# Boosting the sensitivity of high frequency gravitational wave detectors by
PT-symmetry

Chuming Wang  Center for Gravitational Experiment, Hubei Key Laboratory of
Gravitation and Quantum Physics, School of Physics, Huazhong University of
Science and Technology, Wuhan, 430074, P. R. China  Chunnong Zhao  ARC Centre
of Excellence for Gravitational Wave Discovery, The University of Western
Australia, Crawley, Western Australia 6009, Australia  Xiang Li  Burke
Institute for Theoretical Physics, California Institute of Technology,
Pasadena, California 91125, USA  Enping Zhou  Department of Astronomy, School
of Physics, Huazhong University of Science and Technology, Wuhan, 430074, P.
R. China  Haixing Miao  State Key Laboratory of Low Dimensional Quantum
Physics, Department of Physics, Tsinghua University, Beijing, China  Yanbei
Chen  Burke Institute for Theoretical Physics, California Institute of
Technology, Pasadena, California 91125, USA  Yiqiu Ma<EMAIL_ADDRESS>Center for Gravitational Experiment, Hubei Key Laboratory of Gravitation and
Quantum Physics, School of Physics, Huazhong University of Science and
Technology, Wuhan, 430074, P. R. China  Department of Astronomy, School of
Physics, Huazhong University of Science and Technology, Wuhan, 430074, P. R.
China

###### Abstract

The kilo-Hertz gravitational waves radiated by the neutron star merger
remnants carry rich information about the physics of high-density nuclear
matter states, and many important astrophysical phenomena such as gamma-ray
bursts and black hole formation. Current laser interferometer gravitational
wave detectors, such as LIGO, VIRGO, and KAGRA have limited signal response at
the kilo-Hertz band, thereby unable to capture these important physical
phenomena. This work proposes an alternative protocol for boosting the
sensitivity of the gravitational wave detectors at high frequency by
implementing an optomechanical quantum amplifier. With the auxiliary quantum
amplifier, this design has the feature of Parity-Time (PT) symmetry so that
the detection band will be significantly broadened within the kilo-Hertz
range. In this work, we carefully analyze the quantum-noise-limited
sensitivity and the dynamical stability of this design. Based on our protocol,
our result shows that the quantum-noise-limited sensitivity will be improved
by one order of magnitude around 3kHz, which indicates the potential of our
design for a future search of neutron star merger signals.

##  I Introduction

Gravitational waves (GWs) radiated from binary neutron star (BNS) inspirals
have been detected in many events (e.g. GW170817) by the advanced ground-based
laser interferometer GW detectors (LIGO and VIRGO) [1, 2, 3, 4, 5, 6].
However, the GWs radiated by the BNS post-merger remnants, of which the
predicted frequency is around kilo-Hertz (kHz), have not been detected yet,
due to the limitations of sensitivity at high frequencies [7, 8]. These high-
frequency GWs carry rich physics. For example, They may reveal the details of
the center engine of a short gamma-ray burst, the equation of state of ultra-
dense nuclear/quark matter, etc. Upgrading the sensitivity of ground-based GW
detectors at kilo-Hertz frequency is an important experimental task for the
future GW astrophysics [9, 10, 11, 12, 13, 14].

The response of the current LIGO, VIRGO and KAGRA configuration to the high-
frequency GWs is limited by the interferometer’s bandwidth [15]. The amplitude
of GW-induced-sidebands of the main carrier light decreases with the increase
of the GW frequency $\Omega$. The quantum shot noise, which dominates the
noise floor at kHz, has a white spectrum [16, 17, 18, 19, 20, 21]. Various
schemes have been proposed for increasing the signal response at high
frequency. The simplest way is to increase the intra-cavity power (to e.g. 10
MW) [7], at the price of sacrificing the low-frequency sensitivity (e.g. the
proposed new Australia-based instrument NEMO project is targeted on using high
power interferometer [22]). However, there are many technical challenges to
building a high-power Fabry-Perot cavity for GW detection [23, 24, 25]. Other
schemes include: implementing the “white-light-cavity” concept to broaden the
bandwidth for a detuned interferometer at high frequency; using the resonance
created by a long signal recycling cavity; and the design of signal recycling
cavity with internal squeezing [26, 27, 28, 29, 30, 31, Korobko_2019].

Recently, a novel scheme was proposed to boost the GW detector sensitivity by
reshaping its signal response, in which the interferometer mode $\hat{a}$ is
coupled to a quantum parametric amplifier $\hat{c}$ [32, 33]. In the ideal
case, the system Hamiltonian is invariant under the transformation
$\hat{a}\rightarrow\hat{c}^{\dagger}$ or vice versa, that is, the Hamiltonian
has _Parity-Time (PT)-symmetry_. In this case, the GW-induced sideband signal
fields inside the detector response as $\sim\Omega^{-1}$, which means a large
signal boost at low frequency. Compared to the previous white light cavity
design [26], this scheme is dynamically stable since the unstable parametric
process will be balanced by the stabilizing sloshing process, and thereby no
further feedback control is required for stabilization [32, 34]. However, the
$\Omega^{-1}$ signal response means that most of the advantage is obtained at
low frequency, which is easily contaminated by the back-action noise due to
the fluctuating quantum radiation pressure force as well as various classical
noises.

In this paper, we extend this “PT-symmetry” design concept for boosting the
signal response at kilo-Hertz frequency, which could be an alternative
approach to increasing the detector sensitivity for detecting GWs emitted by
BNS post-merger remnants. In this scheme, the main laser is detuned from the
resonance of a signal-recycling laser interferometer, which is coupled to an
oppositely detuned quantum parametric amplifier. We will thoroughly analyze
the conceptual design of this scheme. Our results show that implementing this
PT-symmetry design can significantly boost the sensitivity in a relatively
large searching band at high frequency.

The outline of this paper is organized as follows. Section II gives the basic
configurations and the result in the ideal case by a Hamiltonian approach
based on the single-mode approximation. Then in Section III and IV, we perform
a detailed analysis on the effect of PT-symmetry breaking to the sensitivity
and system dynamical stability. In Section V , we compute the sensitivity
curve using the transfer matrix approach, which is beyond the single-mode and
resolved sideband approximations, considering various noise sources. Finally,
we give the astrophysical implications of our protocol in Section VI.

