[1,2]Yohei Nishino

[1]Department of Astronomy, University of Tokyo, Bunkyo, Tokyo, 113-0033,
Japan

2]Gravitational Wave Science Project, National Astronomical Observatory of
Japan (NAOJ), Mitaka City, Tokyo, 181-8588, Japan

3]Department of Gravitational Waves and Fundamental Physics, Maastricht
University, 6200 MD, Maastricht, The Netherlands

4]Nikhef – National Institute for Subatomic Physics, Science Park 105, 1098 XG
Amsterdam, The Netherlands

5]Department of Applied Physics, The University of Tokyo, Bunkyo, Tokyo,
113-0033, Japan

6]The University of Birmingham, School of Physics and Astronomy, and Institute
of Gravitational Wave Astronomy, University of Birmingham, Edgbaston,
Birmingham, B15 2TT, United Kingdom

# Frequency-dependent squeezing for gravitational-wave detection through
quantum teleportation

<EMAIL_ADDRESS>Stefan Danilishin<EMAIL_ADDRESS>Yutaro
Enomoto<EMAIL_ADDRESS>Teng Zhang<EMAIL_ADDRESS>*  [  [
[  [  [

###### Abstract

Ground-based interferometric gravitational wave detectors are highly precise
sensors for weak forces, limited in sensitivity across their detection band by
quantum fluctuations of light. Current and future instruments address this
limitation by injecting frequency-dependent squeezed vacuum into the detection
port, utilizing narrow-band, low-loss optical cavities for optimal rotation of
the squeezing ellipse at each signal frequency. This study introduces a novel
scheme employing the principles of quantum teleportation and entangled states
of light. It allows achieving broadband suppression of quantum noise in
detuned signal recycled-Fabry-Perot–Michelson interferometers, which is the
baseline design of the low-frequency detector within the Einstein Telescope
xylophone detector, without requiring additional filter cavities or
modifications to the core optics of the main interferometer.

## Introduction

Figure 1: Schematics of frequency-dependent squeezing via quantum
teleportation. We use two-photon formalism, where each beam is described by
quadrature amplitude operators, where subscripts 1 and 2 denote amplitude and
phase quadrature, respectively. All transformations of the beams are
represented by transfer matrices. The Bell observables here are defined as
$\hat{\alpha}_{1}=(\hat{V}_{1}-\hat{A}_{1})/\sqrt{2}$ and
$\hat{\alpha}_{2}=(\hat{V}_{2}+\hat{A}_{2})/\sqrt{2}$. When Victor, Alice and
Bob’s light undergo physical transformations, $\mathbb{U}_{v}$,
$\mathbb{U}_{a}$ and $\mathbb{U}_{b}$, the teleported state is transformed
accordingly to $\mathbb{U}_{b}\mathbb{U}_{a}\mathbb{U}_{v}\ket{\psi}$
accordingly.

In 2015, the Laser Interferometric Gravitational-wave Observatory (LIGO) [1]
achieved a milestone by detecting the first gravitational wave from a binary
black hole (BBH) merger [2]. This heralded the beginning of the gravitational-
wave astronomy, following which the LIGO-Virgo-KAGRA [3, 4] collaboration has
identified over 90 gravitational wave events [5, 6, 7, 8]. Furthermore, the
third generation detectors, i.e. the Cosmic Explorer [9] and the Einstein
Telescope [10], striving for tenfold greater sensitivity, will empower the
exploration of gravitational wave signals from the events spanning the entire
history of the universe [11, 12]. This endeavor will illuminate unresolved
inquiries in fundamental physics and cosmology [13, 14, 15].

Gravitational wave detectors serving as highly precise displacement
measurement instruments, are limited by quantum noise across most of their
frequency band. At low frequencies, the optimal sensitivity of a conventional
detector faces the constraints of the standard quantum limit (SQL) [16], as a
natural consequence of Heisenberg’s uncertainty principle [17]. To overcome
the SQL, scientists have proposed a variety of technologies based on the
quantum non-demolition measurement principle [18, 19]. The mainstream approach
for quantum noise reduction in the existing and future detectors is frequency-
dependent squeezing (FDS) injection [20, 21]. The injected squeezed vacuum is
produced by means of degenerate parametric down-conversion in nonlinear
optical crystals. The squeezing angle at each frequency is optimized by
passive optical cavities to cancel out the rotation of the squeezing ellipse
due to ponderomotive effect within the interferometer.

