Speedmeter scheme for gravitational-wave detectors based on EPR quantum entanglement

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by  E. Knyazev

Speedmeter scheme for gravitational-wave detectors based on EPR
quantum entanglement
E. Knyazev,1 S. Danilishin,2 S. Hild,2 and F.Ya. Khalili1,
∗
1M.V.Lomonosov Moscow State University,
Faculty of Physics, Moscow 119991, Russia
2SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom
WeproposeanewimplementationofaquantumspeedmeterQNDmeasurementscheme.
It employs two independent optical readouts of the interferometer test masses, featuring
strongly different values of the bandwidths γ and of the optical circulating power I ,
1,2 1,2
7 withthespecialrelationshipofI /I =γ /γ . Theoutputsofthesetwopositionmetershave
1 2 1 2
1
tobecombinedbyanadditionalbeamsplitter. Inthisscheme,signalsatthecommonandthe
0
2 differentialoutputsoftheinterferometersetupareproportionaltothepositionandtheveloc-

n ityofthetestmasses,respectively. Theinfluenceofthepositionmeter-likebackactionforce
a
associated with the position signal can be cancelled using the EPR approach by measuring
J
the amplitude quadrature of the beamsplitter common output correlated with this force. In
6
thestandardsignal-recycledMichelsoninterferometertopologyofthemoderngravitational-

]
c wavedetectors,twoindependentopticalpositionmeterscanbeimplementedbytwoorthog-
q
onalpolarisationsoftheprobelight. OuranalysisshowsthattheEPRspeedmeterprovides
-
r significantlyimprovedsensitivityfor allfrequenciesbelow 30Hzcomparedto anequiv-
g
∼
[ alent signal recycled Michelson interferometer. We believe the EPR speedmeter scheme to

1 be very attractive for future upgrades of gravitational wave detectors, because it requires
v onlyminorchangestobeimplementedintheinterferometerhardwareandallowstoswitch
4
betweenthepositionmeterandthespeedmetermodeswithinshorttime-scalesandwithout
9
6 anychangestothehardware.
1
0
.
1
0
7 I. INTRODUCTION
1
:
v The sensitivity of the modern laser-interferometric gravitational-wave (GW) detectors is lim-
i
X ited by quantum fluctuations of the probing light over most of the sensitive frequency range. In
r particular, at higher frequencies their sensitivity is limited by the shot noise (also known in more
a
general context as the measurement noise), created by quantum fluctuations of the phase of the
probing light [1–3]. The resulting sensitivity, about 10 20m/√Hz in units of the equivalent
−
∼
displacement noise, is extremely high and has proved to be sufficient for the direct observation of
gravitationalwavesfromastrophysicalsources[4,5].
At the same time the pair of Advanced LIGO interferometers, which detected the first GW
signals, have not reached yet their design sensitivity, which is planned to provide about a factor
three improvement in astrophysical reach [6]. Suppression of the shot noise, which is necessary
for achieving this goal, will require either an increase of the optical power circulating in the in-
terferometerupto 1MW,ortheapplicationofsqueezedlightstates[7–9],andmostprobablya
∼
combinationofbothapproacheswillbeusedtomaximisethesensitivitygain.
Due to the Heisenberg uncertainty relation, this will lead to the proportional increase of an-
other kind of the quantum noise, namely radiation pressure noise (also known as the quantum
Correspondingauthor: [email protected]
∗
2
back action noise), imposed by the quantum fluctuations of the light power in the interferometer
disturbing the test mass positions. The point of balance between the measurement noise and the
back action noise is known as the Standard Quantum Limit (SQL) [10], and the design sensitivity
oftheAdvancedLIGOinterferometerswilltouchtheSQLatonefrequency.
It has to be emphasized that the SQL is not a truly fundamental limit, and several methods
have been proposed for overcoming the SQL in future GW detectors. A detailed review of these
methods can be found e.g. in [11]. One of the most promising approaches for surpassing the
SQL is based on the quantum speed meter concept, which was first proposed in [12]. The basic
idea of this concept is to measure the velocity of the probe mass(es) instead of their position.