##  II Theoretical principle of the scheme

The basic concept of the scheme can be illustrated (schematically shown in
Fig. 1) by the following idealized mode interaction Hamiltonian [35, 32]:

$\hat{H}_{\rm int}/\hbar=i\omega_{s}(\hat{a}\hat{b}^{\dagger}e^{i\Delta
t}-\hat{a}^{\dagger}\hat{b}e^{-i\Delta
t})+iG(\hat{b}^{\dagger}\hat{c}^{\dagger}e^{i\delta
t}-\hat{b}\hat{c}e^{-i\delta t}).$ (1)

Here the $\hat{a}$ is the annihilation operator of the differential optical
mode of the main signal-recycling interferometer; The $\Delta$ is the detuning
of the main laser beam with respect to the $\hat{a}$ mode, which is introduced
by the signal recycling cavity [35]. This non-zero detuning $\Delta$ creates
an optical resonance at $\Omega=\Delta$, which improves the signal response
around $\Delta$ [35]. At the same time, this detuning also leads to the
optical spring resonance at low-frequency [35]. Since the high-frequency
sensitivity is mainly concerned here, we temporarily ignore the optical spring
effect in this section for simplicity. The GW-induced sideband fields are
extracted from the $\hat{b}$ mode, which is parametrically coupled to the
$\hat{c}$ mode. This parametric interaction could have different realizations,
for example, using an optomechanical device, or a pumped nonlinear crystal
[32, 36]. Unlike the previous work, the frequency matching of the parametric
coupling here is generally not perfect, that is, $\delta\neq 0$.

This Hamiltonian is invariant under the PT-transformation $\hat{a}e^{i\Delta
t}\rightarrow\hat{c}^{\dagger}e^{i\delta t}$ when the PT-symmetry conditions
(incl. $G=\omega_{s}$ and $\delta=-\Delta$) are satisfied. The Heisenberg
equations of motion, considering the coupling between $\hat{a}$ and the GWs
with strength $\alpha$ are:

$\begin{split}&\dot{\hat{a}}(t)=-i\Delta\hat{a}(t)-\omega_{s}\hat{b}(t)+i\alpha
h(t),\\\
&\dot{\hat{c}}^{\dagger}(t)=i\delta\hat{c}^{\dagger}(t)+G\hat{b}(t),\\\
&\dot{\hat{b}}(t)=-\gamma\hat{b}(t)+\omega_{s}\hat{a}(t)+G\hat{c}^{\dagger}(t)+\sqrt{2\gamma}\hat{b}_{\rm
in}(t),\end{split}$ (2)

where $\gamma$ is the coupling rate between mode $\hat{b}(t)$ and external
bath $\hat{b}_{\rm in}(t)$. The outgoing field is $\hat{b}_{\rm
out}(t)=-\hat{b}_{\rm in}(t)+\sqrt{2\gamma}\hat{b}(t)$.

Figure 1: Schematic setup: a PT-symmetric Gravitational Wave (GW) Detector.
The internal signal recycling mirror (iSRM) at the dark port detunes the main
interferometer resonance. An optomechanical device is coupled to the main
interferometer which contributes to the parametric process. The red and blue
dashed lines are the optical fields at the signal and idler channels. Finally,
both of these two channels should be measured to obtain the optimized
sensitivity curve. Figure 2: The idealized kilo-Hertz GW noise spectrum of
gravitational wave detectors enhanced by PT-symmetric configurations, in
comparison to the conventional detuned LIGO configuration. The detector
dynamics here are described by Eq. (2). The parameters we used here is based
on mapping the sample parameters in Tab. 1 of the interferometer to the
effective mode-mode interaction model.

Solving these equations of motion under the PT-symmetry condition leads to the
input-output relation as:

$\begin{split}&\hat{b}_{\rm
out1}(\Omega)=\frac{\Omega-i\gamma}{\Omega+i\gamma}\hat{b}_{\rm
in1}(\Omega)+\frac{2\alpha\sqrt{\gamma}\omega_{s}\Delta}{(\Omega+i\gamma)(\Delta^{2}-\Omega^{2})}h(\Omega),\\\
&\hat{b}_{\rm out2}(\Omega)=\frac{\Omega-i\gamma}{\Omega+i\gamma}\hat{b}_{\rm
in2}(\Omega)+\frac{2\alpha\sqrt{\gamma}\omega_{s}\Omega}{(\Omega+i\gamma)(\Delta^{2}-\Omega^{2})}h(\Omega),\end{split}$
(3)

where we have the amplitude and phase quadrature of optical fields defined in
the two-photon formalism [37] as:

$\begin{split}&\hat{b}_{1}(\Omega)=\frac{1}{\sqrt{2}}[\hat{b}(\Omega)+\hat{b}^{\dagger}(-\Omega)],\\\
&\hat{b}_{2}(\Omega)=\frac{1}{\sqrt{2}i}[\hat{b}(\Omega)-\hat{b}^{\dagger}(-\Omega)].\end{split}$
(4)

Since we are interested in high frequency region where $\Omega\sim\Delta$, the
signal response of $\hat{b}_{\rm out1}$ and $\hat{b}_{\rm out2}$ are roughly
the same and scales approximately as $\sim 1/(\Delta-\Omega)$.

The shot-noise-limited sensitivity of the detuned PT-symmetric scheme,
quantified by the signal referred shot noise spectral density
$S_{hh}(\Omega)$, is given by (suppose the phase quadrature $\hat{b}_{\rm
out2}$ is measured):

$S^{\rm
PT}_{hh}(\Omega)\approx\frac{(\Delta^{2}-\Omega^{2})^{2}(\Omega^{2}+\gamma^{2})}{4\gamma\omega_{s}^{2}\alpha^{2}\Omega^{2}},$
(5)

while for the conventional detuned interferometer (also the phase quadrature
is measured) given by:

$S^{\rm
con}_{hh}(\Omega)=\frac{(\Omega^{2}-\gamma^{2}-\Delta^{2})^{2}+4\gamma^{2}\Omega^{2}}{4\alpha^{2}\gamma(\Omega^{2}+\gamma^{2})}.$
(6)

The comparison between the sensitivities of these two configurations at
$\Omega\sim\Delta$ is shown in Fig. 2. At the resonance point $\Omega=\Delta$,
the PT-symmetry scheme has a larger boost due to its response $\sim
1/(\Omega-\Delta)$ in the ideal case.

