# First detection of collective oscillations of a stored deuteron beam
with an amplitude close to the quantum limit

J. Slim  III. Physikalisches Institut B, RWTH Aachen University, 52056 Aachen,
Germany  N.N. Nikolaev  L.D. Landau Institute for Theoretical Physics, 142432
Chernogolovka, Russia  Moscow Institute for Physics and Technology, 141700
Dolgoprudny, Russia  F. Rathmann  Institut für Kernphysik, Forschungszentrum
Jülich, 52425 Jülich, Germany  A. Wirzba  Institut für Kernphysik,
Forschungszentrum Jülich, 52425 Jülich, Germany  Institute for Advanced
Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany  A. Nass  Institut
für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany  V. Hejny
Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany  J.
Pretz  Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich,
Germany  H. Soltner  Zentralinstitut für Engineering, Elektronik und Analytik,
Forschungszentrum Jülich, 52425 Jülich, Germany  F. Abusaif  Institut für
Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany  A. Aggarwal
Marian Smoluchowski Institute of Physics, Jagiellonian University, 30348
Cracow, Poland  A. Aksentev  Institute for Nuclear Research, Russian Academy
of Sciences, 117312 Moscow, Russia  A. Andres  III. Physikalisches Institut B,
RWTH Aachen University, 52056 Aachen, Germany  L. Barion  University of
Ferrara and Istituto Nazionale di Fisica Nucleare, 44100 Ferrara, Italy  G.
Ciullo  University of Ferrara and Istituto Nazionale di Fisica Nucleare, 44100
Ferrara, Italy  S. Dymov  University of Ferrara and Istituto Nazionale di
Fisica Nucleare, 44100 Ferrara, Italy  Laboratory of Nuclear Problems, Joint
Institute for Nuclear Research, 141980 Dubna, Russia  R. Gebel  Institut für
Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany  M. Gaisser  III.
Physikalisches Institut B, RWTH Aachen University, 52056 Aachen, Germany  K.
Grigoryev  Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich,
Germany  D. Grzonka  Institut für Kernphysik, Forschungszentrum Jülich, 52425
Jülich, Germany  O. Javakhishvili  Department of Electrical and Computer
Engineering, Agricultural University of Georgia, 0159 Tbilisi, Georgia  A.
Kacharava  Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich,
Germany  V. Kamerdzhiev  Institut für Kernphysik, Forschungszentrum Jülich,
52425 Jülich, Germany  S. Karanth  Marian Smoluchowski Institute of Physics,
Jagiellonian University, 30348 Cracow, Poland  I. Keshelashvili  Institut für
Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany  A. Lehrach
Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany  P.
Lenisa  University of Ferrara and Istituto Nazionale di Fisica Nucleare, 44100
Ferrara, Italy  N. Lomidze  High Energy Physics Institute, Tbilisi State
University, 0186 Tbilisi, Georgia  B. Lorentz  GSI Helmholtzzentrum für
Schwerionenforschung, 64291 Darmstadt, Germany  A. Magiera  Marian
Smoluchowski Institute of Physics, Jagiellonian University, 30348 Cracow,
Poland  D. Mchedlishvili  High Energy Physics Institute, Tbilisi State
University, 0186 Tbilisi, Georgia  F. Müller  III. Physikalisches Institut B,
RWTH Aachen University, 52056 Aachen, Germany  A. Pesce  Institut für
Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany  V. Poncza
Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany  D.
Prasuhn  Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich,
Germany  A. Saleev  University of Ferrara and Istituto Nazionale di Fisica
Nucleare, 44100 Ferrara, Italy  V. Shmakova  Institut für Kernphysik,
Forschungszentrum Jülich, 52425 Jülich, Germany  Laboratory of Nuclear
Problems, Joint Institute for Nuclear Research, 141980 Dubna, Russia  H.
Ströher  Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich,
Germany  M. Tabidze  High Energy Physics Institute, Tbilisi State University,
0186 Tbilisi, Georgia  G. Tagliente  Istituto Nazionale di Fisica Nucleare
sez. Bari, 70125 Bari, Italy  Y. Valdau  Institut für Kernphysik,
Forschungszentrum Jülich, 52425 Jülich, Germany  T. Wagner  Institut für
Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany  C. Weidemann
Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany  A.
Wrońska  Marian Smoluchowski Institute of Physics, Jagiellonian University,
30348 Cracow, Poland  M. Żurek  Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA

###### Abstract

We investigated coherent betatron oscillations of a deuteron beam in the
storage ring COSY, excited by a detuned radio-frequency Wien filter. The beam
oscillations were detected by conventional beam position monitors. With the
currently available apparatus, we show that oscillation amplitudes down to
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ can be detected. The interpretation
of the response of the stored beam to the detuned radio-frequency Wien filter
is based on simulations of the beam evolution in the lattice of the ring and
realistic time-dependent 3D field maps of the Wien filter. Future measurements
of the electric dipole moment of protons will, however, require control of the
relative position of counter-propagating beams in the sub-picometer range.
Since here the stored beam can be considered as a rarefied gas of uncorrelated
particles, we moreover demonstrate that the amplitudes of the zero-point
(ground state) betatron oscillations of individual particles are only a factor
of about 10 larger than the Heisenberg uncertainty limit. As a consequence of
this, we conclude that quantum mechanics does not preclude the control of the
beam centroids to sub-picometer accuracy. The smallest Lorentz force exerted
on a single particle that we have been able to determine is
$10\text{\,}\mathrm{a}\mathrm{N}$.

††preprint: Draft for PR AB

##  I Introduction

The approach to the quantum ground state, the observation of quantum effects
in macroscopic systems, and the possibility to detect displacements of
macroscopic bodies on the nanometer scale, are the subject of intense
theoretical and experimental efforts Schreppler  _et al._ (2014); Abbott  _et
al._ (2009); Murch  _et al._ (2008); Biercuk  _et al._ (2010); Rugar  _et al._
(2004). A notable example is the detection of gravitational waves using an
interferometric detector with mirrors in the kilogram range Abbott  _et al._
(2016). In all-electric proton storage rings, coherent beam displacements down
to the picometer range that are caused by Earth’s gravity pull are in
principle accessible using the spin rotations of the proton as a detector
Abusaif  _et al._ (2021). We also mention here the ongoing discussions of the
possibility to detect gravitational waves via perturbations of the beam orbit
in high-energy storage rings ari . Here we report the first detection of
collective oscillations of an intense beam of deuterons in a storage ring with
an amplitude close to the quantum limit. The present study is part of an
international effort to prepare for the search for the permanent electric
dipole moment (EDM) of charged particles. The focus of these studies has been
on systematic effects, e.g., imperfection magnetic fields in storage rings
Saleev  _et al._ (2017), and orbit improvements in a machine using beam-based
alignment Wagner  _et al._ (2021), thereby advancing the high-precision
frontier in spin dynamics in storage rings. A comprehensive description of
this activity and of the proposed stepwise approach leading to a dedicated
proton EDM storage ring is presented in Ref. Abusaif  _et al._ (2021).

Experiments searching for electric dipole moments of charged particles using
storage rings are at the forefront of the incessant quest to find new physics
beyond the Standard Model of particle physics. These investigations bear the
potential to shed light on the origin of the anomalously large matter-
antimatter asymmetry in the Universe Pospelov and Ritz (2005), for which the
combined predictions of the Standard Models of particle physics and of
cosmology fall short of the experimentally observed asymmetry by about seven
to eight orders of magnitude Bernreuther (2002).

The signal for an EDM is the spin precession in electric fields, where it
should be noted that the spins of charged particles can be subjected to large
electric fields only in storage rings. The need to eliminate the
overwhelmingly stronger spin rotations driven by the magnetic moment in
magnetic fields brings to the front an all-electric, so-called frozen spin
proton storage ring Anastassopoulos  _et al._ (2016); Abusaif  _et al._
(2021). An important advantage of such a machine is the ability to
simultaneously store two counter-propagating proton beams. The concurrent
measurement of the EDM-driven spin rotations of the counter-propagating beams
would allow to cancel major systematic effects. To this end, to reach an
ambitious sensitivity to the proton EDM of
$d_{p}\approx${10}^{-29}\text{\,}\mathrm{e}\,\mathrm{c}\mathrm{m}$$, it is
imperative to control the relative vertical displacement of the centers of
gravity of the two beams to an accuracy of about
$5\text{\,}\mathrm{p}\mathrm{m}$ Abusaif  _et al._ (2021). One may wonder
whether such an enormously demanding accuracy is not prohibited by the
Heisenberg uncertainty principle. Towards an ultimate precision search for
EDMs of charged particles, this particular aspect of the systematics of such
measurements had not been investigated so far, and our experiment constitutes
the first step in this direction.

Here we report on the measurement of the amplitude of collectively excited
vertical oscillations of a deuteron beam orbiting in the magnetic storage ring
COSY at a momentum of about
$970\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$ Maier (1997).
The data were taken in 2018 in the course of a dedicated experiment in the
framework of systematic beam and spin dynamics studies for the deuteron EDM
experiment (so-called precursor experiment), presently carried out by the JEDI
collaboration at COSY Rathmann  _et al._ (2013); Morse  _et al._ (2013);
Rathmann  _et al._ (2020). One of the central devices in the precursor
experiment is the radio-frequency (RF) Wien filter (WF), shown in Fig. 1,
which was designed to provide a cancellation of the electric and magnetic
forces acting on the particle. In this operation mode, the Wien filter affects
only the particle spins, but does not perturb the beam orbit Slim  _et al._
(2016, 2017); Slim (2018). A slightly detuned Wien filter, however, exerts a
non-vanishing Lorentz force on the orbiting beam particles. It is shown
collective beam oscillation excited by the WF with amplitudes down to
$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ can be detected with the
currently available equipment. Our approach to measuring ultra-small
displacements complements other measurements of ultra-small forces using
different techniques Schreppler  _et al._ (2014); Abbott  _et al._ (2009);
Murch  _et al._ (2008); Biercuk  _et al._ (2010); Rugar  _et al._ (2004).

