Optimization of quantum noise in space gravitational-wave antenna DECIGO with
optical-spring quantum locking considering mixture of vacuum fluctuations in
homodyne detection
Kenji TsujiA, Tomohiro IshikawaA, Kentaro Komori${}^{B,\ C}$, Koji NaganoD,
Yutaro EnomotoE,
Yuta Michimura${}^{B,\ F}$, Kurumi UmemuraA, Ryuma ShimizuA, Bin WuA, Shoki
IwaguchiA,
Yuki KawasakiA, Akira Furusawa${}^{E,\ G}$, Seiji Kawamura${}^{A,\ H}$

  * A

Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi
464-8602, Japan

  * B

Research Center for the Early Universe (RESCEU), School of Science, University
of Tokyo, Tokyo 113-0033, Japan

  * C

Department of Physics, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan

  * D

LQUOM, Inc., Tokiwadai, Hodogaya, Yokohama city, Kanagawa, 240-8501, Japan

  * E

Department of Applied Physics, School of Engineering, University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

  * F

LIGO Laboratory, California Institute of Technology, Pasadena, California
91125, USA

  * G

Center for Quantum Computing, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198,
Japan

  * H

The Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Nagoya University, Nagoya, Aichi 464-8602, Japan

###### Abstract

Quantum locking using optical spring and homodyne detection has been devised
to reduce quantum noise that limits the sensitivity of DECIGO, a space-based
gravitational wave antenna in the frequency band around 0.1 Hz for detection
of primordial gravitational waves. The reduction in the upper limit of energy
density ${\Omega}_{\mathrm{GW}}$ from $2{\times}10^{-15}$ to
$1{\times}10^{-16}$, as inferred from recent observations, necessitates
improved sensitivity in DECIGO to meet its primary science goals. To
accurately evaluate the effectiveness of this method, this paper considers a
detection mechanism that takes into account the influence of vacuum
fluctuations on homodyne detection. In addition, an advanced signal processing
method is devised to efficiently utilize signals from each photodetector, and
design parameters for this configuration are optimized for the quantum noise.
Our results show that this method is effective in reducing quantum noise,
despite the detrimental impact of vacuum fluctuations on its sensitivity.

## I. INTRODUCTION

Since the first detection of gravitational waves in 2015 [1], the ongoing
observation of gravitational waves resulting from the mergers of binary black
holes and binary neutron stars has marked a momentous juncture in the domain
of gravitational wave astronomy. This continuous endeavor bestows upon us
novel perspectives into astronomical phenomena, catalyzing a revolutionary
transformation within the field of astronomy [2]. Moreover, it is envisaged
that the growing significance of gravitational wave detections will play an
increasingly pivotal role in the realm of astronomy with contributions from
ground-based detectors such as Einstein Telescope [3] and Cosimic Explorer
[4], or space-based detectors such as LISA [5]. This is owing to their unique
ability to discern enigmatic phenomena that pose formidable challenges for
observation through electromagnetic waves.

The focus of our research is primordial gravitational waves. If we can
directly detect these gravitational waves, which originate from quantum
fluctuations in the spacetime during the cosmic inflation, we can gain a more
detailed understanding of the early stages of the universe, including
confirming the occurrence of cosmic inflation [6]. Detectors exhibiting high
sensitivity in the frequency band around 0.1 Hz, which are free of seismic
noise from the Earth and thermal noise from mirror suspension, are required.
However, conventional ground-based detectors, LIGO [7], VIRGO [8] and KAGRA
[9] cannot remove them.

Our project for these detections is called the DECi-hertz Interferometer
Gravitational-wave Observatory (DECIGO) and has been promoted as a space-based
gravitational wave antenna in Japan [10, 11]. DECIGO stands out due to its
distinctive utilization of three drag-free satellites deployed in space and a
Fabry-Perot interferometer configuration, spanning a length of 1000 km, to
mitigate the adverse effects of earth’s seismic noise and thermal noise from
mirror suspensions. This configuration therefore allows us to target a lower
frequency band (in this case the 0.1 Hz band) than the frequency band where
ground-based detectors have high sensitivity, typically around 100 Hz.
Meanwhile, recent analysis of observations by the Planck satellite and others
[12] have shown that the original design of DECIGO was not sensitive enough to
detect primordial gravitational waves. Hence, developing techniques to improve
the sensitivity of DECIGO is required. Previous studies have proposed
optimizing the design parameters of DECIGO [13, 14, 15], as well as a
technique known as quantum locking [16, 17], to enhance sensitivity by
mitigating quantum noise. Quantum noise predominantly affects the low-
frequency band and arises from the quantum fluctuations of laser light.
Notably, quantum locking involves incorporating sub-cavities on both sides
that share a mirror with the 1,000 km primary cavity, resulting in a reduction
of radiation pressure noise. Given the high optical loss and lack of squeezing
feasibility in the main cavity of DECIGO, this method proves to be
exceptionally effective [18], and its utility has been progressively confirmed
through in-principle verification experiments [19]. Additionally, the
configuration termed optical-spring quantum locking, which employs optical
springs [20] and homodyne detection in sub-cavities for quantum locking, has
theoretically exhibited a considerable enhancement in sensitivity [21].

However, prior investigations have not employed homodyne detection based on an
experimental setup that faithfully reflects the actual optical effects, rather
assumed ideal homodyne detection. This limitation arises due to the treatment
of parameter $\eta$, responsible for determining the direction of homodyne
detection, which has not been ascertained through interference light but
rather regarded as an independent variable subject to arbitrary choice during
simulations. In order to faithfully model the system based on the actual
setup, it becomes imperative to account for the influence of the mixture of
vacuum fluctuations. Consequently, the determination of the homodyne detection
direction necessitates the extraction of light from the conventional optical
path to produce interference, wherein the use of beam splitters, acting as
points of interference, is indispensable. Hence, the primary objective of this
paper is the accurate evaluation of the homodyne detection approach, taking
into consideration the mixture of vacuum fluctuations. Furthermore, in this
configuration, an additional photodetector can be employed, and we propose a
method to utilize the signal from this supplementary photodetector. In this
paper, we apply this method while optimizing each design parameter in a manner
akin to previous research [21].