In this case, the measurement noise and the back action noise spectral densities depend on the
observation frequency in such a way that they can provide cancellation of each other by means
of introducing a frequency-independent cross-correlation between them. It can be implemented
simply by using a homodyne detector with the properly set homodyne angle. Note that in the
traditional position-sensitive interferometers, additional long filter cavities are required for this
type of the quantum noises cancellation [13] (4-km cavities were proposed in [13]; it was shown
later that much shorter, but still quite long, tens or hundreds of meters, cavities could be used as
well,buttheycouldprovideonlylimitedsensitivitygain[14–16]).
Several implementations of the quantum speed meter concept suitable for the GW detectors
were proposed, which can be divided into the following two categories: the first one relies on the
ordinary Michelson interferometer topology of the contemporary GW detectors, but requires an
additional long sloshing cavity [17, 18] and therefore does not provide significant advantages in
comparisonwiththefiltercavitiesbasedtopologies. Thesecondcategoryisbasedonthezero-area
Sagnacinterferometertopology[19,20],whichsignificantlydeviatesfromthestandardMichelson
topology. CurrentlyitisasubjectofintenseR&Defforts[21–24].
Here we propose a new kind of the quantum speed meter, the EPR speed meter (from the fa-
mousgedankenexperimentbyEinstein,PodolskyandRosen),whichallowstousetheMichelson
interferometer topology and, at the same time, does not require any additional long-baseline filter
cavitiesorothermajorinfrastructurechanges.
We would like to emphasize that the goal of this short paper is to introduce the concept of
this new speed meter type. Detailed investigations of the technical implementation, such as the
robustness against optical loss, the coupling of laser frequency and amplitude noise, as well as
additional add-on techniques, like the injection of squeezed states will be considered in a follow-
uparticle,currentlyinpreparation.
This paper is organized as follows. In the next section we reproduce the basic analytical treat-
ment of quantum noise in the position meter and speed meter schemes. In Sec.III we present the
conceptoftheEPRspeedmeter. InSec.IVweconsiderapossibleimplementationofourconcept
in a GW wave detector and provide brief estimates of its sensitivity, using parameters similar to
the ones of the envisaged LIGO Voyager GW detectors [25]. The notations and the parameter
valuesusedinthispaperarelistedinTableI.
II. GENERALINTRODUCTIONTOQUANTUMNOISEOFTHEPOSITIONMETERAND
THESPEEDMETER
A. Positionmeter
The (double-sided) power spectral density of the sum of quantum noise components in a po-
sition meter can be presented as follows (a much more detailed analysis of the quantum noise in
3
Quantity Description
c Speedoflight
h¯ ReducedPlankconstants
M=200kg Reducedmassoftheinterferometerequaltothemass
ofeachofthearmcavitiesmirrors[11]
L=4km Lengthoftheinterferometerarmcavities
ω =2πc/1.550µm Resonancefrequencyoftheinterferometerandtheopticalpumpfrequency
o
I =2 3MW Totalopticalpowercirculatinginthebotharmsoftheinterferometer
c
×
4ω I
J= o c =(2π 79Hz)3 Normalizedopticalpowerintheinterferometer
MLc ×
γ Half-bandwidthoftheinterferometer
Ω AudiosidebandfrequencyoftheGWsignal
ζ Homodyneangle
TABLE I. Main notations used in this paper. For the numerical values, we use the ones planned for the
next-generationGWdetectorLIGOVoyager[25].
interferometerscanbefoundin[11]):
2S S
xF FF
S =S + , (1)
PM xx−MΩ2 M2Ω4
where S is the spectral density of the measurement noise, S is the spectral density of the back
xx FF
actionforceandS isthecross-correlationspectraldensityofthesetwonoisesources(weassume
xF
here that S is real in order to avoid subtle but unrelevant to our consideration issues related to
xF
theimaginarypartofS ). ThesespectraldensitiessatisfytheHeisenberguncertaintyrelation
xF
h¯2
S S S2 . (2)
xx FF xF
− ≥ 4
In the rest of this section, we assume this relation is saturated and the interferometer is driven by
thevacuumandlaserfieldsintheminimumuncertaintyquantumstate.
InthemodernGWdetectors,thereisnocross-correlationbetweentheshotnoiseandtheradia-
tion pressure noise, because the resonance-tuned configuration is used in these detectors and only
thephasequadratureoftheoutgoinglightismeasured. Hence
S
FF
S =S + S , (3)
PM xx M2Ω4 ≥ SQL
where
h¯
S = (4)
SQL MΩ2
isthedouble-sidedSQLspectraldensity.