In our experiment, the measurement cycles were much shorter than the intrabeam
interaction time, and the beam attenuation rate was negligibly weak (see
discussion in Weidemann  _et al._ (2015)), therefore we treat the beam as a
rarefied gas of uncorrelated particles. Individual particles undergo stable
betatron oscillations around the equilibrium orbit in the horizontal and
vertical planes, driven by focusing magnetic fields. Apart from their
conventional individual betatron motions, all the particles in a bunch
participate in one and the same collective and coherent oscillation that is
driven by the Wien filter. Therefore, the upper bound of the amplitude of the
collective oscillation of the entire beam corresponds to the upper bound of
the oscillation amplitude of a single particle. In our approach, access to
ultrasmall oscillation amplitudes results from the fact that the measured
signal corresponds to a collective response of the electric charge of about
$N=${10}^{9}$$ deuterons in the bunch.

The beam tracking simulations were carried out to predict the response of the
stored beam to the detuned radio-frequency Wien filter. The simulations use
the elements of the ring lattice and realistic time-dependent 3D field maps of
the Wien filter. These field maps describe the spatial variation of complex
electric and magnetic fields, including the fringe field areas. Furthermore,
the tolerances of the elements of the circuit driving the Wien filter are also
taken into consideration.

As a reference value for the Heisenberg uncertainty relation, we take an
estimate of the amplitude of the single-particle zero-point betatron
oscillation amplitude $Q$. Then, our result for the smallest measured
amplitude of the Wien filter-driven single-particle oscillation is only about
a factor of ten larger than the quantum limit of Heisenberg’s uncertainty
relation for vertical single-particle betatron oscillations. The smallest
detected oscillation amplitude is by three orders of magnitude smaller than
the beam size.

We demonstrate for the first time that the accuracy with which periodic beam
oscillation amplitudes can be measured is vastly higher compared to that of
static beam displacements generated by steerers using the same BPM. In a broad
context, any new precision tool is of interest per se, and the latter point
has important implications, for instance, for all-electric frozen-spin EDM
storage rings. Here, one aims to control the interfering radial magnetic
fields by measuring the vertical spacing of the counter-propagating beams. Our
result complements the potential of using beam oscillations to measure this
distance, as discussed in Ref. Hacıömeroğlu  _et al._ (2019).

One must distinguish the RF-driven collective oscillations above the quantum
limit $Q$ from the quantum uncertainty of the center of mass of the bunch
circulating in a static ring. Specifically, for a rarefied-gas of $N$
uncorrelated particles, the quantum limit of the centroid of the bunch,
detected by the BPMs, amounts to $Q/\sqrt{N}$.

The paper is organized as follows. In Sec. II, the measurement principle is
introduced, followed by a description of the operation of the radio-frequency
Wien filter in Sec. III. The method to determine the beam oscillations is
discussed in Sec. IV, and the evolution of the beam to the combined effect of
the ring lattice and the Wien filter fields are presented in Sec. V. The time-
dependent field maps of the Wien filter are discussed in Sec.V.1, and the
evaluation of the uncertainties is elaborated in Sec. V.2. Experimental
results are presented in Sec. VI, followed by conclusion and outlook in Sec.
VII.

##  II Measurement principle

(a) CAD drawing of the design of the RF Wien filter. 1: RF feed, 2: beam pipe,
3: inner mounting cylinder, 4: inner support structure, 5: lower electrode, 6:
insulator, 7: RF connector, and 8: vacuum vessel. (b) Photograph with a view
along the beam axis showing the gold-plated copper electrodes, which have a
length of $808.8\text{\,}\mathrm{m}\mathrm{m}$.

Figure 1: The waveguide RF Wien filter is mounted inside a cylindrical vessel.
The effective length of the device amounts to
$\ell=$1.16\text{\,}\mathrm{m}$$.The technical details are described in Refs.
Slim  _et al._ (2017, 2016).

The Cooler Synchrotron (COSY) Martin  _et al._ (1985); Maier (1997) at
Forschungszentrum Jülich is a storage ring with a circumference of
approximately $184\text{\,}\mathrm{m}$. Its principal elements used for the
experiments are indicated in Fig. 2. For the investigations presented here,
the two key devices are the RF Wien filter, based on a parallel-plates
waveguide Slim  _et al._ (2016), and a conventional electrostatic beam
position monitor (BPM) that is used to monitor the beam oscillations Forck
_et al._ (2008). The Wien filter generates orthogonal and highly-homogeneous
electric and magnetic fields. In the present experiment, the Wien filter was
operated in the mode with the electric field pointing vertically upward
($y$-direction), whereas the magnetic field points radially outward
($x$-direction), and the beam moves in $z$-direction (see coordinate system in
Fig. 2). The effective length of the Wien filter is
$\ell=$1.16\text{\,}\mathrm{m}$$ (see Refs. Slim  _et al._ (2017, 2016) for
further technical details).

Figure 2: Schematic diagram of the cooler synchrotron and storage ring (COSY)
with the main components, especially the focusing/defocusing magnets
(quadrupoles) and the bending magnets (dipoles). Indicated are the position of
the RF Wien filter and the location of the beam position monitor Böhme  _et
al._ (2018) (BPM 17), used to observe the beam oscillations. Further
components such as the $2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electron
cooler Dietrich  _et al._ (2014), the WASA Adam  _et al._ (2004) and the JEPO
Müller (2019); Müller  _et al._ (2020) polarimeters, and the Siberian snake
Lehrach and Maier (2001) are also shown. The coordinate system used is
indicated.

As a spin rotator for the forthcoming deuteron EDM (precursor) experiment
Rathmann  _et al._ (2013); Morse  _et al._ (2013); Rathmann  _et al._ (2020),
the Wien filter is designed to operate in resonance with the spin precession
of the orbiting deuterons Slim  _et al._ (2016, 2017); Slim (2018); Slim  _et
al._ (2020), and at a vanishing Lorentz force, given by

$\vec{F}=q\left(\vec{E}+\vec{v}\times\vec{B}\right)\,,$ (1)

where $q$ denotes the elementary charge and $\vec{v}$ represents the velocity
of the beam particles. Unlike in conventional DC Wien filters, the crossed
electric field and magnetic fields ($\vec{E}$ and $\vec{B}$) of the RF Wien
filter are generated simultaneously by exciting the transverse electromagnetic
(TEM) mode. The spin resonance tune mapping technique, developed for the Wien
filter operation in the deuteron EDM experiment at COSY, is described in
Rathmann  _et al._ (2020).

When the electric and magnetic fields in the Wien filter are mismatched, i.e.,
when the electric and magnetic forces no longer cancel each other, the RF
fields excite collective beam oscillations at the frequency at which the Wien
filter is operated. In the present experimental set up, a mismatch between
electric and magnetic fields provides a vertically mismatched Lorentz force
(see coordinate system in Fig. 2).

With a vanishing Lorentz force, the beam performs idle vertical (and
horizontal) betatron oscillations

$y(t)=y(0)\sqrt{\frac{\beta_{y}(t)}{\beta_{y}(0)}}\cos\left[\psi_{y}(t)\right]\,,$
(2)

where $\beta_{y}(t)$ is the betatron amplitude function. With the beam
revolution period of $T=2\pi/\omega_{\text{rev}}$, the betatron phase advance
$\psi_{y}(t)$ satisfies $\psi_{y}(t+T)-\psi_{y}(t)=\omega_{y}T=2\pi\nu_{y}$,
where $\nu_{y}$ is the vertical betatron tune given by
$\nu_{y}=\omega_{y}/\omega_{\text{rev}}$.

On the other hand, a mismatched Wien filter exerts stroboscopically, i.e.,
once per turn, a vertical force $F_{y}(n)=F_{y}\cos(n\,\omega_{\text{WF}}T)$
on the stored particle, where $n$ is the turn number and $\omega_{\text{WF}}$
denotes the angular velocity of the RF in the Wien filter (see discussion of
Fig. 12 in Sec. VI). The change of the vertical velocity of the stored
particle, accumulated during the time interval $\Delta t=\ell/v_{z}$ the
particle spends per turn $n$ inside the Wien filter, is given by

$\Delta v_{y}(nT)=\frac{F_{y}(n)\Delta t}{\gamma
m}=-\zeta\omega_{y}\cos(n\,\omega_{\text{WF}}T)\,,$ (3)

where $\gamma$ and $m$ are the Lorentz-factor and the mass of the particle,
respectively. The change $\Delta y$ of the vertical position $y$ in the Wien
filter can be neglected. The coupling of vertical and radial beam oscillations
is negligible (see Sec. IV) and it is sufficient here to treat driven
oscillations in a one-dimensional approximation. Due to the very strong
disparity of synchrotron and fractional WF frequencies, synchro-betatron
coupling can be neglected (see discussion in Syphers  _et al._ (1993)).
Furthermore, beam attenuation either by intrabeam scattering or by interaction
with residual gas during the data acquisition cycle is very small (see
Appendix B), justifying the rarefied gas approximation.

According to Eq. (2), the stroboscopic signal of the betatron motion observed
at any point in the ring, follows the harmonic law with angular velocity
$\omega_{y}$, and we invoke the familiar description of the oscillatory motion
in terms of the complex variable $z=y-iv_{y}/\omega_{y}$. With the initial
condition $z(0)=0$, summing $\Delta v_{y}(kT)$ after $n$ turns, the solution
for $z(n)$ behind the Wien filter reads

$\begin{split}z(n)=\frac{i\zeta}{2}&\cdot\left[\frac{\exp(in\omega_{y}T)-\exp(in\,\omega_{\text{WF}}T)}{\exp(i(\omega_{y}-\omega_{\text{WF}})T)-1}\right.\\\
&+\left.\frac{\exp(in\omega_{y}T)-\exp(-in\,\omega_{\text{WF}}T)}{\exp(i(\omega_{y}+\omega_{\text{WF}})T)-1}\right]\,.\end{split}$
(4)

This expression serves as the initial condition for the idle betatron motion
during the subsequent $(n+1)$ turn and so forth. A similar analytic result
holds also for generic AC dipole-driven betatron oscillations, discussed in a
very different context of machine diagnostics in Ref. Miyamoto  _et al._
(2008) (see also references therein).