In Section II, a more elaborate elucidation of the optical design is
presented, encompassing the mitigation of quantum noise through optical-spring
quantum locking, along with the configuration of homodyne detection
incorporating these constituents. The intricate approach to signal
optimization, achieved through the combination of multiple signals, is
expounded upon in Section III. Furthermore, this section offers a
comprehensive block diagram illustrating the acquisition of these signals.
Sections IV through V showcase simulation results, demonstrating the
parameters that yield the utmost sensitivity for DECIGO under this
configuration, as well as the corresponding obtained signals. Additionally, we
delve into the sensitivity difference of DECIGO in comparison with prior
research.

Figure 1: Standard configuration of quantum locking in DECIGO. The path that
the red laser light passes through represents the main cavity, while the paths
that the green laser lights pass through correspond to the sub-cavities. The
two mirrors of the main cavity are shared with the sub-cavities, and the
photodetector signals from the sub-cavities contain information regarding the
noise induced by the main laser light.

## II. OPTICAL DESIGN

### A. Optical-Spring Quantum Locking

In this subsection, we describe optical-spring quantum locking used to reduce
quantum noise. Initially, we elucidate the methodology employed to address
quantum fluctuations, utilizing a mathematical framework known as quadrature-
phase amplitude [22]. This formalism incorporates the application of creation
and annihilation operators, denoted as $a_{j}$ and $a_{j}^{{\dagger}}$,
respectively, which satisfy the commutation relation (Eq. 1). These operators
play a pivotal role in the analysis of the optical mode within each cavity.

$[a_{j},a_{j^{\prime}}^{{\dagger}}]=2{\pi}{\delta}_{jj^{\prime}}$ (1)

Here, $j$ is an identifier of the three cavities, indicating Main(=m),
Sub1(=s1) or Sub2(=s2). Then, using $a_{j}$ and $a_{j}^{{\dagger}}$ operators,
the amplitude quantum fluctuation $q_{j}$ and phase quantum fluctuation
$p_{j}$ are defined by

$q_{j}={\frac{1}{\sqrt{2}}}(a_{j}{}+a_{j}^{{\dagger}})\hskip 5.69054pt,\hskip
5.69054ptp_{j}={\frac{1}{i\sqrt{2}}}(a_{j}-a_{j}^{{\dagger}}).$ (2)

We now elaborate on the approach of reducing quantum noise through quantum
locking. Figure 1 illustrates the standard configuration of quantum locking
designed for DECIGO. In this configuration, supplementary sub-cavities,
equipped with shorter cavity lengths and shared mirrors, are appended to both
sides of the main optical cavity, which spans a distance of 1,000 km between
DECIGO satellites. Notably, the laser sources employed for the sub-cavities
differ from the one utilized for the main cavity. Based on this setup, crucial
information concerning the amplitude quantum fluctuations of the main laser
light can be extracted from the signals transmitted via the sub-cavity laser
light. By regulating the shared mirror using these signals, it becomes
possible to effectively nullify the radiation pressure noise stemming from the
main cavity, while simultaneously preserving the underlying gravitational wave
signal.

Moreover, we utilize the optical spring for the sub-cavities. The optical
spring is a contrived technology that accomplishes heightened sensitivity by
creating the interaction between mechanical and optical effects. A mirror
positioned slightly away from the resonance point experiences a constant
external force that counterbalances the radiation force within the cavity.
Consequently, when the radiation force changes due to quantum fluctuations,
the mirror behaves akin to one mounted on a spring due to its relationship
with the radiation force.

Figure 2: Concept of homodyne detection. The lower path represents the optical
path for the local oscillator, which undergoes a phase shift by $\xi$. $E_{i}$
is the classical electric field of the laser light. The regions enclosed by
dotted lines are quantum fluctuations of the laser right or vacuum
fluctuations. We use superscript $(l)$ to denote quantum fluctuations of the
laser light and superscript $(v)$ to denote vacuum fluctuations,
distinguishing their respective noise sources.

### B. Homodyne Detection

In addition to employing optical-spring quantum locking, we adopt homodyne
detection to mitigate the influence of quantum noise. The concept of homodyne
detection is elucidated in Figure 2. In this configuration, beam splitter 1
(=BS1) is strategically positioned to extract light as a local oscillator from
the incident light directed towards the cavity. Meanwhile, beam splitter 2
(=BS2) brings the local oscillator light to interference with the light that
either reflects or traverses the cavity. This arrangement allows for detection
along a direction different from that of the normal carrier light, playing a
crucial role in quantum noise reduction. The interfered light is subsequently
captured by two photodetectors, as depicted in the figure. The variables
${\eta}^{(A)}$ and ${\eta}^{(B)}$, which determine the detection axis angle,
are derived through the ensuing steps. To commence, let us delineate the
electric field $E_{1}$ of the light emanating from the laser source as
follows:

$E_{1}=E_{0}e^{i{\omega}_{0}t},$ (3)

where $E_{0}$ is a constant representing the amplitude of the electric field,
and ${\omega}_{0}$ is the angular frequency of the laser light. At the same
time, the local oscillator $E_{2}$ is obtained using the amplitude reflectance
$r^{(h)}_{1}$ of BS1, given by

$E_{2}=r^{(h)}_{1}e^{i{\xi}}E_{0}e^{i{\omega}_{0}t}=r^{(h)}_{1}e^{i{\xi}}E_{1}.$
(4)

Here, $\xi$ is a parameter defined as the relative phase shift associated with
the change in the optical path length. The light directed towards the cavity
combines the light entering the cavity with the light reflected by the input
mirror, and the carrier light $E_{3}$ is expressed as follows:

$E_{3}=\Bigl{[}-r_{1}^{(sc)}+{\frac{{t_{1}^{(sc)}}^{2}r_{2}^{(sc)}e^{i{\phi}^{\prime}}}{1-r_{1}^{(sc)}r_{2}^{(sc)}e^{i{\phi}^{\prime}}}}\Bigr{]}t_{1}^{(h)}E_{1}.$
(5)

Here, $r^{(h)}_{1}$ represents the amplitude transmittance of BS1, while
$r_{1}^{(sc)},t_{1}^{(sc)}$ and $r_{2}^{(sc)},t_{2}^{(sc)}$ correspond to the
amplitude reflectance or the amplitude transmittance of the mirrors used in
the sub-cavities. For the subsequent calculations, we set $r_{2}^{(sc)}$ to 1,
and $t_{2}^{(sc)}$ to 0. Besides, ${\phi}^{\prime}$ is defined as the change
in laser phase when the laser light completes one round trip through the
cavities. Consequently, the field falling on the photodiodes
$E_{\mathrm{PD}}^{(A)},E_{\mathrm{PD}}^{(B)}$ by lights passing through each
optical path is given as follows:

$\begin{split}E_{PD}^{(A)}&=t_{2}^{(h)}E_{3}+r_{2}^{(h)}E_{2}\\\
&=\Bigl{[}t_{1}^{(h)}t_{2}^{(h)}\Bigl{\\{}-r_{1}^{(sc)}+{\frac{{t_{1}^{(sc)}}^{2}r_{2}^{(sc)}}{(1-r_{1}^{(sc)}r_{2}^{(sc)})^{2}+4r_{1}^{(sc)}r_{2}^{(sc)}{\sin^{2}{\frac{{\phi}^{\prime}}{2}}}}}(e^{i{\phi}^{\prime}}-r_{1}^{(sc)})\Bigr{\\}}+r_{1}^{(h)}r_{2}^{(h)}e^{i{\xi}}\Bigr{]}E_{1}\\\
&=\Biggl{[}\Bigl{\\{}t_{1}^{(h)}t_{2}^{(h)}\Bigl{(}-r_{1}^{(sc)}+{\frac{{t_{1}^{(sc)}}^{2}r_{2}^{(sc)}(\cos{{\phi}^{\prime}}-r_{1}^{(sc)})}{(1-r_{1}^{(sc)}r_{2}^{(sc)})^{2}+4r_{1}^{(sc)}r_{2}^{(sc)}{\sin^{2}{\frac{{\phi}^{\prime}}{2}}}}}\Bigr{)}+r_{1}^{(h)}r_{2}^{(h)}{\cos{\xi}}\Bigr{\\}}\\\
&\hskip
170.71652pt+i\Bigl{\\{}t_{1}^{(h)}t_{2}^{(h)}{\frac{{t_{1}^{(sc)}}^{2}r_{2}^{(sc)}{\sin{{\phi}^{\prime}}}}{(1-r_{1}^{(sc)}r_{2}^{(sc)})^{2}+4r_{1}^{(sc)}r_{2}^{(sc)}{\sin^{2}{\frac{{\phi}^{\prime}}{2}}}}}+r_{1}^{(h)}r_{2}^{(h)}{\sin{\xi}}\Bigr{\\}}\Biggl{]}E_{1}\\\
&\equiv(A_{1}+iA_{2})E_{1}\end{split}$ (6)
$\begin{split}E_{PD}^{(B)}&=-r_{2}^{(h)}E_{3}+t_{2}^{(h)}E_{2}\\\
&=\Bigl{[}-t_{1}^{(h)}r_{2}^{(h)}\Bigl{\\{}-r_{1}^{(sc)}+{\frac{{t_{1}^{(sc)}}^{2}r_{2}^{(sc)}}{(1-r_{1}^{(sc)}r_{2}^{(sc)})^{2}+4r_{1}^{(sc)}r_{2}^{(sc)}{\sin^{2}{\frac{{\phi}^{\prime}}{2}}}}}(e^{i{\phi}^{\prime}}-r_{1}^{(sc)})\Bigr{\\}}+r_{1}^{(h)}t_{2}^{(h)}e^{i{\xi}}\Bigr{]}E_{1}\\\
&=\Biggl{[}\Bigl{\\{}-t_{1}^{(h)}r_{2}^{(h)}\Bigl{(}-r_{1}^{(sc)}+{\frac{{t_{1}^{(sc)}}^{2}r_{2}^{(sc)}(\cos{\phi}^{\prime}-r_{1}^{(sc)})}{(1-r_{1}^{(sc)}r_{2}^{(sc)})^{2}+4r_{1}^{(sc)}r_{2}^{(sc)}{\sin^{2}{\frac{{\phi}^{\prime}}{2}}}}}\Bigr{)}+r_{1}^{(h)}t_{2}^{(h)}{\cos{\xi}}\Bigr{\\}}\\\
&\hskip
170.71652pt+i\Bigl{\\{}-t_{1}^{(h)}r_{2}^{(h)}{\frac{{t_{1}^{(sc)}}^{2}r_{2}^{(sc)}{\sin{{\phi}^{\prime}}}}{(1-r_{1}^{(sc)}r_{2}^{(sc)})^{2}+4r_{1}^{(sc)}r_{2}^{(sc)}{\sin^{2}{\frac{{\phi}^{\prime}}{2}}}}}+r_{1}^{(h)}t_{2}^{(h)}{\sin{\xi}}\Bigr{\\}}\Biggr{]}E_{1}\\\
&\equiv(B_{1}+iB_{2})E_{1}.\\\ \end{split}$ (7)

Hence, through the consideration of the electric field $E_{1}$ pertaining to
the laser emission at its immediate inception as the point of reference, each
instance of interference light is detected along an axis rotated at an angle
ascertained by the subsequent equation:

$\displaystyle{\eta}^{(A)}={\arctan{\frac{A_{1}}{A_{2}}}},\hskip
14.22636pt{\eta}^{(B)}={\arctan{\frac{B_{1}}{B_{2}}}}.$ (8)

