# Optomechanically Induced Transparency on Exceptional Surfaces

Y. Pan  School of Physics, Zhengzhou University, Zhengzhou 450001, China
H.-L. Zhang  Academy for Quantum Science and Technology, Zhengzhou University
of Light Industry, Zhengzhou 450002, China  School of Electronics and
Information, Zhengzhou University of Light Industry, Zhengzhou 450001, China
Y.-F. Jiao  Academy for Quantum Science and Technology, Zhengzhou University
of Light Industry, Zhengzhou 450002, China  School of Electronics and
Information, Zhengzhou University of Light Industry, Zhengzhou 450001, China
D.-Y. Wang<EMAIL_ADDRESS>School of Physics, Zhengzhou University,
Zhengzhou 450001, China  S.-L. Su<EMAIL_ADDRESS>School of Physics,
Zhengzhou University, Zhengzhou 450001, China  H. Jing<EMAIL_ADDRESS>Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of
Ministry of Education, Department of Physics and Synergetic Innovation Center
for Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan
410081, China  Academy for Quantum Science and Technology, Zhengzhou
University of Light Industry, Zhengzhou 450002, China

###### Abstract

Exceptional points (EPs) are singularities in non-Hermitian systems, where the
system transmission spectrum varies significantly at the phase transition
point. Here, we propose a practical scheme to study the changes of the
optomechanically induced transparency (OMIT) spectrum on the exceptional
surface (ES), which is formed by designing the structure of the waveguide in a
non-Hermitian cavity optomechanical system. By comparing the transmission
spectra of the system at different normal points, EPs on the same or different
ESs, and exceptional derived points, we find that the peak-valley conversion
of the system transmission spectra is obtained at the phase transition point
and the arbitrary manipulation of the system transmission spectrum can be
realized by moving the system on the same or different ESs. Furthermore, the
phenomena of conversion and enhancement of the fast-slow light in the system
transmission spectra have also been discovered in our researches. Different
from the isolated EP, our proposal can discuss the system properties at
different EPs, can find a richer transmission spectrum, and can provide more
convenient options for experimental implementation, which paves the way for
studying the nature of non-Hermitian systems in a higher dimension.

##  I INTRODUCTION

Cavity optomechanics (COM), as a field exploring the interaction between
optical fields and macroscopic mechanical resonators, has witnessed rapid
development in recent years [1, 2, 3, 4] and yielded numerous research topics,
such as quantum entanglement [5, 6, 7, 8, 9, 10, 11], non-classical mechanical
states [12, 13, 14, 15], ground-state cooling of motion [16, 17],
nonreciprocal optical transmission [18, 19], normal mode splitting [20, 21,
22], optoelectronic quantum transducers [23, 24, 25], phonon lasing [26, 27],
mechanical squeezing [28], weak force detection [29, 30], etc.
Optomechanically induced transparency (OMIT) is similar to electromagnetically
induced transparency [31, 32, 33] and comes from the destructive quantum
interference between the probe field and the anti-Stokes scattered field [34,
35, 36], which has been realized in various experiments [37, 38, 35, 39].
OMIT, as an important phenomenon in quantum information and quantum sensing,
has given rise to numerous practical applications, such as wavelength
conversion [40], single-photon routers [41], light storage [42], charge
measurement [43], slow light effect [44], highly sensitive sensors [45], etc.
What are we interested in here are the changes of OMIT in the non-Hermitian
cavity optomechanical systems [46, 47, 48, 49].

Exceptional Points (EPs), as peculiar singularities in open quantum systems
[50, 51, 52, 53, 54, 55], are the non-Hermitian degeneracy locations and well-
known for their numerous applications [56, 57, 58]. EPs are the singularity
branch points of system in the parameter space where two or more eigenvalues
and corresponding eigenstates coalesce [59, 60, 61, 62]. The phenomenon of
eigenstates coalescing creates a unique eigenstate space and leads to a series
of remarkable experimental findings [63, 64, 65, 66]. However, EPs are
isolated in the parameter space so they are highly sensitive to any
unavoidable perturbation, such as fabrication errors, experimental
uncertainties, and environmental noises. To this end, researchers must
precisely manipulate various parameters to ensure the attainment of EPs in
non-Hermitian systems. For example, two nanopositioners are used to strictly
control the positions of particles with nanometer-level precision to guarantee
the presence of EPs. In addition, it is essential to ensure that the two
particles have extremely small but distinct differences [67, 68].
Alternatively, in $\mathcal{PT}$-symmetric implementation [69, 70, 71, 72],
systemic gain-loss must be strictly balanced while accurately controlling
coupling strength [73]. Once these stringent conditions cannot be satisfied,
the system can not achieve EP, which results in a reversed change in the OMIT
spectrum. To overcome these difficulties and enhance the system robustness,
the concept of exceptional surface (ES) has been proposed [74, 75, 76, 77] and
observed in numerous experiments [78, 79, 80].