Driven by the mismatched Wien filter, all beam particles participate in one
and the same collective and coherent oscillation, and according to Eq. (4),
the beam as a whole exhibits oscillations at the Wien filter frequency
$\omega_{\text{WF}}$. A lock-in amplifier may be used to selectively measure
the corresponding Fourier component of the beam oscillation
$y=\xi_{y}\cos(n\,\omega_{\text{WF}}T)$ from the output of a beam position
monitor. Its amplitude is given by

$\xi_{y}=\frac{\zeta}{2}\cdot\frac{\sin(2\pi\nu_{y})}{\cos(2\pi\nu_{\text{WF}})-\cos(2\pi\nu_{y})}\,,$
(5)

where the vertical betatron tune $\nu_{y}$, and the Wien filter tune
$\nu_{\text{WF}}$, are given by $\nu_{y}=\omega_{y}/\omega_{\text{rev}}$ and
$\nu_{\text{WF}}=\omega_{\text{WF}}/\omega_{\text{rev}}$, respectively. When
the Wien filter tune is close to the vertical betatron tune, a resonant
enhancement of the beam oscillation amplitude $\xi_{y}$ occurs. Equation (5)
describes Hooke’s law, $F_{y}=k_{\text{H}}\xi_{y}$, and Hooke’s constant is
given by

$k_{\text{H}}=\left|\frac{2\gamma m\omega_{y}}{\Delta
t}\cdot\frac{\cos(2\pi\nu_{\text{WF}})-\cos(2\pi\nu_{y})}{\sin(2\pi\nu_{y})}\right|\,.$
(6)

We invoke an approximate description of the betatron motion by a harmonic
oscillator with constant betatron function and evaluate the Heisenberg
uncertainty limit $Q$ for the betatron oscillation amplitude $\xi_{y}$ in
terms of the zero-point oscillator energy $\tfrac{1}{2}\hbar\omega_{y}$, which
yields

$Q^{2}=\frac{\hbar}{m\gamma\omega_{y}}\,\,.$ (7)

For the present experiment, we obtain

$Q=\frac{82}{\sqrt{\gamma\nu_{y}}}\ {\rm nm}\,.$ (8)

With the actual COSY values for the betatron tune $\nu_{y}$ and the Lorentz-
factor $\gamma$ of the beam (see Table 1, Appendix A), the quantum limit of
the vertical betatron oscillations amounts to

$Q\approx$41\text{\,}\mathrm{n}\mathrm{m}$\,.$ (9)

The interpretation of the measured oscillation amplitudes in terms of the Wien
filter parameters requires numerical simulations of the performance of the
Wien filter as an element of the storage ring Slim  _et al._ (2016). The
details relevant to the present study are described below; the corresponding
beam simulations carried out are consistent with the available experimental
results on the properties of COSY Weidemann  _et al._ (2015).

##  III Wien filter operation

The control of the Lorentz force of the waveguide RF Wien filter is based on
the wave-mismatch principle Slim (2018). An impedance mismatch is introduced
at the load part of the device to deliberately create reflections that
generate a standing wave pattern inside the Wien filter Slim  _et al._ (2020).
These standing waves can be represented by the complex-valued field quotient
$Z_{q}$, defined as the ratio of the total electric to the total magnetic
field strength,

$\begin{split}Z_{q}&=\frac{E^{\text{total}}}{H^{\text{total}}}=\frac{E^{+}+E^{-}}{H^{+}-H^{-}}=\frac{E^{+}+\Gamma\cdot
E^{+}}{H^{+}-\Gamma\cdot H^{+}}\\\
&=Z_{\text{w}}\frac{1+\Gamma}{1-\Gamma}=Z_{0}\frac{d}{W}\frac{1+\Gamma}{1-\Gamma}\,,\end{split}$
(10)

where the superscripts ’$+$’ and ’$-$’ refer to the forward and backward
direction of propagation, $Z_{\text{w}}$ is the wave impedance,
$Z_{0}\approx$377\text{\,}\Omega$$ is the vacuum wave impedance,
$d=$100\text{\,}\mathrm{mm}$$ is the distance between the electrodes,
$W=$182\text{\,}\mathrm{mm}$$ is their width Slim  _et al._ (2016), and
$\Gamma$ is the reflection coefficient that controls the amplitude and phase
of the reflected wave. During the measurements described here, the Wien filter
was typically operated at a net input RF power of $600\text{\,}\mathrm{W}$.

The field quotient $Z_{q}$ is controlled via a specially designed RF circuit
Slim  _et al._ (2020). By altering $\Gamma$ via two variable vacuum
capacitors, called $C_{\text{L}}$ and $C_{\text{T}}$, a wide range of $Z_{q}$
values can be covered, and the matching point corresponding to the minimum
induced vertical beam oscillation amplitude may be determined.

##  IV Beam oscillations

In this experiment, the electric field of the Wien filter is oriented
vertically and the magnetic field horizontally. This implies that the
oscillations mainly take place along the $y$-axis [see Eq. (1)]. For the
detection of the vertical beam oscillations, a conventional beam position
monitor has been employed. In order to be most sensitive, BPM 17 located in
the straight section opposite to the Wien filter (see Fig. 2) with a large
vertical $\beta$ function was used, $\beta_{y}^{\rm
BPM}\approx$15.3049\text{\,}\mathrm{m}$$, while at the Wien filter location,
$\beta_{y}^{\rm WF}\approx$2.6784\text{\,}\mathrm{m}$$, as shown in Fig. 3.
The arguments to pick BPM 17 are further discussed below in Sec. V.

Figure 3: Vertical and horizontal beta-functions along the circumference of
COSY Weidemann  _et al._ (2015). The vertical dashed lines mark the location
of the Wien filter and of the beam position monitor used during the
measurement of the beam oscillations.

In order to measure small beam oscillations, a technique based on lock-in
amplifiers111HF2LI 50 MHz Lock-in Amplifier, Zurich Instruments AG, 8005
Zurich, Switzerland, https://www.zhinst.com/others/products/hf2li-lock-
amplifier. was developed Meade (1983). These devices operate in the frequency
domain and lock onto a signal whose frequency is set as a reference, which is
particularly useful in an electromagnetically noisy environment. Each
measurement consisted of two subsequent machine cycles of
$3\text{\,}\mathrm{min}$ duration, as depicted in Fig. 14 in Appendix B.

Figure 4: Readout scheme of the COSY BPM 17. The signals of the four
electrodes are fed into lock-in amplifiers. The differential signal of each
electrode is analyzed at the two reference frequencies given by the COSY RF
and the Wien filter frequency. The resulting Fourier amplitudes of the signals
are recorded in the EPICS333Experimental Physics and Industrial Control
System, https://epics.anl.gov/index.php. archiving system of COSY.

A stored beam bunch circulating at a revolution frequency of $f_{\text{rev}}$
that passes through a beam position monitor induces a voltage signal on all
its four electrodes, as indicated in the readout scheme of the BPM, shown in
Fig. 3. For the detection of vertical beam oscillations, only the voltage
signals $U_{\text{t},\,b}$ from the top (t) and bottom (b) electrodes are
considered. These signals are trains of short pulses with the repetition
frequency $f_{\text{rev}}$. In view of Eq. (2) and as far as the Fourier
spectrum of the beam oscillations is concerned, without loss of generality,
the BPM can be considered to be located right behind the Wien filter, and the
induced voltages can be represented by

$\vspace{0.3cm}U_{\text{t,\,b}}=\left[U_{0}\pm\Delta U\left(\Delta
y\right)\right]\cos(\omega_{\text{rev}}t)\,,$ (11)

where the index ’t’ refers to the $+$ sign and the index ’b’ to the $-$ sign,
respectively. The harmonic factor $\cos(\omega_{\text{rev}}t)$ emphasizes the
pulse repetition frequency, although $\cos(\omega_{\text{rev}}t)=1$ for
$t=nT$. The voltage $U_{\text{t,\,b}}$ is non-zero only at the time the beam
passes through the BPM. Here, $U_{0}$ denotes the voltage proportional to the
beam current that is induced when the beam passes exactly through the center
of the BPM, and $\Delta U\left(\Delta y\right)$ represents the voltage
variation induced by a beam that is vertically displaced by $\Delta y$.

For small beam displacements, the beam position monitor operates in its linear
regime, which implies that the induced voltages take the form

$\Delta U\left(\Delta y\right)=\kappa\cdot\Delta y\cdot U_{0}\,,$ (12)

where $\kappa$ is a calibration factor that needs to be determined. At a
momentum of $970\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$,
the revolution frequency of deuterons orbiting in COSY is
$f_{\text{rev}}\approx$750\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$. The Wien
filter is operated at the $k^{\text{th}}$ sideband of the spin-precession
frequency $f_{s}$, given by

$f_{\text{WF}}=\frac{\omega_{\text{WF}}}{2\pi}=(k+\nu_{s})f_{\text{rev}}=k\cdot
f_{\text{rev}}+f_{s}\,.$ (13)

Here $\nu_{s}=G\gamma$ denotes the spin tune, i.e., the number of spin
precessions per revolution, $G\approx-$0.1430$$ is the magnetic anomaly of the
deuteron, and the spin precession frequency $f_{s}=\nu_{s}f_{\text{rev}}$ Slim
_et al._ (2016). It should be noted that in view of $\omega_{\rm rev}T=2\pi$,
trains of beam oscillation pulses do not depend on the actual choice of the
sideband. In the present experiment the Wien filter was operated at $k=-1$,
which corresponds to
$f_{\text{WF}}\approx$871\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ 444For the
considerations presented in this paper, negative and positive frequencies are
considered equivalent..

