Doctoral Dissertation
博士論文

Frequency-Dependent Squeezed Vacuum Sourcewith Filter Cavity Control using Coherent Control Sidebands
for Gravitational-Wave Detectors(重力波望遠鏡のためのコヒーレントコントロール光によるフィルター共振器制御を用いた周波数依存スクイーズ
ド真空場)

A Dissertation Submitted for the Degree of Doctor ofPhilosophy
December 2020
令和２年１２月博士 (理学) 申請

Department of Physics, Graduate School of Science,
The University of Tokyo東京大学大学院理学系研究科物理学専攻
Naoki Aritomi
有冨尚紀

1

Abstract
The sensitivity of gravitational-wave (GW) detectors is designed to be limited by quantum
noise. The quantum noise originates from the quantum vacuum ﬁeld entering from an output port of the interferometer. The quantum noise is composed of shot noise and radiationpressure noise. The shot noise originates from the phase ﬂuctuation of the vacuum ﬁeld
and dominates at high frequencies of the detector sensitivity, while the radiation pressure
noise originates from the amplitude ﬂuctuation of the vacuum ﬁeld and dominates at low
frequencies. The sensitivity of current GW detectors is dominated by shot noise at highfrequencies. To reduce the shot noise, the frequency-independent squeezing, whose phasequadrature is squeezed, has been injected into the interferometer and the shot noise reduction by ∼3 dB has been achieved in Advanced LIGO and Advanced Virgo. However,
the increase of the radiation pressure noise due to the frequency-independent squeezing
has been observed in Advanced LIGO and Advanced Virgo.
To reduce the shot noise and radiation pressure noise simultaneously, the frequencydependent squeezing, which is phase squeezed at high frequencies and amplitude squeezed
at low frequencies, is required. The most promising way to realize the frequency-dependent
squeezing is reﬂecting a frequency-independent squeezed vacuum ﬁeld oﬀ an optical cavity
called ﬁlter cavity. In the past, the squeezing angle rotation using rigid, meter-scale ﬁlter
cavities has been realized in the MHz and kHz regions. However, frequency-dependent
squeezing with a rotation frequency below 100 Hz, which is required in GW detectors, has
not been demonstrated.
In this thesis, we developed the ﬁrst frequency-dependent squeezed vacuum sourcewith a rotation frequency below 100 Hz by using a 300 m ﬁlter cavity which can ﬁt for
Advanced LIGO and Advanced Virgo. It is shown that our frequency-dependent squeezed
vacuum source can realize the broadband quantum noise reduction in GW detectors.
However, in this measurement, the ﬁlter cavity length noise and the mode mismatch
between the squeezer and the ﬁlter cavity were not satisfying the target value, and thedetuning ﬂuctuation of ∼30 Hz was observed. For further quantum noise reduction, it isnecessary to stabilize the ﬁlter cavity length ﬂuctuation, the alignment ﬂuctuation, andthe detuning ﬂuctuation. In this measurement, the ﬁlter cavity was controlled with theauxiliary green ﬁeld whose frequency is as twice as the squeezing frequency. As the optical
path and the frequency of the green ﬁeld and the squeezed ﬁeld are diﬀerent, the ﬁlter
cavity control with the auxiliary green ﬁeld does not ensure the ﬁlter cavity length andalignment for the squeezed ﬁeld.

2

To solve this problem, I suggested a new control scheme of the ﬁlter cavity using coherent control sidebands, which are already used to control the squeezing angle. As thecoherent control sidebands have the same optical path and almost the same frequency
as the squeezed ﬁeld, the control of the ﬁlter cavity with the coherent control sidebands
ensures the ﬁlter cavity length and alignment for the squeezed ﬁeld. I succeeded in controlling the length of the 300 m ﬁlter cavity with the new control scheme and demonstrated
that the new control scheme reduced the ﬁlter cavity length ﬂuctuation from 3.4 pm to0.75 pm, which satisﬁes the target value of 1 pm. Using this new control scheme, thefrequency-dependent squeezing with a rotation frequency approximately 100 Hz also has
been realized.
The results presented in this thesis are signiﬁcant for improving the sensitivity of thecurrent GW detectors such as Advanced LIGO, Advanced Virgo, and KAGRA as wellas the future third-generation GW detectors such as the Einstein Telescope and Cosmic
Explorer, which require even longer ﬁlter cavities.

Thesis supervisor: Masaki Ando (Associate Professor)
Thesis title: Frequency-Dedenpent Squeezed Vacuum Source with Filter Cavity Control
using Coherent Control Sidebands for Gravitational-Wave Detectors

3

4

Contents
Abstract2

Glossary

9

1 Introduction

12

2 Gravitational Wave
2.1 Derivation of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Linearization of Einstein equation . . . . . . . . . . . . . . . . . . .
2.1.2 Plane wave solution of gravitational waves . . . . . . . . . . . . . . .
2.2 Generation of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Astrophysical sources of gravitational waves . . . . . . . . . . . . . . . . . .
2.4 Observations of gravitational waves . . . . . . . . . . . . . . . . . . . . . . .
2.5 Science from gravitational wave observations . . . . . . . . . . . . . . . . .
2.6 Observation scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15
15
15
16
18
18
20
22
23

3 Gravitational-Wave Detector
3.1 Michelson interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Response to gravitational waves . . . . . . . . . . . . . . . . . . . .
3.1.2 DC readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Fabry-Perot cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Response to gravitational waves . . . . . . . . . . . . . . . . . . . .
3.2.2 Pound-Drever-Hall technique . . . . . . . . . . . . . . . . . . . . . .
3.3 Conﬁguration of GW detectors . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Noise sources in GW detectors . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Sensitivity of GW detectors . . . . . . . . . . . . . . . . . . . . . . . . . . .

25
25
26
27
28
31
32
34
36
36

4 Theory of Quantum States of Light
4.1 Quantization of the electromagnetic ﬁeld . . . . . . . . . . . . . . . . . . . .
4.2 Heisenberg uncertainty principle . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Quantum states of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Fock states and vacuum states . . . . . . . . . . . . . . . . . . . . .
4.3.2 Coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Squeezed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39
39
40
41
41
42
42

5

4.4

Second order nonlinear process . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Phase matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generation of squeezed vacuum states . . . . . . . . . . . . . . . . . . . . .
4.5.1 Equation of motion of OPO with losses . . . . . . . . . . . . . . . .
Balanced homodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . .
Coherent control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Squeezing degradation sources . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.1 Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.2 Classical noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.3 Phase noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.4 Combination of squeezing degradation sources . . . . . . . . . . . . .

43
46
47
47
52
53
56
56
57
58
58

5 Quantum Noise in Gravitational-Wave Detector
5.1 Quantum noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Frequency-independent squeezing . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Frequency-dependent squeezing . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Filter cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Analytical model of squeezing degradation sources . . . . . . . . . . . . . .
5.6 Squeezing degradation budget in 300 m ﬁlter cavity . . . . . . . . . . . . .
5.7 Sensitivity improvement of KAGRA . . . . . . . . . . . . . . . . . . . . . .

61
62
64
66
66
70
73
74

6 Frequency-Independent Squeezing
6.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Laser sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Direct digital synthesizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Phase-locked loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Second harmonic generator . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1 Finesse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.2 Length control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.3 Temperature control . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Green and infrared mode cleaners . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Mach-Zehnder interferometer . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8 Optical parametric oscillator . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8.1 Escape eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8.2 Finesse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8.3 Length control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8.4 Temperature control . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8.5 Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9 Alignment mode cleaner . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Homodyne detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.11 Coherent control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.12 Squeezing spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76
76
77
78
79
80
80
81
82
83
86
88
89
89
89
90
91
92
93
95
98

4.5
4.6
4.7
4.8

6

6.13 Loss and phase noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7 Frequency-Dependent Squeezing with 300 m Filter Cavity
100
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.3 300 m ﬁlter cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.3.1 Green lock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.3.2 Locking accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.3.3 Finesse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.3.4 Round-trip losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.4 Loss and phase noise outside the ﬁlter cavity . . . . . . . . . . . . . . . . . 108
7.5 Frequency-dependent squeezing measurement . . . . . . . . . . . . . . . . . 109
7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8 Control of a Filter Cavity with Coherent Control Sidebands
115
8.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
8.1.1 Filter cavity length signal . . . . . . . . . . . . . . . . . . . . . . . . 116
8.1.2 Filter cavity alignment signal . . . . . . . . . . . . . . . . . . . . . . 119
8.1.3 Experimental setup in GW detectors . . . . . . . . . . . . . . . . . . 123
8.1.4 Coherent control error signal . . . . . . . . . . . . . . . . . . . . . . 124
8.1.5 Reshaping of frequency-dependent phase noise . . . . . . . . . . . . 125
8.2 Noise Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.2.1 Shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.2.2 Backscattering noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.2.3 PLL noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.3 Comparison to the other locking scheme . . . . . . . . . . . . . . . . . . . . 131
8.4 Demonstration of CCFC lock with 300 m ﬁlter cavity . . . . . . . . . . . . 132
8.4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8.4.2 Frequency tuning of CC PLL . . . . . . . . . . . . . . . . . . . . . . 135
8.4.3 CCFC error signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.4.4 Open-loop transfer function of CCFC lock . . . . . . . . . . . . . . . 138
8.4.5 Locking accuracy of CCFC lock . . . . . . . . . . . . . . . . . . . . . 139
8.4.6 Frequency-dependent squeezing with CCFC lock . . . . . . . . . . . 139
8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9 Conclusion

142

A Auto alignment
145
A.1 Wave front sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
A.3 QPD centering loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7

A.4 WFS loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
B List of devices

153

Acknowledgements

155

Bibliography

157

8Glossary
AMC
AOM
AR
AS
BBH
BNS
BS
CBC
CC
CCSB
DDS
DECIGODRFPMI
EOM
ETM
FC
FDS
FIS
FSR
FP
GRMC
GW
HR
IR
IRMC
ITM
LIGO
LO
MI
MZ
NSBH
OPO
PBS
PD
PDH
PLL
PPKTP
PRM

Alignment Mode Cleaner
Acousto-Optic Modulator
Anti-Reﬂection
Anti-Symmetric
Binary Black Hole
Binary Neutron Star
Beam Splitter
Compact Binary Coalescence
Coherent Control
Coherent Control SideBands
Direct Digital Synthesizer
DECi-hertz Interferometer Gravitational-wave Observatory
Dual Recycled Fabry-Perot Michelson Interferometer
Electro-Optic Modulator
End Test Mass
Filter Cavity
Frequency-Dependent Squeezing
Frequency-Independent Squeezing
Free Spectral Range
Fabry-Perot
GReen Mode Cleaner
Gravitational Wave
High Reﬂection
InfraRedInfraRed Mode Cleaner
Input Test Mass
Laser Interferometer Gravitational-wave Observatory
Local Oscillator
Michelson Interferometer
Mach-Zehnder interferometer
Neutron Star Black Hole
Optical Parametric Oscillator
Polarizing Beam Splitter
Photo Detector
Pound-Drever-Hall
Phase-Locked Loop
Periodically Poled Potassium Titanyl Phosphate
Power Recycling Mirror

9

RSE
SHG
SQL
SRM

Resonant Sideband Extraction
Second Harmonic Generation
Standard Quantum Limit
Signal Recycling Mirror

10

11

Chapter 1

Introduction
In 2015, gravitational waves (GWs) from a binary black hole merger were observed by
Advanced LIGO for the ﬁrst time [1]. Thereafter, GW astronomy rapidly became amature research ﬁeld. After the observation of GWs from a binary neutron star merger [2]and the publication of the ﬁrst GW sources catalog [3], the network of the GW detectors(composed of the two Advanced LIGO [4] and Advanced Virgo [5]) performed a thirdobserving run (O3) and detected several candidates per month. The second GW sources
catalog, which contains detections during the ﬁrst half of O3, was recently published [6].
The Japanese detector KAGRA [7] is expected to join the network from the next observing
run (O4) and a ﬁfth detector, LIGO India, is expected to join in 2025 [8]. However, thesensitivity of the current GW detectors is not enough to determine the characteristics ofthe compact objects such as black holes and neutron stars, and to reveal the formation
scenario of the compact binary systems, and to test the general relativity in the stronger
gravity ﬁeld. For the further GW detections and the precise measurement of the GW
waveforms, it is necessary to improve the sensitivity of the current GW detectors.
The principle of the current GW detectors is a Michelson interferometer. As the
motion of the test mass mirrors induced by GWs can be measured by the laser light in theinterferometer, the fundamental sensitivity of the interferometric GW detectors is limitedby the quantum ﬂuctuation of the laser light in the classical picture. Therefore, it isimportant to reduce the quantum noise to improve the fundamental sensitivity of GW
detectors. In the current GW detectors, the quantum noise limits the sensitivity at highfrequencies, while other classical noises limit the sensitivity at low frequencies. In the
future, the other classical noises are expected to be reduced and the quantum noise will
limit the sensitivity at all frequencies.
The quantum noise originates from the quantum ﬂuctuation of light and it is composedof quantum shot noise and radiation pressure noise. The shot noise dominates at highfrequencies of the detector sensitivity, while the radiation pressure noise dominates at low
frequencies. In 1981, Caves reported that quantum noise originates from quantum vacuum
ﬂuctuations entering from an output port of the interferometer [9]. He also suggested thatquantum noise can be reduced by replacing the ordinary vacuum states with so-called
squeezed vacuum states, whose ﬂuctuations are reduced in one quadrature, while they12are increased in the orthogonal quadrature, following Heisenberg’s uncertainty principle.
When the noise reduced quadrature of the squeezed vacuum states is aligned with the GW
signal, the signal to noise ratio of the GW signal is increased and therefore the sensitivityof GW detectors is improved.
In order to enhance the sensitivity of the current GW detectors at high frequencies
where the shot noise mainly dominates, so-called frequency-independent squeezed vacuum sources have been implemented into the GW detectors such as Advanced LIGO andAdvanced Virgo. The squeezed vacuum states of these frequency-independent squeezed
vacuum sources are squeezed in the phase quadrature. Consequently, both Advanced
LIGO and Advanced Virgo successfully reduced the shot noise and their detection rates
were increased by ∼50 % in O3 [10, 11]. However, the frequency-independent squeezed
vacuum currently used in GW detectors cannot reduce the shot noise and radiation pressure noise simultaneously. The reason is the optomechanical coupling between the laser
light and the suspended test masses, which induces a rotation of the squeezing ellipse as afunction of frequency. Therefore, the quadrature whose noise is reduced cannot be alignedwith the GW signal at all frequencies. In fact, the increase in radiation pressure noisedue to the frequency-independent squeezing has recently been observed in both Advanced
LIGO and Advanced Virgo [12, 13]. This is one of the biggest obstacles for advanced GW
detectors to reach their design sensitivity and go beyond.
One of the solutions to reduce the quantum noise at all frequencies is the injectionof so-called frequency-dependent squeezed vacuum, whose squeezing angle rotates by 90◦
as a function of frequency, to counteract the squeezing angle rotation induced by theinterferometer. Such a quantum noise reduction is optimal if the rotation occurs at thecrossover frequency between the shot noise and the radiation pressure noise, which isapproximately 70 Hz for advanced GW detectors. To realize the frequency-dependent
squeezing, it was suggested to reﬂect a frequency-independent squeezed vacuum oﬀ an
optical Fabry-Perot cavity called a ﬁlter cavity [14]. Previously, squeezing angle rotation
using rigid, meter-scale ﬁlter cavities has been realized in the MHz [15] and kHz [16]regions.
In this thesis, we developed the ﬁrst frequency-dependent squeezed vacuum sourcewith the rotation frequency below 100 Hz by using a 300 m ﬁlter cavity which can ﬁt
for Advanced LIGO and Advanced Virgo [17]. It is shown that our frequency-dependent
squeezed vacuum source can realize the broadband quantum noise reduction in GW detectors. However, in this measurement, the ﬁlter cavity length noise and the mode mismatch
between the squeezer and the ﬁlter cavity were not satisfying the target value, and thedetuning ﬂuctuation of ∼30 Hz was observed. For further quantum noise reduction, it isnecessary to stabilize the ﬁlter cavity length ﬂuctuation, the alignment ﬂuctuation, andthe detuning ﬂuctuation. In this measurement, the ﬁlter cavity was controlled with theauxiliary green ﬁeld whose frequency is as twice as the squeezing frequency. As the optical
path and the frequency of the green ﬁeld and the squeezed ﬁeld are diﬀerent, the ﬁlter
cavity control with the auxiliary green ﬁeld does not ensure the ﬁlter cavity length andalignment for the squeezed ﬁeld.
13

To solve this problem, I suggested a new control scheme of the ﬁlter cavity using coherent control sidebands which are already used to control the squeezing angle [18]. As
the coherent control sidebands have the same optical path and almost the same frequency
as the squeezed ﬁeld, the control of the ﬁlter cavity with the coherent control sidebands
ensures the ﬁlter cavity length and alignment for the squeezed ﬁeld. I succeeded in controlling the length of the 300 m ﬁlter cavity with the new control scheme and demonstrated
that the new control scheme reduced the ﬁlter cavity length ﬂuctuation from 3.4 pm to0.75 pm which satisﬁes the target value of 1 pm. Using this new control scheme, thefrequency-dependent squeezing with the rotation frequency around 100 Hz also has beenrealized.
This thesis is organized as follows: in Chapter 2, the theory of gravitational waves andgravitational-wave observations are explained. In Chapter 3, the principle of gravitationalwave detectors is explained. In Chapter 4, the theory of the quantum states of light including squeezed states is described. In Chapter 5, the quantum noise in GW detectors andhow to reduce it are presented. In Chapter 6, the experiment on frequency-independent
squeezing is described. In Chapter 7, the experimental characterization of the 300 m ﬁlter
cavity and the experiment on frequency-dependent squeezing are presented. In Chapter
8, the theoretical suggestion and experimental demonstration of a new control scheme of
the ﬁlter cavity are presented. In Chapter 9, the conclusion is described.

14

Chapter 2

Gravitational Wave
Gravitational waves (GWs) are ripples of spacetime derived from general relativity. As
GWs almost do not interfere with matters, GWs from astrophysical sources can reach
Earth without losing their information. In this chapter, the theory of GWs and recent
GW detections are presented.

2.1

Derivation of gravitational waves

2.1.1

Linearization of Einstein equation

Einstein equation derived from general relativity can be written as follows:
18πGRµν − gµν R = 4 Tµν ,
2
c

(2.1)

where Rµν is the Ricci tensor, gµν is the metric tensor, R is the scalar curvature or Ricci
scalar, Tµν is the energy momentum tensor, G is the gravitational constant, and c is the
speed of light. The greek indices range from 0 to 3. Rµν and R can be written as= Rαµαν ,(2.2)

R = g µν Rµν ,

(2.3)

Rµν

where Rµνρσ is the Riemann curvature tensor:
∂Γµνσ
∂Γµνρ
µ
R νρσ =−
+ Γµαρ Γανσ − Γµασ Γανρ ,
ρ
σ
∂x

∂x

(2.4)

and
1
Γρµν = g ρσ
2∂gσµ ∂gµν
∂gσν
+
−
µ
∂x
∂xν
∂xσ


(2.5)

is the Christoﬀel symbol.
Considering the small perturbation of the metric as follows:gµν = ηµν + hµν ,
15

(2.6)

where ηµν = diag(−1, 1, 1, 1) is the metric in Minkowski spacetime (ﬂat spacetime) andhµν is the small perturbation of the metric. Substituting this into the Einstein equationand considering the ﬁrst order of hµν , the Einstein equation results in the following wave
equation:
16πG□hµν = − 4 Tµν ,
(2.7)
c
1
where hµν ≡ hµν − ηµν η αβ hαβ and the Lorentz gauge
2
µν

∂h
=0
∂xµ

(2.8)

is assumed. This wave equation indicates that the small metric variation propagates thespacetime as waves and these are gravitational waves.

2.1.2

Plane wave solution of gravitational waves

Gravitational waves propagating in the vacuum (Tµν = 0) can be written as□hµν = 0.

(2.9)α

As a solution of this wave equation, hµν = aµν eikα x is considered. Based on the symmetryof hµν , the Lorentz gauge, and wave equation,aµν = aνµ ,(2.10)

aµα k α = 0,(2.11)

α(2.12)

kα k = 0.

Equation (2.11) indicates that the gravitational waves are transverse waves and Equation(2.12) shows that the gravitational waves propagate at the speed of light.
To remove the remaining degrees of freedom for coordinate transformation, Transverse
Traceless gauge (TT gauge) is assumed as follows:
hµ0 = 0,(2.13)

hij ,j = 0,(2.14)

hjj = 0.

(2.15)

Here, the Latin indices range from 1 to 3. Under this gauge, the amplitude of the gravitational wave aµν is written as follows:

0 0
0

0 h+ h×aµν = 
0 h
× −h+

0 0
0

16


0

0
.
0

0

(2.16)

The gravitational waves have two degrees of freedom, which are h+ and h× .
When the plane wave with frequency ω propagates toward the z-axis direction,hTTµν = aµν exp [−iω(t − z/c)].

(2.17)The space line element of the Minkowski spacetime isijdl2 = (δij + hTTij )dx dx= {1 + h+ cos [ω(t − z/c)]}dx2 + {1 − h+ cos [ω(t − z/c)]}dy 2 + dz 2
+ 2h× cos [ω(t − z/c)]dxdy,(2.18)

where the real part of hij is considered.
When the amplitude of the gravitational waves is signiﬁcantly small and the metric is
the Minkowski spacetime, Equation (2.18) indicates that the distance between two points
changes as follows:
dl2 = δij dx′i dx′j ,
h× cos [ω(t − z/c)]
h+ cos [ω(t − z/c)]
′
dx +
dy,
dx =
1+
2
2


h+ cos [ω(t − z/c)]
h× cos [ω(t − z/c)]dy ′ =
1−
dy +
dx.
2
2

(2.19)(2.20)(2.21)

This change is shown in Figure 2.1.

h+h⇥Figure 2.1: + mode and × mode of gravitational waves. Red points represent free masses
and the gravitational waves are injected from the vertical direction to the sheet.

17

2.2

Generation of gravitational waves

The wave equation for GWs (2.7) indicates that GWs are generated from the energy
momentum tensor Tµν . The solution of the wave equation can be written ashµν (t, x) =

4G
c

Z

d3 x′

Tµν (t − |x − x′ |/c, x′ ).
|x − x′ |(2.22)

When the size of the GW source is much smaller than the GW wavelength, the equation
above can be approximated ashij (t, x) =2G
Q̈ij (t − r/c),c4 r(2.23)

where r = |x| andZ
Qij (t) =




1
i j
ρ(t, x) xi xj − δij x x d3 x3

(2.24)

is the quadrupole momentum of the GW source and ρ is the density of the GW source.

2.3

Astrophysical sources of gravitational waves

As the eﬀect of GWs is extremely small, the main GW sources are astrophysical sources
with signiﬁcantly large quadrupole momentums. In this section, the main astrophysicalsources of GWs are introduced.

Compact binary coalescence
The compact binary coalescence (CBC) is the merger the compact binary systems suchas binary black hole (BBH), binary neutron star (BNS). All GW detections by LIGO
and Virgo thus far are GWs from CBC. The CBC is composed of the following threephases: inspiral, merger, and ringdown. In the inspiral phase, the binary system is incircular orbit, and the orbit shrinks by losing its energy through the GWs. In the merger
phase, the binary system merges emitting the largest GWs and forms a perturbed remnant
compact object. In the ringdown phase, the perturbed remnant compact object relaxes tothe stationary state by emitting the GWs. The ringdown of a remnant black hole is called
the quasi normal mode.
When the component masses of CBC are m1 and m2 , the strain of CBC in the inspiralphase is written as [19]h(f ) ≃

1



π 2/3

5
24

1/2

c
r



GMc
c3

5/6

f −7/6 ,(2.25)

whereMc =(m1 m2 )3/5
(m1 + m2 )1/5
18

(2.26)

is called the chirp mass. Here, phase term of h(f ) is ignored.
The inspiral phase ﬁnishes at the innermost stable circular orbit (ISCO). The GW
frequency at ISCO is as follows:1c3√
12 6π GMM⊙
= 2.2 kHz,M

fISCO =

(2.27)(2.28)

where M = m1 + m2 . The most sensitive frequency region of the current GW detectors is∼ 10 Hz -1 kHz and thus the target mass of CBC is ∼ 1-100 M⊙ .
For m1 = m2 = m, the amplitude of GWs from CBC is−24

h(f ) = 4.3 × 10



100 Mpcr

m
1.4 M⊙

5/6 

100 Hzf7/6.

(2.29)

Pulsar
The pulsar is a rotating neutron star emitting periodic electromagnetic pulses. If thepulsar is not axisymmetric around the rotational axis, it generates GWs. The amplitudeof the GWs from the pulsar is [19]h(f ) =

4π 2 G Izz f 2ϵ,c4r(2.30)

where Izz is the moment of inertia around the rotational axis (z axis) of the pulsar, frot is
the rotational frequency of the pulsar, and ϵ = |Ixx − Iyy |/Izz is the ellipticity of the pulsar(Ixx and Iyy are the moments of inertia around the x and y axes, which are orthogonal to
the z axis). The GW frequency from the pulsar can be written as f = 2frot .
The amplitude of the GWs from the pulsar with a typical galactic distance is−25

h = 1.1 × 10



2
 ϵ 
Izz
10 kpcf.
10−61038 kg m2r
1 kHz(2.31)

Supernova
The core collapse supernova (Type II supernova) can generate GWs if there is a nonaxisymmetric mechanism in the supernova. The typical strain and frequency of the corecollapse supernova is estimated to be h ∼ 10−22 −10−21 and ∼ 1 kHz considering a distanceof 10 kpc from the supernova [20].

Early universe
Another important GW source is the stochastic GW background from the early universe.
Quantum ﬂuctuation during inﬂation is one of the candidates that produces the stochastic

19

GW background. The strain of the stochastic GW background can be written as [19]sh(f ) =

3H02 Ωgw (f ),4π 2 f 3(2.32)

where H0 ∼ 68 km s−1 Mpc−1 is the Hubble constant andΩgw (f ) =1 dρgw,
ρc d(log f )(2.33)

where ρgw is the energy density of the stochastic GW background and ρc is the criticalenergy density for a ﬂat universe,
ρc =3c2 H02.
8πG(2.34)As Ωgw is estimated to be ∼ 10−15 [21], the strain of the stochastic GW background ish(f ) ∼ 6 × 10

−25



0.1 Hzf

3/2.

(2.35)The equation above indicates that the amplitude of the stochastic GW background is larger
at lower frequencies. However, this strain sensitivity is diﬃcult to achieve in ground-based
GW detectors because the sensitivity is limited by seismic noise at low frequencies. To
avoid the seismic noise, a spaceborne GW detector called the DECIGO is planned [22].
The ultimate target of DECIGO is the stochastic GW background from the early universe.

2.4

Observations of gravitational waves

In 2015, Advanced LIGO observed gravitational waves from a BBH for the ﬁrst time, as
shown in Figure 2.2 [1].
LIGO and Virgo had observing runs three times thus far. The ﬁrst observing run (O1)
was from September 12, 2015 to January 19, 2016. The second observing run (O2) wasfrom November 30, 2016 to August 25, 2017. The third observing run (O3) was from April
1, 2019 to March 27, 2020. There were 11 events, including a BNS merger, observed in
O1 and O2 [3], and 39 events were observed in the ﬁrst half of O3 (O3a) from April 1,
2019 to October 1, 2019 [6]. Cumulative GW events from O1 to O3a are shown in Figure
2.3. The signiﬁcant increase of GW events from O2 to O3 was achieved by some upgrades,
especially the injection of a squeezed vacuum ﬁeld [10, 11].

Binary black hole merger
There are 46 binary black hole (BBH) mergers observed from O1 to O3a. The BBH merger−3
rate estimated from the results from O1 to O3a is RBBH = 23.9+14.9
yr−1 [24].
−8.6 Gpc

20

Figure 2.2: Observed chirp signals of GW150914 by Hanford and Livingston observatories
of Advanced LIGO [1]. The left and right plots show the GW signals for Hanford and
Livingston, respectively. The ﬁrst and second plots from the top show the measured
GW time series and calculated GW time series with numerical relativity, respectively.
The second from the bottom plots show the residual GW signal after subtracting the
calculated GW signal from the measured GW signal. The bottom plots show the GW
frequency as a function of time. The GW frequency increases over time, which is why this
GW signal is called the chirp signal.

Binary neutron star merger
There are two binary neutron star (BNS) mergers observed from O1 to O3a which are
GW170817 [2] and GW190425 [29]. GW170817 was suﬃciently localized by GW observation, which enabled the observations of the electromagnetic (EM) counterpart to the
GW170817 with multiple wavelengths [30]. On the other hand, GW190425 was not sufﬁciently localized because one of the GW detectors were not working at the time of theevent, and there was no observed EM counterpart to the GW190425 [29].
−3
The BNS merger rate estimated from the results from O1 to O3a is RBNS = 320+490
yr−1−240 Gpc[24].

Neutron star black hole merger
There are two potential neutron star black hole (NSBH) mergers observed from O1 to
O3a, which are GW190426 152155 [6] and GW190814 [25]. The component masses are 5.7
M⊙ and 1.5 M⊙ for GW190426 152155, and 23 M⊙ and 2.6 M⊙ for GW190814. The lower
21

Figure 2.3: Cumulative GW events from O1 to O3a [23].

mass components of these events are either a neutron star or a black hole, and therefore,
these events are either an NSBH or BBH merger.

2.5

Science from gravitational wave observations

Several scientiﬁc milestones were achieved from the GW observations, which will be presented in this section.

Population of compact objects
To understand the formation channel of compact objects and compact binary systems,
the population of compact objects, such as mass and spin, provided crucial information.
From GW observations, the compact objects were found in the wide mass range, which
are not observed by EM. For example, the 2.6 M⊙ compact object in GW190814 is in themass gap region (called lower mass gap) between the neutron stars (< 2.5M⊙ ) and theblack holes (> 5M⊙ ).

Tests of general relativity
GW observations of the compact binary coalescences enable the tests of general relativity
in the strong gravity ﬁeld. By using the binary black holes observed from O1 to O3a, tests
of general relativity were conducted and several constraints on the general relativity weretighten [28]. However, there is still no evidence for violation of the general relativity.

22Multi-messenger observation of a binary neutron star merger
As GW170817 was suﬃciently localized, the EM counterparts to GW170817 were observedwith multiple wavelengths [30]. Approximately 1.7 s after the BNS merger, a short gammaray burst (GRB 170817A) was observed [31]. This conﬁrms that the BNS mergers are oneof the progenitors of the short gamma-ray burst. Using the information on the timedelay of GW170817 and GRB 170817A, the diﬀerence between the speed of gravity andthe speed of light was constrained [31]. Another important EM observation related to
GW170817 was an ultraviolet, optical, and infrared transient called kilonova [30]. The
kilonova was caused by the radioactive decay of the r-process nuclei. This indicates thatthe BNS mergers are progenitors of heavy elements synthesized with the r-process.

Standard siren measurement
GW170817 enabled the ﬁrst measurement of the Hubble constant with gravitational waves[32]. The distance to the source of GW170817 d was obtained from the GW observation,and the Hubble ﬂow velocity of the corresponding GW source vH was obtained from the
EM observation. Using Hubble’s law vH = H0 d, the Hubble constant H0 was determined−1−1
to be 70.0+12.0−8.0 km s Mpc . This measurement is consistent with existing measurements,
while it is independent of them. Future GW observations will improve the precision of the
Hubble constant measurement.

