# A single-atom level mechano-optical transducer for ultrasensitive force
sensing

Yang Liu1,2,∗ Pengfei Lu1,∗ Xinxin Rao1 Hao Wu1 Kunxu Wang1 Qifeng Lao1 Ji
Bian1 Feng Zhu1,2 Le Luo1,2,3

###### Abstract

Using light as a probe to detect a mechanical motion is one of the most
successful experimental approaches in physics. The history of mechanical
sensing based on the reflection, refraction and scattering of light dates back
to the 16th century, where in the Cavendish experiment, the angle of rotation
induced by the gravitational force is measured by the deflection of a light
beam reflected from a mirror attached to the suspension. In modern science,
mechano-optical transducers are such devices that could detect, measure and
convert a force or displacement signal to an optical one, and are widely used
for force detection. Especially, ultraweak force sensor with ultrahigh spatial
resolution is highly demanded for detecting force anomaly in surface science,
biomolecule imaging, and atomtronics. Here we show a novel scheme using a
single trapped ion as a mechano-optical transduction. This method utilizes the
force-induced micromotion, converting the micromotion to a time-resolved
fluorescence signal, in which the ion’s excess micromotion coupled to the
Doppler shift of the scattered photons. We demonstrate the measurement
sensitivity about 600 $\textrm{zN}/\sqrt{\textrm{Hz}}$ (1
$\textrm{zN}=10^{-21}$N). By alternating the detection laser beam in all three
dimensions, the amplitude and the direction of a vector force can be precisely
determined, constituting a 3D force sensor. This mechano-optical transducer
provides high sensitivity with spatial resolution in single-atom level,
enabling the applications in material industry and the search for possible
exotic spin-dependent interactions that beyond the standard model.

School of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519082, China

Center of Quantum Information Technology, Shenzhen Research Institute of Sun
Yat-sen University, Nanshan Shenzhen 518087, China

International Quantum Academy (SIQA), and Shenzhen Branch, Hefei National
Laboratory, Futian District, China

∗These authors contributed equally to this work

<EMAIL_ADDRESS>
Force sensors with high sensitivity have rich applications in fundamental
science and engineering. The ability to measure extremely small forces plays a
vital role in absolute gravimetry and inertial navigation [1, 2, 3],
determination of gravitational constant $G$ [4], gravitational wave detection
[5], force microscopy [6], and spin-resonance imaging [7] et.al. Dating back
in history, since Henry Cavendish’s first measurement of the specific gravity
of the Earth using torsion balance in 1797, a variety of methods of measuring
force have been developed. Among those methods, translating a mechanical
signal to an optical signal have been proved to be one of the most sensitive
ones, especially recent techniques of using both lasers and quantum sensors
for mechano-optical systems, such as micro- and nano-mechanical oscillator [8,
9], optically levitated microspheres [10], and trapped ions [11]. All the
systems have been applied for sensing very weak electric-magnetic, optical,
gravitational and exotic forces. Among these mechano-optical systems, trapped
ions enable exquisite control of both spin and motional degrees of freedom
[12, 13, 14, 15], and thus allowing quantum-enhanced measurement, therefore it
is the most capable platform for detecting ultraweak forces with ultrahigh
spatial resolution.

Currently, there are three typical force-sensing scenarios developed with
trapped ions. The first one is to image the ion’s displacement induced by the
force, such as the ultrafast single-photon counting [14] and super-resolution
imaging [16, 15]. The second one utilizes the Doppler effect, where the force
to be measured affects the frequency of the fluorescence photon,including
phase-coherent Doppler velocimetry [13], quantum lock-in detection of the
fluorescence [17, 18, 19]. The third method employs quantum-enhanced
techniques to measure the change of the spin state induced by the force. The
experiments along this route include measurements of the Fock states overlaps
[20], amplification of coherent displacements by squeezing [21], many-body
spin echo by entangling the motion mode and the collective spin[18, 19]. Based
on these techniques, trapped-ion based force sensor has been proposed for
various applications. In one hand, it has been suggested for a sensitivity in
the yocto-Newton regime, a regime where quantum gravity can be experimentally
tested [22, 23]. On the other hand, instead of perusing the extreme
sensitivity, trapped ions have been proposed for 3D sensing of the force
anomaly, where the information of both direction and frequency of the force
are lacking before the data acquisition. Therefore, most of the schemes
mentioned above are not suitable, since they are designed to measure the
forces that are periodically driven.