In tuned-broadband configurations, a single filter cavity is sufficient as
indicated in prior studies [22, 23]. However, a detuned configuration, such as
the low frequency interferometer of the Einstein Telescope’s xylophone
detector (ETLF), necessitates two filter cavities: one for compensating the
optical detuning resonance in the interferometer, the other for compensating
the so called optical spring resonance from the ponderomotive rigidity effect
[16, 24]. These filter cavities, while presently at a hundred-meter scale in
the second-generation detectors, are slated to reach kilometer scale in the
third-generation observatories. The installation of kilometer-scale filter
cavities entails substantial costs and technological difficulties. To tackle
this challenge, several approaches to achieve the required sensitivity without
the need for filter cavities have been proposed so far, such as the use of
entangled light and negative-mass atomic spin ensembles [25, 26] and the
small-scale optomechanical filters [27]. The EPR (for Einstein-Podolski-Rosen)
or conditional squeezing has been proposed by Ma, et al. [28], and
experimentally demonstrated in [29, 30]. In this scheme, by applying the
concept of EPR steering [31, 32] to gravitational wave detectors, the main
Fabry–Pérot-Michelson interferometer can be repurposed as a single filter
cavity. However, such an EPR squeezing is insufficient for detuned
configurations, which require two filter cavities.

In this paper, we introduce the concept of quantum teleportation (QT)
squeezing, which employs the principle of quantum teleportation to achieve
frequency-dependent squeezing in a detuned dual-recycled Fabry–Pérot-Michelson
interferometer without the use of filter cavities. The quantum teleportation
of an optical state has been a well-established technique in the field of
continuous variable quantum information processing since it was first
demonstrated by Furusawa et al. based on the Braustein–Kimble scheme [33, 34].
The teleportation procedure in the Braustein–Kimble scheme involves Alice, the
sending station; Bob, the receiving platform which shares EPR-entangled
photons [35] and Victor, that brings an unknown quantum state and initiates
the process (see Fig. 1). Alice conducts the Bell measurement involving
Victor’s and her own photons, transmitting the outcome to Bob through the
classical communication channel. Bob then displaces his photon based on
Alice’s information, leading to the successful teleportation of Victor’s
quantum state. By adding physical operations to Alice, Bob, and Victor’s
paths, described as $\mathbb{U}_{a,b,v}$ in Fig 1, one can also manipulate the
final teleported state. In our proposal of QT squeezing, we associate the
main-interferometer response including the ponderomotive squeezing [20] with
$\mathbb{U}_{b}$, and quadrature rotations by passive cavities with
$\mathbb{U}_{v,a}$

Fig. 2 shows the implementation of QT squeezing in a gravitational-wave
detector. Three beams, Victor, Alice and Bob, are injected into the
interferometer from the anti-symmetric port. The interferomter, acting as an
empty cavity for Victor and Alice, provides an optimized frequency-dependent
quadrature rotation by tuning their center frequencies and macroscopic lengths
of the arm and signal extraction cavity, which functions as filter cavities.
Bob’s frequency is matched to the main laser from the symmetric port, then he
sees the interferometer as an active cavity including ponderomotive squeezing
[20]. The output is spectrally separated into two detection ports by the
output mode cleaners; Bob’s beam get collapsed via homodyne detection, while
Victor and Alice are detected through Bell measurement. By applying Wiener
filters to the two outputs of Bell measurement, quantum noise suppression is
achieved as shown in Fig 4. In following sections, we will present the details
of each step mentioned above.

Figure 2: Configuration of QT squeezing. The OPA in the anti-symmetric port is
pumped at two frequencies, $2\omega_{0}+\Delta_{a}$ and
$2(\omega_{0}+\Delta_{v})$, generating entanglement at the sideband
frequencies symmetrically (see the top-right box). The former pumping results
in a two-mode EPR entanglement, Alice and Bob, while the latter forms a
squeezed state, Victor. Three beams are injected through a Faraday isolator.
The central part consists of a signal-recycled Fabry–Pérot Michelson
Interferometer, which includes the beam splitter (BS), input-test mass (ITM),
end-test mass (ETM) and signal-extraction mirror (SEM). The interferometer is
pumped at the frequency $\omega_{0}$, matching Bob’s frequency. The output is
spectrally separated by an output mode cleaner (OMC). Bob’s beam is collapsed
at the homodyne detection, while Victor and Alice is detected through Bell
measurement: two beams are combined with the local oscillator (LO) fields with
the LO angle $\xi_{\mathrm{LO}}$, and subsequently detected by two photo
detectors. The outputs are subtracted from each other and demodulated by two
demodulation angles. The two sets of measurement data are combined using the
optimal filter gain, $(g_{1}\ g_{2})$, and finally we achieve quantum-noise
suppression.

##  1 Quantum state preparation

Figure 3: Schematics of three beams. Top and bottom panels show fields in the
sideband and quadrature pictures, respectively. The OPA is pumped at two
frequencies, $2(\omega_{0}+\Delta_{a})$ and $2\omega_{0}+2\Delta_{v}$,
generating entanglement at the sideband frequencies symmetrically. The former
pumping results in a two-mode EPR entanglement, Alice and bob, while the
latter forms a squeezed state, Victor.

