# Controllable optical sideband generation
by breaking mechanical Parity-Time symmetry

Souvik Mondal<EMAIL_ADDRESS>Kapil Debnath
<EMAIL_ADDRESS>Electronics and Electrical Communication
Engineering Department, IIT Kharagpur, Kharagpur, West Bengal, 721302, India

###### Abstract

The interaction between light and mechanical vibrations in a cavity is often
exploited to produce higher order sidebands (HOS) /combs, which are used in
optical communication networks, spectroscopy, and others. Although a
considerable study of optomechanically induced HOS has been done, its proper
control and manipulation using only continuous wave (CW) laser drive are still
to be explored. Here, we employed mechanical Parity-Time ($\mathcal{PT}$)
Symmetric structure with optically induced gain and loss. It has allowed us to
manipulate the flow of mechanical energies (phonons) between the cavities,
which has consequences on the optical response of the cavities. Based on our
numerical investigations, we found that the higher order optical sidebands
start to emerge by breaking $\mathcal{PT}$ symmetry. We precisely controlled
the number of higher order sideband lines by adjusting the coupling rate
between the cavities with fixed drive power. In addition, we observe that the
Exceptional Point (EP) induces the formation of two synchronized higher order
optical sideband spectra, which opens a promising EP based platform towards
realization of optical readout of various mechanical synchronization
phenomena, memory applications, sensing, synchronization of remote clock time
and others.

††preprint: APS/123-QED

##  I Introduction

Cavity optomechanics has become the subject of extensive study lately. The co-
localization of the optical and the mechanical modes in an optomechanical (OM)
cavity has produced many interesting outcomes ranging from classical to
quantum phenomena [1]. One of them is the self-sustained mechanical
oscillations/phonon lasing in the cavity. Once we drive the cavity with a blue
detuned CW pump laser above a certain threshold power [2], the mechanical
vibrations in the cavity starts to amplify. The interaction of the CW pump
laser with the mechanical vibrations produces Stokes and Anti-Stokes scattered
light. The beating of the CW pump laser with the Stokes/Anti-Stokes scattered
light results in periodic evolution of the intracavity field intensity at the
mechanical resonance frequency, which again drives the mechanical vibrations
coherently in the cavity. In this manner of dynamical backaction, the
mechanical oscillations become self-sustained [3]. The increased driving CW
laser power [4] or the dynamical backaction of large self sustained
oscillation amplitude [5] creates significant modulation in the intracavity
field intensity evolution. As a result, we get higher order sidebands
(HOS)/comb in the optical spectrum with frequency line spacing decided by the
mechanical resonance frequency. Mohammad Ali Miri et al. [6] showed that the
formation of such HOS/comb in an OM cavity is analogous to cascaded four-wave
mixing in microresonator Kerr frequency combs [7]. The HOS can also be
generated if one drives the cavity with a red detuned pump laser and an
external probe field. We require an external probe as the beating of the CW
pump laser with the probe at mechanical resonance frequency ensures coherent
mechanical oscillations [8, 9].

The optical HOS/comb has many exciting applications in spectroscopy [10],
optical clocks [11] and others [12]. So, the control and manipulation of
optomechanically induced HOS/comb become crucial in a low power integrated
structure. In the existing literature, the sidebands in the OM system with
pump-probe drive can be manipulated in many ways. The relative phase and the
power of the probe field is utilized to manipulate the optical spectrum in the
cavity [8]. The photon tunneling rate in coupled passive optical cavities
provides additional means to control the sidebands [13]. Ling Yan-He [14]
showed the use of active-passive coupled optical cavities in $\mathcal{PT}$
symmetry structure to enhance spectral lines. In general, the $\mathcal{PT}$
symmetry structure has been utilized in the field of optomechanics to show the
low-power phonon lasing [15], low-power chaos generation [16], enhanced OMIT
[17], mass sensing [18] and others. Apart from these, hybrid atom-
optomechanical system are also used to enhance and manipulate the sidebands
[19]. But there exists ample scope of investigations about proper control of
sidebands when it comes to driving the cavity with a CW pump laser.
Conventionally, it is done by increasing CW laser drive power, but P. Djorwe
et al. [20] utilized the dynamical attractors to manipulate the optical combs.
The quadratic optomechanical coupling [21] and the dissipation optomechanical
coupling [22] are also being used to manipulate the sidebands.

