$\dagger$$\dagger$footnotetext: These authors contributed equally to this
work.**footnotetext: Current address: School of Physics, University of Sydney,
NSW 2006, Australia$\ddagger$$\ddagger$footnotetext: Corresponding author:
<EMAIL_ADDRESS>
# Ultrasensitive single-ion electrometry in a magnetic field gradient

F. Bonus1,2,3†, C. Knapp1,†, C. H. Valahu1,∗, M. Mironiuc1,2,3, S. Weidt1,3
and W. K. Hensinger1,3,‡ 1Sussex Centre for Quantum Technologies, University
of Sussex, Brighton, BN1 9RH, UK  2Department of Physics and Astronomy,
University College London, London, WC1E 6BT, UK  3Universal Quantum Ltd,
Brighton, BN1 6SB, UK

(3rd September 2024)

###### Abstract

Hyperfine energy levels in trapped ions offer long-lived spin states. In
addition, the motion of these charged particles couples strongly to external
electric field perturbations. These characteristics make trapped ions
attractive platforms for the quantum sensing of electric fields. However, the
spin states do not exhibit a strong intrinsic coupling to electric fields.
This limits the achievable sensitivities. Here, we amplify the coupling
between electric field perturbations and the spin states by using a static
magnetic field gradient. Displacements of the trapped ion resulting from the
forces experienced by an applied external electric field perturbation are
thereby mapped to an instantaneous change in the energy level splitting of the
internal spin states. This gradient mediated coupling of the electric field to
the spin enables the use of a range of well-established magnetometry protocols
for electrometry. Using our quantum sensor, we demonstrate AC sensitivities of
$\mathrm{S^{AC}_{min}}=$960(10)\text{\times}{10}^{-6}\text{\,}\mathrm{V}\mathrm{m}^{-1}\mathrm{Hz}^{-\frac{1}{2}}$$
at a signal frequency of
$\omega_{\mathrm{\epsilon}}/2\pi=$5.82\text{\,}\mathrm{Hz}$$, and DC
sensitivities of
$\mathrm{S^{DC}_{min}}=$1.97(3)\text{\times}{10}^{-3}\text{\,}\mathrm{V}\mathrm{m}^{-1}\mathrm{Hz}^{-\frac{1}{2}}$$
with a Hahn-echo type sensing sequence. We also employ a rotating frame
relaxometry technique, with which our quantum sensor can be utilised as an
electric field noise spectrum analyser. We measure electric field signals down
to a noise floor of
$\mathrm{S_{E}(\omega)}=$6.2(5)\text{\times}{10}^{-12}\text{\,}\mathrm{V}^{2}\mathrm{m}^{-2}\mathrm{Hz}^{-1}$$
at a frequency of $30.0(3)\text{\,}\mathrm{k}\mathrm{Hz}$. We therefore
demonstrate unprecedented electric field sensitivities for the measurement of
both DC signals and AC signals across a frequency range of sub-Hz to
$\sim$500\text{\,}\mathrm{kHz}$$. Finally, we describe a set of hardware
modifications that are capable of achieving a further improvement in
sensitivity by up to six orders of magnitude.

quantum sensing, trapped ions, quantum technologies, atomic and molecular
physics

Precision measurements of electric fields and forces can be used in a wide
range of emergent applications in biological, biomedical, and chemical
research hanlon2020diamond ; aslam2023quantum ; krass2022force ;
yu2021molecular , particle physics yu2021molecular ; carney2021mechanical ;
antypas2022new , gravitational wave detection martynov2016sensitivity , energy
applications crawford2021quantum , and communications meyer2018digital ;
fancher2021rydberg . Consequently, a variety of electrometers based on various
quantum hardware platforms have been developed, including bulk michl2019robust
and single dolde2011electric  NV centres, quantum dots cadeddu2017electric ,
Rydberg atoms sedlacek2012microwave ; facon2016sensitive ; kumar2017atom ;
jing2020atomic , and trapped ions in Penning and Paul traps gilmore2021quantum
; affolter2020phase ; shaniv2017quantum ; biercuk2010ultrasensitive ;
blums2018single .

Existing quantum electrometers have demonstrated ultrasensitive electric field
measurements, however they are restricted to certain frequency bands, with few
sensors being able to measure sub-kHz frequencies jau2020vapor . This is
because commonly used electrometers rely on either near-resonant measurements
of transitions within the quantum system biercuk2010ultrasensitive ;
affolter2020phase ; facon2016sensitive ; jing2020atomic ; liu2021phonon , or
resonant pulse techniques on spin states using phase-coherent sensing
protocols dolde2011electric ; michl2019robust ; shaniv2017quantum . In the
former, the measurement bandwidth is defined by the frequency of available
transitions. In the latter, the lower cut-off frequency of the sensor is
constrained by both the achievable coherence times and the coupling strength
of the quantum states to the electric-field perturbation, while the upper
limit is restricted by the pulse duration of coherent operations on the spin
states.

Access to the frequency band ranging from sub-Hz to several kHz could enable
quantum electrometers to be used for a variety of additional applications,
including medical imaging techniques such as electrical impedance tomography
putensen2019electrical , microscopy qiu2022nanoscale , meteorological
applications such as the long-range geolocation of lightning said2017towards ,
as well as the study of atmospheric phenomena and space weather
fullekrug2009wideband ; silber2017use ; hayakawa2021lithosphere . Geological
prospecting techniques are another use case for a low frequency sensor, where
applications include the detection of a range of subterranean and submarine
features reynolds2011introduction ; steuer2020comparison .

In this work, we report on a novel quantum electric field sensor in which a
magnetic field gradient is used to couple electric field signals to the energy
level separation between the spin states of a two-level system in a single
trapped ion. We experimentally demonstrate DC and low frequency AC electric
field sensitivities that are unmatched by current state-of-the-art
electrometers within our measurement bandwidth. We also demonstrate the
versatility of our sensing scheme by employing a magnetometry technique to
measure electric field noise.

Figure 1: Electric field sensing with a trapped ion in a magnetic field
gradient a. A single ion is confined in an RF Paul trap. Segmented DC
electrodes (blue) provide confinement in the axial (z) direction, whilst the
RF electrodes (red) provide confinement in the radial (x,y) directions. A
magnetic field gradient of
$\frac{\partial\mathrm{B}}{\partial\mathrm{z}}=$22.41(1)\text{\,}\mathrm{T}\text{\,}{\mathrm{m}}^{-1}$$
is present along z. Doppler cooling and re-pump lasers at wavelengths of
$369.52\text{\,}\mathrm{nm}$ and $935.18\text{\,}\mathrm{nm}$ respectively are
indicated by the blue and orange beams. Transitions between the internal spin
states are driven using an external microwave emitter. Electric field signals
are applied to the ion through a DC end-cap electrode (E1), and are generated
using an AWG that is capacitively coupled onto the external signal chain of
E1. Zoom: An external electric field $\vec{\mathrm{E}}$ applies a force
$\vec{\mathrm{F}}$ on the ion, resulting in a displacement $\delta\mathrm{z}$.
The transition frequency of the spin states is then shifted by $\Delta$ due to
the magnetic field gradient. b. Simplified energy level diagram of the
^171Yb^+ ion. Doppler cooling, optical pumping and state detection are carried
out using the standard resonance fluorescence scheme described in
olmschenk2007manipulation . Phase coherent operations on the second order
magnetic field sensitive $\ket{\mathrm{F=0,m_{F}=0}}$ to
$\ket{\mathrm{F=1,m_{F}=0}}$ transition and first order sensitive
$\ket{\mathrm{F=0,m_{F}=0}}$ to $\ket{\mathrm{F=1,m_{F}=+1}}$ transition are
driven by resonant microwave fields. Figure 2: Measuring AC and DC
sensitivities. (a) Bloch sphere representation of the quantum state evolution,
pulse sequence diagram and plot of the evolution of the electric field
amplitude at the ion $\mathrm{E}$ for the AC sensing technique. Blue arrows
and rectangles represent the microwave drive, and the orange rectangles and
lines represent interaction with the electric field. The initial and final
spin states are shown in light and dark green respectively. Each electric
field interaction period of duration $\tau/2$ features a half oscillation of a
signal with frequency $\omega_{\epsilon}/2\pi=1/\tau$. (b) Sensitivity of AC
and DC sensing sequences against shot duration, $\tau+\mathrm{t_{m}}$ for
evolution times ranging from $\tau=$25\text{\,}\mathrm{ms}$$ to
$\tau=$250\text{\,}\mathrm{ms}$$, corresponding to signal frequencies of
$\omega_{\epsilon}/2\pi=$40\text{\,}\mathrm{Hz}$$ to
$\omega_{\mathrm{\epsilon}}/2\pi=$4\text{\,}\mathrm{Hz}$$. Measurements of the
sensitivity near the optimal evolution time
$\tau_{\mathrm{opt}}=$172(2)\text{\,}\mathrm{ms}$$ are indicated by the square
and circular markers in the central insets for DC and AC respectively, and are
sampled with 2950 (DC) and 3750 (AC) shots. The measured probability
$\mathrm{P_{\uparrow}}$ against applied electrode voltage,
$\delta\mathrm{E}/\alpha$ where
$\delta\mathrm{E}=\frac{2}{\pi}\mathrm{E_{PK}}$, for DC (AC) sensitivity
measurements at $\mathrm{S_{min}}$, along with a least squares fit to a sine
wave (solid grey), are shown in the upper (lower) right insets. The dotted
grey lines on the main plot are the theoretically expected curves for DC and
AC sensing from equation 4.

