# Parametrically enhancing sensor sensitivity at an exceptional point

P. Djorwé<EMAIL_ADDRESS>Department of Physics, Faculty of Science,
University of Ngaoundere, P.O. Box 454, Ngaoundere, Cameroon  Stellenbosch
Institute for Advanced Study (STIAS), Wallenberg Research Centre at
Stellenbosch University, Stellenbosch 7600, South Africa  M. Asjad
<EMAIL_ADDRESS>Mathematics Department, Khalifa University of Science and
Technology, 127788, Abu Dhabi, United Arab Emirates  Y. Pennec  Institut
d’Electronique, de Microélectronique et Nanotechnologie, UMR CNRS 8520
Université de Lille, Faculté des sciences et technologies, 59652 Villeneuve
d’Ascq France  D. Dutykh  Mathematics Department, Khalifa University of
Science and Technology, 127788, Abu Dhabi, United Arab Emirates  Causal
Dynamics Pty Ltd, Perth, Australia  B. Djafari-Rouhani  Institut
d’Electronique, de Microélectronique et Nanotechnologie, UMR CNRS 8520
Université de Lille, Faculté des sciences et technologies, 59652 Villeneuve
d’Ascq France

###### Abstract

We propose a scheme to enhance the sensitivity of Non-Hermitian optomechanical
mass-sensors. The benchmark system consists of two coupled optomechanical
systems where the mechanical resonators are mechanically coupled. The optical
cavities are driven either by a blue or red detuned laser to produce gain and
loss, respectively. Moreover, the mechanical resonators are parametrically
driven through the modulation of their spring constant. For a specific
strength of the optical driving field and without parametric driving, the
system features an Exceptional Point (EP). Any perturbation to the mechanical
frequency (dissipation) induces a splitting (shifting) of the EP, which scales
as the square root of the perturbation strength, resulting in a sensitivity-
factor enhancement compared with conventional optomechanical sensors. The
sensitivity enhancement induced by the shifting scenario is weak as compared
to the one based on the splitting phenomenon. By switching on parametric
driving, the sensitivity of both sensing schemes is greatly improved, yielding
to a better performance of the sensor. We have also confirmed these results
through an analysis of the output spectra and the transmissions of the optical
cavities. In addition to enhancing EP sensitivity, our scheme also reveals
nonlinear effects on sensing under splitting and shifting scenarios. This work
sheds light on new mechanisms of enhancing the sensitivity of Non-Hermitian
mass sensors, paving a way to improve sensors performance for better
nanoparticles or pollutants detection, and for water treatment.

Optomechanics, exceptional point, sensor

###### pacs:

42.50.Wk, 42.50.Lc, 05.45.Xt, 05.45.Gg

##  I Introduction

Exceptional Points (EPs), Non-Hermitian degeneracy, are known in physical
systems for their interesting counter-intuitive features and intriguing
effects. Among the fascinating properties of EPs, we can mention
nonreciprocity Thomas et al. (2016); Lau and Clerk (2018), topological
transport phenomena Xu et al. (2023a, b), stopping light Goldzak et al.
(2018), loss-induced suppression and revival of lasing, pump-induced lasing
death and unidirectional invisibility (see Peng et al. (2014) and references
therein). Recently, EPs have been used to enhance sensor sensitivity and
versatile physical systems were proposed for this purpose including
optical/photonic Hodaei et al. (2017); Chen et al. (2017), microwave Nada et
al. (2018), plasmonics Park et al. (2020); Jiang et al. (2022), and
mechanic/acoustic Rosa et al. (2021); Cai et al. (2022). Coupled
optomechanical systems, which result from a coupling between an
electromagnetic field and a mechanical object Djorwé et al. (2013a);
Aspelmeyer et al. (2014), have also been used to engineer EPs owing to their
ability to induce gain and losses depending on their sideband pumping
mechanism. Exceptional points engineering in optomechanics have led to a
number of interesting phenomena such as mass sensing improvement Djorwe et al.
(2019a); Tchounda et al. (2023), topological energy transfer Chen et al.
(2023); Xu et al. (2016), and nonlinearly induced collective phenomena Djorwe
et al. (2018); Djorwé et al. (2020) among others.

Exceptional point sensing is an approach which provides better sensitivity
compared to conventional sensing schemes where the perturbation induces a
shift in the frequency. The resulting shift/signal is generally weak and needs
a pre-amplification process in order to be detected and analyzed Zhu et al.
(2021). For the Non-Hermitian sensors, however, the signal is self-amplified
owing to the abrupt singularity and the topological feature at the EP, leading
to a giant sensitivity enhancement without a pre-amplification requirement of
the detected signal. This new sensing scheme has recently attracted attention
as a means to enhance the responsiveness of sensors, and has started to be
practically implemented. For instance, exceptional point sensing has been used
to monitor internal physiological states via wireless interrogation techniques
Chen and El-Ganainy (2019), and for an efficient wireless power transfer
Assawaworrarit et al. (2017); Song et al. (2021). Beside these interesting
sensing applications probed at EPs, the implementation of EP-sensors is still
in its infancy stage and it seems to rise up scepticism regarding noise
amplification, which induces a degradation of the sensor’s performance
Anderson et al. (2023). This noise issue can be of a fundamental nature owing
to the eigenbasis collapse or of a technical nature related to the
amplification mechanisms used for the realization of EPs. This question has
been a bottleneck of EP-sensors, however, appropriate techniques have started
to be used to handle and suppress this noise issue Kononchuk et al. (2022);
Suntharalingam et al. (2023); Tuxbury et al. (2022). Therefore, tremendous
research activities are still going on to tackle and to get rid of this
limitation Chen and Zhao (2020); Liu et al. (2021); Wu et al. (2023);
Khanbekyan (2023), and this includes nonlinear phenomena Li et al. (2023); Bai
et al. (2022).

The current work introduces a new mechanism to enhance the sensitivity of EP-
sensors, which is based on a mechanical parametric driving effect. Our
proposal consists of two coupled optomechanical systems, where the mechanical
resonators are mechanically coupled. The parametric driving comes from a
nonlinear modulation of the spring constants of the mechanical resonators
Lemonde et al. (2016); Bothner et al. (2020). We investigated two sensing
schemes. The first one is based on the splitting Djorwe et al. (2019a) of the
EP and the second is related to the shifting Wang et al. (2019); Mao et al.
(2023) of the EP. When parametric driving is switched on, we found that i) the
sensitivity is greatly enhanced, ii) the EP is shifted towards low
frequencies, and iii) the strength of the optical driving field corresponding
to the EP increases. Moreover, it has been observed that the overall
sensitivity enhancement in the shifting scenario is weak compared to the
enhancement captured with the splitting scheme. Furthermore, the output
spectra and transmissions of the optical cavities have been investigated under
some system parameters. The effect of perturbations, the parametric driving
impact and thermal noise on the EP were analyzed via the spectra. While the
effect of the other parameters is confirmed as aforementioned, it has been
found that thermal noise amplifies the output signal, leading to better
sensitivity. This work shows how sensing based on the splitting of the EP
outperforms sensors operating in a scenario where the EP is shifted.
Furthermore, it also paves a way towards improvement of sensing through
nonlinear effects in optomechanics Djorwé et al. (2013b); Leijssen et al.
(2017); Djorwe et al. (2019b).

The rest of the work is organized as follows. In Section II we describe the
model and derive the related dynamical equations. The sensitivity schemes
based on splitting and shifting of the EP are presented in Section III. The
transmissions and output spectra of the optical cavities are analyzed in
Section IV, while Section V concludes our work.

##  II Model and dynamical equations

We propose a two-cavity system with mechanically coupled vibrating mechanical
resonators, as shown in /cref[fig:Fig1]. One cavity is driven by a red (blue)
driving field that creates loss (gain) in the system. In addition, the
mechanical resonators are parametrically driven for sensitivity enhancement
prospect as it will be pointed out later on. The Hamiltonian of this system
($\hbar=1$) is given by,

Figure 1: Sketch of our benchmark system. Cavity $1$ ($2$) is driven with red
(blue) electromagnetic field to generate losses (gain). The sprint constants
of the involved mechanical resonators are modulated to induce parametric
driving Lemonde et al. (2016); Bothner et al. (2020).