In this ideal case, the system is also dynamically stable which can be
understood from the following analysis. Adiabatic elimination of the
$\hat{b}-$field (usually has a large bandwidth) leads to the equations of
motion for $\hat{a},\hat{c}$:

$\begin{split}&\left(\frac{d}{dt}+i\Delta+\frac{\omega_{s}^{2}}{\gamma}\right)\hat{a}=-\frac{\omega_{s}G}{\gamma}\hat{c}^{\dagger}-\sqrt{\frac{\omega_{s}^{2}}{2\gamma}}\hat{b}_{\rm
in}+i\alpha h,\\\
&\left(\frac{d}{dt}-i\delta-\frac{G^{2}}{\gamma}\right)\hat{c}^{\dagger}=\frac{\omega_{s}G}{\gamma}\hat{a}+\sqrt{\frac{G^{2}}{2\gamma}}\hat{b}_{\rm
in}.\end{split}$ (7)

There is a damping factor $\omega_{s}^{2}/\gamma$ for the $\hat{a}$ field
while an anti-damping factor $-G^{2}/\gamma$ for the $\hat{c}^{\dagger}$
field. Under the PT-symmetric conditions $G=\omega_{s}$ and $\delta=-\Delta$,
the anti-damping factor and damping factor cancel each other for the effective
mode $\hat{a}+\hat{c}^{\dagger}$:

$\left(\frac{d}{dt}+i\Delta\right)(\hat{a}+\hat{c}^{\dagger})=i\alpha h.$ (8)

Note that in Fig. 2, there is one peak at $\Omega=\Delta$, while we actually
have three different modes $\hat{a},\hat{b},\hat{c}$ in our Hamiltonian, which
means there exists degeneracy due to the PT symmetry [38, 39]. This degeneracy
can be easily understood since $\hat{a}$ is equivalent to $\hat{c}^{\dagger}$,
and the $\hat{b}$-field couples to the $\hat{a},\hat{c}$ fields in such a way
that there is no effect on the resonance frequency of the $\hat{b}$-mode (see
Eq. (8)). Practical imperfections of our system will cause the breaking of the
PT-symmetry, which will affect the detector sensitivity and the system
stability.

This degeneracy can be understood from the algebraic structure of the
Heisenberg equations of motion Eq. (2). If we take the representation which
chooses $\hat{a},\hat{b},\hat{c}^{\dagger}$ to be the basis vectors, the
Heisenberg equation of motion becomes:

$\frac{d}{dt}\hat{\mathbf{v}}=\hat{\mathcal{D}}\hat{\mathbf{v}}+\mathbf{s}(t),$
(9)

where $\hat{\mathbf{v}}\equiv(v_{1},v_{2},v_{3})^{T}$ is a combined mode
consisting $v_{1},v_{2},v_{3}$ of
$\hat{a},\hat{b},\hat{c}^{\dagger}$,respectively; $\mathbf{s}(t)$ is the
vector describes the signal and noise adding to the system:

$\mathbf{s}(t)=(i\alpha h(t),\sqrt{2\gamma}\hat{b}_{in}(t),0)^{T};$ (10)

$\hat{\mathcal{D}}$ is the dynamic matrix:

$\hat{\mathcal{D}}=\left[\begin{array}[]{ccc}-i\Delta&-\omega_{s}&0\\\
\omega_{s}&-\gamma&G\\\ 0&G&i\delta\end{array}\right].$ (11)

The eigenvalues $\lambda_{1,2,3}$ and the corresponding eigenvectors
$\hat{\mathbf{v}}_{1,2,3}$ then describes the poles and its corresponding
modes of the whole system, respectively. The response in the frequency domain
can be written as:

$\mathbf{v}(\Omega)=-\frac{\mathbf{s}(\Omega)}{\lambda+i\Omega}.$ (12)

Satisfying one of the PT-symmetry conditions $\delta=-\Delta$, the eigenvalues
and the corresponding eigenvectors are given by:

$\begin{split}&\lambda_{1}:\quad\mathbf{v}_{1}=(-\frac{G}{\omega_{s}},0,1)^{T},\\\
&\lambda_{2}:\quad\mathbf{v}_{2}=(-\frac{\omega_{s}}{G},-\frac{\gamma-i\Delta}{2\chi}-\frac{\lambda_{3}-\lambda_{2}}{2\chi},1)^{T},\\\
&\lambda_{3}:\quad\mathbf{v}_{3}=(-\frac{\omega_{s}}{G},-\frac{\gamma-i\Delta}{2\chi}+\frac{\lambda_{3}-\lambda_{2}}{2\chi},1)^{T},\\\
\end{split}$ (13)

where

$\begin{split}&\lambda_{1}=-i\Delta,\\\
&\lambda_{2,3}=\frac{1}{2}[-\gamma-i\Delta\mp\sqrt{(\gamma-i\Delta)^{2}+4(G^{2}-\omega_{s}^{2})}].\end{split}$
(14)

The other PT-symmetry condition $G=\omega_{s}$ would lead to a degeneracy
since $\mathbf{v}_{1/3}$ corresponds to the same eigenvalue
$\lambda_{1}=\lambda_{3}=-i\Delta$. Breaking this PT-symmetry condition would
lead to the breaking of this degeneracy, as shown in Fig. 3 where the
eigenvalues are plotted in the complex frequency domain. This figure also
shows that there will be an unstable mode in some parameter regions, of which
the eigenvalue is located on the upper complex plane. This stability issue
will be further explored in Section IV.

Figure 3: The trajectories of the system’s eigenvalues based on the ideal
Hamiltonian with varies $G$ form 0 to 2$\omega_{s}$ and fixed $\omega_{s}$.
Note that these trajectories are symmetric on the left and right plane since
the Hamiltonian is Hermitian. As discussed in the main text, one of the
eigenmodes has a fixed eigenvalue $\lambda_{1}=-i\Delta$, which corresponds to
a fixed pole at frequency $\Omega_{1}=-i\lambda_{1}=\Delta$. The other two
eigenvalues (poles) change with the variation of $G$, they are shown as the
blue dashed lines ($\lambda_{3}$) and the solid lines ($\lambda_{2}$). Under
the PT-symmetry condition $G=\omega_{s}$, the poles of $\lambda_{1}$ and
$\lambda_{3}$ have a degeneracy. Then if $G$ varies across $\omega_{s}$, the
pole of $\lambda_{3}$ gets a positive imaginary part and becomes unstable.
Throughout this variation, $\mathbf{v}_{3}$ is mainly consist of $\hat{a}$
($v_{1}=-\omega_{s}/G$) and $\hat{c}^{\dagger}$ ($v_{3}=1$). Therefore we
denote the mode "$\hat{c}$-like" field after crossing the degeneracy point
since $\omega_{s}<G$, while it’s denoted as "$\hat{a}$-like" field before
since $\omega_{s}>G$.