Next, we consider the quantum state at this juncture. If the quantum state of
the laser light manifests as a coherent state, then the light extracted as a
local oscillator, influenced solely by the amplitude reflectance $r_{1}^{(h)}$
of BS1, also assumes a coherent state. On the contrary, the light emanating
from the cavity to the BS2 exhibits a squeezed state. This arises due to the
amplitude fluctuations of the laser light, which result in mirror
displacements and changes in the optical path length, thereby causing
amplitude fluctuations to manifest as phase fluctuations. This phenomenon is
referred to as ponderomotive squeezing [23]. Consequently, the interference
light resulting from the combination of these two beams is squeezed in its
quantum state. Thus, quantum noise can be ameliorated by judiciously selecting
an appropriate angle for the detection axis, as defined in Eq. 8. The optimal
angle is determined such that the projective component of the quantum
fluctuation along this axis nullifies each other.

Incidentally, it should be noted that this particular configuration embraces
vacuum fluctuations at BS1. Omitting consideration of vacuum fluctuations
would lead to a reduction in the quantum fluctuation of the light transmitted
or reflected by BS1. Hence, we introduce a symmetrical position with respect
to the beam splitter, as illustrated in the figure, wherein vacuum
fluctuations are injected. This serves to compensate for the reduction in
quantum fluctuation of the laser light.

## III. Signal Processing

### A. Block Diagram

We utilize a block diagram to portray the interferometer configuration shown
in Fig. 1, which integrates optical-spring quantum locking and accounts for
the amalgamation of vacuum fluctuations in homodyne detection, as depicted in
Fig. 3. This schematic comprises three distinct sections: the upper and lower
segments represent the sub-cavities, while the central segment denotes the
main cavity. The purple and yellow regions on the left side correspond to
quantum fluctuations of the laser light and the mixed vacuum fluctuations in
homodyne detection, respectively. Within these regions, the upper port
signifies amplitude quantum fluctuations, whereas the lower port pertains to
phase fluctuations. On the right side, the cyan region designates the
detection port.

The amplitude transmittance and amplitude reflectance of a mirror are defined
using the identifier $(k)$, such as (sc) representing the sub-cavity, and the
identifier $n$ representing 1 or 2, as follows

$\bigl{[}{r_{n}^{(k)}}\bigr{]}^{2}+\bigl{[}{t_{n}^{(k)}}\bigr{]}^{2}=1.$ (9)

Note that this equation assumes no loss effects for all mirrors. In addition,
all mirrors used in each cavity have the same mass. In the diagram, the
mirrors of the main cavity, which are shared with the sub-cavities, are
depicted as blocks located in the upper or lower parts of the sub-cavities. In
addition, $G_{ij}$ denotes the matrix of optical-spring effects, which is
determined by the detuning angle $\phi$ and sideband frequency $\Omega$, and
can be expressed as:

$\displaystyle g(\Omega)$
$\displaystyle=\Bigl{[}1-r_{1}r_{2}e^{-i({\phi}+{\frac{2L}{c}}{\Omega})}\Bigr{]}^{-1}$
(10) $\displaystyle G(\Omega)$
$\displaystyle={\frac{1}{2}}\begin{bmatrix}g^{\ast}(\Omega)+g(-{\Omega})&i[g^{\ast}(\Omega)-g(-{\Omega})]\\\
-i[g^{\ast}(\Omega)-g(-{\Omega})]&g^{\ast}(\Omega)+g(-{\Omega})\end{bmatrix},$
(11)

where $c$ is the speed of light taken as $3{\times}10^{8}{\mathrm{\,m/s}}$,
and $L$ is the cavity length. Next, ${\eta}^{\prime}_{A/B}$ represents the
angle between the phase direction of transmitted or reflected light by BS2 in
$E_{3}$ and the axis determined by Eq. 8, and it is defined as follows

${\eta}^{\prime}_{A/B}={\frac{\pi}{2}}+{\eta}^{(A/B)}-{\eta}_{0}^{(A/B)},$
(12)

where ${\eta}_{0}^{(A/B)}$ denotes the angular orientation of each carrier
light with respect to the phase direction of $E_{1}$. The influence of
gravitational waves is introduced into the system within the region enclosed
by the red line positioned at the diagram’s center. This region corresponds to
the displacement of the shared mirror induced by gravitational waves. It is
worth noting that the impact of gravitational waves on the sub-cavities is not
taken into consideration, given their relatively abbreviated cavity lengths,
resulting in negligible alterations in the optical path length caused by
gravitational waves. Hence, gravitational

Figure 3: Block Diagram of the detection system incorporating optical-spring
quantum locking and accounting for the effects of vacuum fluctuations. The
upper and lower sections are the sub-cavities, and the central section is the
main cavity. Purple areas represent quantum fluctuations of the laser light,
yellow areas represent vacuum fluctuations mixed in during homodyne detection,
and cyan areas represent detection ports.

waves are detected in the phase direction by the photodetector situated at the
central portion of the diagram, which captures the main laser light.

Likewise, by utilizing this block diagram, we can ascertain the amplification
or attenuation of individual quantum fluctuations and their subsequent
detection as noise. Table 1 showcases the transfer functions of this system,
elucidating the correlation between mirror displacement caused by
gravitational waves or each quantum fluctuation and the signals obtained by
the five photodetectors. Within this table, the gravitational wave signal is
denoted by $x$, while $p$ and $q$, as defined in the preceding subsection,
represent the bases of the quantum fluctuations. By employing these bases and
transfer functions, $V_{\mathrm{main}}$, the signal acquired from the
photodetector for the main laser light, can be expressed as follows:

$V_{\mathrm{main}}=a\mbox{\boldmath${x}$}+Aq_{m}^{(l)}+Bp_{m}^{(l)}+Cq_{s1}^{(l)}+Dp_{s1}^{(l)}+Cq_{s2}^{(l)}+Dp_{s2}^{(l)}+Eq_{1}^{(v)}+Fp_{1}^{(v)}+Eq_{2}^{(v)}+Fp_{2}^{(v)}.$
(13)

In the same way,
$V_{\mathrm{sub1}}^{(A)},V_{\mathrm{sub1}}^{(B)},V_{\mathrm{sub2}}^{(A)},V_{\mathrm{sub2}}^{(B)}$
which are the signals obtained from the other four photodetectors, are defined
by the combinations shown in the table. Here, note that these signals contain
some common noise components, as the sub-cavities on both sides are assumed to
have the same configuration.

