# Synthetically enhanced sensitivity using higher-order exceptional point and
coherent perfect absorption

Yao-Dong Hu,1 Yi-Pu Wang,1∗ Rui-Chang Shen,1 Zi-Qi Wang,1 Wei-Jiang Wu,1 J. Q.
You1,2† 1 Zhejiang Province Key Laboratory of Quantum Technology and Device,
School of Physics, and State Key Laboratory for Extreme Photonics and
Instrumentation, Zhejiang University, Hangzhou 310027, China  2 College of
Optical Science and Engineering, Zhejiang University, Hangzhou 310027, China
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

Sensors play a crucial role in advanced apparatuses and it is persistently
pursued to improve their sensitivities. Recently, the singularity of a non-
Hermitian system, known as the exceptional point (EP), has drawn much
attention for this goal. Response of the eigenfrequency shift to a
perturbation $\epsilon$ follows the $\epsilon^{1/n}$-dependence at an $n$th-
order EP, leading to significantly enhanced sensitivity via a high-order EP.
However, due to the requirement of increasingly complicated systems, great
difficulties will occur along the path of increasing the EP order to enhance
the sensitivity. Here we report that by utilizing the spectral anomaly of the
coherent perfect absorption (CPA), the sensitivity at a third-order EP can be
further enhanced owing to the cooperative effects of both CPA and EP. We
realize this synthetically enhanced sensor using a pseudo-Hermitian cavity
magnonic system composed of two yttrium iron garnet spheres and a microwave
cavity. The detectable minimum change of the magnetic field reaches $4.2\times
10^{-21}$ T. It opens a new avenue to design novel sensors using hybrid non-
Hermitian quantum systems.

  * November 2023

##  1 Introduction

Non-Hermitian systems are ubiquitous in nature, owing to the flow of energy,
particles, and information to and from the external degrees of freedom that
lie outside the Hilbert space of the considered system [1]. They exhibit
characteristics distinct from the Hermitian counterparts [2, 3, 4, 5]. A
critical advancement of the non-Hermitian physics is related to the
singularity of the system, known as the exceptional point (EP) [6, 7], where
eigenvalues become degenerate and the corresponding eigenvectors coalesce.
This singularity brings new possibilities for intriguing applications,
including the optical isolation [8, 9, 10], band merging [11, 12], loss-
induced transparency [13], asymmetric mode switching [14], single-mode lasers
[15, 16, 17, 18], and topological chirality [19, 20, 21, 22, 23]. Novel
properties at or around the EP have been achieved in a vast array of
platforms, such as microwave [19, 10], photonic [11, 12, 13, 14, 15, 16, 20,
21, 24, 25, 26], magnetic [27, 28], acoustic [29], mechanical [22], and
optomechanical systems [30, 31].

Recently, EP-based sensors have attracted considerable interest due to the
enhanced response to the external perturbation, providing an efficient way to
boost detection sensitivity [32, 33, 34, 35, 36, 37, 38, 39]. The
eigenfrequency splitting at the $n$th-order EP (i.e., EP$n$) follows a
$\epsilon^{1/n}$-dependence (where $\epsilon$ denotes the perturbation
amplitude), which indicates that a higher-order EP is beneficial to higher
sensitivity. Indeed, successful demonstrations of the enhanced sensitivity
have been realized at the EP3 [33, 34], but great difficulties will occur
along the path of further increasing the EP order to enhance the sensitivity.
This is because the achievement of higher-order EPs requires more precise
parameter controls. Thus, to further increase the sensitivity of EP-based
sensors, it is necessary to contemplate new mechanisms. In this work, we
achieve the EP3 of a pseudo-Hermitian Hamiltonian [40, 41], which is
implemented using a cavity magnonic system via precisely adjusting multi-
parameters [27, 42]. The coherent perfect absorption (CPA) [43, 44, 45, 46,
47, 48] is also used to further enhance the sensitivity at the EP3. The
resulting sensitivity is remarkably enhanced owing to the cooperative effects
of both CPA and EP3. A formulated explanation of this synthetic sensitivity
can be characterized by the detectable minimum change of the magnetic field,

$\delta B_{\rm min}=2\pi\frac{\delta A}{\gamma_{e}G_{\rm CPA}G_{\rm EP3}},$
(1)

where $\gamma_{e}$ is the the gyromagnetic ratio of the ferrimagnetic material
that we use ($\gamma_{e}=2\pi*28$ GHz/T), and $\delta A$ represents the
smallest spectrum change that can be resolved in the experiment (e.g.,
$1\times 10^{-13}$ dB). $G_{\rm CPA}$ and $G_{\rm EP3}$ are sensitivity
factors associated with the CPA and EP3, respectively. It is clear that the
sensitivity induced by these two mechanisms is superimposed in a product form,
thus yielding a vast enhancement. Our work paves a novel way to greatly
enhance the sensitivity using both CPA and EP in a single non-Hermitian
system.

##  2 Pseudo-Hermitian cavity magnonic system

###  2.1 Experimental setup

The cavity magnonic system consists of a three-dimensional (3D) rectangular
cavity ($50\times 25\times 7.5~{}\rm{mm}^{3}$) and two 0.5 mm-diameter yttrium
iron garnet (YIG) spheres [figure 1(a)]. The two ports of the cavity are
connected to a vector network analyzer (VNA) for both loading microwave
signals and implementing measurement. The power ratio and phase difference of
the two signals are controlled by a variable attenuator (VA) and a variable
phase shifter (VPS), respectively. Here, the CPA is realized by precisely
adjusting the VA and VPS to achieve zero output signals from the cavity. The
dissipation rates of the cavity ports are tuned by changing the lengths of
antenna pins inserted into the cavity (A).

Figure 1: Schematic diagrams of the experimental setup and parameter
adjustments. (a) The microwave signal generated by a vector network analyzer
(VNA) is divided into two signals by a beam splitter (BS). These two signals
are subsequently tuned by a variable attenuator (VA) and a variable phase
shifter (VPS), respectively, to match the CPA condition. Then, the signals are
loaded into the cavity via two directional couplers (DC), by which the output
signals are sent back to the VNA. Two 0.5 mm-diameter YIG spheres are placed
in the cavity. The magnetic-field distribution of the cavity mode
$\rm{TE}_{102}$ is shown as a colour diagram. (b) The magnetocrystalline
anisotropy field is employed to tune the frequency of the magnon mode. (c) The
frequency of the magnon mode (circles) is measured by rotating the YIG sphere.
(d) The coupling strength between the magnon mode and the cavity mode
(circles) is tuned by moving the YIG sphere from 0 to $L$/4, where $L$ is the
length of the cavity. The curves in (c) and (d) are theoretical results.