On the other hand, if S = 0 and can be made arbitrarily dependent on frequency, then the
xF
(cid:54)
spectraldensity(1)canbeminimized,usingtheexactequalityin(2)andsetting
h¯2 S S
FF FF
S = + , S = , (5)
xx xF
4S M2Ω4 MΩ2
FF
4
whichgives:
h¯2
S = . (6)
PM
4S
FF
In the laser interferometers S is proportional to the optical power inside the interferometer.
FF
It was shown in [26] that this Energetic Quantum Limit actually is a general one for all linear
stationaryinterferometricmeasurements.
The optimized spectral density (6), in principle, can be made arbitrarily small simply by in-
creasing this power. However, conditions (5) can only be satisfied in the given frequency band,
provided that the spectral densities S , S , and S depend on frequency Ω in a rather specific
xx xF FF
waywhichis,sadly,differentfromtheonetheyacquireduetofinitebandwidthofthearmcavities
in the existing GW interferometers. Therefore, to introduce the desired frequency dependence (5)
inabroadband,longadditionalfiltercavitiesarerequired[13].
B. Speedmeter
Inthespeedmeterschemes,thequantumnoisehasthesamegeneralstructure(1),butwiththe
followingpeculiarities:
S
S = vv , S =Ω2S , S = S , (7)
xx FF pp xF vp
Ω2 −
where S is the velocity measurement noise spectral density, S is the momentum perturbation
vv pp
noise spectral density, and S is the corresponding cross-correlation spectral density. It is impor-
vp
tant that S , S , and S can be considered as frequency-independent within the interferometer
vv pp vp
bandwidth[17]. Therelation(2)takesthefollowingform:
h¯2
S S S2 = , (8)
vv pp vp
− 4
andthesumquantumnoisespectraldensityofthespeedmeterreads:
(cid:18) (cid:19)
1 2S S
vp pp
S = S + + . (9)
SM Ω2 vv M M2
IntheparticularcaseofS =0,similaroptimizationasforthePMcanbemade:
vp
h¯2 h¯
S = = , (10)
vv
4S 2M
pp
yieldingthequantumnoiseofthespeedmetertofollowtheSQL:
h¯
S = . (11)
SM MΩ2
Notethatthecorrespondingspectraldensityofthepositionmeter(3)onlytouchestheSQLatone
givenfrequencyandgoesaboveitelsewhere. Therefore,thespeedmeterprovidesbettersensitivity
eveninthissimplecase.
InamoregeneralcaseofS =0,thefollowingoptimization:
vp
(cid:54)
h¯2 S S
pp pp
S = + , S = , (12)
vv vp
4S M2 − M
pp
5
FIG.1. ConceptualschemesoftheEPRspeedmeter. Top: twoopticallyindependentFabry-Perotcavities
sense the position x of the same mass M; their output beams are combined by the beamsplitter, forming
the“+”(position)andthe“ ”(speed)outputs. Bottom: amorepracticalcollinearversion(toleranttothe
−
angular motion of the mass M) of the same scheme; in this case, the signs of the beamsplitter reflectivity
factorshastobeswapped.
gives:
h¯2
S = . (13)
SM 4Ω2S
pp
Similar to the position meter case (6), this spectral density can be arbitrary small, provided that
S issufficientlylarge,whichmeanshighenoughcirculatingopticalpowerintheinterferometer.
pp
Contrarytothepositionmeter,noadditionalelementslikefiltercavitiesarerequiredforthis.
III. IDEAOFTHEEPRSPEEDMETER
ConsidernowtheschemeshowninFig.1(top). HerethemassM formsajointmovablemirror
for two otherwise independent Fabri-Perot cavities having the same eigenfrequency ω , the same
o
lengths L, but different bandwidths γ . The cavities are pumped at the frequency ω and their
1,2 o
outputfieldsarecombinedbythebeamsplitter. Itstwooutputbeamslabeledinthepictureas“+”
and“ ”aremeasuredbythetwohomodynedetectors.