The induced oscillations of amplitude $\xi_{y}$ contribute to Eq. (11) the
harmonic voltage variation $\Delta U\left(y(t)\right)$. The BPM in conjunction
with the lock-in amplifiers is used to measure at times $t=nT$ the beam
positions at the reference frequencies, i.e., at $f_{\text{WF}}$ and at
$f_{\text{rev}}$. Given that $y(t)$ can be evaluated at the spin precession
frequency, the BPM signals of the upper and lower electrodes can be written as
follows

$\begin{split}U_{\text{t,\,b}}(t)=&\left[U_{0}\pm\Delta U\left(\Delta
y\right)\pm\Delta
U\left(y(t)\right)\right]\cos\left(\omega_{\text{rev}}t\right)=\left[U_{0}\pm\Delta
U\left(\Delta
y\right)\pm\kappa\xi_{y}U_{0}\cos\left(\omega_{\text{s}}t\right)\right]\cos\left(\omega_{\text{rev}}t\right)\\\
=&\left[U_{0}\pm\kappa\Delta
yU_{0}\right]\cos\left(\omega_{\text{rev}}t\right)\pm\frac{1}{2}\kappa\xi_{y}U_{0}\cos\left(\omega_{\Delta}t\right)\pm\frac{1}{2}\kappa\xi_{y}U_{0}\cos\left(\omega_{\Sigma}t\right)\,.\end{split}$
(14)

Here $\omega_{\Delta}$ and $\omega_{\Sigma}$ represent sidebands of the Wien
filter frequency at

$\begin{split}\omega_{\Delta}&=\omega_{\text{rev}}-\omega_{\text{s}}=\omega_{\text{WF}}\rvert_{k=1}\,,\text{and}\\\
\omega_{\Sigma}&=\omega_{\text{rev}}+\omega_{\text{s}}=\omega_{\text{WF}}\rvert_{k=-1}\,.\end{split}$
(15)

(a) Magnitude of the field quotient $|Z_{q}|$, evaluated integrally, where
$|Z_{q}|^{\text{int}}=\int|Z_{q}|\text{d}\ell$. Ideally, with $|Z_{q}|$ close
to $176\text{\,}\mathrm{\SIUnitSymbolOhm}$, the electric and magnetic forces
are equal. (b) Phase of the field quotient $\angle Z_{q}$ evaluated
integrally, where $\angle Z_{q}^{\text{int}}=\int\angle Z_{q}\text{d}\ell$. A
non-vanishing $\angle Z_{q}$ implies a phase shift between the electric and
magnetic fields.

Figure 5: Simulated integral magnitude (a) and phase of the field quotient
$Z_{q}$ (b) at each point of the $C_{\text{L}}$ and $C_{\text{T}}$ grid,
indicated by the blue points, $\ell$ denotes the effective length of the Wien
filter. Besides the matching point [see Eq. (19)], $\left(7\times 6\right)$
grid points were investigated.

In order to measure the beam oscillations, four lock-in amplifiers Meade
(1983) were used, two for the horizontal and two for the vertical direction.
For each direction, one lock-in amplifier detects the Fourier amplitudes at
$f_{\text{rev}}\approx$750\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ and a
second one at
$f_{\Sigma}=f_{\text{rev}}+f_{s}\approx$871\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$.
The lock-in amplifiers receive reference frequencies from the signal generator
of the Wien filter and from the master oscillator of COSY. The four Fourier
amplitudes of the top and bottom electrodes are determined practically in
real-time, yielding

$\begin{split}A_{\text{t,\,b}}^{\text{rev}}&=U_{0}\pm\kappa\Delta
yU_{0}\,,\text{and}\\\
A_{\text{t,\,b}}^{\Sigma}&=\mp\frac{1}{2}\kappa\xi_{y}U_{0}\,.\end{split}$
(16)

The amplitude of the vertical oscillation $\xi_{y}$ can then be determined
from

$\frac{A_{\text{t}}^{\Sigma}-A_{\text{b}}^{\Sigma}}{A_{\text{t}}^{\text{rev}}+A_{\text{b}}^{\text{rev}}}=\hat{\xi}_{y}=\kappa\frac{U_{0}}{2U_{0}}\xi_{y}=\frac{1}{2}\kappa\xi_{y}\,.$
(17)

The uncalibrated raw asymmetry of the four Fourier amplitudes is denoted by
$\hat{\xi}_{y}$.

The readout scheme, shown in Fig. 3, was used to concurrently record radial
beam oscillations, and the above analysis has been repeated for the
corresponding $\hat{\xi}_{x}$. The main result is that the coupling of
vertical and radial betatron oscillations is negligibly weak,
$|\hat{\xi}_{x}/\hat{\xi}_{y}|<$2\text{\times}{10}^{-2}$$, which justifies
treating the RF-driven beam oscillations as one-dimensional.

The determination of the calibration constant $\kappa$, required to calibrate
the vertical oscillation amplitude, is described in detail in Appendix B. It
amounts to

$\kappa=\left(5.82\pm
0.43\right)\cdot${10}^{-6}\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}\,.$
(18)

During the experiments, the vertical betatron tune of the machine amounted to
about $\nu_{y}\approx 3.6040$.555The numerical values used for the simulation
calculations are listed in Table 1 of Appendix A. The frequency
$f_{\Sigma}\approx$871\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$, at which the
Wien filter is operated, is well separated from the lowest intrinsic spin
resonances666An intrinsic depolarizing resonance is encountered, when the
betatron motion of the particles is in sync with the spin motion, hence, when
the condition
$f_{s}=\nu_{s}f_{\text{rev}}=f_{y}=(nP\pm\nu_{y}^{\prime})f_{\text{rev}}$ is
fulfilled Huang  _et al._ (2004), where $n\in\mathbb{N}$, $P$ denotes the
superperiodicity of the lattice, and $\nu_{y}^{\prime}$ the fractional tune.
During the experiments described here, $P=1$ (see also Fig. 3). at
$297\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$,
$453\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$,
$1048\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, and
$1204\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$.

The two variable and highly accurate capacitors, $C_{\text{L}}$ and
$C_{\text{T}}$, are driven by stepper motors. They constitute the main
dynamical elements of the driving circuit. Each pair of capacitor values
yields a well-defined field quotient $|Z_{q}|$, as shown in Fig. 5 (a). Away
from the matching point, a phase shift $\angle Z_{q}$ occurs between electric
and magnetic fields, as shown in Fig. 5 (b). The corresponding Lorentz force
leads to the measured beam oscillations, i.e., the function
$\xi_{y}=f\left(C_{\text{L}},C_{\text{T}}\right)$, which can be visualized in
the form of a 2D map, as shown in Fig. 6. The experimental data were taken on
a grid of $(7\times 6)$ points of $C_{\text{L}}$ and $C_{\text{T}}$, with
corresponding grid spacings of $\left(94.5\pm
1.0\right)\,$\mathrm{p}\mathrm{F}$$ for $C_{\text{L}}$ and $\left(95.8\pm
1.0\right)\,$\mathrm{p}\mathrm{F}$$ for $C_{\text{T}}$. Each grid spacing
corresponds to 1000 steps of the corresponding stepper motors. The calibration
of the capacitances $C_{\text{L}}$ and $C_{\text{T}}$ as a function of step
number is discussed in detail in Slim  _et al._ (2020). The grid spans over
$C_{\text{L}}\in[318.88,\,885.58]\,$\mathrm{pF}$$ and
$C_{\text{T}}\in[428.99,\,907.79]\,$\mathrm{pF}$$. The uncertainties of the
grid spacings are systematic ones.777The individually measured uncertainties
of the capacitors are actually much smaller than the stated uncertainty of
$1.0\text{\,}\mathrm{p}\mathrm{F}$. However, other factors, such as the
capacitances and inductances of the connectors and cables and their power
dependencies, also contribute to the aforementioned uncertainties.

The map of the measured and calibrated vertical beam oscillations $\xi_{y}$ is
shown in Fig. 6. The parameters of the matching point are given by

$\begin{split}C_{\textnormal{L}}&=\left(697.1\pm
1.0\right)\,$\mathrm{pF}$\,,\text{ and}\\\ C_{\textnormal{T}}&=\left(503.0\pm
1.0\right)\,$\mathrm{pF}$\,,\end{split}$ (19)

and the corresponding minimal detected beam oscillation amplitude at the
location of BPM 17 amounts to

$\xi_{y}^{\text{min}}\big{\rvert}_{\text{BPM}}=(1.08\pm
0.52)\,$\mathrm{\SIUnitSymbolMicro m}$\,.$ (20)

The above accuracy of $\delta\xi_{y}^{\rm
min}\big{\rvert}_{\text{BPM}}=$0.52\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
can be compared to the accuracy of measurements of static distortions of the
beam orbit, which is about $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (see
Table 2). As expected, the accuracy of the beam oscillation amplitudes is
better by a factor of about 40 compared to the static amplitudes measured
using the same BPM.