$H=H_{\rm{0M}}+H_{\rm{PA}}+H_{\rm{int}}+H_{\rm{drive}},$ (1)

where,

$\displaystyle H_{\rm{0M}}$ $\displaystyle=$
$\displaystyle\sum_{j}\omega_{j}b^{\dagger}_{j}b_{j}+\omega_{c}^{j}a^{\dagger}_{j}a_{j}-g_{j}a^{\dagger}_{j}a_{j}(b^{\dagger}_{j}+b_{j}),$
(2) $\displaystyle H_{\rm{PA}}$ $\displaystyle=$
$\displaystyle\sum_{j}\frac{\chi_{j}}{2}(b^{\dagger
2}_{j}e^{-i\phi_{d}}+b^{2}_{j}e^{-i\phi_{d}})$ (3) $\displaystyle
H_{\rm{int}}$ $\displaystyle=$ $\displaystyle-
J_{m}(b_{1}b_{2}^{\dagger}+b_{1}^{\dagger}b_{2}),$ (4) $\displaystyle
H_{\rm{drive}}$ $\displaystyle=$ $\displaystyle
i\sum_{j}(\rm{E_{j}}a^{\dagger}_{j}e^{-i\omega^{j}_{p}t}-\rm{E^{*}_{j}}a_{j}e^{i\omega^{j}_{p}t}).$
(5)

In Eq. 1, $H_{\rm{0M}}$, $H_{\rm{PA}}$, $H_{\rm{int}}$, and $H_{\rm{drive}}$
are the Hamiltonians describing the optomechanical cavities, the mechanical
parametric amplification (MPA), the mechanical interactions and the driving
fields, respectively. The annihilation (creation) operators related to the
mechanical resonators and optical fields are $b_{j}$ ($b^{\dagger}_{j}$) and
$a_{j}$ ($a_{j}^{\dagger}$), respectively. The other parameters are the
mechanical ($\omega_{j}$) and cavity ($\omega_{c}^{j}$) frequencies, the
optomechanical coupling ($g_{j}$), the phonon hopping coupling rate ($J_{m}$)
and the driving amplitude ($\rm{E_{j}}$). The parametric driving amplitude
$\chi_{j}$ comes from the modulation of the spring constants of the mechanical
resonators at frequency $2\omega_{d}$ and phase $\phi_{d}$ Lemonde et al.
(2016); Bothner et al. (2020). For the sake of simplicity and without loss of
generality, we assume that the optical cavities are driven by the same amount
of driving strength and are degenerated, so that from now on we consider
$\rm{E_{1}}=\rm{E_{2}}=\rm{E}$, $\omega^{1}_{c}=\omega^{2}_{c}=\omega_{c}$,
and $g_{1}=g_{2}=g$. Throughout the work, we also assume $\phi_{d}=0$ and
$\chi_{1}=\chi_{2}=\chi$. In the rotating wave approximation (RWA) and in the
frame rotating at $\omega_{p}^{j}+\omega_{d}$, where $\omega_{p}^{j}$ is the
$j^{th}$ electromagnetic driving frequency, the above Hamiltonian yields:

$\displaystyle H_{\rm{0M}}$
$\displaystyle=\sum_{j}\tilde{\omega}_{j}b^{\dagger}_{j}b_{j}-\Delta_{j}a^{\dagger}_{j}a_{j}-ga^{\dagger}_{j}a_{j}(b^{\dagger}_{j}+b_{j})$
(6a) $\displaystyle H_{\rm{PA}}$ $\displaystyle=\frac{\chi}{2}(b^{\dagger
2}_{j}+b^{2}_{j}),$ (6b) $\displaystyle H_{\rm{int}}$
$\displaystyle=-J_{m}(b_{1}b_{2}^{\dagger}+b_{1}^{\dagger}b_{2}),$ (6c)
$\displaystyle H_{\rm{drive}}$
$\displaystyle=i\sum_{j}(\rm{E}a^{\dagger}_{j}-\rm{E}^{*}a_{j}),$ (6d)

where $\tilde{\omega}_{j}=\omega_{j}-\omega_{d}$ and
$\Delta_{j}=\omega_{p}-\omega_{c}$. In order to diagonalize the quadratic term
in the above Hamiltonian, we introduce the squeezing transformation
$S(r_{j})=\rm{exp}[r_{j}(b^{2}_{j}-b^{\dagger 2}_{j})]$ where the squeezing
parameter is defined as
$r_{j}=\frac{1}{4}\log{\frac{\tilde{\omega}_{j}+\chi}{\omega_{j}-\chi}}$. By
using the Bogoliubov transformation
$b_{s_{j}}=b_{j}\cosh(r_{j})+b^{\dagger}_{j}\sinh(r_{j})$, and invoking the
Heisenberg equation, one gets the following Quantum Langeving Equations
(QLEs),

$\displaystyle\dot{a_{j}}$
$\displaystyle=(i[\Delta_{j}+\tilde{g}(b_{s_{j}}+b^{\dagger}_{s_{j}})]-\frac{\kappa}{2})a_{j}+\sqrt{\kappa}\alpha^{in}_{j}+\sqrt{\kappa}a^{in}_{j},$
(7a) $\displaystyle\dot{b_{s_{j}}}$
$\displaystyle=-(i\Delta_{m}^{j}+\frac{\gamma_{m}}{2})b_{s_{j}}+i\tilde{J}_{m}b_{3-j}+i\tilde{g}a^{\dagger}_{j}a_{j}+\sqrt{\gamma_{m}}\beta^{in}_{j}.$
(7b)

In Eq. 7, the driving amplitude $\rm{E}$ has been replaced by
$\sqrt{\kappa}\alpha^{in}$ with
$\alpha^{in}=\sqrt{\frac{P_{in}}{\hbar\omega_{p}}}$, where $P_{in}$ is the
input power. The optical and mechanical dissipation rates are captured by
$\kappa_{1}=\kappa_{2}=\kappa$ and $\gamma_{1}=\gamma_{2}=\gamma_{m}$,
respectively. Here $a^{in}_{j}$ and $\beta^{in}_{j}$ are input noise operators
of the j-th cavity and mechanical modes with zero mean values.

The nonlinear set of equations displayed in Eq. 7 can be linearized by
splitting the operators in terms of their mean values ($c-$numbers) and some
amount of quantum fluctuations as $a_{j}=\alpha_{j}+\delta\alpha_{j}$ and
$b_{s_{j}}=\beta_{j}+\delta\beta_{j}$, where $\alpha_{j}=\langle a_{j}\rangle$
and $\beta_{j}=\langle b_{s_{j}}\rangle$. This procedure leads to classical
mean equations,

$\displaystyle\dot{\alpha_{j}}$
$\displaystyle=(i\tilde{\Delta}_{j}-\frac{\kappa}{2})\alpha_{j}+\sqrt{\kappa}\alpha^{in},$
(8a) $\displaystyle\dot{\beta_{j}}$
$\displaystyle=-(i\Delta_{m}^{j}+\frac{\gamma_{m}}{2})\beta_{j}+i\tilde{J}_{m}\beta_{3-j}+i\tilde{g}|\alpha_{j}|^{2},$
(8b)

and to the fluctuations equations,

$\displaystyle\delta\dot{\alpha_{j}}$
$\displaystyle=(i\tilde{\Delta}_{j}-\frac{\kappa}{2})\delta\alpha_{j}+i\tilde{G_{j}}(\delta\beta_{j}+\delta\beta^{\dagger}_{j})+\sqrt{\kappa}\delta\alpha^{in}_{j},$
(9a) $\displaystyle\delta\dot{\beta_{j}}$
$\displaystyle=-(i\Delta_{m}^{j}+\frac{\gamma_{m}}{2})\delta\beta_{j}+i\rm{\tilde{J}_{m}\delta\beta_{3-j}}+i\tilde{G_{j}}(\delta\alpha^{\dagger}_{j}+\delta\alpha_{j})+\sqrt{\gamma_{m}}\delta\beta^{in}_{j},$
(9b)

where $\tilde{\Delta}_{j}=\Delta_{j}+2\tilde{g}\Re{\beta_{j}}$,
$\tilde{G_{j}}=\tilde{g}\alpha_{j}$, and we have assumed $\alpha_{j}$ a real
number.

In order to get an effective mechanical model of our proposal based on Eq. 9,
we need to trace out the optical equations. For that purpose, we introduce the
following slowly varying operators with tildes,
$\delta\alpha_{j}=\delta\tilde{\alpha_{j}}e^{i\tilde{\Delta}_{j}t}$,
$\delta\alpha^{in}=\delta\tilde{\alpha^{in}}e^{i\tilde{\Delta}_{j}t}$, and
$\delta\beta_{j}=\delta\tilde{\beta_{j}}e^{-i\Delta_{m}^{j}t}$. By using these
slowly varying operators in Eq. 9, and by neglecting the terms oscillating at
higher frequencies, the set of dynamical equations for the fluctuations
yields,

$\displaystyle\dot{\delta\tilde{\alpha_{1}}}$
$\displaystyle=-\frac{\kappa}{2}\delta\tilde{\alpha_{1}}+i\tilde{G_{1}}\delta\tilde{\beta_{1}}+\sqrt{\kappa}\delta\tilde{\alpha}^{in}_{1},$
(10a) $\displaystyle\dot{\delta\tilde{\alpha_{2}}}$
$\displaystyle=-\frac{\kappa}{2}\delta\tilde{\alpha_{2}}+i\tilde{G_{2}}\delta\tilde{\beta_{2}}^{\dagger}+\sqrt{\kappa}\delta\tilde{\alpha}^{in}_{2},$
(10b) $\displaystyle\dot{\delta\tilde{\beta_{1}}}$
$\displaystyle=-\frac{\gamma_{m}}{2}\delta\tilde{\beta_{1}}+i\tilde{J}_{m}\delta\tilde{\beta}_{2}+i\tilde{G_{1}}\delta\tilde{\alpha_{1}}+\sqrt{\gamma_{m}}\delta\tilde{\beta}^{in}_{1},$
(10c) $\displaystyle\dot{\delta\tilde{\beta_{2}}}$
$\displaystyle=-\frac{\gamma_{m}}{2}\delta\tilde{\beta_{2}}+i\tilde{J}_{m}\delta\tilde{\beta}_{1}+i\tilde{G_{2}}\delta\tilde{\alpha_{2}}^{\dagger}+\sqrt{\gamma_{m}}\delta\tilde{\beta}^{in}_{2}.$
(10d)

The intracavity fields ($\delta\tilde{\alpha_{j}}$) in Eq. 10 can be
integrated out (see Appendix A), and after some arrangements one obtains the
following effective mechanical system,