Each YIG sphere is glued on a plastic rod to adjust its position and crystal
axis orientation. After applying a bias magnetic field B in the $z$-direction,
magnon modes are supported in the sphere. Here we focus on the Kittel mode,
i.e., the uniform precession mode of the spin ensemble. By moving the YIG
sphere along the edge of the cavity ($y$-direction), the coupling strength
between the magnon and cavity modes can be precisely controlled. Meanwhile,
rotating the rod can change the frequency of the magnon mode owing to the
magnetocrystalline anisotropy [42]. Specifically, rotating the rod from 0∘ to
180∘ changes the magnon frequency in a range of 0.3 GHz [figures 1(b) and
1(c)]. When the position of the YIG sphere is tuned from 0 to $\pm L/4$, the
coupling strength between the magnon and cavity modes can be tuned from 0 to
12 MHz [figure 1(d)]. With these control parameters, we can simultaneously
realize the CPA and EP3 in the system.

###  2.2 Pseudo-Hermitian system with a third-order exceptional point

The non-Hermitian Hamiltonian of the system with CPA can be written, in the
rotating frame associated with the cavity frequency (details in B):

$H/\hbar=\left(\begin{matrix}i\kappa_{c}&g_{1}&g_{2}\\\
g_{1}&\Delta_{1}-i\gamma_{1}&0\\\ g_{2}&0&\Delta_{2}-i\gamma_{2}\\\
\end{matrix}\right),$ (2)

where $\kappa_{c}\equiv\kappa_{1}+\kappa_{2}-\kappa_{\rm int}>0$ is the
effective gain of the cavity mode, $g_{1}$ ($g_{2}$) is the coupling strength
between the cavity mode and the magnon mode in the YIG sphere 1 (2), which has
decay rate $\gamma_{1}$ ($\gamma_{2}$), and
$\Delta_{1}~{}(\Delta_{2})=\omega_{1}~{}(\omega_{2})-\omega_{\rm{c}}$ is the
frequency detuning between the cavity mode and the corresponding magnon mode.

In the symmetric case, we have two identical YIG spheres
($\gamma_{1}=\gamma_{2}=\gamma$) and the same coupling strengths
$g_{1}=g_{2}=g$. Also, the effective gain of the cavity is tuned to
$\kappa_{c}=2\gamma$, and $\Delta_{1}=-\Delta_{2}\equiv\Delta$. Then, the
Hamiltonian reduces to

$H/\hbar=\left(\begin{matrix}2i\gamma&g&g\\\ g&\Delta-i\gamma&0\\\
g&0&-\Delta-i\gamma\\\ \end{matrix}\right).$ (3)

The eigenvalues of this Hamiltonian are determined by ${\rm Det}\left(H-\Omega
I\right)=0$, which gives the characteristic polynomial,

$\Omega^{3}+c_{1}\Omega+c_{0}=0,$ (4)

with $c_{0}=2i\gamma(\Delta^{2}+\gamma^{2}-g^{2})$, and
$c_{1}=3\gamma^{2}-2g^{2}-\Delta^{2}$. Here $\Omega\equiv\omega-\omega_{c}$ is
the biased eigenvalue of the system and $I$ is the identity matrix. According
to the Cardano formula, the eigenvalues of the system can be explicitly
written as

$\displaystyle\Omega^{(0)}$
$\displaystyle=\sqrt[3]{\frac{c_{0}}{2}+\sqrt{\frac{c_{0}^{2}}{4}+\frac{c_{1}^{3}}{27}}}+\sqrt[3]{\frac{c_{0}}{2}-\sqrt{\frac{c_{0}^{2}}{4}+\frac{c_{1}^{3}}{27}}},$
(5) $\displaystyle\Omega^{(1)}$
$\displaystyle=\sqrt[3]{\frac{c_{0}}{2}+\sqrt{\frac{c_{0}^{2}}{4}+\frac{c_{1}^{3}}{27}}}\left(-\frac{1}{2}+i\frac{\sqrt{3}}{2}\right)-\sqrt[3]{\frac{c_{0}}{2}-\sqrt{\frac{c_{0}^{2}}{4}+\frac{c_{1}^{3}}{27}}}\left(-\frac{1}{2}+i\frac{\sqrt{3}}{2}\right)^{2},$
$\displaystyle\Omega^{(2)}$
$\displaystyle=\sqrt[3]{\frac{c_{0}}{2}+\sqrt{\frac{c_{0}^{2}}{4}+\frac{c_{1}^{3}}{27}}}\left(-\frac{1}{2}+i\frac{\sqrt{3}}{2}\right)^{2}-\sqrt[3]{\frac{c_{0}}{2}-\sqrt{\frac{c_{0}^{2}}{4}+\frac{c_{1}^{3}}{27}}}\left(-\frac{1}{2}+i\frac{\sqrt{3}}{2}\right).$

In figures 2(a) and 2(b), we plot these three eigenvalues versus both $g$ and
$\Delta$.

(i) When $g=\sqrt{\Delta^{2}+\gamma^{2}}$, i.e., $c_{0}=0$, the three
eigenvalues in equation (5) are reduced to

$\Omega^{(0)}=0,~{}~{}~{}~{}~{}\Omega^{(\pm)}=\pm\sqrt{3g^{2}-4\gamma^{2}},$
(6)

which are drawn as the red curves in figures 2(a) and 2(b). Note that the non-
Hermitian Hamiltonian (3) with $g=\sqrt{\Delta^{2}+\gamma^{2}}$ does not have
the parity-time (PT) symmetry, but can still have three real eigenvalues when
$g>g_{\rm EP3}\equiv 2\gamma/\sqrt{3}$. This is because the non-Hermitian
Hamiltonian becomes pseudo-Hermitian in this case [40, 41, 49, 50]. When
$g<g_{\rm EP3}$, $\Omega^{(\pm)}$ become complex, i.e., one of the three
eigenvalues is real ($\Omega^{(0)}=0$) and the other two, $\Omega^{(\pm)}$,
are a complex-conjugate pair. In particular, when $g=g_{\rm EP3}\equiv
2\gamma/\sqrt{3}$, there is $\Delta=\Delta_{\rm EP3}\equiv\gamma/\sqrt{3}$
owing to the relation $g=\sqrt{\Delta^{2}+\gamma^{2}}$. Now, the three
eigenvalues coalesce to $\Omega^{(\pm)}=\Omega^{(0)}=0$. In figures 2(c) and
2(d), these three eigenvalues of the pseudo-Hermitian Hamiltonian are shown
versus $g$ and $\Delta$, respectively.

