# Enhanced Signal-to-Noise Performance of EP-based Electromechanical
Accelerometers

Rodion Kononchuk1 Jizhe Cai2 Fred Ellis1 Ramathasan Thevamaran2 Tsampikos
Kottos1

###### Abstract

Exceptional points (EP) are non-Hermitian spectral degeneracies where both
eigenvalues and their corresponding eigenvectors coalesce [1, 2, 3, 4].
Recently, EPs have attracted a lot of attention as a means to enhance the
responsivity of sensors, via the abrupt resonant detuning occurring in their
proximity [5, 6, 7, 8, 9, 10, 17, 13, 14, 16, 18, 11, 12, 15, 19, 20]. In many
cases, however, the EP implementation is accompanied by noise enhancement
leading to the degradation of the signal-to-noise performance of the sensor
[17, 16, 18, 15, 19, 20]. The excess noise can be of fundamental nature (due
to the eigenbasis collapse) or of technical nature associated with the use of
amplification mechanisms utilized for the realization of EPs. Here we show,
using an EP-based ${\cal PT}$-symmetric [22, 21] electromechanical
accelerometer, that the enhanced technical noise can be surpassed by the
enhanced responsivity to applied accelerations. The noise due to eigenbasis
collapse is mitigated by exploiting the detuning from a transmission peak
degeneracy (TPD), which forms when the sensor is weakly coupled to
transmission lines, as a sensitivity measurant. These TPDs occur at a
frequency and controlled parameters for which the bi-orthogonal eigen-basis is
still complete and are distinct from the EPs of the ${\cal PT}$-sensor. They
also show a slightly enhanced detuning rate compared to the typically utilized
EPs. Our device demonstrates a three-fold signal-to-noise ratio enhancement
compared to configurations for which the system operates away from the TPD.

Wave Transport in Complex Systems Lab, Department of Physics, Wesleyan
University, Middletown, CT-06459, USA

Department of Engineering Physics, University of Wisconsin-Madison, Madison,
WI-53706, USA

Exceptional points (EP) are spectral degeneracies occurring in the parameter
space of an open (non- Hermitian) system where two $N=2$ (or more) eigen-
frequencies and their corresponding eigenvectors coalesce [1, 2, 3, 4].
Recently, their importance as a means to design hypersensitive sensors has
been highlighted by a number of contributions [5, 6, 7, 8, 15, 9, 10, 17, 13,
14, 16, 18, 11, 12]. At the core of these proposals is the possibility to
induce a sublinear detuning of the resonant frequencies from the EP degeneracy
when the system interacts with a perturbing agent. Specifically, in the
proximity of an $N-$th order EP, the degenerate resonances abide to Puiseux
generalized expansions leading to a resonance detuning $\Delta
f\equiv\left|f-f_{\rm EP}\right|$ that follows an $N-$th root behavior $\Delta
f\sim\varepsilon^{1/N}$ with respect to the perturbation strength
$\varepsilon$ imposed to the system by the presence of the perturbing agent.
Obviously, in the small perturbation limit this sublinear response signifies
an enhanced sensing as compared to a linear response, i.e. $\Delta
f\sim\varepsilon\ll\varepsilon^{1/N}$, utilized in many sensing schemes that
rely on Hermitian degeneracies [23, 24, 25, 26]. In fact, the proposed EP-
based protocols have additional advantages compared to other sensor schemes
(e.g. slow light) whose operational principle relies on abrupt intensity
variations of the measured signal. These type of sensors turned out to be
extremely sensitive at the expense of their dynamic range [27, 28]. The
dynamic range is an important metric in the characteristics of the sensor, and
it is defined as the ratio between the maximum and the minimum perturbation
that a sensor can measure.

Given that efficient sensing is of importance in many fields, the emerging
idea of boosting the sensitivity of a particular system via non-Hermitian
degeneracies could have substantial ramifications across several technological
areas. It is therefore paramount to analyze various aspects of EP-sensing and
deduce the sensing performance of such sensors. While increased sensor
responsivity has been demonstrated in several EP-based sensors [6, 7, 8, 9,
10, 12, 17], the signal-to-noise performance has been controversially debated
in recent theoretical studies [13, 15, 14, 16, 19, 20]. Obviously, it is
imperative to confront these disagreements via a direct experimental
investigation of the effects of noise in EP-sensor performance and identify
platforms and conditions (if any) under which they demonstrate superior
performance. Currently, the only experiment that has analyzed the effects of
noise in the EP-sensitivity (precision of a sensor) has utilized a Brillouin
ring laser gyroscope (RLG) platform [17, 18]. The measurements invoked
ultranarrow spectral line, highly suitable for precise measurements of small
frequency shifts. Unfortunately, the conclusions of this study were
discouraging as far as the performance of EP-sensing is concerned: the
expected boost in responsivity of the gyroscope turned out to be limited by a
broadening of the two laser linewidths due to enhanced noise effects
associated with the collapse of the eigenmode basis.

Figure 1:  $\mathcal{PT}$ symmetric platform for enhanced acceleration
sensing.  a, Schematic of the $\mathcal{PT}$ symmetric electromechanical
sensor. The ${\cal PT}$-symmetric circuit is capacitively coupled to the
transmission lines with a capacitor $C_{e}$. The two circuit tanks are coupled
together with a variable capacitor $C_{c}$ with one plate connected to a test
mass that senses the acceleration. b, The behavior of the sensitivity
enhancement factor $SEF$ (green line), Petermann factor $PF$ (blue line), and
of noise enhancement factors associated with the ${\cal PT}$\- components of
the circuit $NEF^{\cal PT}$ (dark red dotted line) and the transmission line
$NEF^{\rm TL}$ as a function of the applied acceleration $a$ (magenta dotted
line). The black vertical lines indicate the TPD (solid), EP (dotted) and LP
(dashed) points. The highlighted domain indicates the accelerations which are
not captured by the experimental platform. c, The actual acceleration sensing
platform utilizes a micro-fabricated coupling capacitor which is connected to
the test-mass used to sense the applied acceleration.

Here, we show experimentally and theoretically a ten-fold enhancement in
responsivity to small perturbations and a three-fold signal-to-noise ratio
(SNR) improvement in the sensing performance of an EP-based ${\cal
PT}$-symmetric electro-mechanical accelerometer (see Fig. 1a). Our measuring
protocol differs from the previous cases which were analyzing the lasing modes
detuning of a system in the proximity to an EP. Instead, it relies on a
distinction between EPs and the transmission peak degeneracies (TPD) observed
in the transmission spectrum of the ${\cal PT}$-symmetric sensor when it is
interrogated via weakly coupled transmission lines (TL). This fact has been
already recognized in a previous theoretical work [29] for the analogous
concept of transmission deeps. Their distinct position from the position of
the eigenmodes of the (isolated) system is traced to the interference effects
between a direct scattering process and a resonant high-Q mode supported by
the system. We point out that in the absence of loss/gain, these transmission
deeps (or peaks in our case) and their associated degenerate point are related
to the newly established concept of reflectionless scattering modes (RSM). The
RSM frequencies belong to the spectrum of a wave operator with appropriate
boundary conditions which has been shown to exhibit an EP degeneracy [30, 31].
Nevertheless, the transmission deeps (peaks) and their associated TPD in the
case of ${\cal PT}$-symmetric systems are not related to the RSM exceptional
points and the identification of an appropriate wave operator whose spectrum
contains these modes is a topic of an ongoing research.

The transmission peaks frequency detuning from the TPD is influenced by the
underlying EP and follows a square-root behavior with respect to the applied
acceleration $a$. Furthermore, at TPD, the eigenbasis is still complete; thus
the performance of the sensor is not influenced by excess noise effects that
are rooted to the eigenbasis collapse. The latter has been related, in the
platform of Ref. [18], with the so-called Petermann factor ($PF$) which
diverges in the proximity of an EP, while at TPD it remains finite and smaller
than the sensitivity enhancement factor $SEF$ (compare blue and green lines in
Fig. 1b) which measures the responsivity of the system to acceleration
variations (see Supplementary Material). Our theoretical analysis indicates
that $PF$ is directly proportional to the technical noise due to the coupling
with the TL and the noise generated by the gain/loss elements used to create
the EP-singularity. The corresponding noise enhancement factors
$NEF^{TL},NEF^{\cal PT}$, describing the noise power enhancements (see
Supplementary Material), remain finite in the proximity of the TPD and are
surpassed by the SEF of the transmission peaks detuning near the TPD (see
brown and red lines in Fig. 1b).

The sensing device consists of a pair of capacitively coupled parity-time
($\mathcal{PT}$) symmetric RLC resonators [21] with natural frequency
$f_{0}={\frac{1}{2\pi}}{\frac{1}{\sqrt{LC}}}\approx 2.68MHz$ and capacitive
coupling $C_{c}$. Furthermore, one capacitor plate of the coupling capacitor
is connected to a test-mass $m$ which is supported by a spring attached to the
platform (see Fig. 1c and Methods for detailed platform design) and its
displacement from an initial equilibrium position will constitute the
mechanism that will sense acceleration variations (see discussion below). The
$\mathcal{PT}$-symmetry condition is achieved when the gain (implemented using
an amplifier) and loss parameters, namely $-R$ and $R$, are delicately
balanced, and the reactive components, $L$ and $C$, satisfy mirror symmetry:
that is, the impedances of the active and passive circuit tanks, multiplied by
$i$, are complex conjugates of each other at the frequency of interest [21].
When the capacitive coupling is above a critical value $C_{c}^{\rm EP}$, the
system is in the so-called exact phase where the corresponding normal modes
are also eigenvectors of the $\mathcal{PT}$ -symmetric operator. In the
opposite limit (small coupling) the system is in the broken phase where the
normal modes are no longer eigenmodes of the $\mathcal{PT}$ symmetric
operator. In the latter case, the eigenfrequencies appear as a complex
conjugate pair, while in the former domain they are real-valued. The two
phases are separated by an EP singularity of order $N=2$ [22].

We have weakly coupled each RLC resonator to $Z=50$ Ohm transmission lines
(TL) via capacitors Ce. In the density plot shown in Fig. 2a, we report the
measured (normalized) transmittance spectrum $T(f;a)$ versus the applied
differential in-plane acceleration $a$. At $a=0$ the transmission spectrum
demonstrates a transmission peak degeneracy (TPD) which reflects the nearby EP
degeneracy of the eigenfrequencies of the isolated system (corresponding to
$C_{e}=0$). The system is initially set at TPD conditions with the coupling
capacitor plates being at a distance $d\approx 20\mu m$ from one another
corresponding to $C_{c}^{\rm TPD}\approx 50pF$. As the applied differential
in-plane acceleration $a$ increases, the capacitive coupling detunes from its
equilibrium value $C_{c}^{\rm TPD}$ due to a displacement of the test-mass to
a new equilibrium position. The working principle of our accelerometer relies
on detecting concomitant transmission peak shifts from the TPD due to the
capacitive coupling detuning from $C_{c}^{\rm TPD}$. Indeed, from Fig. 2a, we
observe that the transmittance develops two equihight peaks which bifurcate
from the TPD following a characteristic square-root behavior (see dashed green
lines evaluated using a coupled mode theory Eq. (3)), expected from the
Puiseux expansion of resonance frequencies around an EP (see blue lines). In
fact, the responsivity at the TPD detuning, turns out to be slightly stronger
from the analogues one shown by the resonances at the EP (see Eq. (4) below).
The weak coupling with the transmission lines, together with the active nature
of our platform is reflected in the high intensity value of the transmittance
peaks and in the narrow form of the linewidths (see confined dark red area).
These attributes enhance the readout and boost the sensing resolution,
allowing us to easily identify the trajectories of the transmission peaks.
Finally, the highlighted gray area indicates the displacement values of the
capacitor plates for which the transmission peaks acquire extremely large
values triggering saturable nonlinearities.

Figure 2:  Experimentally measured response of the sensor to applied
acceleration. a, Density plot of the measured normalized transmittance
spectrum as a function of acceleration $a$ (top $x$-axis). The bottom $x$-axis
indicates the associated capacitance variation $\epsilon(a)$. The trajectory
of the transmittance peaks is indicated with green dashed lines. At the
proximity of the transmittance peak degeneracy (TPD) the peaks demonstrate a
square-root behavior as expected by the Puiseux series expansion near EPs. The
trajectories of the eigenfrequencies of the isolated dimer are shown by dark
blue lines. They coalesce at $\epsilon=0$ (EP degeneracy). The highlighted
gray area is excluded from our measurements. In this domain, saturable
nonlinearities are triggered. b, The same as in a for the transmittance
spectrum calculated using a couped mode theory (CMT). c, (Top) The
transmission peak versus the capacitance variation $\epsilon(a)$. The red line
indicates the CMT predictions while the dotted light blue line is the measured
data. The inset presents the same data in a double-logarithmic fashion
referring to the capacitance variations with respect to the lasing point. The
dashed and dotted black lines indicates a logarithmic slope of $-1$ and $-2$.
(Bottom) The behavior of half-width-half-maximum linewidth $\Gamma$ versus
$\epsilon$ in the proximity of the transmission peak degeneracy. The red
(dotted light blue) line indicates the CMT (experimental) results.d, The
measured relative transmission peak splitting $\Delta f/f_{0}$ versus the
applied differential acceleration $a$ plotted in double-logarithmic plot. The
dotted (dashed) black line indicates a square-root (linear) scaling with $a$.
Inset shows the same data in linear plot. The red arrow shows the dynamical
range enhancement of the proposed accelerometer with respect to an equivalent
sensor based on linear response. e, The sensitivity of the ${\cal
PT}$-symmetric accelerometer demonstrating an order (red arrow) enhancement in
the proximity of the EP as opposed to a system configuration away from the EP.
In the inset we report the same data in a double-logarithmic plot. Error bars
on panels d and e denote $\pm$ 1 standard deviation obtained from 10
independent measurements.