Based on these above remarkable works [67, 74, 79], we have investigated OMIT
in a non-Hermitian cavity optomechanical system, which consists of a
Whispering Gallery Mode (WGM) microresonator coupled with one single external
nanoparticle to induce symmetric backscattering [79]. Nonreciprocal
transmission is achieved through structural design, thereby realizing
asymmetric coupling [79, 74]. With the combined effect of symmetric and
asymmetric coupling, the system can achieve ES. The formation of ES results in
the continuous connection of EPs in parameter space, which significantly
increases the scheme flexibility and robustness in the parameter selection for
EPs appearing [74]. Due to the numerous EPs provided by ES, we can investigate
the OMIT spectra at different EPs and compare them with the OMIT spectra at
Normal points (NPs). Additionally, we obtained derived points by adjusting the
phase of the unidirectional coupling light [81, 79]. These derived EPs not
only possess the characteristics of EP but also some characteristics of NP.
Furthermore, by adjusting the coupling parameters and phase, we have
successfully controlled the generation of fast light and slow light. Our work
involves multiple EPs and their derived points, which broaden our
understanding of the impact of EPs on OMIT spectra and enrich the OMIT
spectrum series. In addition, our scheme exhibits greater flexibility and
robustness in the parameter selection for EPs. Compared with other schemes,
changing complex optical gain [82] or manipulating the particle position and
size accurately [67, 68], our scheme offers a simpler and more cost-effective
solution to study OMIT in the non-Hermitian systems.

The remaining parts of the paper are organized as follows. In Sec. II, we
present the Hamiltonian of the COM system in the rotating frame and then
derive the conditions for generating ES. Subsequently, we calculate the
transmission coefficient of the probe field. In Sec. III, we discuss the OMIT
spectrum at different points and demonstrate the generation of the fast and
slow light at the local region. In Sec. IV, we briefly discuss the feasibility
of the experiment. The conclusion is presented in Sec. V.

##  II Theoretical model

Figure 1: (a) The schematic of the proposed system in experiments, where PM is
the phasemodulator. (b) Schematic of the non-Hermitian optomechanical system,
where the CW and CCW photon modes are coupled with the symmetric coupling $J$
induced by a nanoparticle and the asymmetric coupling $\lambda$ induced by the
structural design. And the breathing phonon mode $b$ is coupled to the CW and
CCW photon modes with the same optomechanical coupling coefficient $G$. (c)
Conceptional illustration of the ES in a higher dimensional parameter space.
(d) The eigenvalues coalesce in our proposal. The dashed line represents the
general condition when $J=0$. And the solid line corresponds the phase
transition point with $J=t_{0}\sqrt{\gamma_{1}\gamma_{2}}$, which can form the
ES in a higher dimensional parameter space.

Inspired by a recent experiment, which demonstrates sensitivity enhancement in
a non-Hermitian photonic system [79], we consider a non-Hermitian
optomechanical system comprising of a WGM microresonator and a tapered fiber,
as shown in Fig. 1. Due to manufacturing limitations, the WGM microresonator
includes imperfections inevitably, which leads to backscattering of
transmitted light. For simplicity, we can attribute these backscattering to
the coupling of a nanoparticle. At this point, the two transmitted modes of
WGM microresonator, denoted as the clockwise (CW) mode $a_{cw}$ and the
counterclockwise (CCW) mode $a_{ccw}$, are coupled to each other with the
symmetric coupling strength $J$. To study OMIT in a non-Hermitian
optomechanical system, the structure of the fiber is redesigned [74, 79],
which results in the asymmetric one-way coupling between the two transmitted
modes with effective coupling constant $\lambda$.

In our proposal, the microresonator, with resonance frequency $\omega_{0}$ and
intrinsic decay rate $\gamma_{0}$, supports a mechanical mode with mechanical
frequency $\omega_{m}$ and damping rate $\gamma_{m}$. The optomechanical
coupling strength between the mechanical mode and both the optical transmitted
modes is denoted as $G=\omega_{0}/R$, where $R$ is the radius of the
microresonator. The system is driven by a strong pump field with frequency
$\omega_{c}$ and a weak probe field with frequency $\omega_{p}$. The
amplitudes of the pump field and probe field are
$E_{c}=\sqrt{P_{c}/\hbar\omega_{c}}$, $E_{p}=\sqrt{P_{p}/\hbar\omega_{p}}$,
where $P_{c}$ and $P_{p}$ is the laser power of the pump field and probe
field. The Hamiltonian describing the total system can be written as
($\hbar=1$)