Here we present a new schemes based on the mechano-optical transduction
without requiring modulating the force to be measured, in which the exerted
forces acted on a trapped ion can be measured by the fluorescence signal
related to the ion’s micromotion. With the achieved sensitivity of 229.17
$\textrm{zN}/\sqrt{\textrm{Hz}}$, our scheme could enable 3D sensing of the
force anomaly. It is noted that our method is very different with the force
detection using Doppler velocimetry demonstrated in a Penning trap [13], where
the ion’s motion is modulated by AC electric force with the axial center-of-
mass mode frequency. The method presented in this paper does not require any
external modulation of the force. Instead, the force to be measured results in
excess micromotion that intrinsically has the same frequency of the driving
frequency of the RF field used for trapping, so called the RF-photon
correlation [24]. With this technique, by scanning the photon arrival time,
phase information of fluorescence is natually captured, thus providing a lock-
in-like measurement of the force. Different with the standard lock-in method,
in which the signal to be measured must be modulated and the relative phase
between the modulation and demomodulation signal are fixed, the RF-photon
correlation method can measure a unknown force without modulation. This
advantage provide a feasible scheme for sensing of the force anomaly, where
the frequency spectrum of the force is usually unknow before the measurement.

Figure 1: Force sensor based on a trapped ion’s excess micromotion. a, The
basic schematics of force measurement using mechano-optical transducer. With
detection laser along trap axis (or either of the radial principle axis), the
force-induced RF-modulated fluorescence is collected by a photomultiplier tube
(PMT), by which the three-dimensional geometry of the applied force, e.g.
geometry $I$ (ellipsoid in light green) or $II$ (ellipsoid in light purple)
can be found. b, The experimental flow chart of the force sensing using the
mechano-optical transduction scheme. c, The ion’s RF-photon correlated
fluorescence (red circles), where the histogram of photon arrival times
relative to start pulses generated synchronously with the RF driving field is
shown. Photon arrivals are bunched with a periodicity dictated by the driven
RF frequency. The solid red line is an theoretical fit to the data used to
obtain the micromotion index $\beta$. The inset shows the resonance profile of
the electronic transition ${}^{2}S_{1/2}\leftrightarrow$ ${}^{2}P_{1/2}$. The
black square and red line corresponds to the measurement and theoretical fit.
Here, the detuning of the detection laser is chosen as $\Delta=-2\pi\times 20$
MHz $\approx-\Gamma$ marked by the vertical dashed line, in order to obtain a
relatively large photon-correlation signal $\frac{\Delta}{S_{0}}$ [25]. The
power of the detection laser throughout the experiment is kept in the low
intensity limit, i.e. the ion is driven below the saturation
limit$I/I_{sat}\approx 0.5$.

_Near-saddle point case._ The initial ion’s position is deliberately offset
from the saddle point, allowing for the measurement of the force up to a
hundred atto-Newtons. We first carried out the force measurement in geometry
$I$, where the voltage $V_{3}$ on electrode DC3 was varied in small steps. The
micromotion indices, the trapping frequencies, and the ion’s images were
measured for each $V_{3}$, as shown in Figure 2. In Figure 2(a), we calibrated
$\beta_{X}$ to the ion’s axial displacement $r_{X}$ based on the theoretical
relation

$\beta_{X}=\sqrt{\beta^{2}_{X,eff}+\beta_{0}^{2}}=\sqrt{(\pi
q_{X}(r_{X}-r_{X,0})/\lambda)^{2}+\beta^{2}_{0}},$ (1)