2.6

Observation scenario

In the future, the fourth observing run (O4) will start around 2022, and the ﬁfth observing
run (O5) will start around 2025, as shown in Figure 2.4. KAGRA will join the observing
run from O4, and LIGO-India will join from O5 [8]. One of the major upgrades towards O4
and O5 in LIGO, Virgo, and KAGRA is the implementation of the frequency-dependent
squeezing, which is the main topic of this thesis. In particular, LIGO and Virgo plan toinstall a 300 m ﬁlter cavity from O4.
Third-generation GW detectors, such as the Einstein Telescope [33] and Cosmic Explorer [34], are planned to operate in the 2030s. The arm lengths of these detectors are
10-40 km, and the sensitivity is an order of magnitude better than the current GW detectors, which will lead to an increase of approximately 1000 times the GW events. In
third-generation GW detectors, frequency-dependent squeezing with kilometer-scale ﬁlter
cavities is planned to be installed for quantum noise reduction [35, 36].

23

Figure 2.4: Observation scenario of GW detectors [8]. In this ﬁgure, KAGRA was expected
to join O3, but actually KAGRA could not join O3 due to the early end of O3, although
the BNS range of KAGRA reached around 1 Mpc.

24

Chapter 3

Gravitational-Wave Detector
The basic principle of ground-based GW detectors is the Michelson interferometer, which
can measure the diﬀerential motion of test masses and is thus suitable to measure themotion induced by GWs. The conﬁguration of the current GW detectors is called the dual
recycled Fabry-Perot Michelson interferometer (DRFPMI). In this chapter, the principleof the interferometric GW detectors is presented.

3.1

Michelson interferometer

A conﬁguration of the Michelson interferometer is shown in Figure 3.1.

End mirror
Laser
End mirrorBSy<latexit sha1_base64="M6tyLGopi1lL+AzCUVUgcchZyfk=">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</latexit>PDz<latexit sha1_base64="ILW6yb24Z0bPAF6ZMbUmWHM77xw=">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</latexit>x<latexit sha1_base64="kP85kGCnr2++8g5/mBsXzO255iE=">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</latexit>

Figure 3.1: Conﬁguration of the Michelson interferometer.

For simplicity, the splitting ratio of a beam splitter (BS) is assumed to be 50:50, andthe reﬂectivity of the end mirror is 1. In Figure 3.1, when the arm length (distance between
BS and end mirror) in the x-axis (horizontal) direction is Lx and the arm length in the25

y-axis (vertical) direction is Ly , and the carrier laser frequency is ω0 , the round-trip phasebetween the BS and end mirror in x-arm and y-arm are ϕx = 2Lx ω0 /c and ϕy = 2Ly ω0 /c,
respectively. The electric ﬁeld at an anti-symmetric port (AS port) is
1
E0 (e−iϕx − e−iϕy )2
ϕ+ϕ−= −iE0 e−i 2 sin
,
2

EAS =

(3.1)

where E0 is the input electric ﬁeld and ϕ+ ≡ ϕx + ϕy , ϕ− ≡ ϕx − ϕy are deﬁned.
The carrier power at the AS port is
PAS = |EAS |2 = P0 sin2

ϕ−
,
2

(3.2)

where P0 = |E0 |2 . The carrier power at the AS port only depends on the diﬀerential
arm length in the x and y arms. The point where the carrier power at the AS port is 0(ϕ− = 2nπ, n: integer) is called the dark fringe, where all the injected carrier beam is
reﬂected back to the injection port. This condition is necessary for the power recycling
technique, which will be presented in Section 3.3.

3.1.1

Response to gravitational waves

In this section, we consider a response of a Michelson interferometer to GWs. We assume
that GWs only have the plus mode and propagate in the z direction.
The spacetime interval in the TT gauge is
ds2 = −c2 dt2 + (1 + h)dx2 + (1 − h)dy 2 + dz 2 .

(3.3)

As light travels along the null geodesics: ds2 = 0, the light in x arm obeys the following
equation:


dx
c
1
= ±√
≃ ±c 1 − h ,
dt
2
1+h

(3.4)

where ± represents the light propagating in the ±x direction.
The round-trip phase for x arm is



dx −1
dx
dt
roundtripZ
2Lx ω0 ω0 th(t′ )dt′ ,+c2 t−2L/cZ

ϕx (t) = ω0
=

(3.5)

where Lx ≃ L = (Lx + Ly )/2 is assumed. Similarly, the round-trip phase for y arm is
ϕy (t) =2Ly ω0 ω0
−
c
2

Z t

26

t−2L/ch(t′ )dt′ .

(3.6)

Therefore, the electric ﬁeld at the AS port is
EAS =
=

1
E0 (e−iϕx − e−iϕy )21E0 e−iϕx [1 − ei(ϕ− +ϕGW ) ],
2

(3.7)

where
2ω0 (Lx − Ly ),c
Z t
ϕGW (t) = ω0h(t′ )dt′ .
ϕ− =

(3.8)
(3.9)

t−2L/cHere, ϕGW is the phase induced by GWs. Using the Fourier transformation of h(t′ ) in
Equation (3.9),
Z ∞
Z t
′
ϕGW (t) = ω0h(ω)eiωt dωdt′t−2L/c −∞Z ∞HMI (ω)h(ω)eiωt dω,=(3.10)−∞

where
HMI (ω) =2ω0 L sin (ωL/c) −iωL/c
e
c
ωL/c

(3.11)

is the frequency response of a Michelson interferometer to GWs. This response is maximum
when ωL/c = π/2. The optimal arm length for f = 100 Hz of GW frequency is L = 750
km. This arm length is not a realistic value to build it on Earth. To increase the eﬀective
arm length, Fabry-Perot cavities are used in current GW detectors, which will be presented
in Section 3.2. Although the cavity length of the current GW detectors is only 3-4 km,
the eﬀective arm length is signiﬁcantly enhanced by the cavity.
For ω ≪ c/L, which is the case for f = 100 Hz and L = 3-4 km, Equation (3.11) is
HMI (ω) =

3.1.2

2ω0 L
.
c

(3.12)DC readout

From Equation (3.7), the power at the AS port in the dark fringe is
PAS = P0 sin2

ϕGW
.
2

(3.13)In the dark fringe, the GW signal ϕGW is second order at the AS port. To obtain alinear GW signal at the AS port, a DC readout technique [37] is used in the current GW
detectors. In the DC readout, the Michelson interferometer is operated close to the dark
fringe to enable the small carrier light to leak into the AS port. The power at the AS port

27

in the DC readout is
PAS = P0 sin2

ϕ− + ϕGW
P0 2≃(ϕ + 2ϕ− ϕGW ).
2
4 −(3.14)

Therefore, a linear GW signal can be obtained.

3.2

Fabry-Perot cavity

As mentioned in the previous section, Fabry-Perot cavities are used in the current GW
detectors to increase the eﬀective arm length. The Fabry-Perot cavity is composed of two
mirrors that reﬂect the beam back and forth between them and enhance the power inside
the cavity. A conﬁguration of the Fabry-Perot cavity is shown in Figure 3.2.

cavity length

L
Et

E0
<latexit sha1_base64="f8/DuyP0ACOFSHuwjDflV86utQ8=">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</latexit>Erinput mirror

end mirror

rI , tIr E , tE<latexit sha1_base64="ATX1cSB7dDfPUzL8PZx/nQJwbO4=">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</latexit>

Figure 3.2: Conﬁguration of the Fabry-Perot cavity. The Fabry-Perot cavity is composedof an input mirror and end mirror. We assume that the cavity length is L, the amplitude
reﬂectivity and transmissivity of the input mirror are rI and tI , respectively, and theamplitude reﬂectivity and transmissivity of the end mirror are rE and tE , respevtively.

Firstly, we will not consider the eﬀect of optical losses in the cavity. When the input
electric ﬁeld to the input mirror is E0 , the amplitude of the reﬂected ﬁeld from the cavity
is2
Er = E0 rI − E0 t2I rE e−iϕ − E0 t2I rE
rI e−2iϕ − · · ·
∞
X
2
−iϕ
= E0 rI − E0 tI rE e
(rI rE e−iϕ )n


= E0

n=0

t2I rE e−iϕrI −1 − rI rE e−iϕ,(3.15)

where ϕ is the round-trip phase in the cavity and can be written as follows:
ϕ=

2Lω0
.
c

28

(3.16)

Similarly, the amplitude of the transmission ﬁeld is
Et = E0 tI tE e−iϕ/2
= E0

∞
X

(rI rE e−iϕ )n

n=0
−iϕ/2
e

tI tE
.
1 − rI rE e−iϕ(3.17)

The amplitude reﬂectivity and transmissivity of a Fabry-Perot cavity (rFP , tFP ) are
t2I rE e−iϕ,1 − rI rE e−iϕtI tE e−iϕ/2.
1 − rI rE e−iϕrFP (ϕ) = rI −(3.18)tFP (ϕ) =(3.19)The transmission power is
Pt = |Et |2=(tI tE )2P0 .
(1 − rI rE )2 + 4rI rE sin2 ϕ/2(3.20)The power transmissivity |tFP (ϕ)|2 as a function of the round-trip phase is shown in Figure
3.3.
1

power transmissivity

power transmissivity

0.8

0.6

0.4

0.2

0
-3

-2

0

-

2

3

round trip phase (rad)Figure 3.3: Power transmissivity of a Fabry-Perot cavity as a function of the round-trip
phase. rI = rE = 0.89 is assumed.

When the Fabry-Perot cavity resonates, the transmission and the intracavity power
are the maximum. The resonance condition is
ϕ = 2πm(m = 1, 2, · · · ).

29

(3.21)

From Equation (3.16) and ω0 = 2πc/λ0 , this condition can be written as follows:(m = 1, 2, · · · ).

2L = mλ0(3.22)

When the cavity length L is ﬁxed, the transmission power changes periodically as afunction of the laser frequency and this frequency period is called the free spectral range(FSR). From Equation (3.16) and (3.21), the FSR can be written as follows:
fFSR =

c
ωFSR
=
.
2π
2L

(3.23)

From Equation (3.20), the full width at half maximum of the transmission power
fFWHM can be obtained by solving the following equation.
1
1
= .

4rI rE
πLf
2
FWHM
1+
sin2
(1 − rI rE )2cAssuming(3.24)

πfFWHM
πLfFWHM=≪ 1 1 , we can expand the sine term as follows:c
2fFSR
fFWHM =c(1 − rI rE ).
√2πL rI rE(3.25)The ratio of fFSR and fFWHM represents the sharpness of the resonance and is called theﬁnesse, which is√
π rI rE
fFSR=.
(3.26)F≡
fFWHM
1 − rI rEWhen the intracavity loss Tloss is considered, the ﬁnesse (3.26) is
√
π r
F=
,
1−r

(3.27)√
where r = rI rE rloss with rloss = 1 − Tloss .
The cavity decay time is when the stored power inside the cavity is reduced by a factorof e2 after turning oﬀ the input beam to the cavity [38]. The cavity transmission power
after turning oﬀ the input beam can be written as follows:
P (t) = P0 e−2t/τ ,(3.28)

where P0 is the cavity transmission power before turning oﬀ the input beam and τ is thecavity decay time.
The decay time is written as follows:
τ

1

=

F
1
=
.
πfFSR
πfFWHM

This corresponds to F ≫ 1 in Equation (3.26)

30(3.29)

This cavity decay time can be directly measured by turning oﬀ the input beam to the cavity
and measuring the transmission or reﬂection beam from the cavity. This measurement iscalled a ringdown.
The ﬁnesse of a cavity is often measured by scanning the cavity length or the laser
frequency, and measuring the time series of the transmission power from the cavity, as
shown in Figure 3.3. This measurement is called cavity scan. From Equation (3.20), thetime series of the transmission beam from the cavity can be written as follows:
T (t) =a0)
1 + 4( Fπ )2 sin2 ( π(t−ttFSR )(3.30),

where a is the amplitude of the peak height, t0 is the time oﬀset depending on the startingtime of the scan, and tFSR is the time diﬀerence of the free spectral range. By ﬁtting themeasured data with this function, the ﬁnesse can be obtained.

3.2.1

Response to gravitational waves

In this section, we consider a response of a Fabry-Perot cavity to the GWs. For simplicity,
we ignore the losses in the cavity. We assume that GWs only have the plus mode andpropagate in the z direction and the Fabry-Perot cavity is placed along the x direction.
The phase for the beam with round trips in the cavity n times is2Lω0 n ω0
ϕn (t) =
+
c
2

Z t

h(t′ )dt′ .

(3.31)

t−2Ln/c

The amplitude reﬂectivity of the cavity around the resonance isrFP =rI − t2I rE≃ rI − t2I rE

∞
X
n=1
∞
X

n−1 −iϕnerIn−1 rEn−1
rIn−1 rE

n=1

=
=

ω0
1−i
2

Z t

!
h(t′ )dt′

t−2Ln/cZZ ∞∞rI − rEω0 X n−1 n−1 t′
2+ i(tI rE )rI rEh(ω)eiωt dωdt′
1 − rI rE
2
t−2Ln/c −∞
n=1


Z ∞
rI − rEiωt
HFP (ω)h(ω)e dω ,1+i
1 − rI rE−∞(3.32)

where
HFP (ω) =t2I rE
ω0 L
sin (ωL/c) −iωL/cec (rI − rE )(1 − rI rE e−2iωL/c ) ωL/c

(3.33)

is the frequency response of a Fabry-Perot cavity to the GWs. In the case of ωL/c ≪ 1,
rI ≃ 1, and rE = 1,
HFP (ω) = −2ω0 L F1,c π 1 + iω/ωp31(3.34)

where
fp =

ωp
c
fFWHM
=
=
2π
4LF
2

(3.35)

is called the cavity pole. The frequency response of the Fabry-Perot cavity is the ﬁrst-order
low pass ﬁlter with a cutoﬀ frequency fp for GWs (also for cavity length noise and laser
frequency noise). When we compare Equation (3.34) with Equation (3.12), the response of
the Fabry-Perot cavity is enhanced by F/π with respect to the Michelson interferometerwith the same arm length.
In current GW detectors, the Fabry-Perot cavities are used for arms of the Michelsoninterferometer, as shown in Figure 3.4. The response of the Fabry-Perot Michelson interferometer (FPMI) is enhanced by 2F/π compared to the Michelson interferometer withthe same arm length.

ETM

ITM
Laser

ITM

ETM

BS

PD

Figure 3.4: Conﬁguration of a Fabry-Perot Michelson interferometer. There are two FabryPerot cavities used for the arms of the Michelson interferometer to enhance the eﬀective
arm length.

3.2.2

Pound-Drever-Hall technique

The Pound-Drever-Hall (PDH) technique is used to obtain the phase diﬀerence betweena cavity resonance and injection beam and to keep the cavity resonate [39]. An example
conﬁguration of the PDH technique is shown in Figure 3.5.
The phase modulated injection beam is
E = E0 ei(ωt+β sin Ωt)≃ E0 [J0 (β)eiωt + J1 (β)ei(ω+Ω)t − J1 (β)ei(ω−Ω)t ],32(3.36)

LaserEOM

BS

CavityPD

Figure 3.5: Example conﬁguration of the PDH technique. The phase diﬀerence between
the cavity resonance and the injection beam δϕ can be obtained by modulating the phaseof the injection beam using an electro-optic modulator (EOM) and demodulating thereﬂected beam from the cavity at the modulation frequency. The PDH signal is fed backto the cavity length to maintain the cavity resonance.

where β is the modulation depth, Ω is the modulation frequency, and Jn is the n-th order
Bessel function of the ﬁrst kind. From Equation (3.18), the reﬂected power from the cavity
isPr = P0 J0 (β)r(ω)e−iωt + J1 (β)r(ω + Ω)e−i(ω+Ω)t − J1 (β)r(ω − Ω)e−i(ω−Ω)t2= P0 (|J0 (β)r(ω)|2 + |J1 (β)r(ω + Ω)|2 + |J1 (β)r(ω − Ω)|2 )+ 2P0 J0 (β)J1 (β){Re[r(ω)r∗ (ω + Ω) − r∗ (ω)r(ω − Ω)] cos Ωt+ Im[r(ω)r∗ (ω + Ω) − r∗ (ω)r(ω − Ω)] sin Ωt} + (2Ω terms).

(3.37)In Equation (3.37), the phase information of the carrier is included in the Ω terms, which
are the beat notes of the carrier and sidebands. By demodulating the reﬂected signal withsin Ωt, the PDH signal in phase can be obtained as follows:I
Pdemod= P0 J0 (β)J1 (β) Im{r(ω)r∗ (ω + Ω) − r∗ (ω)r(ω − Ω)}.

(3.38)

By demodulating the reﬂected light with cos Ωt, the PDH signal in the quadrature phase
isQ
Pdemod= P0 J0 (β)J1 (β) Re{r(ω)r∗ (ω + Ω) − r∗ (ω)r(ω − Ω)}.
(3.39)
When the modulation frequency is signiﬁcantly larger than the cavity linewidth (Ω ≫
ωFWHM ) and the carrier is around the resonance, the sidebands are oﬀ resonance of thecavity and nearly completely reﬂected by the cavity (r(ω ± Ω) ≃ 1). Therefore,r(ω)r∗ (ω + Ω) − r∗ (ω)r(ω − Ω) ≃ i2 Im{r(ω)}.

33(3.40)

In this case, the Q phase PDH signal (3.39) disappears. The I phase PDH signal (3.38) isI
Pdemod≃ 2P0 J0 (β)J1 (β) Im{r(ω)}
2P0 J0 (β)J1 (β) t2I rE sin ϕ=
2 − 2r r cos ϕ
1 + rI2 rE
I E≃

4P0 J0 (β)J1 (β)t2I rEδϕ,(1 − rI rE )2(3.41)

where ϕ = δϕ ≪ 1 is assumed. The PDH signal is proportional to δϕ around the resonance.
The intracavity power and the normalized PDH signals (In-phase, Quadrature) are
shown in Figure 3.6.

Normalized PDH signal

1

Intracavity power
In-phase
Quadrature

0.5

0

-0.5

-1
- /8

- /16

0

/16

/8

phase (rad)Figure 3.6: Normalized PDH signal of the Fabry-Perot cavity. The red line represents
the intracavity power normalized with respect to its maximum intracavity power. The
blue and green lines represent the PDH signals (in-phase and quadrature) normalizedwith respect to P0 J0 (β)J1 (β). It is assumed that the laser wavelength√ is λ = 1064 nm,
the cavity length is L = 4 cm, the mirrors reﬂectivity is rI = rE = 0.9999, the cavity
linewidth is fFWHM = 8 MHz, the modulation frequency is Ω/2π = 100 MHz, and themodulation depth is β = 0.15 rad.

3.3

Conﬁguration of GW detectors

The standard conﬁguration of the current GW detectors is called the dual recycled FabryPerot Michelson interferometer (DRFPMI). The conﬁguration of DRFPMI is shown in
Figure 3.7. Two additional mirrors, called the power recycling and signal recycling mirrors,
are placed in the FPMI.

34

ETM

ITM
Laser

ITM

PRM

ETM

BS
SRM

PD
Figure 3.7: Conﬁguration of DRFPMI. In addition to the FPMI, there are two additional
mirrors called the power recycling mirror and signal recycling mirror.

Power recycling
The power recycling mirror (PRM) is placed at the injection port of the interferometer andreﬂects back the laser to the interferometer. The PRM and input test mass (ITMs) forma cavity called the power recycling cavity (PRC). When the carrier ﬁeld is anti-resonant
inside the PRC, the power recycling eﬀectively increases the injection laser power.

Signal recycling
The signal recycling mirror (SRM) is placed at the anti-symmetric port of the interferometer and reﬂects the GW signal to the interferometer. The SRM and ITMs form a cavity
called the signal recycling cavity (SRC). When GW sidebands are resonant inside theSRC, the eﬀective bandwidth of the arm cavity is increased because the eﬀective ITM reﬂectivity is reduced for the GW sidebands. This technique is called the resonant sideband
extraction (RSE). The eﬀective bandwidth of the arm cavity with RSE can be written as
follows:
γifo =

1 + rsrγarm ,
1 − rsr(3.42)

where rsr is the amplitude reﬂectivity of the SRM and γarm is the arm cavity half bandwidth. The interferometer bandwidth can be increased with RSE.

35

3.4

Noise sources in GW detectors

There are several noise sources in GW detectors. In this section, the following threemain noise sources in GW detectors will be introduced: quantum noise, seismic noise, andthermal noise.

Quantum noise
The quantum noise is fundamental noise originated from the quantum ﬂuctuation of light.
It is composed of shot noise and radiation pressure noise. In a semi-classical picture, theshot noise is the photon counting noise in a photo detector, and the radiation pressure
noise is the mirror ﬂuctuation owing to the ﬂuctuation of the photon number hitting on the
mirror. In a quantum picture, the origin of the quantum noise is the vacuum ﬁeld entering
the output port of the interferometer [9]. Details of the quantum noise are presented in
Chapter 5.

Seismic noise
The seismic noise originates from the mirror ﬂuctuation owing to the seismic motion of
the ground. Test masses in GW detectors are suspended by multiple pendulums to isolate
the test masses from seismic noise. As the seismic noise below the resonant frequency of asuspension (which is usually an order of 1 Hz) cannot be isolated, it is a signiﬁcant noise
source at low frequencies.

Thermal noise
The thermal noise is composed of the Brownian motion of the mirror substrate, mirror
coating, and suspension wires. The ways to reduce the thermal noise are using high Q
factor coating and suspension wires, and cooling down the mirrors and suspension wires.
LIGO and Virgo operate at room temperature, while test masses are cooled down to 20
K in KAGRA to reduce the thermal noise.

3.5

Sensitivity of GW detectors

The design sensitivities of Advanced LIGO, Advanced Virgo, and KAGRA are shown in
Figure 3.8-3.10. For all cases, quantum noise broadly limits the sensitivity and it is crucial
to reduce the quantum noise to improve the sensitivity of GW detectors.

36

Figure 3.8: Design sensitivity of Advanced LIGO [4].

Figure 3.9: Design sensitivity of Advanced Virgo (AdV) [5].

37seismic+gravity
suspension
mirrorquantum
total (BRSE)

Strain [1/ Hz]

10-21

10-22

10-23

10-24
101

102

103

Frequency [Hz]Figure 3.10: Design sensitivity of KAGRA (broadband RSE) [40].

38Chapter 4

Theory of Quantum States ofLight
Quantum noise in GW detectors originates from the vacuum ﬁeld entering from the output
port of the interferometer. To understand the quantum noise in GW detectors and how
to reduce it, the basics of the quantum states of light are presented in this chapter.

4.1

Quantization of the electromagnetic ﬁeld

In this section, we present the quantization of the electromagnetic ﬁeld. The formalism
used in this chapter (and throughout this thesis) is based on [14].
Quantized electric ﬁeld can be written as follows:
Z ∞rE(t) =
0dω
2πℏωa(ω)e−iωt+ h.c.,
Ac
2π

(4.1)

where A is the eﬀective cross-sectional area of the electric ﬁeld and a(ω) is the annihilation
operator for a photon whose frequency is ω. h.c. indicates the Hermitian conjugate. The
commutation relation of a(ω) is[a(ω), a(ω ′ )] = 0, [a(ω), a† (ω ′ )] = 2πδ(ω − ω ′ ).

(4.2)

We consider a carrier ﬁeld whose frequency is ω0 and its sidebands whose frequencies are
ω0 ± Ω. In GW detectors, the carrier frequency is ω0 /(2π) ∼ 300 THz, while the frequency
of a GW signal is Ω ∼ 10 − 10 kHz, and thus, we can assume Ω ≪ ω0 . In this sideband
picture, Equation (4.1) can be written as follows:rE(t) = e−iω0 t

2πℏω0
Ac

Z ∞
0

[a+ e−iΩt + a− eiΩt ]dΩ+ h.c.,
2π(4.3)

where a+ = a(Ω) and a− = a(−Ω) are annihilation operators for the upper and lower
sidebands. Here, we approximated ω0 ± Ω ≃ ω0 inside the square root. Formalism using
operators a+ , a− is called one-photon formalism as each operator acts on only one photon.
39

Using the annihilation and creation operators for the upper and lower sidebands, we
can deﬁne
a+ + a†−a+ − a†−√a1 (Ω) =, a2 (Ω) = √
.
2
2i

(4.4)

In the time domain,a1 (t) =a(t) + a† (t)a(t) − a† (t)√√, a2 (t) =
.
2
2i

(4.5)

A formalism using operators a1 , a2 is called two-photon formalism because each operator
acts on two photons, which are the upper and lower sidebands [41, 42]. The two-photon
formalism is used throughout this thesis as it is useful for describing the squeezed vacuum
states and calculating the quantum noise in GW detectors.
The commutation relation in the two-photon formalism is[a1 , a†2′ ] = −[a2 , a†1′ ] = i2πδ(Ω − Ω′ ),(4.6)[a1 , a1′ ] = [a1 , a†1′ ] = [a†1′ , a†1′ ] = [a†1′ , a†2′ ] = [a1 , a2′ ] = 0,

(4.7)

and similarly, with 1↔2.
The electric ﬁeld (4.3) can be written in terms of the two-photon formalism as follows:rE(t) =4πℏω0{a1 (t) cos (ω0 t) + a2 (t) sin (ω0 t)},
Ac

(4.8)

where
Z ∞h
a1,2 (t) =0a1,2 (Ω)e−iΩt + a†1,2 (−Ω)eiΩt

i dΩ
2π

.

(4.9)

When the carrier electric ﬁeld is in the cosine quadrature, a1 (t) is the amplitude modulation and a2 (t) is the phase modulation for the carrier. Conventionally, a1 is called theamplitude quadrature and a2 is called the phase quadrature.

4.2

Heisenberg uncertainty principle

For an arbitrary Hermitian operator A and an arbitrary state |ψ⟩, we deﬁne the uncertaintyof the operator as∆A = A − ⟨A⟩ ,40(4.10)

where ⟨A⟩ = ⟨ψ| A |ψ⟩ is the expectation value of A for state |ψ⟩. The variance of theoperator is deﬁned as follows:⟨(∆A)2 ⟩ = ⟨A2 ⟩ − ⟨A⟩2 .

(4.11)

According to the Heisenberg uncertainty principle, for arbitrary Hermitian operators A
and B, and an arbitrary state |ψ⟩,1⟨(∆A)2 ⟩ ⟨(∆B)2 ⟩ ≥ | ⟨[A, B]⟩ |2 .
4

(4.12)As the commutation relation for a, a† in the time domain is [a, a† ] = 1, the commutation
relation for the quadrature operators is [a1 , a2 ] = i. The uncertainty principle for thequadrature operators is1
1⟨(∆a1 )2 ⟩ ⟨(∆a2 )2 ⟩ ≥ | ⟨[a1 , a2 ]⟩ |2 = .
4
4(4.13)

This indicates that the product of variances of the amplitude and phase quadratures has
an unavoidable lower limit for any states.

4.3

Quantum states of light

In this section, the basic quantum states of light, including Fock states, vacuum states,
coherent states and squeezed states, are introduced.

4.3.1

Fock states and vacuum states

Fock states or number states |n⟩ are deﬁned as follows:a |n⟩ =†

a |n⟩ =√√

n |n − 1⟩ ,(4.14)

n + 1 |n + 1⟩ ,(4.15)

N |n⟩ = n |n⟩ ,(4.16)

where N = a† a is the number operator. The Fock states are orthogonal: ⟨n|m⟩ = δnm .
The vacuum states |0⟩ are deﬁned as follows:a |0⟩ = 0.

(4.17)The Fock states can be written in terms of the creation operator as follows:(a† )n|n⟩ = √ |0⟩ .
n!
This indicates that there are n photons in the Fock states |n⟩.

41

(4.18)

The expectation value and variance of a1 and a2 for the Fock states are⟨n| a1 |n⟩ = ⟨n| a2 |n⟩ = 0,1⟨n| (∆a1 )2 |n⟩ = ⟨n| (∆a2 )2 |n⟩ = n + .
2(4.19)(4.20)The vacuum states |0⟩ are the Fock states with no photon. Although the expectation
value of the amplitude and phase quadratures for the vacuum states is 0, the vacuum
states have an uncertainty in amplitude and phase quadratures owing to the Heisenberg
uncertainty principle. This uncertainty of the vacuum states entering from the output
port of the interferometer is the origin of the quantum noise in GW detectors.

4.3.2

Coherent states

Coherent states |α⟩ can be deﬁned as the eigenstates of the annihilation operator as follows:a |α⟩ = α |α⟩ ,(4.21)

where α is an eigenvalue of a. Coherent states in the number state basis can be written as|α⟩ = e−|α|2 /2∞
X
αn√ |n⟩ .
n!
n=0

(4.22)The displacement operator from the vacuum states to the coherent states can be writtenasD(α) |0⟩ = |α⟩ ,D(α) = exp (αa† − α∗ a).

(4.23)(4.24)The expectation value and variance of a1 and a2 for the coherent states are⟨α| a1 |α⟩ =⟨α| a2 |α⟩ =√√2Re[α],(4.25)

2Im[α],(4.26)

1⟨α| (∆a1 )2 |α⟩ = ⟨α| (∆a2 )2 |α⟩ = .
2(4.27)

The coherent states have an equal uncertainty for the amplitude and phase quadrature,and the product of the uncertainty for the amplitude and phase quadrature is the minimum
value allowed by the uncertainty principle.

4.3.3

Squeezed states

The operator for the squeezed states can be written asS(r, ϕ) = exp


1 −2iϕ 2 1 2iϕ †2,rea − re a2242(4.28)

where r is the squeezing factor and ϕ is the squeezing angle. Using the operator for thesqueezed states, the amplitude and phase quadrature evolve as follows:
S(r, ϕ)† a1 S(r, ϕ) = a1 (cosh r − sinh r cos 2ϕ) − a2 sinh r sin 2ϕ,†S(r, ϕ) a2 S(r, ϕ) = a2 (cosh r + sinh r cos 2ϕ) − a1 sinh r sin 2ϕ.

(4.29)(4.30)In the case of ϕ = 0,
S(r)† a1 S(r) = a1 e−r ,(4.31)

S(r)† a2 S(r) = a2 er .

(4.32)The expectation value and variance of a1 and a2 in the squeezed states are as follows:⟨0| S(r)† a1 S(r) |0⟩ = ⟨0| S(r)† a2 S(r) |0⟩ = 0,
1 −2r
⟨0| S(r)† (∆a1 )2 S(r) |0⟩ =
e ,
2
1 2r
⟨0| S(r)† (∆a2 )2 S(r) |0⟩ =e .
2(4.33)(4.34)(4.35)

Compared to the vacuum states, the variance of the amplitude quadrature in the squeezed
states is decreased (or squeezed) by a factor of e2r , while the variance of the phase quadrature is increased (or anti-squeezed) by the same factor. The squeezed states have a minimum uncertainty, however, the variance of the two quadratures are redistributed in thesqueezed states.
The squeezed states for ϕ = 0 can be written in terms of the number state as follows:
∞ p
X
(2n)!
1
S(r) |0⟩ = √(tanh r)n |2n⟩ .
cosh r n=0 2n n!

(4.36)

This indicates that only an even number of photons are present in the squeezed states.
The squeezed states are generated from the pairs of correlated photons, which will bepresented in Section 4.5.

4.4

Second order nonlinear process

The second order nonlinear process plays a key role in generating the squeezed vacuum
states. In this section, the theory of the second order nonlinear process will be presented.
When an electric ﬁeld E is applied to a nonlinear material, it induces a polarization
P as follows:P= ϵ0 (χ(1) E + χ(2) E 2 + ...)= P (1) + P (2) + ...,

(4.37)

where ϵ0 is the free space permittivity and χ(n) is the n-th order optical susceptibility of43

the nonlinear material. In this thesis, only the second order nonlinear term is considered.
When the electric ﬁeld can be written as E = E0 e−iωt + h.c., the second order polarization in Equation (4.37) isP (2) = ϵ0 χ(2) (E02 e−2iωt + h.c.) + ϵ0 χ(2) (2E0 E0∗ ).