Figure 2: Eigenvalues of the three-mode non-Hermitian system. (a) and (b) Real
and imaginary parts of the three eigenvalues versus the coupling strength
$g/2\pi$ and frequency detuning $\Delta/2\pi$, as illustrated by the green,
orange and purple surfaces, respectively. The EP2 lines are indicated as the
black dashed curves, and the intersection point of the two curves is the EP3
(red star). (c) Real and imaginary parts of the eigenvalues versus $g/2\pi$,
corresponding to the side views of (a) and (b) by the “blue” eye. The value of
$g/2\pi$ at EP3 is $g_{\rm EP3}/2\pi=3.46$ MHz. (d) Real and imaginary parts
of the eigenvalues versus $\Delta/2\pi$, corresponding to the side views of
(a) and (b) by the “green” eye. The value of $\Delta/2\pi$ at EP3 is
$\Delta_{\rm EP3}/2\pi=1.73$ MHz. (c) and (d) give the three eigenvalues of
the pseudo-Hermitian Hamiltonian.

Corresponding to the three eigenvalues in equation (6), the pseudo-Hermitian
Hamiltonian, i.e., the non-Hermitian Hamiltonian (3) with
$g=\sqrt{\Delta^{2}+\gamma^{2}}$, have the following eigenvectors:

$\displaystyle\ket{\Phi_{0}}=\left(\begin{array}[]{ccc}1\\\
\frac{-\sqrt{g^{2}-\gamma^{2}}-i\gamma}{g}\\\
\frac{\sqrt{g^{2}-\gamma^{2}}-i\gamma}{g}\end{array}\right),$ (10)
$\displaystyle\ket{\Phi_{+}}=\left(\begin{array}[]{ccc}1\\\
\frac{g}{\sqrt{3g^{2}-4\gamma^{2}}-\sqrt{g^{2}-\gamma^{2}}+i\gamma}\\\
\frac{g}{\sqrt{3g^{2}-4\gamma^{2}}+\sqrt{g^{2}-\gamma^{2}}+i\gamma}\end{array}\right),$
(14) $\displaystyle\ket{\Phi_{-}}=\left(\begin{array}[]{ccc}1\\\
\frac{g}{-\sqrt{3g^{2}-4\gamma^{2}}-\sqrt{g^{2}-\gamma^{2}}+i\gamma}\\\
\frac{g}{-\sqrt{3g^{2}-4\gamma^{2}}+\sqrt{g^{2}-\gamma^{2}}+i\gamma}\end{array}\right).$
(18)

At $g=g_{\rm EP3}\equiv 2\gamma/\sqrt{3}$, these three eigenvectors coalesce
to

$\ket{\Phi_{\pm}}=\ket{\Phi_{0}}=\left(\begin{array}[]{ccc}1\\\
\frac{-1-i\sqrt{3}}{2}\\\ \frac{1-i\sqrt{3}}{2}\end{array}\right),$ (19)

revealing that $g_{\rm EP3}\equiv 2\gamma/\sqrt{3}$ is indeed the third-order
exceptional point (EP3) of the pseudo-Hermitian Hamiltonian.

(ii) The three eigenvalues in equation (5) have EP2 lines when
$27c_{0}^{2}+4c_{1}^{3}=0$. In this case, the three eigenvalues reduce to

$\displaystyle\Omega^{(0)}=2\sqrt[3]{\frac{c_{0}}{2}},$
$\displaystyle\Omega^{(1)}=i\sqrt{3}\sqrt[3]{\frac{c_{0}}{2}},$
$\displaystyle\Omega^{(2)}=-i\sqrt{3}\sqrt[3]{\frac{c_{0}}{2}},$ (20)

Furthermore, when $c_{0}=0$, it follows from $27c_{0}^{2}+4c_{1}^{3}=0$ that
$c_{1}=0$ as well, which corresponds to the critical parameters $g=g_{\rm
EP3}\equiv 2\gamma/\sqrt{3}$, and $\Delta=\Delta_{\rm
EP3}\equiv\gamma/\sqrt{3}$. Now, the three eigenvalues in equation (2.2)
coalesce to $\Omega^{(0)}=\Omega^{(1)}=\Omega^{(2)}=0$. This corresponds to
the case where the EP2 lines insect at the EP3, as shown in figures 2(a) and
2(b).

##  3 Results and discussions

###  3.1 Output spectra at and away from EP3

In the experiment, we utilize a small variation of the external magnetic field
as the perturbation, which results in a frequency change $\Delta_{B}$ for the
Kittel mode of each YIG sphere. The perturbed pseudo-Hermitian Hamiltonian has
been shown in equation (21), where the system has the eigenvalue shift
$\Delta_{\omega}\equiv\Omega^{\prime}-\Omega$ under the perturbation
$\Delta_{B}$. In figures 3(a) and 3(b), we measure the output spectra to
demonstrate the response of the pseudo-Hermitian system to $\Delta_{B}$ at and
away from the EP3. The darkest red areas correspond to the system with (or
nearly with) CPA under zero (or small) perturbation $\Delta_{B}$. It is clear
that each of the three eigenvalue has a frequency shift in the case of
$g>g_{\rm EP3}$ when applying the perturbation $\Delta_{B}$ [figure 3(b)],
while these frequency shifts become the same at the EP3 ($g=g_{\rm EP3}$)
because all eigenvalues coalesce there [figure 3(a)]. Hereafter, we focus on
the central eigenvalue $\Omega=0$ of the pseudo-Hermitian Hamiltonian, so
$\Omega^{\prime}=\Delta_{\omega}$.