Upon rescaling the oscillation amplitudes using Eq. (2) with $\beta$ functions
listed in Table 1 to the WF location, we obtain

$\xi_{y}^{\text{min}}\big{\rvert}_{\rm WF}=(0.45\pm
0.22)\,$\mathrm{\SIUnitSymbolMicro m}$\,,$ (21)

which should be compared to the value of
$Q\approx$41\text{\,}\mathrm{n}\mathrm{m}$$, given in Eq. (9). The largest
measured amplitude of driven beam oscillations at a strongly mismatched point
with $600\text{\,}\mathrm{W}$ of input RF power amounts to

$\xi_{y}^{\text{max}}\big{\rvert}_{\text{BPM}}=(66.2\pm
3.1)\,$\mathrm{\SIUnitSymbolMicro m}$\,.$ (22)

Here one must bear in mind that the sensitivity to a periodic signal scales
inversely with the square root of the observation time.888The relevant
discussion is found in Ref. Bagdasarian  _et al._ (2014). See also the
observation of white noise suppression by two orders in magnitude when using
$5\text{\,}\mathrm{h}$ signal averaging in a test bench experiment with SQUID
BPMs Hacıömeroğlu  _et al._ (2019). The frozen-spin proton EDM experiment aims
at the accumulation of the EDM signal for a duration of about
${10}^{7}\text{\,}\mathrm{s}$ Abusaif  _et al._ (2021). The accuracy
$\delta\xi_{y}^{\text{min}}\big{\rvert}_{\text{BPM}}=$0.52\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ [Eq. (20)] corresponds to an averaging time of $96\text{\,}\mathrm{s}$
[see Fig. (14) in Appendix (B)]. With the currently used BPMs and their
readout electronics, together with an extension of the averaging time to
${10}^{7}\text{\,}\mathrm{s}$, there would be a factor of 320 improvement in
sensitivity to coherent beam oscillations, leading to an accuracy of
$1.6\text{\,}\mathrm{nm}$.

In Fig. 7 (a), the data measured at the matching point [Eq. (19)] are shown.
Each sample was recorded by the lock-in amplifiers with an integration time
set to $0.5\text{\,}\mathrm{s}$, corresponding to an average of $5000$
measurements. A Monte Carlo error propagation model was applied to treat the
uncertainties of the still uncalibrated raw position asymmetries
$\hat{\xi}_{y}$ and the calibration coefficient $\kappa$ Aster  _et al._
(2018). The results are fitted with a normal distribution, as shown in Fig. 7
(b), from which the mean value $\mu_{{\xi}_{y}}$ and the error of the measured
beam oscillations $\sigma_{{\xi}_{y}}$ are estimated. The latter represents
the systematic error of the measurement. It should be noted that the map shown
in Fig. 6 is actually a function of all the circuit elements. The
uncertainties of $\xi_{y}$ are influenced by the uncertainties of all circuit
elements and also by the ones of the BPM itself, which include its readout
electronics, i.e., the lock-in amplifiers.

Figure 6: Measured amplitudes of beam oscillations $\xi_{y}^{\text{exp}}$ at
BMP 17, plotted on a grid as a function of the variable capacitor values
$C_{\text{L}}$ and $C_{\text{T}}$. To avoid crowding up the map, the error
bars of the data points were omitted, these are shown in Fig. 13 instead. The
parameters of the matching point are given in Eq. (19).

(a) Measured oscillation amplitudes $\xi_{y}$ using data samples of
$0.5\text{\,}\mathrm{s}$ duration, each sample reflects the average of $5000$
measurements of the lock-in amplifiers. (b) Probability density distribution
$f_{\xi_{y}}$ of the measured data, fitted with a Gaussian to determine mean
and standard deviation.

Figure 7: Measured beam oscillations at the matching point [Eq. (19)] of the
map shown in Fig. 6. The samples shown in panel (a) were acquired during a
data taking period of $108\text{\,}\mathrm{min}$, using 36 machine fills
(cycles).

To appreciate the result given in Eq. (20), one can compare the oscillation
amplitude to the $1\sigma$ vertical beam size. The latter has been deduced
from the $1\sigma$ beam emittance $\epsilon_{y}$ and the amplitude of the
$\beta$ function at the position of the BPM, yielding

$\sigma_{y}^{\text{BPM}}=\sqrt{\beta_{y}^{\text{BPM}}\epsilon_{y}}\approx$1.4\text{\,}\mathrm{m}\mathrm{m}$\,.$
(23)

In the present experiment, the beam emittance was not monitored. The above
numerical estimate of $\sigma_{y}^{\text{BPM}}$ is based on rescaling the
experimental result for the $2\sigma$ beam emittance of
$49.3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ protons in COSY of
$\epsilon_{y}=(0.92\pm 0.15)\,$\mathrm{\SIUnitSymbolMicro m}$$ Weidemann  _et
al._ (2015) to the conditions of the present experiment. It is noteworthy that
with the present equipment it is possible to access coherent beam oscillations
with amplitudes that are more than three orders of magnitude smaller than the
beam size.

##  V Beam dynamics simulations

To improve our understanding of the measured results, a computer code was
developed to model the beam dynamics in the COSY storage ring. The modeled
storage ring consists of a sequence of drift regions, quadrupole and dipole
magnets, the Wien filter, and beam position monitors. These elements are
represented by transfer matrices, which are well understood and documented in
the literature Wolski (2014). In the model of the ring, the actual settings of
the beam optics elements of COSY were those used at the time when the
experiment took place. Simulations are based on the Hamiltonian formulation as
presented in Ref. Wolski (2014). The Wien filter is modeled by a time-
dependent matrix that also takes into account the arrival time of the
particles.

###  V.1 Time-dependent Wien filter field maps

In order to be able to perform reliable beam simulations, we have placed great
emphasis on good spatial resolution and the accuracy of the 3D field maps
inside the Wien filter999Each field map consists of
$2\text{\times}{10}^{6}$points, 200 points along the $x$ axis
$\left(x\in[$-5\text{\,}\mathrm{m}\mathrm{m}$,$5\text{\,}\mathrm{m}\mathrm{m}$]\right)$,
200 points along the $y$ axis
$\left(y\in[$-5\text{\,}\mathrm{m}\mathrm{m}$,$5\text{\,}\mathrm{m}\mathrm{m}$]\right)$,
and 50 points along the $z$ (Wien filter) axis
$\left(z\in[-\ell/2,+\ell/2]\right)$, where $\ell=$1.16\text{\,}\mathrm{m}$$
is the effective length of the Wien filter., computed using a 3D
electromagnetic simulation tool101010Electromagnetic and circuit simulations
were performed using CST, from Dassault Systèmes, Vélizy-Villacoublay, France,
https://www.3ds.com.. The fringe fields of the Wien filter are included,
because they are of particular importance for the beam oscillations, as will
be discussed later.

(a) 3D electric field distribution of the component $E_{y}$. (b) 3D magnetic
field distribution of the flux density component $B_{x}$.

Figure 8: Examples of the main electric and magnetic field components inside
the waveguide RF Wien filter at the matching point [see Eq. (19)] with an
input RF power of $600\text{\,}\mathrm{W}$. The electric field component in
(a) points vertically upward ($y$-direction), while the component of the
magnetic flux density in (b) points radially outward ($x$-direction).

An example of the computed 3D fields of the Wien filter at the experimentally
determined matching point is shown in Fig. 8. The beam-tracking simulations
use the three vector components of the electric and magnetic fields. The Wien
filter is implemented as an RF kicker, as described by Eq. (3).

Inside COSY there are 32 BPMs available to control the horizontal and vertical
beam position during operation. In order to select one of them with a good
sensitivity to determine the beam oscillations induced by the Wien filter, a
number of particles were tracked, as described above, and the orbit response
induced by a field change at the location of the Wien filter was calculated at
each BPM location111111In the preparatory stage, simulations were carried out
using the Software Toolkit for Charged-Particle and X-Ray Simulations BMAD
Sagan (2006).. As a result, BPM 17, located about $70\text{\,}\mathrm{m}$
downstream of the Wien filter (see Fig. 2), was chosen because it offered good
sensitivity to both radial and vertical beam oscillations.

Figure 8 shows the main field components $E_{y}$ and $B_{x}$. When mismatched,
the Wien filter generates periodic transverse perturbations of the trajectory.
Switching off the Wien filter eliminates such oscillations. The maximum
amplitude of the observed oscillations in the simulation is then taken for
$\xi_{y}$. The two simulated vertical beam oscillation amplitudes of BPM 17
and Wien filter read

$\begin{split}\xi_{y}^{\text{BPM}}&=\left(1.086\pm
0.082\right)\,$\mathrm{\SIUnitSymbolMicro m}$\,,\text{ and}\\\
\xi_{y}^{\text{WF}}&=\left(0.435\pm 0.031\right)\,$\mathrm{\SIUnitSymbolMicro
m}$\,,\end{split}$ (24)

which agree well with the experimentally measured results, given in Eqs. (20)
and (21). A detailed description of the determination of the uncertainties of
the beam simulations is discussed in the next section.

For each and every measured point on the $C_{\text{L}}$ versus $C_{\text{T}}$
grid, a beam dynamics simulation was carried out. For each of these points, a
3D field map of the Wien filter was generated and then used for the beam
tracking simulations. The results of these simulations are shown in Fig. 9,
and are later compared with the results of the measurements.

Figure 9: Simulated amplitudes of beam oscillations $\xi_{y}^{\text{sim}}$ as
a function of the variable capacitor values $C_{\text{L}}$ and $C_{\text{T}}$.
To avoid overcrowding the map, the error bars of the data points were omitted
here and are shown in Fig. 13.

###  V.2 Uncertainty evaluation

The accuracy with which the Lorentz force and the resulting amplitudes of the
beam oscillations can be tuned depends on the accuracy with which the field
quotient $Z_{q}$ can be integrally set to the desired value. $Z_{q}$ depends
on the hardware elements in the driving circuit. In order to evaluate the
effects of uncertainties of these elements, extensive coupled circuit
electromagnetic simulations have been conducted, as discussed in Ref. Slim
_et al._ (2020). The uncertainties involved are listed in Table 6 and shown in
Fig. 16 of Ref. Slim  _et al._ (2020). As far as the Lorentz force is
concerned, most important are the uncertainties of the fixed inductance
$L_{\text{f}}$ and the fixed resistance $R_{\text{f}}$. Once these
uncertainties are known, one can compute the electric and magnetic fields and
the corresponding Lorentz force, including their corresponding errors. Figure
10 shows a few examples of the main components of the electric and magnetic
fields, computed with the above mentioned circuit uncertainties. As will be
explained below, these 3D fields, together with their uncertainties, are
subsequently used as input to the beam simulations.