(4.38)

This equation indicates that the second harmonics of the input electric ﬁeld and theDC ﬁeld are generated. When the input electric ﬁeld is composed of more than one
frequency components, the second order nonlinear optical susceptibility leads to the sum
and diﬀerence frequency generation of the input electric ﬁelds.
For the sum frequency generation, the ﬁeld with a frequency ω3 = ω1 + ω2 is generated
from the input ﬁelds with a frequency ω1 , ω2 , as shown in Figure 4.1.

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!2

(2)

!3 = !1 + !2

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Figure 4.1: Schematic of sum frequency generation. A block in the center is a nonlinear
material with eﬀective nonlinear susceptibility deﬀ = χ(2) /2. The optical axis is deﬁned
as the z axis and the length of the nonlinear material in the z axis is L. A ﬁeld with afrequency ω3 = ω1 + ω2 is generated from the input ﬁelds with a frequency ω1 , ω2 .

The input and generated ﬁelds should satisfy the following wave equation [43]:∇2 En −

1 ∂ 2 PNL
n2 (ωn ) ∂ 2 En
n
=
,
c2
∂t2
ϵ0 c2 ∂t2

(4.39)

where En is the n-th electric ﬁeld, n(ωn ) is the refractive index of the nonlinear material
for the n-th ﬁeld, ωn is the frequency of the n-th ﬁeld, and PNL
n is the nonlinear part ofthe polarization for the n-th ﬁeld.
In the absence of a nonlinear source term, the solution of the wave equation (4.39) is
E3 (z, t) = A3 ei(k3 z−ω3 t) + h.c.,(4.40)

where A3 is the constant amplitude of the generated ﬁeld and
k3 =

n3 ω3
, n3 = n(ω3 ).
c
44(4.41)

When the nonlinear source term is not large, we can assume that the solution to Equation(4.39) remains in the same form as Equation (4.40), except that A3 becomes a slowly
varying function of z. The nonlinear source term in Equation (4.39) can be written as
follows:
P3 (z, t) = P3 e−iω3 t + h.c., P3 = 4ϵ0 deﬀ E1 E2 ,(4.42)

where deﬀ = χ(2) /2.
Substituting Equation (4.40) and (4.42) into the wave equation (4.39),
 2

d A3
n23 ω32 A3 i(k3 z−ω3 t)
dA3
2− k3 A3 ++ 2ik3e+ h.c.
dz 2
dz
c2
4deﬀ ω32=−
A1 A2 ei(k1 +k2 )z−ω3 t + h.c..
c2(4.43)

As k3 = n3 ω3 /c, the third and fourth terms on the left hand side of Equation (4.43) arecancelled out. Assuming a slowly varying amplitude approximation,
d2 A3
dA3
≪ k3
.
2
dz
dz

(4.44)

dA3
2ideﬀ ω32=
A1 A2 ei∆kz ,
dz
k3 c2

(4.45)

Equation (4.43) iswhere ∆k = k1 + k2 − k3 is called the wavevector mismatch. In the simpliﬁed case where
A1 , A2 are constant,
2ideﬀ ω32 A1 A2
A3 =
k3 c2

Z L
ei∆kz dz =0

2ideﬀ ω32 A1 A2 ei∆kL − 1
.
k3 c2
i∆k

(4.46)The intensity of the generated ﬁeld is calculated from the time averaged Poynting vector
as follows:
I3 = 2n3 ϵ0 c|A3 |2 .

(4.47)

Thus, we obtain
8n3 ϵ0 d2eﬀ ω34 |A1 |2 |A2 |2 L2
sinc2
I3 =
k32 c3



∆kL
2


.

(4.48)

I3 is characterized by the sinc function, which is shown in Figure 4.2. I3 is maximum
when the wavevector mismatch is ∆k = 0.

45

1

k L/2)

0.6

sinc 2(

0.8

0.4

0.2

0
-3

-2

-

0

2

3

k L/2

Figure 4.2: sinc2 (∆kL/2) as a function of the wavevector mismatch.

4.4.1

Phase matching

As we discussed in the previous section, the phase matching is important to maximize thenonlinear eﬀect. The perfect phase matching condition ∆k = 0 isn 1 ω1 n 2 ω2
n3 ω3
+
=
,
c
c
c
ω1 + ω2 = ω3 .

(4.49)(4.50)

This condition is typically diﬃcult to achieve because the refractive index is an increasing function of the frequency in the normal dispersion. To achieve the phase matchingcondition, the following three diﬀerent techniques are commonly used [43]. The ﬁrst two
techniques use the birefringence of the nonlinear material, while the third one can be used
for the non-birefringence material.

Type I phase matching
In type I phase matching, the input ﬁelds have the same polarization, while the generated
ﬁeld has the orthogonal polarization with respect to the input ﬁelds. Phase matching
can be done when the refractive indices of the nonlinear crystal for the two orthogonal
polarizations have diﬀerent values. Fine tuning of the refractive index of the crystal forthe input and generated ﬁelds can be done by changing the incident angle of the input
ﬁelds and the temperature of the crystal.
The second harmonic generator (SHG) in our experiment uses MgO : LiNbO3 as a nonlinear crystal. As the refractive index of MgO : LiNbO3 for the IR ordinary ray is approximately the same as that for the green extraordinary ray: no (1064 nm) ≃ ne (532 nm) ≃2.23 [44], phase matching can be done via Type I phase matching.

46

Type II phase matching
In type II phase matching, the input ﬁelds have orthogonal polarizations, and the generated
ﬁeld has the same polarization as either of the injected ﬁelds.

Quasi phase matching
Instead of using the birefringence of the crystal, the quasi-phase matching ﬂips the sign ofthe nonlinearity of the crystal before the phase mismatch between the input and generated
ﬁelds becomes π/2. The quasi-phase matching can be done by inverting the crystal’s
domain periodically with a short length scale. A phase mismatch in the quasi-phasematching can be written as follows:∆kQPM = k1 + k2 − k3 −2π,
Λ(4.51)

where Λ is the crystal inversion period. With quasi-phase matching, deﬀ is replaced withdQPM = π2 deﬀ . Fine tuning of the phase matching condition can be done by tuning therefractive index of the crystal with the crystal temperature.
The optical parametric oscillator (OPO) in our experiment uses periodically oled potassium titanyl phosphate (PPKTP) as a nonlinear crystal, and phase matching is done via
quasi-phase matching.

4.5

Generation of squeezed vacuum states

Squeezed vacuum states can be generated from OPO, which is an optical cavity and has anonlinear crystal inside. OPO converts the pump photon, whose frequency is 2ω0 , to twocorrelated photons whose frequency is ω0 . In this section, the principle of the generationof squeezed vacuum states with OPO is presented.

4.5.1

Equation of motion of OPO with losses

In this section, the calculation of the equation of motion of OPO with losses is presented
based on [45, 46, 47]. The OPO with losses can be modelled, as shown in Figure 4.3. It
is a triangular cavity with the input, output, and loss couplers. We denote the coupler
j = i, o, l for the input, output, and loss couplers, respectively.
The Hamiltonian of the OPO with loss can be written as [48]H = Hcav + Hint + Hbath ,(4.52)

where Hcav is the Hamiltonian for the cavity modes, Hint is the Hamiltonian for theinteraction between the cavity modes and the external modes that couple through couplerj, and Hbath is the Hamiltonian for the external modes,
Hcav = ℏω0 a† a + ℏ(2ω0 )b† b +

47

iℏ †2
(ϵa b − ϵ∗ a2 b† ),2(4.53)

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Ai,out , Bi,out

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Ao,in , Bo,in

Figure 4.3: Model of OPO. a and b are the intracavity ﬁelds at the fundamental anda,ba,b
second harmonic modes. κa,bi , κo , and κl are the decay rates for the input, output, andloss couplers for the fundamental and second harmonic modes. Ai,in and δBi,in , δAo,in andBo,in , and δAl,in and δBl,in are the fundamental and second harmonic ﬁelds entering thecavity via the input, output, and loss couplers, respectively. Ai,out and Bi,out , Ao,out andBo,out , and Al,out and Bo,out are the fundamental and second harmonic ﬁelds exiting thecavity from the input, output, and loss couplers.

Hint = iℏqi
X Z ∞ hq2κaj (A†j a − a† Aj ) + 2κbj (Bj† b − b† Bj ) dω,j(4.54)−∞

Hbath =

XZ ∞
j

−∞

ℏω(A†j Aj + Bj† Bj )dω,(4.55)

where ϵ is the nonlinear coupling parameter and assumed to be a real number (optimalphase matching). a and b are the intracavity ﬁelds for the fundamental and second harmonic modes, respectively. Aj and Bj are the external ﬁelds coupling toq
the coupler j forthe fundamental and second harmonic modes, respectively. κa,b
j = (1 −Rja,b )/τrt is the

decay rate of the coupler j for the fundamental and second harmonic modes, where Rja,bis the power reﬂectivity of the coupler j for the fundamental and second harmonic modePand τrt = L/c is the cavity round-trip time. The total cavity decay rate κa,b = j κa,bj
a,b
corresponds to the cavity pole 2πfp when the total cavity losses are low.
Assuming that the fundamental coherent ﬁeld is injected from the input coupler, andsecond harmonic coherent ﬁeld is injected from the output coupler which is the case forour experiment, the Langevin equation for the fundamental and second harmonic modes48

will be [45]
ȧ =
ḃ =

p
p
p
1
[a, Hcav ] − κa a + 2κai Ai,in eiω0 t + 2κao δAo,in + 2κal δAl,in ,
iℏ
q
q
q
1
b
i2ω0 tbb[b, Hcav ] − κ b + 2κi δBi,in + 2κo Bo,in e+ 2κbl δBl,in ,iℏ(4.56)(4.57)

where Aj,in and Bj,in are the input external ﬁelds from the coupler j for the fundamental
and second harmonic modes.
In a reference frame co-rotating with the injection coherent ﬁeld,pppȧ = ϵa† b − (κa − i∆a )a + 2κai Ai,in + 2κao δAo,in + 2κal δAl,in ,qqqϵḃ = − a2 − (κb − i∆b )b + 2κbi δBi,in + 2κbo Bo,in + 2κbl δBl,in ,2(4.58)(4.59)a , ∆b = 2ω − ω bwhere ∆a = ω0 − ωcav0
cav are the detuning of the injection frequency withrespect to the cavity resonance.
We assume that the cavity operates on the resonance (∆a,b = 0), constant pump(ḃ = 0), and photon number at the fundamental frequency is small (a2 = 0). In thisassumption,pppȧ = ϵba† − κa a + 2κai Ai,in + 2κao δAo,in + 2κal δAl,in ,qqq0 = −κb b + 2κbi δBi,in + 2κbo Bo,in + 2κbl δBl,in .

(4.60)(4.61)Classical calculation
First, we calculate Equation (4.60) classically. Ignoring the quantum ﬂuctuation terms
and considering the steady state ȧ = 0, Equation (4.60) is
p a
2κi Ai,in ,p
0 = |ϵ||b|e−iϕpump a − κa a† + 2κai A†i,in ,0 = |ϵ||b|eiϕpump a† − κa a +(4.62)(4.63)

where ϕpump is the phase of the pump ﬁeld. The a† term in Equation (4.62) represents
the sideband generated by the nonlinear eﬀect. From Equation (4.62), (4.63) and setting
the input ﬁeld is real (Ai,in = A†i,in ),p a2κi 1 + xeiϕpumpa=Ai,in ,
κa1 − x2(4.64)

where x = |ϵ||b|/κa is the OPO nonlinear factor. The OPO nonlinear factor is the ratioof the round-trip gain to the round-trip loss inside the OPO for the fundamental ﬁeld.
When it is above 1, the fundamental intracavity ﬁeld becomes inﬁnity, and the OPO can
produce the fundamental coherent ﬁeld. For the squeezing measurement, the OPO is49

operated below the threshold. The OPO threshold power Pth can be written as follows:
s
x=

Ppump
,
Pth

(4.65)

where Ppump is the injection pump power.
The parametric gain is the ratio of the transmission power (or intracavity power) of thefundamental ﬁeld with/without the pump, which can be obtained from Equation (4.64):
G(ϕpump ) =1 + x2 + 2x cos ϕpump.
(1 − x2 )2(4.66)The injected fundamental ﬁeld is ampliﬁed or de-ampliﬁed, depending on the pump phase.
In particular, the ampliﬁcation factor
G(ϕpump = 0) =1=g
(1 − x)2(4.67)

is called the nonlinear gain.

Quantum calculation
From Equation (4.60), when there is no coherent ﬁeld at the fundamental frequency,δ ȧ = |ϵ||b|eiϕpump δa† − κa δa +Xq
2κaj δAj,in ,(4.68)jδ ȧ† = |ϵ||b|e−iϕpump δa − κa δa† +Xq
2κaj δA†j,in .

(4.69)j†√
From these equations, we can calculate the amplitude and phase quadrature X1a = a+a2†√ ,and X2a = a−a2i

δ Ẋ1a = (−κa + |ϵ||b| cos ϕpump )δX1a + |ϵ||b| sin ϕpump δX2a +XqA2κaj δX1 j,in ,(4.70)

jδ Ẋ2a = (−κa − |ϵ||b| cos ϕpump )δX2a + |ϵ||b| sin ϕpump δX1a +XqA

2κaj δX2 j,in .(4.71)

jObtaining the Fourier transform,(1 − x cos ϕpump + iΩ/κ )δ X̃1a − x sin ϕpump δ X̃2aa=

(1 + x cos ϕpump + iΩ/κa )δ X̃2a − x sin ϕpump δ X̃1a =

X

κa
j
p
X 2κaj
j

50

p a
2κj

κa

A

δ X̃1 j,in ,Aδ X̃2 j,in .

(4.72)(4.73)

From Equation (4.72) and (4.73),
δ X̃1aδ X̃2a!
=×1(1 + x + iΩ/κa )(1 − x + iΩ/κa )1 + x cos ϕpump + iΩ/κax sin ϕpump
x sin ϕpump
1 − x cos ϕpump + iΩ/κa!  P √2κa


j


A
δ X̃1 j,in 

j √κa
2κa
Aj,in
j
j κa δ X̃2

P.

(4.74)The boundary condition for the output coupler is [45]Ao,in + Ao,out =p2κao a.

(4.75)

Converting this to the two-photon formalism and obtaining the Fourier transform,Aδ X̃1,2o,out =pAa2κao δ X̃1,2− δ X̃1,2o,in .

(4.76)The variance of the amplitude and phase quadrature for the transmission ﬁeld can betrans = ⟨0| |δ X̃ Ao,out |2 |0⟩. Using ⟨0| δ X̃ i δ X̃ †j |0⟩ = δ δcalculated as V1,2m
ij nm , where i, j =n1,2Ai,in , Ao,in , Al,in and n, m = 1, 2, the variance of the amplitude and phase quadrature for
the OPO transmission are
2x + (1 + x2 + (Ω/κa )2 ) cos ϕpump
1
+
4xη
esc
! 
((1 − x)2 + (Ω/κa )2 )((1 + x)2 + (Ω/κa )2 ) 
V1trans,=V2trans2a2
2x − (1 + x + (Ω/κ ) ) cos ϕpump
1 + 4xηesc((1 − x)2 + (Ω/κa )2 )((1 + x)2 + (Ω/κa )2 )(4.77)

trans can be written in terms of
where ηesc = κao /κa is called the OPO escape eﬃciency. V1,2
V± as follows:

V1trans
V2trans!

ϕ=cos2 pump2
ϕsin2 pump2

ϕsin2 pump2
ϕcos2 pump
2

!

V+
V−

!
,

(4.78)

where4x,(1 − x)2 + (Ω/κa )24x= 1 − ηesc.
2
(1 + x) + (Ω/κa )2

V+ = 1 + ηesc

(4.79)

V−(4.80)

V+ corresponds to the anti-squeezing quadrature and V− corresponds to the squeezing
quadrature.
When Ω/κa ≪ 1, the variances of the squeezing and anti-squeezing quadrature V± are
V± = 1 ± 4ηescx.
(1 ∓ x)251(4.81)

From Equation (4.81), the nonlinear gain g is related to the generated squeezing level
σdB without losses (ηesc = 1) as follows:
σdB = −10 log10 V− = −20 log10

4.6

1−x
√
= 20 log10 (2 g − 1).
1+x(4.82)

Balanced homodyne detection

To measure the squeezed vacuum states, balanced homodyne detection is widely used, as
shown in Figure 4.4. The local oscillator and the squeezed vacuum states are combinedwith the beam splitter, and the two outputs from the beam splitter are detected withphoto detectors, and the diﬀerence of the two outputs is extracted.

b
c

a
d

Figure 4.4: Conﬁguration of balanced homodyne detection. a, b are input ﬁelds for BS andc, d are output ﬁelds for BS. The output ﬁelds from BS are detected with photo detectors
and then subtracted.

The input-output relation of the beam splitter is
c
d

!

1
=√
2

1 1
1 −1

!

a
b

!
.

(4.83)The photocurrent at each PD is1
Ic ∝ c† c = (a† a + a† b + b† a + b† b),2
1†
Id ∝ d d = (a† a − a† b − b† a + b† b).
2(4.84)(4.85)

When the two signals are balanced, the diﬀerence of these signals is
Idiﬀ = Ic − Id ∝ a† b + b† a.

52(4.86)

When a is the squeezed vacuum ﬁled and b is the local oscillator (LO),a = δaeiϕCC , b = (b + δb)eiϕLO ,(4.87)

where b is the DC amplitude of the LO and δa and δb are ﬂuctuation terms. ϕCC and ϕLO
are the phase of the squeezed vacuum ﬁeld and LO, respectively. ϕCC can be tracked by
the coherent control (CC) ﬁeld, which is presented in the next section.
Ignoring the second order of the ﬂuctuation terms, Equation (4.86) is
Idiﬀ∝ b(δa† ei(ϕLO −ϕCC ) + δae−i(ϕLO −ϕCC ) )√=2b(δX1 cos (ϕLO − ϕCC ) + δX2 sin (ϕLO − ϕCC )).

(4.88)

Normalizing the equation above by the LO power, the variance of the squeezed vacuum
ﬁeld measured by the homodyne detector is
Vmeas = cos2 (ϕLO − ϕCC )V1 + sin2 (ϕLO − ϕCC )V2 .

(4.89)

From Equation (4.78), the amplitude quadrature δX1 is rotated from the anti-squeezing
quadrature δX+ by ϕpump /2. Therefore, we can express the measured quadrature as
follows:


!
ϕpump
ϕpump

 cos
− sin
δX


+
2
2
δXmeas = cos (ϕLO − ϕCC ) sin (ϕLO − ϕCC ) 
ϕpump
ϕpump  δX− .
sin
cos
2
2
(4.90)The variance of the measured quadrature is
Vmeas = cos2 (ϕsqz )V+ + sin2 (ϕsqz )V− ,(4.91)

where
ϕsqz = ϕLO − ϕCC −

ϕpump
2

(4.92)

is called the squeezing angle. ϕsqz = 0 corresponds to the anti-squeezing quadrature andϕsqz = π/2 corresponds to the squeezing quadrature. In this thesis, we deﬁne the homodyne angle as ϕhom = π/2 − ϕsqz . By changing the squeezing angle ϕsqz , any quadratureof the squeezed vacuum ﬁled can be measured. When we want to maintain the same
quadrature, ϕsqz has to be controlled.

4.7

Coherent control

Coherent control is a technique used to control the squeezing angle [49]. The coherent
control ﬁeld, whose frequency is ω0 + Ωcc , is injected to the OPO and copropagates with53

the squeezed vacuum ﬁeld. The coherent control ﬁeld can track the phase of the squeezed
vacuum ﬁeld. The coherent control is composed of two coherent control loops (referred to
as CC1 and CC2 in this thesis). ϕpump is maintained constant by CC1, and the relative
phase between LO and CC (ϕLO − ϕCC ) is maintained constant by CC2 to make thesqueezing angle ϕsqz constant.
For simplicity, we assume that detuning of the CC ﬁeld inside the OPO is signiﬁcantly
smaller than the OPO bandwidth Ωcc ≪ κa . When the CC ﬁeld is injected from the input
port of the OPO, as shown in Figure 4.3, the CC ﬁeld inside the OPO is from Equation(4.62) and (4.63),p a
2κi Acc  i(ω0 +Ωcc )ti(ω0 −Ωcc )t+iϕpump
acav,cc =e+xe,(1 − x2 )κa(4.93)

where Acc is the amplitude of the CC ﬁeld before the OPO. In this formula, the reference
frame is not co-rotating with eiω0 t . Owing to the nonlinear eﬀect of the OPO, an additional
sideband with a frequency of ω0 − Ωcc and phase of ϕpump is generated.

CC1 error signal
The CC1 error signal is obtained by detecting the beat note of the two CC sidebands(CCSB) at the OPO reﬂection. The reﬂected CCSB ﬁelds from the OPO can be calculated
from the boundary condition of the input port as follows:p a2κ acav,cc − Ain,cc i a2κai Acc x i(ω0 −Ωcc )t+iϕpump
2κii(ω0 +Ωcc )t
−
1
A
e
+
e
. (4.94)=cc(1 − x2 )κa(1 − x2 )κaAreﬂ,cc =The beat note of the CCSB at the OPO reﬂection is2κai
4A2cc κai x
PCC1 =−
1cos (2Ωcc t − ϕpump ).
(1 − x2 )κa (1 − x2 )κa(4.95)

Demodulating this signal with cos (2Ωcc + ϕdm,CC1 ) and low-passing it, the CC1 error
signal is
2A2cc κai x
PCC1 =(1 − x2 )κa




2κai
− 1 cos (ϕpump + ϕdm,CC1 ).
(1 − x2 )κa(4.96)As the CC1 error signal is locked at the point where the CC1 error signal (4.96) is 0,
ϕpump = −ϕdm,CC1 +π.
2(4.97)

ϕpump can be ﬁxed to any value by changing the CC1 demodulation phase ϕdm,CC1 .

54

CC2 error signal
The CC2 error signal is obtained from the beat note of CCSB and LO at the homodyne
detector. The CCSB transmit the OPO and reach the homodyne detector with squeezed
vacuum states. When the phase of the CCSB between the OPO and homodyne detector
is ϕCC , the CCSB at the homodyne detector are
iϕCC

e

p

p
2 κai κao Acc  i(ω0 +Ωcc )t+iϕCC
i(ω0 −Ωcc )t+i(ϕCC +ϕpump )a
2κo acav,cc =e+xe. (4.98)(1 − x2 )κa

From Equation (4.86), the beat signal of CCSB and LO (ALO eiω0 t+iϕLO ) at the homodyne
detector isp
4 κai κao Acc ALO
PCC2 ={cos (Ωcc t − ϕLO + ϕCC ) + x cos (Ωcc t + ϕLO − ϕCC − ϕpump )}(1 − x2 )κap
4 κai κao Acc ALO={cos (Ωcc t − ϕsqz − ϕpump /2) + x cos (Ωcc t + ϕsqz − ϕpump /2)}.
(1 − x2 )κa(4.99)
Considering the cosine terms in Equation (4.99),
PCC2 ∝ cos (Ωcc t − ϕsqz − ϕpump /2) + x cos (Ωcc t + ϕsqz − ϕpump /2)= {cos (ϕsqz + ϕpump /2) + x cos (ϕsqz − ϕpump /2)} cos Ωcc t+ {sin (ϕsqz + ϕpump /2) − x sin (ϕsqz − ϕpump /2)} sin Ωcc t
q=
1 + x2 + 2x cos (2ϕsqz ) cos (Ωcc t + ϕCC2 ),(4.100)

where
ϕCC2 = arctan− sin (ϕsqz + ϕpump /2) + x sin (ϕsqz − ϕpump /2)cos (ϕsqz + ϕpump /2) + x cos (ϕsqz − ϕpump /2)

.

(4.101)

Demodulating Equation (4.100) by cos (Ωcc t + ϕdm,CC2 ) and low-passing it, the CC2 error
signal ispp
2 κai κao Acc ALO 1 + x2 + 2x cos (2ϕsqz )PCC2 =cos (ϕCC2 − ϕdm,CC2 ).
κa
1 − x2(4.102)As the CC2 error signal is locked at the point where the CC2 error signal (4.102) is 0,
ϕdm,CC2 = ϕCC2 +π.
2(4.103)The squeezing angle ϕsqz , which is included in ϕCC2 , can be ﬁxed to any value by changing
the CC2 demodulation phase ϕdm,CC2 .

55

CC2 demodulation phase and squeezing angle
The relationship between the squeezing angle and the CC2 demodulation phase is linear
without a nonlinear eﬀect (x = 0), however, it becomes nonlinear with a nonlinear eﬀectof CCSB. The relationship between the CC2 demodulation phase and the squeezing angle(4.103) with ϕpump = 0 is shown in Figure 4.5. The squeezing angle becomes sensitive to
the CC2 demodulation phase around the anti-squeezing quadrature (ϕsqz = 0, π) owing to
the nonlinear eﬀect of CCSB.
180

CC2 demodulation phase [deg]

x=0
x=0.8

135

90

45

0

0

45

90

135

180

Squeezing angle [deg]Figure 4.5: Relation between the CC2 demodulation phase and squeezing angle withϕpump = 0 for x = 0 and 0.8.

4.8

Squeezing degradation sources

In the experiment, there are several sources which degrade the achievable squeezing level.
In this section, the main squeezing degradation sources, including losses, classical noise,and phase noise, are presented.

4.8.1

LossesWhen there are losses in the squeezing path, the squeezed vacuum ﬁeld is lost, and theexternal vacuum ﬁeld comes in from the loss port, and the squeezing level is degraded.
The squeezing and anti-squeezing levels with losses l are
V±l = (1 − l)V± + l,

56

(4.104)

where V± are the squeezing and anti-squeezing levels without any losses,
V± = 1 ± 4x.
(1 ∓ x)2(4.105)

From Equation (4.104) and (4.105), the squeezing and anti-squeezing noise levels withlosses can be written as follows:x,(1 + x)2x= 1 + 4η,(1 − x)2

V−l = 1 − 4η

(4.106)V+l(4.107)

where η = 1 − l is the total eﬃciency, which can be divided into η = ηesc ηloss ηvis ηPD
where ηesc is the escape eﬃciency of OPO, ηloss is the propagation eﬃciency, ηvis is theeﬃciency of the visibility of the LO and squeezed ﬁeld, and ηPD is the eﬃciency of thephoto detectors.
The squeezing and anti-squeezing levels as a function of losses are shown in Figure 4.6.
15

5 dB
10 dB
15 dB

Squeezing ratio (dB)

10
5
0
-5
-10
-15

0

10

20

30

40

50

60

70

80

90

100

loss (%)

Figure 4.6: Squeezing and anti-squeezing levels as a function of losses for 5, 10, and 15 dB
generated squeezing. The squeezing level is degraded more than the anti-squeezing levelby losses.

4.8.2

Classical noise

As the classical noise in the detected signal, such as dark noise of the electronics, increases
both vacuum noise (quantum noise without squeezing) and squeezing noise levels, theclassical noise can be considered as an optical loss [50, 51]. The squeezing level with the57classical noise is
obs
Vsqz
obs
Vvac
Vsqz + Vclass=
V + Vclass
vac

Vsqz
VclassVclass=1−+
Vvac + Vclass Vvac Vvac + Vclass
VclassVclass=1 − obs V− + obs ,
Vvac
Vvac

V−obs =

(4.108)obs and V obs are the observed squeezing ratio, observed squeezing noise
where V−obs , Vsqz
vaclevel, and observed vacuum noise level, respectively. Vclass is the classical noise level. This
formula has the same form as Equation (4.104) and the eﬀect of the classical noise is theobs (ratio of the classical noise and observedsame as the eﬀect of the losses l = Vclass /Vvac
vacuum noise in the power spectral density).

4.8.3

Phase noise

The squeezing angle ﬂuctuation (phase noise) couples the anti-squeezing quadrature to thesqueezing quadrature and degrades the squeezing level. The squeezing and anti-squeezing
levels with the phase noise are
V±θ (t) = V± cos2 θ(t) + V∓ sin2 θ(t).

(4.109)Although the squeezing angle can be controlled by the coherent control, there are residual
ﬂuctuations of the squeezing angle. If we model the ﬂuctuations as normally distributed
around θ = 0 with a small rms ﬂuctuation θrms (< 100 mrad), we can write the averaged
variance of the squeezing and anti-squeezing quadrature as follows:
V±θrmsZ ∞e−θ /2θrms=dθ √(V± cos2 θ + V∓ sin2 θ)2πθrms−∞≃ V± cos2 θrms + V∓ sin2 θrms .
2

2(4.110)The squeezing and anti-squeezing levels as a function of the phase noise are shown in
Figure 4.7.

4.8.4

Combination of squeezing degradation sources

From Equation (4.104) and (4.110), the squeezing and anti-squeezing levels with the loss
and phase noise are
V±l,θrms = (1 − l)(V± cos2 θrms + V∓ sin2 θrms ) + l

58(4.111)

20

Squeezing ratio (dB)

15
10
5
0
-5
-10
5 dB
10 dB
15 dB

-15
-20
100

101

102

phase noise (mrad)

Figure 4.7: Squeezing and anti-squeezing level as a function of phase noise for 5, 10, and15 dB generated squeezing. The squeezing level is degraded by the phase noise, while theanti-squeezing level nearly does not degrade.

From Equation (4.105) and (4.111), the squeezing level with the loss and phase noise canbe written as a function of x as follows: 2
sin θrms cos2 θrms
l,θrms= 1 + 4ηxV−−.
(4.112)(1 − x)2(1 + x)2
When the generated squeezing level is increased, the squeezing level is increased, but
the anti-squeezing level is also increased, which decreases the squeezing level due to phasenoise. Therefore, there is an optimal generated squeezing level for the maximum achievablesqueezing level with loss and phase noise, which can be obtained by solving dV−l,θrms /dx = 0
where
dV−l,θrms(1 + 6x2 + x4 ) cos 2θrms − 4x(1 + x2 )
= −4η
.
dx
(1 − x2 )3(4.113)The optimal value for x is obtained as follows [46]:
x=1−2

p
3/2
2
θrms + 2θrms − 2θrms+ 2θrms.

(4.114)The optimal generated squeezing for a maximum achievable squeezing level depends on
only phase noise.
Although the maximum achievable squeezing level with loss and phase can be obtainedby substituting Equation (4.114) into Equation (4.112), it is easier to obtain it from
Equation (4.111) without using x. From Equation (4.111) and V+ V− = 1, the squeezing
level with loss and phase noise can be written as a function of the anti-squeezing level as59

follows:
V−l,θrms( l,θrms
V+−l1=
tan θrms +2
1−ltan θrms
v

u
2
u
1
1
t tan θ++ tan θrms −rmstan θrms
tan θrms
1−l22

V+l,θrms − l

!2

1−l




− 4 + l,
(4.115)where θrms ≪ 1 is assumed. It is a general formula between the squeezing level and antisqueezing levels with the loss and phase noise and is independent of the generated squeezing
level. From Equation (4.115), the maximum achievable squeezing level for a given loss andphase noise can be obtained. By solving dV−l,θrms /dV+l,θrms = 0, the maximum achievablesqueezing level and the corresponding anti-squeezing level are derived as follows:2(1 − l)+ l,tan θrms + tan 1θrms

V−l,θrms=

V+l,θrms= (1 − l)tan2 θrms + tan21θrms
tan θrms + tan 1θrms(4.116)(4.117)+ l.

The maximum achievable squeezing level (4.116) as a function of loss and phase noise isshown in Figure 4.8.