$\displaystyle H$ $\displaystyle=H_{0}+H_{int}+H_{cw}+H_{ccw},$ (1)
$\displaystyle H_{0}$ $\displaystyle=\
\omega_{0}\hat{a}_{cw}^{\dagger}\hat{a}_{cw}+\omega_{0}\hat{a}_{ccw}^{\dagger}\hat{a}_{ccw}+\frac{\hat{p}^{2}}{2m}+\frac{1}{2}m\omega_{m}^{2}\hat{x}^{2},$
$\displaystyle H_{int}$
$\displaystyle=-G\hat{x}\left(\hat{a}_{cw}^{\dagger}\hat{a}_{cw}+\hat{a}_{ccw}^{\dagger}\hat{a}_{ccw}\right)+J(\hat{a}_{cw}^{\dagger}\hat{a}_{ccw}+\hat{a}_{ccw}^{\dagger}\hat{a}_{cw}),$
$\displaystyle H_{cw}$
$\displaystyle=i\sqrt{\gamma_{1}}\left[\left(E_{c}e^{-i\omega_{c}t}+E_{p}e^{-i\omega_{p}t}\right)\hat{a}_{cw}^{\dagger}-\text{H.c.}\right],$
$\displaystyle H_{ccw}$
$\displaystyle=i\sqrt{\gamma_{2}}\left[\left(t_{1}E_{c}e^{-i\omega_{c}t}+t_{2}E_{p}e^{-i\omega_{p}t}\right)\hat{a}_{ccw}^{\dagger}-\text{H.c.}\right]$
$\displaystyle\quad+\lambda\hat{a}_{cw}\hat{a}_{ccw}^{\dagger}.$

$\lambda$ is the asymmetric one-way coupling constant, which is described by
$\lambda=i\sqrt{\gamma_{1}\gamma_{2}}t_{3}$. Here, $\gamma_{1}$ and
$\gamma_{2}$ are the coupling coefficients between the fiber and the
microresonator at Port 1 and Port 2, respectively. The displacement and
momentum operators of the mechanical oscillator are denoted by $\hat{x}$ and
$\hat{p}$. $m$ is the effective mass of the mechanical oscillator.
$t_{j}=t_{0}e^{i\phi_{j}}$ is the transmission coefficient of different
transmission light between Port 1 and Port 2 in the redesigned fiber, which is
determined by the frequency of the transmission light and the loss coefficient
of the fiber.

Since the single photon optomechanical coupling strength is too small, here we
consider only the non-Hermitian properties of the optical subsystem [83, 84].
We designate the corresponding eigenvalues of the non-Hermitian system as
$\chi^{E}_{\pm}=\omega_{\pm}-i\kappa_{\pm}$, which can be derived as

$\displaystyle\omega_{\pm}=\pm\sqrt{\alpha+\beta},$ (2)
$\displaystyle\kappa_{\pm}=\pm\sqrt{\alpha-\beta},$ (3)

where

$\displaystyle\alpha$
$\displaystyle=\frac{\sqrt{J^{4}+2J^{3}t_{0}\sqrt{\gamma_{1}\gamma_{2}}\sin(\phi_{3})+J^{2}t_{0}^{2}\gamma_{1}\gamma_{2}}}{2},$
(4) $\displaystyle\beta$
$\displaystyle=\frac{J^{2}+Jt_{0}\sqrt{\gamma_{1}\gamma_{2}}\sin(\phi_{3})}{2}.$
(5)

For simplicity, the symmetric coupling $J$ between the two transmitted modes
is considered to be a real number [68, 79]. According to the theory of non-
Hermitian physics, the eigenvalues of system would coalesce at the phase
transition point. Here, $\omega_{+}=\omega_{-}$ and $\kappa_{+}=\kappa_{-}$
mean that the eigenvalues of system coalesce when the system parameters
$\left\\{J,~{}t_{0},~{}\gamma_{1},~{}\gamma_{2},~{}\phi_{3}\right\\}$ satisfy
a specific relationship. At this time, the surface formed by the parameter
relationship is called ES in non-Hermitian physics. For instance, when $J=0$,
meaning the symmetric coupling between the two transmitted modes is
nonexistent, the eigenvalues are $\omega_{\pm}=0$ and $\kappa_{\pm}=0$, as
indicated by the red dashed line in Fig. 1(d). In this article, we mainly
study the more general situation, i.e., $J\neq 0$. When the phase $\phi_{3}$
is tuned to $1.5\pi$, Eqs. (2) and (3) can be simplified as

