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\DOI{10.5802/crphys.264}
\datereceived{2025-05-18}
\daterevised{2025-08-11}
\dateaccepted{2025-09-17}
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\dateposted{2025-10-02}
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\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Measuring short range stray electric fields with a force quantum
sensor}

\alttitle{Mesures de champs \'electriques parasites \`a courte distance
\`a l'aide d'un capteur quantique de force}

\author{\firstname{Yann} \lastname{Balland}\CDRorcid{0009-0005-1832-5780}}
\address{LTE, Observatoire de Paris, Universit\'e PSL, Sorbonne
Universit\'e, Universit\'e de Lille, LNE, CNRS, 61 avenue de
l'Observatoire, 75014 Paris, France}

\author{\firstname{Franck} \lastname{Pereira dos Santos}\CDRorcid{0000-0003-0659-5028}\IsCorresp}
\addressSameAs{1}{LTE, Observatoire de Paris, Universit\'e PSL,
Sorbonne Universit\'e, Universit\'e de Lille, LNE, CNRS, 61 avenue de
l'Observatoire, 75014 Paris, France}
\email[F. Pereira dos Santos]{franck.pereira@obspm.fr}

\begin{abstract}
We use a quantum sensor based on trapped atom interferometry, and
designed for probing short range atom-surface interactions, to
characterize parasitic electric fields produced by adsorbed atoms or
surface charges on a dielectric mirror. Applying controlled external
fields with in-situ electrodes allows measuring electric field
gradients with a relative uncertainty of order of 1\% via variations of
the force induced onto the atoms. More, our sensor can also be
configured as a trapped microwave clock, allowing for direct
measurements of the electric field amplitude via the Stark shift of the
hyperfine transition frequency. Such measurements of the electric field
amplitudes and gradients as a function of the atom-surface distance can
be used to construct a model for the spatial distribution of the atoms
adsorbed onto the surface of the mirror, and to accurately correct
local force measurements, such as related to the Casimir--Polder
interaction, from the detrimental impact of adsorbed atoms or stray
charges. 
\end{abstract}

\begin{altabstract}
Nous utilisons un capteur quantique bas\'e sur l'interf\'erom\'etrie
\`a atomes pi\'eg\'es, et con\c{c}u pour sonder les interactions
atomes-surface \`a courte distance, pour caract\'eriser les champs
\'electriques parasites produits par des atomes adsorb\'es ou des
charges de surface sur un miroir di\'electrique. L'application \`a
l'aide d'\'electrodes in situ de champs externes contr\^{o}l\'es permet
de mesurer les gradients de champ \'electrique avec une incertitude
relative de l'ordre de 1 \% gr\^{a}ce aux variations de la force
induite sur les atomes. En outre, notre capteur peut \'egalement
\^{e}tre configur\'e comme une horloge micro-onde pi\'eg\'ee, ce qui
permet de mesurer directement l'amplitude du champ \'electrique par le
biais du d\'ecalage Stark de la fr\'equence de la transition hyperfine.
Ces mesures des amplitudes et des gradients du champ \'electrique en
fonction de la distance atomes-surface peuvent \^{e}tre utilis\'ees
pour construire un mod\`ele de distribution spatiale des atomes
adsorb\'es sur la surface du miroir et pour corriger avec pr\'ecision
les mesures de force locale, telles que celles li\'ees \`a
l'interaction Casimir--Polder, de l'impact pr\'ejudiciable d'atomes
adsorb\'es ou de charges parasites.
\end{altabstract}

\keywords{\kwd{Quantum sensing}\kwd{Short range forces}\kwd{Electric fields}}

\altkeywords{\kwd{Capteurs quantiques}\kwd{Forces \`a faible distance}\kwd{Champs \'electriques}}

\thanks{European Union's Horizon 2020 Research and Innovation
Programme, Agence Nationale de la Recherche (grant no.
ANR-18-QUAN0015-01)}