18

1

2
3

16

4

total loss (%)

5

14

6

12

7
8

10

1

9

10

10

11

8

12

13

6

14
15

4

16

maximum achievable squeezing level (dB)

102

2
0

10 0
10

101

102

phase noise (mrad)
Figure 4.8: Maximum achievable squeezing level with losses and phase noise.

60

Chapter 5

Quantum Noise in
Gravitational-Wave Detector
Quantum noise in GW detectors originates from the vacuum ﬁeld entering the output
port of the interferometer, as shown in Figure 5.1. Injecting the squeezed vacuum ﬁeld
instead of the vacuum ﬁeld is a key for reducing the quantum noise. In this chapter, abasic summary of the quantum noise and how the squeezed vacuum ﬁeld can improve thequantum noise are presented.

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a1

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Figure 5.1: Injection of vacuum ﬁeld in GW detectors.

61

5.1

Quantum noise

The relationship between the injected vacuum ﬁeld a and the vacuum ﬁeld reﬂected fromthe interferometer b is [14]
b1
b2

!
=e

i2β

1 0
−K 1

!

a1
a2

!

0
√
2K

iβ

+e

!

h
,
2hSQL

(5.1)

where K is the optomechanical coupling factor of the interferometer, h is GW strain signal,
hSQL is the standard quantum limit, and β is the phase shift from the interferometer,



2
γifo
ΩSQL 2K =2 ,ΩΩ2 + γifos8ℏhSQL =,
mΩ2 L2armΩ
β = arctan
,
γifo

(5.2)
(5.3)
(5.4)

where ΩSQL is the approximate frequency at which the quantum noise reaches the standard
quantum limit and γifo is the interferometer bandwidth,ΩSQL =
γifo =

rtsr 8 Parm ω0,
1 + rsr c mTarm
1 + rsr Tarm c
1 + rsr=γarm .
1 − rsr 4Larm
1 − rsr

(5.5)
(5.6)

Here, tsr and rsr are the amplitude transmissivity and reﬂectivity of the signal recycling
mirror, respectively. Parm is the arm intracavity power, m is the test mirror mass, Larm isthe arm cavity length, and ω0 is the carrier frequency. These parameters in KAGRA aresummarized in Table 5.1.
Table 5.1: KAGRA parameter
Parameter
SymbolCarrier ﬁeld frequency
ω0Standard quantum limit frequencyΩSQLInterferometer bandwidthγifo
Arm input mirror transmissivity
Tarm
Signal recycling input transmissivity
t2sr
Arm intracavity power
Parm
Mirror massm
Arm cavity length
Larm

Value
2π × 283 THz
2π × 76.4 Hz
2π × 382 Hz
0.004
0.1536400 kW
22.8 kg
3 km

The GW signal is included in b2 in Equation (5.1):√
b2 = ei2β (a2 − Ka1 ) + eiβ 2K

62

h
.
2hSQL

(5.7)

The ﬁrst term is the quantum noise, and the second term is the GW signal. From thisformula, the quantum noise in terms of the GW strain will behSQL
h = √ eiβ (a2 − Ka1 ).
2K

(5.8)

The power spectral density of quantum noise in GW detectors can be written as follows:
Sh =

h2SQL
2



1
+K
K


≧ h2SQL .

(5.9)

Strain [1/ Hz]The ﬁrst term is shot noise, and the second term is the radiation pressure noise. Quantum√
noise in terms of strain Sh is shown in Figure 5.2. Quantum noise is dominated by shotnoise at high frequencies and dominated by radiation pressure noise at low frequencies.

10

-22

10

-23

shot noiseradiation pressure noise
quantum noise

10-24 1
10

10

2

10

3

Frequency [Hz]Figure 5.2: Quantum noise in GW detectors. In this ﬁgure, the KAGRA parameters
shown in Table 5.1 are assumed.

There is a tradeoﬀ between shot noise and radiation pressure noise in terms of laser
power, and the quantum noise cannot surpass the standard quantum limit (SQL) hSQL ,as shown in Figure 5.3. When K = 1, which corresponds to Ω ≃ ΩSQL with Ω ≪ γifo , thequantum noise reaches the SQL.

Ponderomotive squeezing
Quantum noise in GW detectors can be better understood by considering the amplitude
and phase picture, as shown in Figure 5.4. The vacuum ﬁeld injected into the interferometer is reﬂected by the SRM without entering the interferometer at high frequencies, while at
low frequencies, the vacuum ﬁeld enters the interferometer, and the amplitude ﬂuctuationof the vacuum induces test mass ﬂuctuations through optomechanical coupling, leading to63

10-22Standard Quantum Limit
Parm = 400 kW
Parm = 1600 kW

Strain [1/ Hz]

Parm = 100 kW

10

-23

10-24 1
10

10

2

10

3

Frequency [Hz]Figure 5.3: Quantum noise in GW detectors with various arm intracavity powers and thestandard quantum limit.

the phase ﬂuctuation. This is the origin of the radiation pressure noise. The output fromthe interferometer is squeezed at low frequencies, which is called ponderomotive squeezing.

Input

Outputhigh frequency

a2

phase
a
fluctuation 2

low frequency

gravitational
wave signal

a2
a1

a1

a1
- Ka1

amplitude
fluctuation

phase fluctuation frommirror motion caused byamplitude fluctuation

Figure 5.4: Ponderomotive squeezing.

5.2

Frequency-independent squeezing

To reduce the quantum noise, it was suggested to inject the squeezed vacuum states tothe interferometer instead of the vacuum states as shown in Figure 5.5 [9].

64

frequency
phaseamplitude

Figure 5.5: Injection of frequency-independent squeezing in GW detectors.

Quantum noise with frequency-independent squeezing can be expressed as [14]Ssqz =

h2SQL
2




1
+ K (cosh 2r − sinh 2r cos [2(Φ − ϕsqz )]),
K

(5.10)

where r is the injection squeezing factor, ϕsqz is the injection squeezing angle, and Φ =arccotK.
When the injection squeezing angle ϕsqz = π/2 (phase squeezing), Equation (5.10) is1
Ssqz (ϕsqz = π/2) =

h2SQL
2




1
2r
+ Ke
.
Ke2r

(5.11)

The formula above indicates that the shot noise is reduced by a factor of e2r , while theradiation pressure noise is increased by the same factor. The tradeoﬀ between shot noiseand radiation pressure noise remains, and the SQL cannot be surpassed with the phasesqueezing. Shot noise reduction with phase squeezing was achieved in Advanced LIGO
and Advanced Virgo in O3 [10, 11]. Furthermore, the increase in radiation pressure noisewith frequency-independent squeezing was observed in Advanced LIGO and Advanced
Virgo [12, 13].

1

Here we used, cos2Φ = cos2 Φ − sin2 Φ =K2
1
1 − K2
−
=−
.
2
2
1+K
1+K
1 + K265

5.3

Frequency-dependent squeezing

To reduce the shot noise and radiation pressure noise simultaneously, frequency-dependent
squeezed vacuum states, which are phase squeezed at high frequencies and amplitude
squeezed at low frequencies, are required. When the injection squeezing angle has afrequency dependence, such as ϕsqz (Ω) = Φ(Ω) = arccotK(Ω), the quantum noise withfrequency-dependent squeezing (5.10) can be written as
Ssqz =

h2SQL
2




1
+ K e−2r .
K(5.12)

Strain [1/ Hz]The formula above indicates that both of the shot noise and the radiation pressure noise
can be reduced by a factor of e−2r .
A comparison of the quantum noise with the vacuum, frequency-independent squeezing, and frequency-dependent squeezing are shown in Figure 5.6.

10

-22

10

-23

10

-24

Standard Quantum Limitno squeezing
phase squeezing
amplitude squeezing
frequency-dependent squeezing

101

102

103

Frequency [Hz]Figure 5.6: Quantum noise with vacuum, frequency-independent squeezing, and frequencydependent squeezing.

5.4

Filter cavity

The most promising method to realize the frequency-dependent squeezing is by reﬂecting
the frequency-independent squeezed vacuum ﬁeld oﬀ a Fabry-Perot cavity, called the ﬁlter
cavity [14]. The optical conﬁguration of the frequency-dependent squeezing with the ﬁlter
cavity is shown in Figure 5.7.

66lter cavity
frequency
phase
amplitudefrequency
phase

amplitude

Figure 5.7: Optical conﬁguration of frequency-dependent squeezing with the ﬁlter cavity.
The squeezing angle rotates by 90◦ at low frequencies with the ﬁlter cavity, while thesqueezing angle does not change at high frequencies.

The frequency-independent squeezed ﬁeld is reﬂected by the ﬁlter cavity, acquiring afrequency dependence. This mechanism can be better understood in a sideband picture,
where the squeezing angle at each frequency is determined by the average phase between
the upper and lower sidebands, as shown in Figure 5.8. If the cavity is operating at a
detuned conﬁguration, the symmetric sidebands have a diﬀerential phase change dependingon the cavity dispersion, resulting in a rotation of the squeeze ellipse in the quadratureplane.
Here, we theoretically derive the squeezing angle rotation by the ﬁlter cavity based on[52]. From Equation (3.18), the complex ﬁlter cavity reﬂectivity for a sideband frequencyΩ can be written as follows:
rfc (Ω) = rin −t2in rrt e−iΦ(Ω),
rin 1 − rrt e−iΦ(Ω)(5.13)

where rin is the amplitude reﬂectivity of the input mirror and rrt is the cavity’s round-trip
amplitude reﬂectivity. The cavity round-trip phase Φ(Ω) isΦ(Ω) = (Ω − ∆ωfc )

2Lfc
,
c

(5.14)

where ∆ωfc = ωfc − ω0 is the ﬁlter cavity detuning with respect to the carrier frequency
ω0 and Lfc is the ﬁlter cavity length.
For a high-ﬁnesse cavity near the resonance, we can assume thate−iΦ(Ω) ≃ 1 − iΦ(Ω),
67(5.15)

Normalized sideband frequency: 0
Input: 0

Output: 90

Normalized sideband frequency: 2
Input: 0

Output: 45

-45
-225

Normalized sideband frequency: 3
Input: 0

Output: 12.5

-28
-307

Figure 5.8: Squeezing angle rotation by the ﬁlter cavity. The red, orange, and blue arrows
represent the carrier, upper sideband, and lower sideband, respectively. The left threeplots represent the normalized intracavity power of the ﬁlter cavity√and the phase of theﬁlter cavity reﬂectivity for the normalized sideband frequency 0, 2, 3. The horizontal
axis represents the sideband frequency normalized with respect to ∆ωfc,0 .

andq
rrt ≃ rin ≃ 1 − t2in − Λ2rt ≃ 1 − (t2in + Λ2rt )/2,

(5.16)

where Λ2rt is the round-trip loss in the cavity (not including the input mirror transmission).
Under these assumptions, the complex reﬂectivity of the ﬁlter cavity (5.13) can be
written as follows:
rfc (Ω) ≃ 1 −2−ϵ,
1 + iξ(Ω)(5.17)whereϵ =ξ(Ω) =

fFSR 2Λ ,γfc rtΩ − ∆ωfc,
γfc

with fFSR is the FSR of the ﬁlter cavity and γfc is the ﬁlter cavity half-bandwidth.
68(5.18)(5.19)

The amplitude and phase of the ﬁlter cavity reﬂectivity can be written as follows:sρfc (Ω) = |rfc (Ω)| =1−(2 − ϵ)ϵ,1 + ξ 2 (Ω)(5.20)

αfc (Ω) = arg{rfc (Ω)} = arg{−1 + ϵ + ξ 2 (Ω) + i(2 − ϵ)ξ(Ω)}.

(5.21)We deﬁne the sum and diﬀerence of the amplitude and phase of the ﬁlter cavity reﬂectivity
for the symmetric sidebands as follows:
ρ± = ρfc (±Ω), α± = αfc (±Ω),ρ+ ± ρ−α+ ± α−ρpm =, αpm =.
2
2

(5.22)The transfer matrix of the ﬁlter cavity in the two-photon formalism is
r+ 0
∗
0 r−

Tfc = A2 ·

!
· A−1
2 ,

(5.23)

where r± = rfc (±Ω) and A2 is the conversion matrix from one-photon to the two-photon
formalism,
1 1
−i i

1
A2 = √
2!
.

(5.24)

From Equation (5.22) and (5.23), the transfer matrix of the ﬁlter cavity is
Tfc = eiαm Rαp (ρp I − iρm Rπ/2 ),(5.25)

where Rθ is the 2 × 2 rotation matrix with the rotation angle θ and I is the 2 × 2 identity
matrix. The ﬁrst term represents the sidebands’ phase rotation by αp , which induces thesqueezing angle rotation. The second term represents the eﬀect of the ﬁlter cavity losses.
From (5.21) and (5.22), the squeezing rotation angle by the ﬁlter cavity isαp (Ω) ≃ arctan(2 − ϵ)γfc ∆ωfc
2 − ∆ω 2 + Ω2 ,(1 − ϵ)γfc
fc(5.26)

which holds for the typical ﬁlter cavity parameters, ϵ ≪ 1 or Λ2rt ≪ t2in .
As the frequency region where the squeezing angle rotates is smaller than the interferometer bandwidth Ω ≪ γifo , Equation (5.2) can be approximated as follows:

K≃

ΩSQL
Ω

2
.

(5.27)The required squeezing angle rotation by the ﬁlter cavity isαp ≃ arctan

ΩSQL
Ω

69

2
.

(5.28)

From Equation (5.26) and (5.28), the optimal ﬁlter cavity bandwidth and detuning areγfc√

1 − ϵγfc ,
s
ΩSQL
2
√
√ .
=
(2 − ϵ) 1 − ϵ2∆ωfc,0 =(5.29)(5.30)In the case of KAGRA, ΩSQL = 76 Hz and the ﬁlter cavity half bandwidth should be√γfc ∼ ΩSQL / 2 = 54 Hz. This requires a long and high ﬁnesse ﬁlter cavity.
From Equation (5.18) and (5.30), we can solve for ϵ as follows:
s

ϵ=
2+

4
r

2+2

1+



2ΩSQL
fFSR Λ2rt

4.

(5.31)

From this formula, the eﬀect of the ﬁlter cavity losses ϵ is determined by Λ2rt /Lfc once
ΩSQL is ﬁxed. When we have a longer ﬁlter cavity, the ﬁnesse of the ﬁlter cavity will belower to achieve the same half bandwidth of the ﬁlter cavity, and therefore, the eﬀect of
the ﬁlter cavity losses decreases. As these ﬁlter cavity losses are one of the most dominantsqueezing degradation sources, LIGO and Virgo plan to install a 300 m ﬁlter cavity from
O4.

5.5

Analytical model of squeezing degradation sources

In this section, an analytical model of the squeezing degradation sources and a calculation of quantum noise with the ﬁlter cavity based on [52] is introduced. The squeezing
degradation sources in GW detectors with the ﬁlter cavity are shown in Figure 5.9. There
are mainly three degradation sources: loss, phase noise, and mode mismatch. The loss iscomposed of extra-cavity/intracavity loss of the ﬁlter cavity. The extra-cavity loss can bedivided into the injection/readout loss of the interferometer. The phase noise is composedof an extra-cavity/intracavity phase ﬂuctuation. The mode mismatch is composed of amismatch between the squeezer/ﬁlter cavity and the squeezer/local oscillator.
To calculate the quantum noise with the squeezing degradation sources, we divide thequantum noise into the following three parts: noise due to vacuum ﬂuctuations passing
through the squeezer N1 , noise due to vacuum ﬂuctuations that do not pass throughsqueezer N2 , and noise due to vacuum ﬂuctuations in the readout N3 .
Quantum noise at the interferometer readout is given by the following:
N (ζ) = |bζ · T1 · v1 |2 + |bζ · T2 · v2 |2 + |bζ · T3 · v3 |2≡ N1 + N2 + N3 ,(5.32)√
where vi = 2ℏω0 I (i = 1, 2, 3) is vacuum ﬂuctuation and I is 2 × 2 identity matrix. bζ =ALO (sin ζ cos ζ) is local oscillator and N (ζ = 0) is the quantum noise in the quadrature
containing the interferometer signal.
70

Figure 5.9: Squeezing degradation sources with the ﬁlter cavity in the GW detectors [52].

T1 is written as
T1 = τro Tifo (T00 Tfc + Tmm )Tinj ,!
1 0
Tifo =,−K 1(5.33)

T00 = |t00 |Rarg(t00 ) , Tmm = |tmm |Rarg(tmm ) ,!
eσ0Tinj = τinj Rϕ
R−ϕ ,
0 e−σ

(5.35)(5.34)(5.36)

where Tifo , Tfc , and Tinj are transfer matrices of interferometer, optimally matched ﬁlter
cavity, and injection ﬁeld, respectively. T00 and Tmm are transfer matrices of 00 modeand higher order modes, respectively.
τinj , τro in Equation (5.33) and (5.36) are the injection and readout transmissivity,
respectively, and can be written asq
1 − Λ2inj ,p=
1 − Λ2ro ,τinj =(5.37)

τro(5.38)

where Λ2inj and Λ2ro are injection and readout losses, respectively.
P∗
t00 , tmm in Equation (5.35) can be written as t00 = a0 b∗0 , tmm = ∞n=1 an bn where
an , bn are complex coeﬃcients when we express the squeezed vacuum states and the local
oscillator in the basis of the ﬁlter cavity mode and can be written as
Usqz =
Ulo =

∞
X
n=0
∞
X

an Un , withbn Un , with

n=0

∞
X
n=0
∞
X
n=0

71

|an |2 = 1,

(5.39)|bn |2 = 1,

(5.40)

where Un are the orthogonal basis of spatial modes and U0 is the ﬁlter cavity fundamentalmode.
σ and ϕ in Equation (5.36) are the injection squeezing factor and injection squeezing
angle.
T2 is written as
T2 = τro Tifo Λ2 ,p
Λ2 =
1 − (|τ2 (Ω)|2 + |τ2 (−Ω)|2 )/2,

(5.41)(5.42)

τ2 (Ω) = (t00 rfc (Ω) + tmm )τinj .

(5.43)

T3 is written as
T3 = Λro .

(5.44)

Quantum noise normalized with respect to shot noise level is
N̂ =

N
.
2ℏω0 A2LO(5.45)

Phase noise
Phase noise is the squeezing angle ﬂuctuation that degrades the squeezing level by coupling
to anti-squeezing. Phase noise in the ﬁlter cavity can be divided into two components:
extra-cavity ﬂuctuations (i.e., frequency-independent phase noise) and intracavity ﬂuctuations (i.e., frequency-dependent phase noise).
Assuming multiple incoherent noise parameters Xn in quantum noise N̂ have small
Gaussian-distributed ﬂuctuations with a variance δXn2 , the average readout noise is givenbyN̂avg ≃ N̂ +XnN̂ (Xn + δXn ) + N̂ (Xn − δXn )
− N̂
2

!
.

(5.46)For frequency-independent phase noise, Xn = ϕ, which is the injection squeezing angle.
For frequency-dependent phase noise, Xn = ∆ωfc , which is the ﬁlter cavity detuning. This
detuning noise results from ﬁlter cavity length noise δLfc , driven by seismic motion of thecavity mirrors or sensor noise of the ﬁlter cavity length control loop, according toδ∆ωfc =

ω0
δLfc .
Lfc(5.47)First, the calculation of frequency-independent phase noise is presented. Frequencyindependent phase noise can be represented by variations of injection squeezing angle,
δϕ.
To calculate the eﬀect of only phase noise, we restrict our discussion to an optimally matched ﬁlter cavity and without injection and readout losses. From Equation(5.32)-(5.38) and (5.45), quantum noise owing to vacuum ﬂuctuations passing through the72squeezer, N̂1 can be written as follows:N̂1 (ϕ) = A cos2 ϕ + 2B cos ϕ sin ϕ + C sin2 ϕ,
A =(5.48)(ρ2p e−2σ + ρ2m e2σ )(cos αp + K sin αp )2+ (ρ2p e2σ + ρ2m e−2σ )(K cos αp − sin αp )2 ,(5.49)

B = (e2σ − e−2σ )(ρ2m − ρ2p )× (cos αp + K sin αp )(K cos αp − sin αp ),(5.50)C = (ρ2p e2σ + ρ2m e−2σ )(cos αp + K sin αp )2+ (ρ2p e−2σ + ρ2m e2σ )(K cos αp − sin αp )2 .

(5.51)

When ϕ = 0, N̂1 (ϕ = 0) = A and this represents the quantum noise of an optimally
matched ﬁlter cavity without injection and readout losses, Equation (44) in [52].
From Equation (5.46), the frequency-independent phase noise of N̂1 can be calculated
as follows:N̂1,avg (ϕ) =N̂1 (δϕ) + N̂1 (−δϕ)= A cos2 δϕ + C sin2 δϕ.
2(5.52)

Frequency-dependent phase noise can be calculated by averaging N̂1 (∆ωfc,0 + δ∆ωfc ) andN̂1 (∆ωfc,0 − δ∆ωfc ),N̂1,avg (∆ωfc,0 ) =N̂1 (∆ωfc,0 + δ∆ωfc ) + N̂1 (∆ωfc,0 − δ∆ωfc ).
2(5.53)For the calculation of frequency-dependent phase noise, we can consider only N1 because
the frequency-dependent phase noise for N2 is negligible compared to N1 , and N3 is independent of the ﬁlter cavity.

5.6

Squeezing degradation budget in 300 m ﬁlter cavity

Quantum noise with the squeezing degradation sources in the ﬁlter cavity (5.32) can be
numerically calculated. The realistic squeezing degradation sources in the 300 m ﬁlter
cavity and the corresponding squeezing degradation budget are presented in Table 5.2 andFigure 5.10. The quantum noise can be reduced at all frequencies with a 300 m ﬁlter
cavity. In particular, the quantum noise can be reduced by 4 dB at low frequencies and 6
dB at high frequencies.
As shown in Figure 5.10, the mode mismatch is the most dominant degradation source
at low frequencies, while the injection and readout losses are the most dominant at highfrequencies. As the ﬁlter cavity losses are not the most dominant noise source, increasing
the ﬁlter cavity length more than 300 m will not signiﬁcantly improve the overall sensitivity.

73

Table 5.2: Squeezing degradation sources in 300 m ﬁlter cavity.
Parameter
SymbolValue
Filter cavity losses
Λ2rt
80 ppmInjection losses
Λ2inj
5%
Readout losses
Λ2ro
5%
2
Mode-mismatch squeezer-ﬁlter cavity
ΛmmFC
2%Mode-mismatch squeezer-local oscillator
Λ2mmLO
5%
Frequency-independent phase noise (RMS)δϕ
30 mradFilter cavity length noise (RMS)
δLfc
1 pm
Generated squeezing
σdB
9 dB

0
Ideal system
Injection loss: 0.05 readout loss: 0.05
Filter cavity losses: 80 ppm
Mismatch: SQZ/FC: 0.02 - SQZ/LO: 0.05
Phase noise rms: 30 mradlock accuracy rms: 1 pm
All mechanisms

Quantum noise reduction [dB]

-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
100

101

102

103

Frequency [Hz]Figure 5.10: Squeezing degradation budget in 300 m ﬁlter cavity. The squeezing degradation parameters are listed in Table 5.2.

5.7

Sensitivity improvement of KAGRA

The sensitivity improvement of KAGRA with a 300 m ﬁlter cavity is shown in Figure 5.11.
The KAGRA sensitivity for BRSE without a ﬁlter cavity is 129 Mpc, and the ﬁlter cavity
can improve the sensitivity up to 188 Mpc. The GW event rate will improve by a factor
of 3.

74

seismic+gravity
suspension
mirrorquantum
total (without FC): 129 Mpc
total (with FC): 188 Mpc

Strain [1/ Hz]

10-21

10-22

10-23

10-24
101

102

103

Frequency [Hz]Figure 5.11: Sensitivity improvement of KAGRA with a 300 m ﬁlter cavity. The blacksolid and dashed lines represent the KAGRA sensitivity (BRSE) with and without the
300 m ﬁlter cavity, respectively. In this calculation, the arm intracavity power is 337 kW,
which corresponds to the power at BS of 674 W. The half bandwidth and detuning of theﬁlter cavity are 51 Hz and 47 Hz, respectively. The squeezing degradation parameters are
based on Table 5.2, although the mode mismatch and phase noise are not included in thiscalculation.

75

Chapter 6

Frequency-Independent Squeezing
The two key components to realize the frequency-dependent squeezing are the ﬁlter cavity
and the squeezed vacuum source. To realize frequency-dependent squeezing with rotation below 100 Hz, a frequency-independent squeezed vacuum source capable of producing
squeezing below 100 Hz is necessary. In this chapter, the development of the frequencyindependent squeezed vacuum source is explained. I mainly contributed to the implementation of the homodyne detector and coherent control and realization of the 5.4 dB offrequency-independent squeezing down to 20 Hz in this chapter.

6.1

Experimental setup

The in-air squeezed vacuum source in this experiment follows the design of the GEO600
squeezer [53]. The optical layout of the squeezed vacuum source is shown in Figure 6.1.
The simpliﬁed optical layout is shown in Figure 6.2. The core part is an optical parametricoscillator (OPO) which is a linear cavity hosting a periodically poled potassium titanyl
phosphate (PPKTP) crystal. The squeezed vacuum is generated from a parametric downconversion process in the crystal. This process requires a pump ﬁeld at twice the frequencyof the squeezed ﬁeld and the pump ﬁeld is generated by injecting a 1064 nm laser into
a second harmonic generator (SHG). The main laser, a 2-W 1064-nm Nd:YAG laser, isused to pump the SHG, which generates a green ﬁeld with a wavelength of 532 nm, andis used as a local oscillator (LO) for the balanced homodyne detector to measure thesqueezing. An infrared mode cleaner (IRMC) cavity is installed to clean the beam shapeof the LO before it reaches the homodyne detector. A green mode cleaner (GRMC) cavity
and a Mach-Zehnder interferometer (MZ) are installed along the green path, respectively,
to clean the beam shape and to stabilize the power of the green pump ﬁeld before it
reaches the OPO. Two auxiliary lasers, which are phase-locked to the main laser witha frequency oﬀset, are also used. The ﬁrst one (Auxiliary Laser 1 or coherent control(CC) laser) is injected into the OPO and copropagates with the squeezed vacuum ﬁeld up
to the homodyne detector for CC. The second one (Auxiliary Laser 2 or p-pol laser) is
also injected into the OPO with an orthogonal polarization with respect to the generated
squeezed ﬁeld and is used to control the OPO length.
76

Figure 6.1: Optical layout of frequency-independent squeezed vacuum source.

For the alignment of the squeezed vacuum ﬁeld, a pickoﬀ of the main laser is temporarily injected into the OPO and copropagates with the squeezed vacuum ﬁeld (we referto this beam as a seed beam or bright alignment beam). This beam is also used for thecharacterization of the OPO (such as nonlinear gain) and the ﬁlter cavity (such as ﬁnesse,
locking accuracy, and round-trip losses), which will be explained later.
Two ﬂipping mirrors are placed on the squeezing path so that we can switch between
sending the squeezing directly to the homodyne detector and sending it to the ﬁlter cavity.
A steering mirror placed before the homodyne detector is also a ﬂipping mirror so that
we can switch between sending the squeezing to the homodyne detector and sending it toan alignment mode cleaner (AMC), which will be explained in Section 6.9. It should be
noted that the AMC is not shown in Figure 6.2.

6.2

Laser sources

The laser sources used in this experiment are 1064 nm Nd:YAG lasers. There are mainly
two constraints on the current and temperature of the laser sources. One is the mode
hopping of the lasers and another is the limited frequency range (400 MHz) of the phase-77

Filter cavity

Figure 6.2: Simpliﬁed optical layout of frequency-independent squeezed vacuum source.

locked loop (PLL), which limits the maximum frequency oﬀset that the auxiliary lasers
can have with respect to the main laser. Therefore, the current and temperature of thelasers have to be chosen so that we can avoid the mode hopping of the three lasers andensure that the two beat notes of the main laser/p-pol laser and the main laser/CC laser
are within the PLL frequency range. The current and temperature settings used are listed
in Table 6.1.
Table 6.1: Current and temperature of the laser sources.
Main laser CC laser p-pol laser
Current (A)
1.832
1.185
1.338
Temperature (◦ C)
23.11
38.15
32.49

6.3

Direct digital synthesizer

All the RF sources used in the experiment were generated using direct digital synthesizers(DDS). The amplitude, frequency, and phase of the RF sources were digitally controlled.

78

There were three DDS boards, and each DDS board has four RF channels. The DDS
channel list used in this experiment is shown in Table 6.2. In DDS1, CH0 and CH1 provided
the modulation and demodulation RF sources for SHG/IRMC. CH2 and CH3 in DDS1
provided the modulation and demodulation RF sources for OPO. In DDS2, CH0 provided
the modulation RF sources for FC/GRMC (FC indicates green lock for the ﬁlter cavity,
which will be explained in Chapter 7), and CH1 and CH2 provided the demodulation RF
sources for FC and GRMC. CH3 in DDS2 provided the demodulation RF sources for thetwo quadrant photodiodes (QPDs) for auto alignment. In DDS3, CH0 and CH1 provided
the LO for the CC and p-pol PLL, respectively. CH2 in DDS3 provided the demodulation
RF sources for CC1/CCFC (CCFC is a control loop for the ﬁlter cavity with coherent
control sidebands, which will be explained in Chapter 8). CH3 in DDS3 provided thedemodulation RF source for CC2. CH1 in DDS1 (for SHG/IRMC demodulation), CH3 in
DDS2 (for two QPDs demodulation), and CH2 in DDS3 (for CC1/CCFC demodulation)
were split with 50:50 power splitters (Z99SC-62-S+).
Table 6.2: DDS channel list
DDS1
channelfunctionfrequency
CH0
SHG/IRMC EOM
15.2 MHzCH1
SHG/IRMC demod
15.2 MHzCH2
OPO EOM
87.6 MHzCH3
OPO demod87.6 MHz
DDS2
channelfunctionfrequency
CH0
FC/GRMC EOM
78 MHzCH1
FC demod78 MHzCH2
GRMC demod
78 MHzCH3
AA demod78 MHzDDS3channelfunctionfrequency
CH0
CC PLL LO
7 MHz
CH1
p-pol PLL LO< 400 MHzCH2
CC1/CCFC demod
14 MHzCH3
CC2 demod
7 MHz

6.4

Phase-locked loop

The frequencies of the main laser and the auxiliary lasers are locked by the phase-locked
loop (PLL). The conﬁguration of the PLL is shown in Figure 6.3. The Phase Detector/Frequency Synthesizer (ADF4002 in Analog devices) is used for the PLL. The frequency range that the ADF4002 can handle as a voltage-controlled oscillator (VCO) is
400 MHz.

79

Figure 6.3: Conﬁguration of PLL. Parts of the main laser and the auxiliary lasers are
picked oﬀ and combined with a ﬁber BS. One of the two outputs is fed back to thefrequency of the auxiliary lasers to follow the frequency of the main laser. Another outputof the ﬁber BS is used for monitoring the frequency of the beat note.

6.5

Second harmonic generator

The second harmonic generator (SHG) is a cavity composed of a nonlinear crystal (MgO : LiNbO3 )and a meniscus mirror. The SHG converts 1064 nm of the IR ﬁeld into 532 nm of thegreen ﬁeld through a nonlinear process. The schematic of the cross section of the SHG
cavity used in this experiment is shown in Figure 6.4. The optical parameters of the SHG
are listed in Table 6.3.
Table 6.3: SHG design parameters.
Cavity parameter
Value
MgO : LiNbO3 HR reﬂectivity 99.95 % (IR), 99.8 % (GR)MgO : LiNbO3 AR reﬂectivity< 0.1 % (IR, GR)
Meniscus HR reﬂectivity
92 % (IR), 2 % (GR)MgO : LiNbO3 RoC
12 mm (HR), ∞ (AR)Meniscus RoC
25 mm (HR), 20 mm (AR)Optical cavity length
38 mm
FSR
4 GHz
FWHM
55 MHz (IR), 2.9 GHz (GR)Finesse73 (IR), 1.4 (GR)

6.5.1

FinesseThe IR ﬁnesse of the SHG was measured by the cavity scan. To avoid the conversion fromIR to green in the SHG, the SHG temperature was set in the region where the nonlinear
eﬀect in the SHG is small during the cavity scan. The SHG cavity scan is shown in Figure6.5. The measured IR ﬁnesse of the SHG is 70, which is close to the design ﬁnesse.