$\displaystyle\omega_{\pm}=\pm\sqrt{\frac{|J^{2}-Jt_{0}\sqrt{\gamma_{1}\gamma_{2}}|}{2}+\frac{J^{2}-Jt_{0}\sqrt{\gamma_{1}\gamma_{2}}}{2}},$
(6)
$\displaystyle\kappa_{\pm}=\pm\sqrt{\frac{|J^{2}-Jt_{0}\sqrt{\gamma_{1}\gamma_{2}}|}{2}-\frac{J^{2}-Jt_{0}\sqrt{\gamma_{1}\gamma_{2}}}{2}}.$
(7)

It is easy to observe that when the symmetric coupling satisfied the following
relationship

$J=t_{0}\sqrt{\gamma_{1}\gamma_{2}},$ (8)

the two eigenvalues of the system are same. And the result is shown in the
orange solid line of Fig. 1(d). The point corresponding to the above
relationship in Eq. (8) is the so-called phase transition point where the
first-order quantum phase transition occurs in the non-Hermitian system. The
surface that unfolds in the parameter space according to the relationship is
called ES, which is the main research content in this article.

We have pointed out the formation of ES in the studied non-Hermitian system.
Next, we will derive the transmission spectrum of the system and study the
effect of ES on the transmission spectrum. In the rotating frame with the pump
field frequency,
$\hat{U}=\exp{[i\omega_{c}(\hat{a}^{\dagger}_{cw}\hat{a}_{cw}+\hat{a}^{\dagger}_{ccw}\hat{a}_{ccw})t]}$,
the dynamical quantum Langevin equations of the system can be given by

$\displaystyle\dot{\hat{a}}_{cw}$ $\displaystyle=-(i\Delta-
igx+\gamma)\hat{a}_{cw}-iJ\hat{a}_{ccw}+\sqrt{\gamma_{1}}E_{c}$
$\displaystyle\quad+\sqrt{\gamma_{1}}E_{p}e^{-i\delta
t}-\sqrt{\gamma_{1}\gamma_{2}}t_{3}\hat{a}_{ccw},$
$\displaystyle\dot{\hat{a}}_{ccw}$ $\displaystyle=-(i\Delta-
igx+\gamma)\hat{a}_{ccw}-iJ\hat{a}_{cw}\sqrt{\gamma_{2}}t_{1}E_{c}$
$\displaystyle\quad+\sqrt{\gamma_{2}}t_{2}E_{p}e^{-i\delta
t}+\sqrt{\gamma_{1}\gamma_{2}}t_{3}\hat{a}_{cw},$
$\displaystyle\ddot{\hat{x}}$
$\displaystyle=-\gamma_{m}\dot{\hat{x}}-\omega_{m}^{2}\hat{x}^{2}+\frac{g}{m}\left(\hat{a}_{cw}^{\dagger}\hat{a}_{cw}+\hat{a}_{ccw}^{\dagger}\hat{a}_{ccw}\right),$
(9)

where $\delta=\omega_{p}-\omega_{c}$, $\Delta=\Delta_{a}+J$,
$\Delta_{a}=\omega_{0}-\omega_{c}$, and
$\gamma=(\gamma_{0}+\gamma_{1}+\gamma_{2})/2$. When the system reaches its
steady state, all dynamical variables no longer change with time. Meanwhile,
the steady-state values of the dynamical variables can be given by

$\displaystyle\overline{a}_{cw}$
$\displaystyle=\frac{E_{c}\left(\sqrt{\gamma_{1}}(i\Delta-
ig\overline{x}+\gamma)-t_{1}\sqrt{\gamma_{2}}\left(t_{3}\sqrt{\gamma_{1}\gamma_{2}}+iJ\right)\right)}{(i\Delta-
ig\overline{x}+\gamma)^{2}+\gamma_{1}\gamma_{2}t_{3}^{2}+J^{2}},$
$\displaystyle\overline{a}_{ccw}$
$\displaystyle=\frac{E_{c}\left(t_{1}\sqrt{\gamma_{2}}(i\Delta-
ig\overline{x}+\gamma)+\sqrt{\gamma_{1}}\left(t_{3}\sqrt{\gamma_{1}\gamma_{2}}-iJ\right)\right)}{(i\Delta-
ig\overline{x}+\gamma)^{2}+\gamma_{1}\gamma_{2}t_{3}^{2}+J^{2}},$
$\displaystyle\overline{x}$
$\displaystyle=\frac{g}{m\omega_{m}^{2}}\left\lvert\hat{a}_{cw}^{\dagger}\hat{a}_{cw}+\hat{a}_{ccw}^{\dagger}\hat{a}_{ccw}\right\rvert.$
(10)