The initial quantum states of three beams, Alice, Bob, Victor are prepared
with a multi-mode squeezer [36]. The entangled beams, Alice and Bob, are
centred at frequencies $\omega_{0}+\Delta_{a}$ and $\omega_{0}$, created by
driving the OPA with a pumping beam at frequency $2\omega_{0}+\Delta_{a}$; the
squeezed beam, Victor at frequency $\omega_{0}+\Delta_{v}$, is created by the
second pumping beam at frequency $2(\omega_{0}+\Delta_{v})$ (see Fig. 3). Here
$\Delta_{a}$ is supposed to be within the OPA’s bandwidth, and
$\Delta_{v}-2\Delta_{a}$ equals to the multiples of the free spectral range of
the OPA cavity.

In two-photon formalism [37, 38], the zero-mean random fluctuations of light
field is described by a 2D-vector consisting the operators of amplitude and
phase quadrature, and for Alice, Bob and Victor, there is
$\boldsymbol{\hat{a}}=\\{\hat{a}_{1,\Omega},\,\hat{a}_{2,\Omega}\\}^{\rm
T},\boldsymbol{\hat{b}}=\\{\hat{b}_{1,\Omega},\,\hat{b}_{2,\Omega}\\}^{\rm
T},\boldsymbol{\hat{v}}=\\{\hat{v}_{1,\Omega},\,\hat{v}_{2,\Omega}\\}^{\rm
T}$, where $\Omega$ is sideband frequency, the superscript $\rm T$ stands for
transpose. Below we omit the superscript $\Omega$ for simplicity.

As shown by Duan et al. [39], the strength of the EPR-entanglement of Alice’s
and Bob’s state can be expressed in terms of the spectral densities of the
four EPR-operators, $(\hat{a}_{i}\pm\hat{b}_{i})/\sqrt{2}$, as

$\displaystyle S_{(\hat{a}_{1}\pm\hat{b}_{1})/\sqrt{2}}=e^{\pm 2r},\
S_{(\hat{a}_{2}\pm\hat{b}_{2})/\sqrt{2}}=e^{\mp 2r}$ (1)

where $r$ signifies the squeezing factor. When $r\rightarrow\infty$, the noise
spectra of $S_{(\hat{a}_{1}-\hat{b}_{1})/\sqrt{2}}$ and
$S_{(\hat{a}_{2}+\hat{b}_{2})/\sqrt{2}}$ approach zero, corresponding to the
original EPR entanglement [35]. In a more general situation, with the
quadrature $\hat{a}_{-\theta}=\hat{a}_{1}\cos\theta-\hat{a}_{2}\sin\theta$
measured, the quadrature
$\hat{b}_{\theta}=\hat{b}_{1}\cos\theta+\hat{b}_{2}\sin\theta$ is
conditionally squeezed and vice versa. The spectral density of the conditional
squeezed field reads

$S_{\hat{b}_{\theta}\hat{b}_{\theta}}^{\hat{a}_{-\theta}}=1/{\rm cosh}(2r)\,,\
S_{\hat{b}_{\pi/2+\theta}\hat{b}_{\pi/2+\theta}}^{\hat{a}_{-\theta}}=\rm{cosh}(2r)\,.$
(2)

The amplitude (phase) quadrature of Victor experiences (anti-)squeezing as

$\displaystyle S_{\hat{v}_{1}\hat{v}_{1}}=e^{-2r},\
S_{\hat{v}_{2}\hat{v}_{2}}=e^{2r}.$ (3)

Throughout this work, we assume uniform squeeze factors for each pumping
frequency for the sake of simplicity.

##  2 Noise suppression through quantum teleportation

After passing through the main interferometer, the read out phase quadrature
of Bob represented by the observable $B_{2}$, can be written as,

$\begin{split}\hat{B}_{2}=\Gamma
e^{i\beta_{b}}(\hat{b}_{1}\cos\theta_{b}-\hat{b}_{2}\sin\theta_{b}),\end{split}$
(4)

where $\Gamma$, $\theta_{b}$ and $\beta_{b}$ represent the frequency-dependent
gain, quadrature rotation, and phase shift from the ponderomotive squeezing as
defined in [40] (see also the Methods section). In order to squeeze
$\hat{B}_{2}$, we need to displace its photon at the input stage by applying
quadrature rotation $-\theta_{b}$. This can be prepared by applying a phase
rotation to its entangled pair Alice, $\theta_{a}$ and displacing Victor’s
state with quadrature rotation, $\theta_{v}$, where there needs
$(\theta_{a}+\theta_{v})=-\theta_{b}$. By configuring detunings of Alice and
Victor with respect to the main interferometer, the interferormeter can
function as a empty optical cavity for Victor and Alice and provide desired
quadrature rotations at its output.