##  III Effect of PT-symmetry breaking: sensitivity

The main practical factors that lead to the PT-symmetry breaking can be
generally summarized as follows (also see Fig. 4): (1) the off-resonance
sidebands in the optomechanical filter cavity, which we ignored in the ideal
case in Eq (1); (2) the pondermotive interaction happens inside the
interferometer arm cavities [35]. In this section, we focus on their
influences on the sensitivity of our protocol.

Figure 4: The breaking of the PT-symmetry due to practical system
imperfections: the introduction of the off-resonant sideband fields inside the
optomechanical filter cavity (idler channel) and the pondermotive interactions
in the interferometer arm cavities. The test mass and the idler field do not
couple to the $\hat{a}$ and $\hat{c}$ modes in a symmetric way, thereby
breaking the PT-symmetry of the entire system.

###  III.1 Influence from the off-resonant sidebands

A typical realization of the parametric interaction is to use an
optomechanical device with the interaction Hamiltonian $\hat{H}_{\rm
om}=\hbar\tilde{G}\hat{b}^{\dagger}\hat{b}\hat{x}_{m}$, where
$\tilde{G}=\omega_{0}/L_{b}$ is the single-photon optomechanical coupling
strength and $L_{b}$ is the length of the filter cavity, the $\hat{x}_{m}$ is
the displacement operator of the mechanical oscillator. Supposing the system
is pumped with detuning $\omega_{m}$, the optomechanical interaction can be
written as:

$\begin{split}\hat{H}_{\rm om}/\hbar=G&\left(\hat{b}\hat{c}e^{i\delta
t}+\hat{b}^{\dagger}\hat{c}^{\dagger}e^{-i\delta t}+\right.\\\
&\left.\hat{b}^{\dagger}\hat{c}e^{-i(2\omega_{m}-\delta)t}+\hat{b}\hat{c}^{{\dagger}}e^{i(2\omega_{m}-\delta)t}\right),\end{split}$
(15)

where $G=\tilde{G}x_{\rm zpf}\bar{b}$ with $x_{\rm zpf}$ the zero-point
displacement of the mechanical oscillator and $\bar{b}$ the mean amplitude of
the pumping beam. In the previous section, we ignored the last two terms of
the above Hamiltonian using the resolved sideband limit to obtain Eq. (1).
However, in reality, these far-off resonant sidebands will have a non-
negligible degradation to the detector sensitivity. In this work, we follow
the standard quantum optics terminology and name the optical sideband fields
around $\omega_{0}$ to be the “ _signal channel_ " and those around
$\omega_{0}+2\omega_{m}$ to be the ” _idler channel_ ”, as shown in Fig. 5.
For illustrative purposes, this subsection will study this effect, while
temporarily disregarding the pondermotive effect in analyzing the idler
fields.

Figure 5: Left panel: the structure of the mode-coupling Hamiltonian, where
the $\hat{a}$ and $\hat{c}$ are the main interferometer mode and the
mechanical mode, which are PT-symmetric to each other. For detecting the high-
frequency GWs, detuning should be introduced to these two modes. Right panel:
the sideband fields on-resonance with $\omega_{0}$ are defined as _signal
channel_ , while the far off-resonance sidebands resonance at
$\sim\omega_{0}+2\omega_{m}$ is defined as _idler channel_. The red curve
sketches the main interferometer resonance profile.

Introducing the idler channel around $\omega_{0}+2\Omega_{m}$ will break the
PT-symmetry as shown in Fig. 6, and bring the following two effects.

$\bullet$ Optical spring effect. The optical spring effect in the
optomechanical filter cavity will modify the mechanical frequency to be
$\omega^{\rm opt}_{m}\approx\omega_{m}+\omega_{\rm opt}$, where $\omega_{\rm
opt}=cP_{b}\omega_{0}/2m\omega_{m}^{2}c^{2}L_{b}$. This optical spring effect
must be compensated, otherwise, the PT-symmetry will be broken and the
detector will not be able to reach its optimal sensitivity. This effect is
plotted in the lower panel of Fig. 6, which demonstrates the importance of
this frequency compensation.

$\bullet$ Correction to the GW signal response. The GW response of the
detector will be modified in two ways by introducing the idler channel. First,
the mode degeneracy introduced by PT-symmetry will be broken. This splits the
single-peak at $\Omega=\Delta$ in the sensitivity curve as shown in Fig. 6.
Second, the GW signal sidebands will flow into the idler channels. Therefore
GW information can also be extracted from the detection of the optical
quadratures at the idler channels:

$\begin{split}&\hat{b}_{i1}(\Omega)=\frac{1}{\sqrt{2}}[\hat{b}(2\omega_{m}+\Omega)+\hat{b}^{\dagger}(2\omega_{m}-\Omega)],\\\
&\hat{b}_{i2}(\Omega)=\frac{1}{\sqrt{2}i}[\hat{b}(2\omega_{m}+\Omega)-\hat{b}^{\dagger}(2\omega_{m}-\Omega)].\\\
\end{split}$ (16)

Since the idler channel is separated far from the signal channel in the
frequency domain, this means we can use two homodyne detectors to measure both
the signal channel and idler channel, and then do data-post-processing to
optimize the detector sensitivity, which we will explain in detail later. This
analysis also shows that a full transfer matrix analysis is needed for a more
accurate sensitivity curve.

Figure 6: The detector sensitivity considers purely the effect of idler fields
(i.e. the pondermotive effect is ignored). Upper panel: the sensitivity of
both the idler and signal channels, taking into account the frequency
compensation $\omega_{\rm opt}$. Lower panel: The sensitivity of the signal
channel with/without the frequency compensation. The two dips around the
detuning frequency is a manifestation of degeneracy breaking introduced by the
idler channel.