To study the OMIT in our system, we calculate the optical transmission rate by
linearizing the system operators, i.e., expanding the operator as the sum of
its steady value and a small fluctuation
$\hat{O}(t)=\overline{O}+\delta\hat{O}(t)$ [35]. Specifically, the system
operators are expanded as

$\displaystyle\hat{x}$ $\displaystyle=\overline{x}+\delta\hat{x}e^{-i\delta
t}+\delta\hat{x}^{*}e^{i\delta t},$ (11) $\displaystyle\hat{a}_{i}$
$\displaystyle=\overline{a_{i}}+\delta\hat{a}_{i}^{-}e^{-i\delta
t}+\delta\hat{a}_{i}^{+}e^{i\delta t}\quad(i=cw,ccw).$ (12)

Substituting Eqs. (11) and (12) into Eq. (II), we can obtain

$\displaystyle
f_{1}\delta\hat{a}_{{cw}}^{-}+\left(t_{3}\sqrt{\gamma_{2}}+iJ\right)\delta\hat{a}_{{ccw}}^{-}-ig\overline{a}_{{cw}}\delta\hat{x}$
$\displaystyle=\sqrt{\gamma_{1}}E_{p},$ $\displaystyle
f_{2}\delta\hat{a}_{{cw}}^{+*}+\left(t_{3}^{*}\sqrt{\gamma_{2}}-iJ^{*}\right)\delta\hat{a}_{{cw}}^{+*}+ig\overline{a}_{{cw}}^{*}\delta\hat{x}$
$\displaystyle=0,$ $\displaystyle
f_{1}\delta\hat{a}_{{ccw}}^{-}-\left(t_{3}\sqrt{\gamma_{2}}-iJ\right)\delta\hat{a}_{{cw}}^{-}-ig\overline{a}_{{ccw}}\delta\hat{x}$
$\displaystyle=t_{2}\sqrt{\gamma_{2}}E_{p},$ $\displaystyle
f_{2}\delta\hat{a}_{{ccw}}^{+*}-\left(t_{3}^{*}\sqrt{\gamma_{2}}+iJ^{*}\right)\delta\hat{a}_{{cw}}^{+*}+ig\overline{a}_{{cw}}^{*}\delta\hat{x}$
$\displaystyle=0,$
$\displaystyle\chi^{-1}\delta\hat{x}-g\sum_{i={cw,ccw}}(\overline{a}_{i}^{*}\delta\hat{a}_{i}^{-}+\overline{a}_{i}\delta\hat{a}_{i}^{+*})$
$\displaystyle=0,$ (13)

where $f_{1,2}=\gamma-i\delta\pm i\left(\Delta-g\overline{x}\right)$ and
$\chi^{-1}=m(\omega_{m}^{2}-\delta^{2}-i\delta\gamma_{m})$. The results of
$\delta\hat{a}^{-}_{{cw}}$ and $\delta\hat{a}_{ccw}^{-}$ can be calculated by
solving Eq. (II). However, as the results are too cumbersome, these results
are not shown here. Now, we can derive the output fields at Port 1 and Port 2
using the standard input-output relationships

$\displaystyle s_{out2}$
$\displaystyle=t_{2}s_{in}-t_{3}\sqrt{\gamma_{1}}\delta\hat{a}_{cw}^{-}-\sqrt{\gamma_{2}}\delta\hat{a}_{ccw}^{-},$
(14)

where $s_{in}$ and $s_{out2}$ are the input and out field operators,
respectively. The transmission coefficient is defined by $t=s_{out2}/s_{in}$.
Then, the transmission rate is obtained by

$T=\left\lvert t\right\rvert^{2}=\left\lvert
t_{2}-\frac{t_{3}\sqrt{\gamma_{1}}\delta\hat{a}_{cw}^{-}+\sqrt{\gamma_{2}}\delta\hat{a}_{ccw}^{-}}{E_{p}}\right\rvert^{2}.$
(15)

In this section, we have discussed the appearance of the ES and derived the
transmission spectrum of system. This lays the groundwork for our forthcoming
discussion regarding the effects of ES on OMIT spectrum and the group delay.

##  III Result and discussion

Figure 2: The transmission spectrum as a function of the optical detuning
$\omega_{p}-\omega_{0}$. (a) The transmission spectrum for the different phase
$\phi_{3}$, where we also do not take into account the effects of nanoparticle
($J=0$) and asymmetric coupling ($\gamma_{2}=0$). (b) The effects of
nanoparticle $J=8~{}\mathrm{MHz}$. (c) The effects of asymmetric coupling
$\gamma_{2}=8~{}\mathrm{MHz}$. And the other parameters are chosen as
$\gamma_{1}=8~{}\mathrm{MHz}$ and $t_{0}=1$.