\thanks{\textbf{Note}. Article submitted by invitation}

\maketitle

\end{noXML}

\section{Introduction}\label{sec1}

Quantum sensors based on atom interferometry allow for performing
measurements of inertial forces~\cite{Geiger2020} with excellent
stabilities and accuracies. They now find applications in a variety of
domains, such as geoscience, on the ground~\cite{Stray2022}, onboard
ships and planes~\cite{Bidel2018,Bidel2020} or in space
\cite{Carraz2014,Migliaccio2019,Leveque2021,Hosseiniarani2024,Zingerle2024}, 
inertial navigation~\cite{Darmagnac2024}, fundamental
physics~\cite{Jaffe2017, Sabulsky2019,Tino2021, Struckmann2024} and
metrology~\cite{Thomas2017}. In particular, instruments based on free
falling atoms have reached a level of maturity that allowed for their
transfer to the industry, the development of first commercial
sensors~\cite{Menoret2018} that have been successfully deployed on the
field~\cite{Antoni2022,Guntner2024,Diament2024}, despite their
relatively large size and power consumption. Instruments based on
trapped architectures, on the other hand, despite being less advanced,
promise much greater compactness, which motivates intense research
efforts to further push their performance~\cite{Keil2016, Garrido2019}.
Key challenges in this domain are the preservation of the atomic
coherence despite their trapping~\cite{McDonald2013, Egorov2011,
Hilico2015, Ammar2015, Panda2024}, and the coping with new systematics,
originating from the architecture of the sensor itself~\cite{Pelle2013}
or from its environment~\cite{Haslinger2018, Harber2005}, such as the
proximity of a substrate for atom chips for instance.  Motivated by the
potential of this technology, we have developed a trapped atom sensor
aiming at performing local force measurements \cite{Alauze2018}, and we
have recently shown its ability to resolve tiny short range forces,
with unprecedented stabilities in the quectoNewton range, such as the
Casimir--Polder force between the atoms and the surface of a dielectric
mirror~\cite{Balland2024}. As already demonstrated
in~\cite{McGuirk2004,Harber2005,Obrecht2007}, such measurements are
prone to systematics related to atoms being adsorbed onto the surface
or to stray charges, that produce parasitic forces originating from
electric field gradients. In~\cite{Balland2024}, their effect was
evaluated with an empirical model, based on measurements of the force
at large distances. Instead, we show in this article that our platform
can be used to directly measure locally the electric field in the
vicinity of the surface, with electric field gradients measured with
our force sensor, and electric fields determined via hyperfine clock
measurements. This allows for the characterisation of the distribution
of adsorbed atoms and eventually charges onto the surface, and for the
correction of force measurements from their detrimental impact. 

\section{Principle of the force sensor}\label{sec2}

We briefly recall here the principle of our force sensor, previously
detailed in~\cite{Balland2024}. A sample of ultracold 87Rb atoms
produced in a lower vacuum chamber via evaporative cooling in a crossed
dipole trap is first transported with a Bloch elevator close to the
surface of a dielectric mirror located inside an upper chamber. It gets
recaptured in a shallow vertical optical lattice, produced by a 532~nm
laser retroreflected on the surface of the mirror, with an additional
transverse confinement provided by a progressive wave infrared laser at
1064~nm. After 500 ms trapping time, we are left with a maximum number
of 1000 atoms prepared in the $F=1$, $m_F=0$ state, with rms sizes of
3.5~$\upmu$m along the vertical direction and 50~$\upmu$m in the
transverse directions. Wannier Stark states $|W_m\rangle$, where $m$ is
the well index, are quasi-eigenstates of the system, which form a
ladder of states localized around the different wells. The energy
difference between adjacent Wannier Stark states is given by $\delta
E=E_{m+1}-E_{m}=h \nu_{\mathrm{B}}$, where $\nu_{\mathrm{B}}=m_{\mathrm{Rb}}g\lambda/2h$ is
the Bloch frequency, $m_{\mathrm{Rb}}$ is the mass of a Rb atom, and
$\lambda$ is the wavelength of the lattice laser. Transitions between
distinct Wannier Stark states in different hyperfine states, separated
by $\Delta m$ lattice wells, are then induced using Raman transitions
between the two hyperfine ground states. In particular, a sequence of
two $\uppi/2$ Raman pulses, separated by a free evolution time T,
realizes a Ramsey interferometer. Finally, the populations in the two
output ports of the interferometer are measured with a state selective
fluorescence detection on a CCD camera. 