## Results

We consider a single ion with charge $\mathrm{q}$ confined in a radio-
frequency (RF) Paul trap. A magnetic field gradient is applied at the position
of the ion, as depicted in figure 1. An electric field perturbation,
$\delta\vec{\mathrm{E}}(\mathrm{t})$, will alter the confining potential and
exert a force
$\delta\vec{\mathrm{F}}\mathrm{(t)}=\mathrm{q}\delta\vec{\mathrm{E}}\mathrm{(t)}$
on the ion. This force displaces the ion along the vector
$\vec{\mathrm{r}}=\mathrm{(r_{x},r_{y},r_{z})}$ by an amount (see methods)

$\delta\mathrm{r_{i}(t)}=\frac{\mathrm{q}}{\mathrm{m}\nu^{2}_{\mathrm{i}}}\delta\mathrm{E_{i}(t)},$
(1)

where $\mathrm{i\in\\{x,y,z\\}}$, and $\mathrm{m}$ and $\mathrm{\nu_{i}}$ are
the mass of the ion and its vibrational frequency along the i-axis
respectively. The displacement $\delta\mathrm{r_{i}}$ of the trapped ion
causes a change $\Delta$ in the transition frequency $\omega$ of its spin
states due to the position dependent Zeeman shift. The transduction parameter,
$\gamma_{\mathrm{i}}$, defines the sensitivity of the spin state transition
frequency to changes in the electric field and is given by

$\gamma_{\mathrm{i}}=\frac{\partial\omega}{\partial\mathrm{E_{i}}}=\frac{\partial\omega}{\partial\mathrm{B}}\frac{\partial\mathrm{B}}{\partial\mathrm{r_{i}}}\frac{\partial\mathrm{r_{i}}}{\partial\mathrm{E}_{\mathrm{i}}},$
(2)

where $\frac{\partial\omega}{\partial\mathrm{B}}$ is the sensitivity of the
transition frequency to changes in magnetic field,
$\frac{\partial\mathrm{B}}{\partial\mathrm{r_{i}}}$ is the strength of the
magnetic field gradient along $\mathrm{r}_{\mathrm{i}}$, and
$\frac{\partial\mathrm{r_{i}}}{\partial\mathrm{E}_{\mathrm{i}}}=\mathrm{q}/\mathrm{m}\nu^{2}_{\mathrm{i}}$
is the change in position for a given change in electric field at the ion.
Equation 2 highlights the mechanism of our sensing scheme; the magnetic field
gradient transforms electric fields into magnetic fields in the reference
frame of the ion, which allows for the implementation of a wide range of
magnetometry techniques for electrometry. From equation 2, we see that a
stronger coupling is achieved by lowering the vibrational frequency of the
ion, increasing the strength of the magnetic field gradient, using ions with a
larger charge-to-mass ratio, or by employing transitions with a higher
sensitivity to magnetic fields. Electric field vector sensing is also in
principle possible by tuning the confinement strength of the ion trap to
maximise $\gamma_{\mathrm{i}}$ along one axis, whilst suppressing it along the
others.

All experimental demonstrations of our sensing scheme were conducted using a
single ^171Yb^+ ion confined in a linear RF blade-trap with segmented DC
electrodes lake_2015 . A magnetic field gradient of
$\frac{\partial\mathrm{B}}{\partial\mathrm{z}}=$22.41(1)\text{\,}\mathrm{T}\text{\,}{\mathrm{m}}^{-1}$$
is generated along the axial (z) direction of the trap by a set of samarium-
cobalt magnets. The magnetic field strength at the unperturbed ion position is
$\mathrm{B}_{0}=$8.3767(4)\text{\,}\mathrm{G}$$. Doppler cooling and re-pump
lasers, with wavelengths of $369.52\text{\,}\mathrm{nm}$ and
$935.18\text{\,}\mathrm{nm}$ respectively, are used to cool the ion to near
the Doppler limit, whilst coherent operations on the spin states are realised
by applying microwave fields using an external microwave emitter as shown in
figure 1. Further details of the experimental setup and control techniques can
be found in the methods section. Electric field signals are generated using an
AWG, and injected onto one of the DC end-cap electrodes of the ion trap by
capacitively coupling across a $220\text{\,}\mathrm{p}\mathrm{F}$ capacitor
(see methods). The applied electric field strength is characterised by a
geometric factor, $\mathrm{\alpha_{i}=\frac{\partial E_{i}}{\partial V}}$,
which relates the electric field at the position of the ion to the DC voltage
applied to the electrode. Strong radial confinement
($\mathrm{\nu_{x}\approxeq\nu_{y}}\approx$1.5\text{\,}\mathrm{MHz}$$)
suppresses coupling of radial electric field components to the spin state
transition frequency. The subsequent experiments therefore measure solely the
axial (z) component of the electric field, where we find
$\alpha_{\mathrm{z}}=\alpha=-95.64(4)$ (see methods), and we will drop the
subscript from here on.

### AC and DC Sensing

We use the $\ket{\downarrow}=\ket{\mathrm{F}=0,\mathrm{m_{F}}=0}$ and
$\ket{\uparrow}=\ket{\mathrm{F}=1,\mathrm{m_{F}}=0}$ energy levels of the
${}^{2}\mathrm{S}_{\frac{1}{2}}$ hyperfine manifold of ^171Yb^+ for the
measurements of AC and DC fields (see figure 1(b)). The energy level
separation of the spin states is a function of the magnetic field at the ion,
and is given by $\omega=\omega_{0}+\delta_{2\mathrm{z}}$ where
$\omega_{0}/2\pi\approx$12.64\text{\,}\mathrm{GHz}$$ is the hyperfine
splitting at zero magnetic field and
$\delta_{2\mathrm{z}}/2\pi=$310.8\mathrm{B}^{2}\text{\,}\mathrm{Hz}$$ (B in
Gauss) is the second order Zeeman splitting Vanier1989 . The vibrational
frequency along z is measured to be
$\nu_{z}/2\pi=$161.191(8)\text{\,}\mathrm{kHz}$$, from which we calculate the
transduction parameter
$\gamma=$3998(2)\text{\,}\mathrm{rad}\text{\,}\mathrm{m}\text{\,}{\mathrm{V}}^{-1}$$
(see methods).