(a) Electric field component $E_{y}(z)$ under circuit uncertainties. (b)
Magnetic field component $B_{x}(z)$ under circuit uncertainties.

Figure 10: $200$ examples of the electric and magnetic fields as a function of
$z$ along the beam axis under the circuit uncertainties, specified in the list
of uncertainties in Table 6 of Slim  _et al._ (2020).

The algorithm used to compute the uncertainties of the beam simulations is the
polynomial chaos expansion (PCE), as explained in Refs. Slim  _et al._ (2017,
2020) and in Appendix C. The PCE has been proven in many applications in
science and engineering to be just as accurate as the computationally much
more expensive Monte-Carlo counterpart Smith (2013); Offermann  _et al._
(2015); Adelmann (2018); Slim  _et al._ (2017).

To compute the uncertainties $\sigma_{\xi_{y}}$, the PCE algorithm requires a
random set of the simulated $\xi_{y}$, alongside a set of randomized input
parameters according to their uncertainties to generate the output. The set of
$\xi_{y}$ is produced using a number of beam-tracking simulations, where for
each instance, a 3D field map of the Wien filter is generated, according to
the randomized input parameters. An example of the electric and magnetic
fields evaluated at the center of the Wien filter for the matching case [see
Eq. (19)] is shown in Fig. 10. The magnitudes of the fields vary as a function
of the uncertainties of the driving circuit Slim  _et al._ (2020). The
numerical tracking of the particles through these fields generates a
collection of different $\xi_{y}$ values that the PCE algorithm can use to
project the output onto orthogonal polynomial functions. These functions serve
as basis functions, from which the expansion coefficients are determined that
are used to generate a large sample of outputs to compute the uncertainties of
the beam simulations.

In Fig. 11 (a), the simulated values of $\xi_{y}$ are shown for the matching
case. The detailed steps to achieve this result are discussed in Appendix C.
As shown in Fig. 11(b), fitting these data to a Gaussian yields a standard
deviation of $\sigma_{\xi_{y}}=$0.082\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.
This number is of considerable importance, because, given the uncertainties of
the driving circuit, it sets the lower limit that can be achieved by
minimizing the amplitude of the vertical beam oscillations when more sharply
tuning the driving circuit of the Wien filter. The same procedure is performed
on each point of the map shown in Fig. 9.

(a) Simulated oscillation amplitudes under uncertainties at the matching point
(Eq. (19)). Of the ${10}^{6}$ simulations that were carried out, only
${10}^{4}$ are shown here. (b) Probability density distribution $f_{\xi_{y}}$
of the ${10}^{6}$ simulations from panel (a), fitted by a Gaussian to
determine mean and standard deviation.

Figure 11: Results of the sparce PCE algorithm to compute the uncertainties of
the simulated vertical beam oscillations at BPM 17.

##  VI Comparison of simulation and experimental results

The simulations yield the net Lorentz force exerted by the Wien filter on beam
particles and the corresponding oscillation amplitudes for each measured point
of the $C_{\text{L}}$ versus $C_{\text{T}}$ map, shown in Fig. 6. The only
variables in this case are the field maps of the Wien filter itself. After
$1000$ turns, the beam position is computed at the same location in the ring,
where the measurement using BPM 17 took place (see Fig. 2). The net Lorentz
force is a result of local cancellations between the electric and magnetic
field components, as illustrated in Fig. 12 for the matching point given in
Eq. (19) with the minimal measured oscillation amplitude.

In Fig. 12(a), the local Lorentz force is shown along the trajectory for 5
randomly chosen passes though the Wien filter. The trajectory of the same
particle changes from pass to pass, thereby different Wien filter fields and
consequently different values of the Lorentz force $F_{y}$ will be picked up.
As shown in Fig. 12, even at the matching point, the matching is still
imperfect, and the largest local $F_{y}$ contributions are caused by the
fringe fields at the entrance and exit of the Wien filter. Despite the
different location of the particle in the vertical and horizontal phase space
at the entrance of the Wien filter upon subsequent passes, the integration of
these local forces along the particle trajectories exhibits nevertheless a
perfectly harmonic time dependence with the frequency $f_{s}$, as shown in
Fig. 12(b). The points encircled in blue correspond to the randomly selected
passes through the Wien filter, shown in Fig. 12(a).

(a) Local Lorentz force $F_{y}(z)$ exerted on a single deuteron for different
passes though the Wien filter. The turn numbers used here were randomly
selected between 1 to 100. The fields were evaluated at the crosses and the
interconnecting lines are to guide the eye. (b) Integral Lorentz force
$F_{y}(n)$ evaluated along the trajectory. Each point represents an overall
kick exerted per turn $n$. The points marked in blue correspond to the
integrated local Lorentz force of the individual turns shown in panel (a).

Figure 12:  Simulation of the local and integrated Lorentz force in the Wien
filter at the matching point of Eq. (19). Depending on the initial coordinates
in the vertical and horizontal phase space, the particle travels along
different trajectories, and therefore picks up different field components
$F_{y}$.

In the left panel of Fig. 13, the amplitude of the simulated Lorentz force
$F_{y}^{\text{sim}}$ is plotted versus the simulated oscillation amplitude
$\xi_{y}^{\text{sim}}$ at the Wien filter position. As expected, it exactly
follows Hooke’s law with a spring constant of $k_{\text{H}}=(151.2\pm
0.2)\,$\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{m}^{2}$$. In Fig.
13(b), the measured amplitudes are compared with the ones simulated for the
location of BPM 17. The two sets $\xi_{y}^{\text{exp}}$ and
$\xi_{y}^{\text{sim}}$ are in very good agreement with each other. The
horizontal and vertical error bars are derived from the uncertainties of the
measurements and simulations, represented by the width of the distributions,
as shown in Figs. 7(b) and 11(b). It is important to note that the error bars
refer to systematic uncertainties and should not be confused with statistical
ones. This implies that repetitions of either the measurements or the
simulations will neither reduce the systematic error of the readout
electronics of BPM 17, nor will it affect the uncertainties of the elements of
the driving circuit.

(a) Simulated Lorentz force $F_{y}^{\text{sim}}$ at the Wien filter location
as function of the oscillation amplitude $\xi_{y}^{\text{sim}}$, fitted with
the function $F_{y}^{\text{sim}}=a\cdot\xi_{y}^{\text{sim}}+b$. (b) Simulated
beam oscillation amplitude $\xi_{y}^{\text{sim}}$ versus the measured
oscillation amplitude $\xi_{y}^{\text{exp}}$ at the BPM, fitted with the
function $\xi_{y}^{\text{sim}}=c\cdot\xi_{y}^{\text{exp}}+d$.

Figure 13: (a): Simulated amplitude of the Lorentz force at the Wien filter
location as function of the simulated beam oscillation amplitudes
$\xi_{y}^{\text{sim}}$. (b): Simulated versus measured vertical beam
oscillation amplitudes at the location of BPM 17. The horizontal error bars of
the measured amplitudes $\xi_{y}^{\text{exp}}$ originate from the readout
electronics of BPM 17 and the calibration factor $\kappa$ (see Appendix B),
whereas the vertical ones are determined by the circuit uncertainties using
the PCE method, as described in Appendix C.

The fit shown in Fig. 13 yields $\chi^{2}/\text{ndf}=45.5/41$, very close to
unity York  _et al._ (2004). The linear fit yields a slope of $0.999\pm
0.018$, which is perfectly consistent with unity. The intercept parameter of
the fit yields $($-0.93$\pm 0.31)\,$\mathrm{\SIUnitSymbolMicro m}$$, and
within three standard deviations, it agrees with zero.

The very good agreement between measurements and simulations reflects our good
understanding of both the electromagnetic fields generated in the Wien filter
and of the underlying beam dynamics in the machine. This point is further
substantiated by comparing the simulated amplitudes at the Wien filter and at
the positions of the BPMs with the estimated amplitudes expected from
rescaling based on the $\beta$ functions121212The uncertainty of the $\beta$
functions amounts to about 10%, as discussed in Ref. Weidemann  _et al._
(2015)., taking into account the numerical values, listed in Table 1 of
Appendix A,

$\begin{split}\left.\xi_{y}^{\text{WF}}\right|_{\text{sim}}&=(0.435\pm
0.031)\,$\mathrm{\SIUnitSymbolMicro m}$\,,\\\
\sqrt{\frac{\beta_{y}^{\text{WF}}}{\beta_{y}^{\text{BPM}}}}{\xi_{y}^{\text{BPM}}}|_{\text{sim}}=\left.\xi_{y}^{\text{WF}}\right|_{\text{est}}&=(0.435\pm
0.039)\,$\mathrm{\SIUnitSymbolMicro m}$\,.\end{split}$ (25)

The good agreement between these two numerical values in Eq. (25) indicates
that the observation of the oscillation amplitude at one location in the ring
can be reliably transferred to some other place in the ring by use of Eq. (2).
The above quoted value of
$\left.\xi_{y}^{\text{WF}}\right|_{\text{sim}}=0.435\,$\mathrm{\SIUnitSymbolMicro
m}$$ is about a factor of 10 larger than the quantum limit of the vertical
oscillation amplitude $Q$, given in Eq. (9).

In searches for EDMs in dedicated all-electric storage rings, a continuous
monitoring of the orbits of the two counter-rotating beams is mandatory during
data acquisition within the horizontal spin-coherence time Abusaif  _et al._
(2021). When intrabeam scattering can be neglected Weidemann  _et al._ (2015),
which is arguably justified within the horizontal spin-coherence time, the
beam can be described as a rarefied gas of particles, i.e., the zero-point
oscillations of individual particles are uncorrelated. We repeat the point
from the introduction that in a static regime, the quantum limit of the center
of mass of a bunch with $N$ particles can be estimated via
$Q_{\text{bunch}}=Q/\sqrt{N}$. For a bunch of $N=${10}^{10}$$ stored
particles, one obtains $Q_{\text{bunch}}\simeq$0.4\text{\,}\mathrm{pm}$$. It
follows that Heisenberg’s uncertainty relation does not present an obstacle to
achieving a sensitivity of $5\text{\,}\mathrm{p}\mathrm{m}$ for the vertical
separation of clockwise and counter-clockwise beams – the real challenge is to
develop compact BPMs with a sensitivity improved by a factor of about 300
compared to those used here Böhme  _et al._ (2018).