The appearance of higher order optical sidebands depends on the amplitude of
the steady mechanical oscillation [23] where the oscillations with higher
amplitude results in more spectral lines [5]. In our study, we manipulated the
amplitude of the oscillations to control the sidebands using only a CW pump
laser drive. This is achieved by using mechanical $\mathcal{PT}$ symmetric
structure with optically induced gain and loss. The approach towards physical
realization of this kind of structure can be understood in [24]. Based on our
numerical analysis, we showed that the exception point leads to large self-
sustained oscillations due to the spontaneous localization of the mechanical
energy and increased optomechanical nonlinearity in the cavities [25].
Therefore, the HOS starts to appear from EP onwards and persists in the broken
$\mathcal{PT}$ symmetry phase as well. The switching among $\mathcal{PT}$
phase, EP, and broken $\mathcal{PT}$ phase is done by tuning only the
mechanical coupling factor for fixed driving power. Thus, we demonstrated a
different way of invoking and controlling the sidebands, contrary to
conventional way of adjusting the driving power. Practically, the mechanical
coupling rate can be controlled via the piezoelectric effect [26],
photothermal effect [27] or electrostatic force [28]. The generation of HOS in
the system are robust against non-degenerate mechanical cavities with
different resonance frequencies. This is possible because of the existence of
collective phenomena in the system. In the existing literature, the collective
phenomena in OM system are used in various classical and quantum
synchronization purposes [29, 30, 31]. Thus, we investigated EP induced
synchronization between the two HOS optical spectra.

This paper is organized in the following manner. In Sec II, we showed the
mathematical model and the semiclassical dynamical equations in our coupled OM
system. In Sec III, we demonstrated mechanical $\mathcal{PT}$ symmetry and
behaviour of optical spectra under different situations. In Sec IV, we studied
the synchronization between the two optical HOS in the cavities. Lastly, we
summarize our work in Sec V.

##  II Mathematical model and Dynamical equations

The system, as shown in Fig 1 consists of two optomechanical cavities which
has weak mechanical coupling.

Figure 1: Generic setup of mechanically coupled two OM cavity

The Hamiltonian of the whole system can be represented as the sum of
individual optomechanical cavity terms and the coupling terms (assuming
$\hbar=1$).

Table 1:  Parameters in the coupled OM system Symbols | Meaning
---|---
$E_{1,2}$ | Driving strength of the laser
$\kappa_{1,2}$ | Optical cavity decay rates
$\gamma_{m_{1,2}}$ | Mechanical cavity decay rates
$\omega_{L}$ | Driving laser frequency
$\omega_{0_{1,2}}$ | Optical cavity resonance frequencies
$\omega_{m_{1,2}}$ | Mechanical resonance frequencies
$\omega_{c_{1,2}}$ | Optical resonance frequencies
$\Delta_{1,2}=\omega_{L}-\omega_{c_{1,2}}$ | Detuning of the laser
$J$ | Mechanical coupling rate
$g_{0_{1,2}}$ | Optomechanical coupling rate

$\displaystyle
H_{OM,j}=\Delta_{j}\hat{a}^{\dagger}_{j}\hat{a}_{j}+\omega_{m_{j}}\hat{b}^{\dagger}_{j}\hat{b}_{j}-g_{0_{j}}\hat{a}^{\dagger}_{j}\hat{a}_{j}(\hat{b}^{\dagger}_{j}$
$\displaystyle+\hat{b}_{j})+E_{j}(\hat{a}^{\dagger}_{j}+\hat{a}_{j})$ (1a)
$\displaystyle
H_{coup}=-J(\hat{b}_{j}\hat{b}^{\dagger}_{3-j}+\hat{b}^{\dagger}_{j}\hat{b}_{3-j})$
(1b) $\displaystyle H_{total}=H_{OM_{1}}+H_{OM_{2}}+H_{coup}$ (1c)