The sensitivity to AC signals is characterised using a Hahn-echo type
sequence, where the electric field signal is applied during the free
procession time $\tau$ as described in degen_2017  and illustrated in figure
2(a). We apply an AC electric field with a frequency
$\omega_{\epsilon}=\tau^{-1}$. The pulse sequence maps the electric field
amplitude onto the probability of finding the spin in the $\ket{\uparrow}$
state, $\mathrm{P}_{\uparrow}$. The displacement of the ion in the magnetic
field gradient results in an instantaneous field-induced detuning, $\Delta$,
of the two-level system transition frequency. A superposition of the spin
states will therefore experience a phase shift of
$\mathrm{d}\phi=\Delta\mathrm{(t)dt}$, where
$\Delta\mathrm{(t)}=\gamma\delta\mathrm{E(t)}$ is the detuning of the spin
transition frequency. The total accumulated phase over the signal duration
$\tau$ is
$\phi=\int_{0}^{\frac{\tau}{2}}\Delta\mathrm{(t)dt}-\int_{\frac{\tau}{2}}^{\tau}\Delta\mathrm{(t)dt}$,
which is a function of the electric field amplitude $\delta\mathrm{E(t)}$ and
$\tau$. The electric field amplitude is linearly increased for each
interaction time $\tau$, leading to sinusoidal oscillations in the
$\mathrm{P}_{\uparrow}$. A linear least squares fit is then used to fit an
equation of the form
$\mathrm{P_{\uparrow}}=\frac{1}{2}+\frac{\mathrm{A}}{2}\sin{(\frac{2\pi}{\mathrm{\kappa}}\mathrm{E})}$
to the data. Here $\mathrm{A}$ is the fringe amplitude, $\mathrm{\kappa}$ is
the electric field required to induce a $2\pi$ phase rotation of the spin, and
$\mathrm{E}$ is the electric field at the ion. We extract the resulting
maximal derivative $\frac{\partial\mathrm{P}_{\uparrow}}{\partial\mathrm{E}}$
and use this to calculate the minimum detectable electric field

$\mathrm{E_{min}}=\sigma_{\mathrm{tot}}\left(\frac{\partial\mathrm{P_{\uparrow}}}{\partial\mathrm{E}}\right)^{-1},$
(3)

where $\mathrm{\sigma_{tot}}$ is the total readout uncertainty due to quantum
projection noise ($\sigma^{2}_{\mathrm{quantum}}$) and classical readout noise
($\sigma^{2}_{\mathrm{readout}}$), given by
$\sigma^{2}_{\mathrm{tot}}=\sigma^{2}_{\mathrm{quantum}}+\sigma^{2}_{\mathrm{readout}}\approxeq
1/(\mathrm{4C^{2}N})$ degen_2017 . Here, $\mathrm{C}\approx
1/\sqrt{(1+4\eta)}$ is an overall readout efficiency parameter taylor2008high
, $\mathrm{N}$ is the number of measurements of the spin state, and $\eta$ is
the infidelity associated with state preparation and measurement (SPAM). We
measure a SPAM infidelity of $\eta=1.8\times 10^{-2}$, resulting in
$\mathrm{C}=0.97$. The sensitivity, defined as the minimum detectable signal
measured over one second of averaging, is then calculated as
$\mathrm{S}=\mathrm{E_{min}}\sqrt{\mathrm{t_{exp}}}$. Here,
$\mathrm{t_{exp}}=\mathrm{N}(\tau+\mathrm{t_{m}})$ is the total experimental
duration, where $\mathrm{t_{m}}$ is the overhead associated with
initialisation, manipulation, and readout of the sensor. From degen_2017 , the
optimum sensitivity for a given evolution time $\tau$ is

$\mathrm{S_{min}}=\frac{\mathrm{e}^{\chi(\tau)}\sqrt{\tau+\mathrm{t_{m}}}}{\gamma\mathrm{C}\tau},$
(4)

where $\chi(\tau)$ is the associated decoherence function of the two-level
system. The measured sensitivity for each evolution time is shown in figure
2(b). AC waveforms are applied across the capacitor for various evolution
times. These waveforms are pre-compensated to account for the frequency
dependent phase offset induced by the capacitor (see methods).

Whilst DC signals cannot be injected across the input capacitor, the
sensitivity of the sensor to DC electric fields can still be characterised by
injecting a time varying signal. We also employ a Hahn-echo type sequence for
DC sensing, where the interaction between the electric field and the sensor
only occurs during the first half ($\tau/2$) of the total free evolution time
(see methods). The average electric field over the course of this half
oscillation is given by $\mathrm{\bar{E}=\frac{2}{\pi}E_{PK}}$, where
$\mathrm{E_{PK}}$ is the electric field amplitude. Correspondingly, the sensor
accumulates the same amount of coherent phase $\phi$ as if it were evolved
under a square DC pulse of amplitude $\mathrm{E_{DC}=\bar{E}}$.

The data shown in figure 2 are in good agreement with the theory, which is
plotted from equation 4. For AC sensing, we see a $\sim 5\%$ offset of the
sensitivity relative to the theory near $\tau_{\mathrm{opt}}$. This is due to
higher frequency electric field components capacitively coupling onto the
electrode, which could not be fully eliminated by the pre-compensation
sequence.

The local minimum in sensitivity $\mathrm{S_{min}}$ occurs at an optimal
evolution time $\tau_{\mathrm{opt}}$. This is because the electric field
induced phase accumulation increases linearly with $\tau$, but is counteracted
by the reduction in fringe-contrast of the quantum system due to decoherence
which follows a Gaussian functional form. $\tau_{\mathrm{opt}}$ can therefore
be determined from equation 4. Experimentally, we find the local minimum in
sensitivity to be at $\tau_{\mathrm{opt}}=$172(2)\text{\,}\mathrm{ms}$$ for
$\mathrm{t_{m}}=$66.839\text{\,}\mathrm{ms}$$, and the coherence time
$\mathrm{T}_{2}=$304(3)\text{\,}\mathrm{ms}$$ (see methods). We measure a
minimum AC sensitivity of
$\mathrm{S^{\mathrm{AC}}_{min}}=$960(10)\text{\times}{10}^{-6}\text{\,}\mathrm{V}\mathrm{m}^{-1}\mathrm{Hz}^{-\frac{1}{2}}$$
at a signal frequency of
$\omega_{\mathrm{\epsilon}}=\mathrm{\tau^{-1}_{opt}=$5.82\text{\,}\mathrm{Hz}$}$,
and a minimum DC sensitivity of
$\mathrm{S^{\mathrm{DC}}_{min}}=$1.97(3)\text{\times}{10}^{-3}\text{\,}\mathrm{V}\mathrm{m}^{-1}\mathrm{Hz}^{-\frac{1}{2}}$$.

Figure 3: Minimum detectable signal against measurement time. Measured values
of $\mathrm{E_{min}}$ at fixed measurement times for DC (AC) sensing are shown
in blue (red). The blue (red) lines show the theoretical dependence of
$\mathrm{E_{min}}$, limited only by quantum projection noise. The value of
$\mathrm{E_{min}}$ for a measurement time of $1\text{\,}\mathrm{s}$ (which
defines the minimum sensitivity of the quantum sensor) is also shown (dashed
black). The classical readout error is approximately equivalent for
measurements on the $\ket{\downarrow}$ and $\ket{\uparrow}$ states, meaning it
does not contribute to the experimentally measured standard deviation shown in
this figure. The dotted grey lines represent the magnitude of electric field
emanating from a single elementary charge at the indicated distance.  Figure
4: Rotating frame relaxometry for electric field sensing (a) Periodogram of
the applied noise. White noise with a bandwidth
$\mathrm{B}/2\pi=$3\text{\,}\mathrm{kHz}$$ centred around
$\omega/2\pi=$30\text{\,}\mathrm{kHz}$$ is applied into the system for various
signal PSDs. The Rabi frequency
$\omega_{\mathrm{\epsilon}}=$30.0\text{\,}\mathrm{kHz}$$ of the spin-locking
pulse is indicated by the black dashed line, and the shaded region indicates
the $1\sigma=\pm$300\text{\,}\mathrm{Hz}$$ error of the Rabi frequency
measurement. (b) Pulse sequence diagram and associated Bloch-sphere
representation of the spin-locking sequence. A
$\mathrm{Y}_{\frac{\pi}{2}}$-pulse aligns the spin state with the X-axis. An
X-pulse with Rabi frequency $\Omega_{\mathrm{X}}$ locks the state vector to
the X-axis. Resonant noise at the spin-locking Rabi frequency drives the
$\ket{+}\rightarrow\ket{-}$ transition incoherently. A final
$\mathrm{Y}_{\frac{\pi}{2}}$-pulse transfers state population into the
$\sigma_{z}$-basis for readout. The outer radius of the cone represents all
possible alignments of the final state vector. The measured probability over
many shots $\mathrm{P}_{\uparrow}$ is represented by the projection of the
vector onto the $\mathrm{z}$-axis (white vector). (c) Measurement of decay
rate $\Gamma$ against the resonant voltage power spectral density of applied
noise. Round markers indicate fits of probability measurements to exponential
decay functions, error bars are within the size of the marker. The solid line
is given by equation 6. The left (right) inset shows measurements of the decay
rate and the associated fit for a PSD of
$\mathrm{S_{E}}=$2.770\text{\times}{10}^{-10}\text{\,}\mathrm{V}^{2}\mathrm{m}^{-2}\mathrm{Hz}^{-1}$$
($\mathrm{S_{E}}=$2.689\text{\times}{10}^{-9}\text{\,}\mathrm{V}^{2}\mathrm{m}^{-2}\mathrm{Hz}^{-1}$$)
resulting in a decay rate of $\Gamma=$22(1)\text{\,}{\mathrm{s}}^{-1}$$
($\Gamma=$195(9)\text{\,}{\mathrm{s}}^{-1}$$).