To better understand the outcome of these measurements we have performed a
theoretical analysis using the most general framework of coupled mode theory
(CMT) which was appropriately mapped to describe our $\mathcal{PT}$-symmetric
circuit (see details in the Supplementary Material). Specifically, the
scattering matrix $S$ that describes the open circuit, takes the form

$\mathbf{S}(f)=-\mathbf{I}-i\mathbf{W}\mathbf{G}(f)\mathbf{W}^{T};\quad\quad\mathbf{G}(f)=(\mathbf{H}_{\rm
eff}-f\mathbf{I})^{-1}$ (1)

where, $\mathbf{I}$ is the two-dimensional identity matrix and $G(f)$ is the
Green’s function. The effective Hamiltonian $\mathbf{H}_{\rm eff}$ that
describes the $\mathcal{PT}$-symmetric dimer coupled to the transmission lines
is

$\begin{array}[]{ccc}\mathbf{H}_{eff}=\mathbf{H}_{0}-\frac{i}{2}\mathbf{W}^{T}\mathbf{W};\quad\mathbf{H}_{0}=\left(\begin{array}[]{ccc}f_{0}-\epsilon+i\gamma_{0}&\gamma_{0}+\epsilon\\\
\gamma_{0}+\epsilon&f_{0}-\epsilon-i\gamma_{0}\end{array}\right);\end{array}$
(2)

where the diagonal matrix $\mathbf{W}_{nm}=\delta_{nm}\sqrt{2\gamma_{e}}$
models the coupling of the dimer with the transmission lines and
$\mathbf{H}_{0}$ is the Hamiltonian of the isolated dimer (corresponding to
$\gamma_{e}=0$). For $\epsilon=\epsilon_{\rm EP}=0$ the isolated system forms
a second order ($N=2$) EP degeneracy at frequency $f_{\pm}^{(0)}\equiv f_{\rm
EP}=f_{0}$ (see Supplementary Material).

Using Eq. (1) we have extracted the transmittance
${\mathcal{T}}(f;a)=|S_{21}|^{2}$ and via direct comparison with the
experimental data we were able to identify the various parameters that have
been used in our CMT modeling. Specifically, we have estimated that the
linewidth broadening of the resonances of the individual circuit tank, due to
its coupling with the transmission line is $\gamma_{e}\equiv Z_{0}\sqrt{C\over
L}\left(C_{e}\over C\right)^{2}f_{0}=0.0206\,$MHz. Similarly, we have found
that the variations at the coupling strength between the two resonant modes of
the dimer, due to the displacement of the plates of the capacitor $C_{c}$ when
an acceleration $a$ is imposed to the system, can be modeled by the parameter
$\epsilon(a)=\epsilon_{\rm TPD}+0.0082$ [MHz/g]$\cdot a$. The coupling
strength in the absence of any acceleration is $\epsilon_{\rm TPD}=0.0013$ MHz
- and is associated with the TPD. Finally,
$\gamma_{0}=R^{-1}\sqrt{L/C}f_{0}=0.16$ MHz, is the CMT gain/loss parameter
describing the amplifier/resistor used in the $\mathcal{PT}$ -symmetric
circuit.

Using the above extracted parameters we evaluated the transmittance spectrum
${\mathcal{T}}(f;\epsilon(a))=|S_{21}|^{2}$ from the CMT. The corresponding
density plot is reported in Fig. 2b. The agreement between the theoretical and
experimental results is reassuring for the quality of our fits and allow us to
extract further information for the scattering characteristics of our system.
Specifically, we have calculated the frequencies $f_{\pm}(\epsilon)$ of the
transmittance peaks (green dashed lines in Figs. 2a,b) which take the form

$f_{\pm}(\epsilon)=\left\\{\begin{array}[]{ccc}f_{0}-\epsilon\pm\sqrt{2\gamma_{0}\epsilon+\epsilon^{2}-\gamma_{e}^{2}},&{\rm
for}&\epsilon\geq\epsilon_{\rm TPD}\\\ f_{0}-\epsilon,&{\rm
for}&\epsilon\leq\epsilon_{\rm TPD}\end{array}\right.$ (3)

where $\epsilon_{\rm
TPD}=-\gamma_{0}+\sqrt{\gamma_{0}^{2}+\gamma_{e}^{2}}\neq\epsilon_{\rm EP}$.
Instead, we have that $\epsilon_{\rm
TPD}\approx\frac{\gamma_{e}^{2}}{2\gamma_{0}}\xrightarrow[\gamma_{e}\rightarrow
0]{}\epsilon_{\rm EP}=0$ as expected. Accordingly, the TPD frequency is
$f_{\rm TPD}\equiv f(\epsilon=\epsilon_{\rm
TPD})=f_{-}=f_{+}=f_{0}-\epsilon_{\rm TPD}\xrightarrow[\gamma_{e}\rightarrow
0]{}f_{\rm EP}=f_{0}$. Furthermore, at the vicinity of $\epsilon_{\rm TPD}$
the transmission peaks Eq. (3) scale as

$f_{\pm}\propto f_{\rm
TPD}\pm\sqrt{2}\sqrt[4]{\gamma_{0}^{2}+\gamma_{e}^{2}}\sqrt{\delta\epsilon};\quad{\rm
provided}\,\,{\rm that}\quad\delta\epsilon\ll
2\sqrt{\gamma_{0}^{2}+\gamma_{e}^{2}}$ (4)

where $\delta\epsilon\equiv\epsilon-\epsilon_{\rm TPD}$. It is important to
stress that the above square-root transmission peak splitting for small
$\delta\epsilon$ is a direct consequence of the presence of the EP
degeneracies of the eigenfrequencies $f_{\pm}^{(0)}$ of the isolated system
(e.g. $\gamma_{e}=0$). The latter, can be evaluated by direct diagonalization
of $\mathbf{H}_{0}$, see Eq. (2) and their functional dependence on $\epsilon$
is given by Eqs. (3,4) by substituting $\gamma_{e}=0$. It is interesting to
point out that the presence of $\gamma_{e}$ results in a slight enhancement of
the detuning rate of $f_{\pm}$ as compared to the detuning rate of the
resonant modes $f_{\pm}^{(0)}$.

To guarantee the stability of our system, we have also identified
theoretically the lasing condition of the open circuit (see Supplementary
Material). The latter is associated with the real poles of the scattering
matrix Eq. (1) occurring at $f_{L}=f_{0}-\epsilon_{L}$ where
$\epsilon_{L}=-\gamma_{0}+\sqrt{\gamma_{0}^{2}-\gamma_{e}^{2}}<\epsilon_{\rm
EP}=0<\epsilon_{\rm TPD}$. Its existence can be also recognized by the
divergence of the transmission at this frequency and perturbation values. In
the upper subfigure of Fig. 2c we report the un-normalized transmission peaks
${\mathcal{T}}_{peak}={\mathcal{T}}(f_{\pm};\epsilon)$ extracted from the
measurements (dashed blue line), together with its theoretical values
calculated using the CMT modeling (red line). We find that at the lasing point
${\mathcal{T}}_{peak}(\epsilon\rightarrow\epsilon_{L})\rightarrow\infty$ as
expected. The divergence is characterized by a
${\mathcal{T}}_{peak}(\epsilon)\propto(\epsilon-\epsilon_{L})^{-2}\quad[{\mathcal{T}}_{peak}\propto(\epsilon-\epsilon_{L})^{-1}]$
scaling, which applies in the range $\epsilon<\epsilon_{\rm
TPD}\quad[\epsilon>\epsilon_{\rm TPD}$ and it is nicely confirmed from the
experimental data].

While an enhanced transmitted intensity is an important element for the
readability of the output signal and the identification of the transmission
resonances, the other crucial factor is the sensing resolution. This bound is
proportional to the linewidth $\Gamma$ of the transmission peaks. In the limit
of very strong coupling $\epsilon\gg\gamma_{0}$, we can disregard the
$\mathcal{PT}$ nature of the circuit and we expect that the linewidth $\Gamma$
will be dictated by the coupling of the dimer with the transmission lines. The
latter is characterized by the coupling constant $\gamma_{e}$ which in our
setting takes the value $\gamma_{e}\approx 0.02\,$MHz. At the other limiting
case of $\epsilon\rightarrow\epsilon_{L}$ we expect a narrowing of the
linewidth which in the semiclassical picture (e.g. without taking into
consideration the non-orthogonality of the modes etc) becomes zero. In fact,
the CMT modeling provides us with the possibility to derive an exact
expression for the linewidth $\Gamma(\epsilon)$ for the whole range of
perturbations $\epsilon$ (see Supplementary Material). These CMT results are
shown in the lower subfigure Fig. 2c together with the extracted values from
the measurements which have been evaluated as the half-width at half-maximum
(HWHM) at the transmission peak. Although $\Gamma(\epsilon)$ remains
essentially constant $\Gamma(\epsilon)\approx\gamma_{e}$ at the parameter
range where our experiment is operating, we are able to identify a slight
increase in the proximity of $\epsilon_{\rm TPD}$. Specifically, our CMT
analysis indicates that for $\delta\epsilon\equiv\epsilon-\epsilon_{\rm
TPD}\geq 0$ the linewidth takes the form $\Gamma(\delta\epsilon\rightarrow
0)\sim\sqrt{2}\gamma_{e}-\sqrt{2}\left(\gamma_{e}^{2}+\gamma_{0}^{2}\right)^{1/4}\sqrt{\delta\epsilon}$.
For $\delta\epsilon=0$ we get that
$\Gamma(\delta\epsilon=0)=\sqrt{2}\gamma_{e}$ indicating that the coupling of
the circuit with the leads dictates the minimum measured uncertainty. This
prediction is confirmed by the measurements (see light blue dotted line in
Fig. 2c). On the other side of $\epsilon_{\rm TPD}$ i.e. $\delta\epsilon<0$
the linewidth behaves as
$\Gamma(\epsilon)\approx\sqrt{2}\gamma_{e}+{\sqrt{2(\gamma_{e}^{2}+\gamma_{0}^{2})}\over\gamma_{e}}\delta\epsilon$.
This small $\delta\epsilon$ expansion persists even for
$\epsilon\approx\epsilon_{\rm EP}$ where $\Gamma(\epsilon=\epsilon_{\rm
EP})\approx 0.7\gamma_{e}$. From this analysis we conclude that the
uncertainty in the frequency splitting measurements in the proximity of the
TPD is only slightly enhanced due to the linewidth increase [32].
Nevertheless, it can be always confined to small values when the system
operates in the weak coupling regime. Of course, this conclusion is subject to
the analysis of other characteristics like the sensitivity or other noise
sources that might also affect the measurement process.

The measurement of an individual frequency shift is the most common sensing
scheme for (micro)-electromechanical sensors [33, 34, 35]. However, a
considerable advantage is obtained by measuring the frequency splitting since
it is intrinsically self-referenced i.e. there is no need for an external
reference to suppress or eliminate frequency drift associated with other
sources. From Eq. (4) we have that the transmittance peaks splitting in the
proximity of the TPD, also follows a sublinear square-root behavior with
respect to the perturbation $\delta\epsilon$. Specifically $\Delta f\equiv
f_{+}-f_{-}\propto\sqrt{\delta\epsilon}$. In case of weak coupling to the
transmission lines the upper limit of the sublinear sensing is predominantly
controlled by $\gamma_{0}$, see Eq. (4). We point out that the extent of the
sublinear domain in our system is bounded by nonlinear capacitance effects
occurring at small distances between the plates of the capacitor. Even with
this additional limitation our platform demonstrates an order of magnitude
enhancement of dynamic range, as compared to conventional linear sensors with
the same upper bound of dynamical range (see red double-side arrow in the
inset of Fig. 2d). An overview of the frequency splitting measurements are
shown in Fig. 2d and nicely confirm the square root behavior expected from Eq.
(4).

To further quantify the efficiency of our proposed accelerometer, we have
introduced the sensitivity $\chi=\frac{\partial(\Delta f/f_{0})}{\partial a}$.
In Fig. 2e, we report the experimentally measured sensitivity together with a
theoretically derived curve (black dotted line). We find that indeed the
sensitivity demonstrates an $\chi\sim 1/\sqrt{a}$ divergence in the proximity
of the TPD. A comparison with the conventional linear sensor reveals again a
sensitivity improvement of one order (red double-arrow).

Noise Analysis –  Enhanced eigenvalue splitting is an important aspect of
efficient sensing as it guarantees an enhanced transduction coefficient of the
sensor from the input quantity of interest (e.g. the differential
acceleration) to the output quantity (e.g. $\Delta f$). Another important
aspect of an efficient sensing is the precision of the measurement which is
associated with the smallest measurable change of the input quantity given by
the noise of the sensor output. A cumulative quantification of the noise
effects on the measured frequency splitting $\Delta f$, is obtained by the
analysis of the Allan deviation [36, 37] $\sigma_{\Delta f}$ for various
values of applied differential in-plane acceleration $a$ (see Methods for
details). The Allan deviation describes the stability of the sensor as a
function of the sampling time $\tau$. From our analysis we find that
$\sigma_{\Delta f}$, increases as the applied acceleration $a$ approaches
zero, i.e. in the domain where the transmission peak splitting has a square
-root response. In contrast, we have found that at the domain where the
frequency splitting exhibits a linear sensitivity ($a\geq 0.5$ g) the Allan
deviation $\sigma_{\Delta f}$ remains almost unchanged (compare dark blue
solid line and magenta dash line in Fig. 3a). Such behavior, indicates that
the noise is enhanced at the vicinity of $a=0$ g (where
$\epsilon=\epsilon_{TPD}$) (see red arrow in Fig. 3a).

A more appropriate characterization of the performance of our sensor requires
the analysis of the normalized Allan Deviation $\sigma_{\alpha}=\sigma_{\Delta
f}/\chi$. Such analysis provides an estimate of the total noise effect on the
measured acceleration. In particular, the short -time decay of the normalized
Allan deviation (Velocity Random Walk) behaves like $\sigma_{\rm
VRW}(\tau)=\alpha_{\rm NEA}\cdot\tau^{-1/2}$, with a slope $\alpha_{\rm NEA}$
which decreases from $\alpha_{\rm NEA}=0.0027$g$\cdot$ Hz-1/2 at $a=1.66$g to
a limiting value $\alpha_{\rm NEA}=0.00086$g$\cdot$ Hz-1/2, at $a=0.01$g. The
experimental data shown in Fig. 3b indicates that there is a three-fold SNR
improvement at the vicinity of TPD.