$\displaystyle\dot{\delta\beta_{1}}=-(i\Delta_{m}^{1}+\frac{\rm{\Gamma_{eff}}^{1}}{2})\delta\beta_{1}+i\rm{\tilde{J}_{m}\delta\beta_{2}}+i\sqrt{\Gamma_{1}}\delta\alpha^{in}_{1}+\sqrt{\gamma_{m}}\delta\beta^{in}_{1},$
(11a)
$\displaystyle\dot{\delta\beta_{2}}=-(i\Delta_{m}^{2}+\frac{\rm{\Gamma_{eff}}^{2}}{2})\delta\beta_{2}+i\rm{\tilde{J}_{m}\delta\beta_{1}}+i\sqrt{\Gamma_{2}}\delta\alpha^{in\dagger}_{2}+\sqrt{\gamma_{m}}\delta\beta^{in}_{2},$
(11b)

where the optically induced damping is
$\rm{\Gamma}_{j}=\frac{4|\tilde{G_{j}}|^{2}}{\kappa}$ and the effective
damping are $\rm{\Gamma_{eff}}^{1,2}=\gamma_{m}\pm\rm{\Gamma}_{1,2}$. From now
on, we will assume that $\rm{\Gamma}_{1}=\rm{\Gamma}_{2}=\rm{\Gamma}$. Eq. 11
can be put in its compact form as,

$\dot{\mathcal{O}}=\mathcal{M}\mathcal{O}+i\sqrt{\Gamma}\mathcal{N}+\sqrt{\gamma_{m}}\mathcal{O}^{in},$
(12)

where $\mathcal{O}\equiv(\delta\beta_{1},\delta\beta_{2})^{T}$,
$\mathcal{N}\equiv(\delta\alpha^{in}_{1},\delta\alpha^{in\dagger}_{2})^{T}$,
$\mathcal{O}^{in}\equiv(\delta\beta^{in}_{1},\delta\beta^{in}_{2})^{T}$, and
the matrix

$\mathcal{M}=\begin{bmatrix}-(i\Delta_{m}^{1}+\frac{\rm{\Gamma_{eff}}^{1}}{2})&i\tilde{J}_{m}\\\
i\tilde{J}_{m}&-(i\Delta_{m}^{2}+\frac{\rm{\Gamma_{eff}}^{2}}{2})\end{bmatrix}.$
(13)

##  III Sensitivity through a splitting and shifting of an exceptional point

In order to get the eigenvalues of our effective mechanical system, we rewrite
Eq. 12 in the Schrodinger-like equation, $i\partial_{t}\Psi=\rm{H_{eff}}\Psi$
where $\Psi\equiv(\delta\beta_{1},\delta\beta_{2})^{T}$. By ignoring noise
terms, this leads to an effective Hamiltonian,

$\rm{H_{eff}}=\begin{bmatrix}\Delta_{m}^{1}-i\frac{\rm{\Gamma_{eff}}^{1}}{2}&-\tilde{J}_{m}\\\
-\tilde{J}_{m}&\Delta_{m}^{2}-i\frac{\rm{\Gamma_{eff}}^{2}}{2}\end{bmatrix}.$
(14)

Figure 2: Eigenmodes and the EP splitting (a,b) and shifting (c,d) after a
perturbation strength of $\epsilon=5\times 10^{-3}\omega_{m}$. There is no
parametric drive in both cases ($\chi=0$), and the $\rm{EP}$ happens around
the driving strength $\alpha^{in}=420\omega_{m}^{-1/2}$ for both sensors when
there is no perturbation. The dimensionless used parameters are,
$\kappa=0.1\omega_{m}$, $\Delta_{1}=-\omega_{m}$, $\Delta_{2}=\omega_{m}$,
$\gamma_{m}=10^{-3}\omega_{m}$, $g=2.5\times 10^{-4}\omega_{m}$, and
$J_{m}=2.2\times 10^{-2}\omega_{m}$.

The eigenvalues $\lambda$ of $\rm{H_{eff}}$ are solutions of the following
characteristic equation, $\det(\rm{H_{eff}}-\lambda\mathbb{I}_{2})$. Solving
this equation leads to the eigenvalues,

$\lambda_{\pm}=\frac{1}{2}(\Delta_{m}^{1}+\Delta_{m}^{2})-\frac{i}{4}(\rm{\Gamma_{eff}}^{1}+\rm{\Gamma_{eff}}^{2})\pm\frac{\sigma}{4},$
(15)

where
$\sigma=\sqrt{16\tilde{J}_{m}^{2}+(2\delta{\Delta_{m}}-i\delta\rm{\Gamma_{eff}})^{2}}$
with $\delta{\Delta_{m}}=\Delta_{m}^{1}-\Delta_{m}^{2}$ and
$\delta\rm{\Gamma_{eff}}=\rm{\Gamma_{eff}}^{1}-\rm{\Gamma_{eff}}^{2}$. The
eigenvalues of our system are given by Eq. 15, and are displayed in Fig. 2,
where the eigenfrequencies ($\omega_{\pm}=\operatorname{Re}(\lambda_{\pm})$)
and the dissipations ($\gamma_{\pm}=\operatorname{Im}(\lambda_{\pm})$) are
shown in Fig. 2(a,c) and Fig. 2(b,d), respectively. In this figure, the
parametric driving is off ($\chi=0$) and the solid (dashed) line depicts the
case before (after) the perturbation of the system. In Fig. 2(a,b) the
perturbation induces a splitting of the $\rm{EP}$ while in Fig. 2(c,d), it
instead induces a shift of the $\rm{EP}$. In the former case, any mass
deposition ($\delta m$) acts as a perturbation on the system, and induces a
frequency shift ($\delta\omega$) through the relationship, $\delta
m=\mathcal{R}^{-1}\delta\omega$ where $\mathcal{R}=\frac{\omega_{m}}{2m}$ is
the mass responsibility and $\rm{m}$ is the mass of the resonator supporting
the deposition. Therefore, this mass deposition leads to a frequency shift
that lifts up the $\rm{EP}$-degeneracy, resulting in a sensitivity enhancement
at the $\rm{EP}$ (see double arrow in Fig. 2(a,b)). In the latter case, the
perturbation induces a shift in the dissipation instead, which can be seen as
a dissipation mismatch ($\delta\gamma=|\gamma_{1}-\gamma_{2}|$) between the
mechanical resonators Wang et al. (2019); Mao et al. (2023). This perturbation
can also be thought of as induced by a mismatch between optical damping
($\delta\gamma=|\Gamma_{1}-\Gamma_{2}|$), and may originate from optical
fluctuations within the cavities. This scenario also leads to a sensing scheme
as shown by the double arrow in Fig. 2(c,d). A general observation from Fig. 2
is that after a splitting due to a mass deposition (Fig. 2(a,b)), the
$\rm{EP}$ is destroyed and that limits the ability of the system to perform
multiple sensing scenarios in a time sequence. Indeed, the enhanced
performance is deteriorated after each sensing process since every next
detection is conducted on the basis of being destroyed by the previous
detection. In order to preserve $\rm{EP}$ in the parameter space after a
perturbation, it is preferable to shift the $\rm{EP}$ instead of lifting it
up. Such a sensor based on shifting the $\rm{EP}$ (see Fig. 2(c,d)) will
therefore meet the condition of a non-demolition sensing mechanism. With a
fine adjustment of gain and loss in the system, we expect a linewidth
preservation of the eigenstates after optical perturbation in the sensing
scheme based on shifting the $\rm{EP}$. This may improve the performance of
such sensing scenarios in the perspective of experiments related to the
precision of dispersion measurement Mao et al. (2023). However, we stress that
the sensitivity of this novel sensing scheme is somehow weak compared to the
splitting scheme (compare the double arrows in Fig. 2 (a) and (c)).

In order to enhance the sensitivity in the two aforementioned discussed
sensing schemes, we turn on parametric driving ($\chi\neq 0$). Fig. 3 displays
the enhancement of the splitting when parametric driving is accounted for in
the system. Fig. 3a and Fig. 3c show the frequency splittings in the case of
$\chi=0$ and $\chi=0.3\omega_{m}$, respectively. Their splitting
($\Delta\omega_{\pm}=\omega_{\pm}^{\epsilon}-\omega_{\pm}$) are displayed on
Fig. 3b and Fig. 3d, respectively. It can be seen that the resulting splitting
at the $\rm{EP}$ is greater in Fig. 3d than in Fig. 3b. This reveals the fact
that parametric terms can be used to tune the performance of the sensor.
Another observation is that, the parametric driving shifts the overall
frequencies towards low frequencies (compare $y-axis$ of Fig. 3 (a) and (c)).
The same investigation can also be done in sensing based on shifting, and
similar observations can be pointed out (not shown).

Figure 3: Splitting of the eigenfrequencies for $\chi=0$ (a) and for
$\chi=0.3\omega_{m}$ (c). The solid line is before any perturbation and the
dashed line is after a mass deposition that has induced a shift of
($\delta\omega=2\times 10^{-3}\omega_{m}$ in both cases. (b) and (d) are the
resulting splitting from (a) and (c), respectively. The used parameters are
the same as in Fig. 2.