In order to determine if our quantum sensor is shot noise limited,
$\mathrm{M}=275000$ measurements (shots) are taken at the optimal evolution
time $\tau_{\mathrm{opt}}$ for both AC and DC signals. The electric field
amplitude is set so that a measurement of the quantum system yields a
probability of $\mathrm{P}_{\uparrow}=0.5$. The set of M shots is then
subdivided into $\mathrm{k=M/N}$ sets of $\mathrm{N}$ shots. From this, we
calculate $\mathrm{k}$-individual means, corresponding to the mean probability
of each set of $\mathrm{N}$ shots. Using equation 3 we plot the minimum
electric field $\mathrm{E_{min}}$ calculated using the standard deviation of
each set of $\mathrm{k}$ means (equation 3) against $\mathrm{N}$ in figure 3,
by varying the total measurement duration, $\mathrm{t_{exp}}$ which is a
function of $\mathrm{N}$. The measurement shows that the minimum detectable
electric field follows a $\mathrm{1/\sqrt{t_{exp}}}$ dependence, which is
consistent with a shot noise limited sensor. We find that for one second of
integration time of an AC signal, the quantum electrometer is able to measure
a minimum detectable electric field equivalent to an elementary charge at a
distance of $1.225(6)\text{\,}\mathrm{mm}$.

### Rotating frame relaxometry

In the previous section, we have shown the measurement of DC signals and AC
signals at well-defined frequencies and phases. Our sensor can however also be
employed for the measurement of stochastic signals that feature a
discontinuous phase evolution over the measurement interval. We demonstrate
this by using our sensing scheme to measure power spectral densities of
electric field noise. This is done using a spin-locking sequence. This
technique is well established in magnetometry ithier2005decoherence ;
yan2013rotating , however the gradient mediated coupling of our scheme enables
the first implementation of spin-locking to measure electric field noise. The
pulse sequence, outlined in figure 4(b), begins by initialising the spin state
into the $\ket{\mathrm{+X}}$ eigenstate. A resonant drive of the form
$\frac{\omega_{\epsilon}}{2}\sigma_{\mathrm{x}}$, with Rabi frequency
$\omega_{\epsilon}$, is applied parallel to the orientation of the spin state,
locking the spin along the x-axis of the Bloch sphere. The resonant
interaction lifts the degeneracy of the $\ket{\pm\mathrm{X}}$ eigenstates by
an energy $\epsilon=\hbar\omega_{\epsilon}$, thereby making the two-level
system only sensitive to $\sigma_{\mathrm{Z}}$-type signals oscillating at
angular frequency $\epsilon/\hbar=\omega_{\epsilon}$, effectively creating a
quantum electric field noise spectrum analyser. In the presence of electric
field noise, the resonant drive is applied for a duration $\tau$, after which
the spin state is mapped into the $\sigma_{\mathrm{z}}$ basis for detection.
The measured probability follows an exponential decay in time of the form

$\mathrm{P}_{\uparrow}=\frac{1}{2}\left(1+e^{-\tau\Gamma}\right),$ (5)

where $\Gamma$ is the decay rate of the system. The measured decay is a result
of electric field noise at angular frequency $\omega_{\epsilon}$ being
transformed into $\sigma_{\mathrm{z}}$-noise on the spin states through the
coupling induced by the magnetic-field gradient. We define the power spectral
density (PSD) of electric field noise at an arbitrary angular frequency
$\omega$ as
$\mathrm{S_{\mathrm{E}}(\omega)=\int_{-\infty}^{+\infty}\langle\delta
E(0)\delta E(t)\rangle e^{i\omega t}dt}$. The corresponding PSD of
$\sigma_{\mathrm{z}}$-noise is then related to the PSD of electric field noise
by
$\mathrm{S_{z}}(\omega_{\epsilon})=\gamma^{2}\mathrm{S_{E}}(\omega_{\epsilon})$,
giving a spin-locking decay rate of yan2013rotating

$\Gamma=\frac{1}{2}\mathrm{S_{z}}(\omega_{\epsilon}).$ (6)

Equations 5 and 6 therefore make it possible to extract the PSD of electric
field noise at the angular frequency of the resonant drive
$\omega_{\epsilon}$.

To characterise our sensor experimentally, we capacitively inject electric
field noise into the system for the duration of the spin-locking drive pulse.
The waveform comprises white noise in a $3\text{\,}\mathrm{kHz}$ bandwidth
centred around the resonant drive frequency
$\Omega_{\mathrm{X}}/2\pi=$30.0(3)\text{\,}\mathrm{kHz}$$, as illustrated in
figure 4(a).

For this experiment, we use the first order magnetic field sensitive
$\ket{\downarrow}=\ket{\mathrm{F}=0,\mathrm{m_{F}}=0}$ and
$\ket{\uparrow}=\ket{\mathrm{F}=1,\mathrm{m_{F}}=1}$ spin states, where the
transition frequency is $\omega=\omega_{0}+\delta_{\mathrm{z}}$ and
$\delta_{\mathrm{z}}/2\pi=$1.4\text{\,}\mathrm{MHz}\text{\,}{\mathrm{G}}^{-1}$$
is the first-order Zeeman shift. In addition, we set the axial secular
frequency to $\nu_{z}/2\pi=$264.79(1)\text{\,}\mathrm{kHz}$$, from which we
calculate a coupling strength of
$\gamma=$398.6(2)\text{\times}{10}^{3}\text{\,}\mathrm{rad}\mathrm{m}\mathrm{V}^{-1}$$.
We first verify the relation of equation 6 by characterising the decay rate
$\Gamma$ for various injected noise amplitudes (see figure. 4(c)). This is
done by measuring $\mathrm{P}_{\uparrow}$ as a function of spin-locking drive
durations $\tau$, and fitting the data to equation 5. We then characterise the
minimum detectable signal, which is defined as the electric field PSD for
which the signal-to-noise ratio (SNR) is equal to 1. The SNR is calculated by
measuring the decay rate in the absence of injected noise. From this, we
measure a decay rate $\Gamma_{0}=$0.49(4)\text{\,}{\mathrm{s}}^{-1}$$,
corresponding to a minimum detectable signal of
$\mathrm{S^{min}_{E}}=$6.2(5)\text{\times}{10}^{-12}\text{\,}\mathrm{V}^{2}\mathrm{m}^{-2}\mathrm{Hz}^{-1}$$.

## Discussion

Figure 5: Sensitivity and bandwidth comparison of quantum electrometers.
Comparison of electric field sensitivities and bandwidths of various quantum
sensing hardware platforms. Markers show measured sensitivities as described
in the corresponding reference, and shaded regions illustrate the approximate
bandwidths. The blue square and red circle markers represent the measured
sensitivities described in this work. The lower cutoff frequency of the
experimental system is estimated for a coherence time limited by magnetic
field noise at large $\nu$ corresponding to a cutoff frequency of
$0.25\text{\,}\mathrm{Hz}$. The dashed lines indicate the estimated achievable
sensitivities for a system with $\nu/2\pi=$100\text{\,}\mathrm{kHz}$$ and
$\frac{\partial\mathrm{B}}{\partial\mathrm{z}}=$200\text{\,}\mathrm{T}\mathrm{m}^{-1}$$
using a first order magnetic-field sensitive state of ^171Yb^+ (light red) and
^9Be^+ (light grey). Penning biercuk2010ultrasensitive ; affolter2020phase ;
gilmore2021quantum , Rydberg jing2020atomic ; facon2016sensitive ;
holloway2022rydberg ; kumar2017atom , NV michl2019robust ; qiu2022nanoscale ;
cheng2023radio , Paul shaniv2017quantum ; blums2018single ; wu2023quantum ;
wei2022detection

We report on a novel quantum sensing technique for trapped ions in RF traps. A
magnetic field gradient is used to couple electric field induced displacements
of the ion to its spin state energy level splitting, enabling the use of
magnetometry protocols for electrometry. We demonstrate our scheme with a
single trapped ^171Yb^+ ion by measuring the axial component of electric field
signals emitted by an in-vacuum electrode. We measure AC sensitivities of
$\mathrm{S^{AC}_{min}}=$960(10)\text{\times}{10}^{-6}\text{\,}\mathrm{V}\mathrm{m}^{-1}\mathrm{Hz}^{-\frac{1}{2}}$$
for a signal frequency of $\tau^{-1}=$5.82\text{\,}\mathrm{Hz}$$, and DC
sensitivities of
$\mathrm{S^{DC}_{min}}=$1.97(3)\text{\times}{10}^{-3}\text{\,}\mathrm{V}\mathrm{m}^{-1}\mathrm{Hz}^{-\frac{1}{2}}$$.
In addition, we employ a spin-locking sequence to measure stochastic signals
which feature a discontinuous phase evolution over the measurement time. We
determine a minimum detectable electric field PSD of
$\mathrm{S_{E}(\omega)}=$6.2(5)\text{\times}{10}^{-12}\text{\,}\mathrm{V}^{2}\mathrm{m}^{-2}\mathrm{Hz}^{-1}$$
at a frequency of $\omega/2\pi=$30.0(3)\text{\,}\mathrm{kHz}$$.