Table 1: Transfer functions from the effects of gravitational waves or quantum fluctuations to each photodetector. To From |  ${x}$ (gw) | $q_{m}^{(l)}$ | $p_{m}^{(l)}$ | $q_{s1}^{(l)}$ | $p_{s1}^{(l)}$ | $q_{s2}^{(l)}$ | $p_{s2}^{(l)}$ | $q_{1}^{(v)}$ | $p_{1}^{(v)}$ | $q_{2}^{(v)}$ | $p_{2}^{(v)}$
---|---|---|---|---|---|---|---|---|---|---|---
| Main | $a$ | $A$ | $B$ | $C$ | $D$ | $C$ | $D$ | $E$ | $F$ | $E$ | $F$
Photo | Sub1-A | $-$ | $G$ | $H$ | $I$ | $J$ | $-$ | $-$ | $K$ | $L$ | $-$ | $-$
Detector | Sub2-A | $-$ | $G$ | $H$ | $-$ | $-$ | $I$ | $J$ | $-$ | $-$ | $K$ | $L$
Sub1-B | $-$ | $M$ | $N$ | $O$ | $P$ | $-$ | $-$ | $Q$ | $R$ | $-$ | $-$
| Sub2-B | $-$ | $M$ | $N$ | $-$ | $-$ | $O$ | $P$ | $-$ | $-$ | $Q$ | $R$

### B. Completing the Square

Completing the square in quantum locking is a signal optimization method that
has shown high effectiveness in previous research for reducing quantum noise
[18]. In this approach, a new signal $V=V_{1}+{\chi}V_{2}$ is defined from the
two signals $V_{1}$ and $V_{2}$, with the combination coefficient $\chi$
chosen to minimize the power spectrum. To effectively utilize the signals from
homodyne detection at two ports, we introduce additional combination
coefficients and devise a method to optimize the combination of three signals.
Thus, by using $V_{1}$ ,$V_{2}^{(A)}$ and $V_{2}^{(B)}$ along with the
combination coefficients ${\chi}_{A}$ and ${\chi}_{B}$, a new signal $V$ is
defined as follows:

$V=V_{1}+{\chi}_{A}V_{2}^{(A)}+{\chi}_{B}V_{2}^{(B)}.$ (14)

The power spectrum $S_{V}$ of this signal given by:

$\begin{split}S_{V}&=\left|V_{1}\right|^{2}+\left|V_{2}^{(A)}\right|^{2}\left|{\chi}_{A}+{\frac{V_{2}^{{(A){\dagger}}}V_{1}+{\chi}_{B}V_{2}^{(A){\dagger}}V_{2}^{(B)}}{V_{2}^{(A){\dagger}}V_{2}^{(A)}}}\right|^{2}\\\
&\hskip
14.22636pt+\Biggl{(}\left|V_{2}^{(B)}\right|^{2}-{\frac{\left|V_{2}^{(A)}V_{2}^{(B)}\right|^{2}}{\left|V_{2}^{(A)}\right|^{2}}}\Biggr{)}\left|{\chi}_{B}+{\frac{\left|V_{2}^{(A)}\right|^{2}\Bigl{(}V_{2}^{(B){\dagger}}V_{1}\Bigr{)}-\Bigl{(}V_{2}^{(A)}V_{2}^{(B){\dagger}}\Bigr{)}\Bigl{(}V_{1}V_{2}^{(A){\dagger}}\Bigr{)}}{\left|V_{2}^{(A)}\right|^{2}\left|V_{2}^{(B)}\right|^{2}-\left|V_{2}^{(A){\dagger}}V_{2}^{(B)}\right|^{2}}}\right|^{2}\\\
&\hskip
14.22636pt-{\frac{\left|V_{1}^{{\dagger}}V_{2}^{(A)}\right|^{2}\left|V_{2}^{(B)}\right|^{2}+\left|V_{1}^{{\dagger}}V_{2}^{(B)}\right|^{2}\left|V_{2}^{(A)}\right|^{2}-\Bigl{(}V_{2}^{(B){\dagger}}V_{1}\Bigr{)}\Bigl{(}V_{2}^{(A){\dagger}}V_{2}^{(B)}\Bigr{)}\Bigl{(}V_{1}^{{\dagger}}V_{2}^{(A)}\Bigr{)}-\Bigl{(}V_{2}^{(B)}V_{1}^{{\dagger}}\Bigr{)}\Bigl{(}V_{2}^{(A)}V_{2}^{(B){\dagger}}\Bigr{)}\Bigl{(}V_{1}V_{2}^{(A){\dagger}}\Bigr{)}}{\left|V_{2}^{(A)}\right|^{2}\left|V_{2}^{(B)}\right|^{2}-\left|V_{2}^{(A){\dagger}}V_{2}^{(B)}\right|^{2}}}.\end{split}$
(15)

Therefore, the values of ${\chi}_{A}^{\mathrm{(opt)}}$ and
${\chi}_{B}^{\mathrm{(opt)}}$ that minimize its power spectrum are determined
as follows:

$\displaystyle{\chi}_{A}^{\mathrm{(opt)}}$
$\displaystyle=-{\frac{\left|V_{2}^{(B)}\right|^{2}\Bigl{(}V_{2}^{(A){\dagger}}V_{1}\Bigr{)}-\Bigl{(}V_{2}^{(B)}V_{2}^{(A){\dagger}}\Bigr{)}\Bigl{(}V_{1}V_{2}^{(B){\dagger}}\Bigr{)}}{\left|V_{2}^{(A)}\right|^{2}\left|V_{2}^{(B)}\right|^{2}-\left|V_{2}^{(A){\dagger}}V_{2}^{(B)}\right|^{2}}}$
(16) $\displaystyle{\chi}_{B}^{\mathrm{(opt)}}$
$\displaystyle=-{\frac{\left|V_{2}^{(A)}\right|^{2}\Bigl{(}V_{2}^{(B){\dagger}}V_{1}\Bigr{)}-\Bigl{(}V_{2}^{(A)}V_{2}^{(B){\dagger}}\Bigr{)}\Bigl{(}V_{1}V_{2}^{(A){\dagger}}\Bigr{)}}{\left|V_{2}^{(A)}\right|^{2}\left|V_{2}^{(B)}\right|^{2}-\left|V_{2}^{(A){\dagger}}V_{2}^{(B)}\right|^{2}}}.$
(17)

Finally, we utilize the three signals as follows:

$V_{1}=V_{\mathrm{main}}/a\hskip 2.84526pt,\hskip
14.22636ptV_{2}^{(A)}=\Bigl{[}V_{\mathrm{sub1}}^{(A)}+V_{\mathrm{sub2}}^{(A)}\Bigr{]}/a\hskip
2.84526pt,\hskip
14.22636ptV_{2}^{(B)}=\Bigl{[}V_{\mathrm{sub1}}^{(B)}+V_{\mathrm{sub2}}^{(B)}\Bigr{]}/a,$
(18)

where $a$ is a transfer function from the displacement of the shared mirror
caused by gravitational waves to the main photodetector as shown in Tab. 1.
Each signal is divided by $a$ to calibrate it with the gravitational wave
signal.

## IV. SIMULATIONS

In this section, we delineate the parameter conditions employed in the
simulation, as well as the method utilized for evaluating the sensitivity of
DECIGO. Table 3 exhibits the symbols and ranges/values employed in the
simulation. The upper segment of this table presents the five variable
parameters employed for optimizing sensitivity: $t_{1}^{(h)}$ and
$t_{2}^{(h)}$ represent the amplitude transmittance of each beam splitter.
Detuning angle reflects the effect of the optical spring, and finesse
corresponds to the effective number of light reflections within the cavity.
Additionally, $\xi$ is a parameter introduced to modulate the length of the
optical path taken by the local oscillator. Since the two photodetectors are
symmetrically positioned, the A and B signals can be interchanged by swapping
the amplitude transmittance and reflectance values of BS2 and adjusting the
parameter $\xi$ to become $\xi+\pi$. Note that the range of $\xi$ is therefore
defined as between 0 and $\pi$. The eight parameters presented in the lower
part of the table remain constant, with the exception of the cavity length,
which differs between the main cavity and the sub-cavities.

Table 2: Conditions of main parameters for the optimization of the DECIGO’s
sensitivity.

Meaning | Symbol | Range/Value
---|---|---
Amplitude Transmittance | $t_{1}^{(h)}$ |  $0$ to $1$
$t_{2}^{(h)}$ |  $0$ to $1$
Detuning Angle | ${\phi}_{\mathrm{sub}}$ |  $-{\pi}$ to $\pi$ rad
Finesse | ${\mathcal{F}}_{\mathrm{sub}}$ |  $1$ to $100$
Phase Shift | ${\xi}$ |  $0$ to $\pi$ rad
Cavity Length | $L_{\mathrm{sub}}$ |  $1$ m
$L_{\mathrm{main}}$ |  $1000$ km
Laser Power | $I_{\mathrm{sub}}$ |  $100$ W
$I_{\mathrm{main}}$ |  $100$ W
Laser Wavelength | ${\lambda}_{\mathrm{sub}}$ |  $515$ nm
${\lambda}_{\mathrm{main}}$ |  $515$ nm
Mirror Mass | $M_{\mathrm{sub}}$ |  $100$ kg
$M_{\mathrm{main}}$ |  $100$ kg (shared)

Table 3: Parameters used to calculate the SNR [21].

Meaning | Symbol | Value
---|---|---
Hubble Parameter | $H_{0}$ | $70{\mathrm{\,km{\cdot}sec^{-1}{\cdot}Mpc^{-1}}}$
Time for Correlation | T | 3 years
Frequency | $f$ | 0.1 to 1 Hz
Correlation Function | ${\gamma}$ | 1
Energy Density | ${\Omega}_{\mathrm{GW}}$ | $10^{-16}$
Noise Power Spectral Densities | $P_{1},P_{2}$ |

Next, we adopt Signal-to-Noise Ratio (SNR) as a measure to evaluate the
sensitivity of DECIGO to primordial gravitational waves. The SNR is given by
[24]:

$\mathrm{SNR}={\frac{3H_{0}^{2}}{10{\pi}^{2}}}{\sqrt{T}}\Bigl{[}{\int_{0.1}^{1}}df{\frac{2{\gamma}(f)^{2}{\Omega}^{2}_{\mathrm{GW}}(f)}{f^{6}P_{1}(f)P_{2}(f)}}\Bigr{]}^{\frac{1}{2}},$
(19)

where $P_{1}$ and $P_{2}$ represent spectral densities of noise, as computed
following the methodology outlined in Section III. Notably, in this case,
$P_{1}$ and $P_{2}$ assume equal values. The remaining parameters employed in
the SNR computation are presented in Table 3. In Eq. 19, $T$ signifies the
observation period, which has been set to a duration of 3 years.
${\Omega}_{\mathrm{GW}}$ denotes the energy density of primordial
gravitational waves, and for the purpose of this research, a fixed value of
$10^{-16}$ has been adopted. Furthermore, $\gamma$ denotes the overlap
reduction function [24], assumed to be unity in the configuration of DECIGO.
Additionally, the SNR assessment specifically concentrates on the frequency
band spanning from $0.1$ Hz to $1$ Hz, wherein DECIGO is expected to exhibit
high sensitivity to gravitational waves.