The subscript j=1,2 denotes the left and right optomechanical cavity
respectively. $\hat{a}^{\dagger}_{j}(\hat{a}_{j})$ and
$\hat{b}^{\dagger}_{j}(\hat{b}_{j})$ are the creation(destruction) operators
of the photon and phonon respectively. The details about other parameters are
mentioned in Table 1. The Hamiltonian are written in the rotating frame of the
driving lasers. The dimensionless position and momentum of the mechanical
cavity in terms of $\hat{b}^{\dagger}_{j}$ and $\hat{b}_{j}$ are written as
$\hat{x}_{j}=\frac{\hat{b}_{j}+\hat{b}^{\dagger}_{j}}{\sqrt{2}}$ and
$\hat{p}_{j}=\frac{\hat{b}_{j}-\hat{b}^{\dagger}_{j}}{i\sqrt{2}}$
respectively. Since we require the study of the dynamics of the mean values of
the operators, we can ignore the fluctuations present in the system. Using the
mean field approximation ie.
$\langle{\hat{a}_{j}\hat{x}_{j}}\rangle=\langle{\hat{a}_{j}}\rangle\langle{\hat{x}_{j}}\rangle$
, the dynamical equations of the mean
operators($x_{j}=\langle{\hat{x}_{j}}\rangle$,$p_{j}=\langle{\hat{p}_{j}}\rangle$
and $a_{j}=\langle{\hat{a}_{j}}\rangle$) of the system can be written as

$\displaystyle\frac{da_{j}}{dt}=-i(\Delta_{j}+g_{0_{j}}x_{j})a_{j}-\frac{\kappa_{j}}{2}a_{j}+E_{j}$
(2a) $\displaystyle\frac{dx_{j}}{dt}=\omega_{m_{j}}p_{j}$ (2b)
$\displaystyle\frac{dp_{j}}{dt}=-\omega_{m_{j}}x_{j}-\frac{\gamma_{m_{j}}}{2}p_{j}+Jx_{3-j}+g_{0_{j}}|a_{j}|^{2}$
(2c)

The Eq (2) are coupled and non linear in nature. We evaluated the equations
through numerical simulations since exact analytical derivations are
difficult.We used Runge-Kutta method in obtaining the solution of Eq (2). We
made certain assumptions in the parameters of the dynamical equation. The
mechanical cavities are degenerate i.e.
$\omega_{m_{1}}=\omega_{m_{2}}=\omega_{m}$. We provide identical driving power
$E_{1}=E_{2}=E$ to the cavities. The decay rates and optomechanical coupling
rates are identical ie. $\kappa_{1}=\kappa_{2}=\kappa$ and
$g_{0_{1}}=g_{0_{2}}=g_{0}$. The detunings are in resonance with the
mechanical resonance frequency i.e. $-\Delta_{1}=\Delta_{2}=\omega_{m}$.

##  III $\mathcal{PT}$ Symmetry and Higher Order Sideband Generation

We adiabatically eliminate the optical cavity modes to show mechanical
$\mathcal{PT}$ symmetry. The optomechanical coupling rate $g_{0}$ is assumed
to be weaker than the cavity decay rate $\kappa$ ie. $g_{0}<<\kappa$ for such
condition. Including the effective loss and gain of the mechanical cavities,
the effective non-Hermitian Hamiltonian is written as

