Can the phase of radiation pressure uctuations be ipped in a single path for laser interferometric gravitational wave detectors Tomohiro Ishikawa1 Shoki Iwaguchi1 Bin Wu1 Izumi Watanabe1 Yuki Kawasaki1 Ryuma

2025-04-27 0 0 759.88KB 14 页 10玖币

侵权投诉

Can the phase of radiation pressure ﬂuctuations be ﬂipped in a single

path for laser interferometric gravitational wave detectors?

Tomohiro Ishikawa1,†, Shoki Iwaguchi1, Bin Wu1, Izumi Watanabe1, Yuki Kawasaki1, Ryuma

Shimizu1, Yutaro Enomoto2, Yuta Michimura3, Akira Furusawa2,4, and Seiji Kawamura1,5

1Department of Physics, Nagoya University, Nagoya, Aichi 464-8602, Japan

2Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1

Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

3Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan

4Center for Quantum Computing, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

5The Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,

NagoyaUniversity, Nagoya, Aichi 464-8602, Japan

Abstract

Radiation pressure (RP) noise, one component of quantum noise, can limit the sensitivity of laser

interferometric gravitational wave (GW) detectors at lower frequencies. We conceived a possible RP

noise cancellation method, using phase ﬂipped ponderomotive-squeezed light (FPSL) incident on free-mass

mirrors in interferometers’ arms. This possibility is investigated under the constraint that the method is for

space-based GW detectors in a broad frequency band lower than 1 Hz without using a long optical cavity.

Considering various patterns in a single path small-scale case to generate the FPSL, we proved that no

conﬁguration exists in the single path case.

Keywords: Gravitational waves, laser interferometer, quantum ﬂuctuations, ponderomotive-squeezing

1 Introduction

Gravitational wave (GW) detection is one of the most important tools for current and future astronomy. Direct

observation of astronomical phenomena using GW is remarkably useful because we can investigate even objects

that do not emit electromagnetic waves as long as they move with acceleration. It enables us to observe

and investigate such celestial bodies and their phenomena experimentally, which have been explained only

theoretically. An opportunity revolutionizing the space research was the ﬁrst detection of GWs by LIGO [1]

and Virgo [2]; in 2015, they detected the ﬁrst black-hole binary merger event [3]. Two years later, they also

detected the ﬁrst neutron-star binary merger event [4]. These events provided much information that was hard to

get with electromagnetic observations. At present, many countries plan more sophisticated GW detectors [5, 6].

In particular, detection in the low-frequency band is one of the key factors to develop further GW detectors

because not only the heavenly bodies but also various cosmological events are targeted at the frequencies. As

far as space GW detectors are concerned, there are LISA [7, 8], DECIGO [9, 10], BBO [11], and so on.

In interferometric GW detectors, laser light is used as a probe to sense GWs. It is because very small

distance ﬂuctuations between two free masses, caused by GWs, can be measured as phase ﬂuctuations of the

laser light. However, there is a fundamental noise preventing the detection of the GW signals in the laser light:

quantum noise [12]. It comes from quantum ﬂuctuations of the laser light and/or the vacuum itself, and contains

two components: shot noise and radiation pressure (RP) noise. The shot noise is a sensor noise; ﬂuctuations in

phase quadratures cause ﬂuctuations of photon numbers at a photodetector. On the other hand, the RP noise

is displacement noise; it is generated when the laser light with amplitude ﬂuctuations hits free-mass mirrors in

the interferometers. The linear spectral density for the shot noise, √Sshot, and the RP noise, √SRP, depend

on laser power P0and the GW frequency f, as √Sshot ∝1/√P0and √SRP ∝√P0/f 2, respectively. From the

dependence of the two quantum noises, the GW detector sensitivity is more limited by the RP noise at the

lower frequency band especially when the power is increased.

arXiv:2210.02669v1 [gr-qc] 6 Oct 2022

One promising method of reducing the RP noise in a broad frequency band uses a ﬁlter cavity [13].