## V. RESULT AND DISCUSSION

Figure 4: Optimized sensitivity curves with contributions from noise sources.
The total noise is represented by the thick blue line, while each quantum
noise caused by fluctuations of the laser light or vacuum fluctuations is
shown as thin lines. Especially, quantum noises originating from vacuum
fluctuations are depicted as dashed lines. Additionally, the green or yellow
lines represent signals obtained from only sub-cavity 1. Therefore, note that
when adding the signal from sub-cavity 2 to this figure, the curve will be
scaled up by a factor of $\sqrt{2}$.

Table 4: Optimized parameters. Symbol | Value
---|---
$t_{1}^{(h)}$ | $0.152$
$t_{2}^{(h)}$ | $0.308$
${\phi}_{\mathrm{sub}}$ |  $5.90{\times}10^{-3}$ rad
${\mathcal{F}}_{\mathrm{sub}}$ | $55.3$
${\xi}$ |  $0.260$ rad
SNR | $79.6$

Based on the simulation results, we obtained the optimized sensitivity curves
of DECIGO with optical-spring quantum locking to detect primordial
gravitational waves as shown in Fig. 4. Each sensitivity curve, excluding the
shot noise caused by phase quantum fluctuations of the main laser light,
exhibits a dip around 0.1 Hz, and these dips align with each other. On the
other hand, the shot noise, which has no dip, remains nearly constant within
the target frequency band and contributes to the baseline of DECIGO’s
sensitivity. Furthermore, Table 4 shows the parameters that minimize noise in
relation to the gravitational wave signal, resulting in a calculated SNR of
79.6. The more refined model presented in this paper gives a different result
than less-detailed modeling (SNR=141), as shown in Fig. 5. 111Here, note that
this differs from the results (SNR=214) described in the previous paper [21].
The previous paper used approximations; a constant amplitude of the light is
assumed even in the case of off-resonant condition. In this paper, we
calculated the SNR without any approximations, and obtained SNR of 141.  In
this paper, the local oscillator is extracted from the incoming light directed
towards the cavity. As a result, the laser light going to the sub-cavities are
smaller, and the noises are relatively higher due to the effects entering from
the vacuum fluctuation. In addition, the local light must have a large value
in order to obtain an arbitrary homodyne angle. In the previous research, we
considered a hypothetical situation in which the local laser power is
infinite, independent of the incoming light directed towards the cavity.
However, since it is not possible to do so in reality, we obtained degraded
but realistic sensitivity. In contrast, the SNR without optical-spring quantum
locking is 1.74, indicating that optical-spring quantum locking significantly
helps reduce quantum noise even when the effect of vacuum fluctuations is
considered.

Figure 5: Optimized sensitivity curves considering noise contributions. The
blue line represents the sensitivity curve optimized in this study, which
corresponds to the total noise in Fig. 4. The dashed line represents the
sensitivity curve optimized in the previous research [21], resulting in an SNR
of 141. The red line represents the optimized sensitivity curve when using a
single cavity with a length of 1000 km.

## VI. CONCLUSIONS

To assess the efficacy of homodyne detection in sub-cavities within the
optical-spring quantum locking for mitigating noise in DECIGO, it was
imperative to consider a detector configuration that accounts for the vacuum
fluctuations. In this paper, we investigated the determination of the homodyne
angle by employing extracted light serving as a local oscillator, and
subsequently constructed a comprehensive block diagram incorporating vacuum
fluctuations. Moreover, this particular configuration enables the utilization
of two photodetectors, which sets it apart from previous DECIGO configurations
that allowed only the use of a single photodetector. We presented an
optimization method for processing the signals acquired from each
photodetector to obtain the optimized sensitivity curve. By optimizing the
parameters with these configurations, the impact of vacuum fluctuations was
evaluated. Although the effect of vacuum fluctuation results in a sensitivity
slightly inferior to that of the ideal situation shown in the previous
research, it still exhibits a marked improvement compared to the scenario
without optical-spring quantum locking. Consequently, we demonstrate that
homodyne detection in optical-spring quantum locking is an effective technique
for noise reduction in DECIGO, with the potential to significantly contribute
to the observation of primordial gravitational waves.

Acknowledgements
We would like to thank David H. Shoemaker for commenting on a draft. This work
was supported by JSPS KAKENHI, Grants No. JP19H01924 and No. JP22H01247. This
work was also supported by Murata Science Foundation.