From this calibration, we determined non-zero $q_{X}=0.00234(5)$ and
$\beta_{0}=0.0318(3)$, which were attributed to the non-vanishing RF field
along the trap axis, and $r_{X,0}=-1.35(5)\mu m$ was the initial displacement
from the RF node. It was evident that all $\beta_{X}$ could be well calibrated
by $r_{X}$ according to Eq. 1. It is noted that contrary to $q_{y}$ and
$q_{z}$, which can be deduced from the measurements of radial secular
frequencies $\omega_{y}$ and $\omega_{z}$ with varying RF voltages, $q_{X}$
cannot be obtained from trap frequency measurements, because the RF potential
is never dominant in this direction. By fitting the uncertainties of $q_{X}$,
$\beta_{0}$ and $r_{X,0}$ with this calibration, we estimate that the
corrections for the force component $F_{X}$ along the axial direction of the
linear trap is less than $3.4\%$. In Figure 2(b), by varying voltage on
electrode DC3, the electric forces were attained by following relation
modified from Eq.8:

$F_{X}=\frac{\lambda m\omega^{2}_{X}\sqrt{\beta^{2}_{X}-\beta^{2}_{0}}}{\pi
q_{X}}.$ (2)

Here the trapping frequency $\omega_{X}$ was measured independently. As is
evident from Figure 2(b), when the applied voltage difference was beyond the
range of $\Delta V_{3}=(-0.5V,-0.12V)$, did $\beta_{X}$ and $F_{X}$ start to
increase monotonically. In the range of $5<F_{X}<20$ aN, it clearly
demonstrated very good linear dependence on the applied voltage.

Figure 2: The measurements of electric forces in geometry $I$. (a) The
calibration of $\beta_{X}$ to the ion’s axial displacement $r_{X}$ at various
$\Delta V_{3}$. The black spheres with error bars are experimental data from
independent RF-photon correlation measurements and imaging measurements, while
triangles (diamonds) are projections to the $\Delta V_{3}-r_{X}$
($r_{X}-\beta_{X}$) plane, representing the dependence of the ion’s axial
displacement on the applied voltage (the correspondence of the micromotion
index in trap axis to the ion’s axial displacement). The red line on the
$r_{X}-\beta_{X}$ plane is the theoretical fit to Eq.1. (b) The dependence of
axial electric force $F_{X}$ on the applied voltage and micromotion index
$\beta_{X}$. The triangles, inverted triangles and diamonds represent the
dependence of the micromotion index $\beta_{X}$ on the applied voltage, axial
force $F_{X}$ on the applied voltage and on the micromotion index $\beta_{X}$,
respectively. (c) The calibration of $\beta_{X,Y}$ to the ion’s displacement
$r_{X}$ and $r_{Y}$ at various applied voltages. The blue spheres with error
bars are experimental data, and the cyan surface is the theoretical fit to
Eq.3. (d) The dependence of electric force $F_{X,Y}$ on the applied voltage
and micromotion index $\beta_{X,Y}$. The triangles, inverted triangles and
diamonds represent the dependence of the micromotion index $\beta_{X,Y}$ on
the applied voltage, force $F_{X,Y}$ on the applied voltage and on the
micromotion index $\beta_{X,Y}$, respectively.

In Figure 2(c), the calibration of $\beta_{X,Y}$ to the ion’s displacement
$r_{X}$ and $r_{Y}$ is illustrated, which agrees well with the theoretical fit
according to

$\beta_{X,Y}=\sqrt{(\pi q_{X}(r_{X}-r_{X,0})\sin\phi/\lambda+\pi
q_{y,z}\cos\phi(r_{Y}-r_{Y,0})/\lambda)^{2}+\beta^{2}_{1}},$ (3)

whose derivation can be found in Methods Eq.9. $\phi=14^{\circ}$ is the angle
between the direction of detection laser and $Y$ axis in the lab coordinate,
which was chosen for the minimization of the background scattering from the
vacuum viewport windows. $q_{y,z}=\sqrt{q_{y}q_{z}}=0.209$.
$r_{Y,0}=-0.03(4)\mu m$ is the initial displacement from the RF node.
$\beta_{1}=0.077(19)$ originates from both the non-vanishing RF field along
the trap axis and the residual phase shifts of the RF potential on the
opposing radial electrodes. Thus, based on the measurement of the micromotion
indices and the ion’s position displacements, the electric forces were
attained by