First, we give the selected experimentally feasible parameter values in our
proposal, i.e., $R=34.5~{}\mathrm{\mu m}$, $\omega_{0}=193~{}\mathrm{THz}$,
$\gamma_{0}=6.43~{}\mathrm{MHz}$, $m=50~{}\mathrm{ng}$,
$\omega_{m}=147~{}\mathrm{MHz}$, $P_{c}=1~{}\mathrm{mW}$,
$\gamma_{m}=0.24~{}\mathrm{MHz}$ [85, 68, 67]. To ensure completeness and
facilitate comparison, we begin by considering the standard OMIT spectrum,
which excludes any particles and unidirectional coupling. Initially, we adjust
the phase $\phi_{3}=1.5\pi$, which is consistent with the previous analytic
result. Subsequently, nanoparticle-induced symmetric backscattering and
asymmetric coupling are introduced respectively to stratify the changes of
OMIT. These results are shown in Fig. 2. We can observe that the adjustment of
the phase of the probe field leads to a complete flip and change in the shape
of the OMIT spectrum. When we introduce the external nanoparticle, the
resulting spectrum is consistent with the OMIT spectrum generated by an
external nanoparticle incorporation [67]. Different from the previous schemes,
the introduction of asymmetric coupling results in the non-Hermitian nature of
the system, i.e., $\gamma_{2}\neq 0$. At this time, we can study the change of
the OMIT spectrum on the ES.

To compare the differences between different EPs on the ES and the
distinctions between EPs and NPs, we have selected two different ESs, which
are $t_{0}=1$ and $t_{0}=0.8$, as shown in Fig. 3(a). Additionally, we have
chosen three EPs to compare the differences within the same or different ESs.
Moreover, we have also chosen two NPs to contrast the disparities between NPs
and EPs on the OMIT spectrum.

Figure 3: (a) shows the points appearing in the follow discussions. (b,c) are
the comparison between EPs on the same and different ES, respectively. (d,e)
are the comparison between EPs and NPs. Please refer to Tab. 1 for the data of
all points in Appendix A.

Figure 3 illustrates the variation of transmittance with the detuning of the
probe field at the different points. The trends of the OMIT spectrum for the
three EPs are similar, where two transparent windows are obtained. The left
window is induced by optomechanical coupling, while the right one comes from
the coupling between the two transmitted modes. When the system parameters are
set at the EP, satisfied the relationship in Eq. (8), the right transparent
peak appears, and the results are shown in Figs. 3(b) and (c). On the same ES,
we can see that $\rm EP_{2}$ shows higher transmittance compared to $\rm
EP_{1}$, which is attributed to the higher coupling rate facilitating enhanced
interaction with the cavity. This intensified interaction leads to more
pronounced physical phenomena, resulting in higher transmission at the
transparent windows for $\rm EP_{2}$. Differently, $\rm EP_{1}$ exhibits
higher transmittance globally compared to $\rm EP_{3}$ on the other ES due to
the smaller transmission rate $t_{0}$ at $\rm EP_{3}$. This discrepancy causes
partial light absorption within the fiber, independent of the detection field
frequency, ultimately lowering the overall transmittance of $\rm EP_{3}$
compared to $\rm EP_{1}$. Furthermore, we also selected two additional NPs to
contrast with EPs. Unlike traditional periodic EPs [67], which feature only a
peak-valley conversion pair (a transparent peak turning into an absorption
valley), the ES can exhibit more than one conversion pair. Not only can they
shift from a transparent peak to an absorption valley, but they can also
transition from an absorption valley to a transparent peak [see the
highlighted zone in Figs. 3(d) and (e)]. Those results challenge the
conventional perception of EPs’ impact on the OMIT spectrum and broadens their
properties.

Figure 4: (a) The transmission spectrum at the $\rm EP_{1}$ and its derived
point $\rm EP1^{(1)}$. (b) The comparison between the transmission spectrum at
the $\rm EP1^{(1)}$ and it when the pump field does not exist $E_{c}=0$. (c)
The transmission spectrum at $\rm EP1$ and $\rm EP1^{(2)}$. (d) The
transmission spectrum at $\rm EP_{1}$ and $\rm NP_{3}$ ($\phi_{3}=0.3\pi$).

