\mathcal{P}_{v}$ that a particle with velocity $v$ transfers
an energy $E$ is given by the probability that the particle scatters at a
relative angle $\theta$ corresponding to that energy normalized by the total
probability that the collision occurs. This normalization constant is the
total velocity cross-section $\sigma(v)$. As calculated in Appendix B, the
energy transferred by an incident particle of relative velocity $v$ that
scatters with an angle $\theta$ is given by:

$E_{t}=\dfrac{2\mu^{2}v^{2}}{m}\sin[2](\frac{\theta}{2})$ (9)

Using this, we can write $\mathcal{P}_{v}$ as:

$\mathcal{P}_{v}(E)=\dfrac{\absolutevalue{\mathcal{F}_{v}(\theta)}^{2}}{\sigma(v)}\delta\left(E-\dfrac{2\mu^{2}v^{2}}{m}\sin^{2}\left(\dfrac{\theta}{2}\right)\right)$
(10)

To convert this into the ensemble energy transfer distribution, we must
average over the relative velocity distribution. As shown in Appendix C, this
can be very well approximated by the Maxwell Boltzmann distribution that
describes the background gas in the rest frame. We also account for the
difference in the fluxes in the background gas by adding a velocity weight.
Furthermore, the probability of a collision occurring depends on the velocity,
so we must also add a velocity cross-section weighting which cancels out with
the velocity cross-section in the denominator of Eqn. 10. We normalize by the
total ensemble averaged velocity-weighted cross-section to ensure unit
probability when the distribution is integrated over all energies. This gives
us the expression:

$\displaystyle P_{t}(E)=$
$\displaystyle\dfrac{4\pi}{\langle\sigma(v)v\rangle}\left(\dfrac{m_{b}}{2\pi
k_{\text{\tiny$B$}}T}\right)^{3/2}\int_{0}^{\infty}dv\ \int_{0}^{\pi}d\theta\
\sin\theta\ $ (11) $\displaystyle\times
v^{3}e^{-m_{b}v^{2}/2k_{\text{\tiny$B$}}T}\absolutevalue{\mathcal{F}_{v}(\theta)}^{2}\delta\left(E-\dfrac{2\mu^{2}v^{2}}{m}\sin^{2}\left(\frac{\theta}{2}\right)\right)$

To find the energy transfer distribution function $P_{t}(E)$, for our
comparison with measurements of a 87Rb sensor ensemble exposed to an Ar gas,
we first calculated the scattering amplitudes $\mathcal{F}_{v}(\theta)$ using
the interaction potential for Rb+Ar from Ref. Medvedev  _et al._ (2018) as
input to a single channel quantum scattering computation as described in Ref.
Booth  _et al._ (2019). We then used these values as input to Eqn. 11 to find
$P_{t}(E)$. The results are shown in Figure 3.

###  III.2 Transfer probability from quantum diffractive universality

Another approach to finding the energy transfer distribution function is to
use the universality predictions of Refs. Booth  _et al._ (2019); Shen  _et
al._ (2020). There, it was shown that the loss rate $\Gamma_{\ell}$ of a
particle with zero initial energy as a function of the trap depth $U$
normalized to the total collision rate $\Gamma$, can be described by:

$\dfrac{\Gamma_{\ell}(U)}{\Gamma}=1-\sum_{k=1}^{\infty}\beta_{k}\left(\dfrac{U}{U_{d}}\right)^{k}$
(12)

where the $\beta_{k}$ are species-independent coefficients and $U_{d}$ is a
species-dependent quantum diffraction energy scale. The values of the
coefficients for the first six terms and the experimentally measured values of
$U_{d}$ for 87Rb subject to various background species can be found in Refs.
Booth  _et al._ (2019); Shen  _et al._ (2020).