$\displaystyle F_{X,Y}$ $\displaystyle=m\omega^{2}_{Y}(r_{Y}-r_{Y,0})$ (4)
$\displaystyle=\frac{\lambda m\omega^{2}_{Y}}{\pi
q_{y,z}}\frac{\sqrt{\beta^{2}_{X,Y}-\beta^{2}_{1}}-\sqrt{\beta^{2}_{X}-\beta^{2}_{0}}\sin\phi}{\cos\phi},$

where
$\omega_{Y}=\sqrt{\omega^{2}_{y}\cos^{2}\theta+\omega^{2}_{z}\sin^{2}\theta}$
is the oscillation frequency along the $Y$ axis, as shown in Figure 2(d).
Compared to the axial force $F_{X}$, the radial force $F_{X,Y}$ is more
sensitive and illustrates a nearly symmetric dependence on $\Delta V_{3}$ as
what we expected, despite of relatively large systematic uncertainty. The
minimal detected force is $F_{X}=653.98\pm 743.16$ zN and $F_{X,Y}=119.69\pm
331.91$ zN in 30 s measurement time, corresponding to a measurement
sensitivity about 3582.00 $\textrm{zN}/\sqrt{\textrm{Hz}}$ and 655.57
$\textrm{zN}/\sqrt{\textrm{Hz}}$, respectively [27].

Figure 3: The measurements of electric forces in geometry $II$. (a) The
dependence of the micromotion index $\beta_{X}$ and force component $F_{X}$ on
the $\Delta V_{2,7}$. (b) The dependence of the micromotion index
$\beta_{X,Y}$ and force component $F_{X,Y}$ on the $\Delta V_{2,7}$.

The forces in geometry $II$, generated by varying the voltage on a pair of
axial-symmetric electrodes DC2 and DC7, were also measured. Figure 3 shows the
dependence of the axial force $F_{X}$ and the radial force $F_{X,Y}$ on
$\Delta V_{2,7}$. In Figure 3(a), $F_{X}$ varies nearly symmetric with $\Delta
V_{2,7}$. $F_{X,Y}$ is more sensitive to $\Delta V_{2,7}$ compared with
$F_{X}$. The minimal detected force is $F_{X}=951.13\pm 3030.95$ zN and
$F_{X,Y}=329.39\pm 295.42$ zN in 30 s measurement time, corresponding to a
measurement sensitivity about 5209.55 $\textrm{zN}/\sqrt{\textrm{Hz}}$ and
1804.14 $\textrm{zN}/\sqrt{\textrm{Hz}}$, respectively.

_At saddle point case._ We performed the force measurements exactly at the
saddle point. In order to investigate the best force sensitivity, the ion is
close to the saddle point as possible as we could by by the applying machine
learning approach [26]. By applying voltages on DC2 and DC7, we obtain a
linear dependence of $\beta_{X}$ on the applied voltage $V_{2}$ and $V_{7}$,
shown in Figure 4, while $\beta_{Y}\approx 0$ for all the voltage sets,
indicating that the ion moves along the trap axis $X$. The applied force is
then deduced according to Eq. 2. The inset in Figure 4 shows the histogram
measured at $V_{2}=11.75$ V (red circles) and $V_{2}=12.25$ V (blue triangles)
as well as their theoretical fits (solid lines). The corresponding micromotion
index are $\beta_{X}=0.0030$ and $\beta_{X}=0.0306$. The minimal detected
force is $F_{X}=514.37\pm 560.78$ zN in 30 s measurement time, corresponding
to a measurement sensitivity about 2817.32 $\textrm{zN}/\sqrt{\textrm{Hz}}$,
which is about 2 fold better than near saddle-point case.