###  III.2 Pondermotive effect

Now we discuss the pondermotive effect which has been ignored in the previous
sections. For a detuned main interferometer, the pondermotive interaction
generates an additional stiffness for the test masses [40], which is the
optical spring effect [35]. This optical stiffness comes from the dependence
of radiation pressure force on the test mass displacement. Adding the
pondermotive effect term $+ig\hat{X}$ to the right-hand side of Eq. (2)
($g=\omega_{0}\alpha/L$ is the linearised optomechanical coupling constant in
the arm cavity, $X$ is displacement of test mass.) leads to:

$\dot{\hat{X}}=\hat{P}/M,\quad\dot{\hat{P}}=\hbar
g(\hat{a}+\hat{a}^{\dagger}).$ (17)

The optical spring rigidity $K_{\rm opt}(\Omega)$ under the PT-symmetry
condition can be solved as (where the PT-symmetry conditions $G=\omega_{s}$
and $\delta=-\Delta$ have been used):

$K_{\rm opt}(\Omega)=2\hbar
g^{2}\left[\frac{\Delta}{\Delta^{2}-\Omega^{2}}-\frac{2\Omega\Delta
G^{2}}{(\Omega+i\gamma)(\Delta^{2}-\Omega^{2})^{2}}\right].$ (18)

In the following, it will be written as $K_{\rm opt}(\Omega)=K_{\rm
opt1}(\Omega)+K_{\rm opt2}(\Omega)$, where the $K_{\rm opt1}(\Omega)=2\hbar
g^{2}\Delta/(\Delta^{2}-\Omega^{2})$, which only depends on the optomechanical
coupling $g$ in the arm cavity. This $K_{\rm opt1}(\Omega)$ resembles the
rigidity of a detuned _perfect cavity with zero bandwidth_ , which reflects
the fact that the loss and gain are balanced under the PT-symmetry condition.

The optical rigidity will become very large at $\Omega=\Delta$ due to the
significantly boosted displacement-induced-sideband fields as
$\propto\alpha{\color[rgb]{1,0,0}X}(\Omega)/(\Delta^{2}-\Omega^{2})$. In other
words, at $\Omega=\Delta$, the test masses will become so stiff that the
external force can not drive their motions. Therefore the signal at this
frequency is strongly suppressed and a peak in the sensitivity curve is
expected.

The input-output relations considering the pondermotive effect can be written
as:

$\mathbf{\hat{b}}_{\rm
out}(\Omega)=e^{i\beta(\Omega)}\mathbb{M}(\Omega).\mathbf{\hat{b}}_{\rm
in}(\Omega)+\mathbf{D}(\Omega)h(\Omega),$ (19)

in which we have: $e^{i\beta(\Omega)}=(\Omega-i\gamma)/(\Omega+i\gamma)$ and

$\begin{split}&\mathbb{M}(\Omega)=\\\
&\chi_{M}(\Omega)\left[\begin{array}[]{cc}-\chi^{-1}_{M}(\Omega)+K^{i}_{\rm
opt2}(\Omega)&-(\Delta/\Omega)K_{\rm opt2}^{r}(\Omega)\\\
-(\gamma/\Delta)K^{i}_{\rm opt2}(\Omega)&-\chi^{-1}_{M}(\Omega)+K^{i}_{\rm
opt2}(\Omega)\end{array}\right],\end{split}$ (20)

where $\chi_{M}(\Omega)$ is the test mass mechancial response function
modified by the pondermotive effect:

$\chi_{M}(\Omega)=-\frac{1}{M\Omega^{2}-K_{\rm opt}(\Omega)}.$ (21)

The signal response matrix is:

$\mathbf{D}(\Omega)=-\frac{2ig\sqrt{\gamma}G}{(\Omega+i\gamma)(\Delta^{2}-\Omega^{2})}\chi_{M}(\Omega)M\Omega^{2}L\left[\begin{array}[]{c}\Delta\\\
\Omega\end{array}\right],$ (22)

where at $\Omega=\Delta$, the signal response vanishes, and the sensitivity
will have a very sharp peak as expected. This effect will not happen for the
tuned PT-symmetric interferometer [32] since there is no optical spring effect
when $\Delta=0$. The problem of sensitivity degradation due to this
pondermotive effect can be solved using (1) an optimal combination of
different measurement channels which is discussed in detail in Sec. V, and (2)
negative inertia.

Figure 7: The degradation of detector sensitivity is due to the pondermotive
interaction between the test masses and the optical fields in the main
interferometer. The narrow sensitivity peak at the optical resonance frequency
$3000$ Hz with bandwidth $\sim 10$ Hz comes from the optical stiffness
enhanced by the PT-symmetric configuration. For highlighting the pondermotive
effect, the idler channel effect is ignored here.

###  III.3 Cancelling the pondermotive effect using negative inertia

The pondermotive effect reflects the fact that the system is not fully PT-
symmetric when the test masses dynamics are considered. For achieving a fully
PT-symmetric system, a negative mass term [41, 42, 43, 44, 45, 46] needs to be
introduced with an additional equation of motion:

$\dot{\hat{x}}=-\frac{\hat{p}}{\mu},\quad\dot{\hat{p}}=\hbar\tilde{g}(\hat{c}+\hat{c}^{\dagger}),$
(23)

where $\mu=-M$ and $\tilde{g}=g$ should be satisfied to achieve the PT-
symmetry. In this case, the input-output relation becomes very simple:

$\left[\begin{array}[]{c}\hat{b}_{\rm out1}(\Omega)\\\ \hat{b}_{\rm
out2}(\Omega)\end{array}\right]=e^{i\beta(\Omega)}\left[\begin{array}[]{c}\hat{b}_{\rm
in1}(\Omega)\\\ \hat{b}_{\rm in2}(\Omega)\end{array}\right]+\mathbf{D}_{\rm
sym}(\Omega)h(\Omega),$ (24)

where we define:

$\mathbf{D}_{\rm
sym}(\Omega)=\frac{2ig\sqrt{\gamma}\chi}{(\gamma-i\Omega)(\Delta^{2}-\Omega^{2})}\frac{M\Omega^{2}L}{-M\Omega^{2}+K_{\rm
opt1}(\Omega)}\left[\begin{array}[]{c}\Delta\\\ -i\Omega\end{array}\right].$
(25)

At $\Omega=\Delta$, the signal response is finite. The physical realization of
negative inertia coupled to the mode $\hat{c}$ could be realized by optical
systems, we leave the detailed design to the accompanying paper. In this
paper, we merely introduce this concept without discussing its details. The
main approach to cancel the sensitivity peak at $\Omega=\Delta$ discussed in
this paper is the optimal combination of the different measurement channels
which will be discussed in the next section.

