Broadband quantum noise reduction in future long baselinegravitational-wave detectors via EPR entanglement

Jacob L. Beckey ,1 Yiqiu Ma,2,3 Vincent Boyer ,1 and Haixing Miao41School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom2Center for Gravitational Experiment, School of Physics, Huazhong University of Science and Technology,

Wuhan 430074, China3Theoretical Astrophysics 350-17, Californian Institute of Technology, Pasadena, Californian 91125, USA

4School of Physics and Astronomy and Institute of Gravitational Wave Astronomy,University of Birmingham, Birmingham B15 2TT, United Kingdom

(Received 20 September 2019; published 14 October 2019)

Broadband quantum noise reduction can be achieved in gravitational-wave detectors by injectingfrequency-dependent squeezed light into the dark port of the interferometer. This frequency-dependentsqueezing can be generated by combining squeezed light with external filter cavities. However, in futurelong baseline interferometers (LBIs), the filter cavity required to achieve the broadband squeezing has alow bandwidth—necessitating a very long cavity to mitigate the issue from optical loss. It has been shownrecently that by taking advantage of Einstein-Podolsky-Rosen (EPR) entanglement in the squeezed lightsource, the interferometer can simultaneously act as a detector and a filter cavity. This is an attractivebroadband squeezing scheme for LBIs because the length requirement for the filter cavity is naturallysatisfied by the length of the interferometer arms. In this paper we present a systematic way of finding theworking points for this broadband squeezing scheme in LBIs. We also show that in LBIs, the EPR schemeachieves nearly perfect ellipse rotation as compared to 4-km interferometers which have appreciable erroraround the intermediate frequency. Finally, we show that an approximation for the optomechanicalcoupling constant in the 4-km case can break down for longer baselines. These results are applicable tofuture detectors such as the 10-km Einstein Telescope and the 40-km Cosmic Explorer.

DOI: 10.1103/PhysRevD.100.083011

I. INTRODUCTION

Gravitational-wave (GW) detectors including LIGO andVIRGO,which recentlymade breakthrough discoveries, areMichelson-type interferometers with km size arms [1,2].They are among the largest and most sensitive experimentshumans have ever constructed. To push the limits ofscientific discovery even further, larger, more sensitiveinstruments are already being planned. Two such detectorsare the 10-km Einstein Telescope (ET) [3] and the 40-kmCosmic Explorer [4]. They differ from LIGO in many ways,including scale and configuration, but for our purposes, theycan be treated in a very similar way mathematically.All ground-based GW detectors are plagued by various

noise sources that result from the fact that they are on Earth(e.g., seismic activity). Once these and all other classicalnoise sources are suppressed, the sensitivity of GWdetectors is ultimately limited by the quantum nature oflight. The quantized electromagnetic field is analogous to aquantum harmonic oscillator (position and momentum of amass are replaced by amplitude and phase quadrature oflight). The uncertainty in the amplitude and phase quad-ratures (quantum fluctuations) limits the sensitivity ofinterferometric measurements.

The two primary noise sources in GW interferometersare radiation-pressure noise and photon shot noise. Theformer is due to the quantum fluctuations in the amplitudethat cause random fluctuations in the radiation pressureforce on the mirrors. The latter is due to the uncertainty inthe phase of the light which manifests itself as the statisticalarrival time of photons. These noise sources are modeled asentering through the dark port of the interferometer andcoupling to the coherent laser light. The detector sensitivityachieved when these uncorrelated (random) fluctuationsenter the interferometer is called the standard quantum limit(SQL). It has been shown that this limit could be surpassedif a squeezed vacuum was injected into the interferometer’sdark port instead [5,6]. Depending on the squeezing angleof the injected squeezed vacuum (see Fig. 1), we candecrease the amplitude or phase fluctuations that limit thedetector sensitivity. Amplitude and phase quadratures areconjugate variables (like position and momentum); thus theHeisenberg uncertainty principle states that the product oftheir uncertainties must be greater than some constant.Thus, by decreasing phase fluctuations, we suffer anincrease in amplitude fluctuations. This would not be aproblem if at low gravitational-wave frequencies, the mirror

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suspension systems did not have mechanical resonancesthat amplify these fluctuations and make radiation pressurenoise the limiting noise source. Put simply, at low frequen-cies we need amplitude-squeezed vacuum injection. Oncefar away from these resonances (at higher frequencies), thedetector is then limited by shot noise, and thus a phase-squeezed vacuum is needed. It has been known for sometime that frequency-dependent squeezing would allow oneto surpass the SQL over all frequencies [7]. These pro-posals require additional low-loss filter cavities. It wasshown recently that an alternative approach to achievingfrequency-dependent squeezing without additional cavitiesis to use EPR-entangled signal and idler beams (differentfrequency components in a conventional squeezed lightsource) [8], which has been demonstrated in proof-of-principle experiments [9,10]. In this paper, we present asystematic way of finding the working points for thisbroadband squeezing scheme in LBIs. We also show that inLBIs, the EPR scheme achieves nearly perfect ellipserotation as compared to 4-km interferometers which haveappreciable error. Finally, we show that an approximationfor the optomechanical coupling constant in the 4-km casecan break down for longer baselines.