If the trapped ensemble distribution is localized to a single energy, the
normalized loss rate is the probability that a collision will transfer an
energy greater than $U$. In other words, it is the complement of the
cumulative energy transfer distribution. Thus, we can obtain $P_{t}(E)$ by
differentiating Eqn. 12, which yields:

$P_{t}(E)=\dfrac{1}{U_{d}}\sum_{k=0}^{\infty}(k+1)\beta_{k+1}\left(\dfrac{U}{U_{d}}\right)^{k}$
(13)

This expression is easy to evaluate, and, as shown in Figure 3, the
$\beta_{k}$ and $U_{d}$ values taken from Refs. Booth  _et al._ (2019); Shen
_et al._ (2020) provide a good approximation to the transfer distribution
found using a quantum scattering calculation for Rb+Ar collisions.

##  IV Experimental Measurement

The experimental apparatus used in this study is described in Ref. Booth  _et
al._ (2019). Briefly, it consists of a Rb source, a two-dimensional (2D)
magneto-optical trap (MOT) section, a three-dimensional (3D) MOT section, and
a pump section. The experiment procedure begins by loading Rb atoms into the
2D MOT from the Rb vapor generated by the atom source. A laser beam aligned to
the center of the 2D MOT pushes atoms through a differential pumping tube into
the 3D MOT region where they accumulate. The magnetic field configuration of
the 3D MOT is a spherical quadrupole and is operated with an axial gradient of
13.6 G/cm along the vertical direction. In both the 2D MOT and 3D MOT, the
trapping lasers are detuned 12 MHz below the 87Rb $5^{2}S_{1/2}\to
6^{2}P_{3/2},F=2-3^{\prime}$ transition, and the ‘repump’ lasers are tuned on
resonance with the $5^{2}S_{1/2}\to 5^{2}P_{3/2},F=1-2^{\prime}$ transition.
After the particles are loaded into the 3D MOT, the pump laser frequency is
detuned to 60 MHz below the $F=2\to F=3$ transition for 50 ms to further cool
the atom ensemble. Next, the atoms are optically pumped to the $F=1$ hyperfine
ground state by shutting off the repump laser and leaving the pump laser on
for another 4 ms. Then, we extinguish the pump laser and ramp up the magnetic
field axial gradient to 272 G/cm in 10 ms. Only atoms in the low-field seeking
state, $\left.|F=1,\ m_{F}=-1\right>$ are confined in the magnetic trap (MT).
The atoms are held in the MT for a time $t$ before being recaptured in the MOT
using the same detunings during the stage just before hyperfine pumping. To
count the atom number, we image the MOT ensemble fluorescence onto a
photodetector inducing a voltage, $V$, that is proportional to the atom
number, $N$. The fraction of atoms in the MT recaptured in the MOT is found by
dividing the recaptured atomic fluorescence $V_{\rm{MT}}(t)$ by the initial
MOT fluorescence $V_{\rm{MOT}}$:

$R(t)=\frac{V_{\rm{MT}}(t)}{V_{\rm{MOT}}}$ (14)

To set the magnetic trap depth for the atom ensemble, an RF field is applied
at the end of the hold time. The RF field induces transitions of the atoms
from the trapped state ($\left.|F=1,\ m_{F}=-1\right>$) to the $\left.|F=1,\
m_{F}=0,+1\right>$ states that are not trapped by the MT. During the RF
radiation phase, the RF frequency is swept between $\nu_{\rm{min}}$ and
$\nu_{\rm{max}}$. Atoms with the magnetic energies above
$E_{m}=\hbar\nu_{\rm{min}}$ encounter a resonant RF field and are flipped into
an untrappable spin state and escape from the trap. We therefore define
$E_{m}$ to be the maximum trap depth. The RF radiation is swept sufficiently
for a sufficiently long time in comparison to the motion of the particles in
the trap to ensure that every particle which reaches the magnetic energy
$E_{m}$ during their motion will encounter the RF field. As a result, all
particles that achieve a maximum magnetic energy of $E_{m}$ or higher during
their evolution are ejected.