##  IV Effect of PT-symmetry breaking: Stability

Instability arise when there are external energy sources pump the system
continuously. As shown in Fig.1, the arm cavity and filter cavity are both
pumped in the blue-detuned way. A single optomechanical system pumped by a
blue-detuned coherent field would have instability since the Stokes sideband
would overwhelm the anti-Stokes sideband so that the pumping light is
transferring energy to the mechanical degree of freedom [20, 47]. In our
design protocol, the blue-detuned main interferometer is coupled to the blue-
detuned filter cavity with the sloshing frequency $\omega_{s}$, therefore this
coupling will certainly affect the system dynamics and stability. In Fig. 3,
we have seen that there could be unstable modes when PT-symmetry is broken.

Figure 8: Instabilities induced by the uncompensated resonant frequency drift
introduced by the PT-symmetry breaking. (a) The Stokes sideband of the pumping
light in the filter cavity is a bit off-resonant to the main cavity resonance
due to the optical spring effect $\omega_{\rm opt}$, which breaks the balance
between the parametric and the sloshing process. (b) The Stokes sideband of
the pumping light in the filter cavity is also an off-resonant due to the
small shift of the main cavity resonance due to its coupling with the test
mass, which also breaks the parametric-sloshing balance and creates
instability.

###  IV.1 Instability induced by pondermotive effect

Introducing the pondermotive effect in the arm cavity means the coupling
between the mechanical motion and the optical mode of the main interferometer,
which will shift the optical resonance of the main interferometer [35]. If
this frequency shift was left uncompensated, it will create a suppression of
the Stokes sidebands of the pumping field in the filter cavity when being
sloshed to the main interferometer (see Fig. 8). Therefore, the balance
between the sloshing process and the parametric process is broken and
instability forms. In Fig. 9, we plot the influence of the pondermotive effect
on the trajectories of poles, where the trajectories detour near the detuning
frequency $\Omega=\Delta$. The degenerate point at $\Omega=\Delta$ under the
PT-symmetric condition splits into two different points, one of which is
unstable. Such instability can be removed by carefully tuning the frequency of
the pumping field.

Besides, the pondermotive effect in the arm cavity also generates a dynamical
back-action to the test mass mirrors, which is modified by the coupling to the
auxiliary system as shown in Eq. (18). Using the parameter listed in Tab.1,
these modified dynamics contribute a finite optical spring frequency around
$7.7$ Hz with an anti-damping factor equal to $2\pi\times 0.019$ rad/s to the
test mass. This $7.7$ Hz optical spring resonance manifests as a dip around
$7.7$ Hz in the sensitivity curve Fig. 13. The anti-damping rate $2\pi\times
0.019$ rad/s is well within the LIGO control band.

Figure 9: The influence of the pondermotive interaction inside the arm
cavities on the resonance pole at high frequencies. Upper panel: the global
pole trajectories plotted in the same way as in Fig. 3. Lower panel: zooming-
in the pole trajectories near the detuning point $\Omega=\Delta$, where the
degenerate root (when $\omega_{s}=G$) splits into two different poles and
introduce instability. This instability can be removed by tuning the pumping
field frequency in the filter cavity. The dashed line is the pole trajectory
of the ideal case for comparison.

###  IV.2 Instability induced by the off-resonant sidebands

After removing the instability near $\Omega=\Delta$ induced by the arm cavity
pondermotive effect by tuning the pumping field frequency, the instability
introduced by the idler channel (i.e. the off-resonant sideband fields in the
filter cavity) also needs to be removed. This instability exists because the
introducing of these off-resonant idler fields in the filter cavity brings an
optical spring that modifies the mechanical resonance frequency. The mechanism
is shown in Fig. 8. The Stokes sidebands of the pumping field inside the
filter cavity are off-resonant with the main interferometer optical resonance,
thereby being suppressed when it is sloshed from the filter cavity to the main
interferometer. However, the large bandwidth of the filter cavity indicates
that such a small mechanical resonance shift does not affect the parametric
process. This means that the parametric process will overwhelm the sloshing
process and instability forms. Using the Hamiltonian approach, we can
quantitatively compute the effect of the frequency compensation on the
stability, the result shows that we need to carefully tune the compensation
frequency near the $\omega_{\rm opt}$ to stabilise the system.

We plot the poles trajectories of the modes $\hat{a}$ and $\hat{c}$ on the
complex frequency domain in Fig. 10, when we tune the compensation frequency
under the PT-symmetry condition $\omega_{s}=G$. We found that these two modes
can both be stable only when the compensation frequency is within a small
frequency domain around $\omega_{\rm opt}$.

Figure 10: The poles trajectories of the $\hat{a}$ and $\hat{c}$ modes with
the change of the compensation frequency, when we introduced the idler fields
(after removing the instability at high frequency induced by pondermotive
effect in the arm cavity). The arrows show the direction of compensation
frequency increasing (under the condition of $\omega_{s}=G$). Using our
sampling parameters, these two poles are both stable only when $\omega_{\rm
comp}\in\omega_{\rm opt}+[91.39\rm Hz,91.76\rm Hz]$. The system will be
unstable around 2900 Hz or 3100 Hz if the compensation is out of this region
(corresponding to the red points and blue points).

The above discussions demonstrate that the instabilities which happen around
the detuning frequency $\Omega=\Delta$ can be compensated by the careful
tuning of the pumping frequency in the optomechanical filter cavity, while the
optical spring instability at low frequency can be controlled by
interferometer feedback servo system. For an overall verification, we also
analyzed the stability using the Nyquist criteria [48], which is a
conventional diagrammatic method for studying stability using the open-loop
transfer function. In our analysis, the open-loop transfer function
$G_{o}(\Omega)$ is chosen to describe the response of both the main detuned
signal-recycling interferometer and the pondermotive interaction in the
optomechanical filter cavity, shown in Fig. 1, while the SRM treated as the
feedback kernel. The close loop transfer function is therefore given by

$G_{c}(\Omega)=\frac{G_{o}(\Omega)}{1-r_{\rm SRM}G_{o}(\Omega)},$ (26)

where those transfer functions are calculated using full-transfer matrix
approach without making the resolved-sideband and single-mode approximations
(see Section V). The system is stable only if the contours generated by the
Nyquist map from the complex $\Omega$-plane to the $G_{c}(\Omega)$-plane do
not encircle the origin. In Fig. 11, we plot the winding numbers of the
Nyquist contour around the origin of the complex $G_{c}(\Omega)$-plane when
the above frequency tuning is performed. It finally tells us that the system
is stable since the winding number to the zero point is zero in the case when
we add an extra damping rate equal to $0.02$ Hz to damp the test mass optical
spring instability (using the sampling parameters given in Tab. 1), under the
condition that the filter cavity pumping frequency is carefully tuned.