Figure~\ref{fig:fringes} shows Ramsey fringe patterns obtained by
scanning the frequency difference between the two Raman lasers $\Delta
\nu_{\mathrm{R}}$ across the $\Delta_m=+6$ transition, for two
different atom-mirror distances, of 40 and 500~$\upmu$m. The
measurement parameters are a free evolution time of $T=80$ ms, a
duration of Raman pulses of $\tau=10$ ms and a number of detected atoms
of about 1000. The central fringe is located at $\Delta
\nu_{\mathrm{R}} = \Delta \nu_{\mathrm{HFS}} + \Delta m \nu_{\mathrm{B}}$, with 
$\Delta \nu_{\mathrm{HFS}}$ the hyperfine frequency difference, which
corresponds to the fringe being slightly above 3410~Hz. A clear
frequency shift of the fringes of about 1.5~Hz is observed in between
the two positions. If attributed solely to a change in the force, this
would correspond to a shift of 0.25~Hz in the Bloch frequency and
$6\times10^{-28}$ N in the force. 

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig:fringes}Ramsey fringes at two different distances,
of 40 and 500~$\upmu$m.}
\end{figure}

For the measurements presented below, the frequency of the central
fringe is actually measured using a standard mid-fringe lock method.
Interleaving measurements with $\pm \Delta_m$ allows separating
$\Delta \nu_{\mathrm{HFS}}$, as well as all other clock-type
contributions and their fluctuations, from the force $\Delta m
\nu_{\mathrm{B}}$. With a free separation time of $T=150$ ms, our sensor reaches a
best short term sensitivity on the force measurement of $3.4\times
10^{-28}$ N at 1~s, and averages down to 4~qN (1 quectoNewton (qN) 
$=10^{-30}$ N) after 5 h of averaging time~\cite{Balland2024}.

Figure~\ref{fig:force} displays the results of repeated force
measurements sessions, with the force determined out of interleaved
$\Delta_m=\pm 6$ measurements, as a function of the atom-surface
separation distance, each measurement session being performed over a
few days. 

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig:force}Force as a function of the atom-surface
distance, measured over 4 measurement campaigns spanning across
months.}
\end{figure}

The force displays measurable amplitudes at relatively large range,
much larger than the 10~$\upmu$m range where the stability of our
sensor would allow resolving the expected Casimir force (displayed as a
brown continuous line). This behaviour is attributed to parasitic
forces due to electric field gradients produced by atoms adsorbed on
the surface, as already demonstrated in~\cite{McGuirk2004,
Obrecht2007}. Figure~\ref{fig:force} also shows that this force evolves
from one session to another, indicating that the number and/or the
spatial distribution of the adsorbed atoms evolves with time. Note also
that this evolution is not monotonous with time. This is related to
changes in the status of the experiment in between two consecutive
measurement sessions, being either off or active, but eventually
performing other kinds of measurements, with different measurement
cycles. We actually observed after a cold start that the force
increases over a few days before reaching a steady state when the
experiment performs continuously the same experimental cycle over and
over. But, the force is found to slowly reduce with a time constant of
order of months when letting it off in between measurements separated
by weeks.  

\section{Measuring electric field gradients}\label{sec3}

Parasitic electric fields $E_{\mathrm{p}}$ produced by adsorbed atoms
lead to Stark shifts of the atoms energy levels $\delta
E=-(\alpha_0/2)\mathbf{E}_{\mathbf{p}}^2$, where $\alpha_0$ is the atom
static polarisability, and thus to forces due to their gradients
$\mathbf{F}_{\mathbf{p}} = (\alpha_0/2)\nabla
\mathbf{E}_{\mathbf{p}}^2$ to which our sensor is sensitive. The
amplitude of this force can be modified by applying an additional
electric field $\mathbf{E}_{\mathbf{a}}$, the force then being
$\mathbf{F} = (\alpha_0/2)\nabla (\mathbf{E}_{\mathbf{p}} + 
\mathbf{E}_{\mathbf{a}})^2$. This reduces to $\mathbf{F}_{\mathbf{p}} +
\alpha_0 \sum_{i} E_{\mathrm{a},i}  \nabla E_{\mathrm{p},i}$, when
$\mathbf{E}_{\mathbf{a}}$ is homogeneous. Thus, by measuring the force
as a function of the applied field $E_{\mathrm{a}}$ along one direction
$i$, one can determine the gradient of the amplitude of the parasitic
field $\nabla E_{\mathrm{p},i}$ along that direction.