###  2.1 Bell measurement

The teleportation of Victor’s state to Bob requires a Bell measurement between
Alice and Victor. In the Bell measurement, a local oscillator at frequency
$\omega_{0}+(\Delta_{a}+\Delta_{v})/2$ is combined with Alice and Bob by a
half beam splitter. Subsequently, the two output beams are detected by two
photo detectors and their photon currents are demodulated at frequency
$(\Delta_{v}-\Delta_{a})/2$. By setting the local oscillator angle to $-\pi/2$
and properly choosing the demodulation phase, Bell observables can be carried
out (details are available in the Methods section):

$\displaystyle\boldsymbol{\hat{\alpha}}=\begin{pmatrix}\hat{\alpha}_{1}\\\
\hat{\alpha}_{2}\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}\hat{V}_{1}-\hat{A}_{1}\\\
\hat{V}_{2}+\hat{A}_{2}\end{pmatrix},$ (5)

where
$\hat{A}_{1}=e^{i\beta_{a}}(\hat{a}_{1}\cos\theta_{a}-\hat{a}_{2}\sin\theta_{a})$,
$\hat{A}_{2}=e^{i\beta_{a}}(\hat{a}_{1}\sin\theta_{a}+\hat{a}_{2}\cos\theta_{a})$
and
$\hat{V}_{1}=e^{i\beta_{v}}(\hat{v}_{1}\cos\theta_{v}-\hat{v}_{2}\sin\theta_{v})$,
$\hat{V}_{2}=e^{i\beta_{v}}(\hat{v}_{1}\sin\theta_{v}+\hat{v}_{2}\cos\theta_{v})$
are the quadratures of the Alice and Victor’s beams after passing through the
interferometer. $\beta_{v}$ and $\beta_{a}$ are the averaged phases of both
quadratures.

###  2.2 Post processing

The classical communication channel can be built with displacement operation
or equivalently achieved through post-processing by combining $\hat{B}_{2}$
with $g_{1}\hat{\alpha}_{1}$ and $g_{2}\hat{\alpha}_{2}$, namely,

$\displaystyle\hat{B}_{2}^{\rm
tel}=\hat{B}_{2}-g_{1}\hat{\alpha}_{1}-g_{2}\hat{\alpha}_{2},$ (6)

where $g_{1},g_{2}$ are filter gains, whose optimal values can be derived by

$\displaystyle g_{1}$
$\displaystyle=\frac{S_{\hat{B}_{2}\hat{\alpha}_{1}}S_{\hat{\alpha}_{2}\hat{\alpha}_{2}}-S_{\hat{\alpha}_{2}\hat{\alpha}_{1}}S_{\hat{B}_{2}\hat{\alpha}_{2}}}{S_{\hat{\alpha}_{1}\hat{\alpha}_{1}}S_{\hat{\alpha}_{2}\hat{\alpha}_{2}}-|S_{\hat{\alpha}_{1}\hat{\alpha}_{2}}|^{2}},\quad
g_{2}=\frac{S_{\hat{B}_{2}\hat{\alpha}_{2}}S_{\hat{\alpha}_{1}\hat{\alpha}_{1}}-S_{\hat{\alpha}_{1}\hat{\alpha}_{2}}S_{\hat{B}_{2}\hat{\alpha}_{1}}}{S_{\hat{\alpha}_{1}\hat{\alpha}_{1}}S_{\hat{\alpha}_{2}\hat{\alpha}_{2}}-|S_{\hat{\alpha}_{1}\hat{\alpha}_{2}}|^{2}},$
(7)

(see more details in the Methods section). In an idealized case absence of
imperfections, we can obtain the eventual noise spectrum density of
$B_{2}^{\rm tel}$,

$\begin{split}S^{\mathrm{tel}}_{\hat{B}_{2}\hat{B}_{2}}=|\Gamma|^{2}\frac{1+e^{-2r}\cosh
2r}{e^{-2r}+\cosh 2r}\xrightarrow{r\gg
1}|\Gamma|^{2}\frac{3}{e^{2r}}.\end{split}$ (8)

Taking into account that the gravitational-wave signal sidebands are not
affected throughout the post-processing, Eq.(8) indicates that the strain
sensitivity is improved by a factor of $3/e^{2r}$ across wide range of
frequencies with sufficient squeezing.

##  3 Sensitivity comparison

Figure 4: Top: Quantum-noise limited strain sensitivity of ETLF. The dashed
and solid curves represent the baseline FDS with -10 dB and QT squeezing,
respectively, with different levels of squeezing. The dotted curve is the
baseline FDS with -17 dB squeezing. Bottom : Quantum noise-enhancement in
power-spectral density compared to the non-squeezed case of the baseline FDS
(shown in the dashed-dotted curve in the top panel).