$\displaystyle
H_{eff}=((\omega_{m}-\delta\omega_{m_{1}})-i\gamma_{eff_{1}})\hat{b}^{\dagger}_{1}\hat{b}_{1}$
$\displaystyle+((\omega_{m}+\delta\omega_{m_{2}})+i\gamma_{eff_{2}})\hat{b}^{\dagger}_{2}\hat{b}_{2}-J(\hat{b}_{1}\hat{b}^{\dagger}_{2}+\hat{b}^{\dagger}_{1}\hat{b}_{2})$
(3)

$\delta\omega_{m_{j}}$ is the mechanical resonance frequency shift due to the
optical spring effect. The mechanical vibrations in the cavity driven by red
detuned laser experience extra damping such that the effective decay rate
becomes $|\gamma_{eff_{1}}|=\gamma_{opt_{1}}+\gamma_{m}$. The vibrations in
the blue detuned driven cavity experience gain with the effective rate of
$|\gamma_{eff_{2}}|=\gamma_{opt_{2}}-\gamma_{m}$. $\gamma_{opt_{j}}$ is the
modifications of the decay rates due to the optomechanical coupling. At
$|\Delta|=\omega_{m}$, $\gamma_{opt_{j}}$ is written as [1]

$\displaystyle|\gamma_{opt_{j}}|=\frac{64g_{0}^{2}|a_{j}|^{2}\omega^{2}_{m}}{\kappa(\kappa^{2}+16\omega^{2}_{m})}$
(4)

Applying $\mathcal{P}$ transformation: $i\leftrightarrow i$,
$\hat{b}_{1}\leftrightarrow-\hat{b}_{2}$,
$\hat{b}^{\dagger}_{1}\leftrightarrow-\hat{b}^{\dagger}_{2}$ and $\mathcal{T}$
transformation: $i\leftrightarrow-i$, $\hat{b}_{j}\leftrightarrow\hat{b}_{j}$,
$\hat{b}^{\dagger}_{j}\leftrightarrow\hat{b}^{\dagger}_{j}$ to Eq (3), we can
show that $[H_{eff},\mathcal{PT}]=0$ under the conditions:
$\delta\omega_{m_{j}}$ is negligible and
$|\gamma_{eff_{1}}|=|\gamma_{eff_{2}}|$. The concept of $\mathcal{PT}$
symmetry is also applicable to generalised case where the gain -loss amount
are unbalanced or presence of both loss-loss system. In our system, we operate
in resolved sideband regime ie. $\omega_{m}>>\kappa$ to have minimum optical
spring effect. We adjust the parameters in such a way that the total system
experience net loss. It would ensure that the mechanical dynamics in
$\mathcal{PT}$ phase reaches towards fixed equilibrium value and does not
induce instability. With these considerations, the matrix form of Eq (3) can
be written as

$H_{eff}=\begin{pmatrix}\hat{b}^{\dagger}_{1}&\hat{b}^{\dagger}_{2}\end{pmatrix}\begin{pmatrix}\omega_{m}-i\gamma_{eff_{1}}&-J\\\
-J&\omega_{m}+i\gamma_{eff_{2}}\end{pmatrix}\begin{pmatrix}\hat{b}_{1}\\\
\hat{b}_{2}\end{pmatrix}$ (5)

The eigen-frequencies corresponding to the mechanical supermodes
$\hat{b}_{1}\pm\hat{b}_{2}$ are written as

$\omega_{\pm}=\omega_{m}-i\frac{\gamma_{eff_{1}}-\gamma_{eff_{2}}}{4}\pm\sqrt{J^{2}-(\frac{\gamma_{eff_{1}}+\gamma_{eff_{2}}}{4})^{2}}$
(6)

Figure 2:  Real (a) and Imaginary (b) parts of the eigen-frequencies
$\omega_{\pm}$. The exceptional point occurs at $J=0.0112\kappa$. The
splitting and coalescence of the mechanical spectrum of the right cavity for
the case of (c) $\mathcal{PT}$ Symmetric phase ($J=0.015\kappa$) and (d)
Exceptional point Figure 3:  Phase portraits and dynamics of the mechanical
vibrations in the two cavities. (a) and (c) show the phase portrait and
mechanical oscillations dynamics in the broken $\mathcal{PT}$ phase at
$J=0.002\kappa$ respectively. (b) and (d) also shows the phase portrait and
mechanical oscillations dynamics at EP with $J=0.0112\kappa$ respectively. (e)
shows the decaying $\mathcal{PT}$ phase dynamics at $J=0.015\kappa$. All the
analysis are done under fixed $E=709\kappa$.