Ponderomotively-squeezed light generated in the interferometers is injected into the ﬁlter cavities, which can

modify the level and angle of the squeezed ﬂuctuations in a frequency dependent way. When the modiﬁed

quantum ﬂuctuations after the ﬁlter cavities are detected by the appropriate homodyne detection method, the

RP noise can vanish at all frequencies. This method requires ﬁlter cavities with a similar size as the arm

cavities. However, very long ﬁlter cavities could impair the squeezing quality [14, 15] due to optical diﬀraction

loss. Therefore, space detectors with long arm cavities such as DECIGO cannot use this method.

In this paper, we consider the possibility of another method: a phase ﬂipped ponderomotive-squeezed light

(FPSL) incident on the arms. The FPSL is light in which only the sign of the RP ﬂuctuations is opposite from

that of the conventional ponderomotive-squeezed light without the sign of the amplitude and phase ﬂuctuations

inverting. Using this method for the ground-based detectors, the RP noise at the photodetector could be

completely canceled. In the space-based detectors such as DECIGO, the method of directly injecting the FPSL

to the main arm cavities does not work well due to the large optical diﬀraction loss. Fortunately, a further

improvement of sensitivity was suggested by implementing the quantum locking technique with the help of

sub-cavities inside the same satellite [16, 17]. The FPSL can be injected into these short sub-cavities which

have negligible optical diﬀraction loss. Therefore, the FPSL method could be useful for both ground- and

space-based detectors.

Some previous works have proposed unique conﬁgurations to provide the FPSL [18, 19, 20]. However,

these ideas cannot be applied to DECIGO or similar GW detectors. The techniques in [18, 19] require an

auxiliary cavity that is as long as the main cavity with a high ﬁnesse. The diﬀraction loss of a cavity as long

as DECIGO would greatly impair the squeezing eﬀect of the FPSL. As for the scheme in [20], lowering the

resonant frequency of an atom spin to the GW observation frequency is technically challenging. To ﬁnd a new

conﬁguration providing the FPSL, we searched for a conﬁguration in a single path under the condition that the

scale of the system that could produce the FPSL is much smaller than the wavelength corresponding to the GW

frequency. The single path is deﬁned as a geometry that a laser path is not divided by a beam splitter (BS). It

means that the laser carrier does not contain vacuum ﬂuctuations accompanied with the BS. Optical processes

for the conﬁgurations, which we covered, are listed as follows:

•the conventional ponderomotive-squeezing caused by the reﬂection of the light on a free-mass mirror

including the possibility of non-normal incident angle,

•frequency-independent squeezing with a nonlinear optical medium,

•the ampliﬁcation (or reduction) of the laser amplitude inside an optical cavity, and

•all combinations of the above.

In this paper, we mathematically provide an answer to the question, “Can the phase of radiation pressure

ﬂuctuations be ﬂipped in a single path for laser interferometric gravitational waves detectors?”

This paper is organized as follows. In Sec. 2, we explain the RP noise-canceling strategy, using the FPSL

incident on the interferometer. As a prelude to this description, we discuss the input-output relation of

conventional ponderomotive-squeezing. In Sec. 3, we deﬁne a notation of phase ﬂuctuations coming from

the RP, which will be used in Sec. 4. Also, we discuss an input-output relation for two other optical processes:

frequency-independent squeezing with an optical parametric ampliﬁer (OPA) and changes in the laser eﬀective

amplitude in optical cavities or due to non-normal incident angles. In Sec. 4, we show step by step that we

cannot create the FPSL in the single path case. And ﬁnally, in Sec. 5, we discuss in more detail the reason why

no conﬁguration exists by a case using an optical cavity.

2 Radiation pressure noise-canceling strategy with phase ﬂipped

ponderomotive-squeezed light

In this section, we explain the core strategy in the paper: the RP noise-canceling strategy using the FPSL. Before

considering it, we review the case where conventional vacuum ﬂuctuations are incident from an interferometers’

dark port.