In the top panel of Fig 4, we show the quantum noise-limited sensitivity of
ETLF [10] with the conventional frequency-dependent squeezing (hereafter
called the baseline FDS) and QT squeezing. As a comparison we also plotted the
enhancement factor compared to non-squeezed case of the baseline FDS (denoted
as No SQZ baseline). The parameters considered, including imperfections such
as losses and phase noises, are consistent with the current design of ETLF
employing filter cavities, except for the squeezing level as shown in Table 1.
Detuning frequencies $\Delta_{a}$ and $\Delta_{v}$ are also shown in the
table, while the method of parameter searching and details of macroscopic
length tuning are discussed in the Supplementary materials.

Table 1: Parameters for ETLF Parameters |  Baseline FDS |  QT SQZ
---|---|---
Detuning of the SEC | 0.75 rad
Filter cavity length |  1 km |  -
Arm round trip loss | 45 ppm
SEC loss | 1000 ppm
Injection loss | 4 %
Readout loss | 3 %
FC round-trip loss |  20 ppm |  -
Squeezer noise RMS222Root mean square | 10 mrad
Local oscillator RMS | 10 mrad
SEC length RMS | 1 pm
Filter cavity length RMS |  1 pm |  -
Detuning $\Delta_{a}$ |  - |  $\sim 16.5$ MHz333$768\mathrm{kHz}+11\mathrm{FSR}_{\mathrm{SEC}}$ to be more precise.
Detuning $\Delta_{v}$ |  - |  $\sim 1.59$ GHz444$655\mathrm{kHz}+1059\mathrm{FSR}_{\mathrm{SEC}}$
Squeezing level |  -10 dB |  -17 dB

In general, QT squeezing exhibits sensitivity levels that are inferior to
those of the baseline FDS as shown in most frequency range in the lower panel
of Fig 4. This disparity arises from the 4.8 dB penalty inherent in QT
squeezing as indicated by Eq. (8) and there are also threefold optical losses
summing from three optical paths.

However, we can also notice the performance of QT squeezing surpasses that of
baseline FDS around 8 Hz. The uneven sensitivity enhancement in the baseline
scenarios arises from the dephasing of the squeezed vacuum. As studied in [41,
40], the dephasing of squeezed beam will lead coupling of the noise
fluctuation from anti-squeezed quadrature into the squeezed quadrature. Around
8 Hz, where the effective mechanical susceptibility of test mass strengthens
due to the optical spring effect, the interferometer creates more significant
ponderomotive squeezing onto the quantum fields interacting with the mirrors.
Therefore, the sensitivity is more susceptible to dephasing effect in
particular at this frequency band, and the optimal input squeezing level is
limited to $\sim\ 10$ dB (see the dashed and dotted curves in Fig 4). Such
dephasing can stem from length fluctuation of optical paths and optical losses
in detuned cavities. In the baseline FDS, the filter cavity length is
constrained to 1 km due to infrastructure limitations, as also employed in
[42].

In contrast, QT squeezing leverages the stability and length of the 10-km-long
arm cavities as filter cavities. The length fluctuation of the arm cavities is
well-suppressed by multi-stage suspension and control systems.

As the filter cavity, the 10 km interferometer has lower effective loss
compared with 1 km filter cavity, evaluated through the term loss per unit
length [41]. We take the 1000 ppm loss from SEC into account, and it turns out
the SEC loss is mitigated by the low transmissivity of the input-test masses
(see the Supplementary Materials for more details). Those result in less
effective loss and dephasing, thus allows higher input squeezing level.
Consequently, by accommodating -17 dB squeezing, we can achieve $\sim$ 5 dB
sensitivity improvement over the whole frequency band.

As a figure of merit, We plot the detection horizon of equal-mass non-spinning
compact binary coalescence in Fig 5. The detection criterion is set at a
signal-to-noise ration of 8. The overall power spectral densities integrate
classical and quantum noise of ETLF, as well as the total noise of the high-
frequency part of ET, covering the frequency range above 20 Hz. The horizon
plot indicates that the QT squeezing achieves performance almost equivalent to
the baseline FDS; more precisely, QT squeezing shows slightly better maximum
horizon at the cost of the mass-range below $20\ M_{\odot}$. This is a
consequence of the local improvement of the quantum noise around 8 Hz.

Figure 5: Detection horizon of ET with two squeezing schemes in ETLF, i.e.,
the baseline FDS and QT squeezing.

##  4 Discussions

Our scheme attains frequency-dependent squeezing by replacing the external
filter cavities, which would otherwise require kilometer-scale additional
vacuum tunnel and suspension systems, with the main interferometer itself
through the application of the quantum-teleportation technique. It offers a
clear sensitivity advantage benefiting from the long effective length of the
arm cavity as the filter cavities.