In order to get more insight into the performance of these sensing schemes, we
plotted in Fig. 4 the splitting and the enhancement factor. Fig. 4(a,b) show
these quantities for the sensing based on splitting while Fig. 4(c,d) display
the same quantities for the sensing based on shifting effect. The splitting
represented here is the quantity
($\Delta\omega_{\pm}=|\omega_{\pm}^{\epsilon}-\omega_{\pm}|$) at the
$\rm{EP}$, while the enhancement factor is defined as being this splitting
over the perturbation strength
$\eta=|\frac{\Delta\omega_{\pm}}{\delta\omega}|$. In all these figures, the
full line is without parametric driving ($\chi=0$), while the dashed-dotted
and dashed lines are for $\chi=0.5\omega_{m}$ and $\chi=0.7\omega_{m}$,
respectively. It can be seen that parametric driving improves both the
sensitivity and the enhancement factor. Furthermore, one can observe that
these quantities scale faster in the splitting sensing scheme as compared to
shifting scenario sensing (compare Fig. 4a to Fig. 4c or Fig. 4b to Fig. 4d).
One benefit of sensors operating under splitting mechanisms over those based
on shifting is that the former ones perform very well for small perturbations,
while the latter do not. Indeed, Fig. 4(a,b) show giant enhancement
sensitivity for small perturbations, and that is owing to the square root
topological feature at the $\rm{EP}$. However, such an enhancement for small
perturbations is missing in Fig. 4(c,d), and is even zero (see inset). One
reason behind this lack of sensitivity to small perturbations is that the
shift is negligible for tiny disturbances

Figure 4: (a,c) Sensitivities versus the perturbation for different values of
$\chi$. (b,d) Enhancement factors versus the perturbation for different values
of $\chi$. (a,b) are from the splitting sensing scheme, while (c,d) result
from the sensing approach based on shifting the $\rm{EP}$. The rest of the
parameters are the same as in Fig. 2.

To understand the performance of these two sensing scenarios regarding the
parametric term, one evaluates the quantity
$\Delta\lambda_{\pm}=\lambda_{\pm}^{\epsilon}-\lambda_{\pm}$. This term is the
difference between the perturbed eigenvalue $\lambda_{\pm}^{\epsilon}$ and the
non-perturbed one, where $\epsilon$ is the strength of the perturbation. As
stated above, our proposal can experience sensing through both splitting and
shifting of the $\rm{EP}$. As we are interested in quantifying the sensitivity
at the $\rm{EP}$, we can derive this eigenvalue’s difference at the $\rm{EP}$
which yields (see Appendix B),

$\Delta\lambda_{\pm}^{\rm{EP}}=\frac{\Delta\omega^{\epsilon}}{2}-\frac{i|\delta\gamma|}{4}\pm\frac{\sigma_{\rm{EP}}^{\epsilon}}{2},$
(16)

where

$\Delta\omega^{\epsilon}=\frac{1}{2}\left[\frac{\omega_{m}\delta\omega}{\Delta_{m}^{2}}-\frac{1}{2}\left(\frac{\chi_{1}^{2}}{\omega_{m}}+\frac{\chi_{2}^{2}}{\omega_{m}}\right)\right],$
(17)

and

$\sigma_{\rm{EP}}^{\epsilon}=\sqrt{(8J_{m}r)^{2}+4\nu^{2}-4i\nu|\delta\gamma|+4\nu\mu-2i\mu|\delta\gamma|},$
(18)

with
$\nu=\frac{1}{2}\left(\frac{\chi_{2}^{2}}{\omega_{m}}-\frac{\chi_{1}^{2}}{\omega_{m}}\right)-\frac{\omega_{m}\delta\omega}{\Delta_{m}^{2}}$
and $\mu=-i\delta\Gamma_{\rm{eff}}$. In these equations (Eq. 17 and Eq. 18),
all scenarios of perturbations allowed in our system have been considered. By
considering that there is no parametric driving ($\chi_{j}=0$), and that there
is no perturbation of the dissipation ($\delta\gamma=0$), the system reduce to
the one investigated in Djorwe et al. (2019a), and the expressions in Eq. 17
and Eq. 18 reduce to $\Delta\omega^{\epsilon}=\frac{\delta\omega_{2}}{2}$ and
$\sigma_{\rm{EP}}^{\epsilon}=(1+i)\sqrt{\frac{\delta\Gamma_{\rm{eff}}\delta\omega_{2}}{8}}$
as expected. This sensing scenario is based on a splitting at the $\rm{EP}$ as
it can be seen in Fig. 2(a,b). In that case, the amplification factor is given
by:

$\eta=\left|\frac{\Re{(\Delta\lambda_{\pm}^{\rm{EP}})}}{\delta\omega}\right|=\sqrt{\frac{\delta\Gamma_{\rm{eff}}}{8\delta\omega}}=\sqrt{\frac{m\delta\Gamma_{\rm{eff}}}{4\omega_{m}\delta
m}}.$ (19)

When the perturbation is mainly from the dissipation ($\chi_{j}=0$,
$\delta\omega=0$, $\delta\gamma\neq 0$), which can be caused either by the
mismatch damping of the two mechanical resonators or the fluctuation of the
gain and loss induced by a driving fluctuation, both sensitivity and
amplification factor are reduced to,

$\Delta\lambda_{\pm}^{\rm{EP}}=-\frac{i}{4}|\delta\gamma|\pm\frac{i}{4}\sqrt{2|\delta\gamma|\delta\Gamma_{\rm{eff}}},$
(20)

and

$\eta=\left|\frac{\Re{(\Delta\lambda_{\pm}^{\rm{EP}})}}{\delta\omega}\right|=\sqrt{\frac{\delta\Gamma_{\rm{eff}}}{8|\delta\gamma|}},$
(21)

which also scales as a square root of the perturbation. It should be
highlighted that $\delta\gamma$ can be negative in order to make sense of the
enhancement factor definition. Moreover, Eq. 20 shows that there is no effect
of the perturbation on the reference frequency while there is a negative shift
of $\frac{1}{4}|\delta\gamma|$ in the reference dissipation as seen in Fig.
2(c,d). When parametric mechanical driving is involved ($\chi_{j}\neq 0$), we
expect to point out some improvement as depicted in Fig. 4. By considering the
sensing based on splitting scheme for instance ($\chi_{j}\neq 0$,
$\delta\omega\neq 0$, $\delta\gamma=0$), one deduces from Eq. 16 the
expression for the frequency difference at the $\rm{EP}$ as,

$\Delta\lambda_{\pm}^{\rm{EP}}=\frac{1}{2}\left(\frac{\omega_{m}}{\Delta_{m}^{2}}\delta\omega-\frac{\chi^{2}}{\omega_{m}}\right)\pm\frac{1}{2}\sqrt{(8J_{m}r)^{2}+i\frac{\omega_{m}}{\Delta_{m}^{2}}\delta\omega\delta\Gamma_{\rm{eff}}}.$
(22)

In this expression, it results that the parametric driving shifts the
reference frequency through $\frac{\chi^{2}}{\omega_{m}}$ as it appears in
Fig. 3c (compare $\rm{y-axis}$ of (a) and (c)), and also enhances the
sensitivity. Indeed, the term ($(8J_{m}r)^{2}$) which comes from the expansion
of $\rm{\cosh{2r}}$ not only shifts the $\rm{EP}$ from the driving strength
$\alpha^{in}\sim 420\omega_{m}^{1/2}$ (see Fig. 3a) to $\alpha^{in}\sim
500\omega_{m}^{1/2}$ (see Fig. 3c), but also contributes to the enhancement of
the sensitivity as depicted in Fig. 4(a,c). Furthermore, it should also be
underlined that $\chi$ induces an amount of sensitivity improvement through
the the term $\frac{\omega_{m}}{\Delta_{m}^{2}}$ under the square-root in Eq.
22. Similar explanations regarding the effects of parametric driving on the
sensitivity can be done for the sensing scenario based on the shifting
mechanism. In fact, the expression of the frequency difference at the
$\rm{EP}$ in this case is derived as,

$\Delta\lambda_{\pm}^{\rm{EP}}=-\frac{\chi^{2}}{2\omega_{m}}-\frac{i}{4}|\delta\gamma|\pm\frac{1}{4}\sqrt{(8J_{m}r)^{2}-2|\delta\gamma|\delta\Gamma_{\rm{eff}}}.$
(23)

From this expression, we can see that $\chi$ induces a shift of the reference
frequency through $\frac{\chi^{2}}{2\omega_{m}}$ as seen while comparing the
$\rm{y-axis}$ of Fig. 3a to Fig. 3c. As in Fig. 2d, the shift in the
dissipation is still captured by $\frac{|\delta\gamma|}{4}$. More importantly,
in Eq. 23, the sensitivity is mainly enhanced by the parametric driving
through $(8J_{m}r)^{2}$. Therefore, these analytical expressions predict the
improvement of both sensing schemes exhibited by our proposal, and are in good
agreement with our findings.