II. THEORY

A. EPR entanglement in squeezed-light source

In this section, we illustrate the EPR entanglementgenerated in the squeezed-light source, which consists ofa nondegenerate optical parametric amplifier with a non-linear electric susceptibility (χð2Þ in our case). Such a devicetakes in two modes and a pump beam (energy source) and

produces two amplified modes which we call the signal andidler beams. In frequency space, we can visualize theoptical parametric amplifier taking in uncorrelated side-bands and entangling (correlating) them. In fact, anyfrequency modes ω1, ω2 within the squeezing bandwidththat suffice ωp ¼ ω1 þ ω2 will be entangled with eachother. In our proposed scheme [8,11], we detune the pumpfield by an amount Δ (of order MHz) such thatωp ¼ 2ω0 þ Δ, where ω0 is the interferometer carrierfrequency. This creates correlated sidebands around thefrequencies ω0 and ω0 þ Δ (Fig. 2). The correspondingamplitude and phase quadratures defined with respect to ω0

and ω0 þ Δ are therefore entangled [12,13].In Caves and Schumaker’s two-photon formalism [14],

the amplitude and phase quadratures are written in terms ofsidebands.

a1ðΩÞ ¼aðω0 þ ΩÞ þ a†ðω0 −ΩÞffiffiffi

2p ; ð1Þ

a2ðΩÞ ¼aðω0 þΩÞ − a†ðω0 − ΩÞ

iffiffiffi2

p ; ð2Þ

b1ðΩÞ ¼bðω0 þ Δþ ΩÞ þ b†ðω0 þ Δ −ΩÞffiffiffi

2p ; ð3Þ

b2ðΩÞ ¼bðω0 þ ΔþΩÞ − b†ðω0 þ Δ −ΩÞ

iffiffiffi2

p : ð4Þ

The general quadratures for the signal and idler beams canthen be written as

aθ ¼ a1 cos θ þ a2 sin θ; ð5Þ

bθ ¼ b1 cos θ þ b2 sin θ: ð6Þ

In the high squeezing regime, the fluctuations in the jointquadratures, a1 − b1 and a2 þ b2, are simultaneously wellbelow the vacuum level. This is the experimental signatureof theEPRentanglement. This does not violateHeisenberg’suncertainty principle because ½a1 − b1; a2 þ b2� ¼ 0. Inanalogy to the original EPR paper [12,15], a−θ is maximallycorrelated with bθ. This correlation allows us to reduce our

FIG. 1. LBI setup and sensitivity curves for various squeezingschemes. A fixed squeezing angle only surpasses the SQL (blackdotted line) over a narrow frequency band. Our broadbandsqueezing scheme (purple curve) is achieved by rotating thenoise ellipse in a frequency-dependent way as shown below theplot. Acronyms used are as follows: end test mirror (ETM),input test mirror (ITM), power recycling mirror (PRM), signalrecycling mirror (SRM), and output mode cleaner (OMC).

FIG. 2. Visualization of the frequency-mode entanglement inthe pumped optical parametric amplifier in our proposed scheme.

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uncertainty in a−θ bymaking ameasurement on bθ. This keytheoretical result that enables our conditional squeezingscheme has recently been realized experimentally [16].

B. Interferometer as detector and filter

The signal and idler beams enter the dark port of theinterferometer and couple to the laser that enters the brightport. The interferometer we consider has both signal andpower recycling cavities to increase sensitivity as shown inFig. 1. To the signal beam, the interferometer looks like aresonant cavity. The input-output relation for the phasequadrature of the signal beam (which will contain our GWsignal) is [7]