However, we also need to note two drawbacks of the QT squeezing: (1) a factor
of 3 (4.8 dB) higher injected squeezing required to reach the same level of
detected squeezing at the readout port as achieved by the baseline FDS. (2)
threefold noise contributions from input and output losses, which limit the
sensitivity across the entire frequency band (see the figure of noise budget
in a Supplementary Materials).

In addition, it is essential to highlight the technical flexibility of our
scheme. The configuration illustrated in Fig. 2 allows for seamless
transitions between detuned and tuned configurations. This transition can be
achieved by turning off the pumping laser beam for Victor and adjusting the
pumping frequency of the other beam, along with the macroscopic lengths of the
arm and signal extraction cavity,effectively reverting back to the EPR
squeezing in [28]. Furthermore, in the case of detuned configurations, QT
squeezing provides the capability to address variations in the SEC detuning by
optimizing the parameters of pumping frequencies and macroscopic length
tuning. This means we do not need to replace the input mirror of filter cavity
itself corresponding to a change of the value of detuning.

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Acknowledgments Authors are grateful to S. Hild for stimulating discussions
and insight into potential benefits our scheme might bring for Einstein
Telescope. Research by Y. N. is supported by JSPS Grant-in-Aid for JSPS
Fellows Grant Number 23KJ0787. T. Z. acknowledges the support of the Institute
for Gravitational Wave Astronomy at the University of Birmingham. S. D. is
grateful for support to the Faculty of Science and Engineering of Maastricht
University and to the European Research Council (within the frames of the
ERC-2020-AdG agreement No. 101019978).

##  5 Methods

###  5.1 Derivation of Wiener filters

In the ideal lossless case, response of a differential mode cavity of the
detuned interferometer for the carrier field can be written as below according
to the scaling-law theorem [43]:

$\displaystyle\begin{pmatrix}\hat{B}_{1}\\\ \hat{B}_{2}\end{pmatrix}$
$\displaystyle=\frac{1}{\tilde{M}}\begin{pmatrix}C_{11}&C_{12}\\\
C_{21}&C_{22}\end{pmatrix}\begin{pmatrix}\hat{b}_{1}\\\
\hat{b}_{2}\end{pmatrix},$ (9)

where

$\displaystyle\tilde{M}=\\{(\gamma-i\Omega)^{2}+\delta^{2}\\}\Omega^{2}-\delta\Theta,$
$\displaystyle
C_{11}=C_{22}=\Omega^{2}(\Omega^{2}-\delta^{2}+\gamma^{2})+\delta\Theta,$
$\displaystyle\quad
C_{12}=2\delta\gamma\Omega^{2}-2\gamma\Theta,C_{21}=-2\delta\gamma\Omega^{2}.$

Here $\delta$ and $\gamma$ are the effective detuning and half-bandwidth.
$\Theta=\frac{8\omega_{0}P_{c}}{McL}$ is the normalized optical power with the
arm circulating power $P_{c}$, reduced mass $M$ and arm length $L$. From these
relations, $\Gamma$, $\theta_{b}$ and $\beta_{b}$ in Eq. (4) can be derived
as:

$\displaystyle\Gamma=\frac{\sqrt{C_{21}^{2}+C_{22}^{2}}}{|\tilde{M}|},\quad\theta_{b}=-\arctan\left({\frac{C_{22}}{C_{21}}}\right),\quad\beta_{b}=\arg{\tilde{M}^{*}}.$
(10)

Combining the measurement data with filter gains $(g_{1}\ g_{2})$ such that
$\hat{B}_{2}^{g}=\hat{B}_{2}-g_{1}\hat{\alpha}_{1}-g_{2}\hat{\alpha}_{2}$, we
have the noise spectrum:

$\displaystyle S_{\hat{B}_{2}^{g}\hat{B}_{2}^{g}}$
$\displaystyle=S_{\hat{B}_{2}\hat{B}_{2}}+|g_{1}|^{2}S_{\hat{\alpha}_{1}\hat{\alpha}_{1}}+|g_{2}|^{2}S_{\hat{\alpha}_{2}\hat{\alpha}_{2}}$
$\displaystyle-
g_{1}^{*}S_{\hat{B}_{2}\hat{\alpha}_{1}}-g_{1}S_{\hat{\alpha}_{1}\hat{B}_{2}}-g_{2}^{*}S_{\hat{B}_{2}\hat{\alpha}_{2}}-g_{2}S_{\hat{\alpha}_{2}\hat{B}_{2}}$
$\displaystyle+g_{1}g_{2}^{*}S_{\hat{\alpha}_{1}\hat{\alpha}_{2}}+g_{1}^{*}g_{2}S_{\hat{\alpha}_{2}\hat{\alpha}_{1}},$
(11)