Figure 3: Output spectra and sensitivity factors. (a) Total output spectrum at
the EP3 ($g=g_{\rm EP3}=2\pi\times 3.46$ MHz) versus the perturbation
$\Delta_{B}/2\pi$, where the dashed and dotted vertical lines correspond to
$\Delta_{B}/2\pi=0$ and $0.025$ MHz, respectively. (b) Total output spectrum
away from the EP3 ($g=2\pi\times 4.59~{}{\rm MHz}>g_{\rm EP3}$) versus the
perturbation $\Delta_{B}/2\pi$. Dashed curves in (a) and (b) are
eigenfrequency shifts $\Delta_{\omega}=\Omega^{\prime}-\Omega$ obtained by
diagonalizing the Hamiltonian, i.e. equation (21). (c) Log-log plot of the
frequency shift $\Delta_{\omega}/2\pi$ of the central eigenvalue versus
$|\Delta_{B}|/2\pi$ at $g/2\pi=3.46$ and $4.59$ MHz, respectively. The slopes
1/3 and 1 are fitted using equations (27) and (30), respectively. (d)
Sensitivity factor at the EP3 versus $|\Delta_{B}|/2\pi$. (e) Total output
spectra extracted at $\Delta/2\pi=0$ and $0.025$ MHz, corresponding to the two
vertical lines in (a). The spectral dip at $\Delta_{B}=0$ has the minimum
value $-91.5$ dB. (f) CPA-related sensitivity factor $G_{\rm CPA}$ versus the
eigenfrequency shift $\Delta_{\omega}/2\pi$.

###  3.2 Perturbation-induced eigenfrequency shift

The perturbation of the external magnetic field can induce frequency change of
the Kittel mode in each YIG sphere. In our experiment, we use two identical
YIG spheres, so these frequency changes can be nearly identical. By
considering the magnetic-field perturbation, the pseudo-Hermitian Hamiltonian,
i.e., the Hamiltonian (3) with $\Delta=\sqrt{g^{2}-\gamma^{2}}$, can be
converted to

$H^{\prime}/\hbar=\begin{pmatrix}2i\gamma&g&g\\\
g&\sqrt{g^{2}-\gamma^{2}}+\Delta_{B}-i\gamma&0\\\
g&0&-\sqrt{g^{2}-\gamma^{2}}+\Delta_{B}-i\gamma\\\ \end{pmatrix},$ (21)

where $\Delta_{B}$ is the frequency change of the Kittel mode induced by the
magnetic-field perturbation. From the eigenvalue equation ${\rm
Det}\left(H^{\prime}-\Omega^{\prime}I\right)=0$, we can obtain the relation
between the eigenvalue
$\Omega^{\prime}(\Delta_{B})\equiv\omega^{\prime}(\Delta_{B})-\omega_{c}$ and
the frequency change $\Delta_{B}$, which is

$\frac{2\Delta_{B}-\Omega^{\prime}}{g^{2}}+\frac{4\left(g^{2}-\gamma^{2}\right)\left(\Omega^{\prime}-\Delta_{B}\right)}{\left[{\left(\Omega^{\prime}-\Delta_{B}\right)^{2}+g^{2}}\right]^{2}-4\left(g^{2}-\gamma^{2}\right)\left(\Omega^{\prime}-\Delta_{B}\right)^{2}}=0,$
(22)

where
$\Omega^{\prime}(\Delta_{B})\equiv\omega^{\prime}(\Delta_{B})-\omega_{c}$
reduces to the eigenvalue $\Omega\equiv\omega-\omega_{c}$ of the pseudo-
Hermitian Hamiltonian when $\Delta_{B}=0$.

For a sufficiently small perturbation $\Delta_{B}$, we can ignore the higher-
order terms $O\left(\Delta_{B}^{2}\right)$ and write the relation between
$\Delta_{B}$ and $\Omega^{\prime}$ as

$\Delta_{B}=\frac{\Omega^{\prime 3}\left(\Omega^{\prime
2}+g^{2}\right)-\left(3g^{2}-4\gamma^{2}\right)\left(\Omega^{\prime
2}+g^{2}\right)\Omega^{\prime}}{g^{4}+2\Omega^{\prime 2}\left(\Omega^{\prime
2}+2g^{2}\right)-\left(3g^{2}-4\gamma^{2}\right)\left(4\Omega^{\prime
2}+g^{2}\right)}.$ (23)

Based on equation (23), we can study the response of the eigenfrequency to the
perturbation.

We define the perturbation-induced eigenfrequency shift as

$\Delta_{\omega}\equiv\omega^{\prime}\left(\Delta_{B}\right)-\omega=\Omega^{\prime}\left(\Delta_{B}\right)-\Omega.$
(24)

Here we focus on the central branch of the eigenvalues of the pseudo-Hermitian
Hamiltonian, i.e., $\Omega=\Omega^{(0)}=0$ [cf. equation (6)].

(i) At the EP3, i.e., $g=g_{\rm EP3}\equiv 2\gamma/\sqrt{3}$, equation (23)
reduces to

$\Delta_{B}=\frac{\Omega^{\prime 3}\left(\Omega^{\prime 2}+g_{\rm
EP3}^{2}\right)}{g_{\rm EP3}^{4}+2\Omega^{\prime 2}\left(\Omega^{\prime
2}+2g_{\rm EP3}^{2}\right)}.$ (25)

For a small perturbation, we can assume that $\Omega^{\prime}$ is close to
$\Omega=\Omega^{(0)}=0$ such that $\Omega^{\prime}\ll g$. Then, it follows
from equation (25) that

$\Delta_{B}=\frac{\Omega^{\prime 3}}{g_{\rm
EP3}^{2}}=\frac{\Delta_{\omega}^{3}}{g_{\rm EP3}^{2}},$ (26)

i.e.,

$\Delta_{\omega}=g_{\rm EP3}^{2/3}{\Delta_{B}}^{1/3},$ (27)

where the ${\Delta_{B}}^{1/3}$ dependence is characteristic of the EP3.

From equation (27), we obtain the sensitivity factor at the EP3:

$G_{\rm EP3}\equiv\left(\frac{\Delta_{\omega}}{\Delta_{B}}\right)_{\rm
EP3}=\frac{g_{\rm EP3}^{2/3}}{{\Delta_{B}}^{2/3}}.$ (28)

It tends to infinity when $\Delta_{B}\rightarrow 0$, indicating the singular
behavior at the EP3. In figure 3(d), we numerically show the relation between
the sensitivity factor ${G}_{\rm EP3}$ and the perturbation $\Delta_{B}$.