It should be noted that the maximum magnetic energy achieved by the particles
is not necessarily the total energy of the particle. Through the course of a
particle’s trajectories, its energy is divided between magnetic potential
energy, gravitational potential energy, and kinetic energy. It is possible
that at the time where the particle reaches its maximum magnetic energy, it
has nonzero gravitational and kinetic energies. However, in the limit of large
magnetic field gradients, these energies will be negligible relative to the
magnetic energy. We therefore conduct our experiments at a gradient that is
sufficiently high such that the maximum magnetic energy of the particle is
essentially the same as the total energy of the particle. This limit is
reached when the trapping gradient is much larger than the minimum gradient
required to magnetically levitate the atoms and when the initial kinetic
energy of the atoms (set by the MOT) is much smaller than their magnetic
potential energy set by the initial cloud size and the MT gradient. The study
of the relationship between the total energy and the maximum magnetic energy
outside of this limit is beyond the scope of this work and is the topic of
future work.

To measure the time-dependent cumulative energy distribution function (CDF),
$F(E_{m},t)=\int_{0}^{E_{m}}f(E,t)dE$, of the atoms in the magnetic trap, we
measure the fraction $R(t)$ of atoms in the magnetic trap that are recaptured
in the MOT after the hold time $t$ as a function of the trap depth $E_{m}$. To
observe the time evolution of the cumulative energy distribution, we repeat
the CDF measurement for a series of hold times
$t_{\text{\tiny$1$}},...,t_{k}$. The measured time-dependent CDFs are
presented as discrete dots in Fig. 4.

##  V Validation of the Model

For the comparison of the theoretical model and experiment, we flow pure Ar
gas into the experiment chamber through the pump section in the apparatus. We
maintain the Ar pressure at $10^{-6}\;\rm{Pa}$, which is two orders of
magnitude higher than the initial ambient gas pressure. Therefore, we can
approximate the entirety of the background to be Ar gas.

Since our experimental measurements are of the CDF of the trapped ensemble, we
integrate Eq. 6 to find the time-dependent cumulative energy distribution $F$
of the trapped ensemble:

$\displaystyle F(E,t)=\int_{0}^{E}$ $\displaystyle dE_{0}\
\Theta(U-E_{0})\sum_{k=0}^{\infty}\dfrac{(\Gamma t)^{k}e^{-\Gamma t}}{k!}$
(15)
$\displaystyle\times\prod_{j=1}^{k}\left(\int_{0}^{U}dE_{j}P_{t}(E_{j-1}-E_{j})\right)f_{0}(E_{k})$

where $U$ is the maximum possible energy a trapped particle can have. To make
the comparison, we must first measure the initial distribution $f_{0}(E)$ at
time $t_{0}$, then evaluate the expression for $F(E,t)$ and compare it to the
experimentally measured recapture fractions $R(E,t+t_{0})$ at various hold
times. The only free parameter in our procedure is the total collision rate
$\Gamma$, which is determined from the observed loss rate and the ensemble
distribution changes. This method provides a much more precise determination
of the total collision rate than the method previously used – an extrapolation
of the exponential decay fits back to zero trap depth.

One challenge with the proposed procedure is finding the initial energy
distribution. We experimentally measure CDFs, but our model uses its
derivative, the probability density function (PDF) $f_{0}(E)$. When the CDF
measurements are differentiated, the underlying measurement noise can lead to
un-physical (negative values) and jagged features in the PDF. We resolved this
issue by using Gaussian Process Regression (GPR) to calculate a smooth
function to describe the CDF data with the constraint that the resultant CDF
be monotonically increasing. This gives us a smooth, positive-definite
probability distribution. We used the gpytorch Gaussian process (GP) library
provided by Ref. Gardner  _et al._ (2018). To account for the uncertainty in
the experimental measurements, we re-sampled each experimental data point used
to calculate the loss in each iteration of the algorithm from a Gaussian
distribution centered on the measurement with a standard deviation equal to
the uncertainty of the measurement. This has the added benefit of combining
the fitting uncertainty and experimental uncertainty for the initial
distribution that can be used for measurement uncertainty analysis. The
results of the GP interpolations to all of the experimental CDFs are shown in
Fig. 4.