In $\mathcal{PT}$ symmetry phase, the real part of eigen-frequencies of the
supermodes are split by amount
$\sqrt{J^{2}-(\frac{\gamma_{eff_{1}}+\gamma_{eff_{2}}}{4})^{2}}$ with a
decaying component of $\frac{\gamma_{eff_{1}}-\gamma_{eff_{2}}}{4}$. In the
broken $\mathcal{PT}$ phase, the splitting of the real part of eigen-
frequencies vanishes. In this scenario, one of the supermodes will experience
gain while other losses. The phase transition from $\mathcal{PT}$ phase to
broken $\mathcal{PT}$ phase occurs at
$J=\frac{\gamma_{eff_{1}}+\gamma_{eff_{2}}}{4}$ which is known as the
exceptional point (EP). Now, we set the driving amplitude as $E=709\kappa$ for
the rest of the study and it is low enough to avoid formation of chaos in the
system. We set the intrinsic mechanical decay rate in one cavity higher than
the other i.e. $\gamma_{m_{1}}=0.5\times 10^{-2}\kappa$ and
$\gamma_{m_{2}}=10^{-4}\kappa$. In this manner we ensure that the coupled
mechanical cavities experience net loss in $\mathcal{PT}$ dynamics. The
optomechanical coupling rate and the mechanical resonance frequency are set as
$g_{0}=10^{-3}\kappa$ and $\omega_{m}=10\kappa$ respectively. The parameters
mentioned here can be achieved in experiments too [32, 33]. Fig 2(a)-(b) shows
the real and imaginary parts of the eigen-frequencies respectively, under the
variation of mechanical coupling rate $J$. The $\mathcal{PT}$ symmetric phase
is observed at high coupling rate where the Real $\omega_{\pm}$ are split
about $\omega_{m}$. The EP occurs at $J=0.0112\kappa$ with the two eigen-
frequencies overlapping with each other. Lowering the coupling rate will push
the system into the broken $\mathcal{PT}$ phase, where one supermode
experience loss while other gain. The Fig 2(c)-(d) show the nature of
mechanical spectrum $|x_{2}(\omega)|$ of the right cavity for $\mathcal{PT}$
phase and EP respectively.

Figure 4:  (a) Normalized self sustained oscillation amplitude
($\varepsilon_{j}$) vs the mechanical coupling rate $J$. The EP invokes abrupt
increase of $\varepsilon_{j}$. (b)-(d) The steady evolution of the intracavity
photon number $|a_{2}(t)|^{2}$ for $\mathcal{PT}$ phase ($J=0.015\kappa$), EP
($J=0.0112\kappa$) and broken $\mathcal{PT}$ phase ($J=0.002\kappa$)
respectively.  Figure 5:  The output optical spectrum $S_{out_{2}}(\omega)$ of
the cavity with blue detuned driving laser corresponding to the (a)
$\mathcal{PT}$ phase ($J=0.015\kappa$) (b) Exceptional Point
($J=0.0112\kappa$) and (c) broken $\mathcal{PT}$ phase ($J=0.002\kappa$).