(ii) Away from the EP3, there is $g\neq g_{\rm EP3}\equiv 2\gamma/\sqrt{3}$.
For a small perturbation, we can still assume that $\Omega^{\prime}\ll g$ and
then equation (23) is reduced to

$\Delta_{B}=-\frac{\left(3g^{2}-4\gamma^{2}\right)\Omega^{\prime}}{g^{2}-\left(3g^{2}-4\gamma^{2}\right)},$
(29)

which gives a linear dependence of the eigenfrequency shift on the
perturbation $\Delta_{B}$,

$\Delta_{\omega}=\left(1-\frac{g^{2}}{3g^{2}-4\gamma^{2}}\right)\Delta_{B}.$
(30)

Figure 3(c) shows the frequency shift $\Delta_{\omega}$ of the central
eigenvalue versus $|\Delta_{B}|$. Theoretically, at the EP3, this
eigenfrequency shift is $\Delta_{\omega}=g^{2/3}\Delta_{B}^{1/3}$, while
$\Delta_{\omega}\sim\Delta_{B}$ away from the EP3. These theoretical results
fit well with the experimental data. The sensitivity factor at the EP3 [figure
3(d)] can be defined by

$G_{\rm EP3}\equiv\left(\frac{\Delta_{\omega}}{\Delta_{B}}\right)_{\rm
EP3}=g^{2/3}\Delta_{B}^{-2/3}.$ (31)

Compared with the case away from the EP3, the eigenvalue shift
$\Delta_{\omega}$ shows a steep slope at the EP3, yielding a more profound
amplification of the eigenfrequency shift under the same perturbation
$\Delta_{B}$.

###  3.3 Enhanced sensing using CPA

In our experiment, the parameters of the system are chosen to achieve the
pseudo-Hermitian Hamiltonian, i.e., the Hamiltonian (3) with
$g=\sqrt{\Delta^{2}+\gamma^{2}}$, but the CPA condition is not obeyed in some
cases for comparison. With these parameters, it follows from equation (C) in C
that the total output spectrum is given by

$\displaystyle\left|S_{\rm
tot}(\Omega)\right|^{2}=\frac{\left({p+1}\right)\left|{\left[{m(\Omega)+{{2\sqrt{\kappa_{1}\kappa_{2}}e^{i\phi}}/\sqrt{p}}+2\kappa_{1}}\right]+in(\Omega)}\right|^{2}}{{m(\Omega)}^{2}+{n(\Omega)}^{2}}$
$\displaystyle+\frac{\left({\sqrt{p}e^{-i\phi}-\sqrt{\kappa_{1}/\kappa_{2}}}\right)k(\Omega)}{{m(\Omega)}^{2}+{n(\Omega)}^{2}},$
(32)

with

$\displaystyle
k(\Omega)=4\left({\sqrt{\kappa_{1}\kappa_{2}}+\kappa_{2}e^{i\phi}/\sqrt{p}}\right)^{2}\left({\sqrt{p}e^{-i\phi}-\sqrt{\kappa_{1}/\kappa_{2}}}\right)$
$\displaystyle+4\left(\sqrt{\kappa_{1}\kappa_{2}}+\kappa_{2}e^{i\phi}/\sqrt{p}\right)\left[{m(\Omega)+2\sqrt{\kappa_{1}\kappa_{2}}e^{i\phi}/\sqrt{p}+2\kappa_{1}}\right],$
(33) $\displaystyle
m\left(\Omega\right)=2\gamma-2\kappa_{1}-2\kappa_{2}-\frac{g^{2}\gamma}{\left({\Omega-\Delta}\right)^{2}+\gamma^{2}}-\frac{g^{2}\gamma}{\left({\Omega+\Delta}\right)^{2}+\gamma^{2}},$
(34) $\displaystyle
n\left(\Omega\right)=\Omega-\frac{g^{2}\left({\Omega-\Delta}\right)}{\left({\Omega-\Delta}\right)^{2}+\gamma^{2}}-\frac{g^{2}\left({\Omega+\Delta}\right)}{\left({\Omega^{\prime}+\Delta}\right)^{2}+\gamma^{2}},$
(35)

where $g=\sqrt{\Delta^{2}+\gamma^{2}}$, and $\Omega=\omega-\omega_{c}$, while
$p$ and $\phi$ denote the power ratio and phase difference between the two
input field, respectively. When the perturbation is applied, the output
spectrum $|S^{\prime}_{\rm tot}\left(\Omega^{\prime}\right)|^{2}$ has the same
form as equation (3.3), but $\Omega$ is replaced by
$\Omega^{\prime}=\omega^{\prime}-\omega_{c}$ and $\pm\Delta$ in equation.
(3.3) are replaced by $\Delta+\Delta_{B}$ and $-\Delta+\Delta_{B}$,
respectively.

In the presence of CPA, it follows from equation (45) that
$\sqrt{p}e^{-i\phi}=\sqrt{\kappa_{1}/\kappa_{2}}$. The output spectrum in
equation (3.3) is reduced to

$|{{\rm S}_{\rm tot}\left(\Omega\right)}|_{\rm
CPA}^{2}={{\left({{\kappa_{1}/\kappa_{2}}+1}\right)\frac{{\Omega}^{2}\left({\Omega}^{2}-3g^{2}+4\gamma^{2}\right)^{2}}{\left({\Omega}^{2}-g^{2}\right)^{2}+4{\Omega}^{2}\gamma^{2}}}/\left[{m\left(\Omega\right)}^{2}+{n\left({\Omega}\right)}^{2}\right]}.$
(36)

Here $|S_{\rm tot}\left(\Omega\right)|_{\rm CPA}^{2}$ reaches its minimum
value 0 when $\Omega=0$ or $\Omega=\pm\sqrt{3g^{2}-4\gamma^{2}}$, which are
exactly the eigenvalues of the pseudo-Hermitian Hamiltonian given in equation
(6).