Figure 4: Electric force Detection at the saddle point. The dependence of
$\beta_{x}$ on $\Delta V_{2,7}$. The micromotion is minimized at the saddle
point with the micromotion indices of $\beta_{X}=0.0027(6)$ and
$\beta_{Y}=0.0308(28)$, and $q_{y}=0.1827$ and $q_{z}=0.1840$ obtained from
the trapping frequency dependence on the applied RF voltage. $V_{2}$ and
$V_{7}$ were varied simultaneously in small voltage step $\Delta V=50$ mV, and
their relation was $V_{7}=V_{2}+0.25V$. In the inset, the red circle (blue
triangle) and solid line correspond to the histogram measured at
$V_{2}=11.75V$ ($V_{2}=12.25V$) and its fitting, respectively. The
corresponding micromotion index is $\beta_{X}=0.0030$ ($\beta_{X}=0.0306$).

We have demonstrated a proof-of-principle experiment for measuring extremely
weak electric forces using a mechano-optical transducer with a single trapped
ion. Comparing with other sensing techniques with trapped ions, our method of
mechano-optical transduction can measure a unknown force without having the
knowledge of their frequency spectrum, thus distinguishing itself for 3D
sensing of force anomaly. For this application, electric forces in two
different geometries have been accurately measured by alternating the
detection laser between $X$ and $Y$ axes, illustrating an excellent 3D force
sensor given full optical access in three dimensions. Furthermore, electric
forces down to $F_{X}=514.37\pm 560.78\textrm{zN}$ and $F_{X,Y}=119.69\pm
331.91\textrm{zN}$ have been acquired in 30 s data acquisition time with a
sensitivity of 2817.32 $\textrm{zN}/\sqrt{\textrm{Hz}}$ and
655.57$\textrm{zN}/\sqrt{\textrm{Hz}}$ in $X$ and $Y$ axis, respectively. The
experimental uncertainty includes statistical fluctuations in averaging over
the ion fluorescence, the uncertainty in the fitted micromotion index, the
uncertainty in the calibration of the applied electric field due to relatively
large coma abberations in the ion image, the uncertainty in the measured
trapping frequencies, and the uncertainty in the trap stability q-parameter.

Further improvements are possible for a better sensitivity. First, detection
efficiency can be enhanced by realistic modifications and update of current
experimental setup, for instance, larger solid angle for fluorescence light
collection enabled by a binary phase Fresnel lens with $NA=0.64$ [15] or by
integration with a glass vacuum cell [28], using a superconducting single
photon detectors with $\approx 80\%$ detection efficiency [29, 30] compared to
current photon multiplier tube with a nominal 32$\%$ at 370 nm. Adapting both
improvements would result in a factor of $\times 4$ enhancement in detection
efficiency, providing up to 2-fold increase in the force detection
sensitivity. According to Eq. 8 and the pseudopotential approximation [31],

$\displaystyle F_{i}$ $\displaystyle\approx\frac{\lambda
m\beta_{i}\Omega^{2}}{4\pi}(\frac{a_{i}}{q_{i}}+\frac{q_{i}}{2})$ (5)
$\displaystyle\geq\frac{\lambda mq_{i}\beta_{i}\Omega^{2}}{4\pi},$

where $\Omega$ is the RF driving frequency. The minimal detectable electric
force can be achieved for $q_{i}=\sqrt{2a_{i}}$. Therefore, additional
optimization of the trapped-ion setup, such as trapping a different species
with smaller mass and reducing the driving RF frequency could allow us to
further enhance the sensitivity by more than four orders of magnitude. As an
exanple, with single trapped ${}^{9}Be^{+}$ [32], the RF frequency
$\Omega=2\pi\times$ 1.4 MHz, $q_{i}=0.017$, $\beta_{i}=0.001$ for a segmented
linear RF trap [33], it is possible to discriminate electric force of well
below 1 $yN$ (1 $yN=10^{-24}N$) with a sensitivity $\approx$ 2 $yN/\sqrt{Hz}$.
Such a single-atom level mechano-optical transducer would provide
opportunities to search for the fifth-force beyond the standard model [34],
including exotic spin-dependent force [35, 36, 37, 38, 39, 40] and other types
of the new forces [41, 42].