In figure 5 we compare the sensitivity and bandwidth of our scheme with
current state-of-the-art quantum electrometers. Current quantum hardware
platforms use a variety of measurement schemes for electrometry, resulting in
a range of achievable bandwidths and measurable sensitivities. Single
qiu2022nanoscale ; cheng2023radio  and bulk NV centres michl2019robust
utilise resonant pulse schemes on their spin transition frequency and are able
to operate at ambient conditions, allowing highly increased flexibility in
sensor placement krass2022force . However, coherence times and coupling
strengths limit both the achievable sensitivities and the bandwidth. Rydberg
atoms measure Stark shifts on internal transitions induced by near-resonant
fields, enabling high sensitivity electrometry in the
$100\text{\,}\mathrm{MHz}$ to $500\text{\,}\mathrm{GHz}$ range
facon2016sensitive ; sedlacek2013atom . Ion crystals in Penning traps are
sensitive to electric fields at or near the motional resonances of the
crystal, which are typically in the $50\text{\,}\mathrm{kHz}$ to
$10\text{\,}\mathrm{MHz}$ range biercuk2010ultrasensitive ; affolter2020phase
; gilmore2021quantum . Rydberg and Penning trap architectures have also
demonstrated electric field sensitivities below the standard quantum limit
(SQL) through entanglement-based schemes gilmore2021quantum ;
facon2016sensitive . Finally, there exist a variety of sensors based on RF
Paul traps, which implement both fluorescence based schemes to measure DC
electric fields blums2018single ; wei2022detection ; liu2022single , and
resonant pulse schemes for Doppler shift measurements shaniv2017quantum .

The achieved minimum sensitivities discussed in this work are unmatched by
existing sensing hardware platforms across the measurement bandwidth of our
sensor. Our sensing scheme can be used for highly sensitive electric field
measurements in the DC and sub-Hz to $\sim$500\text{\,}\mathrm{kHz}$$
frequency range. The lower cut-off frequency is limited by the coherence time
of the two-level system, whilst the upper cut-off frequency is a function of
the maximal achievable Rabi frequency of the refocussing $\pi$-pulses. Our
experimentally measured optimal sensitivity is limited by both classical
noise, and hardware constraints specific to the experimental system. Voltage
noise on the electrodes of the ion trap directly couples to the spin states,
which limits the $\mathrm{T_{2}}$ coherence time. Previous measurements on our
particular experimental setup have shown that the coherence time of our system
is dominated by voltage noise on the trapping electrodes and scales as
$\mathrm{T_{2}}\propto\nu^{4}\left(\frac{\partial\mathrm{B}}{\partial\mathrm{z}}\right)^{-2}$valahu2021robust
. Equations 2 and 4 therefore show that the sensitivity in the current
implementation of our electrometer becomes independent of both the secular
frequency and the magnitude of the magnetic field gradient. However, this is
not a universal scaling law, meaning modifications to the hardware of the
sensor would improve the measured $\mathrm{S_{min}}$ and further increase the
bandwidth of the sensor. These include reducing the PSD of voltage noise on
the electrodes, replacing the existing low-pass filter (LPF) with one that has
a larger roll-off rate and a lower cutoff frequency, or by using a voltage
source that enables a different scaling of $\mathrm{T_{2}}$ with
$\frac{\partial\mathrm{B}}{\partial\mathrm{z}}$ and $\nu_{\mathrm{z}}$.
Additionally, the time penalty associated with phase matching electric field
signals across the input capacitor leads to an increase in $\mathrm{t_{m}}$,
increasing the minimum achievable sensitivity in the current experimental
hardware (see methods). Using an in-vacuum antenna rather than a DC electrode
as electric field source would avoid the need for capacitive coupling of
electric field signals, thus leading to immediate improvements of
$\mathrm{S_{min}}$.

Measured sensitivities can be further improved through hardware modifications
of the quantum sensor. Extending the coherence time by reducing the voltage
noise on the electrodes, in combination with dynamical decoupling techniques,
would enable the use of first order magnetic field sensitive transitions as
well as larger magnetic field gradients. Additionally, using a trapped ion
with a large charge-to-mass ratio such as ^25Mg^+ or ^9Be^+ instead of
^171Yb^+ would further improve achievable sensitivities. For example, using
the first order magnetic field sensitive $\ket{\mathrm{F=2,m_{F}=-2}}$ to
$\ket{\mathrm{F=1,m_{F}=-1}}$ transition in the $\mathrm{S_{\frac{1}{2}}}$
hyperfine manifold of ^9Be^+, in a system with
$\frac{\partial\mathrm{B}}{\partial\mathrm{z}}=$200\text{\,}\mathrm{T}\mathrm{m}^{-1}$$,
would result in AC sensitivities of
$<$5\text{\times}{10}^{-9}\text{\,}\mathrm{V}\mathrm{m}^{-1}\mathrm{Hz}^{-\frac{1}{2}}$$
for an evolution time of $\tau_{\mathrm{opt}}=$170\text{\,}\mathrm{ms}$$ (and
$\mathrm{T}_{2}=2\tau_{\mathrm{opt}}$). A further reduction in sensitivity by
a factor of $\frac{1}{\sqrt{\mathrm{N}}}$ can also be achieved by increasing
the number of quantum systems, N, within the sensor and evolving these in
parallel.

Miniaturisation, portability, and hardware complexity are also important
considerations for deploying quantum sensors in the field schwindt2016highly ,
and to ensure optimal positioning of the sensor relative to electric field
sources. As the sensor presented in this work operates in-vacuum, sensor
placement relative to a signal source may be more challenging for some
applications. However, the development of compact ion-trapping systems is a
well established area of research, with significant advancements being made in
vacuum system miniaturisation aikyo2020vacuum ; Fernandez2023fibre .
Additionally, our scheme does not require cryogenic cooling of the hardware,
which reduces portability constraints.

In addition to improving sensitivities and portability, hardware modifications
can broaden the range of applications of the sensor. A system that allows for
independent tuning of the confinement strength along each axis of vibration
can be used for the vector sensing of electric fields. Switchable static
magnetic field gradients as described in siegele2022fabrication  could also be
used to realise a hybrid magnetic field and electric field sensor, where this
sensor has an identical measurement bandwidth for both magnetic and electric
fields. Furthermore, our electric field sensor is compatible with
entanglement-enhanced sensing techniques. Static magnetic field gradient
entanglement schemes for trapped ions using long-wavelength radiation
mintert2001ion ; weidt2016trapped  can be implemented, and could allow for the
sensor to reach sensitivities below the standard quantum limit.

## Acknowledgements

This work was supported by the U.K. Engineering and Physical Sciences Research
Council via the EPSRC Hub in Quantum Computing and Simulation (EP/T001062/1),
the U.K. Quantum Technology hub for Networked Quantum Information Technologies
(No. EP/M013243/1), the European Commission’s Horizon-2020 Flagship on Quantum
Technologies Project No. 820314 (MicroQC), the U.S. Army Research Office under
Contract No. W911NF-14-2-0106 and Contract No. W911NF-21-1- 0240, the Office
of Naval Research under Agreement No. N62909-19-1- 2116 and the University of
Sussex. F.B. and M.M. acknowledge the support from the Engineering and
Physical Sciences Research Council (EP/S021582/1) via the Centre for Doctoral
Training in Delivering Quantum Technologies at the University College London.