80MacorholderMgO:LiNbO3
AR HR

HR

Meniscus
AR

CopperPZT
holder Peltier Thermistor
heaterFigure 6.4: Schematic of cross section of SHG. The injected IR ﬁeld is p-polarized andgenerated green ﬁeld is s-polarized. The conversion eﬃciency of the SHG from IR togreen is 35 %. The SHG cavity length is controlled with a Piezoelectric (PZT) actuator
attached to the meniscus mirror. The PDH signal of the SHG cavity is obtained at the
SHG transmission and fed back to the PZT actuator. The temperature of the crystal ismonitored with the thermistors and controlled with the Peltier heaters.

4

SHG transmission
fitting: finesse = 70.30(5)SHG transmission (V)

3.5
3
2.5
2
1.5
1
0.5
0
-0.5
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

time (s)Figure 6.5: SHG cavity scan. The blue and red lines represent the SHG transmission curveand its ﬁtting curve, respectively. The oﬀset was removed from the data. The ﬁnesse value
obtained from the ﬁtting is 70.30(5) which is close to the expected value.

6.5.2

Length control

The open-loop transfer function of SHG cavity length control is shown in Figure 6.6.

81

gain (dB)

SHG gain

0
-50
-100
2
10

10

3

10

4

phase (deg)200

SHG phase

0

-200
2
10

10

3

10

4

frequency (Hz)Figure 6.6: Open-loop transfer function of SHG cavity length control. The UGF is 2 kHz
and the phase margin is 56 deg.

6.5.3

Temperature control

The temperature of the nonlinear crystal inside the SHG is controlled for phase matching.
The temperature control is composed of thermistors (103JT-025), Peltier heaters (CP0.831-06L), and a temperature controller (Thorlabs TED200C). According to the manual of
the thermistor and temperature controller, the relation between the resistance R and thetemperature T of the thermistor is
T (R) =BT0,
T0 ln RR0 + B(6.1)

where R0 = 10 kΩ is the nominal resistance of the thermistor at the nominal temperatureof T0 = 298.15 K and B = 3435 K is the energy constant of the thermistor. We can set
the target resistance of the thermistor in the temperature controller and the diﬀerence
between the target resistance and current resistance is sent to the Peltier heaters.
To determine the optimal temperature for the SHG, we measured the generated green
power from the SHG as a function of the SHG temperature as shown in Figure 6.7. The
shape of the generated green power is not symmetric around the optimal temperature
unlike Figure 4.2, which is the case for the single-pass SHG. This asymmetry is causedby the phase shift of the fundamental and second harmonics in the forward and backward
passes in the SHG cavity (double pass) [54, 55, 56]. The generated second harmonic powerwith the double pass can be written as
P2 ∝

Z L"
0

#
2
i∆kz
ei∆kz
e
iϕ+i∆kL+edz ,0
1 − i z−z
1 − i z−(L−z0 )
zR
zR

82

(6.2)

where L is the crystal length, z0 is the distance between the entrance surface of the crystal
and the beam waist, zR = kw02 /2 is the Rayleigh range, and w0 is the radius of the beam
waist. ϕ = ϕ2 − 2ϕ1 where ϕ1 and ϕ2 are the phases of the fundamental and second
harmonics, respectively.
For the plane wave (zR → ∞), the generated second harmonic power isP2 ∝ sinc

2

∆kL
2




cos

2


ϕ + ∆kL.
2(6.3)

There is an additional factor of cos2 ( ϕ+∆kL) compared with the single pass, which causes
2
the asymmetry around the optimal temperature, as shown in Figure 6.7.
60

green power (mW)

50
40
30
20
10
0
-10
330

330.5

331

331.5

332

332.5

SHG temperature (K)Figure 6.7: Generated green power from SHG as a function of SHG temperature.

6.6

Green and infrared mode cleaners

The GRMC and IRMC are triangular cavities used to clean the beam shapes of the green
pump ﬁeld and the LO, respectively. The conﬁgurations of GRMC and IRMC are shownin Figure 6.8. They have the same conﬁgurations, apart from wavelength.

83281 mm
22 mm

Figure 6.8: Conﬁguration of GRMC. The conﬁguration of IRMC is the same as GRMC
apart from wavelength. The triangular cavity is composed of one top mirror with the PZT
actuator and two bottom mirrors (input, output mirrors). The PDH signal of the cavity
is obtained at the cavity reﬂection and fed back to the PZT actuator of the top mirror.

The optical parameters of the green and infrared mode cleaners are shown in Table 6.4.
The optical parameters of the GRMC/IRMC are also the same, apart from the wavelength.
The nominal GRMC/IRMC transmissivity is 89 % without any losses.
Table 6.4: Design parameters of GRMC and IRMC.
GRMC/IRMC
top mirror RoC
1mbottom mirrors RoC∞
top mirror reﬂectivity
99.9 % (0◦ AOI)
bottom mirrors reﬂectivity
99.2 % (s-pol)nominal cavity transmissivity
89 %cavity length
0.584 mFSR
513 MHz
FWHM
1.39 MHzﬁnesse368

The ﬁnesse of GRMC and IRMC are measured by a cavity scan. The GRMC and
IRMC cavity scans are shown in Figure 6.9 and 6.10, respectively. The measured ﬁnesseof GRMC and IRMC are 323 and 291, respectively.

84

1.2

GRMC transmission
fitting: finesse = 322.9(3)GRMC transmission (V)

1
0.8
0.6
0.4
0.2
0
-0.2
0.08

0.085

0.09

0.095

0.1

0.105

0.11

time (s)Figure 6.9: GRMC cavity scan. The blue and red lines represent the GRMC transmission
curve and its ﬁtting curve, respectively. The oﬀset was removed from the data. The ﬁnesse
value obtained from the ﬁtting is 322.9(3) which is close to the expected value.

1.4

IRMC transmission
fitting: finesse = 290.6(1)IRMC transmission (V)

1.2
1
0.8
0.6
0.4
0.2
0
0.08

0.085

0.09

0.095

0.1

0.105

0.11

0.115

time (s)Figure 6.10: IRMC cavity scan. The blue and red lines represent the IRMC transmission
curve and its ﬁtting curve, respectively. The oﬀset was removed from the data. The ﬁnesse
value obtained from the ﬁtting is 290.6(1) which is close to the expected value.

The open-loop transfer functions of GRMC and IRMC cavity length control are shownin Figure 6.11 and 6.12.

85

gain (dB)

20

GRMC gain

0
-20
-40
-60
10

2

10

3

10

4

phase (deg)200

GRMC phase

0

-200
2
10

10

3

10

4

frequency (Hz)Figure 6.11: Open-loop transfer function of GRMC cavity length control. The UGF is 1.9
kHz and the phase margin is 52 deg.

gain (dB)50

IRMC gain

0

-50
2
10

10

3

10

4

phase (deg)200

IRMC phase

0

-200
102

103

104

frequency (Hz)Figure 6.12: Open-loop transfer function of IRMC cavity length control. The UGF is 2.2
kHz and the phase margin is 60 deg.

6.7

Mach-Zehnder interferometer

The Mach-Zehnder interferometer (MZ) is placed on the path of the green pump after the
SHG before the GRMC to stabilize the green pump power injected into the OPO. The86

conﬁguration of the MZ is shown in Figure 6.13. The open-loop transfer function of the
MZ is shown in Figure 6.14.

offsetFigure 6.13: Conﬁguration of Mach-Zehnder interferometer. One of the mirrors in the MZ
has a PZT actuator for MZ control. The error signal is obtained by picking oﬀ the GRMC
transmission and fed back to the PZT actuator in the MZ. The green pump power after
the MZ can be controlled by changing the oﬀset in the MZ control loop.

gain (dB)50

MZ gain

0

-50
102

103

104

phase (deg)200

MZ phase

0

-200
102

103

104

frequency (Hz)Figure 6.14: Open-loop transfer function of MZ control loop. The UGF is 2 kHz and thephase margin is 89 deg.

87

6.8

Optical parametric oscillator

The OPO is a cavity composed of a nonlinear crystal (PPKTP) and a meniscus mirror.
The schematic cross section of OPO used in this experiment is shown in Figure 6.15. The
OPO design parameters are listed in Table 6.5.

Macor
holder
HR

PPKTP
AR HR

Meniscus
AR

CopperPZT
holder Peltier Thermistor
heaterFigure 6.15: Schematic cross section of OPO. The pump, squeezing, and CC ﬁelds are spolarized, while the length control ﬁeld is p-polarized. The OPO cavity length is controlledwith the PZT actuator attached to the meniscus mirror. A PDH signal of the OPO cavity
is obtained by picking oﬀ the length control ﬁeld with PBS at OPO transmission and fed
back to the PZT actuator. The temperature of the crystal is monitored with thermistors
and controlled with Peltier heaters.

Table 6.5: OPO design parameters
Cavity parameter
Value
PPKTP HR reﬂectivity
99.975 % (IR, GR)
PPKTP AR reﬂectivity< 0.2 % (IR, GR)
Meniscus HR reﬂectivity
92 % (IR), 20 % (GR)PPKTP RoC
8 mm (HR), ∞ (AR)Meniscus RoC
25 mm (HR), 20 mm (AR)Escape eﬃciency
95 %Optical cavity length
38 mm
FSR
4 GHz
FWHM
56 MHz (IR), 1.1 GHz (GR)Finesse72 (IR), 3.8 (GR)88

6.8.1

Escape eﬃciency

The OPO escape eﬃciency can be written asηesc =

κao
T
,
=
a
κ
T + Lcav(6.4)

where T = 8 % is the transmissivity of the meniscus mirror and Lcav is the intracavity
round-trip loss of the OPO (not including T ). Because the PPKTP AR reﬂectivity is
0.2 % and the PPKTP HR transmissivity is 0.025 %, the intracavity round-trip loss of
the OPO is Lcav = 0.2 · 2 + 0.025 = 0.425 %. Therefore, the design escape eﬃciency isηesc = 8/(8 + 0.425) = 95 %.

6.8.2

FinesseTo measure the IR ﬁnesse of the OPO cavity, the OPO cavity length was scanned and the
IR transmission was measured as shown in Figure 6.16. The measured IR ﬁnesse of the
OPO is 69.
2

OPO transmission
fitting: finesse = 68.76(5)OPO transmission (V)

1.5

1

0.5

0

-0.5
0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

time (s)Figure 6.16: OPO cavity scan. The blue and red lines represent the OPO IR transmission
curve and its ﬁtting curve, respectively. The oﬀset was removed from the data. The ﬁnesse
value obtained from the ﬁtting is 68.76(5), which is close to the expected value.

6.8.3

Length control

The OPO cavity length is controlled with a p-polarized laser so as not to interfere with thesqueezed vacuum ﬁeld, which is s-polarized. The frequency of the p-polarized laser has tobe chosen such that both the p- and s-polarized beams are resonant inside the OPO. The
open-loop transfer function of the OPO cavity length control is shown in Figure 6.17.

89

gain (dB)50

OPO gain

0

-50
2
10

10

3

10

4

phase (deg)200

OPO phase

0

-200
2
10

10

3

10

4

frequency (Hz)Figure 6.17: Open-loop transfer function of OPO cavity length control. The UGF is 3.6
kHz and the phase margin is 57 deg.

6.8.4

Temperature control

To ﬁnd the optimal temperature for OPO, we injected the seed IR ﬁeld into the OPO
and measured the generated green power. The generated green power from the OPO asa function of the OPO temperature is shown in Figure 6.18. Similar to the SHG, it is not
symmetric around the optimal temperature owing to the double pass conﬁguration.
120

green power (mW)

110
100
90
80
70
60
50
304

305

306

307

308

309

310

311

312

313

OPO temperature (K)Figure 6.18: Generated green power from OPO as a function of the OPO temperature.

90

6.8.5Threshold

The OPO threshold power can be measured from the parametric ampliﬁcation and deampliﬁcation of the IR seed beam injected into the OPO. The parametric ampliﬁcationand de-ampliﬁcation can be measured from the maximum and minimum values of the
OPO transmission of the IR seed beam, while the phase of the pump ﬁeld is scannedwith a phase shifter, which will be explained in Section 6.11. From Equation (4.66), theparametric gain G is written as
G= 
1∓

1
q

2 ,

(6.5)

Ppump
Pth

where − and + correspond to the parametric ampliﬁcation and deampliﬁcation, respectively.
The measured parametric (de)ampliﬁcation as a function of the pump green power isshown in Figure 6.19. From this measurement, the OPO threshold power is determined
as 80.6(2) mW.

parametric amplification: threshold = 80.56 +/- 0.18
parametric deamplification

parametric gain

102

101

100

10-1

0

10

20

30

40

50

60

70

pump power (mW)

Figure 6.19: Parametric (de)ampliﬁcation as a function of the pump green power. The
measured parametric ampliﬁcation is used for ﬁtting.

The expected generated squeezing as a function of the pump power with Pth = 80.6
mW is shown in Figure 6.20.

91

25

squeezing (dB)

20

15

10

5

0
0

10

20

30

40

50

60

pump power (mW)

Figure 6.20: Expected generated squeezing as a function of the pump power with Pth =80.6 mW.

6.9

Alignment mode cleaner

To overlap LO and squeezed ﬁeld completely, an alignment mode cleaner (AMC) was put
next to the homodyne detector as shown in Figure 6.21. The speciﬁcation of the AMC is
the same as the IRMC. The overlapping of LO and squeezed ﬁeld is achieved by aligning
the LO and the seed beam to the AMC.

Squeezing(Seed beam)

Homodyne

Alignmentmode cleaner

Local oscillator
Figure 6.21: Conﬁguration of AMC.

The nonoptimal overlapping of the LO and a squeezed ﬁeld introduces losses. The
overlapping of the LO and the squeezed ﬁeld is called visibility, which can be determinedby measuring the maximum and minimum values of the beat note between the LO and92

the seed beam, as follows:
Vexp =

Vmax − Vmin.
Vmax + Vmin

(6.6)

When the LO power is P1 and the seed beam power is P2 , the optimal visibility is√
2 P1 P2
Vopt =.
P1 + P2(6.7)

The optimal visibility is 1 when P1 = P2 , and it is smaller than 1 otherwise. The eﬃciency
2 /V 2 . The typical eﬃciency of the visibility is ηof the visibility is ηvis = Vexp
vis = 0.98
opt
when the seed beam is directly sent to the homodyne detector without the ﬁlter cavity.

6.10

Homodyne detector

The homodyne detector in this experiment was designed and developed in the Albert Einstein Institute (AEI) in Germany. A picture of the homodyne detector is shown in Figure
6.22. The homodyne detector has DC and RF outputs. The DC output is proportional tothe diﬀerence of photocurrents in each PD and is used for measuring squeezing. The RF
output is the demodulated signal at 7 MHz and is used for the CC2 control loop.

Figure 6.22: A picture of homodyne detector. The homodyne detector has two photodiodes, and one of the photodiodes is covered with an aluminum cover in the picture. The
photodiodes of the homodyne detector are vertically tilted by 20◦ for higher quantum
eﬃciency with photon recycling [57]. Photon recycling has not been implemented in thisexperiment.

The dark noise of the homodyne detector and the LO shot noise are shown in Figure6.23. The dark noise of the homodyne detector is a noise source for squeezing and it isequivalent to the optical losses of 1.6 %. Other noise sources for squeezing are coherent
control sidebands (CCSB), and the fundamental noise from the CCSB is their shot noise.
93Because the LO power is 1.78 mW and the CCSB power is around 10 µW before the BS
for the homodyne detector, the CCSB shot noise is more than 20 dB lower than the LO
shot noise. The corresponding loss from the CCSB shot noise is less than 1 %.

power spectrum (dBVrms/ Hz)-120

Homodyne dark noise
shot noise with 1.78 mW LO

-125
-130
-135
-140
-145
-150
-155
101

102

103

104

105

frequency (Hz)Figure 6.23: Dark noise of the homodyne detector and LO shot noise. The black line isthe dark noise of the homodyne detector and the red line is the shot noise with 1.78 mW
LO (0.89 mW LO in each PD). The clearance between the dark noise and shot noise with
obs = 10−18/10
the 1.78 mW LO is about 18 dB. The loss from the dark noise is l = Vclass /Vvac= 1.6 %.

Alignment procedure of LO and squeezed ﬁeld
For common mode rejection of the classical noise of the LO, the LO power on each homodyne PD should be very well balanced. In addition, the LO and the squeezed ﬁeld shouldbe aligned accurately to the AMC for overlapping. Thus, the LO and the squeezed ﬁeld
should be properly aligned as follows.
1. DC balance of LO
The condition of balance for the LO ﬁeld is achieved when the homodyne DC signal is
0. The homodyne DC signal for the LO is set to 0 by tweaking two lenses before thehomodyne detector. If tweaking the lenses is not enough for DC balance, the BS for thehomodyne detector is tweaked so that the DC signal of the homodyne detector becomes0.
2. Alignment of LO and seed beam into AMC
The LO is aligned to the AMC with two steering mirrors before the AMC. Then, the seedbeam is aligned to the AMC with steering mirrors on the path of the squeezing before theseed beam combines with the LO.

94

6.11

Coherent control

To control the squeezing angle, the CC laser, which has an oﬀset frequency of 7 MHz withrespect to the main laser, is injected into the OPO. CC is composed of two control loops(CC1 and CC2). The CC1 error signal is obtained at the OPO reﬂection by detecting thebeat notes of the two sidebands (CCSB) at 14 MHz, and the CC2 error signal is obtainedat the homodyne detector by detecting the beat notes of the CCSB and LO at 7 MHz.
To control the green pump phase for CC1 and the LO phase for CC2, the optical paths ofthe green and LO ﬁelds are changed using phase shifters (see the PZT actuated mirror as
shown in Figure 6.24.) located before the GRMC on the green path and before the IRMC
on the LO path.

Figure 6.24: Phase shifter for coherent control. The mirror position can be adjusted by a
PZT actuator which is ﬁxed on a rigid metal mount.

The open-loop transfer functions of the CC1 and CC2 are shown in Figure 6.25 and
6.26.

95

CC1 gain

gain (dB)

20
0
-20
-40
10

2

10

3

10

4

phase (deg)200

CC1 phase

0

-200
2
10

10

3

10

4

frequency (Hz)Figure 6.25: Open-loop transfer function of CC1. The UGF is 2.5 kHz and the phasemargin is 51 deg.

gain (dB)50

CC2 gain

0

-50
2
10

10

3

10

4

phase (deg)200

CC2 phase

0

-200
102

103

104

frequency (Hz)Figure 6.26: Open-loop transfer function of CC2. The UGF is 4 kHz and the the phasemargin is 80 deg

The phase noise of CC1 and CC2 can be estimated from the in-loop CC1, 2 error
signals. To calibrate the phase noise, the peak-to-peak values of CC1, 2 error signals were
measured by scanning the phase shifter for CC1, 2. The CC1, 2 error signals can be96written as follows:Vppsin θ,
2

Verr =(6.8)

where Verr are the CC1, 2 error signals, Vpp are the peak-to-peak values of the CC1, 2
error signals, and θ are CC1, 2 phase noise. From this equation, the conversion factor
from the CC1, 2 error signals (V) to the CC1, 2 phase noises (rad) is determined as 2/Vpp
when θ ≪ 1. The measured CC1, 2 phase noises are shown in Figure 6.27 and 6.28. The
CC1, 2 in-loop rms phase noises are 84.2 and 17.3 mrad, respectively.
100

CC1 in loop phase noise
CC1 in loop rms phase noise: 84.2 mrad

phase noise (rad/ Hz)

10-1

10-2

10-3

10-4

10-5 2
10

103

104

frequency (Hz)Figure 6.27: CC1 in-loop phase noise.

10-1

CC2 in loop phase noise
CC2 in loop rms phase noise: 17.3 mrad

phase noise (rad/ Hz)

10-2

10-3

10-4

10-5

10-6 0
10

101

102

103

104

frequency (Hz)Figure 6.28: CC2 in-loop phase noise.

97

105

6.12

Squeezing spectrum

The measurement of the squeezing and anti-squeezing spectra are shown in Figure 6.29.
The measured squeezing and anti-squeezing levels are 5.4 dB and 14.6 dB, respectively,
between 100 Hz and 100 kHz. This measurement was performed with the pump green
power of 40 mW, which is half of the OPO threshold. Around 5 dB of squeezing wasachieved between 20 Hz and 100 Hz, where the frequency-dependent squeezing rotation
occurs. Some peaks in the spectrum are harmonics of 50 Hz, which may originate fromsome electronic components.
25

shot noisesqueezing is 5.41dB
anti-squeezing is 14.62dB

squeezing (dB)

20
15
10
5
0
-5
-10
1
10

10

2

10

3

10

4

10

5

frequency (Hz)Figure 6.29: Squeezing and anti-squeezing spectrum. The measured squeezing level is 5.4
dB and anti-squeezing level is 15.6 dB.

6.13

Loss and phase noise

The loss and phase noise in the frequency-independent squeezed vacuum source can be
estimated by measuring the squeezing and anti-squeezing levels with diﬀerent pump green
powers as shown in Figure 6.30. By ﬁtting this data with Equation (4.115), we determine
the loss as 25.2(2) % and the phase noise as 21.4(5) mrad. The current squeezing level ismainly limited by optical losses.

Loss budget
The loss budget of the frequency-independent squeezed vacuum source is summarized inTable 6.6. The classical noise listed in the Table includes dark noise of the homodyne
detector (1.6 %) and shot noise of the CCSB (1 %). The unknown loss is around 9 %and the origin of the unknown loss is under investigation. However, we notice that the IR

98

10

green power = 20,25,30,35,40,45,50,55,60 mW
loss = 25.2(2) %, phase noise = 21.4(5) mradno loss, no phase noise
loss = 25.2(2) %, no phase noisesqueezing (dB)

8

6

4

2

0
0

5

10

15

20

25

30

anti-squeezing (dB)Figure 6.30: Measured squeezing and anti-squeezing levels with diﬀerent pump green
powers. The pump green power was increased from 20 mW to 60 mW and the antisqueezing level increased accordingly. The error bars of the data are determined fromthe standard deviation of the data. From this measurement, the loss and phase noise are
estimated to be 25.2(2) % and 21.4(5) mrad, respectively.

transmission from the OPO is decreased by around 7 % after propagation in free space.
This reduction could be an indication of some scattering present inside the OPO.
Table 6.6: Loss budget of frequency-independent squeezed vacuum source.
loss sourceloss
OPO escape eﬃciency (design)5%
mirror, lens, Faraday isolator
6%
quantum eﬃciency of photodiodes
1%
visibility
2%classical noise
2.6 %unknown
8.6 %total
25.2 %

6.14

Conclusion

To realize frequency-dependent squeezing below 100 Hz, a frequency-independent squeezed
vacuum source capable of producing squeezing below 100 Hz is necessary. In this chapter,the development of the frequency-independent squeezed vacuum source was presented, and
5.4 dB of squeezing down to 20 Hz was achieved. The current squeezing level is limited by
optical losses, and a part of their origin is still under investigation. It is expected that an
OPO with escape eﬃciency as high as 99% [58] will signiﬁcantly reduce the optical losses.
99Chapter 7

Frequency-Dependent Squeezing
with 300 m Filter Cavity
To realize frequency-dependent squeezing with rotation below 100 Hz, a 300 m-long ﬁlter
cavity in the former TAMA interferometer was used. Combining the 300 m ﬁlter cavity
and the squeezed vacuum source presented in the previous chapter, the ﬁrst demonstration of the frequency-dependent squeezing with the rotation frequency below 100 Hz has
been realized [17]. In this chapter, the experimental characterization of the 300 m ﬁlter cavity and the result of the frequency-dependent squeezing are presented. I mainly
contributed to the realization and characterization of the frequency-dependent squeezingin this chapter, while the simulation and characterization of the ﬁlter cavity mirrors andinitial measurement of round-trip losses were done before I joined this experiment and arereported in [62, 63, 64].

7.1

Overview

The ﬁlter cavity used in this experiment is one of the 300 m arms of the former TAMA300
interferometer [59] in National Astronomical Observatory of Japan (NAOJ). The cavity
mirrors, with 10 cm in diameter, are suspended with a double pendulum system [60]placed on a vibration isolation multilayer stack [61]. Mirror motion is sensed by opticallevers and controlled using coil-magnet actuators to align the cavity and to damp themechanical resonances of the suspension [64]. To realize a squeezing rotation around 70
Hz, a storage time of about 3 ms in a ﬁlter cavity is necessary, which requires either along or a very high-ﬁnesse cavity. As the longer cavity is more robust to length-dependent
squeezing degradation sources such as cavity losses and cavity length noise, the use of
several 100 m-long ﬁlter cavities have been considered for advanced detectors [52, 62].
The squeezing rotation frequency by the ﬁlter cavity depends on the cavity bandwidth,
which is inversely proportional to its length and ﬁnesse. Given the 300 m length of thecavity, the ﬁnesse should be 4360 at a wavelength of 1064 nm to provide the squeeze
angle rotation at approximately 75 Hz, which corresponds to an optimal quantum noise
reduction for KAGRA.
100

Because the ﬁlter cavity losses are one of the major limitations to squeezing, as discussed in Section 5.6, the requirements of surface quality of the ﬁlter cavity mirrors wereﬁrstly determined by numerical simulations [62]. After the cavity assembly, the round-trip
losses were measured as 50-90 ppm, which is consistent with the requirements [63]. The
ﬁlter cavity parameters are listed in Table 7.1.
Table 7.1: Summary of the ﬁlter cavity parameters. The mirror transmissivity and radiusof curvature have been measured at the Laboratoire des Matériaux Avancés (LMA). It
should be noted that the round-trip losses include the end mirror transmissivity [63].
Cavity parameter
Value
Length300 m
Mirror diameter
10 cm
Input mirror radius of curvature
438 m
End mirror radius of curvature
445 m
Input mirror transmissivity (1064 nm) 0.136%
End mirror transmissivity (1064 nm) 3.9 ppm
Finesse (1064 nm)
4360
Input mirror transmissivity (532 nm)
0.7%
End mirror transmissivity (532 nm)
2.9%
Finesse (532 nm)
172
Beam diameter at waist
1.68 cmBeam diameter at the mirrors
2.01 cm

7.2

Experimental setup

The experimental setup used to achieve the frequency-dependent squeezing is shown in
Figure 7.1. It is composed of two parts: a frequency-independent squeezed vacuum source
and a ﬁlter cavity whose mirrors are suspended with a double pendulum for seismic noise
isolation. These two parts are connected through an in-vacuum injection system, which
includes two suspended mirrors.

101

Squeezed Vacuum

532 nm

OPO length control

Coherent Control

Faraday Isolator

Suspended Mirror

EOM

In-vacuum 300 m long Filter Cavity

1064 nmPZT actuated mirror

Fiber based PLL

Photodiode

BS

PBS

In-vacuum Injection Telescope
Homodyne

AOM

GRMC

OPO

MZ

CC1
IRMC

SHG

PLL

PLL

Auxiliary Laser 1

Auxiliary Laser 2

CC2

Main Laser

In-air Filter cavity control and Squeezed Vacuum Source

Figure 7.1: Schematic view of the experimental setup. The squeezed vacuum ﬁeld is generated by the OPO and injected into the ﬁlter cavity using an in-vacuum injection telescope.
The frequency-dependent squeezed vacuum ﬁeld from the ﬁlter cavity is reﬂected by thein-vacuum Faraday isolator and measured by the homodyne detector. The green ﬁeld
generated by the SHG is used to pump the OPO and to control the ﬁlter cavity.

102

7.3

300 m ﬁlter cavity

To realize frequency-dependent squeezing with a rotation frequency around 75 Hz, a crucial
parameter is the ﬁlter cavity half bandwidth (or ﬁlter cavity ﬁnesse). As we discussed in
Section 5.5, the ﬁlter cavity losses and length noise, which are the main degradation
sources for squeezing, are also important parameters. In this section, the characterizationof the 300 m ﬁlter cavity, such as ﬁlter cavity length noise, ﬁnesse, and round-trip losses,
is presented.

7.3.1

Green lock

The length of the ﬁlter cavity is controlled using an auxiliary green ﬁeld that is picked oﬀ
the SHG output. The cavity ﬁnesse for the auxiliary green ﬁeld is about 25 times smaller
than that for the infrared ﬁeld. The green ﬁeld is superposed to the squeezed ﬁeld at an
in-vacuum dichroic mirror, as shown in Figure 7.1. The PDH signal is obtained from thereﬂected green ﬁeld, and the correction signal is sent to the PZT actuator of the main laser
to act on its frequency. An acousto-optic modulator (AOM) is placed on the green ﬁeld
path before injection into the ﬁlter cavity. By driving the AOM at diﬀerent frequencies,
it is possible to control the detuning of the carrier with respect to the cavity resonance bychanging the relative frequencies of the green and squeezed ﬁelds.
Table 7.2 and 7.3 show the zeros, poles, and frequency dependence of the open-looptransfer function for the green lock.
Table 7.2: Zeros and poles of open-loop transfer function for green lock.
zeros
Qquantity origin
145 Hzsimple1servo
2.68 kHz simple
4
servo
88 kHzsimple
1
servo
poles
Q
quantity origin
0.0001 Hz simple
5
servo
1.45 kHz simple
1
cavity90 kHzsimple1piezo
142 kHz
0.78
2
servo
329 kHz simple
1
servo
380 kHz simple
1
servo
658 kHz simple
1
servo
675 kHz simple
1
servo

The measured open-loop transfer function of the green lock for the ﬁlter cavity is shownin Figure 7.2.

103

Table 7.3: Frequency dependence of the open-loop transfer function for green lock.
frequency
frequency dependence of OLTF
DC < f < 145 Hz
f −5
145 Hz < f < 1.45 kHz
f −4
1.45 kHz < f < 2.68 kHz
f −5
2.68 kHz < f < 88 kHz
f −1

gain (dB)100

FC green gain

50

0
10

3

10

4

phase (deg)200

FC green phase

0

-200
10

3

10

4

frequency (Hz)Figure 7.2: Open-loop transfer function of the green lock for the ﬁlter cavity. The UGF
is 14 kHz and the phase margin is 54 deg.