## References

  * [1] LIGO Scientific Collaboration and Virgo Collaboration (Abbott, B. P.; Abbott, R.; Abbott, T. D.; Abernathy, M. R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R. X.; et al.) _Observation of Gravitational Waves from a Binary Black Hole Merger_ _Phys. Rev. Lett._ 2016, _116_ , 061102.
  * [2] LIGO Scientific Collaboration and Virgo Collaboration (Abbott, R.; Abbott, T. D.; Abraham, S.; Acernese, F.; Ackley, K.; Adams, A.; Adams, C.; Adhikari, R. X.; Adya, V. B.; Affeldt, C.; et al.) _GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo during the First Half of the Third Observing Run_ _Phys. Rev. X_ 2021, _11_ , 021053.
  * [3] Punturo, M.; Abernathy, M.; Acernese, F.; Allen, B.; Andersson, N.; Arun, K.; Barone, F.; Barr, B.; Barsuglia, M.; Beker, M.; et al. _The third generation of gravitational wave observatories and their science reach_ _Class. Quant. Grav._ 2010, _27_ , 084007.
  * [4] Dwyer, S.; Sigg, D.; Ballmer, S. W.; Barsotti, L.; Mavalvala, N.; Evans, M.; _Gravitational wave detector with cosmological reach_ _Phys. Rev. D_ 2015, _91_ , 082001.
  * [5] Amaro-Seoane, P.; Audley, H.; Babak, S.; Baker, J.; Barausse, E.; Bender, P.; Berti, E.; Binetruy, P.; Born, M.; Bortoluzzi, D.;et al. _Laser Interferometer Space Antenna_ _arXiv_ 2017, 1702.00786.
  * [6] Kuroyanagi, S.; Tsujikawa, S.; Chiba, T.; Sugiyama, N. _Implications of the $B$-mode polarization measurement for direct detection of inflationary gravitational waves_ _Phys. Rev. D_ 2014, _90_ , 063513.
  * [7] LIGO Scientific Collaboration (Aasi, J.; Abbott, B. P.; Abbott, R.; Abbott, T; Abernathy, M. R.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R. X. et al.) _Advanced LIGO_ _Class. Quant. Grav._ 2015, _32_ , 074001.
  * [8] Acernese, F.; Agathos, M; Agatsuma, K.; Aisa, D.; Allemandou, N.; Allocca, A.; Amarni, J.; Astone, P.; Balestri, G.; Ballardin, G.; et al. _Advanced Virgo: a second-generation interferometric gravitational wave detector_ _Class. Quant. Grav._ 2014, _32_ , 024001.
  * [9] Somiya, K.; _Detector configuration of KAGRA–the Japanese cryogenic gravitational-wave detector_ _Class. Quant. Grav._ 2012, _29_ , 0124007.
  * [10] Seto, N.; Kawamura, S.; Nakamura, T. _Possibility of Direct Measurement of the Acceleration of the Universe Using 0.1 Hz Band Laser Interferometer Gravitational Wave Antenna in Space_ _Phys. Rev. Lett._ 2001, _87_ , 221103.
  * [11] Kawamura, S.; Ando, M.; Seto, N.; Sato, S.; Musha, M.; Kawano, I.; Yokoyama, J.; Tanaka, T.; Ioka, K.; Akutsu, T.; et al. _Current status of space gravitational wave antenna DECIGO and B-DECIGO_ _Progress of Theoretical and Experimental Physics_ 2021, _2021_ ; 05A105.
  * [12] Akrami Cheghasiahi, Y.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A. J.; Barreiro, R.; Bartolo, N.; Basak, S.; et al. _Planck 2018 results: X. Constraints on inflation_ _Astronomy and Astrophysics (A & A)_ 2020, _641_.
  * [13] Iwaguchi, S.; Ishikawa, T.; Ando, M.; Michimura, Y.; Komori, K.; Nagano, K.; Akutsu, T.; Musha, M.; Yamada, R.; Watanabe, I.; et al. _Quantum Noise in a Fabry-Perot Interferometer Including the Influence of Diffraction Loss of Light_ _Galaxies_ 2021, _9_.
  * [14] Ishikawa, T.; Iwaguchi, S.; Michimura, Y.; Ando, M.; Yamada, R.; Watanabe, I.; Nagano, K.; Akutsu, T.; Komori, K.; Musha, M.; et al. _Improvement of the Target Sensitivity in DECIGO by Optimizing Its Parameters for Quantum Noise Including the Effect of Diffraction Loss_ _Galaxies_ 2021, _9_.
  * [15] Kawasaki, Y.; Shimizu, R.; Ishikawa, T.; Nagano, K.; Iwaguchi, S.; Watanabe, I.; Wu, B.; Yokoyama, S.; Kawamura, S. _Optimization of Design Parameters for Gravitational Wave Detector DECIGO Including Fundamental Noises_ _Galaxies_ 2022, _10_.
  * [16] Courty, J.-M.; Heidmann, A.; Pinard, M.; _Quantum Locking of Mirrors in Interferometers_ _Phys. Rev. Lett._ 2003, _90_ , 083601.
  * [17] Heidmann, A.; Courty, J.-M.; Pinard, M.; Lebars, J.; _Beating quantum limits in interferometers with quantum locking of mirrors_ _Phys. Rev. Lett._ 2004, _6_ , 5684.
  * [18] Yamada, R.; Enomoto, Y.; Nishizawa, A.; Nagano, K.; Kuroyanagi, S.; Kokeyama, K.; Komori, K.; Michimura, Y.; Naito, T.; Watanabe, I.; et al. _Optimization of quantum noise by completing the square of multiple interferometer outputs in quantum locking for gravitational wave detectors_ _Physics Letters A_ 2020, _384_ , 126626.
  * [19] Ishikawa, T.; Kawasaki, Y.; Tsuji, K.; Yamada, R.; Watanabe, I.; Wu, B.; Iwaguchi, S.; Shimizu, R.; Umemura, K.; Nagano, K.; Enomoto, Y.; Komori, K.; Michimura, Y.; Furusawa, A.; Kawamura, S. _First-step experiment for sensitivity improvement of DECIGO: Sensitivity optimization for simulated quantum noise by completing the square_ _Phys. Rev. D_ 2023, _107_ , 022007.
  * [20] Braginskiǐ, V. B.; Manukin, A. B.; Tikhonov, M. Yu.; _Investigation of dissipative ponderomotive effects of electromagnetic radiation_ _Soviet Journal of Experimental and Theoretical Physics_ 1970, _31_ , 829.
  * [21] Yamada, R.; Enomoto, Y.; Watanabe, I.; Nagano, K.; Michimura, Y.; Nishizawa, A.; Komori, K.; Naito, T.; Morimoto, T.; Iwaguchi, S.; et al. _Reduction of quantum noise using the quantum locking with an optical spring for gravitational wave detectors_ _Physics Letters A_ 2021, _402_ , 127365.
  * [22] Schleich, W. P. _Quantum optics in phase space_ ; John Wiley & Sons, 2011; _9783527602971_.
  * [23] Caves, C. M.; _Quantum-mechanical noise in an interferometer_ _Phys. Rev. D_ 1981, _23_ , 1693.
  * [24] Allen, B.; Romano, J. D. _Detecting a stochastic background of gravitational radiation: Signal processing strategies and sensitivities_ _Phys. Rev. D_ 1999, _59_ , 102001.