where

$\displaystyle S_{\hat{B}_{2}\hat{B}_{2}}$ $\displaystyle=\Gamma^{2}\cosh 2r,$
$\displaystyle S_{\hat{\alpha}_{1}\hat{\alpha}_{1}}$
$\displaystyle=\frac{e^{-2r}\cos^{2}\theta_{v}+e^{2r}\sin^{2}\theta_{v}+\cosh
2r}{2},$ $\displaystyle S_{\hat{\alpha}_{2}\hat{\alpha}_{2}}$
$\displaystyle=\frac{e^{-2r}\sin^{2}\theta_{v}+e^{2r}\cos^{2}\theta_{v}+\cosh
2r}{2},$ $\displaystyle S_{\hat{\alpha}_{1}\hat{\alpha}_{2}}$
$\displaystyle=\frac{(e^{-2r}-e^{2r})\sin\theta_{v}\cos\theta_{v}}{2},$
$\displaystyle S_{\hat{B}_{2}\hat{\alpha}_{1}}$
$\displaystyle=S^{*}_{\hat{\alpha}_{1}\hat{B}_{2}}=-\frac{\Gamma
e^{i(\beta_{b}-\beta_{a})}\cos\theta_{v}\sinh 2r}{\sqrt{2}},$ $\displaystyle
S_{\hat{B}_{2}\hat{\alpha}_{2}}$
$\displaystyle=S^{*}_{\hat{\alpha}_{2}\hat{B}_{2}}=-\frac{\Gamma
e^{i(\beta_{b}-\beta_{a})}\sin\theta_{v}\sinh 2r}{\sqrt{2}}.$

$S_{\hat{B}_{2}^{g}\hat{B}_{2}^{g}}$ takes its minimum when $g_{1}$ and
$g_{2}$ are Wiener filters determined as follows:

$\displaystyle g_{1}$
$\displaystyle=\frac{S_{\hat{B}_{2}\hat{\alpha}_{1}}S_{\hat{\alpha}_{2}\hat{\alpha}_{2}}-S_{\hat{\alpha}_{2}\hat{\alpha}_{1}}S_{\hat{B}_{2}\hat{\alpha}_{2}}}{S_{\hat{\alpha}_{1}\hat{\alpha}_{1}}S_{\hat{\alpha}_{2}\hat{\alpha}_{2}}-|S_{\hat{\alpha}_{1}\hat{\alpha}_{2}}|^{2}}=-\frac{\sqrt{2}\Gamma
e^{i(\beta_{b}-\beta_{a})}\sinh{2r}\cos\theta_{v}}{\cosh{2r}+e^{-2r}},$
$\displaystyle g_{2}$
$\displaystyle=\frac{S_{\hat{B}_{2}\hat{\alpha}_{2}}S_{\hat{\alpha}_{1}\hat{\alpha}_{1}}-S_{\hat{\alpha}_{1}\hat{\alpha}_{2}}S_{\hat{B}_{2}\hat{\alpha}_{1}}}{S_{\hat{\alpha}_{1}\hat{\alpha}_{1}}S_{\hat{\alpha}_{2}\hat{\alpha}_{2}}-|S_{\hat{\alpha}_{1}\hat{\alpha}_{2}}|^{2}}=-\frac{\sqrt{2}\Gamma
e^{i(\beta_{b}-\beta_{a})}\sinh{2r}\sin\theta_{v}}{\cosh{2r}+e^{-2r}}.$

Note that we used the relation $\theta_{b}+\theta_{a}=-\theta_{v}$.
Substituting those into Eq. (11) leads to the noise spectrum density shown in
Eq. (8)

###  5.2 Mathematical description of Bell measurement

We describe the Bell measurement procedure, with specific reference to section
II.C in [44]. Bell measurement utilizes a coherent laser as the Local
Oscillator (LO) to measure the quadratures $\hat{V}_{1}-\hat{A}_{1}$ and
$\hat{V}_{2}+\hat{A}_{2}$ (see Fig 2). The frequency of the LO is precisely
tuned to match the central frequency of Victor and Alice. In our specific
experimental context, it is necessary to control the LO frequency to
$\omega_{0}+(\Delta_{a}+\Delta_{v})/2$, a frequency regime in the radio
frequencies (RF) domain, typically in the megahertz (MHz) range.