The noise-equivalent acceleration (NEA) $\alpha_{\rm NEA}$ can be further
expressed as a sum of various terms associated with a different noise sources
that might affect the precision of the measurements. Specifically we have:

$\alpha_{\rm NEA}^{2}(\epsilon)\equiv\alpha_{\rm th}^{2}+\alpha_{\rm
TL}^{2}+\alpha_{\mathcal{PT}}^{2}+\alpha_{\rm add}^{2},\quad\left[{\rm
units}\,\,{\rm of}\,{\rm g}\cdot{\rm Hz}^{-1/2}\right].$ (5)

The first term arises from a thermal Brownian motion of the test mass and is
given by $\alpha_{\rm th}=\sqrt{4k_{\textrm{B}}T\omega_{n}/mQ}$. In this
expression $k_{\textrm{B}}$ is the Boltzmann’s constant, $T=293$ K is the
ambient temperature, $m=27$gr is the mass, $\omega_{n}=2\pi f_{n}=2513\,$
rad/sec is the frequency and $Q=40$ is the quality factor of the resonant mode
of the mechanical test-mass. The reduction of the thermal noise equivalent
acceleration is fundamental in minimizing the NEA towards its standard quantum
limit corresponding to the position uncertainty in the quantum ground state of
the mechanical mass. Typically it is overrun by the other terms in Eq. (5).
The other two terms arise from the thermal noise of the transmission lines
($\alpha_{\rm TL}$), the $\mathcal{PT}$-circuit noise
($\alpha_{\mathcal{PT}}$) produced by the gain-loss elements of the platform.
The remaining term is the added noise ($\alpha_{\rm add}$) associated with the
fluctuations due to the Brownian motion of the plates of the coupling
capacitors $C_{c2}$ and $C_{v}$ (see Methods) and the thermal noise produced
from the movement of various mechanical parts of the experimental setup (see
Methods). Both $\alpha_{\rm th}$ and $\alpha_{\rm add}$ are not associated
with (and do not depend on) parameters of the $\mathcal{PT}$ symmetric circuit
itself and thus, do not depend on $\epsilon$.

Figure 3:  Measured Allan Deviation. a, Measured Allan deviation
$\sigma_{\Delta f}$ of the transmission peaks splitting as a function of the
sampling time $\tau$ for various values of the applied acceleration $a$ (see
legend). The enhancement of $\sigma_{\Delta f}$ at the vicinity of the TPD at
$a=0$ g is shown by red arrow. b, Measured normalized Allan deviation
$\sigma_{\alpha}=\sigma_{\Delta f}/\chi$ as a function of the sampling time
$\tau$ for various values of the applied acceleration $a$ (see legend). The
suppression of $\sigma_{\alpha}$ due to enhanced sensitivity at the vicinity
of $a=0$ g is shown by red arrow. The black dotted lines on panels (a,b)
indicate a power decay $\tau^{-1/2}$, associated with a VRW regime, while the
black dashed lines are proportional to $\tau$ indicating the presence of a RR
regime. c The normalized noise equivalent acceleration $\alpha_{\rm
NEA}/\alpha_{\rm NEA}(a=0.8$g) (black line and circles) and the normalized
$FOM/FOM(a=0.8$g) (red line and crosses) versus the applied differential
acceleration. The black/red arrows indicate a threefold reduction/enhancement
of the noise/FOM in the proximity of the TPD.

One can further understand the demonstrated sensing efficiency by realizing
that $\alpha_{\mathcal{PT}}\equiv{\widetilde{\sigma}}_{\mathcal{PT}}/\chi$ and
$\alpha_{\rm TL}\equiv{\widetilde{\sigma}}_{\rm TL}/\chi$ where
${\widetilde{\sigma}}_{\mathcal{PT}}$ and ${\widetilde{\sigma}}_{\rm TL}$ are
the spectral densities of variances in the acceleration measurements due to
noise sources associated with the amplification/attenuation elements of the
isolated circuit and with the transmission lines respectively. These spectral
densities are proportional to the resonance linewidth $\Gamma$ and to the
noise enhancement factors associated with the corresponding noise sources
while they are inverse proportional to the transmitted signal through the
electronic circuit (see Supplementary Material). The theoretical analysis
based on CMT indicates that both ${\widetilde{\sigma}}_{\mathcal{PT}}$ and
${\widetilde{\sigma}}_{\rm TL}$ do not experience strong variations with
respect to $\epsilon$ and both saturate to a finite value at
$\epsilon=\epsilon_{\rm TPD}$. Since $\chi$ diverges at $\epsilon_{\rm TPD}$
(see green line in Fig. 1b), the NEA will decreases as $\epsilon$ approaches
$\epsilon_{\rm TPD}$ towards its noise floor level
$\alpha_{NEA}^{2}(\epsilon\rightarrow\epsilon_{TP})\rightarrow\alpha_{th}^{2}+\alpha_{\rm
add}^{2}$. To appreciate further the enhancement of SNR in the proximity of
the $\epsilon_{\rm TPD}$ we have normalized the NEA with respect to its value
at large accelerations e.g. $a=0.8\,$g. The normalized data (black circles)
are shown in Fig. 3c and demonstrate clearly a three-fold noise reduction in
the proximity of TPD. The CMT results are also indicated in this figure with a
black solid line and nicely match the experimental findings. Furthermore, the
CMT analysis shows that the noise associated with the TL coupling and
gain/loss elements used to create the sensor singularity are equivalent to
voltage (or current) noise sources in the circuit. These are physically
distinct from the parametrically coupled signal transduction sensing
mechanism, and are transformed differently by the transmission singularity,
toward a favorable signal-to-noise ratio.

We conclude that in the specific case of accelerometers the lower limit of NEA
is dictated by the Brownian motion of the test mass and the capacitor plates
which scales proportionally to the sensitivity $\chi$. We stress that on the
modeling level, the Brownian motion has to be modeled as a fluctuation of the
Hamiltonian matrix elements (i.e. coupling coefficient) while the electronic
thermal noise due to amplifiers, resistors and transmission lines is described
by additive stochastic Langevin terms in the Hamiltonian matrix. The
electronic thermal noise, does not scale with $\chi$; it rather abides to a
noise-specific transfer function which in the case of the PT-symmetric circuit
is proportional to the $PF$. Therefore, whenever the sensitivity $\chi$
overwhelms the noise-specific transfer function, the limiting value of
$\alpha_{\rm NEA}$ is the same for any sensing (linear or sub-linear) scheme.
An example of a linear sensing protocol that demonstrates extreme sensitivity
is associated with slow light sensors, utilizing high-Q resonant modes like
the ones occurring at the band-edges of a Fiber Bragg grating (FBG) [27, 28].
Such sensors rely on the abrupt intensity variations of the transmission of a
CW when a perturbation (e.g. stress) modifies the effective refractive index
of the FBG, thus inducing a spectral shift of the high-Q Lorentzian resonance.
This enhanced sensitivity is offset by an extremely short dynamic range that
these sensors demonstrate: while the large sensitivity $\chi$ leads to a
reduction of the lower sensing bound by minimizing the NEA, it simultaneously
lowers the upper sensing bound, which is typically limited by the maximum
measured signal that the sensor can output. In contrast, the enhanced
sensitivity demonstrated by the TPD sensing scheme reduces, from one hand the
lower sensing bound, while from the other hand, is not affecting the upper
sensing bound of the measured accelerations. Additionally, in linear sensors
the high sensitivity is inversely proportional to the mechanical resonant
frequency of the test mass $f_{n}$. The latter dictates the operational
bandwidth of the sensor - if the frequency of the applied acceleration signal
exceeds the $f_{n}$ this signal will not be detected by the sensor. As a
result, the high sensitivity of a linear sensor also comes at a cost of its
bandwidth. Based on these key figure of merits, the performance of our
proposed TPD-based proof-of-concept accelerometer compares favorably with the
available on-market on-chip accelerometers such as NXP MMA1270KEG [39]. For
example, they both demonstrate an $\alpha_{\rm NEA}=0.00086$g$\cdot$ Hz-1/2,
for a measurement range up to ($\geq 2$) g, while our prototype has a 6.5
times larger operational bandwidth.

An alternative way to reach the ultimate bound of the SNR of a linear sensor
is by increasing the power of the input signal to “infinite” levels. Of
course, this is practically challenging – if not impossible. In our case, a
finite power consumption enhancement is automatically taking place when the
system operates at the proximity of the TPD. It is therefore imperative to
take also this effect into consideration when comparing the efficiency of a
sensor. This point has been recently raised by Alu and coworkers which have
introduced an additional Figure of Merit (FOM) that takes into consideration
the consumed power $U$ used for the sensing [38]. We have expressed FOM in
terms of $\alpha_{\rm NEA}$ as $FOM=(\alpha_{\rm NEA}^{2}P)^{-1}$ and plotted
in Fig. 3c the rescaled FOM (with the value $FOM(a=0.8\,$g) extracted from the
CMT (red solid line) and the experiment (red crosses) using the measured
$\alpha_{\rm NEA}$. The analysis indicates an improved FOM of the ${\cal
PT}$-symmetric sensor in the proximity of TPD.

Let us also discuss the opposite limit of long-time growth of
$\sigma_{\alpha}(\tau)$. This analysis allows us to extract the so-called
drift rate ramp (DRR), describing systematic (deterministic) errors due to
temperature fluctuations. By definition $\sigma_{DRR}(\tau)=\alpha_{\rm
DRR}\cdot\tau$, with best fit $\alpha_{\rm DRR}\approx 0.0007$ g$\cdot$s-1 for
almost all acceleration values. Finally, the saturation value
$\sigma_{BI}(\tau)=\alpha_{\rm BI}\cdot\tau^{0}$ of the Allan deviation (black
dashed line) is indicative of the Bias Instability (BI), and sets the smallest
possible reading of our sensor due to the random flickering of electronics or
other components. The extracted value ranges from $\alpha_{BI}\approx 0.0009$
g (accelerations closer to TPD) to $\alpha_{BI}\approx 0.0014$ g
(accelerations away from TPD).

Conclusions – We have studied theoretically and experimentally the sensing
performance of a ${\cal PT}$-symmetric electromechanical accelerometer in the
proximity to an exceptional point (EP) degeneracy. When the system is weakly
coupled to transmission lines, the spectrum demonstrates transmission peak
degeneracies (TPDs) which are influenced by the presence of the EP but occur
at distinct parameter values. Our findings indicate that the TPDs inherit the
enhanced sensitivity of the EP which, however, is not offset by the slight
noise enhancement. We measured a three-fold signal-to-noise ratio enhancement
at the vicinity of TPD. Although our analysis was focusing on avionic sensing,
our platform can also be used for supersensitive microfluid flow sensors,
pressure sensors or for the detection of magnetic field variations. Our
results, show promise for the use of EP-based platforms as a novel scheme for
enhanced sensing without the concerns associated with the noise amplification.

##  1 Methods

###  1.1 Transmittance spectrum and frequency splitting measurements

The transmitted signal through the electronic circuit was collected for
different applied in-plane projections of gravity acceleration. The device was
mounted on a rotational stage Newport ESP100BCC which was electronically
controlled via a Newport ESP301 controller. The setup allows to control the
horizontal tilt of the device with minimal increment of 0.2 mdeg and, thus,
vary the in-plane projection of the gravity acceleration in a range of 2g ([-1
g :1 g]). At each specific tilting angle the transmission spectrum was
collected using an ENA network analyzer Keysight E5080A. The individual
frequency sweeps contain 401 points in a range of 2.58 MHz - 2.74 MHz, with
individual frequency bandwidth of 1.5 MHz. Single measurement is obtained from
the collected spectrum which was averaged over 100 consecutive individual
sweeps for each applied acceleration, which resulted in a sampling time of
$\tau\approx 25$ ms. The frequencies of the transmittance peaks $f_{\pm}$ were
then identified from the resulted spectrum, which allows to calculate the
frequency splitting $\Delta f$.

###  1.2 Allan deviation measurement

The Allan deviation $\sigma_{\Delta f}(\tau)$ of the transmission peak
splitting is defined as

$\sigma_{\Delta f}(\tau)=\sqrt{{1\over
2(M-1)}\sum_{n=1}^{M-1}(\overline{\Delta f}_{n+1}-\overline{\Delta
f}_{n})^{2}}$ (6)

where $\tau$ is the sampling time, $M$ is the total number of frequency
measurements, and $\overline{\Delta f}_{n}$ indicates the average frequency
splitting during the sampling time interval $[n\tau,(n+1)\tau]$. For the
extraction of Allan deviation the transmittance peak splittings $\Delta f$
were sampled with frequency of 400 Hz ($\tau_{min}=2.5$ ms) over a period of
90 s.

###  1.3 Figure of Merit (FOM) and consumed power

From the CMT analysis (see Supplementary material) one can show that:

$|S_{21}|^{2}+|S_{11}|^{2}+2\gamma_{2}|a_{2}|^{2}-2\gamma_{1}|a_{1}|^{2}-1=0;\quad{\bf
a}=-i{\bf GW}^{T}(1,0)^{T},$ (7)

where the $\gamma_{1}/\gamma_{2}$ is the gain/loss of the first/second cavity.
In our settings $\gamma_{1}=\gamma_{2}=\gamma_{0}$. The total power $P$
consumed by the platform is

$P=\int_{f_{min}}^{f_{max}}\left(|S_{21}|^{2}+|S_{11}|^{2}+2\gamma_{2}|a_{2}|^{2}\right)|\mathbf{a}_{{\rm
TL}_{1}}^{in}|^{2}\,df=\int_{f_{min}}^{f_{max}}\left(2\gamma_{1}|a_{1}|^{2}+1\right)|\mathbf{a}_{{\rm
TL}_{1}}^{in}|^{2}df.$ (8)

In our experimental platfrom $f_{min}=2.58$ MHz and $f_{max}=2.74$ MHz, which
are the lower/upper bound of frequency range used to perform the measurements.
Taking into account the fact that the input signal $\mathbf{a}_{{\rm
TL}_{1}}^{in}$ has a uniform spectral density, the $FOM$ introduced in
Ref.[38] can be written as

$FOM=(\alpha^{2}_{NEA}P)^{-1}\propto\int_{f_{min}}^{f_{max}}(\alpha_{NEA}\left(|S_{21}|^{2}+|S_{11}|^{2}+2\gamma_{2}|a_{2}|^{2}\right))^{-1}\,df.$
(9)

We used the $S_{21}$ measurements to fit the measurements with CMT parameters,
allowing us to estimate the total power $P$ using Eq.8. Finally, along with
experimental data of $\alpha_{NEA}$ this allows to estimate the $FOM$ using
Eq.9.