##  IV Transmissions and output spectra

The current section analyzes the transmission and output spectra of the cavity
fields. The effects of thermal noise, and the different involved perturbations
will be investigated in order to figure out their impact on the EP feature.
For that purpose, we start from Eq. 10 (or its analogous without tilde), and
by moving to the frequency domain we derive the following scattering
coefficients (see details in Appendix C),

$\displaystyle S_{11}(\omega)$
$\displaystyle=1-\dfrac{2\tilde{G}_{1}\xi_{2}(\omega)\sqrt{\Gamma_{1}}}{\sqrt{\kappa}\xi_{eff}(\omega)},\,\,S_{12}(\omega)=-\dfrac{2i\tilde{G}_{1}\tilde{J}_{m}\sqrt{\Gamma_{2}}}{\sqrt{\kappa}\xi_{eff}(\omega)},$
(24a) $\displaystyle S_{13}(\omega)$
$\displaystyle=\dfrac{2i\tilde{G}_{1}\xi_{2}(\omega)\sqrt{\gamma_{m}}}{\sqrt{\kappa}\xi_{eff}(\omega)},\,\,S_{14}(\omega)=\dfrac{2\tilde{G}_{1}\tilde{J}_{m}\sqrt{\gamma_{m}}}{\sqrt{\kappa}\xi_{eff}(\omega)},$
(24b) $\displaystyle S_{21}(\omega)$
$\displaystyle=\dfrac{2i\tilde{G}_{2}\tilde{J}_{m}\sqrt{\Gamma_{1}}}{\sqrt{\kappa}\xi_{eff}(\omega)},\,\,S_{22}(\omega)=1+\dfrac{2\tilde{G}_{2}\xi_{1}(\omega)\sqrt{\Gamma_{2}}}{\sqrt{\kappa}\xi_{eff}(\omega)},$
(24c) $\displaystyle S_{23}(\omega)$
$\displaystyle=\dfrac{2\tilde{G}_{2}\tilde{J}_{m}\sqrt{\gamma_{m}}}{\sqrt{\kappa}\xi_{eff}(\omega)},\,\,S_{24}(\omega)=-\dfrac{2i\tilde{G}_{2}\xi_{1}(\omega)\sqrt{\gamma_{m}}}{\sqrt{\kappa}\xi_{eff}(\omega)},$
(24d)

where $\xi_{eff}(\omega)=\xi_{1}(\omega)\xi_{2}(\omega)+\tilde{J}^{2}_{m}$
with $\xi_{1}(\omega)=\Gamma^{1}_{eff}/2+i(\Delta^{1}_{m}-\omega)$ and
$\xi_{2}(\omega)=\Gamma^{2}_{eff}/2+i(\Delta^{2}_{m}-\omega)$. The
coefficients $S_{11}$ and $S_{22}$ stand for the transmission of the first and
second cavity, respectively.

Figure 5: (a,b) Transmission $S_{11}$ and $S_{22}$ of the first and second
cavities, respectively. (c,d) Output spectra $S_{1}^{out}$ and $S_{2}^{out}$
of the first and second cavities, respectively. The full line depicts the case
at the EP for $\alpha^{in}=420\omega_{m}^{1/2}$ (see Fig. 2 (a)). The dashed
and dash-dotted lines correspond to cases away from the EP, and the other
parameters are the same as in Fig. 2

Owing to our assumption that the cavities are degenerated, these two
coefficients become similar when the driving is strong enough (see full lines
in Fig. 5 (a) and (b)). For weak driving strength ($\alpha^{in}\lesssim
90\omega_{m}^{1/2}$), however, $S_{11}$ shows two dips while $S_{11}$ exhibits
two peaks because they are far from the EP (not shown). This weak driving
regime is interesting for nonreciprocal phenomena, and will be developed in
our future investigations. The full lines in Fig. 5 show the shape of
transmissions (Fig. 5 (a) and (b)) and output spectra (Fig. 5 (c) and (d)) at
the EP. The output spectra can be expressed in terms of the given coefficients
in Eq. 24 and yield (see Appendix C),

$\displaystyle S^{(1)}_{out}(\omega)$
$\displaystyle=\left|S_{11}(\omega)\right|{}^{2}+\left|S_{12}(\omega)\right|{}^{2}+\left(2N_{s_{1}}+1\right)\left|S_{13}(\omega)\right|{}^{2}$
$\displaystyle+\left(2N_{s_{2}}+1\right)\left|S_{14}(\omega)\right|{}^{2},$
(25a) $\displaystyle S^{(2)}_{out}(\omega)$
$\displaystyle=\left|S_{21}(\omega)\right|{}^{2}+\left|S_{22}(\omega)\right|{}^{2}+\left(2N_{s_{1}}+1\right)\left|S_{23}(\omega)\right|{}^{2}$
$\displaystyle+\left(2N_{s_{2}}+1\right)\left|S_{24}(\omega)\right|{}^{2}.$
(25b)

where
$N_{s_{1,2}}=(n_{th_{1,2}}+1)\sinh^{2}(r_{1,2})+n_{th_{1,2}}\cosh^{2}(r_{1,2})$,
and $n_{th}$ the thermal population. As it can be seen, the shapes of these
spectra are mainly depending on those of the transmission ones. The dashed and
dash-dotted lines in Fig. 5 depict the situations away from the EP, and we can
see two peaks which are reminiscent of the eigenfrequencies shown in Fig. 2(a)
for instance. In what follows, we will only consider the spectrum
$S_{1}^{out}$ and will analyze the effects of some parameters on it as
displayed in Fig. 6. The effect of thermal noise $n_{th}$ on the output
spectrum is shown in Fig. 6(a), where we see that the EP is kept even though
the output signal is amplified. This means that any output signal from our EP-
sensor will be amplified, and this is due to the gain in our system. Such an
amplification phenomenon is well-known in optomechanical systems, and seems to
be beneficial for Non-Hermitian sensing. Moreover, this thermal noise effect
can be monitored by tuning the squeezing parameter $r$ through $N_{s}$. Fig.
6(b) depicts the effect of parametric driving on the output spectrum, and it
can be observed that $\chi$ shifts the spectrum’s shape towards low-frequency
values as expected (see Fig. 3(c) for instance). Moreover, the driving
strength at which EP happens has been increased. Owing to these changes in the
input driving, the EPs corresponding to both the dashed and the dash-dotted
lines are missing in Fig. 6(b). To capture these EPs and to quantify how they
have been shifted, one needs to adjust the driving strength to
$\alpha^{in}\sim 130\omega_{m}^{1/2}$ for the dashed line and $\alpha^{in}\sim
170\omega_{m}^{1/2}$ for the dash-dotted line. Fig. 6(c) and Fig. 6(d) display
the effects of the perturbations on the frequency and the dissipation (or
quality factor), respectively. The former corresponds to a sensing scheme
based on the splitting of the EP, while the latter relies on sensing upon
shifting of the EP. As aforementioned in Section III, it can be seen that the
EP is destroyed in Fig. 6(c) while it is preserved and shifted (down) in Fig.
6(d). Moreover, the preserved peak of the EP in Fig. 6(d) has been de-
amplified and we stress that this may impair the performance of the sensing
process.

Figure 6: Dynamics of the EP in the output spectrum versus some system
parameters. The effect of the thermal noise in (a), and the parametric driving
in (b), the frequency shift perturbation in (c), and dissipation mismatch
perturbation in (d). The full lines captures the EP at
$\alpha^{in}=420\omega_{m}^{1/2}$ as in Fig. 2(a), while the dashed and dash-
dotted lines correspond to the perturbed cases depending on the studied
parameter. The other used parameters are the same as in Fig. 2.

##  V Conclusion

We investigated the effects of mechanical parametric driving on sensing
performance at an exceptional point. This parametric behaviour comes from the
modulation of the mechanical spring constant of the involved resonators. Two
sensing schemes have been put forward. The sensing upon splitting of the EP as
the frequency undergoes a perturbation, and the sensing through shifting of
the EP that is induced by a damping mismatch for instance. It has been pointed
out that mechanical parametric driving enhances the sensitivity of both
schemes. Moreover, this parametric driving shifts the frequency at which EP
happens towards low values. Furthermore, we figured out the fact that the
sensitivity enhancement is more efficient under the splitting scenario than
what is realized while shifting the EP. These results have also been confirmed
through output spectra and transmission analysis of the optical cavities.
These parametric effects on sensing reveal how nonlinearities can be used as
requirement tools for better performances of sensors. This work provides a new
way of improving EP-sensors and sheds light on the beneficial effects of
nonlinear phenomena on Non-Hermitian sensing.

## Acknowledgments

This work has been carried out under the Iso-Lomso Fellowship at Stellenbosch
Institute for Advanced Study (STIAS), Wallenberg Research Centre at
Stellenbosch University, Stellenbosch 7600, South Africa. P. Djorwe
acknowledges the receipt of a grant from the APS-EPS-FECS-ICTP Travel Award
Fellowship Programme (ATAP), Trieste, Italy. M. Asjad and D. Dutykh have been
supported by the Khalifa University of Science and Technology under Award No.
FSU- 2023-014.