A2 ¼ e2iβða2 −Ka1Þ þffiffiffiffiffiffiffi2K

p hhSQL

eiβ; ð7Þ

where hSQL ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8ℏ=ðmΩ2L2

armÞp

is the square root of theSQL, β is a phase shift given as β≡ arctanΩ=γ, with γbeing the detection bandwidth, and K is the optomechan-ical coupling constant that determines the couplingbetween the light and the interferometer mirrors. Forcurrent GW detectors, the signal-recycling cavity lengthis of the order of 10 meters, and the SRM transmission isquite high (tens of percent). In this case, we can effectivelyview the signal-recycling cavity (SRC) formed by SRMand ITM as a compound mirror by ignoring the propagationphase ΩLSRC=c picked up by the sidebands, as firstmentioned by Buonanno and Chen [17]. When the SRCis tuned, the corresponding optomechanical coupling con-stant K is the same as that given by Kimble et al. [7]:

KKLMTV ¼ 32ω0Parm

mL2armΩ2ðΩ2 þ γ2Þ ; ð8Þ

where Parm is the arm cavity power, m is the mirror mass,and γ is equal to cTSRC=ð4LarmÞ, with TSRC being theeffective power transmission of the SRC when viewed as acompound mirror.For LBIs, the definition of such a coupling constant can

differ from that given by Kimble et al. This is because theSRC length is longer and also because the SRM trans-mission becomes comparable to that of ITM in order tobroaden the detection bandwidth in the resonant-sideband-extraction mode; i.e., the effective transmissivity TSRC ofSRC approaches 1. The approximation for defining γapplied in Ref. [17], which assumes TSRC ≪ 1, starts tobreak down. We also need to take into account thefrequency-dependent propagation phase of the sidebands,which leads to the following expression for the couplingconstant [18]:

KLBIs ¼2h2SQLLarmω0Parmγsω

2s

ℏc½γ2sΩ2 þ ðΩ2 − ω2sÞ2�

: ð9Þ

Here Larm is the interferometer arm length; ωs is a resonantfrequency that arises from the coupling between the signalrecycling and arm cavities. The frequency and bandwidthfor such a resonance are given by

ωs ¼cTITM

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLarmLSRC

p ; γs ¼cTSRM

4LSRC: ð10Þ

The idler beam, however, is far away from the carrierfrequency and does not produce a noticeable radiationpressure effect on the test mass. As such, it sees theinterferometer as a simple detuned cavity as shown inFig. 3, the same as was done in Ref. [17]. As such, the idlerbeam simply experiences a frequency-dependent ellipserotation. This can be seen in the idler input-output relationwhich is given as

FIG. 3. The signal recycling interferometer can be mapped to athree-mirror cavity. The signal recycling cavity can then bemapped into a single mirror with effective transmissivities andreflectivities [17]. This final, two-mirror cavity is resonant for thesignal beam (at ω0) but detuned for the idler beam (at ω0 þ Δ);thus, the idler simply experiences a frequency-dependent ellipserotation. This allows us to use the interferometer itself as a filtercavity. The single-trip propagation phase ϕSRC is equal to theinteger number of π for the carrier in the resonant-sideband-extraction mode, and it is equal to some specific number for theidler, as explained in the text.

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B2 ¼ eiαð−b1 sinΦrot þ b2 cosΦrotÞ; ð11Þ

where α is an unimportant overall phase and the importantrotation angle Φrot is given by [7,11]

Φrot ¼ arctan

�Ωþ δfγf

�þ arctan

�−Ωþ δf

γf

�: ð12Þ

Here δf and γf are the effective detuning and bandwidth ofthe interferometer with respect to the idler beam. They aredefined as

2ðωidler þ δfÞðLarm=cÞ þ argðridlerSRCÞ ¼ 2nπ; ð13Þγf ≡ cjtidlerSRCj2=ð4LarmÞ; ð14Þ

where ωidler ¼ ω0 þ Δ, n is an integer number, and ridlerSRC

and ridlerSRC are the effective amplitude reflectivity and trans-missivity of the SRC for the idler beam:

ridlerSRC ¼ffiffiffiffiffiffiffiffiffiffiRITM

pþ TITM

ffiffiffiffiffiffiffiffiffiffiffiRSRM

p

1 −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRITMRSRM

pe2iϕ

idlerSRC

; ð15Þ

tidlerSRC ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTSRMTITM

peiϕ

idlerSRC

1 −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRITMRSRM

pe2iϕ

idlerSRC

: ð16Þ

The phase ϕidlerSRC ¼ ΔLSRC=c by assuming ω0LSRC=c is

equal to the integer number of π as in the resonant-sideband-extraction mode. Note that the issue of thecompound-mirror approximation for the carrier mentionedearlier does not occur for the idler beam. Because Δ ≫ Ω,the sideband propagation phase inside SRC can be ignored,and also the effective SRC transmissivity for the idlerT idlerSRC ¼ jtidlerSRCj2 is much smaller than 1, which makes γf