Using the shortest hold time ($t_{0}=0.7$ seconds) CDF interpolation shown in
Fig. 4 as the initial distribution, we used Eqn. 15 to numerically calculate
the CDF predicted by the model for different times $t=t_{e}-t_{0}$, where the
$t_{e}$ are the run times of the experimental data. We evaluated the series in
Eqn. 15 up to the fourth order heating term, which has a contribution of less
than 0.1% for the time scales considered. Higher order terms are even smaller
and were neglected.

Figure 4: Experimental and interpolated cumulative energy distributions of an
87Rb atomic ensemble in a magnetic trap after different hold times to a
background gas of Ar. The ensemble lifetime was approximately 1.5 seconds and
the hold times were 0.7 s (blue), 1.2 s (orange), 1.8 s (green), 2.3 s (red),
2.8 s (cyan), 3.4 s (brown), and 3.9 s (pink). Experimentally measured values
(dots) and Gaussian Process Regression (GPR) interpolations (lines) are shown
with shaded regions that indicate the combined uncertainties in the GPR
interpolation and the experimental data. To achieve this, between every
iteration of the GPR, we re-sampled each data point against which the GPR
interpolation curve is compared from a Gaussian distribution centered on the
measured value with standard deviation equal to the uncertainty in the
measurement.

Given $f_{0}(E)$ and $P_{t}(E)$, the only free parameter remaining in the
model is $\Gamma$, the total collision rate. To find it, we perform a global
mean square error fit. Namely, we use Eq. 15 to compute the predicted CDFs and
then evaluate the global sum of the squared residuals between all of the CDFs
calculated by the model and the experimental CDFs. To account for the
measurement uncertainty in the experimental points, we sample each
experimental data comparison point from a Gaussian distribution centered on
the GPR interpolated curves (shown in Fig. 4) with a standard deviation equal
to the standard deviation of the interpolation at this point. We then find the
value of $\Gamma$ that minimizes the global sum of the squared residuals. This
final minimization step is repeated for 200 different sample sets providing
200 distinct values of $\Gamma$. The mean of these values is our reported
$\Gamma$, and the standard deviation tells us the propagation of the
measurement uncertainty into our value of the collision rate. The fit
uncertainty for each sample set is several orders of magnitude smaller than
the variation of the $\Gamma$ values arising from the measurement uncertainty,
so we do not consider it. We find a collision rate of $\Gamma=0.646(1)\
\text{s}^{-1}$, which is 10.8% higher than what would be obtained by fitting
the highest energy points in the CDFs to an exponential decay law and is 2.8%
higher than the result obtained by extrapolating the loss rates to zero trap
depth. A plot of the model CDF fits with this value of the collision rate
compared to the experimental data is shown in Figure 5. The uncertainty
$\Delta\Gamma\approx 1.5\times 10^{-4}\Gamma$, which an improvement in
precision from our previous analysis methods by over a factor of five. We note
that this increase in precision is not due to using the GPR interpolation to
increase our sample size to which we can fit. The fitting of the model to the
GPR interpolated data uses fewer points than actually gathered in the
experiment, demonstrating that the precision is a product of the method and
not the sample size.

Figure 5: Experimental and model-predicted cumulative particle energy
distributions at different times. The points are experimental data (the same
as in Fig. 4), and the solid lines are the distributions predicted by the
model (Eqn. 15). The shaded region is the uncertainty in the model that
propagates from the uncertainty in the initial distribution. The highest
particle number curve (blue) was taken at the experimental hold time
$t_{0}=0.7\ \text{s}$, and it is used as an initial distribution for the
model. The subsequent curves are obtained by evaluating the model at times
$t=t_{e}-t_{0}$, where $t_{e}$ is the experimental hold time used for the
corresponding CDF. The values of $t_{e}$ for the curves shown here are 1.2 s
(orange), 1.8 s (green), 2.3 s (red), 2.8 s (cyan), 3.4 s (brown), and 3.9 s
(pink). The fit of the model to the experimental data provides the collision
rate. We find a value of $\Gamma=0.646(1)\ \text{s}^{-1}$. By construction, we
have perfect agreement between the experiment and model at $t=0$, but we see
that the curves agree for longer hold times as well.