One of the advantages of ES is its wide range of adjustable parameters. In
addition to the aforementioned coupling coefficient and transmission rate
parameters $\left\\{J,~{}t_{0},~{}\gamma_{1},~{}\gamma_{2}\right\\}$, there is
a more extensive set of parameters available for adjustment, such as phase
($\phi_{1},~{}\phi_{2},~{}\phi_{3})$. By tuning the phase, we can not only
move the system away from or towards the EP, but also generate more derived
points of the original EP, such as $\rm EP^{(1)}_{1},\rm EP^{(2)}_{2}$ with
different phases [see Fig. 4]. Surprisingly, these derived points of EP not
only exhibit similar physical properties as $\rm NP_{1}$ [see Fig. 4(c)], but
they can even eliminate the transparency windows caused by optomechanical
interaction [see Fig. 4(b)]. For $\phi_{1}=0.55\pi$, the left transmission
peak disappears [see the blue dotted curve in Fig. 4(a)], which is almost
completely overlapped with the transmission spectrum when the pump light is
turned off. In other words, manipulating the phase of the pump light allows us
to offset the optomechanical interaction influence. That is because the
introduction of an extra phase causes the destructive interference between the
pump light entering the cavity from Port 1 and that from Port 2. Consequently,
this interference prevents the effective delivery of driving power to the
cavity. In Fig. 4(d), when we change $\phi_{3}$, the parameter relationship in
Eq. (8) will no longer be satisfied, which represents the system is out of ES.
Meanwhile, if $\phi_{3}=0.3\pi$, the transmission spectrum will be flipped.
The original transparency window becomes an absorption valley, while the
absorption valley turns into a transmission window, as shown in Fig. 4(d).

We have previously discussed the impact of EP and NP on transmission spectra.
There are two methods to shift the system from the EP in the ES to the NP. The
first method involves adjusting the coupling strength, e.g.,
$J,~{}\gamma_{1},~{}\gamma_{2}$, so that they no longer satisfy the
relationship $J=t_{0}\sqrt{\gamma_{1}\gamma_{2}}$. The second method entails
altering the phase $\phi_{3}$, thereby disrupting the effectiveness of the
equation $J=t_{0}\sqrt{\gamma_{1}\gamma_{2}}$ in forming an ES. We find that
both of these methods can induce a peak-valley conversion pair in the
transmission spectrum.

Figure 5: Group delay of the probe light $\tau_{p}$ as a function of the
detuning $\delta_{p}$. (a) The group delay of the two points in Fig. 3(d). (b)
The group delay of the two points in Fig. 4(d).

In addition to discussing the effect of ES on the transmission spectra, the
phenomenon of group velocity dispersion, i.e., the slow-light effects, is also
affected by ES. Here, the optical group delay is characterized by

$\tau_{g}=\frac{d\>\rm arg(t)}{d\delta_{p}}.$ (16)

For the first case (changing the coupling strength), there are two peak-valley
conversion pairs for $\rm EP_{1}$ and $\rm NP_{1}$, as shown in Fig. 3(d). The
first peak-valley conversion pair occurs at $\omega_{0}-\omega_{p}\approx 0$,
where the transmission is converted to absorption when the system changes from
$\rm EP_{1}$ to $\rm NP_{1}$. However, the second peak-valley conversion pair
is the opposite, which occurs at $\omega_{0}-\omega_{p}\approx
18~{}\mathrm{MHz}$. At this time, the fast-slow light effects in both places
are also completely different, as shown in Fig 5. We find that both $\rm
EP_{1}$ and $\rm NP_{1}$ exhibit slow light behavior for the first peak-valley
conversion location, which suggests the formation of slow-fast light is not
significantly influenced by the peak-valley conversion in the transmission
spectrum at $\omega_{0}-\omega_{p}\approx 0$. On the other hand, the fast-slow
light effects are opposite for the second peak-valley conversion location when
the system changes from $\rm EP_{1}$ to $\rm NP_{1}$. Finally, we also discuss
the influence of phase on the fast-slow light effect and the results are shown
in Fig 5(b). We can see that the slow light effect is enhanced at both peak-
valley conversion locations when the system changes from EP
($\phi_{3}=1.5\pi$) to NP ($\phi_{3}\neq 1.5\pi$).

##  IV Experimental feasibility

In this section, the focus is primarily on discussing the feasibility of
theoretical models in experiments. The cavity optomechanical system of the WGM
type has been extensively studied experimentally [82, 68, 86, 85, 87].
Therefore, the theoretical model consists of two crucial components to be
experimentally implemented. The first part involves the coupling between the
transmitted modes of WGM and the external nanoparticle. The second part deals
with controlling the transmission rate and phase of light. Based on these
experimental and theoretical studies [85, 68, 67], we have chosen the feasible
parameters according to those WGM experiments. The chosen parameters are
$R=34.5~{}\mathrm{\mu m}$, $\omega_{0}=193~{}\mathrm{THz}$,
$\gamma_{0}=6.43~{}\mathrm{MHz}$, $m=50~{}\mathrm{ng}$,
$\omega_{m}=147~{}\mathrm{MHz}$, $P_{c}=1~{}\mathrm{mW}$,
$\gamma_{m}=0.24~{}\mathrm{MHz}$ throughout the discussions. Next, let’s focus
on the key parts of this proposal.