properly defined.The rotation angle Φrot needs to be equal to arctanK to

achieve the required frequency-dependent squeezing. Thisusually cannot be realized exactly with a single cavity, andtwo cavities are required. Indeed, for LIGO implementa-tion of such an idea [8], the rotation in the intermediatefrequency deviates from the ideal one by a noticeableamount. As we will see, for LBIs, the broadband operationmode can make such a deviation negligible because thetransition frequency from the radiation-pressure-noisedominant regime to the shot-noise dominant regime ismuch lower than the detection bandwidth; a single filtercavity is close to being sufficient and ideal [19]. Thecorresponding required detuning and bandwidth given inEq. (9) and ωs ≫ Ω, following a similar derivation asRefs. [8,19], are

γf ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΩ2KLBIs

2

r ����ωs≫Ω

≈

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ω0ParmTSRM

mc2T2ITM

s; ð17Þ

δf ¼ −γf: ð18Þ

From Eq. (46) in [7], one can show that the sensitivity of theinterferometer with an imperfect rotation angle is

Sh ≈h2SQL

2 cosh 2r

�Kþ 1

K

�þ h2SQL

2

sinh2 2rcosh 2r

�Kþ 1

K

�δΦ2

ð19Þ

where r is the squeezing factor and δΦ ¼ Φrot−arctanK ≪ 1. The first term in Eq. (19) is the sensitivitywhen the rotational angle is realized exactly and the secondterm is the degradation in sensitivity due to the error in therotational angle. In the case of a 15-dB squeezing injection asconsidered in [8], r ¼ 1.73, so the ratio of the correction termto the exact term is ≈249δΦ2. So, if we want to keep therelative correction to less than 10%, we will need the error inthe rotation angle δΦ < 0.02 rad [i.e., 249ð0.02 radÞ2×100% ¼ 9.96% < 10%]. So, as long as the proposed schemekeeps the overall error in the rotation angle to less than0.02 rad, we will suffer no more than a 10% degradation innoise reduction. This requirement turns out to be easilysatisfied in the broadband detection mode of LBIs due to thereason mentioned earlier.

III. NUMERICAL RESULTS

In this section, we show the systematic approach tofinding the working points for implementing this idea. Theprocedure was outlined in Ref. [8], which showed oneworking point out of many for LIGO parameters. Thetunable parameters are the detuning frequency of the pumpΔ, the small change of the arm length δLarm, and the SRClength δLSRC with respect to their macroscopic values. Weshow how the relevant domains for the various tunableparameters in our scheme were derived. Then, we presentthe result of our search for solutions to the resonancecondition within these bounds.For illustration, we choose the detector parameters on a

scale similar to the Einstein Telescope. To highlight that theEPR squeezing scheme is not restricted to one particular setof detector design parameters, we allow the macroscopicarm length Larm and LSRC to vary from the nominal valuesoutlined in the Einstein Telescope design study [3]. Thedetector parameters are outlined in the following table.

Parameters Name Value

Larm Arm cavity length [9995, 10005] mLSRC Signal recycling cavity length [100, 200] mm Mirror mass 150 kgTITM ITM power transmissivity 0.04TSRM SRM power transmissivity 0.04Parm Intracavity power 3 MWr Squeezing parameter 1.73 (15 dB)

To start, we equate Eq. (14) with Eq. (17) and solve forϕSRC. This is the exact phase accumulated by the idler after

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one round trip in the SRC, so we denote it ϕexactSRC . Next, for

the effective cavity to have a detuning frequency satisfyingEq. (18), δf ¼ −γf, we tune the idler detuning Δ, δLarm,and δLSRC to find solutions to Eq. (13). The idler detuningΔ has to be in the lowMHz regime because if it was lower itwould interfere with the carrier, but if it were too high,electronics would not work optimally. The allowable rangeis taken to be

Δ2π

∈ ½5; 50� MHz: ð20Þ

Since we want to keep γf fixed while we tuneΔ to make theresonance condition Eq. (13) satisfied for δf ¼ −γf, we canonly alter Δ by integer numbers n of the free spectral rangeof the SRC, namely, Δ ¼ ðϕexact

SRC þ nπÞc=LSRC. The mini-mum allowed detuning is 5 MHz, and this will occur whenLSRC is at its maximum, i.e., 200 m. This will correspond tothe minimum allowed n. Similarly, the max detuningoccurs at the minimum SRC length and the max n. Wefind the relevant values of n to be

n ∈ ½7; 33�: ð21Þ

We find that the overall rotation angle Φrot is not verysensitive to changes in the SRC phase. It is acceptable tohave ϕSRC slightly deviated from the exact value ϕexact