## Author Contributions

F.B. and C.K. contributed equally to this work, performed the experiments and
analysed the data. C.H.V. conceived the idea. C.H.V. and M.M. wrote the
experimental control software. S.W. and W.K.H. supervised this work. F.B.,
C.K. and C.H.V. wrote the manuscript. All authors discussed the results and
contributed to the manuscript.

## Methods

### Transduction parameter

We consider the dynamics of a string of $\mathrm{N}$ trapped ions perturbed by
an external electric field, which results in a force
$\delta\mathrm{F_{j}(t)}=-\mathrm{q}\delta\mathrm{E_{j}(t)}$ on ion
$\mathrm{j}$. Restricting ourselves to a single direction without loss of
generality, the Lagrangian of this system is james1997quantum

$\displaystyle\mathrm{L}=$
$\displaystyle\frac{\mathrm{m}}{2}\left(\sum_{\mathrm{p}=1}^{\mathrm{N}}(\dot{\mathrm{Q}}_{\mathrm{p(t)}})^{2}-\nu_{\mathrm{p}}^{2}\mathrm{Q_{p}}^{2}(\mathrm{t})\right)+$
$\displaystyle\mathrm{q}\mathrm{Q_{p}(t)}\sum_{\mathrm{j}=1}^{\mathrm{N}}\mathrm{b_{j}^{(p)}}\delta\mathrm{E_{j}(t)},$
(7)

where $\nu_{\mathrm{p}}$ are the normal mode frequencies and
$\mathrm{b_{j}^{(p)}}$ describes how strongly ion $\mathrm{j}$ couples to the
mode $\mathrm{\mathrm{p}}$. The normal modes of motion, $\mathrm{Q_{p}(t)}$,
are related to small displacements of the ion, $\delta\mathrm{r(t)}$ of Eq.1,
via

$\mathrm{Q_{p}(t)}=\sum_{\mathrm{j}=1}^{\mathrm{N}}\mathrm{b_{j}}^{(\mathrm{p})}\delta\mathrm{r(t)}.$
(8)

The equation of motion of the $\mathrm{p^{th}}$ normal mode is found from the
Lagrangian using the relation
$\frac{\mathrm{d}}{\mathrm{dt}}(\frac{\partial\mathrm{L}}{\partial\dot{\mathrm{Q}}_{\mathrm{p}}})=\frac{\partial\mathrm{L}}{\partial\mathrm{Q_{p}}}$,
resulting in

$\ddot{\mathrm{Q}}_{\mathrm{p}}\mathrm{(t)}+\nu_{\mathrm{p}}^{2}\mathrm{Q_{p}(t)}=\frac{\mathrm{e}}{\mathrm{M}}\sum_{\mathrm{j}=1}^{\mathrm{N}}\mathrm{b_{j}^{(p)}}\delta\mathrm{E_{j}(t)}.$
(9)

Without loss of generality, we restrict ourselves to a single ion chain,
$\mathrm{N}=1$, and consider the centre-of-mass motion along the z-axis. After
setting $\mathrm{p}=\mathrm{z}$ and $\mathrm{b}_{1}^{(1)}=1$, Eq. 9 becomes

$\ddot{\mathrm{Q}}_{\mathrm{z}}(\mathrm{t})+\nu_{\mathrm{z}}^{2}\mathrm{Q_{z}(t)}=\frac{\mathrm{e}}{\mathrm{M}}\delta\mathrm{E(t)}.$
(10)

This corresponds to the equation of a driven harmonic oscillator. Taking the
Fourier transform, Eq. 10 becomes

$\hat{\mathrm{Q}}_{\mathrm{p}}(\omega)=\frac{\mathrm{e}}{\mathrm{m}(\nu_{\mathrm{z}}^{2}-\omega^{2})}\delta\hat{\mathrm{E}}(\omega),$
(11)

where $\hat{\cdot}$ denotes the Fourier transform. For $\mathrm{N}=1$ ion, we
find $\mathrm{Q_{p}(t)=\delta r(t)}$ and Eq. 11 becomes

$\delta\hat{\mathrm{r}}(\omega)=\frac{\mathrm{e}}{\mathrm{m}(\nu_{\mathrm{z}}^{2}-\omega^{2})}\delta\hat{\mathrm{E}}(\omega).$
(12)

In the limit $\nu_{\mathrm{z}}\gg\omega$, Eq. 12 reduces to

$\delta\hat{\mathrm{r}}(\omega)=\frac{\mathrm{e}}{\mathrm{m}\nu_{\mathrm{z}}^{2}}\delta\hat{\mathrm{E}}(\omega),$
(13)

from which one can retrieve the expression of Eq. 1.

### Experimental setup

Figure 10 shows a schematic of the experimental setup used in this work. The
ion trap is mounted inside a vacuum chamber maintained at an average pressure
of $2.4\text{\times}{10}^{-11}\text{\,}\mathrm{mbar}$. The ion is Doppler
cooled using a $369.52\text{\,}\mathrm{nm}$ laser that is red-detuned from the
$\mathrm{{}^{2}S_{\frac{1}{2}}\ket{F=1}}$ to
$\mathrm{{}^{2}P_{\frac{1}{2}}\ket{F=0}}$ transition. The laser beam is
double-passed through an acousto-optic modulator (AOM) to allow for fine
frequency and amplitude control via FPGA. An electro-acoustic modulator (EOM)
is also utilised to generate $2.11\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$
sidebands for state preparation. These sidebands allow the population to be
driven into the $\mathrm{{}^{2}P_{\frac{1}{2}}\ket{F=1}}$ state via optical
pumping, after which it decays into the
$\ket{\downarrow}=\mathrm{{}^{2}S_{\frac{1}{2}}\ket{F=0}}$ ground state.
Population that is off-resonantly driven into the
$\mathrm{{}^{2}S_{\frac{1}{2}}\ket{F=0}}$ during Doppler cooling is returned
to the cooling cycle by continuously applied microwaves near
$12.64\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. Population can also exit the
Doppler cooling cycle by decaying into the ${}^{2}\mathrm{D}_{\frac{3}{2}}$
manifold, where a $935.18\text{\,}\mathrm{nm}$ re-pump laser applied on the
${}^{2}\mathrm{D}_{\frac{3}{2}}$ to ${}^{3}\mathrm{D[3/2]}_{\frac{1}{2}}$
transition returns population into the
$\mathrm{{}^{2}S_{\frac{1}{2}}\ket{F=1}}$ state. The re-pump laser is also
modulated by an EOM at $3.07\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ to
improve re-pumping efficiencies. The microwaves are generated by a Keysight
E8267D PSG Vector signal generator (VSG), which produces a carrier signal of
$12.54\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. This is then mixed with radio
frequency (RF) pulses near $100\text{\,}\mathrm{MHz}$ generated via a Keysight
M8190A two-channel arbitrary waveform generator (AWG), which is then amplified
and emitted via an external microwave emitter to allow for coherent
manipulation of the spin state. Measurement of the spin is carried out using a
state dependent fluorescence scheme as described in olmschenk2007manipulation
, where the average SPAM error is $\eta=1.8\times 10^{-2}$. The voltage
signals used for measurements of the AC and DC sensitivity are directly
applied into the capacitor from the second channel of the M8190A AWG. For
electric field noise measurements the white-noise waveform is generated with
an Agilent 33522A AWG. The white noise signal is attenuated by two
$30\text{\,}\mathrm{dB}$ RF attenuators and its output is controlled with an
external RF switch.

### Gradient measurement

The magnetic field gradient strength along the axial direction can be
calculated by measuring the transition frequencies of two co-trapped ^171Yb^+
ions. As the splitting of the ^171Yb^+ spin states is dependent on the
strength of the magnetic field at the position of the ion, the magnetic field
gradient in the axial direction is given by

$\frac{\partial\mathrm{B}}{\partial\mathrm{z}}=\frac{\mathrm{B_{2}-B_{1}}}{\delta\mathrm{Z}},$
(14)

where $\mathrm{B_{1}}$ and $\mathrm{B_{2}}$ are the magnetic field strengths
at the location of each ion, and $\delta\mathrm{Z}$ is the ion separation (see
figure 6). The ion separation is a result of the mutual coulomb repulsion
between the ions and the oppositely acting axial confinement force.
$\delta\mathrm{Z}$ given by james1997quantum ,

$\delta\mathrm{Z}=\left(\frac{\mathrm{e}^{2}}{4\pi\epsilon_{0}\mathrm{m}\nu^{2}_{z}}\right)^{2}\frac{2.018}{\mathrm{N}^{0.559}},$
(15)

where $\nu_{\mathrm{z}}$ is the axial vibrational centre of mass (COM)
frequency, m is the mass of a single charged particle, and N is the number of
ions in the crystal. We measure
$\nu_{\mathrm{z}}/2\pi=$161.191(8)\text{\,}\mathrm{kHz}$$ via the “tickling”
method  $-$ an AC electric field is applied to the trap using an external RF
coil, which excites the axial motion of the ion crystal when the applied
frequency is resonant with the axial vibrational frequency, leading to a
measurable decrease in ion fluorescence due to the Doppler shift. We then
compute $\mathrm{\delta Z=$12.64(1)\text{\,}\mathrm{\SIUnitSymbolMicro m}$}$
from equation 15.