In order to apply such homogeneous additional fields, electrodes have
been placed inside the vacuum chamber, their arrangement being depicted
in Figure~\ref{fig:electrodes}. Four longitudinal linear rods along an
horizontal direction $x$, placed at equal distances from the atoms, and
two circular ones allow to produce electric fields aligned respectively
in the ($y,z$) plane and along $x$. In particular, applying a
difference of potential of $V=1.25$ kV between the upper and lower
pairs of rods allow to produce a vertical electric field of 30 kV/m,
with a good homogeneity at the atoms position.

\begin{figure}
{\vspace*{-3pt}}
\includegraphics{fig03}
{\vspace*{-3pt}}
\caption{\label{fig:electrodes}Arrangement of the electrodes around the
mirror of interest, placed in its ceramic support, at the center of the
upper vacuum chamber.}
\end{figure}

Figure~\ref{fig:grad} displays the measurements by our sensor of the
force experienced by the atoms as a function of the electric field
applied in the vertical direction, for two different atom-surface
distances of 30 and 50~$\upmu$m. We observe as expected a linear
behaviour, and a linear fit to the data allows extracting the gradient
of the parasitic electric field along the vertical direction (of order
of 3 kV/cm$^2$ at these distances) with an uncertainty of order of 20
V/cm$^2$.

\begin{figure}
\includegraphics{fig04}
{\vspace*{-3pt}}
\caption{\label{fig:grad}Force versus applied electric field, for two
different atom-surface distances.}
{\vspace*{-3pt}}
\end{figure}

A series of such measurements can then performed at different
atom-surface distances, interleaved with measurements of the force in
the absence of applied electric field. Figure~\ref{fig:figmeasmodel}
displays the results of a campaign of such concurrent measurements,
with the measurements of the force at the top of the figure and of the
electric field gradients in the middle. 

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig:figmeasmodel}Measurements and models of the force
(top) and the electric field gradients (middle) as a function of the
atom-surface distance, and the corresponding reconstructed electric
field (bottom). Measurements are displayed by symbols, and models by
continuous lines. Shaded areas represent the uncertainties in the
models. The measurements are displayed as differential, taking the
measurement at the largest distance of about 400~$\upmu$m as a
reference.}
\end{figure}

It also displays, at the bottom and as crossed symbols, the
reconstructed amplitudes of the electric field along the vertical
direction, obtained via the integration of the gradient measurements.
Finally, the force due to stray electric fields can be calculated out
of the reconstructed electric field, and the results are displayed as
green crosses at the top of the figure. This clearly shows that the
force at large distances (larger than about 20~$\upmu$m) is dominated
by the effect of stray \mbox{electric} fields. As for the deviation at the
shortest distance between the direct force \mbox{measurements} and the force
due to stray electric fields, it is related to the Casimir--Polder
force~\cite{Balland2024}. Note that differential measurements are
displayed in all plots, the figures displaying the difference in the
force, the gradient and the electric field with respect to reference
values obtained very far from the surface, at a distance of about 
400~$\upmu$m. 

\section{Model of the adsorbed atom distribution}\label{sec4}

Similar to what was performed in~\cite{Obrecht2007}, and more recently
in~\cite{Balland2024}, we first tried to reproduce the dependence of
the bare force measurements (i.e.\ with no applied electric field) as a
function of the distance by a model of the effect of adsorbed atoms,
with a Gaussian density distribution at the surface with a rms width of
the order of the transverse size of the atomic sample in the trap. The
results of such a model is displayed on Figure~\ref{fig:figmeasmodel}
as a purple line. While this model captures reasonably well the
dependence of the bare force, it fails reproducing the measured
electric field gradients, as shown in Figure~\ref{fig:figmeasmodel},
middle. 

A better agreement is found by modelling the adsorbed atom distribution
by a bimodal double gaussian distribution, one gaussian distribution
function having a transverse size of 31(3)~$\upmu$m, of the order of
the atomic sample, and the second one having a much larger size, of
410(20)~$\upmu$m, which we attribute to adsorbed atoms that have
diffused across the surface. By contrast with the first approach, we
actually adjusted the model onto the measured gradients rather than the
force, with four fit parameters, two rms sizes and two amplitudes,
taking distributions co-centered with the atomic cloud. 

The results of this last model, which is displayed in
Figure~\ref{fig:figmeasmodel} as a red line, allows reproducing both
the measured electric field gradients and the corresponding (bare)
force. We understand from this bimodal distribution that our
measurements were performed at a time after starting the experiment
where adsorbed atoms had started diffusing slowly across the surface,
leading to a wider density distribution that may not have reached its
stationary state.