Acknowledgements This work is supported by the Key-Area Research and
Development Program of Guang Dong Province under Grant No.2019B030330001, the
National Natural Science Foundation of China under Grant No.11774436,
No.11974434 and No. 12074439. Le Luo received supports from Guangdong Province
Youth Talent Program under Grant No.2017GC010656, Sun Yat-sen University Core
Technology Development Fund. Yang Liu acknowledges the support from Natural
Science Foundation of Guangdong Province under Grant 2020A1515011159, and
Science and Technology Program of Guangzhou, China 202102080380.

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Methods In a typical Paul trap, when the ion is disturbed by an additional
static electric field $E_{dc}$, its equation of motion can be written as [24]:

$\ddot{u}_{i}+[a_{i}+2q_{i}cos(\Omega
t)]\frac{\Omega^{2}}{4}u_{i}=\frac{qE_{i}}{m}.$ (6)

For $|q_{i}|\ll 1$ and $|a_{i}|\ll 1$, the solution to lowest order in $q_{i}$
and $a_{i}$ is

$u_{i}(t)\cong[r_{i}+s_{i}cos(\omega_{i}t+\phi_{si})][1+\frac{q_{i}}{2}cos(\Omega
t+\phi_{i})],$ (7)

where
$r_{i}\cong\frac{4eE_{i}}{m(a_{i}+\frac{q_{i}^{2}}{2})\Omega^{2}}\cong\frac{eE_{i}}{m\omega_{i}^{2}}$
is a displacement of the equilibrium position from the rf node for a trapped
ion with charge $e$ and mass $m$. $s_{i}$ is the secular motion amplitude with
the frequency of
$\omega_{i}=\frac{\Omega}{2}\sqrt{a_{i}+\frac{q_{i}^{2}}{2}}$. $\phi_{si}$ is
the phase of secular motion determined by the initial conditions of the ion
position and velocity. $\phi_{i}$ is the phase of the micromotion.

This solution describes that for a trapped ion near the saddle point of the RF
trapping potential, a weak electric force $q\vec{E}_{i}$ would not only shift
the ion’s equilibrium position by $r_{i}$, but also induces an excess
micromotion modulated by the RF frequency of $\Omega$ with an amplitude of
$\frac{1}{2}r_{i}q_{i}$ along $\hat{u}_{i}$ at position $r_{i}$. With the
assumption of zero phase difference between two ac electrodes $\phi_{ac}=0$,
the micromotion index is described by
$\beta_{i}=\frac{1}{2}k_{i}q_{i}|r_{i}|$, where $k_{i}$ is the component of
wave vector on axis $i$. Since $q_{i}$ is proportional $V_{RF}/\Omega^{2}$ and
is dependent on the trap’s geometric configuration, it would not change by
varying the DC voltages, therefore the micromotion index is proportional to
the ion’s displacement $\beta_{i}\propto r_{i}$. Because of the coefficient
$kq_{i}/2=\pi q_{i}/\lambda\gg 1$ for a typical Paul trap (in our case,
$q_{x}\sim 4\times 10^{-4}$ and $q_{y,z}\sim 0.2$) and ultraviolet (or
visible) lasers, it would bring a large amplification effect on measuring a
weak force. Thus, an extremely weak electric force which cannot be detected by
displacement measurement from imaging method, can be precisely measured
through measurement of $\beta_{i}$ using RF-photon correlation technique [25].

In our experiment, weak unknown force is generated by applying a small
electric DC voltage to the DC electrodes of the trap. Then the excess
micromotion generates a fluorescence signal modulated by the RF frequency. The
fluorescence is detected by the photomultiplier tube (PMT) and is successively
analyzed by the time-to-digital converter (TDC). The experimental schematics,
detailed experimental sequences and demodulated ion’s fluorescence are shown
in Figure 1. The micromotion index $\beta_{i}$ is inferred by fitting the
recorded fluorescence histogram to the model of damped harmonic oscillator
[25, 26]. The force related to the micromotion is determined by

$F_{i}=\frac{\lambda m\omega^{2}_{i}\beta_{i}}{\pi q_{i}}\propto\Delta V,$ (8)

since $\beta_{i}=\frac{1}{2}k_{i}q_{i}r_{i}=\pi q_{i}eE_{i}/\lambda
m\omega^{2}_{i}$. Therefore, a three-dimensional force can be precisely
determined from micromotion indices, which can be measured simply by
alternating the detection laser between three axes, i.e. the three principal
axes of the trap. Such a method have a intrinsical lock-in function, where the
modulated and demodulated signal are mechanical and optical, respectively.