Figure 11: The winding number: the system is stable with no poles on the upper
side of the complex plane (i.e. the winding number is zero) when we add a
damping rate of 0.02 Hz to compensate the optical spring instability. The
contour in the complex $\Omega$-plane is chosen to be the upper half-circle
with a radius equal to $2\pi\times 2\times 10^{4}$ rad/s. We homogeneously
choose the data points in this half-circle contour and map it to the
$G_{c}(\Omega)$-plane. The winding turns are defined as $\theta(\Omega)/2\pi$
explained in the upper pannel, where the shape peaks/dips indicates a close-
pass of the contour near the origin.

##  V Full analysis of an optomechanical realisation using exact transfer
matrix approach

In the discussion above, we have used the following approximations: (1)
single-mode approximation for each optical system so that $\Omega L_{\rm
arm}/c\ll 1$ and $2\omega_{m}L_{\rm SRC}/c\ll 1$, (2) the resolved sideband
approximation that allows to ignore the idler mode in the parametric process.
In this section, we show the exact case, especially to make use of the ignored
idler mode to compensate for the pondermotive degradation of sensitivity. Our
numerical calculation of the sensitivity curve follows the standard transfer
matrix approach, which was briefly summarised in [32].

The method of Wiener filtering can be used to combine the signal and idler
channels. The detailed derivation of this multi-channel Wiener filtering
method is discussed in the appendix of [49]. Denoting the two Wiener filter
functions as $K_{1,2}(\Omega)$, and the combined output $\hat{y}(\Omega)$ can
be written as:

$\hat{y}(\Omega)=K_{1}(\Omega)\hat{y}_{s}(\Omega)+K_{2}(\Omega)\hat{y}_{i}(\Omega),$
(27)

where $\hat{y}_{s/i}$ is the measured signal/idler field quadrature operator.
These operators can be formally represented as:

$\hat{y}_{s}=\mathbf{n}_{s}\cdot\mathbf{a}_{s}+d_{s}h,\quad\hat{y}_{i}=\mathbf{n}_{i}\cdot\mathbf{a}_{i}+d_{i}h,$
(28)

where $\mathbf{n}_{s/i}$ and $d_{s/i}$ are the transfer functions for noise
field $\mathbf{a}_{s/i}=(a_{1s/i},a_{2s/i})^{T}$ and the GW signal $h$.

Optimizing the sensitivity by varying these two filter functions leads to the
minimum noise given as the inverse of the largest eigenvalue of the following
matrix:

$\mathbf{M}=\mathbf{N}^{-1}\cdot\textbf{{\hbox{\Sigma}}}.$ (29)

where the noise and signal matrix are given by:

$\mathbf{N}=\left[\begin{array}[]{cc}\mathbf{n}_{s}\mathbf{S}_{a_{s}a_{s}}\mathbf{n}_{s}^{\dagger}&\mathbf{n}_{s}\mathbf{S}_{a_{s}a_{i}}\mathbf{n}_{i}^{\dagger}\\\
\mathbf{n}_{i}\mathbf{S}_{a_{i}a_{s}}\mathbf{n}_{s}^{\dagger}&\mathbf{n}_{i}\mathbf{S}_{a_{i}a_{i}}\mathbf{n}_{i}^{\dagger}\end{array}\right],$
(30)

and

$\Sigma=\left[\begin{array}[]{cc}|d_{s}|^{2}&d_{s}d_{i}^{*}\\\
d_{i}d_{s}^{*}&|d_{i}|^{2}\end{array}\right].$ (31)

The combined sensitivity is shown in Fig. 12, where the Wiener filtering
process favours the idler channel near 3 kHz. This combined sensitivity has a
much larger bandwidth and sensitivity than the conventional configuration at
around 3 kHz. Moreover, we also provide the behaviour of the sensitivity curve
in a larger frequency band, say $[1,10^{4}]$ Hz, in Fig. 13. Besides, the
Wiener filtering combined sensitivity has almost no difference from the signal
channel since the signal channel is favoured outside the frequency band around
3 kHz. The low-frequency peak corresponds to the optical spring effect due to
the main interferometer detuning.

Figure 12: The quantum-noise-limited sensitivity around 3000 Hz obtained by the full transfer matrix simulation. Upper panel: The detector sensitivity obtained from the signal/idler channels and their Wiener-filtering combination. Lower panel: comparing the Wiener-filtering combined sensitivity curve with that of the conventional detector and idealized PT-symmetric configuration. Figure 13: The quantum-noise-limited sensitivity obtained by the full transfer matrix simulation in frequency band [$1$,$10^{4}$] Hz. Upper panel: The detector sensitivity obtained from the signal/idler channels (phase quadrature) and their Wiener-filtering-combination. Lower panel: The detector sensitivity obtained from the signal/idler channels (amplitude quadrature) and their Wiener-filtering-combination. Note that the dashed line is almost overlapped with the signal channel except at around 3000 Hz, which can be hardly distinguished in the broad frequency band. Parameters | Symbols | Values
---|---|---
Arm cavity length | $L_{a}$ | 4 km
ITM mirror transmissivity | $T_{\rm ITM}$ | 0.028
Arm cavity power | $P_{c}$ | 1.5 MW
Arm cavity loss | $\epsilon_{\rm ETM}$ |  $10$ ppm
Arm cavity mirror mass | $M$ |  $100$ kg
iSRM transmissivity | $T_{\rm iSRM}$ |  $573.36$ ppm
SR cavity length | $L_{c}$ | 56 m
SR cavity loss | $\epsilon_{\rm SRC}$ |  $150$ ppm
Oscillator mass | $m$ |  $10$ mg
Mechanical frequency | $\omega_{m}$ |  $0.1$ MHz
Mechanical quality factor | $Q_{m}$ |  $10^{10}$ [50]
SRM transmissivity | $T_{\rm SRM}$ | $0.02$
Filter cavity round trip loss | $\epsilon_{f}$ |  $10$ ppm
Filter cavity power | $P_{f}$ | 7481.25 W
Filter cavity length | $L_{f}$ |  $40$m
Filter cavity environmental temperature | $T_{\rm en}$ |  $1$ K
Input squeezing level | $r_{\rm sq}$ |  $10$ dB
Table 1: Sample parameters for the detuned PT-symmetric gravitational wave
detector.