##  VI Conclusion

We have presented a mathematically rigorous model for the time evolution of
the energy distribution of a trapped particle ensemble as a result of
collisions with a thermal background gas. We have also described an analysis
method that allows us to compare cumulative distributions functions predicted
by our model to experimental data. This procedure involves finding the
collision-induced transfer energy distribution, using a Gaussian process
regression to interpolate the data, and performing a single-parameter fit that
extracts the collision rate. We have used this procedure and found good
agreement between our model and experimental data obtained for trapped 87Rb
subject to a thermal argon background. We obtained a total collision rate,
using information about heating and trap loss, of $0.646(1)\ \text{s}^{-1}$.
This is an uncertainty of below $0.2\%$, which is significantly more precise
than any previous measurements.

This work has several implications for future studies in the area of particle
sensors for vacuum metrology. Our work suggests that previous collision rate
measurements need to be corrected to account for the change in the
distribution, and we have presented a theory to perform this correction that
provides the total collision rate with a low uncertainty. This theory is
presently being applied to reanalyze both our previous measurements (Refs.
Booth  _et al._ (2019); Shen  _et al._ (2020, 2021)) and measurements taken by
other groups Eckel (2023). We note that this theory can be applied to other
species (e.g. molecules), trap geometries, and experimental procedures, as it
does not depend on the distribution of the particles loaded into the trap or
the trapping potential that is used. It only depends on one CDF measurement
used as an initial distribution $f_{0}$ to evolve and knowledge of the
transfer energy distribution; no additional information about the experimental
details are required. We further comment that our method is robust against
noise in the measurement of $f_{0}$ and prior heating that has already changed
$f_{0}$ from the distribution first loaded into the trap. We can also
generalize the evolution model to systems with multiple background particles
by adding an additional set of iterated integrals for each species present.

The analysis techniques we have discussed provides new ways to understand and
study diffractive collision physics. In particular, our formulation of the
cumulative distribution functions in Eqn. 15 suggests a way to measure the
transfer distribution using experimental data. As a preliminary test, we found
that repeating the analysis described in Section III using the universal
polynomial form of the transfer distribution shown in Eqn. 13 yields a total
collision rate of $0.642(1)\ \rm{s^{-1}}$. This differs by our result computed
for the Medvedev transfer function by 0.61%, demonstrating the effectiveness
of quantum diffractive universality in describing the low energy sector of the
transfer energy distribution for 87Rb+Ar collisions. We intend to extend this
analysis to other species and refine our understanding of the uses and limits
of universality Booth  _et al._ (2019).

##  VII Appendix

###  VII.1 Single Particle Heating

In this section, we show our derivation of the energy transferred to a trapped
particle after $k$-collisions as shown in Eqn. 1. We consider a cold trapped
gas of particle mass $m$ in a quadrupole trap subject to the potential $U$
that is only a function of the coordinate x. We subject the trapped gas to a
background thermal gas of particle mass $m_{b}$. Consider a trapped particle
with momentum p at a position x that collides with a background particle of
mass $m_{b}$, and let $\textbf{p}_{t}$ be the momentum transferred to the
trapped particle by the collision. The final energy $E_{f}$ of the particle
will be:

$E_{f}=\dfrac{(\textbf{p}+\textbf{p}_{t})^{2}}{2m}+U(\textbf{x})$ (16)

We now define the transfer energy $E_{t}$ to be:

$\displaystyle E_{t}=\dfrac{p_{t}^{2}}{2m}$ (17)

We can rewrite the final energy in terms of the initial energy $E_{i}$ and the
tranfer energy in the following way:

$E_{f}=E_{i}+E_{t}+\dfrac{\textbf{p}\cdot\textbf{p}_{t}}{m}$ (18)

Next, suppose that the particle with momentum $\textbf{p}_{0}$ undergoes $n$
collisions with transfer momenta $\textbf{p}_{1}$,…,$\textbf{p}_{n}$. Applyin
induction to Equation 18 tells us that the final energy $E_{f}$ of the
particle is given by:

$E_{f}=E_{i}+\sum_{k=1}^{n}\left(E_{k}+\dfrac{\textbf{p}_{k}}{m}\cdot\sum_{i=0}^{k-1}\textbf{p}_{i}\right)$
(19)

###  VII.2 Single Particle Energy Transfer

Here, we shall calculate how the energy transferred in a collision event
depends on the relative velocities of the particles and the scattering angle
as shown in Eqn. 9. Suppose that the background particle has incoming momentum
$\textbf{p}_{i}$ in the rest frame of the trapped particle, that the
background particle scatters with outgoing momentum $\textbf{p}_{f}$, and the
trapped particle scatters with outgoing momentum $\textbf{q}_{f}$. The
conservation of momentum tells us that:

$\textbf{p}_{i}=\textbf{p}_{f}+\textbf{q}_{f}$ (20)

As the trapped particle is initially stationary, the center of mass velocity
$\textbf{v}_{c}$ is given by:

$\textbf{v}_{c}=\dfrac{\textbf{p}_{i}}{m+m_{b}}$ (21)

We can boost into the center of mass frame through the Galilean transformation
$\textbf{v}\mapsto\textbf{v}-\textbf{v}_{c}$. Let $\textbf{p}_{i}^{\prime}$
and $\textbf{p}_{f}^{\prime}$ denote the initial and final momenta of the
background particle in the center of mass frame and let
$\textbf{q}_{i}^{\prime}$ and $\textbf{q}_{f}^{\prime}$ denote the initial and
final momenta of the trapped particle in the center of mass frame. By
definition, we know that:

$\textbf{p}_{i}^{\prime}=\textbf{p}_{i}-m_{b}\textbf{v}_{c}$ (22)

Substituting in our expression for $\textbf{v}_{c}$ and simplifying, we find:

$\textbf{p}_{i}^{\prime}=\dfrac{m\textbf{p}_{i}}{m+m_{b}}$ (23)

The conservation of energy tells us that:

$\dfrac{q_{i}^{\prime 2}}{2m}+\dfrac{p_{i}^{\prime
2}}{2m_{b}}=\dfrac{q_{f}^{\prime 2}}{2m}+\dfrac{p_{f}^{\prime 2}}{2m_{b}}$
(24)

In the center of mass frame, the total momentum of the system vanishes. Thus,
$\textbf{p}_{i}^{\prime}=-\textbf{q}_{i}^{\prime}$ and
$\textbf{p}_{f}^{\prime}=-\textbf{q}_{f}^{\prime}$. Substiting these results
into our expression for the conservation of energy, we have:

$\dfrac{p_{i}^{\prime 2}}{2m}+\dfrac{p_{i}^{\prime
2}}{2m_{b}}=\dfrac{q_{f}^{\prime 2}}{2m}+\dfrac{q_{f}^{\prime
2}}{2m_{b}}\implies p_{i}^{\prime}=q_{f}^{\prime}$ (25)

This tells us that $\textbf{q}_{f}^{\prime}$ is equal to
$\textbf{p}_{i}^{\prime}$ up to some rotation. We can now choose our
coordinates such that $\textbf{p}_{i}$ lies along the $x$-axis and
$\textbf{q}_{f}$ lies in the $xy$-plane. Then, $\textbf{p}_{i}^{\prime}$ lies
along the $x$-axis and $\textbf{q}_{f}^{\prime}$ is also in the $xy$-plane.
Let $\theta$ be the angle between $\textbf{p}_{i}^{\prime}$ and
$\textbf{p}_{f}^{\prime}$. As $\textbf{p}_{f}^{\prime}$ and
$\textbf{q}_{f}^{\prime}$ have opposite directions, the angle between
$\textbf{p}_{i}^{\prime}$ and $\textbf{q}_{f}^{\prime}$ will be $\theta+\pi$.
We can summarize these relationships in terms of a rotation matrix in the
following way:

$\textbf{q}_{f}=\left(\begin{array}[]{rr}\cos(\pi+\theta)&-\sin(\pi+\theta)\\\
\sin(\theta+\pi)&\cos(\theta+\pi)\end{array}\right)\textbf{p}_{i}^{\prime}+m\textbf{v}_{c}$
(26)