##  Appendix A Effective mechanical Hamiltonian

Some details about derivation of the effective mechanical Hamiltonian are
presented in this Appendix. In the rotating wave approximation (RWA) and in
the frame rotating at $\omega_{p}^{j}+\omega_{d}$, where $\omega_{p}^{j}$ is
the $j^{th}$ driving frequency, our system’s Hamiltonian yields:

$\displaystyle H_{\rm{0M}}$
$\displaystyle=\sum_{j}\tilde{\omega}_{j}b^{\dagger}_{j}b_{j}-\Delta_{j}a^{\dagger}_{j}a_{j}-ga^{\dagger}_{j}a_{j}(b^{\dagger}_{j}+b_{j})+\frac{\chi}{2}(b^{\dagger
2}_{j}+b^{2}_{j})$ (26a) $\displaystyle H_{\rm{int}}$
$\displaystyle=-J_{m}(b_{1}b_{2}^{\dagger}+b_{1}^{\dagger}b_{2}),$ (26b)
$\displaystyle H_{\rm{drive}}$
$\displaystyle=iE\sum_{j}(a^{\dagger}_{j}-a_{j}),$ (26c)

where $\tilde{\omega}_{j}=\omega_{j}-\omega_{d}$ and
$\Delta_{j}=\omega_{p}-\omega_{c}$. This Hamiltonian can be diagonalized by
introducing the squeezing transformation $S(r)=\rm{exp}[r(b^{2}-b^{\dagger
2})]$, where $\rm{r}$ will be figured out latter on. This operator transforms
through the Bogoliubov transformation $b$ to $b_{s}$ as,

$b_{s}=b\cosh(r)+b^{\dagger}\sinh(r).$ (27)

From Eq. 27, it is straightforward to get,

$\displaystyle b_{j}$
$\displaystyle=b_{s_{j}}\cosh(r_{j})-b^{\dagger}{s_{j}}\sinh(r_{j})$ (28a)
$\displaystyle b^{\dagger}_{j}$
$\displaystyle=b{\dagger}_{s_{j}}\cosh(r_{j})-b_{s_{j}}\sinh(r_{j}),$ (28b)

which can be used in Eq. 26, and that leads to the following transformed
Hamiltonian,

$\displaystyle H_{\rm{0M}}$
$\displaystyle=\sum_{j}\Delta_{m}^{j}b^{\dagger}_{s_{j}}b_{s_{j}}-\Delta_{j}a^{\dagger}_{j}a_{j}-\tilde{g}a^{\dagger}_{j}a_{j}(b^{\dagger}_{s_{j}}+b_{s_{j}})$
(29a) $\displaystyle H_{\rm{int}}$
$\displaystyle=-\tilde{J}_{m}(b_{s_{1}}b_{s_{2}}^{\dagger}+b_{s_{1}}^{\dagger}b_{s_{2}}),$
(29b) $\displaystyle H_{\rm{drive}}$
$\displaystyle=iE\sum_{j}(a^{\dagger}_{j}-a_{j}),$ (29c) $\displaystyle
H_{\rm{diss}}^{S}$
$\displaystyle=-i\frac{\kappa}{2}\sum_{j}a^{\dagger}_{j}a_{j}-i\frac{\gamma_{m}}{2}\sum_{j}b^{\dagger}_{s_{j}}b_{s_{j}},$
(29d)

where the Hamiltonian $H_{\rm{diss}}$ captures the optical ($\kappa$) and the
mechanical ($\gamma_{1}=\gamma_{2}=\gamma_{m}$) dissipation in the system.
Furthermore, the driving amplitude has been expressed as
$E=\sqrt{\kappa}a^{in}$, where $\alpha^{in}$ is related to the input power
$P_{in}$ through $\alpha^{in}=\sqrt{\frac{P_{in}}{\hbar\omega_{p}}}$. The
following parameters have been also defined,

$\displaystyle\tilde{g}$ $\displaystyle=ge^{-r},$ (30a)
$\displaystyle\Delta_{m}^{j}$
$\displaystyle=\sqrt{\tilde{\omega}_{j}^{2}-\chi^{2}},$ (30b)
$\displaystyle\tilde{J}_{m}$ $\displaystyle=J_{m}\cosh(2r).$ (30c)

The Quantum Langeving Equations (QLEs) of the Hamiltonian given in Eq. 29 can
be derived using the Heisenberg equation
$\dot{\mathcal{O}}=i[H,\mathcal{O}]=\mathcal{N}$, where $H$ is a Hamiltonian,
$\mathcal{O}$ is an operator ($\mathcal{O}\equiv b_{s_{j}},a_{j}$), and
$\mathcal{N}$ the quantum noise related to the operator. Following this
procedure, the QLEs of our system yield,

$\displaystyle\dot{a}_{j}$ $\displaystyle=$
$\displaystyle(i[\Delta_{j}+\tilde{g}(b_{s_{j}}+b^{\dagger}_{s_{j}})]-\frac{\kappa}{2})a_{j}+\sqrt{\kappa}\alpha^{in}_{j}+\sqrt{\kappa}a^{in}_{j},$
(31a) $\displaystyle\dot{b}_{s_{j}}$ $\displaystyle=$
$\displaystyle-(i\Delta_{m}^{j}+\frac{\gamma_{m}}{2})b_{s_{j}}+i\tilde{J}_{m}b_{3-j}+i\tilde{g}a^{\dagger}_{j}a_{j}+\sqrt{\gamma_{m}}\beta^{in}_{j}.$
(31b)

The nonlinear set of equations displayed in Eq. 31 can be linearized by
splitting the operators in term of their mean values ($c-$numbers) and some
amount of quantum fluctuations as $a_{j}=\alpha_{j}+\delta\alpha_{j}$ and
$b_{s_{j}}=\beta_{j}+\delta\beta_{j}$, where $\alpha_{j}=\langle a_{j}\rangle$
and $\beta_{j}=\langle b_{s_{j}}\rangle$. This procedure leads to the
classical mean equations,

$\displaystyle\dot{\alpha_{j}}$
$\displaystyle=(i\tilde{\Delta}_{j}-\frac{\kappa}{2})\alpha_{j}+\sqrt{\kappa}\alpha^{in}_{j},$
(32a) $\displaystyle\dot{\beta_{j}}$
$\displaystyle=-(i\Delta_{m}^{j}+\frac{\gamma_{m}}{2})\beta_{j}+i\tilde{J}_{m}\beta_{3-j}+i\tilde{g}|\alpha_{j}|^{2},$
(32b)

and to the fluctuations equations,

$\displaystyle\delta\dot{\alpha}_{j}$
$\displaystyle=(i\tilde{\Delta}_{j}-\frac{\kappa}{2})\delta\alpha_{j}+i\tilde{G_{j}}(\delta\beta_{j}+\delta\beta^{\dagger}_{j})+\sqrt{\kappa}\delta\alpha^{in}_{j},$
(33a) $\displaystyle\rm{\delta\dot{\beta}_{j}}$
$\displaystyle=\rm{-(i\Delta_{m}^{j}+\frac{\gamma_{m}}{2})\delta\beta_{j}}\rm{+i\tilde{J}_{m}\delta\beta_{3-j}}\rm{+i\tilde{G_{j}}(\delta\alpha^{\dagger}_{j}+\delta\alpha_{j})}$
$\displaystyle+\rm{\sqrt{\gamma_{m}}\delta{\beta}^{in}_{j}},$ (33b)

with $\tilde{\Delta}_{j}=\Delta_{j}+2\tilde{g}\Re{\beta_{j}}$, and
$\tilde{G_{j}}=\tilde{g}\alpha_{j}$ where the intracavity field amplitude
$\alpha_{j}$ is assumed to be real. In order to get an effective mechanical
model describing Eq. 33, we need to trace out the optical equations. For that
purpose, we introduce the following slowly varying operators with tildes,
$\delta\alpha_{j}=\delta\tilde{\alpha_{j}}e^{i\tilde{\Delta}_{j}t}$,
$\delta\alpha^{in}=\delta\tilde{\alpha^{in}}e^{i\tilde{\Delta}_{j}t}$, and
$\delta\beta_{j}=\delta\tilde{\beta_{j}}e^{-i\Delta_{m}^{j}t}$. By using these
slowly varying operators in Eq. 33, one get

$\displaystyle\dot{\delta\tilde{\alpha_{j}}}$
$\displaystyle=-\frac{\kappa}{2}\delta\tilde{\alpha_{j}}+i\tilde{G_{j}}(\delta\tilde{\beta_{j}}e^{-i(\Delta_{m}^{j}+\tilde{\Delta})t}+\delta\tilde{\beta_{j}}^{\dagger}e^{i(\Delta_{m}^{j}-\tilde{\Delta}_{j})t})$
(34a) $\displaystyle+\sqrt{\kappa}\delta\tilde{\alpha}^{in}_{j},$ (34b)
$\displaystyle\dot{\delta\tilde{\beta_{j}}}$
$\displaystyle=-\frac{\gamma_{m}}{2}\delta\tilde{\beta_{j}}+i\tilde{J}_{m}\delta\tilde{\beta}_{3-j}+i\tilde{G_{j}}(\delta\tilde{\alpha_{j}}^{\dagger}e^{i(\Delta_{m}^{j}-\tilde{\Delta}_{j})t}$
(34c)
$\displaystyle+\delta\tilde{\alpha_{j}}e^{i(\Delta_{m}^{j}+\tilde{\Delta}_{j})t})+\sqrt{\gamma_{m}}\delta\tilde{\beta}^{in}_{j}.$
(34d)

As gain and losses are involved in our system, we assume that the first cavity
is driven at the red sideband ($\tilde{\Delta}_{1}=-\Delta_{m}^{j}$) while the
second one is driven at the blue sideband
($\tilde{\Delta}_{2}=\Delta_{m}^{j}$). Therefore, on the red sideband we get
($j=1$),