7.3.2

Locking accuracy

Locking accuracy of green ﬁeld
The locking accuracy of the green ﬁeld is evaluated using the in-loop green PDH signal.
The calibration of the green PDH signal is achieved by injecting a line at 28 kHz, which isabove the UGF of the green lock ∼ 14 kHz, and therefore, the eﬀect of the noise suppression
by the control loop is small. The line was injected into the PZT actuator of the main laser
and measured at the PZT monitor channel and the PDH signal, which are respectively
before and after the servo for the green lock. By comparing the two line amplitudes, thecalibration factor from the error signal to the locking accuracy (Hz/V) can be obtained.
The amplitude of the 28 kHz line in Hz is
SHz = Vin (Vrms ) ·√2 · 100 · 2 · 106 (Hz/V) = 5827 (Hz)

(7.1)

where Vin = 20.6 · 10−6 Vrms has been measured at the PZT monitor channel and the√factor of 2 is the conversion factor from Vrms to V. The factor of 100 is the attenuation

104

factor of the PZT monitor channel. 106 Hz/V is the gain of the PZT actuator of the main
laser and the factor of 2 represents the frequency doubling in the SHG.
Considering the cavity pole of the ﬁlter cavity and the open-loop transfer function of
the green lock, the injected line in the error signal isVerr =

1
SHz
1
·r
·
 2
|1 + GGR (f )| K (Hz/V)
1 + ffp(7.2)√
where Verr = 1.3 Vrms = 2 · 1.3 V is the measured amplitude of the injected line in theerror signal, K is the calibration factor (Hz/V), GGR is the open-loop transfer functionof the green lock at the frequency of the injected line, f = 28 kHz is the frequency of theinjected line, and fp = 1.45 kHz is the cavity pole for the green ﬁeld. From Figure 7.2, thegain and phase of GGR at 28 kHz are −5.8 dB and −125 deg, respectively. From Equation(7.1) and (7.2), the calibration factor is K = 2.0 × 102 Hz/V.
The measured green locking accuracy is shown in Figure 7.3. The green locking accuracy (rms) is 20.3 Hz.

green error signal (Hz/ Hz)102

green error signal
green error signal (rms) is 20.3 Hz

101
100
10-1
10-2
10-3
10-4 2
10

103

104

105

frequency (Hz)Figure 7.3: Locking accuracy of green ﬁeld with green√ lock. The spectrum of the inloop green PDH signal is calibrated in units of Hz/ Hz using the calibration factor of
K = 2.0 × 102 Hz/V. The green locking accuracy (rms) is 20.3 Hz.

Locking accuracy of IR ﬁeld
The locking accuracy of the IR ﬁeld is a signiﬁcant parameter because it represents theﬁlter cavity length noise for squeezing. To evaluate the locking accuracy of the IR ﬁeld, theIR seed beam was injected into the ﬁlter cavity and the reﬂected seed beam was detectedusing an auxiliary RFPD. The reﬂected IR signal was demodulated at 15.2 MHz to obtain
the IR PDH signal. To calibrate the IR PDH signal, the ﬁlter cavity was locked with thegreen ﬁeld and the frequency of the AOM was scanned around the IR resonance. The
105

calibration factor from the IR PDH signal to frequency (Hz/V) is
cPDH =sAOM tpp
2Vpp

(7.3)

where sAOM is the scan speed of the AOM for IR (Hz/s), Vpp is the peak-to-peak valueof the IR PDH signal (V), and tpp is the time diﬀerence of the peak-to-peak of IR PDH
signal (s). It should be noted that the scan speed of the AOM for IR is half of that for
the green ﬁeld. The factor of 2 in Equation (7.3) is used because the slope of the error
signal around the resonance is as about twice as the slope of the peak-to-peak value.
To measure the locking accuracy, the IR seed ﬁeld was set on resonance by adjusting
the AOM driving frequency and the IR PDH signal was measured. The measured IR error
signal is shown in Figure 7.4. The IR locking accuracy with green lock (rms) is 3.2 Hz.
The conversion from frequency noise δf to length noise δL in the 300 m cavity is
δL
Lfc
Lfc λ
=
=
= 1.06 pm/Hz.
δf
f0
c

(7.4)

Therefore, the frequency noise of 3.2 Hz corresponds to the ﬁlter cavity length noise of 3.4
pm, while the target ﬁlter cavity length noise is 1 pm. The IR locking accuracy below 10
Hz is dominated by the ﬁlter cavity length noise and it is dominated by the laser frequency
or the phase noise above 10 Hz. Although the IR locking accuracy (rms) above 10 Hz is ∼1 Hz, the ﬁlter cavity length noise below 10 Hz contributes to the IR locking accuracy of∼ 2 Hz. This means that the green lock is not enough to stabilize the IR locking accuracy
below 1 pm.

IR locking accuracy (Hz/ Hz)101

IR locking accuracy
IR locking accuracy (rms) is 3.2 Hz

100

10-1

10-2

10-3

10-4 -1
10

100

101

102

103

104

105

frequency (Hz)Figure 7.4: IR locking accuracy with the green lock. The IR locking accuracy with thegreen lock (rms) is 3.2 Hz, which corresponds to the ﬁlter cavity length noise of 3.4 pm.
It should be noted that this is an out-of-loop estimation of the IR locking accuracy.

1067.3.3FinesseThe ﬁnesse of the ﬁlter cavity was estimated by measuring the decay time of the cavity.
The ﬁlter cavity was set on resonance for an IR seed ﬁeld and, by driving the AOM, it was
suddenly driven out of resonance. The IR transmission was monitored during this process.
The ringdown measurement is shown in Figure 7.5. By ﬁtting the ringdown measurementwith Equation (3.28), the decay time is determined as τ = 2.680(1) ms. Using Equation(3.29), the ﬁnesse is estimated to be F = πfFSR τ = 4210(2), which is close to the design
value of 4360.
2

IR trasmission
fitting: finesse = 4210(2)IR transmission (V)

1.5

1

0.5

0
0

0.002 0.004 0.006 0.008

0.01

0.012 0.014 0.016 0.018

0.02

time (s)Figure 7.5: Finesse measurement with ringdown. The measured decay time is τ = 2.680(1)ms, and the corresponding ﬁnesse is 4210(2). The oﬀset was removed from the data.

7.3.4

Round-trip losses

The round-trip losses of the ﬁlter cavity can be measured from the ratio of the IR reﬂected
power from the cavity when it is on/oﬀ resonance. From Equation (3.18), this ratio canbe written as follows:
#2
"
p


Pres
rI − rE 2
rI − 1 − Λ2rtp
Rcav ==,≃Pin
1 − rI rE
1 − rI 1 − Λ2rt

(7.5)

where Pres is the IR reﬂected power on resonance and Pin is the input power, which canbe measured from the IR reﬂected power when it is oﬀ resonance. Here it is assumed thatqprE = 1 − TE ≃ 1 − Λ2rt .

(7.6)

This means that the transmissivity of the end mirror is included in the round-trip losses.

107

By solving Equation (7.5), we obtain the following:
TI (1 − Rcav )Λ2rt = √.
( Rcav rI + 1)2(7.7)

The ﬁrst round-trip loss measurement was done in 2018 before I joined this experiment.
The round-trip losses measured at that time were 50-90 ppm [63]. After I joined thisexperiment, the round-trip losses were measured again in 2020. The result is shown in
Table 7.4. The result indicates that the round-trip losses are a bit larger than before.
One possible explanation is that the ﬁlter cavity mirrors might be contaminated becase
the vacuum chamber was opened several times and no cleaning was done during the last
two years. We also measured frequency-dependent squeezing during a similar period in
2020, and therefore, we estimate that the round-trip losses are 120(30) ppm during thefrequency-dependent squeezing measurement.
Table 7.4: Measurement of round-trip losses in 2020.
Measurement date Round-trip losses
2020.01.23
148 ppm
122 ppm
2020.02.07
2020.02.10
95 ppm

7.4

Loss and phase noise outside the ﬁlter cavity

In addition to the ﬁlter cavity losses and length noise, loss and phase noise outside theﬁlter cavity are also the main squeezing degradation sources. The loss and phase noise
outside the ﬁlter cavity was determined by measuring the squeezing and anti-squeezing
levels with diﬀerent pump green powers in a frequency region where the squeezing is oﬀ
resonance of the ﬁlter cavity. Under this condition, the squeezed ﬁeld sees the ﬁlter cavity
as a perfect mirror, and therefore, the loss and phase noise outside the ﬁlter cavity can bemeasured. The result is shown in Figure 7.6. From this measurement, we ﬁnd that theloss and phase noise outside the ﬁlter cavity are 40(1) % and 30(5) mrad, respectively.
The optical loss is increased by 15 % as compared with that without the ﬁlter cavity, as
shown in Figure 6.30. The increase in the optical loss is mainly due to the in-vacuum
Faraday isolator with ∼ 15 % loss for double pass.

108

10

green power = 18,21,25,33,39,50,57 mW
loss = 40(1) %, phase noise = 30(5) mradno loss, no phase noise
loss = 40(1) %, no phase noisesqueezing (dB)

8

6

4

2

0
0

5

10

15

20

25

30

anti-squeezing (dB)Figure 7.6: Squeezing and anti-squeezing levels with diﬀerent pump green powers. The
pump green power was increased from 18 mW to 57 mW and the anti-squeezing level
increased accordingly. The error bars of the data are determined from the standard deviation of the data. From this measurement, the loss and phase noise outside the ﬁlter
cavity are estimated to be 40(1) % and 30(5) mrad, respectively.

7.5

Frequency-dependent squeezing measurement

The measurement of the frequency-dependent squeezing was done by measuring the quantum noise of the LO at the homodyne detector, changing the relative phase of the LO andthe squeezing angle (homodyne angle). The theoretical frequency-dependent squeezing
spectra for diﬀerent homodyne angles are shown in Figure 7.7.

109

Homodyne
angle

0

30
60
90

Figure 7.7: Theoretical quantum noise spectra of the frequency-dependent squeezing fordiﬀerent homodyne angles. The degradation parameters in Table 7.5 are assumed.

The measured frequency-dependent squeezing spectra for diﬀerent homodyne angles
are shown in Figure 7.8. The measured squeezing level was 3.4 ± 0.4 dB above the rotation
frequency and ∼1 dB below the rotation frequency. The measured spectra are ﬁtted witha theoretical model that uses the squeezing degradation parameters listed in Table 7.5.
These parameters are the same for every curve, while the homodyne angle and the cavity
detuning are ﬁtted for each curve. The average of the ﬁtted cavity detuning in Figure
7.8 was about 63 Hz, which corresponds to the squeezing rotation frequency of about√
63 × 2 = 90 Hz.
Although it is possible to operate the cavity at the optimal detuning, the detuningin Figure 7.8 was set to have a squeezing rotation at a slightly higher frequency thanthe optimal one (about 90 Hz instead of 75 Hz) because the spectrum is dominated by
the backscattering noise below ∼70 Hz. The backscattering noise comes from a leaked
LO which is reﬂected by the ﬁlter cavity and injected into the homodyne detector. Thisbackscattering noise depends on the mirror motion and alignment condition of the ﬁlter
cavity.

110

Homodyne angle: 0.5 0.3 deg, Cavity detuning: 47.6
0.6 Hz
Homodyne angle: 14.8 0.1 deg, Cavity detuning: 68.6
0.3 Hz
Homodyne angle: 27.7 0.1 deg, Cavity detuning: 63.2
0.4 Hz
Homodyne angle: 39.4 0.1 deg, Cavity detuning: 60.6
0.4 Hz
Homodyne angle: 61.1 0.2 deg, Cavity detuning: 69.1
0.4 Hz
Homodyne angle: 92.9 1.4 deg, Cavity detuning: 70.5
1.3 Hz

Noise relative to coherent vacuum [dB]

12
10
8
6
4
2
0
-2
-4
50

100

500

Frequency [Hz]Figure 7.8: Measured noise spectra of the frequency-dependent squeezing for diﬀerent
homodyne angles. The measured squeezing level was 3.4 ± 0.4 dB above the rotation
frequency and ∼1 dB below the rotation frequency. Each curve above 70 Hz has beenﬁtted with a theoretical model assuming the degradation parameters listed in Table 7.5,
and the homodyne angle and the cavity detuning are extracted from the ﬁtting. Eachspectrum has a resolution of 0.5 Hz and is averaged 100 times, leading to an acquisition
time of 200 s.

Table 7.5: Current and target squeezing degradation parameters.
Parameter
Current
TargetFilter cavity losses
120 ± 30 ppm 80 ppm
Propagation losses
36 % ± 1%
10 %Mode-mismatch squeezer-ﬁlter cavity
6 % ± 1%2%Mode-mismatch squeezer-local oscillator
2 % ± 1%5%Frequency-independent phase noise (rms) 30 ± 5 mrad 30 mrad
Filter cavity length noise (rms)
3 ± 0.5 pm
1 pm
Generated squeezing
8.3 ± 0.1 dB
9 dB

To check the reproducibility of the frequency-dependent squeezing spectra, the noise
spectra for homodyne angles around 90 deg were measured as shown in Figure 7.9. Although the noise level below ∼70 Hz ﬂuctuated due to the backscattering noise, each
spectrum has a rotation frequency around 100 Hz and agrees with the theoretical curve
above 70 Hz.

111

Noise relative to coherent vacuum [dB]10
Homodyne angle: 85 0.6 deg, Cavity detuning: 62.1
0.8 Hz
Homodyne angle: 100.5 0.8 deg, Cavity detuning: 77.6
0.7 Hz
Homodyne angle: 80.2 0.4 deg, Cavity detuning: 56.6
0.6 Hz
Homodyne angle: 92.9 1.4 deg, Cavity detuning: 70.5
1.3 Hz

8

6

4

2

0

-2

50

100

500

Frequency [Hz]Figure 7.9: Measured noise spectra of the frequency-dependent squeezing for homodyne
angles around 90 deg. The measured noise relative to the coherent vacuum were around±1.5 dB below the rotation frequency due to the backscattering noise. Each curve above 70
Hz has been ﬁtted with a theoretical model assuming the degradation parameters listed
in Table 7.5, and the homodyne angle and the cavity detuning are extracted from theﬁtting. Each spectrum has a resolution of 0.5 Hz and is averaged 100 times, leading to an
acquisition time of 200 s.

7.6

Discussion

The achievable quantum noise reduction in GW detectors with the ﬁlter cavity described inthis chapter is shown in Figure 7.10. Here, a squeezing angle rotation at 75 Hz, which is the
optimal frequency for KAGRA, is assumed. This plot also assumes a lossless matchingwith the interferometer. Because the loss values used in Figure 7.10 are larger thanthose for advanced detectors in the next observation runs (including the losses due to a
non-optimal coupling between the squeezer and the interferometer), the quantum noise
reduction expected in advanced detectors will be higher than that shown in Figure 7.10
[65, 66].
The drift of the alignment condition of the suspended mirrors, such as cavity mirrors
and mirrors used to inject the squeezing into the cavity, limits the mode matching between
the squeezed ﬁeld and the cavity to ∼94 %. Therefore, the auto alignment of the ﬁlter
cavity is necessary to improve the mode matching.
Propagation losses for the squeezed ﬁeld are dominated by the low OPO escape eﬃciency (∼90 %) as well as the in-vacuum Faraday isolator loss (∼15 % for double pass).
Both components will be replaced with lower loss components. In particular, Faraday
isolators with losses below 1% [67] and the OPO with an escape eﬃciency as high as 99 %[58] have been realized. It is expected that the integration of a Faraday isolator with better
112

Quantum noise reduction [dB]8

Ideal system
Phase noise
Propagation losses
Filter cavity losses
Mode mismatch
Filter cavity length noise
All mechanisms
Freq. independent squeezing

6
4
2
0
-2
-4
-6
-8
10

0

10

1

10

2

10

3

Frequency [Hz]Figure 7.10: Estimated degradation budget for the frequency-dependent squeezing source.
The squeezing degradation parameters used in this plot are listed in Table 7.5. The
solid black curve represents the expected quantum noise reduction for a GW detector
such as KAGRA. The dashed black curve represents the case for a frequency-independent
squeezing (with the same squeezing level as the frequency-dependent case). As expected,
quantum noise is signiﬁcantly reduced at low frequencies owing to frequency-dependent
squeezing as compared with frequency-independent squeezing.

isolation factor and a better vibration isolation system will also reduce the backscattering
noise to below ∼50 Hz.
Because the ﬁlter cavity was controlled and aligned with respect to the auxiliary green
ﬁeld, any relative misalignment of the squeezed ﬁeld with respect to the green ﬁeld will
not be corrected. In fact, a slow drift of the squeezed ﬁeld axis with respect to thegreen one was observed. This drift limits the long-term operation of the ﬁlter cavity. A
correlation was also observed between the alignment condition of the squeezed ﬁeld and its
detuning with respect to the cavity resonance. This relation is suspected to be the causeof the cavity detuning ﬂuctuation observed in Figure 7.8. The mechanisms that couple thealignment ﬂuctuation with the detuning ﬂuctuation are still under investigation. To solvethese problems, I proposed a new control scheme which uses the already present coherent
control ﬁeld for both length and alignment control [18]. This scheme will be explained in
Chapter 8. A similar scheme, using an additional ﬁeld at near the squeezing frequency,
has been demonstrated for the length control of a 16 m ﬁlter cavity at MIT [68].

7.7Conclusion

In this chapter, the characterization of the 300 m ﬁlter cavity and the ﬁrst demonstrationof frequency-dependent squeezing with the squeezing angle rotation below 100 Hz were113

reported.
Regarding the characterization of the 300 m ﬁlter cavity, the squeezing degradation
sources were characterized and the method to reduce them was discussed. In particular,
it is revealed that the conventional control scheme for the ﬁlter cavity with an auxiliary
green ﬁeld is not enough to stabilize the locking accuracy of the ﬁlter cavity.
Regarding the demonstration of frequency-dependent squeezing, the rotation frequencyof the squeezing angle was below 100 Hz, which is necessary for broadband quantum
noise reduction in current GW detectors. It is indicated that the frequency-dependent
squeezing source reported in this chapter can reduce the quantum noise in GW detectors
at all frequencies. Because ﬁlter cavities whose length is comparable with 300 m, as in ourdemonstration, are planned to be installed soon in Advanced LIGO, Advanced Virgo, and
KAGRA, the results presented in this chapter play a key role in realizing this technology
in advanced detectors.
This result is not only a step toward for the integration of ﬁlter cavities with the current
GW detectors in planned upgrades, but also a demonstration of the technology planned in
the third-generation detectors such as the Einstein Telescope and Cosmic Explorer, which
will use the frequency-dependent squeezing with kilometer-scale ﬁlter cavities [35, 36].

114

Chapter 8

Control of a Filter Cavity with
Coherent Control Sidebands
One of the main challenges in the generation of frequency-dependent squeezing by usingﬁlter cavities is the length and alignment control of the ﬁlter cavity. Because squeezed
vacuum has no coherent amplitude, it cannot provide the error signals necessary for thecontrol. Therefore the use of auxiliary ﬁelds is necessary. In the experiments described in
the previous chapter, the ﬁlter cavity was controlled using an auxiliary green ﬁeld witha wavelength of 532 nm, while the wavelength of the squeezed ﬁeld is 1064 nm. Because
the optical paths for the green and squeezed ﬁelds are diﬀerent and the relative alignment
between them can drift, the length and alignment control of the ﬁlter cavity with the green
ﬁeld do not ensure the alignment of the squeezed ﬁeld to the ﬁlter cavity. In addition,
the ﬂuctuation of the relative phase delay between the green and squeezed ﬁelds inducedby the anisotropies or temperature dependency of the cavity mirror coating can lead to a
detuning ﬂuctuation [69].
In this chapter, I study a new scheme for the length and alignment control for theﬁlter cavity whose length and alignment error signals can be provided by the coherent
control (CC) ﬁeld [18]. The CC ﬁeld is present in all the squeezed vacuum sources for GW
detectors and it is used to control the squeezing angle [49]. Because the coherent control
sidebands (CCSB) are generated inside the OPO together with the squeezed vacuum ﬁeld,
they have the same mode matching conditions and almost the same frequency. The relative
frequency of the squeezed ﬁeld and CCSB can be controlled accurately using a PLL andcan be tuned so that the squeezed ﬁeld is properly detuned in the ﬁlter cavity. Thisfrequency diﬀerence is only a few MHz which makes any possible phase delay eﬀect causedby the coating negligible. Therefore, the length and alignment controls with CCSB ensure
the appropriate detuning and alignment of the ﬁlter cavity for the squeezed vacuum ﬁeld.
This new control scheme was also implemented in the 300 m ﬁlter cavity for thelength control and an improvement of the locking accuracy from 3.4 pm to 0.75 pm was
successfully demonstrated. The experimental demonstration of the new control scheme isalso presented in this chapter.

115

8.1

Principle

8.1.1

Filter cavity length signal

From Equation (4.93), the CC ﬁeld passing through the OPO can be written asEcc =p2κao acav,cc = acc

1xei(ω0 +Ωcc )t + accei(ω0 −Ωcc )t+iϕpump ,2(1 − x )(1 − x2 )(8.1)

where acc is the amplitude of the CC ﬁeld at the OPO transmission without the pump
ﬁeld,
p a a
κi κo Acc
acc =
.
κa
2

(8.2)

It should be noted that Equation (8.1) assumes that Ωcc is signiﬁcantly smaller than the
OPO bandwidth κa , Ωcc ≪ κa . In this chapter, we assume that ϕpump is maintained at aconstant value equal to 0 by CC1, but has residual noise, ϕpump = δϕpump ≪ 1.
To obtain frequency-dependent squeezing from a ﬁlter cavity, the resonance of theﬁlter cavity must be properly detuned from the carrier. By choosing the frequency of theCC ﬁeld (Ωcc ) as follows, the CC ﬁeld can be resonant inside the ﬁlter cavity, while theresonance of the ﬁlter cavity is properly detuned from the carrier (Figure 8.1):Ωcc = n × ωFSR + ∆ωfc,0 ,(8.3)

where n is an integer number, ωFSR = 2πfFSR = πc/Lfc is the FSR of the ﬁlter cavity, Lfc
is the ﬁlter cavity length, and ∆ωfc,0 is the optimal ﬁlter cavity detuning with respect tothe carrier (5.29). Under this condition, the CC sideband at −Ωcc is detuned by −2∆ωfc,0
with respect to the ﬁlter cavity resonance. Thus, this sideband is almost reﬂected by theﬁlter cavity. The phase of the ﬁlter cavity reﬂectivity is shown in Figure 8.2. Because thephase of the ﬁlter cavity reﬂectivity for the CC ﬁeld, which is on resonance, is sensitive tothe ﬁlter cavity length change compared with the other sideband, which is oﬀ resonance,
the ﬁlter cavity length signal can be obtained by detecting the beat notes of CCSB.
The CCSB reﬂected by the ﬁlter cavity can be written as
Ecc = a+ r+ ei(ω0 +Ωcc )t + a− r− ei(ω0 −Ωcc )t+iδϕpump ,where a± is
a+ = acc1x, a− = acc,(1 − x2 )(1 − x2 )

(8.4)

(8.5)

and r± (∆ωfc ) = rfc (±∆ωfc,0 , ∆ωfc ) is the complex reﬂectivity of the ﬁlter cavity for the
CCSB. From Equation (5.17), the reﬂectivity can be written as
rfc (±∆ωfc,0 , ∆ωfc ) ≃ 1 −2−ϵ,
1 + iξ(±∆ωfc,0 , ∆ωfc )

116

(8.6)

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!fc,0

Figure 8.1: Frequency relationship inside the ﬁlter cavity. The red dashed line represents
a carrier, and the orange and blue lines represent the CCSB.

whereξ(±∆ωfc,0 , ∆ωfc ) =±∆ωfc,0 − ∆ωfc
.
γfc

(8.7)

It should be noted that the sideband frequency Ω is ﬁxed at ±∆ωfc,0 for the CCSB under
the condition expressed in Equation (8.3) and ∆ωfc is the variable representing the actual(nonoptimal) ﬁlter cavity detuning.
Amplitude and phase of the ﬁlter cavity reﬂectivity for the CCSB can be written as
ρ± (∆ωfc ) = |rfc (±∆ωfc,0 , ∆ωfc )|s(2 − ϵ)ϵ=,1−
2
1 + ξ (±∆ωfc,0 , ∆ωfc )(8.8)

α± (∆ωfc ) = arg{rfc (±∆ωfc,0 , ∆ωfc )}= arg{−1 + ϵ + ξ 2 (±∆ωfc,0 , ∆ωfc ) + i(2 − ϵ)ξ(±∆ωfc,0 , ∆ωfc )}. (8.9)
The ﬁlter cavity length signal can be obtained by detecting the beat notes of the CCSB
as follows:Pcc =a+ r+ ei(ω0 +Ωcc )t + a− r− ei(ω0 −Ωcc )t+iδϕpump2∗ i(2Ωcc t−δϕpump )= (DC term) + 2a+ a− Re{r+ r−e}.

(8.10)

Demodulating this signal by sin (2Ωcc t − α− (∆ωfc,0 )) (in-phase) and cos (2Ωcc t − α− (∆ωfc,0 ))(quadrature) and low-passing it, we can obtain the ﬁlter cavity length signal as a function

117

Phase of filter cavity reflectivity (rad)

0

-

-2

-3

-2

-1

0

1

2

3

4

5

Normalized sideband frequency

Figure 8.2: Phase of ﬁlter cavity reﬂectivity. The horizontal axis is the sideband frequency
normalized with respect to ∆ωfc,0 . The black dashed lines represent the sideband frequencyof the CCSB.

of the ﬁlter cavity detuning ∆ωfc as follows:PI= −a+ a− ρ+ (∆ωfc )ρ− (∆ωfc )× sin (α+ (∆ωfc ) − α− (∆ωfc ) + α− (∆ωfc,0 ) − δϕpump ),(8.11)

PQ = a+ a− ρ+ (∆ωfc )ρ− (∆ωfc )× cos (α+ (∆ωfc ) − α− (∆ωfc ) + α− (∆ωfc,0 ) − δϕpump ).

(8.12)The relative phase noise of the CCSB δϕpump is a noise source for the ﬁlter cavity length
signal. Assuming δϕpump = 0, the normalized ﬁlter cavity length signals (8.11) and (8.12)
with respect to a+ a− are shown in Figure 8.3. The parameters used in this calculation
are listed in Table 8.1.
The ﬁlter cavity length noise δLfc causes detuning noise δ∆ωfc as follows:δ∆ωfc =

ω0
δLfc .
Lfc(8.13)

When ∆ωfc ≡ ∆ωfc,0 + δ∆ωfc is deﬁned, the phase response of the ﬁlter cavity reﬂectivity
to the detuning noise can be calculated from Equation (8.9) as follows:dα(Ω, ∆ωfc )
δ∆ωfc
d∆ωfc
∆ωfc =∆ωfc,0
−1
(Ω − ∆ωfc,0 )2
8F+1≃
δLfc ,
2λγfc

δα(Ω) =

(8.14)

where α(Ω, ∆ωfc ) = arg{rfc (Ω, ∆ωfc )}, λ is the wavelength of the carrier, and F is theﬁlter cavity ﬁnesse. Here, it is assumed that ϵ ≪ 1, which is true for the parameters listed118Normalized filter cavity error signal

1

0.5

0

-0.5

-1
-3

CCSB intracavity power
Carrier intracavity power
Error signal (In-phase)
Error signal (Quadrature)

-2

-1

0

1

2

3

4

5

Normalized filter cavity detuning with respect to carrier

Figure 8.3: Filter cavity error signal normalized with respect to a+ a− . The horizontal
axis is the ﬁlter cavity detuning normalized with respect to ∆ωfc,0 . The red solid anddashed lines represent the intracavity power of the CCSB and carrier in the ﬁlter cavity
normalized with respect to their maximum intracavity powers, respectively. The blue andgreen lines represent the ﬁlter cavity length signal (in-phase and quadrature). The ﬁlter
cavity length signal (in-phase) becomes 0 when ∆ωfc = ∆ωfc,0 .

in Table 8.1. The ﬁlter cavity length signal (In-phase) (8.11) isPIa+ a− ρ+ ρ−= sin(α+ (∆ωfc,0 + δ∆ωfc ) − α+ (∆ωfc,0 ) − α− (∆ωfc,0 + δ∆ωfc ) + α− (∆ωfc,0 ))≃ δα(∆ωfc,0 ) − δα(−∆ωfc,0 )




1064 nmF
δLfc= 26 mrad
.
λ
4360
1 pm

(8.15)

Because the relative phase noise of the CCSB (δϕpump ) can be stabilized by CC1 below
1.7 mrad [58], the residual ﬁlter cavity length signal (8.15) can be obtained with a goodenough SNR. The phase noise of an RF source for the demodulation also becomes a noise
source for the ﬁlter cavity length signal. The typical phase noise of an RF source for the
demodulation is several tens of µrad and, therefore, signiﬁcantly smaller than the residual
ﬁlter cavity length signal (8.15).

8.1.2

Filter cavity alignment signal

The CCSB can be also used to control the alignment of the ﬁlter cavity using the wave
front sensing (WFS) technique [70]. The principle of wave front sensing is explained inAppendix A.
The misalignment of the ﬁlter cavity axis with respect to the input beam axis as wellas that of the immediately reﬂected beam axis with respect to the ﬁlter cavity axis can be

119Table 8.1: Parameters for 300 m ﬁlter cavity [62].
Parameter
SymbolValue
Filter cavity length
Lfc
300 m
Filter cavity half-bandwidth
γfc
2π × 57.3 Hz
Filter cavity detuning∆ωfc,02π × 54 Hz
Filter cavity ﬁnesse
F
4360
Filter cavity input mirror transmissivity
t2in0.00136
Filter cavity round-trip losses
Λ2rt
80 ppm2
Injection lossesΛinj
5%
Readout losses
Λ2ro
5%
2
Mode-mismatch losses
ΛmmFC2%(squeezer-ﬁlter cavity)
Mode-mismatch losses
Λ2mmLO5%(squeezer-local oscillator)
Frequency-independent phaseδϕ
30 mradnoise (rms)δLfc
Filter cavity length noise (rms)
1 pm
Generated squeezing
σdB
9 dB
Nonlinear gaing
3.6

represented in terms of the dimensionless coupling factors γ and γr as follows:γ = δx/w0 + iδθ/θ0 ,′′γr = δx /w0 + iδθ /θ0 ,(8.16)(8.17)

where w0 is the beam radius at the waist position and θ0 = λ/πw0 is the beam divergence.
δx and δx′ represent the shift in the x-axis direction measured at the waist position of theﬁlter cavity axis with respect to the input beam axis and the immediately reﬂected beamaxis with respect to the ﬁlter cavity axis, respectively. δθ and δθ′ represent the tilt aroundthe y-axis of the ﬁlter cavity axis with respect to the input beam axis and the immediately
reﬂected beam axis with respect to the ﬁlter cavity axis, respectively. Here, the z-axis isthe beam axis and the y-axis is orthogonal to the x- and z-axis. z = 0 is the beam waist
position. γ and γr can be expressed in terms of the input and end mirror misalignment of
the ﬁlter cavity as follows:R
R δθI + δθEδθI − δθE+i2w0
2R − Lfc
θ0
R
R
2δθI − 2R−L(δθI − δθE )Lfc δθI − 2 (δθI + δθE )
fc
+i
,
=
w0
θ0


Lfc
2
=
+i
δθI − γ,
w0
θ0

γ =
γr

(8.18)(8.19)

where R is the radius of curvature of the input and end mirrors, δθI and δθE are the angular
misalignments of the input and end mirrors, respectively. The directions of the input andend mirror misalignments are deﬁned so that the positive direction of the misalignment
120

causes the shift of the ﬁlter cavity axis in the positive direction of the x-axis.
We treat only the x-axis misalignment (HG 10 mode), and the calculations of they-axis misalignment (HG 01 mode) are entirely analogous.
From Equation (A.21) and (A.22), the 2Ωcc term of the ﬁlter cavity error signal (8.10)
can be written as∗ i2Ωcc te}
Pcc (2Ωcc ) = 2a+ a− Re{r+ r−∗ ∗∗ ∗ i2Ωcc t= 2a+ a− Re{(U00 rc0 − U10 C)(U00
rs0 − U10
S )e
}
∗
∗
∗
= 2a+ a− Re{(U00 U00
rc0 rs0− U00 U10
rc0 S ∗∗∗∗− U00
U10 rs0
C + U10 U10
CS ∗ )ei2Ωcc t }.

(8.20)The WFS signal (W ) is expressed as the sum of the second and the third terms of∗ yields
Equation (8.20). Deﬁning U = U00 U10
W

∗ ∗
∗
∗
= 2a+ a− Re({−U rc0 (rs0γr + rs1γ) − U ∗ rs0(rc0 γr + rc1 γ ∗ )}ei2Ωcc t )∗∗∗= 2a+ a− Re({−rc0 rs0(U γr∗ + U ∗ γr ) − rc0 rs1
U γ − rc1 rs0
U ∗ γ ∗ }ei2Ωcc t ). (8.21)

The diﬀerential signal of W in the x-axis direction with a QPD is
Z Z
Wdiﬀ =
dxdy{W (x > 0) − W (x < 0)}.