SRC .This makes it easier for us to find solutions to our resonancecondition Eq. (13). As noted before, we must keep the errorin the overall rotation angle to less than 0.02 rad. That is,we need jδΦj ¼ jΦrot − arctanKj < 0.02 rad where Φrot isgiven in Eq. (12). To ensure the error in rotation angle isless than 0.02 rad over the whole positive frequencydomain, we require

maxΩ

����ΔϕSRCdΦrot

dϕSRC

���� < 0.02: ð22Þ

Using the parameters listed in the table, one finds

jΔϕSRCj < 0.002: ð23Þ

To show the working points for different macroscopicarm lengths and SRC lengths, we scan the length by 1-mstep size. Additionally, we sweep ϕapprox

SRC in between½ϕexact

SRC − 0.002;ϕexactSRC þ 0.002� with a step size of 0.0001.

There are 1.2 million combinations of these parameterswith the given step sizes. We take advantage of the fact thateach combination is independent and can thus be checkedin parallel. We define the working points as those thatrequire microscopic changes of arm length ΔLarm and SRClength δLSRC smaller than 1 cm. Our search resulted in3444 working points summarized in Fig. 5. We pick one of

the many working points for illustration, and the resultingsensitivity curve is given in Fig. 4. The EPR schemeachieves almost the ideal frequency-dependent rotation ofthe squeezing quadrature angle. This is the result of usheavily restricting our parameter space to bound the error.The step sizes were chosen with computational expenses

in mind, so the resolution is not particularly high. As such,Fig. 5 shows several “dead zones” as well as a couple of“hot points.” The natural question to ask is whether theseare real or whether they are a by-product of our numericalprecision. Zooming in around two such points, we pro-duced the subfigures on the right-hand side of Fig. 5. In thecase of the dead zone, we see that there are actuallyworking points where there appeared to be none. This ispromising, as it points to the conclusion that a workingpoint can be found given precise fine-tuning. Similarly, wezoomed in on a hot point (top right panel of Fig. 5), andinterestingly we still see a line structure where there are asmany as 35 working points surrounded by areas thatapparently have zero working points. So, to check whetherthis was a real feature of the system or an issue of numericalprecision, we again zoomed in around the hot points. Whatwe found, once again, was that the dead zones must simplybe due to the numerical precision chosen. With more timeor computational power (or both), one could map arelatively smooth landscape of working points for theET. In our case, our goal was to simply show that thisEPR-based squeezing scheme is not very sensitive to theactual arm length and SRC length, and that we can alwaysfind some working points for given a set of parameters.

FIG. 4. Approximate sensitivity (dotted curve) plotted along-side the sensitivity when ideal ellipse rotation is achieved (redcurve). For this plot, we use Larm ¼ 10003 m, LSRC ¼ 100 m,Δ=ð2πÞ ¼ 1.04 MHz, and ϕapprox

SRC ¼ 0.16828.

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IV. CONCLUSIONS

We have shown that EPR entanglement-based squeezingcan be implemented in LBIs. We derived the relevantbounds on the tunable parameters to ensure that ourapproximate ellipse rotation scheme very nearly matchesthe exact rotation achievable through the use of externalfilter cavities. The goal of the project was to map theinterferometer working points for this squeezing scheme inLBIs like the ET and Cosmic Explorer. We accomplishedthis at a rather low resolution of the parameter space.Zooming in around areas that had very many workingpoints or very few showed that the landscape of workingpoints seems to be quite smooth. In other words, if an areaappears to have no working points, it is likely because thestep size used to iterate through the parameter space was

too low. This is ideal for experimental implementation ofsuch a scheme, for if we cannot fulfill the requirementgiven the nominal parameters, there should be anotherworking point less than a few centimeters away. As such,we conclude that EPR-based squeezing is an appealingalternative to other broadband quantum noise reductionschemes that require additional filter cavities.

ACKNOWLEDGMENTS

Wewould like to thank P. Jones and A. Freise for fruitfuldiscussions. J. B. was supported by a Fulbright-Universityof Birmingham postgraduate grant during the completionof this research. H. M. has been supported by the UK STFCErnest Rutherford Grant No. ST/M005844/11.

FIG. 5. The left panel shows all the working points found at the resolution set by the step size of the arm cavity length and SRC lengthto be 1 m. The color of the point indicates how many working points exist at a given combination of LSRC and Larm. The right column ofplots are zoomed in considerably. The existence of working points in these plots indicates that with sufficient fine-tuning, workingpoints for this squeezing scheme can always be found.

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