$\displaystyle\delta\tilde{\dot{\alpha_{j}}}$
$\displaystyle=-\frac{\kappa}{2}\delta\tilde{\alpha_{j}}+i\tilde{G_{j}}(\delta\tilde{\beta_{j}}+\delta\tilde{\beta_{j}}^{\dagger}e^{2i\Delta_{m}^{j}t})+\sqrt{\kappa}\delta\tilde{\alpha^{in}_{j}},$
(35a) $\displaystyle\delta\tilde{\dot{\beta_{j}}}$
$\displaystyle=\rm{-\frac{\gamma_{m}}{2}\delta\tilde{\beta_{j}}+i\tilde{J}_{m}\delta\tilde{\beta}_{3-j}+i\tilde{G_{j}}(\delta\tilde{\alpha_{j}}^{\dagger}e^{2i\tilde{\Delta}_{j}t}+\delta\tilde{\alpha_{j}})+\sqrt{\gamma_{m}}\delta\tilde{\beta}^{in}_{j}},$
(35b)

while on the blue sideband we have ($j=2$),

$\displaystyle\delta\tilde{\dot{\alpha_{j}}}$
$\displaystyle=-\frac{\kappa}{2}\delta\tilde{\alpha_{j}}+i\tilde{G_{j}}(\delta\tilde{\beta_{j}}e^{-2i\Delta_{m}^{j}t}+\delta\tilde{\beta_{j}}^{\dagger})+\sqrt{\kappa}\delta\tilde{\alpha}^{in}_{j},$
(36a) $\displaystyle\delta\tilde{\dot{\beta_{j}}}$
$\displaystyle=\rm{-\frac{\gamma_{m}}{2}\delta\tilde{\beta_{j}}+i\tilde{J}_{m}\delta\tilde{\beta}_{3-j}+i(\tilde{G_{j}}\delta\tilde{\alpha_{j}}^{\dagger}+\tilde{G_{j}}^{\ast}\delta\tilde{\alpha_{j}}e^{2i\tilde{\Delta}_{j}t})+\sqrt{\gamma_{m}}\delta\tilde{\beta}^{in}_{j}}.$
(36b)

In the RWA, the terms oscillating with high frequency can be ignored and the
set of dynamical equations describing the system reduces to,

$\displaystyle\dot{\delta\tilde{\alpha_{1}}}$
$\displaystyle=-\frac{\kappa}{2}\delta\tilde{\alpha_{1}}+i\tilde{G_{1}}\delta\tilde{\beta_{1}}+\sqrt{\kappa}\delta\tilde{\alpha}^{in}_{1},$
(37a) $\displaystyle\dot{\delta\tilde{\alpha_{2}}}$
$\displaystyle=-\frac{\kappa}{2}\delta\tilde{\alpha_{2}}+i\tilde{G_{2}}\delta\tilde{\beta_{2}}^{\dagger}+\sqrt{\kappa}\delta\tilde{\alpha}^{in}_{2},$
(37b) $\displaystyle\dot{\delta\tilde{\beta_{1}}}$
$\displaystyle=-\frac{\gamma_{m}}{2}\delta\tilde{\beta_{1}}+i\tilde{J}_{m}\delta\tilde{\beta}_{2}+i\tilde{G_{1}}^{\ast}\delta\tilde{\alpha_{1}}+\sqrt{\gamma_{m}}\delta\tilde{\beta}^{in}_{1},$
(37c) $\displaystyle\dot{\delta\tilde{\beta_{2}}}$
$\displaystyle=-\frac{\gamma_{m}}{2}\delta\tilde{\beta_{2}}+i\tilde{J}_{m}\delta\tilde{\beta}_{1}+i\tilde{G_{2}}\delta\tilde{\alpha_{2}}^{\dagger}+\sqrt{\gamma_{m}}\delta\tilde{\beta}^{in}_{2}.$
(37d)

In order to get the effective mechanical system, we need to trace out the
intracavity field ($\delta\tilde{\alpha_{j}}$) in Eq. 37. For this purpose,
let us introduce
$\delta\tilde{\alpha_{j}}=\delta\alpha_{m}^{j}e^{-\frac{\kappa}{2}t}$, which
after using them in Eq. 37 leads to,

$\displaystyle\dot{\delta\alpha_{m}^{1}}$
$\displaystyle=(i\tilde{G_{1}}\delta\tilde{\beta_{1}}+\sqrt{\kappa}\delta\tilde{\alpha^{in}_{1}})e^{\frac{\kappa}{2}t},$
(38a) $\displaystyle\dot{\delta\alpha_{m}^{2}}$
$\displaystyle=(i\tilde{G_{2}}\delta\tilde{\beta_{2}}^{\dagger}+\sqrt{\kappa}\delta\tilde{\alpha^{in}_{2}})e^{\frac{\kappa}{2}t}.$
(38b)

These equations can be integrated,

$\displaystyle\delta\alpha_{m}^{1}$
$\displaystyle=\int_{-\infty}^{t}(i\tilde{G_{1}}\delta\tilde{\beta_{1}}+\sqrt{\kappa}\delta\tilde{\alpha^{in}_{1}})e^{\frac{\kappa}{2}\tau}d\tau,$
(39a) $\displaystyle\delta\alpha_{m}^{2}$
$\displaystyle=\int_{-\infty}^{t}(i\tilde{G_{2}}\delta\tilde{\beta_{2}}^{\dagger}+\sqrt{\kappa}\delta\tilde{\alpha^{in}_{2}})e^{\frac{\kappa}{2}\tau}d\tau,$
(39b)

where in the weak coupling regime ($\kappa\gg\tilde{G_{j}}$), can be
adiabatically integrated and yields,

$\displaystyle\delta\tilde{\alpha}_{1}$
$\displaystyle=\frac{2}{\kappa}(i\tilde{G_{1}}\delta\tilde{\beta_{1}}+\sqrt{\kappa}\delta\tilde{\alpha}^{in}_{1}),$
(40a) $\displaystyle\delta\tilde{\alpha}_{2}$
$\displaystyle=\frac{2}{\kappa}(i\tilde{G_{2}}\delta\tilde{\beta_{2}}^{\dagger}+\sqrt{\kappa}\delta\tilde{\alpha}^{in}_{2}).$
(40b)

By using Eq. 40 back in Eq. 37, we obtain the following effective mechanical
system,

$\displaystyle\dot{\delta\beta_{1}}$
$\displaystyle=-(i\Delta_{m}^{1}+\frac{\rm{\Gamma_{eff}}^{1}}{2})\delta\beta_{1}+i\tilde{J}_{m}\delta\beta_{2}+i\sqrt{\Gamma_{1}^{\ast}}\delta\alpha^{in}+\sqrt{\gamma_{m}}\delta\beta^{in}_{1},$
(41a) $\displaystyle\dot{\delta\beta_{2}}$
$\displaystyle=-(i\Delta_{m}^{2}+\frac{\rm{\Gamma_{eff}}^{2}}{2})\delta\beta_{2}+i\tilde{J}_{m}\delta\beta_{1}+i\sqrt{\Gamma_{2}}\delta\alpha^{in\dagger}+\sqrt{\gamma_{m}}\delta\beta^{in}_{2},$
(41b)

where optical induced damping is
$\rm{\Gamma}_{j}=\frac{4|\tilde{G_{j}}|^{2}}{\kappa}$ and the effective
damping are $\rm{\Gamma_{eff}}^{1,2}=\gamma_{m}\pm\rm{\Gamma}_{1,2}$. From now
on, we will assume that $\rm{\Gamma}_{1}=\rm{\Gamma}_{2}=\rm{\Gamma}$. Eq. 41
can be put in its compact form as,

$\dot{\mathcal{O}}=\mathcal{M}\mathcal{O}+i\sqrt{\mathcal{K}}\mathcal{N}+\sqrt{\gamma_{m}}\mathcal{O}^{in}$
(42)

where $\mathcal{O}\equiv(\delta\beta_{1},\delta\beta_{2})^{T}$,
$\mathcal{K}\equiv(\Gamma_{1}^{\ast},\Gamma_{1})^{T}$,
$\mathcal{N}\equiv(\delta\alpha^{in},\delta\alpha^{in\dagger})^{T}$
,$\mathcal{O}^{in}\equiv(\delta\beta^{in}_{1},\delta\beta^{in}_{2})^{T}$, and
the matrix

$\mathcal{M}=\begin{bmatrix}-(i\Delta_{m}^{1}+\frac{\rm{\Gamma_{eff}}^{1}}{2})&i\tilde{J}_{m}\\\
i\tilde{J}_{m}&-(i\Delta_{m}^{2}+\frac{\rm{\Gamma_{eff}}^{2}}{2})\end{bmatrix}.$
(43)

##  Appendix B Splitting vs shifting sensing schemes

This Appendix is meant to provide a comparison between the two sensing schemes
investigated in the main text. We have shown the possibility of sensing based
either the splitting or the shifting of the $\rm{EP}$, but a straight
comparison between the two is missing. A general eigenvalue difference
evaluated at the $\rm{EP}$ can be given by,