The magnetic field at each ion is calculated by measuring the magnetic field
dependent transition frequency of each ion, shown in the inset plots of figure
6. From these measurements we find
$\mathrm{B}_{1}=$7.1328(8)\text{\,}\mathrm{G}$$ and
$\mathrm{B}_{2}=$9.9655(5)\text{\,}\mathrm{G}$$. Finally, from equation 14,
the magnetic field gradient strength is
$\frac{\partial\mathrm{B}}{\partial\mathrm{z}}=$22.41(1)\text{\,}\mathrm{T}\text{\,}{\mathrm{m}}^{-1}$$.

Figure 6: Measurement of the magnetic field gradient. The blue markers
represent two co-trapped ^171Yb^+ ions that form an ion crystal in the
$\mathrm{y}-\mathrm{z}$-plane. The left y-axis represents the radial
($\mathrm{y}$) position of the ions. The axial ($z$) separation of the ions is
$12.64(1)\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and is symmetric about the
single ion equilibrium location in the axial direction of the trap. The
increasing value of absolute magnetic field with axial position is represented
by the threshold between the white and purple shaded regions. Values of
magnetic field for a given axial position are displayed on the right y-axis.
The insets show measurements and least-squares fits of the spin state
transition frequency of each ion. The horizontal axis of the insets indicate
the detuning of the applied microwave pulse, relative to probing frequencies
near $12.64\text{\,}\mathrm{GHz}$ corresponding to the spin transition of ion
1, $\omega_{1}$ (left inset), and ion 2, $\omega_{2}$ (right inset).

### Calibrating $\alpha$ and $\gamma$

The geometric factor of an electrode, $\alpha$, relates the electric field at
the position of the ion to the voltage applied to the electrode, and is
defined as

$\alpha=\frac{\partial\mathrm{E}}{\partial\mathrm{V}}=\frac{\partial\omega}{\partial\mathrm{V}}\frac{\partial\mathrm{E}}{\partial\mathrm{z}}\left(\frac{\partial\mathrm{B}}{\partial\mathrm{z}}\frac{\partial\omega}{\partial\mathrm{B}}\right)^{-1},$
(16)

where
$\frac{\partial\mathrm{E}}{\partial\mathrm{z}}=\frac{\mathrm{m}\nu^{2}_{\mathrm{z}}}{\mathrm{e}}$.
We calibrate $\alpha$ by first measuring the change in magnetic field at the
ion due to a change in the voltage applied to the E1 electrode
($\frac{\partial\mathrm{B}}{\partial\mathrm{V}}$) using the second order
sensitive spin state transition frequency and
$\nu_{z}/2\pi=$161.191(8)\text{\,}\mathrm{kHz}$$ (see figure 7). The
measurement is performed with a single ^171Yb^+ ion by applying a voltage
$\mathrm{V_{0}+\delta V}$ to the electrode, where
$\mathrm{V_{0}=$1.75\text{\,}\mathrm{V}$}$ is the static voltage contributing
to the axial confining potential and $\delta\mathrm{V}$ is an offset that is
varied from $-$50\text{\,}\mathrm{mV}$$ to $+$50\text{\,}\mathrm{mV}$$. We
extract the value of $\frac{\partial\mathrm{B}}{\partial\mathrm{V}}$ from a
least squares fit to a straight line of the magnetic field measurements for
each voltage offset. From this, we then determine
$\frac{\partial\omega}{\partial\mathrm{V}}=\frac{\partial\mathrm{B}}{\partial\mathrm{V}}\frac{\partial\omega}{\partial\mathrm{B}}=-$382\text{\times}{10}^{3}\text{\,}\mathrm{rad}\text{\,}{\mathrm{V}}^{-1}$$.
The geometric factor is finally calculated from equation 16, giving
$\alpha=$-95.64(4)\text{\,}{\mathrm{m}}^{-1}$$.

Figure 7: Calibrating the geometric factor. Measurement of the shift in
resonance frequency of the two-level system after applying a static voltage
offset $\mathrm{\delta V}$ to the E1 electrode. The red circles indicate the
magnetic field determined from spin state transition frequency measurements at
different values of $\mathrm{\delta V}$, whilst the yellow square represents
the magnetic field corresponding to a measurement of the unperturbed
($\delta\mathrm{V}=0$) transition frequency. The data are fitted to a straight
line using a least-squares fit, shown in red. The top axis shows the axial
displacement of the ion for a given $\mathrm{\delta V}$, which is calculated
using the previously measured value of $\alpha$.

The transduction parameter is found using
$\gamma=\frac{1}{\alpha}\frac{\partial\omega}{\partial\mathrm{V}}=\left(\frac{\partial\mathrm{V}}{\partial\mathrm{E}}\frac{\partial\omega}{\partial\mathrm{V}}\right)$.
For the second order magnetic field sensitive transition we measure
$\gamma=$3998(2)\text{\,}\mathrm{rad}\text{\,}\mathrm{m}\text{\,}{\mathrm{V}}^{-1}$$.

### Electric field sensing protocol

For the sensing of AC fields, we follow the pulse sequence protocol outlined
in degen_2017  and illustrated in figure 8. The AC sensing sequence is
realised by first initialising the two-level system into the
$\ket{+}=\frac{1}{\sqrt{2}}(\ket{\downarrow}+\ket{\uparrow})$ state using a
$\frac{\pi}{2}$-pulse. The superposition state then evolves under an electric
field perturbation for a time $\frac{\tau}{2}$. A $\pi$-pulse reorients the
spin along the equator of the Bloch sphere, before the quantum state again
evolves under the electric field perturbation for a time $\frac{\tau}{2}$. A
final $\frac{\pi}{2}$-pulse maps the state population into the
$\sigma_{\mathrm{z}}$-basis for detection. Using this pulse-sequence, the
sensitivity of the spin state transition frequency is maximised for AC signals
oscillating at a frequency of $\tau^{-1}$.

The DC sensing experiments also use an echo-type pulse sequence, whose
benefits are twofold. Firstly, the coherence time of the sensor is greatly
extended when compared to that of the Ramsey type sequence, which allows for
increased sensitivities. Secondly, the refocusing $\pi$-pulse also compensates
for detuning errors in the microwave pulses. The pulse sequence is also
illustrated in figure 8. The pulse sequence begins with a
$\frac{\pi}{2}$-pulse to initialise the spin into the
$\ket{+}=\frac{1}{\sqrt{2}}(\ket{\downarrow}+\ket{\uparrow})$ state. DC
signals cannot be applied through a capacitor. The low-pass filter (LPF)
signal chain of the DC electrode is also not suitable for a fast application
of DC square pulses during the sensing pulse sequence, as the LPF would
significantly attenuate and distort the signal. Therefore, in order to
quantify the sensor’s response to DC signals, we apply an AC signal of
frequency $\tau^{-1}$ for the duration of the first $\frac{\tau}{2}$ delay
time. This corresponds to an equivalent DC voltage on the electrode of
$\mathrm{V_{DC}=\frac{2}{\pi}V_{PK}}$ where $\mathrm{V_{PK}}$ is the amplitude
of the applied signal. $\frac{2}{\pi}\mathrm{V_{PK}}$ is the average voltage
over the half-oscillation of the AC waveform. The applied time-varying pulse
therefore causes the spin state to accumulate the same amount of phase,
$\phi$, as a square DC pulse of amplitude $\mathrm{\frac{2}{\pi}V_{PK}}$
applied for a duration $\frac{\tau}{2}$ based on the equation relating phase
accumulation to the detuning of the spin transition:
$\phi=\int_{0}^{\frac{\tau}{2}}\gamma\alpha\delta\mathrm{V}\mathrm{(t)dt}$.
The refocusing $\pi$-pulse is then applied, followed by the second
$\frac{\tau}{2}$ delay time, during which no additional voltage signals are
applied to the electrode, followed by a final $\frac{\pi}{2}$-pulse.