We study the dynamics of the mechanical oscillations in different conditions
by solving Eq (2) with the initial conditions of all the dynamical variables
set to zero. Fig 3(e) shows the mechanical dynamics in $\mathcal{PT}$ phase
for the coupling rate of $J=0.015\kappa$. We observe a periodic amplitude
variation dynamics in both the cavities, which depicts strong coupling between
the cavities. The mechanical dynamics has decaying envelope because of the net
loss faced by the system. In the mechanical phase portrait, the system reaches
towards a fixed equilibrium point as the system has decaying dynamics.
Lowering the coupling rate to $J=0.0112\kappa$ takes the system at EP (Fig
3(d)). At this point, the coupling is weak enough to trigger spontaneous
localization of the mechanical energy in both the cavities. It is manifested
as the increase of the amplitude of oscillations in both the cavities. The
amplitude will not grow forever as the optomechanical non-linearity
($g_{0}x_{j}$) becomes prominent. Thus the oscillations will eventually settle
into a self sustained oscillations with amplitude $A_{j}$. In mechanical phase
portrait diagram, the system shows limit cycle behaviour (Fig 3(b)). Lowering
the coupling rate even further at $J=0.002\kappa$, pushes the system to the
broken $\mathcal{PT}$ symmetric phase as shown in Fig 3(c). Here, the dynamics
show self sustained oscillations in both the cavities similar to that of EP.
The mechanical oscillations in the left cavity experience optically induced
loss. Lowering the coupling, reduces the supply of mechanical energy to this
cavity from the optically induced gain cavity. So we observed significantly
reduced steady amplitude of oscillations as shown in Fig 3(c). These are
reflected in the phase portrait (Fig 3(a)) for the broken $\mathcal{PT}$ phase
where the size of the limit cycle in the left cavity is significantly lesser
than other.

Fig 4(a) shows the changes in the normalized amplitude $\varepsilon_{j}$
($\varepsilon_{j}=\frac{g_{0}A_{j}}{\omega_{m}}$) under the variation of $J$.
The amplitude are obtained by observing the mechanical dynamics. In the
$\mathcal{PT}$ phase, we have $\varepsilon_{j}=0$. In the broken
$\mathcal{PT}$ phase, we have a finite and varying $\varepsilon_{j}$. The
magnitude of $\varepsilon_{j}$ is determined by the balance of mechanical
losses with the driving laser power and the amount of coupling between the
cavities. We observe a continuous increase of $\varepsilon_{2}$ towards the
value $3.2$ (i.e. for the case of $J=0$) as we further reduce the coupling,
while $\varepsilon_{1}$ decreases towards 0. For the remaining part of this
section, we will only focus on the affect of vibrations on the optical
response of the right cavity. Fig 4(b)-(d) shows the evolution of the
intracavity photon number $|a_{2}(t)|^{2}$ in the right cavity for different
conditions. In $\mathcal{PT}$ phase, $|a_{2}(t)|^{2}$ decays to steady value
of around $5\times 10^{3}$. Once EP is reached, $|a_{2}(t)|^{2}$ deviates away
from steady fixed value as the finite self sustained oscillation modulates the
intracavity optical field. The large self sustained mechanical oscillations
(high $\varepsilon_{j}$) in the broken $\mathcal{PT}$ phase strongly modulates
the intracavity field. Analytically, we can write the intracavity light field
as infinite sum of frequency components $\pm k\omega_{m}$, if we assume
$x_{j}(t)=\varepsilon_{j}cos(\omega_{m}t)$.

$\displaystyle
a_{j}(t)=e^{i\varepsilon_{j}sin(\omega_{m}t)}\sum_{k=-\infty}^{\infty}a_{j}^{k}e^{ik\omega_{m}t}$
(7)

In terms of $\varepsilon_{j}$, $a_{j}^{k}\propto J_{k}(-\varepsilon_{j})$,
where $J_{k}$ is the kth order first kind Bessel function [23].