\section{UV illimination}\label{sec5}

UV illumination was shown effective to force the desorption of atoms
adsorbed onto surfaces~\cite{Meucci1994, Torralbo2015, Klempt2006}.
This motivated us to expose the surface of the mirror to the UV light
produced by 800 mW LED at 370~nm. Several sessions of illuminations
over a couple hours have been performed, with differential measurements
of the electric field gradients and the force at distances of 40 and
400~$\upmu$m in between. While we observed no significant variations
over the first sessions, a dramatic change occurred in the force after
some time, which actually changed sign, while leaving the electric
field gradients unchanged. Such a behaviour could be explained by the
appearance of an additional electric field, with an amplitude larger
than the one produced by the adsorbed atoms, but with an opposite sign.
To confirm this hypothesis, we have performed direct measurements of
the electric field, via clock-type measurements of the differential
Stark shift of the hyperfine transition $(\delta\alpha_0/2)|E|^2$. For
that, we performed Ramsey spectroscopy of that transition, using a
sequence of two $\uppi/2$ microwave pulses separated by a free
evolution time of 300~ms. We then measure the position of the central
fringe of the Ramsey fringe pattern, which is shifted in the presence
of stray electric field shift. To amplify the effect of the field
$E_{\mathrm{p}}$ we want to determine~\cite{Lodewyck2012}, we added an
applied field $E_{\mathrm{a}}$ of 30 kV/m and performed measurements
with alternated signs $\pm E_{\mathrm{a}}$. The corresponding
difference between the differential Stark shifts in the two different
orientations of the applied field is then given by $2\Delta\alpha_0
E_{\mathrm{p}} E_{\mathrm{a}}$. This differential measurement allows
for separating this quantity from all other shifts of the hyperfine
transition. Finally, this method allows us to determine the electric
field with an uncertainty of order of 300 V/m for half an hour
averaging time. 

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig:postuv}Concurrent measurements of the electric
field, its gradient and the force versus the atom-surface distance
after UV illumination of the mirror.} 
\end{figure}

The results of these measurements, concurrent with electric field
gradients and force measurements are displayed on
Figure~\ref{fig:postuv}. One sees at the bottom of the figure that the
force indeed changed sign with respect to before UV illumination. More,
we clearly observe the presence of a large electric field of order of
${-}$15, ${-}$20 kV/m, rather constant with the atom-surface distance. We
attribute this field to stray charges, created by the UV illumination,
and distributed rather \mbox{homogeneously} across the surface, thus inducing
no significant change to the gradient. Finally, the UV illumination
turned out to be detrimental, resulting in a long-standing charging of
the surface, the force remaining repulsive over the following months. 

Similar charging effects have actually already been observed with
semiconducting or dielectric surfaces (see for
instance~\cite{McGuirk2004,Harlander2010}) and more recently in the
context of electrometry with Rydberg atoms, where lasers in the blue
and green spectral regions were shown to induce surface charging in
glass cells filled with hot alkali vapours~\cite{Ma2020,Patrick2025}.

\section{Conclusion}\label{sec6}

We have used a highly sensitive local force sensor to measure forces
exerted onto atoms close to the surface of a mirror by adsorbed atoms
that generate electric field gradients. Applying external fields with
electrodes allows measuring electric field gradients via force
measurements, and even electric fields directly via microwave clock
measurements. These measurements allow to build models of the
distribution of parasitic sources of electric fields, being either
adsorbed atoms or charges. For a more complete picture though, 3D
measurements, rather the 1D measurement performed here, would need to
be performed, which can be done in our setup by applying external
fields along all three directions. The complete knowledge of the
electric gradients can then be exploited to accurately correct local
force measurements from their detrimental effects, and will be
instrumental for improving the measurement of the Casimir--Polder
force~\cite{Balland2024}, and for performing tests of gravity at short
range~\cite{Chen2016} and of dark energy
theories~\cite{Jaffe2017,Sabulsky2019,Chen2024}.

\section*{Acknowledgements}

We thank Xiaobing Deng and Luc Absil for early contributions. This
research has been carried out in the frame of the QuantERA project
TAIOL, funded by the European Union's Horizon 2020 Research and
Innovation Programme and the Agence Nationale de la Recherche
(ANR-18-QUAN0015-01).

\CDRGrant[ANR]{ANR-18-QUAN0015-01}

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

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