Furthermore, we also considered the effect of various losses in this protocol,
such as the noise introduced by the optical loss in the signal recycling
cavity (formed by iSRM, beam-splitter and the two ITMs in Fig. 1), the optical
loss in the arm-cavity (which is attributed to the ETM loss) and the optical-
loss/thermal noise in the optomechanical filter cavity. The corresponding
sampling parameters are listed in Tab. 1. The resultant noise budget is shown
in Fig. 14, where the sensitivity is plotted in both the frequency range of
$1-10^{4}$ Hz and around 3000 Hz. As also discussed in [51], the thermal noise
in the filter cavity behaves similarly to the optical loss in the filter
cavity in the broad frequency range, while a bit differently around the
detuning frequency $\Omega=\Delta$ due to the suppression of the filter’s
optical loss noise near the detuning frequency [26]. The noise components of
the noise budget in Fig. 14 are plotted using the Wiener combination
coefficient computed based on the total noise of idler and signal channels,
and we also assumed an injection of 10 dB phase squeezing field.

Figure 14: The noise budget of the detuned PT-symmetric gravitational wave
detector, where the noise due to the signal recycling cavity loss, filter
cavity loss, ETM loss and the thermal noise of the mechanical oscillator in
the filter cavity is considered. The upper panel is the noise budget from 1 Hz
to $10^{4}$ Hz, while the lower panel is the noise budget around 3000 Hz.

##  VI Astrophysical implications

Figure 15: The sensitivity curves with GW signal emitted by neutron star
merger remnants with two different equation of states. The black line is the
sensitivity of the conventional detuned interferometer, while the blue line is
the sensitivity of our protocol. These sensitivity curves are plotted using
the parameters in Tab. 1. The orange line is the BNS merger waveform of the
APR4-q10-M1375 model calculated in [52] and the skyblue line is the waveform
of 15H model listed in the data bank [53].

GW observations of binary neutron star (BNS) inspiral have already been
achieved and various constraints on the equation of state (EoS) of neutron
stars (NS) have been interpreted from the observation of GW170817 [54, 55, 56,
57, 58, 59]. Boosting the sensitivity of GW detectors around the frequency of
3 kHz could open up the possibility of direct detection of the merger and
post-merger GW signals of BNS merger events, with which the understanding of
the EoS of dense matter at supranuclear densities could be pushed to a new
level.

Based on BNS merger simulations carried out by various groups, it is widely
accepted that the fate of the post-merger remnant is determined by the ratio
between the maximum mass of a cold non-rotating NS (i.e., $M_{\mathrm{TOV}}$)
and the total mass of the binary (which could be accurately measured by
inspiral GW signal) [60]. Therefore, simply a non-/detection of the GW signal
from the post-merger remnant could tell that the remnant experiences
prompt/delayed-collapse to a black hole (BH). This already allows for an
independent constraint on $M_{\mathrm{TOV}}$. Furthermore, it has been found
that the post-merger GW signals could be used to constrain crucial properties
of the merging NSs such as compactness and tidal deformability [61, 9, 11, 62,
63]. However, the most relevant frequency range of the post-merger GW signals
lies in the range from 2 kHz to 3 kHz, which is too high for the current
generation GW detectors to resolve even for close sources such as GW170817.
Our design could make it possible for a measurement of the frequency peaks in
the post-merger phase and hence make complementary constraints on NS
properties.

Moreover, the density of the BNS merger remnant could be several times higher
than the inspiral stage. Strong interaction phase transition is suggested to
be possible under such conditions and could leave detectable features in the
post-merger GW signals [64, 65]. Compared the cases without a phase-
transition, the peak frequency of the post-merger GW is found to shift to a
even higher value. Having a detector with a sufficiently broad sensitivity and
frequency resolution at roughly 3 kHz could identify such a shift and
constrains the density range of the strong interaction phase transition, see
Fig. 15.

##  VII Dicussion and Conclusion

In this work, we provided an alternative design protocol based on the PT-
symmetry to the high-frequency gravitational wave detector, targeted at the
physics of binary neutron star coalescence. We have analyzed in detail the
optomechanical realization of a detuned PT-symmetric interferometer, targeted
at improving the sensitivity to the kilo-Hertz GWs using single-mode
approximation and full transfer matrix simulation. The effects of the idler
field and pondermotive interactions are analyzed in the single-mode
approximation, which provides physical insight for understanding the
sensitivity curves obtained using transfer matrix simulation. We showed that
using the same parameter setting as in the tuned case, after (1) compensating
the optical spring in the optomechanical quantum amplifier and (2) the Wiener-
filtering combination of the measurement data obtained from both signal
channel and idler channel, we can in principle achieve a significant boost of
the sensitivity around $3$ kHz. This sensitivity is 10-times better than the
conventional design. The dynamical instability of the system is induced by the
optical spring effect, which can be controlled in the detector feedback servo
system. Our work focuses on the conceptual designs of this protocol while
leaving the more practical and technical considerations to future studies.

##  VIII Acknowledgements

The authors thank our colleagues AEI-Caltech-MSU MQM discussion group and also
Huan Yang for fruitful discussions. Y. M. is supported by the start-up fund
provided by Huazhong University of Science and Technology. C. Z. acknowledges
the support of Australian Research Council through Grants No. DP170104424 and
No. CE170100004. X. L. and Y. C.’s research is funded by the Simons Foundation
(Grant No. 568762). H. M. acknowledge support from the Birmingham Institute
for Gravitational Wave Astronomy and the UK EPSRC New Horizons award (Grant
No. EP/V048872/1). H. M. has also been supported by a UK STFC Ernest
Rutherford Fellowship (Grant No. ST/M005844/11), and the start-up fund
provided by Tsinghua University and the State Key Laboratory of Low
Dimensional Quantum Physics of Tsinghua University.

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