Without the perturbation $\Delta_{B}$, our system has been tuned to exhibit
the CPA, and it can exhibit nearly CPA under a small $\Delta_{B}$. In
decibels, the output spectrum of the system with CPA has an extremely sharp
dip, but a less sharp dip occurs in the system nearly with CPA [cf. figure
3(e)].In the present experiment, we focus on the central branch of the three
eigenvalues, i.e., $\Omega=\Omega^{(0)}=0$. We can define a sensitivity factor
related to the CPA:

$G_{\rm{CPA}}\equiv 2\pi\frac{\Delta|S_{\rm tot}^{({\rm
min})}|^{2}}{\Delta_{\omega}}=\frac{10{\rm log_{10}}|{S^{\prime}}_{\rm
tot}^{({\rm min})}(\Delta_{\omega})|^{2}-10{\rm log_{10}}|S_{\rm tot}^{({\rm
min})}(0)|^{2}_{\rm CPA}}{(\Delta_{\omega}/2\pi)},$ (37)

where $|S_{\rm tot}^{({\rm min})}(0)|^{2}_{\rm CPA}$ is the minimum value of
the output spectrum in the presence of CPA, where $\Omega=0$, and
$|{S^{\prime}}_{\rm tot}^{({\rm min})}(\Delta_{\omega})|^{2}$ is the minimum
value of the corresponding output spectrum under the perturbation
$\Delta_{B}$, where $\Omega^{\prime}=\Delta_{\omega}$. Because
$|{S^{\prime}}_{\rm tot}^{({\rm min})}(\Delta_{\omega})|^{2}\neq 0$ and
$|S_{\rm tot}^{({\rm min})}(0)|^{2}_{\rm CPA}=0$, $10\log_{10}|S_{\rm
tot}^{({\rm min})}(0)|^{2}_{\rm CPA}\rightarrow-\infty$, giving rise to
$G_{\rm{CPA}}\rightarrow\infty$ in the ideal case. However, in a practical
experiment, $10\log_{10}|S_{\rm tot}^{({\rm min})}(0)|^{2}_{\rm{CPA}}$ cannot
reach $-\infty$, but it can reach $-91.5$ dB here at the EP3, resulting from
the limited measurement precision of the VNA and the limited adjusting
accuracy of the system parameters. Nevertheless, this minimum value can still
produce a significantly large enhancement factor $G_{\rm CPA}$ at the EP3 [cf.
figure 3(f)], especially in the case of small $\Delta_{\omega}$.

###  3.4 Synthetic sensitivity in the presence of both CPA and EP3

Based on the experimental data of the output spectra, we obtain the observed
changes of the minimum values of the output spectra $\Delta|S_{\rm tot}^{({\rm
min})}|^{2}$ at the EP3 versus the perturbation $\Delta_{B}$, as shown in
figure 4(a). From this relationship between $\Delta|S_{\rm tot}^{({\rm
min})}|^{2}$ and $\Delta_{B}$, we can define the synthetically enhanced
sensitivity of the system in the presence of both CPA and EP3, $G_{\rm
syn}\equiv 2\pi\Delta|S_{\rm tot}^{({\rm min})}|^{2}/\Delta_{B}$, which is
readily given by $G_{\rm syn}=G_{\rm CPA}G_{\rm EP3}$. Figure 4(b) shows the
obtained $G_{\rm syn}$ versus the perturbation $\Delta_{B}$, in comparison
with the corresponding $G_{\rm CPA}$ and $G_{\rm EP3}$. Because the
enhancements related to the CPA and EP3 are combined in a product form, the
synthetic enhancement factor becomes significantly greater than the respective
enhancement factors [figure 4(b)]. This clearly demonstrates the distinct
advantage of using both CPA and EP3 in a single non-Hermitian system to
produce a sensor with high sensitivity.

Figure 4: Synthetically enhanced sensitivity in the presence of both CPA and
EP3. (a) Minimum-value changes of the output spectra $\Delta|S_{\rm
tot}^{({\rm min})}|^{2}$ at the EP3 versus the perturbation $\Delta_{B}/2\pi$.
(b) Synthetic sensitivity factor $G_{\rm syn}$ in comparison with the
respective sensitivity factors $G_{\rm CPA}$ and $G_{\rm EP3}$. Curves are the
corresponding fitting results.

###  3.5 Detectable minimum change of the magnetic field

Finally, we discuss the limit of the minimum magnetic-field change that can be
detected by the current synthetically enhanced sensor. For the YIG sphere, the
relation between the applied magnetic field and the frequency of the uniform
magnon mode is

$\omega_{m}=\gamma_{e}|\vec{B}|+\omega_{m,0},$ (38)

where $\gamma_{e}/2\pi=28$ GHz/T denotes the gyromagnetic ratio and
$\omega_{m,0}$ is determined by the anisotropy field. In the synthetically
enhanced sensor, the adjustment of the magnetic field is limited by the
smallest change of the current supply, i.e., $1\times 10^{-4}$ A, in our
electromagnet. The measured frequency change of the magnon mode is
$\Delta_{B}/2\pi=(\omega^{\prime}_{m}-\omega_{m})/2\pi\approx 0.025~{}{\rm
MHz}$, where $\omega^{\prime}_{m}$ ($\omega_{m}$) is the frequency of the
magnon mode with (without) the perturbation. It follows from
$\Delta_{B}=\gamma_{e}\delta{B}$ that the corresponding magnetic-field change
is $\delta{B}\approx 9\times 10^{-7}$ T.

According to equation (27), the frequency shift of the eigenvalue at EP3 is

$\Delta_{\omega}={g_{\rm
EP3}}^{2/3}{\Delta_{B}}^{1/3}=\left(2\gamma/\sqrt{3}\right)^{2/3}\times(2\pi\times
0.025)^{1/3}\approx 2\pi\times 0.67~{}({\rm MHz}),$ (39)

where $\gamma/2\pi\approx 3$ MHz is the measured damping rates of the two
nearly identical YIG spheres.