Bell measurement involves several key steps. Firstly, a half beam splitter
(HBS) is employed to combine the two idlers with the LO, expressed as:

$\displaystyle E_{r,t}(t)=\frac{S(t)\pm L(t)}{\sqrt{2}}.$ (12)

Here $E_{r,t}$ are the reflection and transmission of the HBS. $S(t)$ and
$L(t)$ are the output from the interferomter and the LO field, respectively,
expressed in the sideband picture as:

$\displaystyle S(t)$
$\displaystyle=\int_{-\Lambda}^{\Lambda}\frac{\mathrm{d}\Omega}{2\pi}\\{\hat{A}_{\omega_{0}+\Delta_{a}+\Omega}e^{-i(\omega_{0}+\Delta_{a}+\Omega)t}+\hat{V}_{\omega_{0}+\Delta_{v}+\Omega}e^{-i(\omega_{0}+\Delta_{v}+\Omega)t}+\mathrm{h.c.}\\}$
(13) $\displaystyle L(t)$
$\displaystyle=De^{i\\{\omega_{0}+(\Delta_{a}+\Delta_{v})/2\\}}+\mathrm{h.c.},$
(14)

where $\Lambda\lesssim(\Delta_{a}+\Delta_{v})/2$ is the demodulation
bandwidth, $D$ is the complex amplitude, and h.c. represents Hamiltonian
conjugate.

Secondly, two fields are detected by the photodetectors, combining two outputs
to reject classical and quantum fluctuations in the LO field. After the
combination, the photocurrent is proportional to the square of the field Eq.
(12):

$\displaystyle i(t)$ $\displaystyle\propto E_{r}^{2}-E_{t}^{2}\propto
S(t)L(t)$
$\displaystyle=D\int_{-\Lambda}^{\Lambda}\frac{\mathrm{d}\Omega}{2\pi}\\{A_{\omega_{0}+\Delta_{a}+\Omega}e^{i\\{(\Delta_{v}-\Delta_{a})-\Omega\\}t}+V_{\omega_{0}+\Delta_{v}+\Omega}e^{-i\\{(\Delta_{v}-\Delta_{a})+\Omega\\}t}\\}+\mathrm{h.c.}$
$\displaystyle\quad+[\mathrm{irrelevant\ terms\ at\ high\ frequencies}]$ (15)

Finally, by mixing $\cos\\{(\Delta_{v}+\Delta_{a})t/2+\xi_{\mathrm{d}}\\}$ and
applying a low-pass filter with the cut-off of $\Lambda$, one obtains:

$\displaystyle
O(\xi_{\mathrm{d}};\Omega)=D\int_{-\Lambda}^{\Lambda}\frac{\mathrm{d}\Omega}{2\pi}\\{A_{\omega_{0}+\Delta_{a}+\Omega}e^{-i\xi_{\mathrm{d}}}e^{-i\Omega
t}+\mathrm{h.c.}\\}$
$\displaystyle+D\int_{-\Lambda}^{\Lambda}\frac{\mathrm{d}\Omega}{2\pi}\\{V_{\omega_{0}+\Delta_{v}+\Omega}e^{i\xi_{\mathrm{d}}}e^{-i\Omega
t}+\mathrm{h.c.}\\}.$ (16)

In the quadrature picture, the quadrature operator $A_{\zeta}$ is defined as:

$\displaystyle A_{\zeta}=A_{1}\sin\zeta+A_{2}\cos\zeta,$ (17)

where

$\displaystyle
A_{1}=\frac{A_{\omega_{0}+\Omega}+A^{\dagger}_{\omega_{0}-\Omega}}{\sqrt{2}},\
A_{1}=\frac{A_{\omega_{0}+\Omega}-A^{\dagger}_{\omega_{0}-\Omega}}{i\sqrt{2}}.$
(18)

Using those relations, Eq. (16) leads to:

$\displaystyle
O(\xi_{\mathrm{d}};\Omega)=|D|\int_{0}^{\Lambda}\frac{\mathrm{d}\Omega}{2\pi}e^{-i\Omega
t}\\{A_{\zeta_{A}}(\Omega)+V_{\zeta_{V}}(\Omega)\\},$ (19)

where $\zeta_{A}=\frac{\pi}{2}+\arg\\{{De^{-i\xi_{\mathrm{d}}}}\\}$ and
$\zeta_{V}=\xi_{\mathrm{d}}+\frac{\pi}{2}+\arg D$ (see also Eqs. (9)-(12) in
[44]). In the frequency domain, one obtains:

$\displaystyle
O(\xi_{\mathrm{d}};\Omega)=|D|\frac{A_{\zeta_{A}}(\Omega)+V_{\zeta_{V}}(\Omega)}{\sqrt{2}}$
(20)

The LO angle $\arg D$ is considered as a free parameter, determined
experimentally, while the demodulation angle $\xi_{\mathrm{d}}$ can be
adjusted after detection. By setting $\arg D=\pi/2$, the outputs become
$(V_{1}-A_{1})/\sqrt{2}$ and $(V_{2}+A_{2})/\sqrt{2}$ with $\xi_{\mathrm{d}}$
of $-\pi/2$ and $\pi$, denoted as I and Q phase in Fig 2, respectively.