(8.22)

Asr

Z Z
dxdy{U (x > 0) − U (x < 0)} =2 −iη(z)e,π(8.23)

and rc1 , rs1 ≃ 1 because of the Gouy phase separation in the cavity, the WFS signal canbe written asr2∗ −iη ∗∗ iη ∗ i2Ωcc t
Wdiﬀ = 2a+ a−Re({−r+ r−(e γr + eiη γr ) − r+ e−iη γ − r−e γ }e),π(8.24)
where r± = r± (∆ωfc,0 ) and η represents the Gouy phase. Demodulating (8.24) bysin (2Ωcc t − α− (∆ωfc,0 )) (In-phase) and low-passing it, the ﬁrst term of Equation (8.24),
which is proportional to the ﬁlter cavity length signal, will disappear.
The WFS signal after the demodulation is
r

WI

2
a+ a−
π
∗ iη ∗
× {Re(r+ e−iη γ + r−e γ ) sin α− (∆ωfc,0 )=+∗ iη ∗Im(r+ e−iη γ + r−e γ ) cos α− (∆ωfc,0 )}.

(8.25)The WFS signal as a function of the Gouy phase is shown in Figure 8.4. The displacement121normalized WFS signal

0.5

shift
tilt

0

-0.5

0

/2

3 /2

2

gouy phase (rad)Figure 8.4: Filter cavity WFS signal as a function of the Gouy phase normalized withq2
π a+ a− . The red and blue lines represent the shift signal with δx = w0 and the tilt
signal with δθ = θ0 , respectively.

and tilt signals of the ﬁlter cavity can be obtained with two diﬀerent Gouy phases.
The fourth term in Equation (8.20) is the beat note of the CCSB HG10 mode andbecomes a noise source for the ﬁlter cavity length signal, which is the ﬁrst term in Equation(8.20). The fourth term in Equation (8.20) is
2a+ a− |U10 |2 Re{CS ∗ ei2Ωcc t }
∗
∗
∗ ∗ ∗
∗
= 2a+ a− |U10 |2 Re{(rc0 rs0|γr |2 + rc0 rs1
γr γ + rc1 rs0γ γr + rc1 rs1|γ|2 )ei2Ωcc t }.

(8.26)
Demodulating by sin (2Ωcc t − α− (∆ωfc,0 )), the ﬁrst term in Equation (8.26), which is
proportional to the ﬁlter cavity length signal, will disappear. After integration with respect
to x, y, Equation (8.26) will become−a+ a−{∗ ∗ ∗Re(rc0 γr γ + rs0
γ γr ) sin α− (∆ωfc,0 )∗ ∗ ∗+ Im(rc0 γr γ + rs0
γ γr ) cos α− (∆ωfc,0 )+ |γ|2 sin α− (∆ωfc,0 )},(8.27)

where rc1 , rs1 ≃ 1 is used. From Equation (8.18) and (8.19), Equation (8.27) can be
numerically calculated in terms of the input and end mirror misalignment δθI , δθE asshown in Figure 8.5. Here it is assumed that R = 415 m and w0 = 0.825 cm [63].
Equation (8.27) normalized with a+ a− should be smaller than the residual ﬁlter cavity
length signal normalized with a+ a− which is PI /a+ a− = 23 mrad. It should be notedthat the ﬁlter cavity length signal in Equation (8.15) is normalized with a+ a− ρ+ ρ− . We
2 = θ2consider the maximum angular motion of the ﬁlter cavity mirrors θmax as δθI2 +δθE
max
122

end mirror misalignment (urad)

5

0.05

0.04

0

0.03

0.023

-5
-5

0

5

0.01

input mirror misalignment (urad)

Figure 8.5: Coupling from input and end mirror misalignment of the ﬁlter cavity to
the ﬁlter cavity length signal (8.27) normalized with respect to a+ a− . As the couplingshould be smaller than the ﬁlter cavity length signal ∼ 0.023, the input and end mirror
misalignment should be inside the ellipse corresponding to 0.023.

which corresponds to a circle with radius θmax in Figure 8.5. This circle should be smaller
than the ellipse corresponding to 0.023. θmax can be determined from the semi-minor
axis of the ellipse corresponding to 0.023 in Figure 8.5. Thus, the requirement of θmax is
obtained as θmax = 2.7 µrad, which is achievable through double pendulum suspensions in
KAGRA [71]. By numerically calculating |γ|2 as a functions of the input and end mirror
misalignment, this requirement corresponds to |γ|2 = 0.01. Because this requirement
is more stringent than the requirement of mode-mismatch losses (squeezer-ﬁlter cavity),
which is 2 %, the requirement of γ is set as |γ|2 = 0.01.

8.1.3

Experimental setup in GW detectors

An example of the experimental implementation of this scheme in GW detectors is shownin Figure 8.6. There are three control loops related to the CC ﬁeld which are CC1, CC2,
and the ﬁlter cavity control loop with CCSB (hereafter, we refer to it as CCFC).
The CCFC error signal can be obtained at an output mode cleaner (OMC) reﬂection
because the CCSB are almost fully reﬂected by the OMC, while the carrier is almost
entirely transmitted. The CCFC error signal is obtained by demodulating the OMC reﬂection at a frequency of 2Ωcc and is fed back to the ﬁlter cavity length. The demodulation
phase can be determined by injecting the bright carrier ﬁeld to the ﬁlter cavity and simultaneously checking the carrier transmission and CCFC error signal, as shown in Figure8.3. The demodulation phase can be ﬁne-tuned by optimizing the GW sensitivity. The
CC1 error signal for controlling the relative phase between the green pump ﬁeld and theCC ﬁeld can be obtained at the OPO reﬂection and it is fed back to the optical path
length of the pump ﬁeld. The CC2 error signal for controlling the relative phase between
the carrier and the CC ﬁeld is obtained at the OMC transmission and fed back to the PLL

123

between the interferometer and squeezer lasers [72].

Interferometer
Interferometer Laser

Filter Cavity

PLL
CCFC

2

OMC

cc

Squeezer Laser

SHG

Coherent Control Laser

OPO

PLL
cc

cc

CC2

2

cc

CC1

Figure 8.6: Example of experimental implementation of a CCFC scheme in GW detectors.
The red solid, red dashed and orange lines represent the carrier, squeezed ﬁeld, and CC
ﬁeld, respectively. The green line represents the green pump ﬁeld, which is generated by
the SHG and injected into the OPO.

8.1.4

Coherent control error signal

As either of CCSB enters the ﬁlter cavity and senses the ﬁlter cavity length noise, the ﬁlter
cavity length noise appears in the CC2 loop, which controls the relative phase between
the LO and CCSB. In the case of GW detectors, the LO is the interferometer laser.
In this section, the CC2 error signal, which includes the phase noise coming from theﬁlter cavity, is calculated. For simplicity, we write ρ± (∆ωfc ) = ρ± , α± (∆ωfc ) = α± and
α± (∆ωfc,0 ) = α±,0 .
The power at the OMC transmission is
PCC2 =+

a0 ei(ω0 t+ϕLO ) + τcc a+ ρ+ ei(ω0 +Ωcc )t+i(ϕCC +α+ )τcc a− ρ− ei(ω0 −Ωcc )t+i(ϕCC +α− +δϕpump )2= 2τcc a0 a+ ρ+ cos (Ωcc t − ϕLO + ϕCC + α+ )+ 2τcc a0 a− ρ− cos (Ωcc t + ϕLO − ϕCC − α− − δϕpump )+ (DC term) + (2Ωcc term),(8.28)

where a0 is the amplitude of the LO and τcc is the transmissivity of the CCSB from the
OPO to the OMC transmission.
124

Demodulating (8.28) by cos (Ωcc t + θdm,CC2 ) and low-passing it, where the demodulation phase θdm,CC2 isθdm,CC2 =α+,0 − α−,0,2(8.29)

we ﬁndπPI = τcc a0 a+ ρ+ sin −ϕLO + ϕCC + αp,0 + δα+ +2π− τcc a0 a− ρ− sin ϕLO − ϕCC − αp,0 − δα− − − δϕpump ,2(8.30)

where
α+,0 + α−,0,2= α± − α±,0 .

αp,0 =(8.31)

δα±(8.32)

When the squeezing angle ϕsqz is changed from π/2 (squeeze quadrature) by δϕsqz , Equation (4.92) can be written as
ϕsqz = ϕLO − ϕCC =π+ δϕsqz .
2(8.33)

Assuming δϕsqz , δα± , δϕpump ≪ 1, the CC2 error signal (8.30) can be written as
PI = τcc a0 a+ ρ+ [(1 + aρ) sin αp,0+{−(1 + aρ)δϕsqz + 2δαp (∆ωfc , a, ρ) + aρδϕpump } cos αp,0 ],(8.34)

where a = a− /a+ = x is the unbalance of the amplitude of the CCSB and ρ = ρ− /ρ+ isthe unbalance of the ﬁlter cavity reﬂectivity of the CCSB. δαp (∆ωfc , a, ρ) becomesδαp (∆ωfc , a, ρ) =

δα+ + aρδα−.
2(8.35)

The ﬁrst term in Equation (8.34) is the constant oﬀset, the second term is the relative
phase noise between CC and LO, which does not include the phase noise coming from theﬁlter cavity (frequency-independent phase noise), the third term is the phase noise of the
CCSB coming from the ﬁlter cavity length noise (frequency-dependent phase noise at the
detuning frequency), and the fourth term is the phase noise coming from the relative phasenoise of the CCSB. The constant oﬀset in Equation (8.34) should be removed to obtain
δϕsqz . δαp (∆ωfc , a, ρ) is the coupling from the CCFC loop which reshapes frequencydependent phase noise as explained in the following section.

8.1.5

Reshaping of frequency-dependent phase noise

The CC2 error signal calculated in Section 8.1.4 reshapes the frequency-dependent phasenoise, which originates from the ﬁlter cavity length noise. This is caused by the coupling125

between the CCFC and CC2 loops as shown in Figure 8.7.

Filter cavity length control loop (CCFC)
Phase noise of CCSB

2⌦cc
<latexit sha1_base64="lEigyDw+d9fp2dXZIy2JeLYDCsE=">AAACfHichVHdShtBGD1Ztaap1tgiCN5IE0UQw5cIqXol9MY7o2k0YCTsjpO4uH/sTgK6+AJ9AS+8slDw5zG88QV6kUcovUzBGyn9drO0lFD7DTNz5sx3vjkzY3iWGSiiXkobGR17MZ5+mXk1Mfl6Kjv9Zi9wO76QNeFarl839EBapiNrylSWrHu+1G3DkvvGyYdof78r/cB0nY/q1JOHtt52zJYpdMVUMztTyje2bdnWmw1bV8e+HQpxnm9mc1SgOOaHQTEBOSRRcbPXaOAILgQ6sCHhQDG2oCPgdoAiCB5zhwiZ8xmZ8b7EOTKs7XCW5Ayd2RMe27w6SFiH11HNIFYLPsXi7rNyHgv0lW6oTw90R9/o6Z+1wrhG5OWUZ2OglV5z6tNs9fG/KptnheM/qmc9K7SwFns12bsXM9EtxEDfPbvoVzd2F8JF+kzf2f8V9eieb+B0f4gvO3L3kqtn4i9Yj6L8+8GHwV6pUFwtrO6UcpuUfEYac3iHJX7x99jEFiqo8ckhrnCLu9RPLa8tayuDVC2VaN7ir9DKvwBl85Jy</latexit>

differential

FC error signal

FC lengthcommonCC2 error signal
CC2 phasecommon⌦cc<latexit sha1_base64="haTUItVd5lF0Y2E+qwEF2/NuNKg=">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</latexit>

Coherent control loop (CC2)common= Frequency-dependent phase noise at the detuning
frequency

common= Frequency-independent phase noise

Figure 8.7: Coupling between CCFC and CC2 loops.

The ﬂuctuation of the ﬁlter cavity length causes both diﬀerential and common phasenoise of the CCSB. The diﬀerential phase noise of the CCSB is the CCFC error signal,
while the common phase noise of the CCSB is the frequency-dependent phase noise at
the detuning frequency, which is the second term in Equation (8.34). This frequencydependent phase noise at the detuning frequency couples to the CC2 loop and is suppressedby the CC2 feedback loop, while the frequency-dependent phase noise increases at highfrequencies. In this section, the calculation of the frequency-dependent phase noise withthe feedback of the CC2 loop is explained.
The frequency-dependent phase noise when the CCSB are oﬀ resonance of the ﬁlter
cavity can be calculated as Equation (5.53). However, with the CCFC scheme, the detuning noise of the ﬁlter cavity (8.35) is fed back by the CC2 loop. As shown in Figure
8.6, this feedback from the CC2 loop is sent to the squeezer laser, which leads to a change
in the injection squeezing angle ϕ. Therefore, the frequency-dependent phase noise of N̂1
with feedback from the CC2 loop can be calculated as follows:N̂1,avg (ϕ, ∆ωfc,0 )
1n=N̂1 (−δαp (∆ωfc,0 + δ∆ωfc , a, ρ), ∆ωfc,0 + δ∆ωfc )2
o+ N̂1 (−δαp (∆ωfc,0 − δ∆ωfc , a, ρ), ∆ωfc,0 − δ∆ωfc ) .

(8.36)It should be noted that frequency-dependent phase noise δαp is low above the cavity poleof the ﬁlter cavity ∼ 57 Hz, and it is assumed that the gain of the CC2 loop below thecavity pole is large enough such that the feedback of the CC2 loop is perfect.
Figure 8.8 and 8.9 show quantum noise relative to the coherent vacuum with ﬁlter
cavity length noise δLfc = 1 pm and 3 pm. The frequency-dependent phase noise with126

the CCFC scheme and the conventional scheme are represented as purple and dotted
purple lines, which are almost overlapping in Figure 8.8. The parameters used in thiscalculation are listed in Table 8.1. The unbalance of the ﬁlter cavity reﬂectivity for the
CCSB is ρ = 1.1 and the unbalance of the amplitude for the CCSB is a = 0.47 (g = 3.6).
As shown in Figure 8.8, the frequency-dependent phase noise with the CCFC scheme
and conventional schemes are low and almost the same with δLfc = 1 pm. However, as
indicated in Figure 8.9, the frequency-dependent phase noise with the CCFC scheme issuppressed at low frequencies by the feedback from the CC2 loop, while it is increased athigh frequencies.

Quantum noise relative to coherent vacuum [dB]0
Ideal system
Injection/Readout losses
Filter cavity losses
Mode mismatch (worst case)Frequency-independent phase noise
Frequency-dependent phase noise (CCFC scheme)
Frequency-dependent phase noise (conventional scheme)
All mechanisms (CCFC scheme)
All mechanisms (conventional scheme)

-1
-2
-3
-4
-5
-6
-7
-8
-9
-10

10 1

10 2

10 3

Frequency [Hz]Figure 8.8: Quantum noise relative to coherent vacuum with δLfc = 1 pm. The solid
purple and black lines represent the frequency-dependent phase noise and total noise with
the CCFC scheme, respectively. The dotted purple and black lines represent the frequencydependent phase noise and total noise with the conventional scheme, respectively. The
solid lines and dotted lines are almost overlapping because the eﬀect of the frequencydependent phase noise is small with δLfc = 1 pm.

Eﬀective phase noise at high frequencies with feedback from the CC2 loop (δαp in
Equation (8.36)) can be calculated from Equation (8.14) and (8.35) as follows:δα(∆ωfc,0 ) + aρδα(−∆ωfc,0 )
 2



1064 nmF
δLfc≃ 18 mrad
.
λ
4360
1 pm

|δαp (∆ωfc,0 ± δ∆ωfc , a, ρ)| =(8.37)The squeezing angle is aﬀected by the misalignment of LO and CC. The squeezing angle
ﬂuctuation at the OMC transmission, including the misalignment of the LO and CC, can

127Quantum noise relative to coherent vacuum [dB]0
Ideal system
Injection/Readout losses
Filter cavity losses
Mode mismatch (worst case)Frequency-independent phase noise
Frequency-dependent phase noise (CCFC scheme)
Frequency-dependent phase noise (conventional scheme)
All mechanisms (CCFC scheme)
All mechanisms (conventional scheme)

-1
-2
-3
-4
-5
-6
-7
-8
-9
-10

10 1

10 2

10 3

Frequency [Hz]Figure 8.9: Quantum noise relative to coherent vacuum with δLfc = 3 pm. The solid
purple and black lines represent the frequency-dependent phase noise and total noise with
the CCFC scheme, respectively. The dotted purple and black lines represent the frequencydependent phase noise and total noise with the conventional scheme, respectively. The
frequency-dependent phase noise with the CCFC scheme is reduced at low frequencies,
while it is increased at high frequencies compared with that with the conventional scheme.

be written as [73]δϕalignment ≃XLO
Aij ρCCij ρij sin ϕij ,(8.38)

ij

where Aij ∼ 1/100 is the attenuation factor of higher order modes caused by the OMCLOand ρCC
ij and ρij are the relative amplitudes of the CC and LO TEM ij modes withCC
LO CC
respect to the TEM 00 mode. ϕij = ϕLOij − ϕij and ϕij , ϕij are the relative phases of
the CC and LO TEM ij modes with respect to the TEM 00 mode. Considering only the2
LO 2−2 for i + j = 1, the squeezing
HG10/01 mode and assuming that |ρCC
ij | = |ρij | = 10
angle ﬂuctuation will be δϕalignment ∼ O(10−4 ) rad, which is low enough compared withthe frequency-independent/dependent phase noise.

8.2

Noise Calculation

The requirements of length and alignment control for the ﬁlter cavity are δLfc = 1 pm and|γ|2 = 0.01, respectively. In this section, it is shown that shot noise and PLL noise satisfy
these requirements. It is also illustrated that backscattering noise caused by the leaked
carrier from the interferometer to the ﬁlter cavity does not spoil the quantum noise above
10 Hz, where the quantum noise limits the GW sensitivity.

1288.2.1

Shot noise

Shot noise for length control
We assume that the power of the CC ﬁeld after passing through the OPO is P+ = a2+ = 1µW and the power of the lower CC sideband is P− = a2 P+ = 0.22 µW. When the ﬁlter
cavity length signal is obtained at the OMC reﬂection, the junk light at the OMC reﬂection,
which includes higher order modes and other RF sidebands, contributes to the shot noise.
The shot noise of the CCSB and the junk light at the OMC reﬂection is written asqp
Pshot = 2ℏω0 (ρ2+ P+ + ρ2− P− + Pjunk ) ≃ 2ℏω0 Pjunk ,(8.39)

where it is assumed that the frequencies of the carrier, CCSB, and junk light are approximately the same and ρ2± P± ≪ Pjunk . This shot noise within the ﬁlter cavity control
bandwidth becomes the ﬁlter cavity length noise owing to the control loop. This shot noiseis the most fundamental limit for the ﬁlter cavity length signal. From Equation (8.15),
we can calculate the maximum power of the junk light at the OMC reﬂection so as not to
spoil the ﬁlter cavity length signal as follows,
ρ+ ρ−

p
p
P+ P− {δα(∆ωfc,0 ) − δα(−∆ωfc,0 )} >2ℏω0 Pjunk ∆f ,P+ P− δL2fc 2Fℏω0 ∆f λ2= 15 W

 
2 
 

P+ 2
F
δLfc 2 20 Hz
×
,
1 µW
4360
1 pm
∆f

(8.40)

Pjunk < 15

(8.41)

where ∆f is the ﬁlter cavity length control bandwidth which is set by the requirementof backscattering noise [65]. According to Equation (8.41) and assuming the parameterslisted in Table 8.1, Pjunk < 15 W should be satisﬁed so as not to spoil the ﬁlter cavity
length signal. This requirement is satisﬁed [74].
Shot noise for alignment control
In this section, we consider only the shot noise of the shift signal of the ﬁlter cavity. The
calculation for the tilt signal is entirely analogous. From Equation (8.25), we can calculate
the maximum power of the junk light at the OMC reﬂection so as not to spoil the ﬁlter
cavity alignment signal as follows:
r

2p
δx p
> 2ℏω0 Pjunk ∆fWFS ,P+ P− aWFS
π
w0

129

(8.42)

Pjunk

 2
P+ P−
δx
< 0.056ℏω0 ∆fWFS w0= 660 W P+ 2 (δx/w0 )2
1 Hz
,
×
1 µW
0.01
∆fWFS

(8.43)

where aWFS = 0.42 is the maximum amplitude of the normalized WFS signal in Figure
8.4 and ∆fWFS is the ﬁlter cavity alignment control bandwidth. According to Equation(8.43), Pjunk < 660 W should be satisﬁed so as not to spoil the ﬁlter cavity alignment
signal and this requirement is also satisﬁed.

8.2.2

Backscattering noise

The backscattering noise originates from the leaked carrier from the interferometer to theﬁlter cavity. The leaked carrier is injected into the ﬁlter cavity and scattered by the ﬁlter
cavity length noise and reinjected into the interferometer. Because this leaked carrier has
the same frequency as the squeezed ﬁeld, the ﬂuctuation of the leaked carrier must be lower
than the vacuum ﬂuctuation so as not to spoil the quantum noise of the interferometer[65]. Considering the safety factor (Csafe = 1/10) and the squeezing enhancement factor(Csqz ≃ 1/2 for 6 dB of quantum noise enhancement),
δα(0)

p
p
Pleak < Csafe Csqz 2ℏω0 ,−10Pleak < 3.1 × 10
4360 2×F(8.44)

W√ !2
10−16 m/ Hz,
δLfc (f )(8.45)

where δα(0) is the phase response of the carrier to the ﬁlter cavity length noise (8.14) andPleak is the power of the leaked carrier from the interferometer to the ﬁlter cavity. It is√
assumed that δLfc (f ) = 10−16 m/ Hz above 10 Hz, which can be realized in Advanced
LIGO [65]. As the carrier output from the interferometer is Pcarrier = 35 mW in Advanced
LIGO [65], the isolation factor of 81 dB is required from the output of the interferometerto the ﬁlter cavity. Recently, a Faraday isolator with an isolation factor higher than 40
dB and less than 1 % of loss [67] has been developed. Using two Faraday isolators, anisolation factor higher than 80 dB can be achieved with a loss less than 2 %, and thebackscattering noise can satisfy the requirement.

8.2.3

PLL noise

The PLL that controls the relative frequency between the squeezer laser and the CC laser
can cause the detuning noise. The PLL frequency noise reﬂected by the ﬁlter cavity canbe written as
SPLL (ϕ)f
SPLL,fc (f ) = p,(8.46)
1 + (f /ffc )2130

√where ffc = 57.3 Hz is the ﬁlter cavity half bandwidth and SPLL (ϕ) = 5 µrad/ Hz is the
PLL phase noise within the PLL control bandwidth ∼ 40 kHz. The PLL phase noise has
been chosen such that the rms value of the PLL phase noise, δϕPLL = 1 mrad. The rms
value of the PLL frequency noise within the ﬁlter cavity control bandwidth iss
Z ∆f
δfPLL,fc =0

2
df SPLL,fc
(f )

s
Z ∆f
= SPLL (ϕ)
df
0f2
1 + (f /ffc )2s∆f= SPLL (ϕ)ffc∆f − ffc arctan.
ffc(8.47)The rms value of the PLL frequency noise is δfPLL,fc = 0.25 mHz, which corresponds to
the rms value of the ﬁlter cavity length noise δLPLL,fc = 2.7 × 10−16 m. Therefore, the
PLL noise satisﬁes the requirement.

8.3

Comparison to the other locking scheme

Other than the CCFC scheme, a ﬁlter cavity locking scheme using the resonant locking
ﬁeld (RLF) has been demonstrated at MIT for the length control [68]. The conﬁgurationof the CCFC and RLF schemes in GW detectors is shown in Figure 8.10. RLF is phaselocked to the CC ﬁeld and detuned with respect to it to resonate inside the ﬁlter cavity.
RLF is injected into the OPO together with the CC ﬁeld and shares the same optical pathwith it, except that the RLF is resonant inside the ﬁlter cavity, while the CC ﬁeld is not.
The ﬁlter cavity length signal is obtained from the beat note of the RLF and LO from theinterferometer at the OMC transmission.
Compared with the RLF scheme, the CCFC scheme has the advantage in that it uses
the already present CC ﬁeld and does not require any modiﬁcation of the optical setup.
However, the CCFC scheme is worse in terms of the shot noise level. Here, we compare
the SNR in the CCFC and RLF schemes in terms of shot noise.
SNR in the CCFC scheme in terms of the shot noise is
SNRCCFC ∼ p

PCC
δαm ,
2ℏω0 Pjunk(8.48)

where PCC is the CC power at the OMC reﬂection and δαm is the relative phase changeof the CCSB by the ﬁlter cavity length noise.
SNR in the RLF scheme in terms of shot noise is√√
PLO APRLF
APRLF
SNRRLF ∼ √δαp = √
δαp ,(8.49)
2ℏω0 PLO2ℏω0
where PLO is the LO power from the interferometer at the OMC transmission, PRLF is the
RLF power before OMC, A ∼ 1/100 is the attenuation factor of the RLF by the OMC,
131

RLF

CCFC

Coherent control field

Filter Cavity

OMC

2

OPO

cc

Resonant locking fieldrlf

ccFilter Cavity

OMC

filter cavity control
Coherent control field

OPO
ccsqueezing angle control

ccfilter cavity control

squeezing angle control
rlf

cc

Figure 8.10: Conﬁguration of CCFC (left) and RLF (right) schemes in GW detectors.
The solid red, orange, purple, and dashed red lines represent the carrier ﬁeld, CC ﬁeld,
RLF, and squeezed ﬁelds, respectively. The CC and RLF are detuned by Ωcc and Ωrlf
with respect to the carrier, respectively.

and δαp is the common phase change of the CCSB by the ﬁlter cavity length noise.
The ratio of the SNRs in the CCFC and RLF schemes iss
SNRCCFC
PCC
1
PCC
∼p
∼ 0.03,(8.50)∼√SNRRLF
Pjunk APRLF
A Pjunk
where we assumed that δαp ∼ δαm and PCC = PRLF = 1 µW, Pjunk = 100 mW which are
typical values in GW detectors. This ratio indicates that the SNR in the CCFC scheme isworse than that in the RLF scheme by a factor of ∼ 30. Nevertheless, the shot noise levelin the CCFC scheme can satisfy the requirement, as it was calculated in Section 8.2.1.

8.4

Demonstration of CCFC lock with 300 m ﬁlter cavity

The CCFC lock was implemented for the length control of the 300 m ﬁlter cavity and an
improvement of the locking accuracy of the ﬁlter cavity was demonstrated. In this section,
the experimental result of the CCFC lock for a 300 m ﬁlter cavity is described.

8.4.1

Experimental setup

The experimental setup for the demonstration of the CCFC lock is shown in Figure 8.11.
The setup is almost the same as that for the frequency-dependent squeezing measurement,
but a 50:50 BS is added to the IR reﬂection path of the ﬁlter cavity to pick oﬀ the CCSB
132

at the expense of introducing an additional loss of 50 %. In the real GW detectors, the
CCSB can be obtained at the OMC reﬂection without introducing the pick oﬀ mirror as
explained in Section 8.1.3. The CCSB are detected with RFPD and demodulated at 14
MHz to obtain the CCFC error signal. The CCFC error signal is added to the ﬁlter cavity
green error signal and fed back to the main laser frequency. The block diagram of the
CCFC and green locks is shown in Figure 8.12. We set the green pump power to 42 mW
to obtain a large enough CCFC error signal. The 42 mW of pump power corresponds to15.8 dB of generated squeezing with 80.6 mW of the OPO threshold.

133

Squeezed Vacuum

532 nm

OPO length control

Coherent Control

Faraday Isolator

Suspended Mirror

EOM

In-vacuum 300 m long Filter Cavity

1064 nmPZT actuated mirror

Fiber based PLL

Photodiode

BS

PBS

In-vacuum Injection Telescope
Homodyne

CCFC

AOM

GRMC

OPO

MZ

CC1
IRMC

SHG

PLL

PLL

Auxiliary Laser 1

Auxiliary Laser 2

CC2

Main Laser

In-air Filter cavity control and Squeezed Vacuum Source

Figure 8.11: Schematic of the experimental setup for CCFC lock. The setup is almost thesame as that for the frequency-dependent squeezing measurement, but we put a 50:50 BS
in the IR reﬂection path of the ﬁlter cavity to pick oﬀ the CCSB. The CCSB are detectedwith RFPD and demodulated at 14 MHz to obtain the CCFC error signal. The CCFC
error signal is added to the ﬁlter cavity green error signal and fed back to the main laser
frequency.

134greenMain laser

SHG

PLL
CCSB
OPO

CC laser

Filter Cavity
CCFC
error signal

green
error signal
IR Filter

FilterFigure 8.12: Block diagram of CCFC lock and green locks. Information of the ﬁlter inthis block diagram is summarized in Table 7.2. The IR ﬁlter for the CCFC lock has a ﬁrst
order low pass ﬁlter with a cut oﬀ frequency of 30 Hz and gain of 500.

8.4.2

Frequency tuning of CC PLL

In the CCFC lock, the CC PLL frequency has to be tuned so that either of CCSB is onresonance in the ﬁlter cavity and the separation frequency of the CCSB is 2∆ωfc,0 = 108
Hz as shown in Figure 8.1. We measured the CC separation frequency as a function of the
CC PLL frequency by changing the AOM frequency and looking at CCSB transmission
from the ﬁlter cavity. The measured CC separation frequency is shown in Figure 8.13.

8000

CC separation frequency (Hz)

6000
4000
2000
0
-2000
-4000
-6000
measured
fitting: CC separation frequency (Hz) = -1907605 * CC PLL frequency (MHz) + 13347486

-8000
6.993

6.994

6.995

6.996

6.997

6.998

6.999

7

CC PLL frequency (MHz)Figure 8.13: Relation between CC frequency separation and CC PLL frequency.

135

From the ﬁtting of the measurement, the relation between the CC separation frequency
and the CC PLL frequency is
CC separation frequency (Hz) = a × CC PLL frequency (MHz) + b,

(8.51)

a = −1907605 ± 36859, b = 13347486 ± 257882.
In principle, we can estimate the optimal CC PLL frequency for the optimal CC separation
frequency 2∆ωfc,0 = 108 Hz from this relationship. However, the error of the ﬁtting
parameters is very large with respect to 108 Hz, and it is impossible to extract the optimal
CC PLL frequency from Equation (8.51) with good enough precision.
To ﬁne-tune the CC PLL frequency, we have to check the shape of the CCFC error
signal. After a certain amount of tuning of the CC PLL frequency, we set the CC PLL
frequency to 6.99701259 MHz for the CCFC lock. The PLL frequencies for the CC1/CCFC
and CC2 demodulation are also tuned accordingly. The PLL settings for the CCFC lockare listed in Table 8.2. The binary number in the PLL settings is used for ﬁne tuning of
the PLL frequencies.
Table 8.2: PLL setting for CCFC lock.
ChannelFrequency (MHz)Binary number
CC PLL/CC2 demod
6.99701259
11 10010101 00011100 11001000
CC1/CCFC demod
13.99402518
111 00101010 00111001 10010001

8.4.3

CCFC error signal

In Section 8.1, we derived the theoretical CCFC error signal (8.11) and (8.12). In thiscalculation, we demodulated the beat notes of the CCSB (8.10) by sin (2Ωcc t − α− (∆ωfc,0 ))and assumed that ϕpump = δϕpump ≪ 1. In general, the demodulation phase α− (∆ωfc,0 )and the pump phase δϕpump can assume any value. In this case, the CCFC error signal(8.11) can be written as
PCCFC = −a+ a− ρ+ (∆ωfc )ρ− (∆ωfc ) sin (α+ (∆ωfc ) − α− (∆ωfc ) + ϕdm,CCFC − ϕpump ),(8.52)
where ϕdm,CCFC is an arbitrary CCFC demodulation phase and ϕpump is an arbitrary pumpphase. For the case of ϕdm,CCFC = α− (∆ωfc,0 ) + δϕdm,CCFC and ϕpump = π, the theoretical
CCFC error signal (8.52) for diﬀerent demodulation phases δϕdm,CCFC is shown in Figure
8.14.