Consequently, this system constitutes a mechano-optical transducer at a single
atom level. Such a transducer not only has extremely high spatial resolution,
but also has the advantages over standard lock-in technique, because it
continuously scans the phase between modulated and demodulated signal, thus
enabling the detection of force of unknown frequency spectrum. For developing
a force sensor, there usually has to be a compromise between detection
sensitivity and measurement range. In our system, the best force sensitivity
can be achieved when the ion’s unperturbed position exactly at the saddle
point, while the larger force range can be obtained when the ion’s unperturbed
position have a limited offset to the saddle point.

In this method, since the fluorescence is modulated by the RF frequency and
its second harmonics [24, 25], any noise which is hardly in phase with the RF
frequency, and averaging over $N\gg 1$ times, would greatly enhance the
contrast of the desired signal relative to the background noise by a factor of
$N^{1/2}$, similar to the phase-coherent Doppler velocimetry [13].

Practically, in order to realize 3D force sensing, the force measurement were
carried out in two different force geometries by varying the voltages of the
different DC electrodes. In the force detection for both geometries, the axial
and radial forces were measured by alternately switching the detection laser
between axis $X$ (the trap axis) and $Y$ in lab coordinate. In geometry $I$,
the forces are applied by only varying the voltage on electrode DC3, while in
geometry $II$, the forces are applied by only varying the voltage on
electrodes DC2 and DC7. The ion’s displacements are also measured by recording
the ion’s images at various voltage $V_{i}$ for calibration purposes. From
these measurements, the dependence of the $\beta_{i}$ on the applied voltage
$V_{i}$, as well as the dependence of the ion’s displacement $r_{i}$ on
$V_{i}$, which provides a direct calibration of the relation between the
micromotion index $\beta_{i}$ and the ion’s displacement $r_{i}$ using two
independent methods, are obtained.

The calibration of $\beta_{X,Y}$ to the ion’s displacement $r_{X}$ and $r_{Y}$
is given by

$\begin{split}\beta_{X,Y}&=\sqrt{{{({\beta_{X,eff}}\sin\phi+{\beta_{y,eff}}\cos\phi\cos\theta+{\beta_{z,eff}}\cos\phi\sin\theta)}^{2}}+\beta_{1}^{2}}\\\
&=\sqrt{\begin{array}[]{l}(\pi{q_{X}}({r_{X}}-{r_{X,0}})\sin\phi/\lambda+\pi{q_{y}}({r_{y}}-{r_{y,0}})\cos\phi\cos\theta/\lambda\\\
+\pi{q_{z}}({r_{z}}-{r_{z,0}})\cos\phi\sin\theta/\lambda{)^{2}}+\beta_{1}^{2}\end{array}}\\\
&\simeq\sqrt{\begin{array}[]{l}(\pi{q_{X}}({r_{X}}-{r_{X,0}})\sin\phi/\lambda+\pi{q_{y,z}}\cos\phi[({r_{y}}-{r_{y,0}})\cos\theta\\\
+({r_{z}}-{r_{z,0}})\sin\theta]/\lambda{)^{2}}+\beta_{1}^{2}\end{array}}\\\
&=\sqrt{(\pi q_{X}(r_{X}-r_{X,0})\sin\phi/\lambda+\pi
q_{y,z}\cos\phi(r_{Y}-r_{Y,0})/\lambda)^{2}+\beta^{2}_{1}}\end{split}$ (9)

where $\theta\approx 53^{\circ}$ is the angle between $Y$ axis and principal
axis $y$ in the radial plane. Its variation with the applied voltage was much
smaller than one degree and thus was ignored in this analysis. $r_{y,0}$ and
$r_{z,0}$ is the initial displacement in principal axes $y$ and $z$,
respectively. $q_{y}$ and $q_{z}$ are linearly proportional to the RF voltage
theoretically, and $q_{y}\approx 0.203(2)$ and $q_{z}\approx 0.215(3)$ were
obtained from measurements of radial trapping frequency as a function of RF
voltage.