In experiments, the coupling strength between the fiber and the WGM
transmitted mode can reach up to tens of MHz at its highest and can be
controlled by varying the distance [88, 79], which completely meets the range
of our model (changing from $1.23~{}\mathrm{MHz}$ to $16.4~{}\mathrm{MHz}$).
The coupling coefficient $J$ between the nanoparticle and the WGM transmitted
mode depends on the volume of the particle in the WGM evanescent field, which
can be controlled by adjusting the size of the particle or their distance
[67]. According to those existing experiments [68], $J$ can be varied from
$6.4~{}\mathrm{MHz}$ to $8~{}\mathrm{MHz}$, which can be fully achievable in
our proposal. The unidirectional coupling can be achieved by adding an
isolator at the end of the waveguide in experiments[78, 79, 89]. Furthermore,
the phase of light fields can be adjusted by the phase modulator(PM). The
transmission coefficient $t_{0}$ of the waveguide can be changed by changing
the waveguide length in actual experiment [79]. Anyway, the experimental
parameters and techniques involved in our protocol are fully feasible under
the current experimental conditions.

##  V CONCLUSION

In conclusion, we have proposed a scheme to study the behavior of OMIT on ES
in a non-Hermitian WGM-type optomechanical system, where the non-Hermitian
nature of the system comes from the ingenious structural design of waveguide.
Compared to a single EP, ES is not only able to provide an infinite number of
EPs to discuss the effect of phase transition point on the system OMIT, but
also is more robust for parameter selection, which is more friendly for
experimental design [74, 79]. We respectively study the influence of NP, EP
(on the same or different ESs), and the exceptional derived points on the
system OMIT. And we find some interesting phenomena in the transmission
spectrum of the system, such as peak-valley conversion induced by phase
transition and arbitrary manipulation of transmission spectra by using
different EPs. Finally, we study the fast-slow light effects of transmission
spectroscopy at different exceptional and normal points and find the
conversion and enhancement effects of the fast-slow light of the system. Our
research can provide theoretical guidance for information transmission in non-
Hermitian quantum systems, and it is expected to realize ES-enhanced robust
quantum sensing.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
under Grants (No. 12204424, No. 12147149, No. 12147156, No. 12274376, No.
12074346, No. U21A20434, No. 11935006, No. 11774086), the China Postdoctoral
Science Foundation (No. 2022M722889), the Natural Science Foundation of Henan
Province under Grant (No. 212300410085), the Hunan Provincial Major Sci-Tech
Program (Grant No. 2023ZJ1010), and the Science and Technology Innovation
Program of Hunan Province (Grant No. 2020RC4047).

##  Appendix A The data of EPs and NPs

Table 1: The data of EPs and NPs | $J$ | $\gamma_{1}$ | $\gamma_{2}$ | $t_{0}$ | $\phi_{1}$ | $\phi_{2}$ | $\phi_{3}$
---|---|---|---|---|---|---|---
$\rm EP_{1}$ | 8 MHz | 8 MHz | 8 MHz | 1 | 0 | 0 | 1.5$\pi$
$\rm EP_{2}$ | 12 MHz | 12 MHz | 12 MHz | 1 | 0 | 0 | 1.5$\pi$
$\rm EP_{3}$ | 6.4 MHz | 8 MHz | 8 MHz | 0.8 | 0 | 0 | 1.5$\pi$
$\rm NP_{1}$ | 16.4 MHz | 8 MHz | 8 MHz | 1 | 0 | 0 | 1.5$\pi$
$\rm NP_{2}$ | 1.23 MHz | 8 MHz | 8 MHz | 0.8 | 0 | 0 | 1.5$\pi$
$\rm EP^{(1)}_{1}$ | 8 MHz | 8 MHz | 8 MHz | 1 | 0.55$\pi$ | 0 | 1.5$\pi$
$\rm EP^{(2)}_{1}$ | 8 MHz | 8 MHz | 8 MHz | 1 | 0 | $\pi$ | 1.5$\pi$
$\rm NP_{3}$ | 8 MHz | 8 MHz | 8 MHz | 1 | 0 | 0 | 0.3$\pi$

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