The output light field from the cavities is obtained from input-output
formalism [34] i.e. $s_{out_{j}}(t)=E(t)-\kappa a_{j}(t)$. We obtained the
output optical spectrum by taking the Fourier transform
$S_{out_{j}}(\omega)=\int_{-\infty}^{\infty}s_{out_{j}}(t)e^{-i\omega t}dt$.
Fig 5 shows the output optical spectrum from the right cavity under different
conditions. We took Fast Fourier Transform (FFT) of the steady response of
$s_{out_{2}}(t)$ to show the optical spectrum of the right cavity. As, we are
studying in the rotating frame of driving laser, the actual spectrum is
shifted by $\omega_{L}$. In $\mathcal{PT}$ phase, the spectrum has no
sidebands (Fig 5(a)). We see a sudden transition in the nature of output
spectrum at EP (Fig 5(b)) as the sidebands starts to emerge with a frequency
spacing of $\omega_{m}$. Fig 5(c) shows significant increase of spectral lines
in the broken $\mathcal{PT}$ phase. Thus, we achieved a proper control of the
output spectrum from the right cavity just by varying the mechanical coupling
rate.

##  IV Synchronization between two spectra

Till now, we observed the output optical spectrum from the blue detuned cavity
with frequency spacing of $\omega_{m}$. The sidebands would also emerge in the
red detuned cavity since there exists finite oscillation amplitude
$\varepsilon_{1}$ from EP onwards. We show, in this section, the output HOS
exists in both the optical cavities in a “synchronized” manner.

Figure 6:  Demonstration of the collective effects in our system with a
frequency mismatch of $\omega_{m_{2}}-\omega_{m_{1}}=0.0018\omega_{m}$. (a)
and (b) Spectral profile and intracavity photon number at $J=0.07\kappa$
(synchronized case) respectively. The inset in (a) shows the 1st order
spectral component coincide each other at a slight offset frequency from
$\omega=\omega_{m}$. (c) and (d) Spectral profile and intracavity photon
number at $J=0.008\kappa$ (non-synchronized case) respectively. The other
parameters are same as in Fig 3.

Here, we define synchronization in terms of (i) locking of the spectral peaks
to a common frequency, if there exists any frequency mismatch and (ii) nature
of spectral envelope. Since, the nature of two spectral envelope are
determined by $\varepsilon_{j}$ (Eq (7)), they become similar when the
difference $|\varepsilon_{2}-\varepsilon_{1}|$ remain below a certain level.

We set the resonance frequencies in the mechanical cavities non degenerate ie.
$\omega_{m_{1}}\neq\omega_{m_{2}}$ and included the optical spring effect on
the mechanical vibrations such that
$\omega_{eff_{1,2}}=\omega_{m_{1,2}}\mp\delta_{\omega_{m_{1,2}}}$. This new
frequency is replaced in the matrix (Eq (5)) to obtain the eigenvalues of the
supermodes as

$\displaystyle\omega_{\pm}=\frac{\omega_{eff_{1}}+\omega_{eff_{2}}}{2}-i\frac{\gamma_{eff_{1}}-\gamma_{eff_{2}}}{4}$
$\displaystyle\pm\sqrt{J^{2}+\left(\frac{\Delta\omega}{2}+i\frac{\gamma_{eff_{1}}+\gamma_{eff_{2}}}{4}\right)^{2}}$
(8)

where $\Delta\omega=\omega_{eff_{2}}-\omega_{eff_{1}}$. In this case, the
system is no longer $\mathcal{PT}$ symmetric and the EP is determined by
complex interplay of coupling rate, frequency mismatch and the effective decay
rates.

Figure 7:  Possible regimes in our system in the parameter space of resonance
frequency mismatch $(\omega_{m_{2}}-\omega_{m_{1}})/\omega_{m}$ and mechanical
coupling rate $J$. The solid line denotes the EP and the dotted line
differentiates synchronized from the non-synchronized regimes. The other
parameters are same as in Fig 3.