136

Normalized CCFC error signal

1

demodulation phase = 0 deg
demodulation phase = 30 deg
demodulation phase = 60 deg
demodulation phase = 90 deg

0.5

0

-0.5

-1

-8

-6

-4

-2

0

2

4

6

8

Normalized filter cavity detuning with respect to carrier

Figure 8.14: Theoretical CCFC error signal normalized with respect to a+ a− for diﬀerent
demodulation phases δϕdm,CCFC . The horizontal axis is the ﬁlter cavity detuning normalized with respect to the optimal detuning ∆ωfc,0 . The dashed black line represents the
optimal detuning.

The mode mismatch between the CCSB and the ﬁlter cavity aﬀects the CCFC error
signal. The CCFC error signal with mode mismatch normalized with respect to a+ a− canbe written as
PCCFC /(a+ a− ) = (1 − amm )2 ρ+ ρ− sin (α+ − α− + α−,0 + δϕdm,CCFC )+ (1 − amm )amm ρ+ sin (α+ + α−,0 + δϕdm,CCFC )+ (1 − amm )amm ρ− sin (−α− + α−,0 + δϕdm,CCFC )+ a2mm sin (α−,0 + δϕdm,CCFC ),(8.53)

where amm (≥ 0) is the amplitude of mode mismatch between the CCSB and the ﬁlter
cavity.
We measured the CCFC error signal by scanning the ﬁlter cavity with AOM. The
measured CCFC error signals with diﬀerent demodulation phases are shown in Figure
8.15. The measured data was ﬁtted with the theoretical CCFC error signal (8.53). The
ﬁtting parameters are the demodulation phase δϕdm,CCFC , CC detuning with respect tothe carrier ∆ωfc,0 , and starting time of the scan. The mode matching is ﬁxed to 90%,
which corresponds to amm = 0.05. The amplitude of the measured CCFC error signal isnormalized with respect to 28 mV, which corresponds to a+ a− in Equation (8.52). Thisamplitude can be measured from the CCFC error signal when the CCSB are oﬀ resonanceof the ﬁlter cavity (α+ (∆ωfc ) = α− (∆ωfc ) = 0, ρ+ (∆ωfc ) = ρ− (∆ωfc ) = 1) and the green
pump phase ϕpump is scanned.

137

demodulation phase: 5.2 deg, CC detuning: 53.4 Hz
demodulation phase: 37.7 deg, CC detuning: 58.8 Hz
demodulation phase: 62.6 deg, CC detuning: 57.9 Hz
demodulation phase: 94.3 deg, CC detuning: 56.4 Hz

Normalized CCFC error signal

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-8

-6

-4

-2

0

2

4

6

8

Normalized filter cavity detuning with respect to carrier

Figure 8.15: Measured CCFC error signals normalized with respect to 28 mV for diﬀerent
demodulation phases δϕdm,CCFC . The horizontal axis is the ﬁlter cavity detuning normalized with respect to the optimal detuning 54 Hz. The dashed black lines represent theﬁtting of the CCFC error signals. The ﬁtting parameters are the demodulation phaseδϕdm,CCFC , the CC detuning with respect to carrier ∆ωfc,0 , and the starting time. The
mode matching is ﬁxed to 90% which corresponds to amm = 0.05. The solid black line
represents the optimal detuning.

The measured CCFC error signal is consistent with the theoretical one. The blue curvein Figure 8.15, which is close to in-phase (δϕdm,CCFC = 90 deg), crosses zero around the
optimal detuning at 54 Hz. Using this in-phase signal, we can lock the ﬁlter cavity aroundthe optimal detuning as well as adjust the detuning by changing the CC PLL frequency
or the CCFC demodulation phase. In real GW detectors, this tuning will be done by
optimizing the sensitivity of the GW detectors.

8.4.4

Open-loop transfer function of CCFC lockTo measure the open-loop transfer function of the CCFC lock, a signal was injected before
the IR ﬁlter, as shown in Figure 8.12, and the transfer function was measured between the
points before and after the signal injection point. This transfer function corresponds to
GCCFC,
1 + GGR(8.54)

where GCCFC and GGR are open-loop transfer functions of the CCFC and green locks,
respectively. Because GGR is known, we can obtain the information on GCCFC from thismeasurement.
The measured GCCFC /(1 + GGR ) and estimated open-loop transfer function of the
CCFC lock from it are shown in Figure 8.16.
138

Ggreen200

gain (dB)

GCCFC
GCCFC/(1+Ggreen)

100

GCCFC/(1+Ggreen) measured

0
-100
1

10

10 1

10 2

phase (deg)

10

2

3

10

4

10

10 3

10 4

10 5

10

5

100

0

-100

frequency (Hz)Figure 8.16: Measured GCCFC /(1 + GGR ) and estimated open-loop transfer function of
CCFC lock. The UGF of the CCFC lock is about 6 kHz. The crossover frequency of the
CCFC and green lock is about 1.7 kHz.

8.4.5

Locking accuracy of CCFC lock

The IR locking accuracy with the CCFC lock is obtained from the in-loop CCFC error
signal. The measured locking accuracy of the CCFC lock is shown in Figure 8.17. The
IR locking accuracy with the CCFC lock is improved below 400 Hz as compared with thegreen lock. The IR locking accuracy (rms) with the CCFC lock is improved from 3.2 Hz
to 0.7 Hz, which corresponds to a ﬁlter cavity length noise of 0.75 pm and satisﬁes thetarget value of 1 pm.

8.4.6

Frequency-dependent squeezing with CCFC lock

The frequency-dependent squeezing was measured using the with CCFC lock for diﬀerent
homodyne angles. The measured spectra using the CCFC lock are shown in Figure 8.18.
The average of the ﬁtted cavity detuning in Figure 8.18 was about 94 Hz, which corre√
sponds to the squeezing rotation frequency of about 94 × 2 = 133 Hz. The squeezing
level at high frequencies is ∼ 1.5 dB due to a large optical loss from the BS, which is used
to pick oﬀ the CCFC error signal. The detuning ﬂuctuation obtained from the measurement is about 10 Hz and this value might be better than that without the CCFC lock,which is shown in Figure 7.8. Further investigation of the detuning ﬂuctuation and the
reproducibility of the measurement shown in Figure 8.18 is a future work.
Although the detuning ﬂuctuation with the CCFC lock might be better than thatwithout the CCFC lock, the detuning ﬂuctuation still exists even with the CCFC lock.
One possibility for this ﬂuctuation might be the alignment ﬂuctuation of the ﬁlter cavity.

139

102

green lock (rms: 3.2 Hz)
CCFC lock (rms: 0.7 Hz)

IR locking accuracy (Hz/ Hz)

101
100
10-1
10-2
10-3
10-4
10-5 -1
10

100

101

102

103

104

105

frequency (Hz)Figure 8.17: IR locking accuracy of CCFC lock. The black and red curves are IR locking
accuracy with green lock and CCFC lock, respectively. The black curve is the same as
Figure 7.4. The IR locking accuracy (rms) improved from 3.2 Hz to 0.7 Hz with the CCFC
lock. It should be noted that the IR lock accuracy with the green lock (black curve) is an
out-of-loop measurement, while the IR locking accuracy with the CCFC lock (red curve)
is an in-loop measurement.

The alignment ﬂuctuation may introduce a certain oﬀset in the CCFC error signal andchange the zero crossing point of the CCFC error signal. Auto alignment of the ﬁlter cavity
is expected to stabilize the detuning ﬂuctuation caused by the alignment ﬂuctuation.
Table 8.3: Squeezing degradation parameters with CCFC lock.
Parameter
Current
Filter cavity losses
120 ± 30 ppm
Propagation losses
68 % ± 1%
Mode-mismatch squeezer-ﬁlter cavity
6 % ± 1%Mode-mismatch squeezer-local oscillator
2 % ± 1%Frequency-independent phase noise (rms) 30 ± 5 mrad
Filter cavity length noise (rms)
1 ± 0.5 pm
Generated squeezing
15.8 ± 0.1 dB

8.5

Conclusion

In this chapter, I suggested a new length and alignment control scheme of a ﬁlter cavity
with coherent control sidebands, which are already used to control the squeezing angle.
This scheme ensures accurate length and alignment of the ﬁlter cavity with respect tothe squeezed vacuum ﬁeld because the CCSB are copropagating ﬁelds at almost the same
140

20

Squeezing [dB]15

Homodyne angle: 4.1 0.2 deg, Cavity detuning: 98
1.1 Hz
Homodyne angle: 24.6 0.1 deg, Cavity detuning: 95.2
0.8 Hz
Homodyne angle: 51.3 0.2 deg, Cavity detuning: 93.4
0.7 Hz
Homodyne angle: 85 1.4 deg, Cavity detuning: 87.8
1.5 Hz

10

5

0

-5

102

Frequency [Hz]Figure 8.18: Noise spectra of the frequency-dependent squeezing with CCFC lock for different homodyne angles. Each curve has been ﬁtted, assuming the degradation parameterslisted in Table 8.3, to extract the homodyne angle and cavity detuning. Each spectrum
has a resolution of 1 Hz and is averaged 50 times, leading to an acquisition time of 50 s. It
should be noted that this measurement was done before the detuning was optimized andtherefore the cavity detuning obtained from this measurement is a bit diﬀerent from theoptimal detuning.

frequency as the squeezed ﬁeld. I implemented the new control scheme in 300 m ﬁlter
cavity for the length control and demonstrated that the new control scheme improved thelocking accuracy of the ﬁlter cavity from 3.4 pm to 0.75 pm, which satisﬁes the target
value of 1 pm. Frequency-dependent squeezing with the ﬁlter cavity length control by the
CCFC lock was also realized. The detuning ﬂuctuation was about 10 Hz, which might bebetter than that without the CCFC lock.
The current squeezing level in the CCFC scheme is limited by losses. A loss of 50 % isintroduced from the BS to pick oﬀ the CCFC error signal, and this loss can be avoided in
the real GW detectors becase the CCFC error signal can be obtained at the OMC reﬂection
without introducing the pick-oﬀ BS. In the future, alignment control of the ﬁlter cavity
using the CCFC scheme will be implemented to improve the mode matching between
the squeezer and the ﬁlter cavity. This implementation is also expected to stabilize thedetuning ﬂuctuation, which enables the stable operation of the ﬁlter cavity. The CCFC
scheme developed in this chapter will be tested in the real GW detectors in the near future.

141

Chapter 9

Conclusion
The sensitivity of GW detectors is designed to be limited by quantum noise. The quantum
noise originates from the quantum vacuum ﬁeld entering from an output port of theinterferometer. The quantum noise is composed of shot noise and radiation pressure noise.
The shot noise originates from the phase ﬂuctuation of the vacuum ﬁeld and dominates athigh frequencies of the detector sensitivity, while the radiation pressure noise originatesfrom the amplitude ﬂuctuation of the vacuum ﬁeld and dominates at low frequencies. The
sensitivity of current GW detectors is dominated by shot noise at high frequencies. To
reduce the shot noise, the frequency-independent squeezing, whose phase quadrature issqueezed, has been injected into the interferometer and the shot noise reduction by ∼3 dB
has been achieved in Advanced LIGO and Advanced Virgo. However, the increase of theradiation pressure noise due to the frequency-independent squeezing has been observed in
Advanced LIGO and Advanced Virgo.
To reduce the shot noise and radiation pressure noise simultaneously, the frequencydependent squeezing which is phase squeezed at high frequencies and amplitude squeezed
at low frequencies is required. The most promising way to realize the frequency-dependent
squeezing is reﬂecting the frequency-independent squeezed vacuum ﬁeld oﬀ an opticalcavity called the ﬁlter cavity. In the past, the squeezing angle rotation using rigid, meterscale ﬁlter cavities has been realized in the MHz and kHz regions. However, frequencydependent squeezing with a rotation frequency below 100 Hz, which is required in GW
detectors, has not yet been demonstrated.
In this thesis, we developed the ﬁrst frequency-dependent squeezed vacuum source witha rotation frequency below 100 Hz by using a 300 m ﬁlter cavity which can ﬁt for Advanced
LIGO and Advanced Virgo. It is shown that our frequency-dependent squeezed vacuum
source can realize the broadband quantum noise reduction in GW detectors. For further
quantum noise reduction, it is necessary to stabilize the ﬁlter cavity length ﬂuctuation, thealignment ﬂuctuation, and the detuning ﬂuctuation. In this measurement, the ﬁlter cavity
was controlled with an auxiliary green ﬁeld whose frequency is as twice as the squeezing
frequency. As the optical path and the frequency for the green ﬁeld and the squeezed ﬁeld
are diﬀerent, the ﬁlter cavity control with the auxiliary green ﬁeld does not ensure theﬁlter cavity length and alignment for the squeezed ﬁeld.
142

To solve this problem, I suggested a new control scheme of the ﬁlter cavity using coherent control sidebands which are already used to control the squeezing angle. As thecoherent control sidebands have the same optical path and almost the same frequency
as the squeezed ﬁeld, the control of the ﬁlter cavity with the coherent control sidebands
ensures the ﬁlter cavity length and alignment for the squeezed ﬁeld. I succeeded in controlling the length of the 300 m ﬁlter cavity with the new control scheme and demonstrated
that the new control scheme reduced the ﬁlter cavity length ﬂuctuation from 3.4 pm to0.75 pm which satisﬁes the target value of 1 pm. Using this new control scheme, thefrequency-dependent squeezing with a rotation frequency around 100 Hz also has beenrealized.
Frequency-dependent squeezed vacuum source with the several hundred-meter-long ﬁlter cavities is planned to be installed in current GW detectors such as Advanced LIGO,Advanced Virgo, and KAGRA in the near future. The new control scheme for the ﬁlter
cavity developed in this thesis could be integrated in these detectors. Using these technologies, the sensitivity of the current GW detectors will be improved at all frequencies
and the detection range will be improved by ∼50 %, which increases the detection rate bymore than three times. The result in this thesis is also a big step toward for the sensitivity
improvement of the future third-generation GW detectors such as the Einstein Telescope
and Cosmic Explorer, which require even longer ﬁlter cavities and the higher squeezing
levels. Therefore, the result in this thesis will be signiﬁcant for the development of GW
astronomy.

143

144

Appendix A

Auto alignment
For the stable operation of the ﬁlter cavity, it is essential to control the angles of the ﬁlter
cavity mirrors. The auto alignment of the ﬁlter cavity can be achieved using a wave front
sensing (WFS) technique [70]. In this appendix, the theory of WFS and the experimentof WFS with the green ﬁeld in TAMA are presented.

A.1

Wave front sensing

A ﬁlter cavity input beam, which includes the Hermite-Gaussian (HG) 10 mode can be
written as
!

 a
0
Ein = U00 U10(A.1)
E0 eiωt ,a1
where a0 and a1 are the amplitudes of the HG 00 and 10 modes, respectively.
We recall the HG mode expression [76]:Umn (x, y, z) = Um (x, z)Un (y, z) exp[−ikz + i(m + n + 1)η(z)],

(A.2)

where

Um (x, z) =2
πw2 (z)1/4 r

1
Hm
m
2 m!

√

2x
w(z)

!

" 
#
2
kx2
x
−i
exp −
,
w(z)
2R(z)

(A.3)

where Hm is the m-th order Hermite polynomial, w(z) is the beam radius, η(z) is the
Gouy phase, and R(z) is the beam radius of curvature:q
1 + z 2 /z02 ,(A.4)

η(z) = arctan (z/z0 ),(A.5)

z(1 + z02 /z 2 ),(A.6)

w(z) = w0

R(z) =

z0 = kw02 /2,
where z0 is the Rayleigh range and w0 is the beam radius at the waist position.
145

(A.7)

A ﬁlter cavity mode that has a shift δx and a tilt δθ with respect to the input beam
can be written as
!


a0
Efc =
E0 eiωt ,(A.8)
U00 U10 M (γ)
a1!
1γ
M (γ) =,
(A.9)
−γ ∗ 1whereγ = δx/w0 + iδθ/θ0 .

(A.10)

Here θ0 = λ/(πw0 ) is the beam divergence, and the ﬁrst order of γ is considered.
The reﬂection matrix for an optimally aligned ﬁlter cavity can be written as
align
Rfc
=

rc0 0
0 rc1

!
,

(A.11)

where rc0 and rc1 are the cavity reﬂectivities for the HG 00 and HG 10 mode, respectively:t2I rE e−iϕ,1 − rI rE e−iϕt2I rE e−i(ϕ+ηrt ).
= rI −
1 − rI rE e−i(ϕ+ηrt )rc0 = rI −(A.12)

rc1(A.13)

Here ϕ is the round-trip phase for the HG 00 mode and ηrt is the round-trip Gouy phase:ϕ=2Lω, ηrt = arctan(2L/z0 ).
c(A.14)The reﬂection matrix of a misaligned ﬁlter cavity in the ﬁrst order of γ, γr can be written
as
align
mis
Rfc
(γ, γr ) = M ∗ (γr )Rfc
M (γ)

=

rc0
rc0 γ + rc1 γr∗−(rc0 γr + rc1 γ ∗ )
rc1

!
,

(A.15)

where we deﬁne:
γr = δx′ /w0 + iδθ′ /θ0 ,(A.16)

where δx′ and δθ′ are the shift and tilt of the promptly reﬂected beam axis with respect
to the cavity axis.
Assuming that a0 = 1, a1 = 0 for the input beam, the beam reﬂected from a misaligned
ﬁlter cavity can be written as
Eref = [U00 rc0 − U10 (rc0 γr + rc1 γ ∗ )]E0 eiωt = [U00 rc0 − U10 C]E0 eiωt ,146(A.17)

where, for compactness, we deﬁned
C = rc0 γr + rc1 γ ∗ .

(A.18)

This can be generalized to the case of a sideband:
Eref = [U00 rs0 − U10 (rs0 γr + rs1 γ ∗ )]E0 ei(ω+Ω)t = [U00 rs0 − U10 S]E0 ei(ω+Ω)t ,(A.19)

where rs0 and rs1 are the cavity reﬂectivities for the HG 00 and HG 10 modes of the
sideband andS = rs0 γr + rs1 γ ∗ .
(A.20)
It is useful to introduce a compact form that ”hides” the spatial modes:cEref= [U00 rc0 − U10 C]E0 eiωt = rc E0 eiωt ,(A.21)s
Eref= [U00 rc0 − U10 S]E0 ei(ω+Ω)t = rs E0 ei(ω+Ω)t ,(A.22)

to recall the usual expression for the in-phase PDH signal (3.38):
I= P0 J0 (β)J1 (β)Im(rc rs∗ − rc∗ rs ),Pdemod(A.23)

andrc rs∗ − rc∗ rs = [U00 rc0 − U10 C][U00 rs0 − U10 S]∗ − [U00 rc0 − U10 C]∗ [U00 rs0 − U10 S]
∗
∗
∗
∗
∗
∗
∗
= U00 U00
(rc0 rs0− rc0
rs0 ) + U00 U10(rs0 C ∗ − rc0 S ∗ ) + U00
U10 (rc0
S − rs0
C)∗+U10 U10(CS ∗ − C ∗ S).

(A.24)

The ﬁrst term is the usual PDH signal, which is spatially symmetric and cancelled out in
the WFS. For the same reason, the fourth term, which contains only the HG10 modes,
will cancel out. Therefore, we retain only the terms that mix the HG00 and HG10 modes.
∗ , the WFS signal is
By deﬁning U = U00 U10
W

∗ ∗
∗
∗ ∗
∗
= U [rs0 (rc0γr + rc1γ) − rc0 (rs0γr + rs1γ)]∗∗+ U ∗ [rc0(rs0 γr + rs1 γ ∗ ) − rs0(rc0 γr + rc1 γ ∗ )]= −(rc0 rs1 − rc1 rs0 )(U γ − U ∗ γ ∗ )


∗ δθ
∗ δx
+ (U + U )i.
= −(rc0 rs1 − rc1 rs0 ) (U − U )w0
θ0(A.25)

Here, we assumed that all the reﬂectivities are real numbers. Using Equation (A.3), it canbe proven thatU − U ∗ = −2iU1∗ U0∗ U0 U0 sin η,(A.26)

U + U ∗ = 2U1∗ U0∗ U0 U0 cos η.

(A.27)

147

Equation (A.23) will become
I
Pdemod= P0 J0 (β)J1 (β)ImW= 2P0 J0 (β)J1 (β)U1∗ U0∗ U0 U0 (rc0 rs1 − rc1 rs0 )




δx
δθ
sin η −cos η .(A.28)
w0
θ0

The WFS signal is obtained by considering the diﬀerence between the two halves of the
QPD. It can be proven that:Z Z

rdxdy{U1∗ U0∗ U0 U0 (x > 0) − U1∗ U0∗ U0 U0 (x < 0)} =2.
π(A.29)

Finally, we can write the WFS signal as follows:r
I
PWFS,demod

=



8
δx
δθ
P0 J0 (β)J1 (β)(rc0 rs1 − rc1 rs0 )sin η −cos η . (A.30)
πw0
θ0

The relation between the cavity axis shift, tilt and input, end mirror misalignment, δθIand δθE can be expressed as follows:
δx
δθ

!

!
!
RI (RE − LE ) RE (RI − LI )
δθI
,
RI
−RE
δθE

1
=
RI + RE − L(A.31)

where RI and RE are the input and end mirror radius of curvature, respectively, and LI
and LE are the distances between the waist and the input and end mirrors:
LI

=

LE =

L(L − RE ),2L − RI − RE
L(L − RI ).
2L − RI − RE

(A.32)(A.33)

Using the values listed in Table A.1 for the TAMA ﬁlter cavity, we ﬁnd that:
δx
δθ

!
=

!
!
224
217
δθI
.
0.75 −0.76
δθE(A.34)

This yields
δx = 224 δθI + 217 δθE ,δθ = 0.75 δθI − 0.76 δθE .
When the carrier is on resonance and the sidebands are located far from the resonance,
we have
rc0 = 0.61, rc1 = rs0 = rs1 ≃ 1.

148

(A.35)

Table A.1: Filter cavity parameters for wavefront sensing with green ﬁeld.
Parameter
LengthInput mirror radius of curvature
End mirror radius of curvature
Input mirror transmissivity (532 nm)
End mirror transmissivity (532 nm)
Finesse (532 nm)
Modulation depth
Modulation Frequency
Beam radius at waist (532 nm)
Beam divergence (532 nm)
Rayleigh range (532 nm)
Round-trip Gouy phase (532 nm)
Power on WFS

Symbol
L
RI
RE
TI
TE
F
β
Ω
w0
θ0
z0
ηrt
P0

Real value
300 m
438 m
445 m
0.7%
2.9%
172
0.185 rad78 MHz0.0059 m
2.85 · 10−5 m
209 m
1.236 rad2.7 mW

This means that the WFS for the misalignment of the input mirror isr

8
PI [W] =P0 J0 (β)J1 (β)(rc0 rs1 − rc1 rs0 )π= (3.4 sin η − 2.4 cos η)δθI ,




224
0.75
sin η −cos η δθI
w0
θ0
(A.36)and that for the end mirror is
r


8
217
0.76
PE [W] =P0 J0 (β)J1 (β)(rc0 rs1 − rc1 rs0 )sin η +cos η δθEπw0θ0
= (3.3 sin η + 2.4 cos η)δθE .
(A.37)The WFS signal as a function of the Gouy phase is shown in Figure A.1.

A.2

Experimental setup

Becase there was not enough space to accommodate the optics for auto alignment, a second
layer optical table was constructed on the optical bench as shown in Figure A.2.
Two QPDs are placed with 90 degree of Gouy phase separation. The input and end
mirrors of the ﬁlter cavity are controlled with WFS signals from the two QPDs. There isalso a QPD for the green ﬁeld at the ﬁlter cavity transmission, which is used to control the
pointing of the injection green ﬁeld. The pointing signal is fed back to a steering mirror
at the BS chamber.

149

8
ITM
ETM

6

WFS signal [W/rad]

4
2
0
-2
-4
-6
-8

0

1

2

3

4

5

6

7

gouy phase [rad]Figure A.1: WFS signal as a function of the Gouy phase for the green ﬁeld. The blue
and red curves are the WFS signals for the input and end mirrors of the ﬁlter cavity,
respectively.

Figure A.2: Second layer optical table for auto alignment.

A.3

QPD centering loop

For WFS, it is necessary to center the beam in the QPD. To achieve this, a beam centeringsystem was developed in TAMA [77]. The QPD DC signals are used to extract the pitch
and yaw motion of the beam and fed back to actuators called galvo. The galvo has twogalvanometer scanners mounted orthogonally, each of which is used to rotate a small mirror
as shown in Figure A.3.
150

Figure A.3: Galvo actuator for auto alignment.

Using the beam centering system, we successfully locked the QPD centering loops for
pitch and yaw in the both QPDs. The servo for the beam centering system can measure
the closed-loop transfer function by measuring ADD OUT/ADD IN in the servo [77].
The open-loop transfer function can be obtained from the closed-loop transfer function byusing the following formula:
GOL = −1 +

1
,
GCL

(A.38)

where GOL andGCL are the open- and closed-loop transfer functions.
The measured open-loop transfer function of the QPD centering loop is shown in Figure
A.4. It should be noted that the UGF can be increased up to around 2 kHz.

151

gain (dB)

QPD2 DC pitch gain

20
0
-20

phase (deg)

10

1

10

2

10

3

10

4

QPD2 DC pitch phase

100
0
-100
10

1

10

2

10

3

10

4

frequency (Hz)Figure A.4: Open-loop transfer function of the beam centering system. The UGF is 800
Hz and the phase margin is 30 deg.

A.4

WFS loop

Currently, the WFS loop is working and the ﬁlter cavity alignment for the green ﬁeld is
stabilized by WFS. A characterization of the WFS loop is ongoing.

152

Appendix B

List of devices
In this appendix, the list of devices used in this experiment is summarized. Table B.1 andB.2 are lists of EOM and PD.

typeresonant (15.2 MHz)
resonant (78 MHz)
resonant (87.6 MHz)

type
resonant (15.2 MHz)
broadbandbroadbandbroadbandbroadband

Table B.1: List of EOM.
functionname
SHG/IRMC (15.2 MHz) Model 4003
FC/GRMC (78 MHz)EO-78K3-VIS
OPO (87.6 MHz)EO-88K3-NIR

Table B.2: List of photo detector.
functionname
SHG/IRMC (15.2 MHz), 8880 (TAMA PD)
CC1/CCFC (14 MHz)
FC/GRMC (78 MHz),PD100-NIR
OPO (87.6 MHz)PDA05CF2
MZ
PDA100A-EC
PLL
DET01CFC(/M)

153

manufacturer
New focus
Qubig
Qubig

manufacturer
Clear-pulse
Qubig
Thorlabs
Thorlabs
Thorlabs

154

Acknowledgements
This experiment was supported by many people. I could not achieve this experiment
without them.
First, I thank my supervisor, Masaki Ando, who is an associate professor of the University of Tokyo. He supported me to focus on the experiment in NAOJ. I thank his advice
in the weekly meeting and the rehearsal of the JPS meeting and the thesis defense and soon.
I thank Raﬀaele Flaminio, a professor of LAPP. He was my former supervisor in
NAOJ and he is the principal investigator of our ﬁlter cavity experiment. I am gratefulthat he started this interesting experiment and warmly accepted me at NAOJ. His advice,
particularly, at the ﬁlter cavity meeting, was really helpful.
I thank Yoichi Aso, an associate professor of NAOJ. He is my supervisor in NAOJ. He
helped the development of the digital system and suspensions in TAMA.
I thank Kentaro Somiya, an associate professor of Tokyo Tech. I thank him a lot for
his ﬁnancial support by JST CREST.
I thank Matthew Evans, an associate professor of MIT. I visited MIT for 2 weeks
to work on a 16 m ﬁlter cavity experiment. He kindly accepted me at MIT although I
adjusted my visit schedule several times.
I thank Yuta Michimura, an assistant professor of the University of Tokyo. I thank him
for his ﬁnancial support of my travel. I learned a lot from him since I was an undergraduate
student. I am always surprised by his various works on KAGRA, DECIGO, Lorentz
invariance test, optical levitation, axion dark matter search, and so on.
I thank all the collaborators. Ryutaro Takahashi, an assistant professor of NAOJ, tookcare of the suspensions and vacuum system in TAMA. Chien-Ming Wu, a researcher in
National Tsing Hua University in Taiwan, is an expert on squeezing ; he visited us twiceand made incredible progress. I am impressed that he found our problem about alignmentof the homodyne detector and solved it soon, improving the squeezing level and bandwidth
within a few days. Pierre Prat, an engineer in APC, developed an automatic analog servo
for the squeezer and ﬁlter cavity used in our experiment. We would have wasted a lotof time without the servo. Henning Vahlbruch, a researcher in MPG-AEI, developed our
homodyne detector and provided us some essential advice about our squeezer. Eleonora
Polini, a PhD student in LAPP, designed the injection and reﬂection telescope of our
ﬁlter cavity. Ayaka Shoda, who was an appointed assistant professor in NAOJ, helped our
digital system and suspensions in TAMA. For the digital system, we got substantial help

155

from KAGRA staﬀ; Osamu Miyakawa, Takahiro Yamamoto, and Shoichi Oshino. Yuefan
Guo, a PhD student in Nikhef, designed the optical conﬁguration for auto alignment.
Martin van Beuzekom, an engineer in Nikhef, developed QPDs for the auto alignment.
There were also many visitors from all over the world to help our experiment. I thank allof them.
I also thank the Ando lab members for their useful comments in the weekly meetingand so on.
I thank the LIGO lab members in MIT, particularly, Matthew Evans, Lee McCuller,
Kentaro Komori, Christopher Whittle, and Dhruva Ganapathy. I sometimes talked with
Kentaro Komori about our own experiments and I was able to understand the ﬁlter cavity
in MIT and LIGO much better thanks to him.
I thank the following oﬃcers for their support: Mayuko Niwata in the University ofTokyo, Mizuho Yoshizumi, Mie Ueda, Megumi Ohyama, Mikiko Harada in NAOJ, Fumiko
Miyama in Tokyo Tech, Marie Woods in MIT, and Felicia Armijo and Nately Sych in
Caltech.
Lastly, I thank Yuhang Zhao, Eleonora Capocasa, and Matteo Leonardi with whom
I worked most of the time during my PhD. I would not have been able to complete thisexperiment without their great work. Yuhang Zhao, a researcher in NAOJ, proceeded
the experiment step by step and made steady progress. Eleonora Capocasa, an assistantprofessor in APC, designed and characterized our ﬁlter cavity. She also developed thedigital system for our ﬁlter cavity. Matteo Leonardi, an assistant professor of NAOJ, is
my vice supervisor in NAOJ and I learned a lot of things from him. Every time we had
troubles, he always led us in the right direction. I cannot count the number of questions
I have asked him thus far, and discussions with him always deepens my understanding
much better. I am honored to have a great supervisor like him.
This work was supported by JSPS Grant-in-Aid for Scientiﬁc Research (Grants No.
18H01224, No. 18H01235 and No. 18K18763) and JST CREST (Grant No. JPMJCR1873).

156

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