$\Delta\lambda_{\pm}^{\rm{EP}}=\frac{\Delta\omega^{\epsilon}}{2}-\frac{i|\delta\gamma|}{4}\pm\frac{\sigma_{\rm{EP}}^{\epsilon}}{2},$
(44)

where

$\Delta\omega^{\epsilon}=\frac{1}{2}\left[\frac{\tilde{\omega}_{2}\delta\omega_{2}}{\Delta_{m}^{2}}-\frac{\chi_{2}\delta\chi_{2}}{\Delta_{m}^{2}}-\frac{\chi_{1}\delta\chi_{1}}{\Delta_{m}^{1}}-\frac{1}{2}\left(\frac{\chi_{1}^{2}}{\tilde{\omega}_{1}}+\frac{\chi_{2}^{2}}{\tilde{\omega}_{2}}\right)\right],$
(45)

and

$\sigma_{\rm{EP}}^{\epsilon}=\sqrt{(8J_{m}r)^{2}+4\nu^{2}-4i\nu|\delta\gamma|+4\nu\mu-2i\mu|\delta\gamma|},$
(46)

with
$\nu=\frac{1}{2}\left(\frac{\chi_{2}^{2}}{\tilde{\omega}_{2}}-\frac{\chi_{1}^{2}}{\tilde{\omega}_{1}}\right)+\frac{\chi_{2}\delta\chi_{2}}{\Delta_{m}^{2}}-\frac{\tilde{\omega}_{2}\delta\omega_{2}}{\Delta_{m}^{2}}-\frac{\chi_{1}\delta\chi_{1}}{\Delta_{m}^{1}}$
and $\mu=(2\delta\tilde{\omega}-i\delta\Gamma_{\rm{eff}})$. In these
expressions, $\delta\omega_{2}$ and $\delta\gamma$ are the two types of
perturbation used in the main text. The former is induced by a mass deposition
on the second mechanical resonator, while the latter comes from optical
fluctuations or a mismatch damping of the two mechanical resonators. We stress
that there can be also a perturbation induced by the fluctuation of the
parametric driving which can be captured through $\delta\chi_{j}$ in Eq. 46.
However, we have not consider such a perturbation in our investigation in the
main text. We have also made some assumptions for a seek of simplicity,
$\tilde{\omega}_{1}=\tilde{\omega}_{2}=\omega_{m}$ and
$\chi_{1}=\chi_{2}=\chi$. With these approximations, we were able to derive
the different expressions for sensitivity and enhancement factor presented in
the main text, which were qualitatively in good agreement with our results. In
order to give a direct comparison between the two sensing scenarios, we
plotted both their sensitivity and enhancement factor in Fig. 7. The full line
is from the sensing based on the shifting mechanism while the dashed line
refers to splitting at the $\rm{EP}$. It can be clearly seen that the scheme
based on the splitting performs better as discussed in the main text.

Figure 7: Comparison of the sensitivities and the enhancement factors for the
two sensing schemes when $\chi=0$. The used parameters are those as in Fig. 7.

##  Appendix C Transmission and output spectra

The measurement of cavity field fluctuations is commonly performed more
conveniently in the frequency domain rather than the time domain
experimentally. Hence, employing the Fourier transform definition for an
operator $\mathcal{O}(\omega)=\int^{\infty}_{-\infty}\mathcal{O}(t)e^{-i\omega
t}dt$, Eq.(42) can be written in Fourier domain as

$\mathcal{O}(\omega)=\mathcal{D}(\omega)\left[i\sqrt{\mathcal{K}}\mathcal{N}(\omega)+\sqrt{\gamma_{m}}\mathcal{O}^{in}(\omega)\right],$
(47)

where $\mathcal{D}(\omega)=(-\mathcal{M}-i\omega)^{-1}$. We are interested to
obtain the spectra two optical modes at the output of the opto-mechanical
cavity. According to the input-output theory, the operators for the output
fields are related to the cavity and to the input noise operators by the
relation
$\delta\alpha^{out}_{1}(\omega)=\sqrt{\kappa}\delta\alpha_{1}(\omega)-\delta\alpha^{in}_{1}(\omega)$
and
$\delta\alpha^{out}_{2}(\omega)=\sqrt{\kappa}\delta\alpha_{2}(\omega)-\delta\alpha^{in}_{2}(\omega)$.
Then by using Eq.(40) and Eq.(47), the expressions for the two transmitted
fields in Fourier domain are given by,

$\displaystyle\delta\alpha^{out}_{1}(\omega)$
$\displaystyle=S_{11}(\omega)\delta\alpha^{in}_{1}(\omega)+S_{12}(\omega)\delta\alpha^{in^{\dagger}}_{2}(\omega)+S_{13}(\omega)\delta\beta^{in}_{1}(\omega)$
$\displaystyle+S_{14}(\omega)\beta^{in}_{2}(\omega),$ (48a)
$\displaystyle\alpha^{out^{\dagger}}_{2}(\omega)$
$\displaystyle=S_{21}(\omega)\delta\alpha^{in}_{1}(\omega)+S_{22}(\omega)\delta\alpha^{in^{\dagger}}_{2}(\omega)+S_{23}(\omega)\delta\beta^{in}_{1}(\omega)$
$\displaystyle+S_{24}(\omega)\beta^{in}_{2}(\omega)$ (48b)

where

$\displaystyle
S_{11}(\omega)=1-\dfrac{2\tilde{G}_{1}\xi_{2}(\omega)\sqrt{\Gamma_{1}}}{\sqrt{\kappa}\xi_{eff}(\omega)},\,\,S_{12}(\omega)=-\dfrac{2i\tilde{G}_{1}\tilde{J}_{m}\sqrt{\Gamma_{2}}}{\sqrt{\kappa}\xi_{eff}(\omega)},$
(49a) $\displaystyle
S_{13}(\omega)=\dfrac{2i\tilde{G}_{1}\xi_{2}(\omega)\sqrt{\gamma_{m}}}{\sqrt{\kappa}\xi_{eff}(\omega)},\,\,S_{14}(\omega)=\dfrac{2\tilde{G}_{1}\tilde{J}_{m}\sqrt{\gamma_{m}}}{\sqrt{\kappa}\xi_{eff}(\omega)},$
(49b) $\displaystyle
S_{21}(\omega)=\dfrac{2i\tilde{G}_{2}\tilde{J}_{m}\sqrt{\Gamma_{1}}}{\sqrt{\kappa}\xi_{eff}(\omega)},\,\,S_{22}(\omega)=1+\dfrac{2\tilde{G}_{2}\xi_{1}(\omega)\sqrt{\Gamma_{2}}}{\sqrt{\kappa}\xi_{eff}(\omega)},$
(49c) $\displaystyle
S_{23}(\omega)=\dfrac{2\tilde{G}_{2}\tilde{J}_{m}\sqrt{\gamma_{m}}}{\sqrt{\kappa}\xi_{eff}(\omega)},\,\,S_{24}(\omega)=-\dfrac{2i\tilde{G}_{2}\xi_{1}(\omega)\sqrt{\gamma_{m}}}{\sqrt{\kappa}\xi_{eff}(\omega)}$
(49d)

where $\xi_{eff}(\omega)=\xi_{1}(\omega)\xi_{2}(\omega)+\tilde{J}^{2}_{m}$
with $\xi_{1}(\omega)=\Gamma^{1}_{eff}/2+i(\Delta^{1}_{m}-\omega)$ and
$\xi_{2}(\omega)=\Gamma^{2}_{eff}/2+i(\Delta^{2}_{m}-\omega)$. The input
quantum noise operators cavity and mechanical modes are characterized by the
following correlations

$\displaystyle\langle\delta\alpha^{in}_{j}(t),\delta\alpha^{in^{\dagger}}_{j}(t^{\prime})\rangle$
$\displaystyle=\delta(t+t^{\prime}),$ (50a)
$\displaystyle\langle\delta\beta^{in^{\dagger}}_{j}(t^{\prime}),\delta\beta^{in}_{j}(t)\rangle$
$\displaystyle=N_{s}\delta(t+t^{\prime}),$ (50b)
$\displaystyle\langle\delta\beta^{in}_{j}(t),\delta\beta^{in}_{j}(t^{\prime})\rangle$
$\displaystyle=M_{s}\delta(t+t^{\prime}),\qquad\text{for j=1, 2}$ (50c)

where $N^{j}_{s}=(n^{j}_{m}+1)\sinh^{2}(r_{j})+n^{j}_{m}\cosh^{2}(r)$ and
$M^{j}_{s}=(2n^{j}_{m}+1)\sinh(r_{j})\cosh(r_{j})$ with
$n^{j}_{m}=[e^{\hbar\omega_{j}/K_{B}T}-1]^{-1}$ is the mean thermal occupation
number of the mechanical modes. Here we assumed that both mechanical
resonators are at same temperature $T$. By using the expressions from Eq. 48
and Eq. 50, one can derive the output spectra defined as

$\displaystyle S^{(1)}_{out}(\omega)$ $\displaystyle=\int[\langle
a^{out\dagger}_{1}(\omega)\,\,a^{out}_{1}(\omega^{\prime})\rangle+\langle
a^{out}_{1}(\omega)\,\,a^{out\dagger}_{1}(\omega^{\prime})\rangle]d\omega^{\prime},$
$\displaystyle=\left|S_{11}(\omega)\right|{}^{2}+\left|S_{12}(\omega)\right|{}^{2}+\left(2N_{s_{1}}+1\right)\left|S_{13}(\omega)\right|{}^{2}$
$\displaystyle+\left(2N_{s_{2}}+1\right)\left|S_{14}(\omega)\right|{}^{2},$
(51a) $\displaystyle S^{(2)}_{out}(\omega)$ $\displaystyle=\int[\langle
a^{out\dagger}_{2}(\omega)\,\,a^{out}_{2}(\omega^{\prime})\rangle+\langle
a^{out}_{2}(\omega)\,\,a^{out\dagger}_{2}(\omega^{\prime})\rangle]d\omega^{\prime},$
$\displaystyle=\left|S_{21}(\omega)\right|{}^{2}+\left|S_{22}(\omega)\right|{}^{2}+\left(2N_{s_{1}}+1\right)\left|S_{23}(\omega)\right|{}^{2}$
$\displaystyle+\left(2N_{s_{2}}+1\right)\left|S_{24}(\omega)\right|{}^{2}.$
(51b)

where
$N_{s_{1,2}}=(n_{m_{1,2}}+1)\sinh^{2}(r_{1,2})+n_{m_{1,2}}\cosh^{2}(r_{1,2})$.

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