We know that the EP results in coalescence of the two mechanical supermodes,
which leads to the mechanical vibrations in the coupled non degenerate
cavities readjust itself to lock to a certain phase and amplitude, and to a
common frequency [35, 36]. Such phenomenon has its impact on the output
optical response of the cavities. The HOS spectra in the two cavities will
emerge from EP onwards with a common frequency spacing and certain spectral
envelope. Fig 6(a) shows the nature of the two output spectra. By setting the
mismatch as $\omega_{m_{2}}-\omega_{m_{1}}=0.0018\omega_{m}$ and coupling as
$J=0.07\kappa$ we observe that both the spectra has similar spectral envelope,
as the difference $|\varepsilon_{2}-\varepsilon_{1}|$ is not significant. Even
though the spectral envelope is similar, the lines for the left cavity is more
concentrated towards $\omega>0$, while the other towards $\omega<0$. It arises
because of the resolved sideband operation and the detuning of the driving
laser [1]. Besides, both the HOS spectra has same uniform frequency spacing
with the inset of Fig 6(a) showing the zoomed out version of the 1st order
component around $\omega=\omega_{m}$. It is interesting to see the nature of
evolution of the intracavity field intensity corresponding to it. We observe
in Fig 6(b), that the intensity in left cavity follows the right cavity with a
certain DC offset between the two. So, based on all these discussed points, we
say that the two output spectra in Fig 6(a) are synchronized. Now, keeping the
same frequency mismatch, but reducing the coupling to $J=0.008\kappa$, will
vastly change the nature of spectrum in both the cavities (Fig 6(c)). Even
though the sidebands in both the spectra locked at the common frequency
spacing, the blue detuned cavity has more spectral lines than the other one
since $|\varepsilon_{2}-\varepsilon_{1}|$ become significant. These are also
reflected in intracavity intensity dynamics in Fig 6(d), where the evolution
of $|a_{2}(t)|^{2}$ significantly becomes different than other. We say, in
this scenario, that the two spectra are non synchronized.

We provided a region of synchronization in the parameter space of frequency
mismatch and the coupling rate in Fig 7 with fixed $E=709\kappa$. We
determined the region by observing the spectra, mechanical dynamics and
intracavity intensity dynamics. We see that for larger mismatch, higher
coupling strength is required to induce the synchronization between the
spectra. Thus EP occurs at higher $J$ for high $\omega_{m_{1}}-\omega_{m_{2}}$
(solid black line in Fig 7). On the other hand, the dotted line marks the
transition between the synchronized and non-synchronized spectra. In the non-
synchronized regime, $\varepsilon_{2}\gg\varepsilon_{1}$ and thus, we get
results similar to the one in Fig 6 (c)-(d). But, this dotted line does not
increase appreciably with the increase in mismatch. Thus, the region of
synchronization becomes wider for larger mismatches. So, the process of
synchronization of the spectrum is much more sensitive to EP for weak
mismatches, as the synchronization region is much narrow near EP.

##  V Summary

In summary, we theoretically investigated the formation, control and
synchronization of optical sideband spectra in weak mechanically coupled
optomechanical cavities. The study provided rich insights behind the interplay
of mechanical gain-loss structure to that of optomechanical non-linearity. The
work has been done with fixed CW driving laser power (i.e. the amount of
mechanical gain/loss kept fixed) with varying mechanical coupling rate. We
obtained the mechanical dynamics along with the corresponding intracavity
intensity dynamics under different circumstances. Consequently, we found an
abrupt emergence of sidebands in the spectrum at EP by performing FFT of the
steady output light field. The number of spectral lines went higher by tuning
the system into deep broken $\mathcal{PT}$ phase. Thus, our scheme allows
precise control of the optical spectral lines by engineering the mechanical
coupling rate in a low power integrated chip-scale device. We also showed
collective behaviour (synchronization) between two spectra under varied
mechanical resonance frequency mismatches. Thus the study proved to be robust
against the strict condition of $\mathcal{PT}$ symmetry and found the
synchronization process sensitive to EP for a weak mismatch. So, the EP based
generation of synchronized wideband spectra with equidistant lines provides a
promising platform for optical communication technology, optical readout of
mechanical synchronization process [30], memory applications, remote clock
timing synchronization [37] and others.

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