In addition to the electric field interaction time $\tau$ the second relevant
time parameter from equation 4 is $\mathrm{t_{m}}$ which breaks down as
follows for our experimental implementation: DC offset application delay time:
$50\text{\,}\mathrm{ms}$ (see below section), Doppler cooling and detection:
$14.599\text{\,}\mathrm{ms}$, state preparation and microwave pulses:
$2.155\text{\,}\mathrm{ms}$, data processing and FPGA delays:
$85\text{\,}\mathrm{\SIUnitSymbolMicro s}$, total:
$\mathrm{t_{m}}=$66.839\text{\,}\mathrm{ms}$$.

Figure 8: AC, DC and spin-locking (SL) pulse sequence diagram and time
evolution of input signal. Blue lines represent the AWG voltage output, whilst
red lines show the voltage evolution on the in-vacuum electrode. Note that the
voltage on the electrode is attenuated and phase-offset relative to the AWG
voltage. The AC sensing technique is characterised by applying a full
oscillation of an AC signal at frequency $\frac{1}{\tau}$ onto the electrode.
Sensitivity to DC signals is characterised by applying a half-oscillation of
an AC signal at frequency $\frac{1}{\tau}$ onto the E1 electrode of the ion
trap. For spin-locking, noise resonant with the spin-locking Rabi frequency is
applied onto the electrode. The input signal exhibits a frequency dependent
phase shift and a frequency dependent attenuation across the input capacitor.
For AC and DC sensing the offsets are pre-compensated, as can be seen by the
dark-blue line. For spin-locking a continuous signal is switched into, thus
pre-compensation is not applied.

### Capacitive coupling of AC signals

Due to the absence of an in-vacuum antenna, the electric field signals
measured by the trapped ion are emitted from an in-vacuum end-cap electrode,
which also generates a DC confinement electric field. Voltage waveforms are
generated using an AWG, and capacitively coupled onto the electrode across a
$220\text{\,}\mathrm{p}\mathrm{F}$ capacitor. Due to their frequency dependent
impedance, capacitors act as high-pass filters, thereby attenuating lower
frequency signals more strongly. The fixed response time of a capacitor will
also shift the phase of AC signals that are applied across it. The shift in
phase of the AC signal can, if unaccounted for, affect the total coherent
phase $\phi$ that is accumulated by the spin states. In order to achieve an
optimal measurement of the sensitivity of our experimental system, it is
necessary for the electric field signal at the ion to be in-phase with the
Hahn-echo sensing pulse sequence. This is because $\phi$ is the difference
between the coherent-phase accrued in the first and second interaction time
$\frac{\tau}{2}$. An electric field signal that is not in phase with the Hahn-
echo sequence will therefore lead to a reduction in measured sensitivity.
References michl2019robust  and degen_2017  provide further information on
this effect.

We measure the phase shift on signals applied across the capacitor for the
span of frequencies used in the AC and DC sensing experiments using an
oscilloscope. Based on these measurements, we then pre-compensate the signal
that is applied across the capacitor by applying an inverse phase shift,
negating the effect of the capacitor on the phase of the voltage waveform.
This ensures that the voltage on the electrode, and therefore the electric
field signal at the ion, is in phase with the Hahn-echo sequence.

Shifting the phase of the voltage waveform introduces a discontinuity into the
signal. This manifests as a sudden change in the voltage across the capacitor
from $0$ to $\mathrm{V_{\Phi}=V_{A}\sin{\Phi}}$ where $\Phi$ is the phase of
the AC voltage signal. Given that the current across a capacitor is defined as
$\mathrm{I=C\frac{dV}{dt}}$, where $\mathrm{C}$ is the capacitance of the
capacitor, the high rate of change of voltage induces a large current flow
across the capacitor, which introduces additional coherent phase offsets of
the superposition state. To suppress this unwanted perturbation, we apply a DC
voltage offset of $\mathrm{V}_{\Phi}$ into the capacitor in the time prior to
the initialisation of the $\ket{+}$ state, which minimises the sudden voltage
spike across the capacitor from the phase-shifted AC voltage waveform. To
ensure that the sensor reaches steady state prior to the application of the AC
electric field signal, an additional $50\text{\,}\mathrm{ms}$ delay is added
between the application time of the DC offset and the first resonant microwave
pulse. This makes up the majority of the $\mathrm{t_{m}}$ time, which is
broken down in the previous section. The pre-compensation technique for the AC
and DC sensing pulse sequences is visualised in figure 8, which illustrates
both the AWG and in-vacuum electrode voltage evolution throughout the
experimental pulse-sequence.

We also measure the frequency dependent attenuation of the capacitor using an
oscilloscope. We determine the transfer function of the capacitor by fitting a
Butterworth high-pass filter to these data. We then determine the total
attenuation of the electric field signal for a given frequency $\tau^{-1}$.

### Determination of the coherence time

We measure the coherence time of the two-level system using a Hahn-echo
experiment. The spin is initialised in the $\ket{\downarrow}$ state, after
which a $\frac{\pi}{2}$-pulse rotates the spin into the $\ket{+\mathrm{X}}$
eigenstate. A refocusing $\pi$-pulse is applied in-between two free evolution
periods of duration $\frac{\tau}{2}$. A final $\frac{\pi}{2}$-pulse maps the
state into the $\sigma_{\mathrm{z}}$-basis for detection. Varying the phase of
the final pulse from $-2\pi$ to $2\pi$ results in sinusoidal fringes in the
probability of measuring $\ket{\uparrow}$. As the free evolution time is
increased, decoherence leads to a reduction in the amplitude of these fringes,
where the coherence time $\mathrm{T}_{2}$ is then given by the point at which
the fringe contrast reaches $\mathrm{e}^{-1}$. As the AC and DC sensing
experiments are also based on the Hahn-echo sequence, the fringe amplitudes
from these experiments can also be utilised for the coherence time
measurement. The fringe amplitudes of these three experiments against the free
evolution time are shown in figure 9. These data are then aggregated and
fitted to a Gaussian decay function of the form
$\chi^{-1}(\mathrm{t})=\exp(-\frac{\mathrm{t}^{2}}{\mathrm{T}_{2}^{2}})$ using
a least squares fit, yielding a coherence time of
$\mathrm{T}_{2}=$304(3)\text{\,}\mathrm{ms}$$.

Figure 9: Coherence time measurement of the second order sensitive clock
states. The spin states used for the experimental demonstration of AC and DC
sensing are defined by the $\mathrm{\ket{\downarrow}=\ket{F=0,m_{F}=0}}$ and
$\mathrm{\ket{\uparrow}=\ket{F=1,m_{F}=0}}$ energy levels. The fringe
contrasts associated with each of the AC sensing, DC sensing, and Hahn-echo
experiments are shown for a range of free evolution times $\tau$. The black
dashed line indicates the $1/\mathrm{e}$ threshold. The grey line is a least
squares fit of these measurements to a Gaussian decay function, corresponding
to a coherence time of $\mathrm{T}_{2}=$304(3)\text{\,}\mathrm{ms}$$. Figure
10: Experimental Setup. Electric field sensing configurations, coherent
control, optical and electric field signal chains for the operation of the
quantum sensor. Coherent control is achieved using triggered microwave pulses
generated by amplitude modulation of an RF signal from a two-channel AWG with
a microwave carrier using a VSG. The microwave tone is amplified and emitted
into the vacuum chamber using a microwave horn. The second channel of this AWG
provides the electric field signals for AC and DC sensing. These signals are
synchronously coupled into the quantum sensor in configuration II.
Configuration I shows the setup for rotating frame relaxometry. Here, a signal
is continuously output using a second AWG. Interaction with the spin state is
toggled using an RF switch. The signal is then attenuated and capacitively
coupled onto the electrode. Doppler cooling, optical pumping and state
detection of the ion are achieved by modulating a $369.52\text{\,}\mathrm{nm}$
laser beam using an acousto-optic modulator (AOM) and an electro-optic
modulator (EOM). An EOM in the $935.18\text{\,}\mathrm{nm}$ beam allows for
efficient repumping. The photo-multiplier tube (PMT) is used for fluorescence
detection of the spin state.

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