2.1 Quantum ﬂuctuations and GW signals at the dark port using the conventional

vacuum ﬂuctuations as input

We consider a Michelson type laser interferometer, whose fringe at the anti-symmetric port is set as completely

dark, as shown in Fig. 1. In this geometry, the quantum ﬂuctuations at the photodetector are obtained only from

the ﬂuctuations incident to the interferometer from the dark port: vacuum ﬂuctuations. We ignore quantum

ﬂuctuations inherent in the laser light, and only discuss a relation between the vacuum ﬂuctuations before

incident to the interferometer, (ˆx, ˆp), and the ﬂuctuations returned from the interferometer, (ˆx0,ˆp0).

Free-mass mirror

Source

Photodetector

Free-mass mirror

!𝑥, !𝑝 !𝑥!, !𝑝!

!𝑥"#, !𝑝"#

!𝑥$%&, !𝑝$%&

Figure 1: Schematic geometry of a Michelson laser interferometer, showing the notations of quantum ﬂuctuations

in some parts of the interferometer. The operators (ˆx, ˆp) and (ˆx0,ˆp0) represent the amplitude and phase

components of the vacuum ﬂuctuations injected into the interferometer and reﬂected from the interferometer

at the BS along the dark port. The operators (ˆxin,ˆpin) and (ˆxout,ˆpout) show the ﬂuctuations that leave the BS

and head for the BS inside the arm of the interferometer.

When the laser light enters the interferometer, the vacuum ﬂuctuations from the dark port combine with

the laser light at the BS, with opposite signs on the two arms; thus, laser light in each arm of the interferometer

contains the vacuum ﬂuctuations (ˆx, ˆp). Here we describe the ﬂuctuations in each interferometer arm as the

one-sided electric ﬁeld operator:

E+(z, t) = r2π~ω0

Ace−iω0(t−z/c)Z∞

0hˆa+e−iΩ(t−z/c)+ ˆa−e+iΩ(t−z/c)idΩ

2π,(1)

where ω0is the carrier frequency, Ω (ω0) is the measurement frequency, and Ais an eﬀective cross-sectional

area [13]. Also ˆa+,ˆa−are annihilation operators and deﬁned using ω0,Ω:

ˆa+≡ˆaω0+Ω,ˆa−≡ˆaω0−Ω.(2)

These two annihilation operators ˆa±do not represent the amplitude and phase quadratures of the vacuum

ﬂuctuations directly; thus, we deﬁne operators ˆxin, ˆpin as follows:

ˆxin ≡1

√2ˆa++ ˆa†

−,ˆpin ≡ − i

√2ˆa+−ˆa†

−.(3)

Also, we can deﬁne the time-varying operators ˆxin(t),ˆpin(t) by taking the Fourier transform of ˆxin,ˆpin in Eq. (3):

ˆxin(z, t) = Z∞

dΩ

2πˆxine−iΩ(t−z/c)+ ˆx†

ineiΩ(t−z/c),

ˆpin(z, t) = Z∞

dΩ

2πˆpine−iΩ(t−z/c)+ ˆp†

ineiΩ(t−z/c).

(4)

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Canthephaseofradiationpressureuctuationsbeippedinasinglepathforlaserinterferometricgravitationalwavedetectors?TomohiroIshikawa1,,ShokiIwaguchi1,BinWu1,IzumiWatanabe1,YukiKawasaki1,RyumaShimizu1,YutaroEnomoto2,YutaMichimura3,AkiraFurusawa2,4,andSeijiKawamura1,51DepartmentofPhysics,NagoyaUniversity,N...

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Can the phase of radiation pressure uctuations be ipped in a single path for laser interferometric gravitational wave detectors Tomohiro Ishikawa1 Shoki Iwaguchi1 Bin Wu1 Izumi Watanabe1 Yuki Kawasaki1 Ryuma

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