We can write $\textbf{p}_{i}^{\prime}$ and $\textbf{v}_{c}$ in terms of
$\textbf{p}_{i}$, which yields:

$\textbf{q}_{f}=\left(\begin{array}[]{rr}-\cos\theta&\sin\theta\\\
-\sin\theta&-\cos\theta\end{array}\right)\dfrac{m\textbf{p}_{i}}{m+m_{b}}+\dfrac{m\textbf{p}_{i}}{m+m_{b}}$
(27)

In these coordinates, $\textbf{p}_{i}$ lies along the $x$-axis. Then, this
expression becomes:

$\textbf{q}_{f}=\dfrac{mp_{i}}{m+m_{b}}\left(\begin{array}[]{c}1-\cos\theta\\\
-\sin\theta\end{array}\right)$ (28)

The energy $E_{t}$ transferred to the trapped particle in the frame where it
is at rest is merely the final kinetic energy of the particle, which is
$q_{f}^{2}/2m$. Then, our previous expression tells us that:

$E_{t}=\dfrac{1}{m}\left(\dfrac{mp_{i}}{m+m_{b}}\right)^{2}(1-\cos\theta)$
(29)

Now, we define the reduced mass of the system $\mu$ to be:

$\mu=\dfrac{mm_{b}}{m+m_{b}}$ (30)

Writing the transfer energy expression in terms of the reduced mass yields:

$E_{t}=\dfrac{2\mu^{2}p_{i}^{2}}{mm_{b}^{2}}\sin[2](\frac{\theta}{2})$ (31)

Notice that $p_{i}/m_{b}$ is merely the relative velocity of the background
particle. We denote this velocity to be $v_{r}$. In terms of this velocity, we
conclude that:

$E_{t}=\dfrac{2\mu^{2}v_{r}^{2}}{m}\sin[2](\frac{\theta}{2})$ (32)

###  VII.3 The Background Gas Distribution

In this section, we shall justify our approximation of the relative speed
distribution of the background particles using the lab frame speed
distribution of the background gas. In the lab frame, the background obeys a
Maxwell-Boltzmann velocity distribution $d(v)$ with some temperature $T$.
However, as the trapped particle is not at rest the relative velocity
distribution is some other function $d^{\prime}(v_{r})$. Let
$\boldsymbol{\epsilon}$ be the velocity of the trapped particle. The deviation
is maximized when $\boldsymbol{\epsilon}$ is collinear with the velocity of
incoming background particle, because this maximizes the difference between
the magnitudes of the background particle velocity and the relative velocity.
We shall then take $\boldsymbol{\epsilon}$ to be in the same direction as the
background particle velocity, which reduces this to a one dimensional problem.
The relative speed distribution $d^{\prime}$ is:

$d^{\prime}(v)=4\pi\left(\dfrac{m_{b}}{2\pi
k_{\text{\tiny$B$}}T}\right)^{3/2}(v+\epsilon)^{2}e^{-m_{b}(v+\epsilon)^{2}/2k_{\text{\tiny$B$}}T}$
(33)

where $m_{b}$ is the mass of the background particle. Expanding about
$\epsilon=0$, we find:

$\displaystyle d^{\prime}(v)=4\pi$ $\displaystyle\left(\dfrac{m_{b}}{2\pi
k_{\text{\tiny$B$}}T}\right)^{3/2}v^{2}e^{-m_{b}v^{2}/2k_{\text{\tiny$B$}}T}$
(34)
$\displaystyle\times\left(1+\dfrac{2\epsilon}{v}-\dfrac{m_{b}v\epsilon}{k_{\text{\tiny$B$}}T}-\dfrac{5m\epsilon^{2}}{2k_{\text{\tiny$B$}}T}+\cdots\right)$

where we have dropped terms of order $\epsilon^{k}/v^{k}$,
$\epsilon^{k}/T^{k}$ and $v^{k}\epsilon^{k}/T^{k}$ for $k\geq 2$. The velocity
$\epsilon$ is of order $\sqrt{3k_{\text{\tiny$B$}}T_{t}/m_{t}}$ where $T_{t}$
is the trap temperature and $m_{t}$ is the trapped particle mass. Similarly,
$v$ is of order $\sqrt{3k_{\text{\tiny$B$}}T/m_{b}}$. Then, we can write:

$\displaystyle\left(1+\dfrac{2\epsilon}{v}-\dfrac{m_{b}v\epsilon}{k_{\text{\tiny$B$}}T}-\dfrac{5m_{b}\epsilon^{2}}{2k_{\text{\tiny$B$}}T}+\cdots\right)$
(35) $\displaystyle\hskip
56.9055pt\sim\left(1-\dfrac{15m_{b}T_{t}}{m_{t}T}-\sqrt{\dfrac{m_{b}T_{t}}{m_{t}T}}\right)$

For our system, $T_{t}\sim 100\ \mathrm{\mu K}$, $T$ is $294\ \mathrm{K}$, and
$m_{b}/m_{t}=0.46$. Then, the correction is of order $1\times 10^{-6}$.

This analysis is not comprehensive, as the background particle and trapped
particle velocities are both drawn from probability distributions that can
deviate from the mean value estimates. However, even if we consider $0\
\text{m}/\text{s}<\epsilon<0.47\ \text{m}/\text{s}$ and $80\
\text{m}/\text{s}<v<1000\ \text{m}/\text{s}$, which includes 99% of sampled
$\epsilon$ and $v$, the maximum value that the correction term reaches is
$0.011$. This also does not account for the fact that the trapped particle
velocity is randomly oriented, so the projection of $\boldsymbol{\epsilon}$
along v will be smaller than predicted by the speed distribution. This
justifies our approximation that $d^{\prime}(v)\approx d(v)$.

###  VII.4 Extrapolating the loss rate

Figure 6: The ensemble loss rate versus trap depth extracted from the data in
Fig. 4. The cumulative particle fraction at a fixed cut energy and different
hold times were fit to a simple exponential function to obtain an estimate for
the loss rate at different trap depths. These loss rate values were then
extrapolated back to find the value at zero trap depth. The extrapolated value
is $\Gamma=0.664(4)\;\mathrm{s}^{-1}$ if the values at small depths and with
large uncertainties are excluded from the fit. If all points are included, the
value is $\Gamma=0.4(3)\;\mathrm{s}^{-1}$.

An alternative approach to the one presented here for obtaining the total
collision rate from the observed loss rate is to fit the recaptured fraction
as a function of hold time to a simple exponential decay law at progressively
smaller and smaller trap depths. The results can then be extrapolated to zero
trap depth to infer the total collision rate.

To illustrate the limits of this approach, the ensemble loss rate versus trap
depth was extracted from the data in Fig. 4. The cumulative particle fraction
at a fixed cut energy and different hold times was fit to a simple exponential
function to obtain the loss rate at that trap depth. The results are shown in
Fig 6. The resulting loss rates were then extrapolated back to infer the loss
rate at zero trap depth. The extrapolated value at zero trap depth is
$\Gamma=0.664(4)\;\mathrm{s}^{-1}$ if the values at small trap depth and with
large uncertainties are excluded from the fit. This value is larger by 2.8 %
than the value extracted from the rigorous analysis of the CDFs.

There are several problems with this naive approach. One is that the ensemble
loss rate is not a simple exponential and actually increases with time as the
ensemble heats. The second problem is that, because the atoms are not at zero
energy, the effective trap depth of the ensemble is not the same as the cut
energy at which the CDF is taken. A manifestation of this is that, because the
energy distribution of the atoms has a non-zero width and may also include an
offset below which there are no atoms, the trap depth dependence of the loss
rate becomes flat at the lowest cut energies. Evidently, another consequence
of this difference between the cut energy and the effective trap depth is that
an overestimate results when extrapolating the trap loss rate back to zero cut
energy using only data at higher trap depths where the SNR is high.

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