In our experiment, the minimum value of the total output spectrum changes 21.5
dB around the EP3 when, e.g., $\Delta_{B}/2\pi=0.025$ MHz, as illustrated in
figure 3(e). The synthetic sensitivity factor is $G_{\rm syn}=21.5/0.025=860$.
Correspondingly, the CPA-related sensitivity factor and the sensitivity factor
at the EP3 are $G_{\rm CPA}=32.1$ and $G_{\rm EP3}=26.8$, respectively.
Indeed, as shown in figure 4(b), the synthetic sensitivity factor is much
larger than the respective sensitivity factors. Note that the smallest
spectrum change that can be resolved by the vector network analyzer used in
our experiment (KEYSIGHT PNA-L Network Analyzer N5232B) is $\delta A\sim
1\times 10^{-13}$ dB. From the definition of the synthetic sensitivity factor,
it follows that the detectable minimum change of the magnetic field $\delta
B_{\rm min}$ is determined by

$G_{\rm syn}=2\pi\frac{\delta A}{\gamma_{e}\delta B_{\rm min}}.$ (40)

Thus, we have

$\delta B_{\rm min}=\frac{2\pi\times 10^{-13}}{(2\pi\times 28\times
10^{3})\times 860}\approx 4.2\times 10^{-21}~{}({\rm T}),$ (41)

which indicates that the detectable minimum change of the magnetic-field can
reach the magnitude $\sim 10^{-21}$ T.

##  4 Conclusions

In summary, we have demonstrated a synthetically enhanced magnetic sensor
using both CPA and EP3 in a pseudo-Hermitian cavity magnonic system. The
sensitivity enhancements associated with these two mechanisms are superimposed
in a product manner, yielding a greatly amplified synthetic sensitivity. This
synthetically enhanced sensitivity offers a new way to explore promising
applications of precision metrology via hybrid non-Hermitian quantum systems.

This work is supported by the National Key Research and Development Program of
China (No. 2022YFA1405200), National Natural Science Foundation of China (No.
$92265202$, No. $11934010$, and No. $12174329$), and the Fundamental Research
Funds for the Central Universities (No. $2021{\rm FZZX}001$-$02$).

##  Appendix A Dissipation rates of the cavity ports

The dissipation rates of the cavity ports are tuned by changing the lengths of
antenna pins inserted into the cavity. The shorter the antenna pin is inserted
into the cavity, the smaller the dissipation rate of the corresponding port
is.

Figure 5: Precise adjustments of the dissipation rates of the two ports, i.e.,
$\kappa_{1}$ and $\kappa_{2}$, by tuning the length decrements of the antenna
pins. The shorter the pins are inside the cavity, the dissipation rates will
be smaller. When the length decrements are larger than 3 mm, the dissipation
rates of the two ports become nearly unchanged. This corresponds to the
intrinsic dissipation rate $\kappa_{\rm int}$ of the cavity. The length
decrements of the two pins used in our experiment are marked by the red and
blue arrows.

In figure A1, we gradually reduce the length of the antenna pin inserted into
the cavity, and finally the cavity loss converges to a finite value that
corresponds to the intrinsic dissipation rate of the cavity, which is about 2
MHz in our experiment. Besides, the dissipation rates of the two ports satisfy
$(\kappa_{1}+\kappa_{\rm int})/2\pi=6$ MHz and $(\kappa_{2}+\kappa_{\rm
int})/2\pi=6$ MHz, as is illustrated with arrows in figure A1.

##  Appendix B Non-Hermitian Hamiltonian of the system in the presence of
coherent perfect absorption

In the rotating frame associated with the cavity frequency $\omega_{c}$, the
Hamiltonian can be written as

$H_{s}/\hbar=\Delta_{1}b_{1}^{\dagger}b_{1}+\Delta_{2}b_{2}^{\dagger}b_{2}+g_{1}\left(a^{\dagger}b_{1}+ab_{1}^{\dagger}\right)+g_{2}\left(a^{\dagger}b_{2}+ab_{2}^{\dagger}\right),$
(42)

where $\Delta_{1(2)}=\omega_{1(2)}-\omega_{c}$ is the frequency detuning
between the Kittel mode in the YIG sphere 1(2) and the cavity mode. The
Langevin equations of the coupled system can now be written as

$\displaystyle\dot{a}$
$\displaystyle=-\frac{i}{\hbar}\left[a,H_{s}\right]-\left(\kappa_{1}+\kappa_{2}+\kappa_{\rm
int}\right)a+\sqrt{2\kappa_{1}}a_{1}^{({\rm
in})}+\sqrt{2\kappa_{2}}a_{2}^{({\rm in})},$ $\displaystyle\dot{b}_{1}\\!$
$\displaystyle=-\frac{i}{\hbar}\left[b_{1},H_{s}\right]-\gamma_{1}b_{1},$ (43)
$\displaystyle\dot{b}_{2}\\!$
$\displaystyle=-\dfrac{i}{\hbar}\left[b_{2},H_{s}\right]-\gamma_{2}b_{2},$

where $\kappa_{1}$, $\kappa_{2}$ and $\kappa_{\rm int}$ are the dissipation
rates of the cavity mode associated with port 1, port 2 and the intrinsic loss
of the cavity, respectively, while $\gamma_{1}$ and $\gamma_{2}$ are the
dissipation rates of the Kittel modes in the two YIG spheres. The input and
output fields of the $i$th port, $a_{i}^{({\rm in})}$ and $a_{i}^{(\rm out)}$,
obey the input-output relation

$a_{i}^{({\rm in})}+a_{i}^{({\rm
out})}=\sqrt{2\kappa_{i}}a,~{}~{}~{}~{}i=1,2.$ (44)

When the coherent perfect absorption (CPA) occurs, $a_{1}^{({\rm
out})}=a_{2}^{({\rm out})}=0$, so

$a_{1}^{({\rm in})}=\sqrt{2\kappa_{1}}a,~{}~{}~{}~{}a_{2}^{({\rm
in})}=\sqrt{2\kappa_{2}}a.$ (45)

The quantum Langevin equations in equation (B) become

$\displaystyle\dot{a}$
$\displaystyle=-\frac{i}{\hbar}\left[a,H_{s}\right]+\left(\kappa_{1}+\kappa_{2}-\kappa_{\rm
int}\right)a,$ $\displaystyle\dot{b}_{1}\\!$
$\displaystyle=-\frac{i}{\hbar}\left[b_{1},H_{s}\right]-\gamma_{1}b_{1},$ (46)
$\displaystyle\dot{b}_{2}\\!$
$\displaystyle=-\frac{i}{\hbar}\left[b_{2},H_{s}\right]-\gamma_{2}b_{2},$

which can be rewritten as

$\displaystyle\dot{a}$ $\displaystyle=-\frac{i}{\hbar}\left[a,H_{\rm
eff}\right],$ $\displaystyle\dot{b}_{1}\\!$
$\displaystyle=-\frac{i}{\hbar}\left[b_{1},H_{\rm eff}\right],$ (47)
$\displaystyle\dot{b}_{2}\\!$
$\displaystyle=-\frac{i}{\hbar}\left[b_{2},H_{\rm eff}\right],$