###  1.4 Circuit design and fabrication

n The detailed schematics of the ${\cal PT}$-symmetric circuit is shown in the
Extended data Fig. Extended Data Figure 1. The main elements of the circuit
are a pair of RLC resonators, where resistor, inductor and capacitor are
connected in parallel to ground. The inductor L in each unit is a Murata
11R103C with inductance of $L=10$ $\mu$H. The total capacitance consists of a
pair of in-parallel connected capacitors C and Cv, where the former one is
Murata GRM21A5CE271JW01D with fixed value of $C=270$ pF, while the latter one
is a voltage controlled capacitor Murata LXRW0YV33-0-056 with variable
capacitance $C_{v}=16.5-33$ pF. It is used to precisely control the resonant
frequency of each resonator. External voltage was used in order to control the
$C_{v}$. It was applied via an EG$\&$G 7265 DSP Lock-in amplifier, connected
via a BNC port through a resistor Rv Yageo RC0402FR-75KL with resistance
$R_{v}=5$ kOhm and grounded $10$ $\mu$F capacitance Cv1 Murata
GCM32EL8EH106KA7. The total resistance in each resonator unit consists of a
fixed resistor R1 Bourns CR1206FX-1101ELF with resistance $R_{1}=1.1$ kOhm,
which is connected in-series with a mechanically controlled variable resistor
R2 Bourns 3296’w-1-20RLF with resistance $R_{2}=0-200$ Ohm.

In the first resonator these resistors are connected to the output voltage
port of an operational amplifier (Op Amp) in order to produce gain. The Op Amp
used in the circuit is one of the three on-chip Op Amps of the Analog Devices
ADA4862-3 triple amplifier. The internal resistance between the inverting port
to ground is $R_{G1}=550$ Ohm . This is the same as the resistance between the
output port and the inverting input port $R_{G2}=550$ Ohm (see extended data
Fig. Extended Data Figure 1). The non-inverting input of the amplifier unit is
connected to the node of the first resonator via the resistor Rs, Bourns
RC0402FR-075KL with resistance $R_{s}=133$ Ohm. The second unit of the same
ADA4862-3 triple amplifier is connected with non-inverting input to the node
of the first resonator through the same resistance $R_{s}=133$ Ohm. The
resistor $R_{G2}$ is left floating in order to obtain a unity gain. The
tripple amplifier is powered via the voltage source through ports -Vs and +Vs,
as shown in extended data Fig. Extended Data Figure 1. The output port of the
Op Amp unit is connected with the probing input BNC port via the capacitor Ce,
Murata GCM1887U1H103JA6J with capacitance $C_{e}=10$ nF. This capacitance was
used to block the DC-signals from entering the circuit, while the coupling of
the input probing signal is achieved via parasitic capacitive coupling between
the input BNC port and the circuit elements. The resistors Re1, Bourns
CRM1206FX2700ELF with resistance $R_{e1}=270$ Ohm and Re2, Bourns
CR1206FX60R4ELF with resistnace $R_{e1}=50$ Ohm are connected as shown in the
extended data Fig. Extended Data Figure 1 in order to match the circuit
impedance with the 50 Ohm coaxial cable connected to the BNC port.

In the second resonator unit the resistors R1 and R2 are connected to ground
in-parallel to the capacitors C and Cv. The probing output BNC port is
connected to the node of the second resonator through the Op Amp unit of
another ADA4862-3 triple amplifier. The Op Amp unit operates in a unity gain
regime (similar to the second Op Amp connected to the node of the first
resonator).

The two resonator units are capacitevely coupled together using a trimmed
capacitor Cc2 Knowles JZ300HV with variable capacitance $C_{c2}=5.5-30$ pF.
The spring-mass is coupled to the circuit producing an extra coupling
capacitance $C_{cv}\approx 35$ pF, when the spring-mass is at rest. When an
in-plane acceleration is applied, the mass with the attached capacitor plate
moves toward the stationary plate, resulting in an increase of the capacitance
$C_{cv}$. The circuit was built on a prototype board and was mounted on top of
the spring-mass as it is seen in Fig. 1. The electrodes of the spring mass
were connected to the circuit with 0.18 mm wires which provides enough
flexibility during the motion of the test mass. The wire were glued to the
electrodes using silver filled epoxy adhesive MG Chemicals 8331-14G.

###  1.5 Spring-mass capacitive sensor design and fabrication

The spring-mass capacitive sensor consists of two main pieces. The first one
is a base plate shown in Figs. Extended Data Figure 2a and b. It consists of a
56 x 25 mm x 5 mm copper plate with one side having a 20 mm long ledge with 6
mm height. The ledge is used to mount the stationary capacitor plate. The
second main piece is a 0.027 kg, 5 mm thick copper mass which is suspended on
four 10 x 0.3 x 2 mm bendable beams attached to stationary stands (see Figs.
Extended Data Figure 2c and d). The mass is suspended 1 mm above the bottom
level of the stationary stands. The mass and base plates were micro machined
using electrical discharge machining (EDM) technique. The pair of 23 x 0.5 x 5
mm glass substrates were used to sputter (PVD) a 10 nm thin Ti layer followed
by a 100 nm thin Au layer which acts as electrodes for the capacitor plates.
One of the glass plates was glued to the ledge of the base plate (as it is
shown in Fig. Extended Data Figure 3a) at the noncoated side. In a similar
manner the second plate was glued to the back vertical surface of the copper
mass at the noncoated side (see Fig.Extended Data Figure 3b). The copper mass
with the glass plate was then mounted on the copper base and glued with a
conductive adhesive (see Fig. Extended Data Figure 3c). The assembly was
performed under a microscope with a custom built micro-translational stage to
achieve the 20$\mu$m gap between the two gold-metalized surfaces of the glass
plates (see inset in Fig. Extended Data Figure 3c). The spacing between the
pair of glass plates with gold nanolayers forms a capacitance of about 35 pF,
when the mass is at rest. When an acceleration is applied to the platform, the
plates are moving closer to one another, leading to an increase of the
capacitance. The open parts of the metalized surfaces were connected to the
circuit as it is discusses above, while the platform and the mass are
grounded. The pair of glass plates form the parasitic capacitances $C_{p1}$
and $C_{p2}$ between the gold electrodes and the grounded test mass. This
parasitic capacitances are shown in Fig. Extended Data Figure 1 and originally
result in a frequency missmach between the LC resonators. The tuning
capacitors Cv were used to compensate the associated frequency mismatch
between the resonators.

Figure Extended Data Figure 1:  Extended Data Figure 1: Schematic of the
circuit diagram. The blue elements are ADA4862-3 amplifiers, while the green
element indicates the spring-mass which provides the acceleration dependent
capacitance Ccv.  Figure Extended Data Figure 2:  Extended Data Figure 2:
Details of the mechanical sensor elements. a,b, Drawings of the copper
platform. c,d, Drawings of the test-mass.  Figure Extended Data Figure 3:
Extended Data Figure 3: Assembly process of the acceleration capacitive
sensor. a, Deposition of the gold nanofilms on a glass substrate that create
conductive electrodes which form the capacitors plates. Attachment of the
glass plates to the stationary platform and test-mass. b, Placement of the
test mass with glass plate on top of the copper base. c, Assembled capacitive
inertial sensor. The inset shows the magnified view of the area between the
capacitor plates which is about 20 um.

Acknowledgments:  We acknowledge partial support from NSF-CMMI-1925543, NSF-
CMMI-1925530, ONR N00014-19-1-2480 and from a grant from Simons Foundation for
Collaboration in MPS No. 733698. RT and JC also acknowledge the partial
support for this research provided by the University of Wisconsin-Madison,
Office of the Vice Chancellor for Research and Graduate Education with funding
from the Wisconsin Alumni Research Foundation.

Author contributions:  R.K., J.C., F. E. and R.T. designed the mechanical
device. R.K. and F. E. designed and fabricated the electronic circuit. J.C.
and R.T. fabricated the mechanical device. R.K. performed the characterization
and data processing of the accelerometer and developed the theory with the
support of T.K. All authors discussed the results. T.K. conceived the project.
R.K. and T.K. wrote the manuscript with input from all authors.

Competing interests: The authors declare that they have no competing
interests.

Data availability : Data are available from the corresponding author upon
reasonable request.

Code availability : All codes are available from the corresponding author upon
reasonable request.

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Supplementary Material

##  S1 CMT modeling of the electronic circuit

The electronic circuit of Fig. 1a can be modeled using a CMT picture, see Fig.
S1. This mapping allows us to extend the conclusions of our investigations to
a broader family of systems where CMT is applicable. At the same time it
provides a simple mathematical description of our experimental set-up.

Let us first start from the description of an $LC$ circuit, which is described
by a set of equations:

$\centering\Bigg{\\{}\begin{array}[]{ccc}\mathcal{V}=L\frac{d\mathcal{I}}{dt}\\\
\mathcal{I}=-C\frac{d\mathcal{V}}{dt}.\end{array}\@add@centering$ (S1)

The above equations can be re-written in the form of a second order
differential equation for the voltage:

$\mathcal{\ddot{V}}+\omega_{0}^{2}\mathcal{V}=0.$ (S2)

which can be solved easily, giving

$\centering\Bigg{\\{}\begin{array}[]{ccc}\mathcal{V}=|V|cos(\omega_{0}t+\phi)\\\
\mathcal{I}=\sqrt{\frac{C}{L}}|V|sin(\omega_{0}t+\phi),\end{array}\@add@centering$
(S3)

where $|V|$ is the voltage amplitude of the LC resonator, $\phi$ is the
associated phase and $\omega_{0}=\frac{1}{\sqrt{LC}}$ is the natural frequency
of the circuit.

It is useful to change variables and define the complex mode amplitude $a(t)$
and its conjugate $a^{*}(t)$ requiring the normalization:
$|a(t)|^{2}=W=\frac{C}{2}|V|^{2}$, where $W$ is an energy stored in the
circuit. This yields that

$\centering\begin{array}[]{ccc}a=\sqrt{\frac{C}{2}}(\mathcal{V}-i\sqrt{\frac{L}{C}}\mathcal{I})=\sqrt{\frac{C}{2}}(\mathcal{V}-\frac{\dot{\mathcal{V}}}{i\omega_{0}})=\sqrt{\frac{C}{2}}|V|e^{-i\omega_{0}t},\\\
a^{*}=\sqrt{\frac{C}{2}}(\mathcal{V}+i\sqrt{\frac{L}{C}}\mathcal{I})=\sqrt{\frac{C}{2}}(\mathcal{V}+\frac{\dot{\mathcal{V}}}{i\omega_{0}})=\sqrt{\frac{C}{2}}|V|e^{i\omega_{0}t},\end{array}\@add@centering$
(S4)

The voltage $\mathcal{V}(t)$ and its derivative $\mathcal{\dot{V}}(t)$ can
then be expressed in terms of the complex mode amplitudes as:

$\centering\begin{array}[]{ccc}\mathcal{V}=\frac{1}{\sqrt{2C}}(a+a^{*})\\\
\mathcal{\dot{V}}=\frac{-i\omega_{0}}{\sqrt{2C}}(a-a^{*}).\end{array}\@add@centering$
(S5)

from where the expression for the current follows immediately using the second
equation of Eq. (S3).

We are now moving forward with the description of the $\mathcal{PT}$ symmetric
system of coupled $RLC$ circuits (Fig.1a of the main text) using the complex
amplitudes representation. First, we write Kirchhoff’s laws for the voltages
at the nodes of the $RLC$ resonators. They take the form

$\centering\Bigg{\\{}\begin{array}[]{ccc}(1+\kappa)\mathcal{\ddot{V}}_{1}-\gamma_{1}\omega_{0}\mathcal{\dot{V}}_{1}+\omega_{0}^{2}\mathcal{V}_{1}-\kappa\mathcal{\ddot{V}}_{2}=0\\\
(1+\kappa)\mathcal{\ddot{V}}_{2}+\gamma_{2}\omega_{0}\mathcal{\dot{V}}_{2}+\omega_{0}^{2}\mathcal{V}_{2}-\kappa\mathcal{\ddot{V}}_{1}=0,\end{array}\@add@centering$
(S6)

where $\gamma_{1,2}=\frac{1}{R_{1,2}}\sqrt{\frac{L}{C}}$ is the gain/loss of
the first/second $LC$ resonator due to the negative/positive resistance $\mp
R_{1,2}$ and $\kappa=\frac{C_{c}}{C}$ is the ratio of the coupling capacitor
$C_{c}$ to the capacitance $C$, which represents the coupling strength between
the two resonators. Equation (S6) can be rewritten in terms of the complex
amplitudes $a_{1},a_{2}$ and their conjugates $a_{1}^{*},a_{2}^{*}$ as [1, 2]

$\centering\Bigg{\\{}\begin{array}[]{ccc}\frac{d}{dt}a_{1}\approx-i\omega_{0}a_{1}+\frac{1}{2}\omega_{0}\gamma_{1}(a_{1}-a_{1}^{*})+\frac{1}{2}i\omega_{0}\kappa(a_{1}-a_{1}^{*}-a_{2}+a_{2}^{*})\\\
\frac{d}{dt}a_{2}\approx-i\omega_{0}a_{2}-\frac{1}{2}\omega_{0}\gamma_{2}(a_{2}-a_{2}^{*})+\frac{1}{2}i\omega_{0}\kappa(a_{2}-a_{2}^{*}-a_{1}+a_{1}^{*}),\end{array}\@add@centering$
(S7)

where it was assumed that the coupling between the resonators is weak i.e.
$\kappa\ll 1$. We can simplify further the above equation by invoking a
rotating-wave approximation. This allows us to decouple $a$ and $a^{*}$ and
re-write Eq. (S7) in a CMT form:

$\centering\frac{d}{dt}\mathbf{a}\approx-i\mathbf{H}_{0}\mathbf{a},\,\,\,\mathbf{H}_{0}=\omega_{0}\mathbf{\widetilde{H}}_{0}=\omega_{0}\
\left(\begin{array}[]{ccc}1-\frac{\kappa}{2}+i\frac{\gamma_{1}}{2}&\frac{\kappa}{2}\\\
\frac{\kappa}{2}&1-\frac{\kappa}{2}-i\frac{\gamma_{2}}{2}\end{array}\right),\,\,\,\mathbf{a}=(a_{1},a_{2})^{T}\@add@centering$
(S8)

where $\mathbf{\widetilde{H}}_{0}$ is the (dimensionless) Hamiltonian of the
two-mode system (below a tilde will indicate dimensionless operators).