Experimental Setup Our force detection is performed on a single 171Yb+ ion
confined in a linear Paul trap similar to Ref [26]. It consists of four gold-
plated ceramic blade electrodes, where two opposite blades labelled as RF1 and
RF2 are driven with an RF potential, creating the transverse (y-z) quadrupole
confinement potential and other two opposite blades each have five segments
serving to confine the ion along the x-axis. The RF voltages are fed through a
pair of home-made helical resonator (quality factor Q$\approx$ 300) and a
directional coupler (Mini Circuits ZEDC-15-2B) and a RF amplifier (Mini
Circuits LZY-22+) to the RF source (Rigol DSG821). Each electrode is connected
to a programmable precision DC voltage power supply (BS1-16-14, Stahl-
electronics). The gap between the electrodes is 470 $\mu m$ and 220 $\mu m$ in
Y and Z direction, respectively. The distance from the trap center to the
electrodes R $\approx$ 259 $\mu m$ . The cooling laser is 369.5 nm from a
frequency-doubled Ti: Saphire laser (M Squared Lasers Ltd, ECD-F) with its
fundamental frequency is stabilized using acousto-optic modulation transfer
spectroscopy of Iodine, and the repumping lasers are 935 nm and 638 nm lasers
(Toptica, DLC DL PRO).

The ion images in x-y lab coordinate plane are recorded with an electron-
magnified charge-coupled device (Andor, ixon-ultra-897 EMCCD). In our imaging
optics, the isotropic fluorescence from the ion at $\lambda$=369.5 nm is
transmitted through a vacuum viewport and collected by an objective lens of
numerical aperture NA = 0.397 with $\times$ 10 magnification. The intermediate
image passes through a pinhole for spatial filtering and subsequently
magnified by another lens with $\times$ 50 magnification in total.

The ion fluorescence passes the same imaging ion optics, and the temporal
profile of ion fluorescence due to the applied drive is detected by recording
scattered-photon arrival times using a photomultiplier tube (H10682-210,
Hamamatsu) relative to a homemade edge-triggered D-type flip-flop electronics.
Photon arrival times are then determined over $N$ iterations of the experiment
through demodulation using a START-STOP time-to-digital converter (HRM-TDC,
SensL).

For the force detection near saddle-point, the initial voltages on electrodes
DC1, DC2, $\cdots$ DC12 were 9.69 V, 9.88 V, 1.867 V, 10.05 V, 10.05 V, 9.69
V, 9.88 V, 2.0 V, 10.05 V, 10.05 V, 0.186 V and 0.01 V, respectively. The
amplitude of RF voltage was 206.48 V with the frequency of $2\pi\times 21.250$
MHz. The corresponding trapping frequencies were $\omega_{x}\sim 2\pi\times
721.66$ kHz, $\omega_{y}\sim 2\pi\times 1.184$ MHz and $\omega_{z}\sim
2\pi\times 1.602$ MHz, respectively.

While for the force detection at saddle-point, the applied voltages were
$(V_{1},V_{2}...V_{12},V_{RF})$=(11.75 V, 11.75 V, 1.754 V, 9.35 V, 9.35 V, 12
V, 12 V, 1.9 V, 9.6 V, 9.6 V, -0.079 V, -0.22 V, 227.84 V) at 0.45W RF power
and $\Omega=2\pi\times 22.625$ MHz RF frequency. The corresponding trapping
frequencies were $(\omega_{x},\omega_{y},\omega_{z})=(2\pi\times
0.748,2\pi\times 1.382,2\pi\times 1.655)$ MHz.