where

$H_{\rm
eff}/\hbar=i\kappa_{c}a_{1}^{\dagger}a_{1}+(\Delta_{1}-i\gamma_{1})b_{1}^{\dagger}b_{1}+(\Delta_{2}-i\gamma_{2})b_{2}^{\dagger}b_{2}+g_{1}\left(a^{\dagger}b_{1}+ab_{1}^{\dagger}\right)+g_{2}\left(a^{\dagger}b_{2}+ab_{2}^{\dagger}\right),$
(48)

with $\kappa_{c}\equiv\kappa_{1}+\kappa_{2}-\kappa_{\rm int}$. When
$\kappa_{c}>0$, the CPA provides an effective gain to the cavity. In the
matrix form, this non-Hermitain Hamiltonian can be written as

$H_{\rm eff}/\hbar=\begin{pmatrix}{i\kappa_{c}}&g_{1}&g_{2}\\\
g_{1}&{\Delta_{1}-i\gamma_{1}}&0\\\ g_{2}&0&{\Delta_{2}-i\gamma_{2}}\\\
\end{pmatrix},$ (49)

.

##  Appendix C Formulism of the output spectrum

The quantum Langevin equations of the cavity magnonic system have been
illustrated in equation (B). It notes that when applying the Fourier
transformations

$x(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\\!\\!x(\omega)e^{-i\omega
t}d\omega,~{}~{}~{}x=a,b_{1},b_{2},$ (50)

we obtain the Langevin equations in the frequency domain,

$\displaystyle-i\omega
a(\omega)=-\frac{i}{\hbar}\left[{a(\omega),H_{s}(\omega)}\right]-\left({\kappa_{1}+\kappa_{2}+\kappa_{\rm
int}}\right)a(\omega)$ $\displaystyle+\sqrt{2\kappa_{1}}a_{1}^{({\rm
in})}(\omega)+\sqrt{2\kappa_{2}}a_{2}^{({\rm in})}(\omega),$
$\displaystyle-i\omega
b_{j}(\omega)=-\frac{i}{\hbar}\left[b_{j}(\omega),H_{s}(\omega)\right]-\gamma_{j}b_{j}(\omega),~{}~{}j=1,2.$
(51)

Using the input-output relation

$a_{j}^{({\rm in})}(\omega)+a_{j}^{({\rm
out})}(\omega)=\sqrt{2\kappa_{j}}a(\omega),~{}~{}j=1,2,$ (52)

we can obtain the general relations between the output fields, i.e.,
$a_{1}^{({\rm out})}$ and $a_{2}^{({\rm out})}$, and the input fields, i.e.,
$a_{1}^{({\rm in})}$ and $a_{2}^{({\rm in})}$:

$\displaystyle a_{1}^{({\rm out})}=r_{11}a_{1}^{({\rm in})}+t_{12}a_{2}^{({\rm
in})},$ $\displaystyle a_{2}^{({\rm out})}=r_{22}a_{2}^{({\rm
in})}+t_{21}a_{1}^{({\rm in})},$ (53)

with the transmission and reflection coefficients

$\displaystyle
t_{12}(\omega)=t_{21}(\omega)=-\frac{2\sqrt{\kappa_{1}\kappa_{2}}}{m+in},$
$\displaystyle r_{jj}(\omega)=-1-\frac{2\kappa_{j}}{m+in},~{}~{}j=1,2,$ (54)

where

$\displaystyle m=-\left({\kappa_{1}+\kappa_{2}+\kappa_{\rm
int}}\right)-\sum_{j=1}^{2}\frac{g_{j}^{2}\gamma_{j}}{\left({\omega-\omega_{j}}\right)^{2}+\gamma_{j}^{2}},$
$\displaystyle
n=\left({\omega-\omega_{c}}\right)-\sum_{j=1}^{2}\frac{g_{j}^{2}\left({\omega-\omega_{j}}\right)}{\left({\omega-\omega_{j}}\right)^{2}+\gamma_{j}^{2}}.$
(55)

Note that the output field of each port is the sum of the reflection signal at
that port and the transmission signal from the other port. The amplitudes of
the output fields at the two ports of the cavity are

$\displaystyle{\rm
S}_{1}(\omega)=r_{11}(\omega)\sqrt{p}e^{-i\phi}+t_{12}(\omega),$
$\displaystyle{\rm
S}_{2}(\omega)=r_{22}(\omega)+t_{21}(\omega)\sqrt{p}e^{-i\phi},$ (56)

where $p$ and $\phi$ denote, respectively, the power ratio and phase
difference between the two input fields.

The total output spectrum is obtained by summing the output-field intensities
at the two ports:

$\displaystyle\left|{\rm S_{\rm tot}(\omega)}\right|^{2}$
$\displaystyle\equiv\left|{\rm S_{1}(\omega)}\right|^{2}+\left|{\rm
S_{2}(\omega)}\right|^{2}$
$\displaystyle=\frac{\left({p+1}\right)\left|{\left[{m(\omega)+{{2\sqrt{\kappa_{1}\kappa_{2}}e^{i\phi}}/\sqrt{p}}+2\kappa_{1}}\right]+in(\omega)}\right|^{2}}{{m(\omega)}^{2}+{n(\omega)}^{2}}$
$\displaystyle+\frac{\left({\sqrt{p}e^{-i\phi}-\sqrt{\kappa_{1}/\kappa_{2}}}\right)k(\omega)}{{m(\omega)}^{2}+{n(\omega)}^{2}},$
(57)

where

$\displaystyle k(\omega)$
$\displaystyle=4\left({\sqrt{\kappa_{1}\kappa_{2}}+\kappa_{2}e^{i\phi}/\sqrt{p}}\right)^{2}\left({\sqrt{p}e^{-i\phi}-\sqrt{\kappa_{1}/\kappa_{2}}}\right)$
$\displaystyle+4\left(\sqrt{\kappa_{1}\kappa_{2}}+\kappa_{2}e^{i\phi}/\sqrt{p}\right)\left({m+2\sqrt{\kappa_{1}\kappa_{2}}e^{i\phi}/\sqrt{p}+2\kappa_{1}}\right).$
(58)

.

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