Next, we incorporate in our modeling the (weak) coupling of the $LC$
resonators to the transmission lines occurring via the capacitance
$C_{e}=\kappa_{e}C$, $\kappa_{e}\ll 1$. At the transmission lines the voltage
$\mathcal{V}_{\rm TL}$ and the current $\mathcal{I}_{\rm TL}$ are represented
as a superposition of a forward (+) and a backward (-) propagating wave i.e.
$\mathcal{V}_{\rm TL}=\mathcal{V}_{\rm TL}^{+}+\mathcal{V}_{\rm TL}^{-}$,
$\mathcal{I}_{\rm TL}=\mathcal{I}_{\rm TL}^{+}+\mathcal{I}_{\rm
TL}^{-}=\frac{1}{Z_{0}}(\mathcal{V}_{\rm TL}^{+}-\mathcal{V}_{\rm TL}^{-})$,
where $Z_{0}$ is a characteristic impedance of the transmission line. At the
same time we express the voltages in terms of complex wave amplitudes as:
$\mathcal{V}_{\rm TL}^{\pm}=\pm\sqrt{\frac{Z_{0}}{2}}(S^{\pm}+S^{\pm*})$,
where $S^{\pm}=|S^{\pm}|e^{-i\omega_{0}t}$. The Kirchhoff’s equations that
describe the voltage and current at the junction between the transmission line
and a resonator with loss given by $\gamma_{0}$ is:

$\frac{1}{C}\frac{d}{dt}\mathcal{I}_{\rm
TL}=\mathcal{\ddot{V}}+\gamma_{0}\omega_{0}\mathcal{\dot{V}}+\omega_{0}^{2}\mathcal{V};\quad\mathcal{I}_{\rm
TL}=C_{e}\frac{d}{dt}\left(\mathcal{V}_{\rm TL}-\mathcal{V}\right).$ (S9)

Furthermore, assuming weak coupling $\kappa_{e}\ll 1$ and considering a
rotating-wave approximation, we can re-write the above set of equations in the
form:

$\centering\Bigg{\\{}\begin{array}[]{ccc}\frac{d}{dt}a\approx-i\omega_{0}\left(1-\frac{1}{2}\gamma_{e}\right)a-\frac{1}{2}\omega_{0}\gamma_{0}a-\frac{1}{2}\omega_{0}\gamma_{e}a-i\sqrt{\omega_{0}\gamma_{e}}S^{+}\\\
S^{-}\approx S^{+}-i\sqrt{\omega_{0}\gamma_{e}}a.\end{array}\@add@centering$
(S10)

where $\gamma_{e}=\frac{Z_{0}}{\sqrt{L/C}}\kappa_{e}^{2}$.

Combining together Eqs. (S8,S10) allows us to describe our RLC circuit in a
temporal CMT form, where after redefinition of phase factors of the propagated
waves in the transmission lines as $\mathcal{S}^{\pm}\xrightarrow[]{}\pm
i\mathcal{S}^{\pm}$. we get:

$\centering\Bigg{\\{}\begin{array}[]{ccc}\frac{d}{dt}\mathbf{a}=-i(\mathbf{H}_{0}-\frac{i}{2}\mathbf{W}^{T}\mathbf{W})\mathbf{a}+\mathbf{W}^{T}\mathbf{\mathcal{S}}^{+}\\\
\mathbf{\mathcal{S}}^{-}=-\mathbf{\mathcal{S}}^{+}+\mathbf{Wa}.\end{array},\,\,\,\,\,\,\mathbf{W}\equiv\sqrt{\omega_{0}}\mathbf{\widetilde{W}}=\left(\begin{array}[]{ccc}\sqrt{\omega_{0}\gamma_{e}}&0\\\
0&\sqrt{\omega_{0}\gamma_{e}}\end{array}\right),\,\,\mathcal{S}^{\pm}=(S^{\pm
1}_{1},S^{\pm}_{2})^{T}\@add@centering$ (S11)

Using Eq. (S8) and Eq. (S11) we are now able to map the electronic circuit
used in our experiment to the CMT formalism which allows to generalize the
analysis of our platform the broad range of frameworks. In the sections below
we proceed with the general CMT analysis. For convenience we have redefined
the various parameters associated with the LRC circuit as:
$\omega_{0}(1-\frac{\kappa}{2}-\frac{\kappa_{e}}{2})\rightarrow\omega_{0}$,
$\kappa\rightarrow 2\kappa$, $\gamma_{1,2}\rightarrow 2\gamma_{1,2}$ and
$\gamma_{e}\rightarrow 2\gamma_{e}$. Moreover, since $\gamma_{e}$ and
$\kappa_{e}$ are directly related (see above), we will be referring to the
isolated ${\cal PT}$-dimer (zero coupling with the transmission lines i.e.
$\kappa_{e}=0$) with the equivalent condition $\gamma_{e}=0$.

Figure S1:  Schematic of the coupled mode theory setup:  The platform consists
of two resonators with resonant frequency $\omega_{0}$ and balanced gain
$+\gamma_{0}$ (red circle), and loss $-\gamma_{0}$ (green circle) elements.
The system is prepared in a way that the coupling strength between the two
resonators is $\kappa=\kappa_{\rm EP}=\gamma_{0}$. The variations in the
coupling (due to e.g. differential acceleration) are probed by sending an
input signal $\mathbf{a}_{{\rm TL}_{1}}^{in}$ from the left transmission line
and measuring the outgoing signal $\mathbf{a}_{{\rm TL}_{2}}^{out}$ at right
transmission line. The coupling strength between the coupled mode system and
the transmission lines is denoted by $\gamma_{e}$. The resonant modes are
affected by an ambient noise associated with the right and the left
transmission lines $\mathbf{n}_{{\rm TL}_{1}/{\rm TL}_{2}}^{in}$ and by the
noise $\mathbf{n}_{1,2}^{in}$ generated internally due to the presence of the
gain (left) and the loss (right) mechanisms. The system emits noise to the
left/right transmission lines $\mathbf{n}_{{\rm TL}_{1}/{\rm TL}_{2}}^{out}$
and to the left/right internal reservoirs $\mathbf{n}_{1/2}^{out}$. When the
system experiences an external perturbation (e.g. acceleration) the coupling
constant between the two resonators is modified as
$\kappa\rightarrow\kappa_{\rm EP}+\epsilon$; thus producing a detuning of the
resonant frequencies $\omega_{0}\rightarrow\omega_{0}-\epsilon$ of the
individual resonators.

##  S2 Analysis of the CMT Model and Scattering Approach

We first analyze the eigen-spectrum of the Hamiltonian
$\mathbf{\widetilde{H}}_{0}$ (see previous section) which describes the
isolated system (i.e. $\gamma_{e}=0$) :

$\mathbf{\widetilde{H}}_{0}\equiv\frac{\mathbf{H}_{0}}{\omega_{0}}\ =\
\left(\begin{array}[]{ccc}1+i\gamma_{1}&\kappa\\\
\kappa&1-i\gamma_{2}\end{array}\right);$ (S12)

where $\gamma_{1,2}$ are the dimensionless linewidths of the two resonant
modes due to the presence of the gain/loss elements at the first and the
second LC resonator respectively. In case of ${\cal PT}$ symmetric
configurations the gain/loss elements are perfectly balanced and therefore
$\gamma_{1}=\gamma_{2}=\gamma_{0}$.

Direct diagonalization of Eq. (S12) allows us to evaluate the (dimensionless)
eigenfrequencies and the eigenvectors of the coupled mode system:

$\widetilde{\omega}_{\pm}^{(0)}=\frac{\omega_{\pm}^{(0)}}{\omega_{0}}=1\pm\sqrt{\kappa^{2}-\gamma^{2}_{0}};\quad\quad{\bf\upsilon}_{\pm}^{(0)}=\left(\frac{i\gamma_{0}\pm\sqrt{\kappa^{2}-\gamma^{2}_{0}}}{\kappa},1\right)^{T}$
(S13)

From Eq. (S13) we conclude that the two modes of the ${\cal PT}$ circuit
demonstrate an exceptional point (EP) degeneracy when the coupling constant
$\kappa$ becomes equal to the gain/loss strength i.e.
$\kappa_{EP}=\gamma_{0}$. For this coupling strength, the value of the
degenerate frequency becomes
$\widetilde{\omega}_{\pm}^{(0)}=\widetilde{\omega}_{EP}^{(0)}=1$.

Next, we assume that the couple resonators are initially positioned at the EP
configuration while a small perturbation $\epsilon$, associated with a small
variation at the coupling capacitor due to an applied acceleration, is imposed
to the system. Such perturbations will affect the coupling rate as
$\kappa=\kappa_{EP}\rightarrow\kappa_{EP}+\epsilon$. At the same time they
will induce a shift at the natural frequencies of the resonators
$\omega_{0}\rightarrow\omega_{0}(1-\epsilon)$ (see Eq. (S8)). Finally, we take
into account the linewidth broadening of the natural frequencies of the
individual resonators due to their coupling to the transmission lines (see Eq.
(S11)). Incorporating these perturbation effects in Eq. (S12), we come up with
the following (dimensionless) effective Hamiltonian (see Eq. (S11))

$\begin{array}[]{ccc}\widetilde{\mathbf{H}}_{\rm
eff}=\mathbf{\widetilde{H}}_{0}-\frac{i}{2}\widetilde{\mathbf{W}}^{T}\widetilde{\mathbf{W}}=\left(\begin{array}[]{ccc}1-\epsilon-i\gamma_{e}+i\gamma_{0}&\gamma_{0}+\epsilon\\\
\gamma_{0}+\epsilon&1-\epsilon-i\gamma_{e}-i\gamma_{0}\end{array}\right);\end{array}$
(S14)

where the (dimensionless) linewidth broadening term $\gamma_{e}$ is introduced
via the matrix $\widetilde{\mathbf{W}}$, see Eq. (S11). A direct
diagonalization of $\widetilde{\mathbf{H}}_{\rm eff}$ gives us the eigenmodes
of the open system

$\widetilde{\omega}_{\pm}^{\rm
eff}(\epsilon;\gamma_{e})=1-\epsilon-i\gamma_{e}\pm\sqrt{2\gamma_{0}\epsilon+\epsilon^{2}};\quad\mathbf{\upsilon}_{\pm}^{\rm
eff}=\left(\frac{i\gamma_{0}\pm\sqrt{2\gamma_{0}\epsilon+\epsilon^{2}}}{\gamma_{0}+\epsilon},1\right)^{T}$
(S15)

which, similar to the isolated system (see Eq. (S13)), they also have an EP
degeneracy at $\epsilon=\epsilon_{EP}=0$. Notice that for $\gamma_{e}=0$, the
eigenmodes of Eq. (S15) coincide with the eigenmodes
$\widetilde{\omega}_{\pm}^{(0)}(\epsilon)$ of the Hamiltonian
$\mathbf{\widetilde{H}}_{0}(\epsilon)=\mathbf{\widetilde{H}}_{\rm
eff}(\epsilon;\gamma_{e}=0)$ describing the isolated ${\cal PT}$-symmetric
dimer in the case that a perturbation $\epsilon$ has modified the coupling
between the two resonators and their natural frequencies as discussed above.

The scattering matrix that describes the transport properties of our system is
evaluated from Eq. (S11) and takes the form

$\mathbf{S}(\widetilde{\omega})=-\mathbf{I}-i\widetilde{\mathbf{W}}\mathbf{G}(\widetilde{\omega})\widetilde{\mathbf{W}}^{T};\quad\mathbf{G}(\widetilde{\omega})=(\widetilde{\mathbf{H}}_{\rm
eff}-\widetilde{\omega}\mathbf{I})^{-1}.$ (S16)

where $\mathbf{I}$ is an identity matrix and $G(\widetilde{\omega})$ is the
Green’s function associated with the effective Hamiltonian
$\widetilde{\mathbf{H}}_{\rm eff}$ of Eq. (S14). A direct substitution of Eq.
(S14) into Eq. (S16) allows us to evaluate the individual Green’s function
matrix elements:

$\begin{array}[]{cc}G_{11}(\widetilde{\omega})=\frac{1-\epsilon-i\gamma_{e}-i\gamma_{0}-\widetilde{\omega}}{(-1-\widetilde{\omega}-i\gamma_{e})^{2}+2(1-\widetilde{\omega}-i\gamma_{e}-\gamma_{0})\epsilon},\\\
G_{12}(\widetilde{\omega})=G_{21}(\widetilde{\omega})=\frac{\gamma_{0}+\epsilon}{-(-1-\widetilde{\omega}-i\gamma_{e})^{2}+2(1-\widetilde{\omega}-i\gamma_{e}-\gamma_{0})\epsilon},\\\
G_{22}(\widetilde{\omega})=\frac{1-\epsilon-i\gamma_{e}+i\gamma_{0}-\widetilde{\omega}}{(-1-\widetilde{\omega}-i\gamma_{e})^{2}+2(1-\widetilde{\omega}-i\gamma_{e}-\gamma_{0})\epsilon}.\end{array}$
(S17)

and via Eq. (S16) the transmittance. The latter takes the form

$\mathcal{T}(\widetilde{\omega})\equiv|S_{21}(\widetilde{\omega})|^{2}=\frac{4\gamma^{2}_{e}(\gamma_{o}+\epsilon)^{2}}{(-1+\widetilde{\omega}-\epsilon)^{4}+2(\gamma^{2}_{e}-2\gamma_{0}\epsilon-\epsilon^{2})(-1+\widetilde{\omega}+\epsilon)^{2}+(\gamma^{2}_{e}+2\gamma_{0}\epsilon+\epsilon^{2})^{2}}.$
(S18)

which allow us to extract analytically the trajectories of the frequencies
associated with the transmission peaks. The later are the physical observables
that have been used in our sensing study (see main text) and they take the
form:

$\widetilde{\omega}_{\pm}=\left\\{\begin{array}[]{cc}1-\epsilon;&\epsilon\leq\epsilon_{\rm
TPD}\\\
1-\epsilon\pm\sqrt{2\gamma_{0}\epsilon+\epsilon^{2}-\gamma_{e}^{2}};&\epsilon\geq\epsilon_{\rm
TPD}\end{array}\right.$ (S19)

where $\epsilon_{\rm
TPD}=-\gamma_{0}+\sqrt{\gamma_{0}^{2}+\gamma_{e}^{2}}(\neq\epsilon_{\rm
EP}=0)$ is their coalescence point. At this perturbation strength we have that
$\widetilde{\omega}_{\pm}=\widetilde{\omega}_{\rm TPD}=1-\epsilon_{\rm
TPD}(\neq\widetilde{\omega}_{\rm EP}=1)$. From the above equation it is
straightforward to show that the transmission peak frequencies around the
transmission peak degeneracy (TPD) point, follow a sublinear Puiseux expansion
with a fractional $1/2$ power. This is a consequence of the EP degeneracy of
the eigenfrequencies of the underlying isolated system. It is crucial to point
out that while $\widetilde{\omega}_{\pm}$ follow closely the trajectories of
the corresponding eigenfrequencies $\widetilde{\omega}_{\pm}^{(0)}$, their
degeneracy occurs at a different parameter value i.e. $\epsilon_{\rm
TPD}>\epsilon_{\rm EP}$. As a result, at the TPD the bi-orthogonal basis of
the effective Hamiltonian Eq. (S14) does not collapse and, therefore, the
associated Petermann factor does not diverge. This divergence of the Petermann
factor was considered the source of the sensitivity limitations imposed to the
Brillouin ring laser gyroscope in Ref.[18]. Furthermore, the separation of
$\epsilon_{\rm TPD}$ and $\epsilon_{\rm EP}$ guarantees the analyticity of the
various physical observables in the proximity of TPD.