Finally, a satisfactory agreement has been achieved between Hooke’s constant,
simulated using the electromagnetic fields in the Wien filter and the $\beta$
functions of the COSY lattice, and the theoretical approximation of the no-
lattice model assuming constant $\beta$ functions of Eq. (6), yielding

$\begin{split}k_{\text{H}}^{\text{sim}}&=(151.2\pm
0.2)\,$\mathrm{MeV}\text{\,}{\mathrm{m}}^{-2}$\,,\text{ and}\\\
k_{\text{H}}^{\text{th}}&=$207\text{\,}\mathrm{MeV}\text{\,}{\mathrm{m}}^{-2}$\,.\end{split}$
(26)

The theoretical estimate of $k_{\text{H}}^{\text{th}}$, calculated using the
numerical values listed in Table 1 of Appendix A, is about a factor of 1.4
larger than the simulated one. The given uncertainty of
$k_{\text{H}}^{\text{sim}}$ does not include the systematic scale uncertainty
of the BPM calibration factor $\kappa$ [see Eq. (29) in Appendix A]. At the
matching point [see Eq. (19)], the Lorentz force amounts to

$F_{y}^{\text{WF}}=k_{\text{H}}^{\text{sim}}\cdot\left.\xi_{y}^{\text{WF}}\right|_{\text{sim}}\approx
66\,$\mathrm{eV}\text{\,}{\mathrm{m}}^{-1}$={10.6}\,$\mathrm{aN}$\,,$ (27)

where the intercept parameter has been ignored because of its smallness.

##  VII Conclusion and outlook

As part of several studies to investigate the performance of the waveguide RF
Wien filter, exploratory data were taken to provide a benchmark on the
sensitivity to very weak collective vertical beam oscillations of deuterons
stored in the COSY ring. To a good approximation, the beam can be viewed as a
rarefied gas of uncorrelated particles, and the sensitivity limit is
applicable to the classical motion of individual particles, propagating along
the ring circumference in the confining oscillatory potential. Simulations of
the beam dynamics in the COSY ring equipped with an RF Wien filter suggest
that with the present apparatus, the sensitivity to collective beam
oscillations on the sub-micron level is only a factor of about 10 larger than
the amplitude of single-particle zero-point quantum oscillations of the stored
deuterons. From the perspective of future EDM experiments, our finding
confirms that, as far as the Heisenberg uncertainty relation is concerned, a
separation of the centroids of two counter-propagating beams may be determined
to sub-picometer accuracy.

The reported excellent agreement between simulated and experimentally observed
vertical beam oscillations at COSY suggests that a further increase in
sensitivity to collective beam oscillations is possible. Specifically, the
simulation on finer capacitor grids indicates that by further optimization of
the Wien filter settings to $C_{\textnormal{L}}=\left(692.76\pm
1.00\right)$\mathrm{p}\mathrm{F}$$ and $C_{\textnormal{T}}=\left(495.77\pm
1.00\right)$\mathrm{p}\mathrm{F}$$, an oscillation amplitude at the Wien
filter location of $\xi_{y}=\left(0.077\pm
0.032\right)$\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ may be achieved. Thus in
that case, the vertical oscillation amplitude would only be about a factor of
2 away from the quantum limit, with a corresponding Lorentz force of
$F_{y}\sim$3\text{\,}\mathrm{aN}$$.

## Acknowledgments

We would like to thank I. Bekman, B. Breitkreutz, J. Hetzel and R. Stassen for
their support in setting up the COSY accelerator for the experiment. The work
presented here has been performed in the framework of the JEDI collaboration
and is supported by an ERC Advanced Grant of the European Union (proposal
number 694340). In addition, it was supported by the Russian Fund for Basic
Research (Grant No. 18-02-40092 MEGA) and by the Shota Rustaveli National
Science Foundation of the Republic of Georgia (SRNSFG Grant No. DI-18-298:
High precision polarimetry for charged-particle EDM searches in storage
rings).

##  Appendix A Quantities used in beam simulations

In order to provide a consistent calculation of all effects in the storage
ring, the beam simulations were carried out using the set of quantities given
in Table 1 as an input. The vertical machine tune $\nu_{y}$ is a result of
simulations with the known COSY lattice, reflecting the actual currents of the
magnetic elements in the machine at the time when the experiment was
conducted. The simulations provide the uncalibrated parameters of the vertical
beam oscillations to about per mill accuracy, and giving the kinematic, ring,
and Wien filter parameters to four digits appears therefore sufficient. It
should be noted that within the simulation calculations carried out in the
context of the present work, all quantities have been computed to double
precision (machine epsilon of $1.11\text{\times}{10}^{-16}$). Of the physical
quantities, the highest sensitivity to the vertical betatron tune is exhibited
by the theoretical estimate for Hooke’s constant,
$\text{d}k_{\text{H}}^{\text{th}}/\text{d}\nu_{y}\approx$2\text{\times}{10}^{3}\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{m}^{2}$$.
The largest uncertainty contributing to the error of the detected oscillation
amplitudes arises from the calibration factor $\kappa$ of the beam position
monitor, given in Eq. (29). It amounts to about $7.3\%$ and is considered a
systematic scale-factor uncertainty (see Appendix B).

Table 1:  Numerical values used for the beam simulations. The genuinely independent input parameters are listed in bold face. The derived quantities are displayed in normal font and are truncated to four decimal places. Quantity | Symbol | Value
---|---|---
deuteron beam momentum | $\bm{p}$ | $970.0000\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$
deuteron mass | $\bm{m}$ | $1875.6128\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}^{2}$
deuteron $G$ factor | $\bm{G}$ | $-0.1430$
Lorentz factor | $\beta$ | $0.4594$
Lorentz factor | $\gamma$ | $1.1258$
COSY circumference | $\bm{L_{\text{COSY}}}$ | $183.4728\text{\,}\mathrm{m}$
revolution frequency | $f_{\text{rev}}$ | $750\,603.7600\text{\,}\mathrm{H}\mathrm{z}$
vertical machine tune | $\nu_{y}$ | $3.6040$
vertical $\beta$ function at BPM 17 | $\beta_{y}^{\text{BPM}}$ | $15.3049\text{\,}\mathrm{m}$
vertical $\beta$ function at WF | $\beta_{y}^{\text{WF}}$ | $2.6784\text{\,}\mathrm{m}$
effective length WF | $\bm{\ell}$ | $1.1600\text{\,}\mathrm{m}$
frequency WF | $\bm{f_{\text{WF}}}$ | $871\,000.0000\text{\,}\mathrm{H}\mathrm{z}$
tune WF | $\nu_{\text{WF}}$ | $1.1604$

##  Appendix B Calibration of beam position monitor

The complex amplitudes measured by the lock-in amplifiers describe the
magnitude and phase of each signal, and are here expressed by the
corresponding real and imaginary components, denoted by $X$ and $Y$,
respectively, i.e., $A=X+iY$. Examples of the data recorded at the sum
frequency $f^{\Sigma}$ and at the revolution frequency $f^{\text{rev}}$ are
shown in Fig. 14. The observed weak attenuation of the beam current during a
measurement cycle by less than 7% clearly indicates a weak beam loss by
intrabeam or residual gas interactions, thus justifying our treatment of the
beam as a rarefied gas. The effect of switching on the power amplifiers of the
Wien filter at $t=$60\text{\,}\mathrm{s}$$ is clearly visible. In both panels,
one observes a separation of the quantities recorded by the top and bottom
electrodes in the $\mathrm{\SIUnitSymbolMicro V}$ range for both frequencies
after the Wien filter is switched on. This separation is much more pronounced
at the Wien filter frequency than at the revolution frequency.

(a) Real ($X^{\Sigma}$) and imaginary part ($Y^{\Sigma}$) of the complex
Fourier amplitudes $A^{\Sigma}$ at the Wien filter frequency. (b) Real
($X^{\text{rev}}$) and imaginary part ($Y^{\text{rev}}$) of the complex
Fourier amplitudes $A^{\text{rev}}$ at the revolution frequency.

Figure 14: Fourier amplitudes $A=X+iY$ for the top and bottom electrodes of BPM 17 recorded by the lock-in amplifier as a function of time in the cycle at a strongly mismatched point ($C_{\text{L}}=$907.79\text{\,}\mathrm{p}\mathrm{F}$$ and $C_{\text{T}}=$885.58\text{\,}\mathrm{p}\mathrm{F}$$), at the Wien filter frequency (a), and at the revolution frequency (b). In both panels, the stored beam current is shown in black. The cycle starts right after injection is completed at $t=$0\text{\,}\mathrm{s}$$, beam preparation continues until $t=$55\text{\,}\mathrm{s}$$, and the Wien filter is switched on and data acquisition starts at $t=$60\text{\,}\mathrm{s}$$. At $t=$156\text{\,}\mathrm{s}$$ the Wien filter is switched off and data acquisition stops. Table 2: Current $I$ (in $\%$ of the maximum admissible current) in the vertical steerers to generate bumps and the corresponding position change of the vertical orbit $y$ by $\Delta y$ at the location of BPM 17. $I$ (steerer) [%] |  $y$ [$\mathrm{m}\mathrm{m}$] |  $\Delta y$ [$\mathrm{m}\mathrm{m}$]
---|---|---
$-5$ | $-$7.756$\pm$0.030$$ | $-$7.466$\pm$0.030$$
$-4$ | $-6.684\pm 0.038$ | $-$6.395$\pm$0.038$$
$-3$ | $-5.629\pm 0.016$ | $-$5.339$\pm$0.016$$
$-2$ | $-4.518\pm 0.020$ | $-$4.229$\pm$0.020$$
$-1$ | $-3.489\pm 0.018$ | $-$3.119$\pm$0.018$$
$\phantom{+}0$ | $-2.439\pm 0.029$ | $-$2.150$\pm$0.029$$
$+1$ | $-1.429\pm 0.020$ | $-$1.140$\pm$0.020$$
$+2$ | $-0.288\pm 0.028$ | $\phantom{+}$0.000$\pm$0.000$$
$+3$ | $+0.798\pm 0.044$ | $+$1.085$\pm$0.044$$
$+4$ | $+1.872\pm 0.014$ | $+$2.160$\pm$0.014$$
$+5$ | $+2.928\pm 0.069$ | $+$3.211$\pm$0.069$$
Figure 15: Calibration curve of BPM 17. The ratio $R$, defined in Eq. (28),
depends on the introduced vertical beam displacement $\Delta y$ at the beam
position monitor.