Furthermore, from Eqs. (S18,S19) we can calculate the scaling of the
transmittance peak $\mathcal{T}_{peak}$ as follows:

$\mathcal{T}_{peak}=\left\\{\begin{array}[]{cc}\frac{4\gamma_{e}^{2}(\gamma_{0}+\epsilon)^{2}}{\left(\gamma_{e}^{2}+2\gamma_{0}\epsilon+\epsilon^{2}\right)^{2}};&\epsilon\leq\epsilon_{\rm
TPD}\\\
\frac{(\gamma_{0}+\epsilon)^{2}}{2\gamma_{0}\epsilon+\epsilon^{2}};&\epsilon\geq\epsilon_{\rm
TPD}\end{array}\right.$ (S20)

This implies that at the point of transmittance peak coalescence
$\epsilon=\epsilon_{TPD}$, the transmittance peak approaches the value
$\mathcal{T}_{peak}(\epsilon_{TPD})=1+\left(\frac{\gamma_{0}}{\gamma_{e}}\right)^{2}$.

Finally, from Eq. (S16,S17) we have identified the lasing conditions of the
open system, as the frequency $\widetilde{\omega}_{L}$ and perturbation
$\epsilon_{L}$ values for which the scattering matrix diverges. The
corresponding values for lasing action are
$\epsilon_{L}=-\gamma_{0}+\sqrt{\gamma_{0}^{2}-\gamma_{e}^{2}}$, and
$\omega_{L}=1-\epsilon_{L}$. We have, therefore, that:
$\epsilon_{L}<\epsilon_{EP}=0<\epsilon_{\rm TPD}$.

A panorama of the transmittance spectrum versus the perturbation strength
$\epsilon$ is shown as density plots for various representative values of
$\gamma_{e}>\gamma_{0}$ in Figs. S2a-d. At the same figures we also report the
trajectory of the transmission peaks (green dashed lines) Eq. (S19), together
with the eigenmodes ${\cal R}e\left(\widetilde{\omega}_{\pm}^{\rm
eff}(\epsilon;\gamma_{e})\right)$ of the Hamiltonian
$\widetilde{\textbf{H}}_{\rm eff}$, see Eq. (S15). We point out that these
modes are the same with the eigenmodes of
$\widetilde{\textbf{H}}_{0}(\epsilon)$, and therefore we will not distinguish
them in this work.

##  S3 Quantifications of Sensitivity Enhancement and Non-Orthogonality of
modes

In order to quantify the non-orthogonal nature of the eigenmodes of our
system, and specifically its proximity to an EP, we are introducing the so-
called Petermann factor ($PF$) [18]. Specifically:

$PF(\epsilon)=\frac{1}{2}\left(1+\frac{Tr(\mathbf{H}_{tr}^{+}\mathbf{H}_{tr})}{\left|Tr(\mathbf{H}_{tr}^{2})\right|}\right)=\frac{(\gamma_{0}+\epsilon)^{2}}{2\gamma_{0}\epsilon+\epsilon^{2}},$
(S21)

where $\mathbf{H}_{tr}=\mathbf{H}_{\rm eff}-\frac{1}{2}Tr(\mathbf{H}_{\rm
eff})$ is a traceless part of the Hamiltonian $\mathbf{H}_{\rm eff}$. Notice
that the $PF(\epsilon)$ evaluated in Eq. (S21) is the same one associated with
the $\mathbf{H}_{\rm
eff}(\epsilon;\gamma_{e}=0)\equiv\mathbf{H}_{0}(\epsilon)$ and therefore we do
not distinguish these two below. We will show that the Petermann factor $PF$
is related with the so-called noise enhancement factor (NEF) of our ${\cal
PT}$-symmetric circuit.

Furthermore, in order to quantify the sensitivity enhancement associated with
the square-root degeneracy of the eigenfrequencies of the system ($SEF_{\rm
mode}$), we have intrioduced the sensitivity enhancement factor

$SEF_{mode}(\epsilon)\equiv\left|\frac{\partial(\Delta\widetilde{\omega}^{\rm
eff})}{2\partial\epsilon}\right|^{2}=\frac{(\gamma_{0}+\epsilon)^{2}}{2\gamma_{0}\epsilon+\epsilon^{2}},$
(S22)

where $\Delta\widetilde{\omega}^{\rm eff}=(\widetilde{\omega}_{+}^{\rm
eff}-\widetilde{\omega}_{-}^{\rm eff})$. The final equality in Eq. (S22)
requires to substitute the expressions for the resonant frequencies of the
$\widetilde{\mathbf{H}}_{\rm eff}$ from Eq. (S15). It turns out that both
$PF(\epsilon)$ and $SEF_{\rm mode}(\epsilon)$ are given by the same
expressions.

Since our sensing protocol utilizes the transmission peak frequencies
$\widetilde{\omega}_{\pm}$ for measuring the differential acceleration, it is
natural to introduce an associated sensitivity enhancement factor which is
given by:

$SEF(\epsilon)=\left|\frac{\partial(\Delta\widetilde{\omega})}{2\partial\epsilon}\right|^{2}=\frac{(\gamma_{0}+\epsilon)^{2}}{2\gamma_{0}\epsilon+\epsilon^{2}-\gamma_{e}^{2}}.$
(S23)

where
$\Delta\widetilde{\omega}=\widetilde{\omega}_{+}-\widetilde{\omega}_{-}$. From
the above expression we see that the $SEF$ diverges at $\epsilon_{\rm
TPD}=-\gamma_{0}+\sqrt{\gamma_{0}^{2}+\gamma_{e}^{2}}$ where the transmission
peaks coalesce.

These three quantities are reported in Fig. S2 for three representative values
of linewidth broadening $\gamma_{e}$ due to the coupling of the ${\cal
PT}$-symmetric circuit to the transmission lines.

##  S4  Thermal noise associated with the transmission line and the gain/loss
elements of the $\mathcal{PT}$-symmetric circuit

The transport properties our ${\cal PT}$-symmetric circuit (see Fig. S1) in
the presence of noise sources can be analyzed using a temporal CMT [1, 3]:

$\centering\Bigg{\\{}\begin{array}[]{ccc}\frac{d\mathbf{a}}{d\widetilde{t}}=-i\widetilde{\mathbf{H}}_{\rm
eff}\mathbf{a}+\mathbf{K}^{T}\mathbf{n}^{in}\\\
\mathbf{n}^{out}=\mathbf{Ka+Cn}^{in},\end{array}\@add@centering$ (S24)

where $\widetilde{t}=\omega_{0}\cdot t$ is the dimensionless time,
$\widetilde{\mathbf{H}}_{\rm eff}$ is the effective Hamiltonian given by Eq.
(S14), $\mathbf{K}$ is a matrix modeling the coupling between the system and
the various noise sources involved in the problem and $\mathbf{C}$ is a matrix
that describes a direct scattering process and has to satisfy the relation
$\mathbf{CK}^{*}=-\mathbf{K}$ [3]. In the scenario that we consider here, the
coupling matrices $\mathbf{K}$ and $\mathbf{C}$ have the form:

$\centering\mathbf{K}=\left(\begin{array}[]{ccc}\sqrt{2\gamma_{0}}&0\\\
0&\sqrt{2\gamma_{0}}\\\ \sqrt{2\gamma_{e}}&0\\\
0&\sqrt{2\gamma_{e}}\end{array}\right),\mathbf{C}=\left(\begin{array}[]{cccc}-1&0&0&0\\\
0&-1&0&0\\\ 0&0&-1&0\\\ 0&0&0&-1\end{array}\right).\@add@centering$ (S25)

Finally, in Eq. (S24) the state vector $\mathbf{a}$ describes the excitation
inside the scattering domain, and $\mathbf{n}^{in}(\mathbf{n}^{out})$ are
state vectors describing incoming (outgoing) excitations from the noise
sources to the system (from the system to the noise sources, including the
transmission lines).

We assume that there are four noise sources whose inputs are described by the
vector $\mathbf{n}^{in}$ $=(n_{1}^{in},n_{2}^{in},n_{{\rm
TL}_{1}}^{in},n_{{\rm TL}_{2}}^{in})^{T}$. The vector components
$n_{1/2}^{in}$ describe noise signals from the gain/loss sources which are
connected to the left/right resonator of the system respectively, while the
vector components $n_{{\rm TL}_{1}/{\rm TL}_{2}}^{in}$ describe incident noise
signals associated with the left/right transmission lines. The properties of
the noise sources are described by their correlations

$\displaystyle\langle n_{{\rm TL}_{l}}^{*in}(\widetilde{\omega})n_{{\rm
TL}_{m}}^{in}(\widetilde{\omega}^{\prime})\rangle$ $\displaystyle=$
$\displaystyle 4k_{b}T_{{\rm
TL}_{l}}Z\Delta_{f}\delta_{lm}\delta(\widetilde{\omega}-\widetilde{\omega}^{\prime})$
(S26) $\displaystyle\langle
n_{l}^{*in}(\widetilde{\omega})n_{m}^{in}(\widetilde{\omega}^{\prime})\rangle$
$\displaystyle=$ $\displaystyle
4k_{b}T_{l}R\Delta_{f}\delta_{lm}\delta(\widetilde{\omega}-\widetilde{\omega}^{\prime})$

where $k_{b}$ is the Boltzmann cnstant, $T_{{\rm TL}_{l}}$ is the temperature
of the noise reservoir associated with the transmission line $l$, $T_{l}$ is
the temperature of the noise source associated with the gain/loss elements at
the resonator $l$, $Z$ is the impedance of the transmission lines, $R$ is the
resistance responsible for the amplification/attenuation mechanisms associated
with the two modes of the system, and $\Delta_{f}$ \- is the frequency
bandwidth over which the signal at each frequency is measured (i.e.
$\Delta_{f}=1/\tau$ is inversely proportional to the sampling time $\tau$ over
which the signal is measured and averaged). Finally, $\delta_{ij}$ and
$\delta(x)$ are the Kronecker delta and the Dirac delta functions
respectively.

The output noise signal $\textbf{n}^{out}$ is evaluated from Eq. (S24) and
takes the form

$\mathbf{n}^{out}=\left(\begin{array}[]{cccc}n_{1}^{out}\\\ n_{2}^{out}\\\
n_{{\rm TL}_{1}}^{out}\\\ n_{{\rm
TL}_{2}}^{out}\end{array}\right)=-i\mathbf{KGK}^{T}+\mathbf{Cn}^{in},$ (S27)

where $\mathbf{G}$ is the Green’s function given by Eq. (S16). The vector
component $n_{l}^{out}$ describes the noise output detected at the
$l$-resonator of the system and $n_{{\rm TL}_{l}}^{out}$ describes the noise
output emitted at the $l-$th transmission line (see Fig. S1). Below we analyze
the effects of the output noise signal at the right transmission line $n_{{\rm
TL}_{2}}^{out}$, on the transmission measurements (we assume that the incident
signal is from the left transmission line). The right noise output is given by

$n_{{\rm TL}_{2}}^{out}=-n_{{\rm
TL}_{2}}^{in}-i\sqrt{2\gamma_{e}}\left(\sqrt{2\gamma_{e}}G_{21}n_{{\rm
TL}_{1}}^{in}+\sqrt{2\gamma_{e}}G_{22}n_{{\rm
TL}_{2}}^{in}+\sqrt{2\gamma_{0}}G_{21}n_{1}^{in}+\sqrt{2\gamma_{0}}G_{22}n_{2}^{in}\right),$
(S28)

where $G_{lm}$ are the Green’s function elements given by Eq. (S17). Armed
with this knowledge we are now able to evaluate the noise power associated
with measurements at the right transmission line:

$\mathcal{S}_{{\rm TL}_{2}}^{C}(\widetilde{\omega})=\langle\left|n_{{\rm
TL}_{2}}^{out}(\widetilde{\omega})\right|^{2}\rangle=\mathcal{S}_{{\rm
TL}_{2}}^{{\cal PT}}(\widetilde{\omega})+\mathcal{S}_{{\rm
TL}_{2}}^{TL}(\widetilde{\omega})$ (S29)

where the super-index $C$ indicates the cumulative circuit noise from the
gain/loss elements of the circuit (denoted with the super-index ${\cal PT}$)
and the ambient noise originating from the transmission lines (denoted with
the super-index $TL$). The contribution from the internal (external) noise
sources $n_{1,2}^{in}(n_{{\rm TL}_{1,2}}^{in})$ are denoted as
$\mathcal{S}_{{\rm TL}_{2}}^{{\cal PT}}(\mathcal{S}_{{\rm TL}_{2}}^{TL})$ and
are given by:

$\begin{array}[]{cccc}\mathcal{S}_{{\rm TL}_{2}}^{{\cal
PT}}=4k_{B}TR\Delta_{f}\left(4\gamma_{e}\gamma_{0}\Big{(}|G_{22}|^{2}+|G_{21}|^{2}\right)\Big{)}\\\
\mathcal{S}_{{\rm
TL}_{2}}^{TL}=4k_{B}TZ\Delta_{f}\Big{(}1-4\gamma_{e}\mathcal{I}m(G_{22})+4\gamma_{e}^{2}\left(|G_{22}|^{2}+|G_{21}|^{2}\right)\Big{)}.\end{array}$
(S30)

Since our sensing scheme relies on measuring the frequency splitting between
the transmission peaks $\Delta\widetilde{\omega}$, it is important the analyze
the noise power at frequencies $\widetilde{\omega}_{\pm}$, for
$\epsilon\geq\epsilon_{\rm TPD}$ (see Eq. (S19)), where its contribution will
be detrimental. In this respect, we evaluate at $\widetilde{\omega}_{\pm}$ the
Green’s function elements from Eq. (S17) and using Eq. (S30), we get that:

$\mathcal{S}_{{\rm TL}_{2}}^{{\cal
PT}}(\epsilon)=4k_{B}TR\Delta_{f}\left(\frac{2\gamma_{0}}{\gamma_{e}}\frac{(\gamma_{0}+\epsilon)^{2}-\gamma_{0}\gamma_{e}}{2\gamma_{0}\epsilon+\epsilon^{2}}\right);\quad\mathcal{S}_{{\rm
TL}_{2}}^{TL}(\epsilon)=4k_{B}TZ\Delta_{f}\left(\frac{(\gamma_{0}+\epsilon)^{2}+\gamma_{0}^{2}}{2\gamma_{0}\epsilon+\epsilon^{2}}\right)$
(S31)

which allow us to conclude that both $\mathcal{S}_{{\rm TL}_{2}}^{{\cal PT}}$
and $\mathcal{S}_{{\rm TL}_{2}}^{TL}$ saturate to a final value at
perturbations $\epsilon=\epsilon_{\rm TPD}$ where the transmission peaks
coalesce and the SEF diverges (see Eq. (S23)).

Next, we evaluate the noise enhancement factors $NEF_{{\rm TL}_{2}}^{{\cal
PT}}(\epsilon)$ and $NEF_{{\rm TL}_{2}}^{TL}(\epsilon)$, defined as the ratio
of the noise power at a perturbation $\epsilon$ to the noise power evaluated
at the $\epsilon\rightarrow\infty$ limit. In the domain
$\epsilon\geq\epsilon_{\rm TPD}$ we get

$NEF_{{\rm TL}_{2}}^{{\cal PT}}(\epsilon)\equiv\frac{\mathcal{S}_{{\rm
TL}_{2}}^{{\cal PT}}(\epsilon)}{\mathcal{S}_{{\rm TL}_{2}}^{{\cal
PT}}(\epsilon\rightarrow\infty)}=\frac{(\gamma_{0}+\epsilon)^{2}-\gamma_{e}\gamma_{0}}{2\gamma_{0}\epsilon+\epsilon^{2}};\quad
NEF_{{\rm TL}_{2}}^{TL}(\epsilon)\equiv\frac{\mathcal{S}_{{\rm
TL}_{2}}^{TL}(\epsilon)}{\mathcal{S}_{{\rm
TL}_{2}}^{TL}(\epsilon\rightarrow\infty)}=\frac{(\gamma_{0}+\epsilon)^{2}+\gamma_{0}^{2}}{2\gamma_{0}\epsilon+\epsilon^{2}}$
(S32)

where the asymptotic noise power values $\mathcal{S}_{{\rm TL}_{2}}^{{\cal
PT}}(\epsilon\rightarrow\infty)=\frac{2\gamma_{0}}{\gamma_{e}}\cdot
4k_{B}TR\Delta_{f}$ and $\mathcal{S}_{{\rm
TL}_{2}}^{TL}(\epsilon\rightarrow\infty)=4k_{B}TZ\Delta_{f}$ have been
evaluated directly from Eq. (S31).

For the specific experimental parameters used in our set-up ($R\approx 1200$
Ohm, $Z\approx 50$ Ohm, and $\gamma_{0}>\gamma_{e}$) we can make further
progress in the evaluation of the total noise enhancement factor.
Specifically, we have that $\mathcal{S}_{{\rm TL}_{2}}^{\cal
PT}\gg\mathcal{S}_{{\rm TL}_{2}}^{TL}$ (see Eq. (S31)), indicating that the
noise enhancement factor $NEF_{{\rm
TL}_{2}}^{C}(\epsilon)\equiv\left(\mathcal{S}_{{\rm TL}_{2}}^{\cal
PT}(\epsilon)+\mathcal{S}_{{\rm
TL}_{2}}^{TL}(\epsilon)\right)/\left(\mathcal{S}_{{\rm TL}_{2}}^{\cal
PT}(\epsilon\rightarrow\infty)+\mathcal{S}_{{\rm
TL}_{2}}^{TL}(\epsilon\rightarrow\infty)\right)$ can be approximated as

$NEF_{{\rm TL}_{2}}^{C}(\epsilon)\approx NEF_{{\rm TL}_{2}}^{\cal
PT}=\frac{2\gamma_{e}^{2}((\gamma_{0}-\gamma_{e})^{2}+(\gamma_{0}+\epsilon)^{2})}{(\gamma_{e}^{2}+2\gamma_{0}\epsilon+\epsilon^{2})^{2}}=PF(\epsilon)-\frac{\gamma_{e}\gamma_{0}}{2\gamma_{0}\epsilon+\epsilon^{2}};\quad\epsilon\geq\epsilon_{\rm
TPD}$ (S33)

where $PF(\epsilon)$ is the Petermann factor, see Eq. (S21). In fact, Eq.
(S33) can be further approximated at the weak coupling limit to the
transmission lines $\gamma_{e}\ll\gamma_{0}$. In this case, the noise
enhancement factor approaches the Petermann factor $PF$ (see Fig. S2), i.e.

$NEF_{{\rm TL}_{2}}^{C}(\epsilon\geq\epsilon_{\rm
TPD},\gamma_{0}\gg\gamma_{e})\approx
PF(\epsilon);\quad\epsilon\geq\epsilon_{\rm TPD}$ (S34)

For the completeness of the study, we also report the noise enhancement factor
in the parameter range $\epsilon<\epsilon_{\rm TPD}$. In this case we have
that the noise power (see Eq. (S30)) becomes:

$\displaystyle\mathcal{S}_{{\rm TL}_{2}}^{\cal PT}$ $\displaystyle=$
$\displaystyle
4k_{B}TR\Delta_{f}\left(\frac{4\gamma_{0}\gamma_{e}((\gamma_{0}-\gamma_{e})^{2}+(\gamma_{0}+\epsilon)^{2})}{(\gamma_{e}^{2}+2\gamma_{0}\epsilon+\epsilon^{2})^{2}}\right)$
(S35) $\displaystyle\mathcal{S}_{{\rm TL}_{2}}^{TL}$ $\displaystyle=$
$\displaystyle
4k_{B}TZ\Delta_{f}\left(1+\frac{4\gamma_{0}\gamma_{e}(2\gamma_{0}-\gamma_{e}+\epsilon)(\gamma_{0}+\epsilon)}{(\gamma_{e}^{2}+2\gamma_{0}\epsilon+\epsilon^{2})^{2}}\right)$

and consequently the corresponding noise enhancement factors are

$NEF_{{\rm TL}_{2}}^{\cal
PT}(\epsilon)=\frac{2\gamma_{e}^{2}((\gamma_{0}-\gamma_{e})^{2}+(\gamma_{0}+\epsilon)^{2})}{(\gamma_{e}^{2}+2\gamma_{0}\epsilon+\epsilon^{2})^{2}};\quad
NEF_{{\rm
TL}_{2}}^{TL}(\epsilon)=1+\frac{4\gamma_{0}\gamma_{e}(2\gamma_{0}-\gamma_{e}+\epsilon)(\gamma_{0}+\epsilon)}{(\gamma_{e}^{2}+2\gamma_{0}\epsilon+\epsilon^{2})^{2}}$
(S36)

The dependence of $NEF_{{\rm TL}_{2}}^{\cal PT}$ and $NEF_{{\rm TL}_{2}}^{TL}$
from the perturbation $\epsilon$ is shown in Fig. S2e-h for three typical
cases of $\gamma_{e}<\gamma_{0}$. In all cases, the noise enhancement factor
does not offsets the sensitivity enhancement factor $SEF$ at the proximity of
the $\epsilon_{\rm TPD}$ in agreement with the conclusions of our experimental
analysis (see main text).

Figure S2:  Spectral response of a $\mathcal{PT}$ symmetric dimer operating
below the lasing threshold.  a-d. Normalized transmission spectra as a
function of the coupling variation $\epsilon$ for various values of coupling
strength between the system and transmission lines $\gamma_{e}$. The blue
solid lines denote the eigen-frequencies of the system ${\cal
R}e\left(\omega_{\pm}^{\rm eff}\right)$ (see Eq. (S15)) and the green dotted
lines denote the frequencies of the transmission peaks $\omega_{\pm}$. e-h.
The corresponding peak transmittance ${\mathcal{T}}_{peak}$ (red dotted line),
$SEF$ (blue solid line), $NEF_{{\rm TL}_{2}}^{\cal PT}$ (dark red solid line)
and $NEF_{{\rm TL}_{2}}^{TL}$ (magenta solid line) as a function of the
coupling variation $\epsilon$

##  S5 Uncertainty in the measurement of transmission peaks splitting due to
cumulative circuit noise

To properly estimate the sensor performance, one needs to study the
amplification of the transmitted signal, in conjunction with the noise
generated from the various sources. First we realize that the uncertainty
$\sigma_{\Delta\widetilde{\omega}}^{C}$ in the measurement of the transmission
peaks splitting $\Delta\widetilde{\omega}$, due to the cumulative circuit
noise generated by the gain/loss elements of the circuit and the ambient noise
of the transmission lines, is proportional to the spectral width of the
transmission peaks $\Gamma$. Of course, the proportionality factor between
$\sigma_{\Delta\widetilde{\omega}}^{C}$ and $\Gamma$ depends on a number of
other parameters associated with the individual frequency sweeps (e.g.
frequency step, spectral bandwidth, number of points, individual frequency
bandwidth, period of individual frequency sweep, delay between the spectral
sweeps, etc) during which a single transmission spectrum measurement is
collected. Since we are not able to estimate all these contributions we are
proceeding by assuming that this proportionality factor is mainly dominated by
the noise produced by the gain/loss elements of the circuit and the ambient
noise of the transmission lines. A closer inspection of Eq. (S18) reveals that
the spectral width is

$\begin{array}[]{cccc}\Gamma(\epsilon\geq\epsilon_{TPD})&=&\sqrt{-\gamma_{e}^{2}+2\gamma_{0}\epsilon+\epsilon^{2}+2\gamma_{e}\sqrt{2\gamma_{0}\epsilon+\epsilon^{2}}}-\sqrt{-\gamma_{e}^{2}+2\gamma_{0}\epsilon+\epsilon^{2}}\\\
\Gamma(\epsilon\leq\epsilon_{TPD})&=&\sqrt{-\gamma_{e}^{2}+2\gamma_{0}\epsilon+\epsilon^{2}+\sqrt{2}\sqrt{\gamma_{e}^{4}+\epsilon^{2}(2\gamma_{0}+\epsilon)^{2}}}.\end{array}$
(S37)

The uncertainty in the measurement of the transmission peak splitting
$\Delta\widetilde{\omega}$, due to the generated cumulative circuit noise, is
then evaluated using the results from Eq. (S29):

$\left(\sigma^{C}_{\Delta\widetilde{\omega}}\right)^{2}\propto\Gamma^{2}(\epsilon)\frac{\mathcal{S}_{{\rm
TL}_{2}}^{\cal PT}+\mathcal{S}_{{\rm
TL}_{2}}^{TL}}{|S_{21}(\widetilde{\omega}=\widetilde{\omega}_{\rm
TPD})|^{2}|\mathbf{a}_{{\rm
TL}_{1}}^{in}|^{2}}=\left(\sigma_{\Delta\widetilde{\omega}}^{\cal
PT}(\epsilon)\right)^{2}+\left(\sigma_{\Delta\widetilde{\omega}}^{TL}(\epsilon)\right)^{2},$
(S38)

where we have introduced the partial uncertainties
$\sigma_{\Delta\widetilde{\omega}}^{\cal PT}(\epsilon)$ and
$\sigma_{\Delta\widetilde{\omega}}^{TL}(\epsilon)$ associated with the
gain/loss power noise from the circuit and with the ambient noise from the
transmission lines. The expression in the denominator in Eq. (S38) is the
output field intensity $\mathbf{a}_{L2}^{out}=S_{21}\mathbf{a}_{L1}^{in}$ at
the right transmission line, where we have assumed an input signal from the
left transmission line $\mathbf{a}_{L1}^{in}$ and $S_{21}$ is the element of
the scattering matrix given by Eq. (S18). Obviously the above expression Eq.
(S38) applies for the perturbation range $\epsilon\geq\epsilon_{\rm TPD}$
where the transmission spectrum demonstrates two peaks. In this domain the
noise powers $\mathcal{S}_{L2}^{\cal PT}$ and $\mathcal{S}_{L2}^{TL}$ are
given by Eq. (S31). Finally, the value of the transmission coefficient at the
position of the peaks in this parameter domain, is found from Eq. (S18) to be:

$|S_{21}(\widetilde{\omega}=\widetilde{\omega}_{\pm})|^{2}=\frac{(\gamma_{0}+\epsilon)^{2}}{2\gamma_{0}\epsilon+\epsilon^{2}};\quad\epsilon\geq\epsilon_{\rm
TPD}$ (S39)

which indicates that the signal transmitted from the left to the right
transmission line is enhanced in a same manner as a Petermann Factor (PF) of
the isolated system (see Eq. (S21) and Fig. S2). Furthermore, substituting in
Eq. (S38) the expressions for $\mathcal{S}_{L_{2}}^{\cal
PT},\mathcal{S}_{L_{2}}^{TL}$ from Eq. (S31) we get

$\begin{array}[]{cccc}\left(\sigma_{\Delta\widetilde{\omega}}^{\cal
PT}(\epsilon)\right)^{2}\propto\Gamma^{2}(\epsilon)\frac{4k_{B}TR\Delta_{f}}{|\mathbf{a}_{L1}^{in}|^{2}}\cdot\frac{2\gamma_{0}}{\gamma_{e}}\left(1-\frac{\gamma_{0}\gamma_{e}}{(\gamma_{0}+\epsilon)^{2}}\right)\\\
\left(\sigma_{\Delta\widetilde{\omega}}^{TL}(\epsilon)\right)^{2}\propto\Gamma^{2}(\epsilon)\frac{4k_{B}TZ\Delta_{f}}{|\mathbf{a}_{L1}^{in}|^{2}}\cdot\left(1+\frac{\gamma_{0}^{2}}{(\gamma_{0}+\epsilon)^{2}}\right).\end{array}$
(S40)

One can also introduce the uncertainty enhancement factor which describes the
degree of enhancement in the uncertainty measurements of the transmission peak
frequency splittings with respect to system configurations away from the TPD.
We have

$\begin{array}[]{cccc}\left(\Sigma_{\Delta\widetilde{\omega}}^{\cal
PT}(\epsilon)\right)^{2}=\left(\frac{\sigma_{\Delta\widetilde{\omega}}^{\cal
PT}(\epsilon)}{\sigma_{\Delta\widetilde{\omega}}^{\cal
PT}(\epsilon\rightarrow\infty)}\right)^{2}\propto\left(\frac{\Gamma^{2}(\epsilon)}{\gamma^{2}_{e}}\right)\left(1-\frac{\gamma_{0}\gamma_{e}}{(\gamma_{0}+\epsilon)^{2}}\right)\\\
\left(\Sigma_{\Delta\widetilde{\omega}}^{TL}(\epsilon)\right)^{2}=\left(\frac{\sigma_{\Delta\widetilde{\omega}}^{TL}(\epsilon)}{\sigma_{\Delta\widetilde{\omega}}^{TL}(\epsilon\rightarrow\infty)}\right)^{2}\propto\left(\frac{\Gamma^{2}(\epsilon)}{\gamma^{2}_{e}}\right)\left(1+\frac{\gamma_{0}^{2}}{(\gamma_{0}+\epsilon)^{2}}\right).\end{array}$
(S41)

where the “asymptotic” uncertainties have been calculated from Eq. (S40) to be
$\left(\sigma_{\Delta\widetilde{\omega}}^{\cal
PT}(\epsilon\rightarrow\infty)\right)^{2}\propto\gamma_{e}^{2}\frac{4k_{B}TR\Delta_{f}}{|\mathbf{a}_{L_{1}}^{in}|^{2}}\cdot\frac{2\gamma_{0}}{\gamma_{e}}$
and
$\left(\sigma_{\Delta\widetilde{\omega}}^{TL}(\epsilon\rightarrow\infty)\right)^{2}\propto\gamma_{e}^{2}\frac{4k_{B}TZ\Delta_{f}}{|\mathbf{a}_{L_{1}}^{in}|^{2}}$.

Since our experimental platform operates under weak coupling conditions
$\gamma_{e}\leq\gamma_{0}\ll 1$ (units of $\omega_{0}$), we deduce from the
above equation that both enhancement factors
$\Sigma_{\Delta\widetilde{\omega}}^{\cal PT}(\epsilon)$ and
$\Sigma_{\Delta\widetilde{\omega}}^{TL}(\epsilon)$ experience small variations
in the proximity of $\epsilon\approx\epsilon_{\rm TPD}$, see Fig. S3a. We
conclude, therefore, that the noise enhancement of the proposed platform
leaves unaffected the uncertainty of the measurement of the transmission peak
frequency splitting in the vicinity of $\epsilon=\epsilon_{\rm TPD}$ and does
not offset the sensitivity enhancement, which diverges at the same point, see
Eq. (S23). This can be contrasted with the results of Ref. [17, 18] where they
have found that the improved responsivity of a laser gyroscope operating near
an EP is precisely compensated by increased laser noise.

##  S6 Uncertainty in the measurement of transmission peak splitting due to
fluctuations of the coupling strength between the two modes

Apart from the noise originated from the transmission lines and the gain/loss
elements of the electronic circuit, our sensing platform suffers also from
noise associated with fluctuations at the variation $\epsilon$ of the coupling
constant $\kappa$ between the two resonators. These fluctuations are
characterized by an uncertainty $\sigma_{\epsilon}^{\kappa}$ in the coupling
variation $\epsilon$. Their physical origin ranges from the thermal motion of
the test-mass (which is used to perturb the coupling strength – coupling
capacitance in our circuit) when external acceleration is applied, to
fluctuations of the coupling capacitance due to thermal expansion of the glass
plates used in our platform, or to variations in their dielectric properties.
This uncertainty will be enhanced by the SEF given by Eq. (S23), thus
resulting in a cumulative coupling uncertainty
$\left(\sigma_{\Delta\widetilde{\omega}}^{\kappa}(\epsilon)\right)^{2}=4\cdot
SEF(\epsilon)\left(\sigma_{\epsilon}^{\kappa}\right)^{2}$. We infer,
therefore, that the uncertainty due to fluctuations of the coupling strength
sets the floor level for our measurements. At the same time, we are pointing
out, that this type of noise is not dictated by the presence of the
exceptional points. This is not the case with the current optical EP-based
lasing platforms whose noise-floor is inherently associated with the formation
of EP itself.

Next we estimate the uncertainty $\sigma_{\epsilon}^{\kappa}$ pertaining to
the above analysis. It can be decomposed to the thermal noise
$\sigma_{\epsilon,th}^{\kappa}$ associated with the Brownian motion of the
test-mass and the remaining noise- sources. We will refer to these sources as
added noise sources and they will be characterized by
$\sigma_{\epsilon,add}^{\kappa}$. Typically, these noise sources prohibit our
system from reaching the floor noise level which is dictated by Brownian
thermal noise. The uncertainty associated with the latter one is:

$\left(\sigma_{\epsilon,th}^{\kappa}\right)^{2}=\left(\frac{\partial{\epsilon}}{\partial{a}}\right)^{2}\cdot\alpha^{2}_{th}\Delta_{f},$
(S42)

where $a$ is the applied acceleration, and
$\alpha_{th}=\sqrt{\frac{4k_{b}T\omega_{n}}{mQ}}$ is the thermal noise
equivalent acceleration (see main text), where $\omega_{n}=2\pi\cdot f_{n}$
and $f_{n}$ is the natural frequency of the mechanical degree of freedom
(spring-mass) of our platform, $m$ is the mass of the spring-mass and $Q$\- is
the quality factor of the mechanical spring-mass resonator. The sensitivity
parameter $\partial\epsilon/\partial a=Const$ that appears in Eq. (S42)
describes the sensitivity of the coupling strength to the applied acceleration
and is determined by the design of the electronic circuit and the spring-mass.

##  S7 Cumulative characterization of uncertainty in the presence of noise
sources

We are now ready to describe in a compact manner all the above noise sources.
The cumulative uncertainty is given as a sum of the partial uncertainties i.e.
$\sigma_{\Delta\widetilde{\omega}}^{2}=\left(\sigma_{\Delta\widetilde{\omega}}^{C}\right)^{2}+\left(\sigma_{\Delta\widetilde{\omega}}^{\kappa}\right)^{2}$.
Using Eqs. (S23,S38,S40,S42) we have

$\begin{array}[]{cccc}\sigma_{\Delta\widetilde{\omega}}^{2}(\epsilon)\propto
4\Delta_{f}\left(\frac{k_{B}T\Gamma^{2}(\epsilon)}{|\mathbf{a}_{L1}^{in}|^{2}}\right)\left(2R\frac{\gamma_{0}}{\gamma_{e}}\cdot\left(1-\frac{\gamma_{0}\gamma_{e}}{(\gamma_{0}+\epsilon)^{2}}\right)+Z\cdot\left(1+\frac{\gamma_{0}^{2}}{(\gamma_{0}+\epsilon)^{2}}\right)\right)+\\\
+4\Delta_{f}\cdot\frac{(\gamma_{0}+\epsilon)^{2}}{2\gamma_{0}\epsilon+\epsilon^{2}-\gamma_{e}^{2}}\left(\frac{4k_{b}T\omega_{n}}{mQ}\left(\frac{\partial{\epsilon}}{\partial{a}}\right)^{2}+\left(\sigma_{\epsilon,add}^{\kappa}\right)^{2}\right)\end{array}$
(S43)

At this point, it is possible to estimate the noise equivalent acceleration
$a_{NEA}^{2}(\epsilon)={\frac{1}{\Delta_{f}\cdot\chi^{2}(\epsilon)}}\sigma^{2}_{\Delta\widetilde{\omega}}(\epsilon)$,
where the sensitivity of the frequency splitting between the transmission
peaks with respect to an applied acceleration $a$ is
$\chi^{2}(\epsilon)=4SEF(\epsilon)\cdot(\partial{\epsilon}/\partial{a})^{2}$.
We have

$\begin{array}[]{cccc}\alpha_{NEA}^{2}(\epsilon)\propto\left(\frac{k_{B}T\Gamma^{2}(\epsilon)}{|\mathbf{a}_{L1}^{in}|^{2}}\right)\cdot\left(\frac{\partial{\epsilon}}{\partial{a}}\right)^{-2}\cdot\left(2R\frac{\gamma_{0}}{\gamma_{e}}\cdot\left(\frac{(\gamma_{0}+\epsilon)^{2}-\gamma_{0}\gamma_{e}}{\left(2\gamma_{0}\epsilon+\epsilon^{2}-\gamma_{e}^{2}\right)^{-1}}\right)+Z\cdot\left(\frac{(\gamma_{0}+\epsilon)^{2}+\gamma_{0}^{2}}{\left(2\gamma_{0}\epsilon+\epsilon^{2}-\gamma_{e}^{2}\right)^{-1}}\right)\right)+\\\
+\frac{4k_{b}T\omega_{n}}{mQ}+\left(\sigma_{\epsilon,add}^{\kappa}\right)^{2}\left(\frac{\partial{\epsilon}}{\partial{a}}\right)^{-2}\end{array}$
(S44)

where the first term describes $\alpha_{\cal PT}^{2}$, the second is
$\alpha_{TL}^{2}$, the third is $\alpha_{th}^{2}$ and the last term is
associated with $\alpha_{add}^{2}$ (see Eq. (5) of the main text).

From Eq. (S44) we can easily deduce that when $\epsilon=\epsilon_{\rm
TPD}=-\gamma_{0}+\sqrt{\gamma_{0}^{2}+\gamma_{e}^{2}}$ the noise contributions
$\alpha_{\cal PT}^{2}+\alpha_{TL}^{2}$ to $\alpha_{NEA}$ go to zero (see blue
dashed line on Fig. S3b). Such behavior indicates that the proposed sensing
platform is capable of mitigating these noise effects at the vicinity of
$\epsilon=\epsilon_{\rm TPD}$. Consequently, at the vicinity of
$\epsilon=\epsilon_{\rm TPD}$, the noise equivalent acceleration $a_{NEA}$ is
suppressed (see dark red line on Fig. S3b) to the value determined by the
joint contributions of the thermal noise equivalent acceleration $\alpha_{th}$
and the added noise $\alpha_{add}^{2}$ (see green dotted line on Fig. S3b).
Specifically, we will have:

$\alpha_{NEA}^{2}(\epsilon\rightarrow\epsilon_{\rm
TPD})=\alpha_{th}^{2}+\alpha_{add}^{2}=\frac{4k_{b}T\omega_{n}}{mQ}+\left(\sigma_{\epsilon,add}^{\kappa}\right)^{2}\left(\frac{\partial{\epsilon}}{\partial{a}}\right)^{-2}.$
(S45)

Figure S3:  Noise at the vicinity of $\epsilon=\epsilon_{\rm TPD}$.  a.
Enhancement of the uncertainty of the measured transmission peaks splitting at
the vicinity of $\epsilon=\epsilon_{\rm TPD}$ due to internal noise sources
$\left(\Sigma_{\Delta\widetilde{\omega}}^{\cal PT}\right)^{2}$ (light red
solid line) and external noise sources
$\left(\Sigma_{\Delta\widetilde{\omega}}^{TL}\right)^{2}$ (dark red solid
line). The sensitivity enhancement factor is shown by blue dashed solid line.
b. Suppression of the uncertainty in the measured acceleration due to the
internal and external noise sources (dashed blue line). The red solid line is
the spectral density of the noise variance in the measured acceleration (noise
equivalent acceleration $\alpha_{NEA}^{2}$) as a function of the proximity to
the $\epsilon_{\rm TPD}$. The spectral density of the variance in the measured
acceleration due to the internal and the external noise sources ($\alpha_{\cal
PT}^{2}+\alpha_{TL}^{2}$) is shown by a blue dashed line, while the sum of the
thermal noise $\alpha_{th}^{2}$ and the added noise $\alpha_{add}^{2}$ is
shown by a green dotted line. In the calculations we have assumed that
$f_{0}=2.68$MHz, $\gamma=0.16$ MHz , $\gamma=0.02$ MHz,
$\frac{\partial\epsilon}{\partial a}=0.0082$ MHz/g, $R=26\cdot Z$ and that
$\left(\alpha_{th}^{2}+\alpha_{add}^{2}\right)\cdot(\frac{\partial\epsilon}{\partial
a})^{2}=0.0002\cdot\frac{4k_{B}TZ}{|\mathbf{a}_{{\rm TL}_{1}}^{in}|^{2}}$

## References

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