The quantities $A_{\text{t}}^{\text{rev}}$ and $A_{\text{b}}^{\text{rev}}$,
given in Eqs. (16), are related to a vertical beam displacement $\Delta y$ in
the following way,

$\displaystyle
R=\frac{A_{\text{t}}^{\text{rev}}-A_{\text{b}}^{\text{rev}}}{A_{\text{t}}^{\text{rev}}+A_{\text{b}}^{\text{rev}}}=\kappa\frac{2U_{0}\Delta
y}{2U_{0}}=\kappa\Delta y\,.$ (28)

The calibration constant $\kappa$ is experimentally determined by introducing
local vertical beam bumps in the ring at the location of BPM 17. The orbit
positions $y$ and the orbit displacements $\Delta y$, listed in Table 2,
differ by the position of the unperturbed orbit, and are generated by altering
the current of a set of vertical steerers.

The steerer magnets have well-known conversion factors from current to
magnetic field. The calibration factor $\kappa$ is obtained by fitting the
ratio $R$ from Eq. (28) as a function of the vertical orbit variation $\Delta
y$, exhibiting the nearly linear relationship shown in Fig. 15. The slope
corresponds to

$\kappa=\left(5.82\pm
0.43\right)\cdot${10}^{-6}\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}$\,.$
(29)

##  Appendix C Simulation uncertainties

The uncertainties of the simulated amplitudes of the beam oscillations are
computed using the Polynomial Chaos Expansion (PCE) algorithm. The
functionality of the algorithm is explained below for one of the simulated
data points of the map shown in Fig. 9.

The PCE algorithm offers an alternative to the well-known Monte-Carlo (MC)
method without compromising the intended accuracy. It uses orthogonal
polynomials to represent randomly changing variables to describe observables
by means of a finite (truncated) series (for more details, see, e.g., Ref.
Slim  _et al._ (2017)). When the defined criteria of convergence are met, the
expansion coefficients can be used to generate an arbitrarily large sample of
observables, from which the uncertainties can be computed to the desired
statistical accuracy.

The PCE algorithm has been compared with the MC method in many applications
and has been shown to provide very reliable results Smith (2013). The PCE
requires much fewer simulations to converge compared to the MC method. For
instance, for the present case, 200 beam tracking simulations per point in the
2D the map of beam oscillations, shown in Fig. 9 were sufficient to reach
convergence.

Data: Generates Gaussian-distributed ensemble of uncertain circuit parameters
using the Latin Hypercube Sampling (LHS) scheme $X_{i}$

Result: Compute uncertainty of $\xi_{y}$

Given $X_{i}$, run full wave simulations;

Generate 3D electric and magnetic fields;

Run beam tracking simulations to compute ${\xi_{y}}_{i}$;

Standardize input data $X_{i}\longrightarrow\tilde{X}_{i}$;

Guess hyperbolic truncation norm, $q$-norm;

Start with lowest possible expansion order $p$;

Generate basis functions ${H}_{p}({\tilde{X}_{i}})$ ($p^{\text{th}}$-order
Hermite polynomials);

Generate hyperbolically truncated set of basis functions
${H}_{p}^{q}({\tilde{X}_{i}})$;

Apply Least-Angle Regression (LAR) algorithm;

Estimate optimum sparse set of basis functions
${H}_{p}^{q*}({\tilde{X}_{i}})$;

Compute expansion coefficients $C_{j}$, given
${\xi_{y}}_{i}=\sum_{j}C_{j}{H}_{p}^{q*}({\tilde{X}_{i}})$;

Compute leave-one-out error $\text{LOO}_{\text{err}}$;

Check convergence condition( $\text{LOO}_{\text{err}}<10^{-2}$);

while  _not convergent_ do

Enhance model (vary $p$ and $q$);

if  _convergent_ then

Generate large sample of $\xi_{y}$;

Estimate statistical parameters;

Terminate algorithm;

else

Enrich input samples $X_{i}$;

Repeat algorithm;

Algorithm 1 Sparse Polynomial Chaos Expansion Slim  _et al._ (2017).

In cases where the number of random input variables $m$ is larger than $10$,
the PCE method offers clear advantages over the MC method. The reason is that
the number of basis functions in the PCE method increases enormously as a
consequence of the tensor product of the involved polynomials. Therefore, the
algorithm has been improved further to allow for a reduction of the number of
simulations required. Such an approach is also adopted here, as described in
Algorithm 1. The hyperbolic truncation scheme together with the Least-Angle
Regression (LAR) method form a sparse version of the original algorithm.

An $m$-dimensional set is first created, representing $N$ combinations of
simultaneous random variables. Many methods can be used to generate such sets,
and here the Latin-hypercube sample scheme is adopted Slim  _et al._ (2020).
Subsequently, the set is standardized for convergence reasons. Depending on
the distribution of the data, the basis functions, here Hermite polynomials,
are determined. The number of basis functions restricts the lower limit of the
number of simulations (full-wave and tracking) which are usually
computationally expensive. As a rule of thumb, with $N$ basis functions, the
PCE algorithm requires at least $1.5\times N$ (in this case, full-wave)
simulations to converge. The number of basis functions itself can, however, be
reduced by the hyperbolic truncation scheme that eliminates higher-order terms
that do not have a significant impact on the observation objects Sudret and
Der Kiureghian (2000); Sudret (2008). Furthermore, by applying the LAR
algorithm, the number of remaining basis functions can be further reduced
substantially, whereby the problem becomes computationally solvable in a very
efficient fashion.

The matching point, specified in Eq. (19), yields the minimum measurable beam
oscillations, as given by Eq. (20). This experimental result can be estimated
using the beam-tracking calculations. Subsequently, the concrete steps of the
application of the PCE algorithm are discussed.

All the reasonable sources of uncertainties of the circuit are represented by
15 random parameters that are allowed to vary simultaneously. At first, a
sample of ($200\times 15$) entries is generated using the Latin-hypercube
sampling scheme. As an example of this sample, the variation of the three
circuit elements $C_{\textnormal{L}}$, $C_{\textnormal{T}}$, and the load
resistor $R_{\text{f}}$ is shown in Fig. 16(a).

(a) Sample of $C_{\text{L}}$, $C_{\text{T}}$, and $R_{\text{f}}$ used in the
PCE calculations showing a subset of the 15-dimensional input of random
circuit uncertainties. (b) Truncation schemes of the PCE algorithm. (c)
Expansion coefficients on a semi-log scale. 91 coefficients have been selected
after applying the LAR algorithm to the matching point. (d) Comparison between
the tracking results and the PCE with respect to the oscillation amplitude,
determined using the expansion coefficients of (c).

Figure 16: Intermediate results of the PCE algorithm applied at the matching
point [see Eq. (19)]. Quantitative results of the PCE algorithm are summarized
in Table 3.

All the uncertain parameters in the electromagnetic circuit simulations are
used to generate the electric and magnetic fields shown in Fig. 10. These are
subsequently used in the beam-tracking calculations. For the matching point of
the map [Eq. (19)], $N=200$ full-wave simulations were conducted. The import
of these field maps into the beam-tracking calculations resulted in a set of
$N=200$ values of $\xi_{y}$. This set is not directly used to conduct the
statistical analysis. Instead, in conjunction with the input samples, these
data are used as input to the sparse PCE algorithm.

The optimum set of basis function is determined using the LAR algorithm, as
shown in Fig. 16(b). With an expansion order of $p=6$ and a truncation norm
$q=0.35$, executing the PCE algorithm required $91$ basis functions to
converge, reflected by the low value of the leave-one-out error
$\text{LOO}_{\text{err}}=$1.7\text{\times}{10}^{-4}$$. Subsequently, the
expansion coefficients are computed, qualitatively depicted in Fig. 16(c). It
is shown in Fig. 16(d) that the PCE algorithm perfectly reproduces the
tracking results using these expansion coefficients. Finally, these
coefficients are used to reconstruct a larger sample of $\xi_{y}$ to estimate
the error $\sigma_{\xi_{y}}$. Figure 11 shows ${10}^{4}$ of the ${10}^{6}$
reconstructed samples. The PCE parameters used are summarized in Table 3.

The fitting of these results with a Gaussian, as depicted in panel (b) of Fig.
11, yields a standard deviation of
$\sigma_{\xi_{y}}=$0.082\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$. The
same technique is repeated for each point in the map.

Table 3: PCE simulation parameters of the matching point in Eq. (19). Parameter | Value
---|---
order of expansion $p$ | $6$
dimension $m$ | $15$
hyperbolic truncation $q$ | $0.35$
leave-one-out error $\text{LOO}_{\text{err}}$ | $1.71\text{\times}{10}^{-4}$
number of used basis functions $P^{\text{LAR}}$ | 91
number of used full-wave simulations $N$ | 200

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