−
Using the two-photon amplitudes notations of [27, 28], the input/output relations for these
cavitiescanbewrittenas(seee.g. [29]):
(cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33)
bˆc aˆc 0
1,2 =R 1,2 +G , (14)
bˆs j aˆs j xˆ
1,2 1,2
c,s
where j =1,2 is the cavity number, aˆ are the cosine and the sine quadratures of the input field
j
ofthecavity j,bˆc,s arethecorrespondingoutputfieldquadratures,
j
γ +iΩ
j
R = (15)
j
γ iΩ
j
−
6
arethefrequency-dependentreflectivitiesofthecavitiesforthecavitysidebandfields,
(cid:114)
2√2ω E γ
o j j
G = (16)
j
γ iΩ cL
j
−
are theoptomechanical transferfunctions, E are theclassical amplitudesof theintracavity fields,
j
normalizedasfollows:
h¯ω E2 =I , (17)
o j j
c,s
and I is the optical power, circulating in the cavity j = 1,2. Note that if aˆ correspond to the
j j
c,s c,s
vacuum input fields, then the same is true for R aˆ . Therefore, below we absorb R into aˆ in
j j j j
ordertosimplifytheequations.
Thebeamsplittertransformstheoutputfieldsasfollows:
(cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33)
bˆc aˆc 0
= +G , (18)
bˆ±s aˆ±s ± xˆ
± ±
where
c,s c,s
aˆ aˆ
aˆc,s = 1 ± 2 (19)
± √2
aretheneweffectiveinputvacuumfieldsand
G G
1 2
G = ± . (20)
± √2
arethetransferfunctionsforthe“+”and“ ”channels.
−
In order to create the speed meter type frequency dependence of the optomechanical coupling,
weproposetoexploitthedifferenceinthefrequencydependenceofG . Notethatifγ =γ and
1,2 1 2
(cid:54)
E E
1 2
= , (21)
√γ √γ
1 2
then the DC values of G and G are equal to each other, but the AC values are different. In
1 2
this case, the leading term of the factor G is proportional to Ω, as it is required for the speed
−
measurement. Morespecifically,inthiscase
2ω E 2γ γ iΩ(γ +γ )
o 1 1 2 1 2
G = − , (22a)
+
√γ cL (γ iΩ)(γ iΩ)
1 1 2
− −
2ω E iΩ(γ γ )
o 1 2 1
G = − . (22b)
− √γ1cL (γ1 iΩ)(γ2 iΩ)
− −
Takingalsointoaccountthatthe“+”and“ ”noisecomponentsin(18)areuncorrelated,itfollows
−
from Eqs.(22) that the “+” and “ ” ports of the beamsplitter correspond to two independent
−
meters: the position one for the “+” port (note that G is flat at low frequencies) and the speed
+
meterforthe“ ”port.
−
According to the Heisenberg uncertainty relation, each of these meters create the back action
force of its own. Indeed, it is easy to show that the fluctuational component of the radiation
pressureforceactingonthemassM isequalto(seeagain[29]):
Fˆfl =h¯(G1∗aˆc1+G2∗aˆc2)=Fˆfl++Fˆfl , (23)
−
7
where
Fˆfl =h¯G∗aˆc . (24)
± ± ±
ItiseasytoseethattheforceFˆ hasthespeedmetertypefrequencydependence.
fl
−
In principle, a more practical version with the collinear placement of the elements, which is
tolerant to the angular motion of the test mass, is also possible, see Fig.1(bottom). Evidently, in
this case the signs of reflectivities of the two beamsplitter surfaces has to be swapped. In all other
aspects,thefeaturesofthisversionareidenticaltotheoneconsideredbefore.
In general the two outputs of the proposed two-meters setup can be combined in a variety of
ways. Here we mainly focus at the simplest pure speed meter case, which can be implemented
by “switching off” the position-sensitive “+” channel. This can be done by measuring the ampli-
tude (cosine) quadrature of the “+” output field, which does not contain any information on the
mechanical motion, but instead it is correlated with radiation pressure force Fˆ . Therefore, it
fl+
is entangled with the mechanical motion. The measurement of aˆc removes this entanglement in
+
the EPR way and projects the mass M into the conditional state with effectively eliminated per-
turbation component created by the “+” channel. Of course, physically this perturbation remains
unchanged, but it becomes known to the experimenter and can be eliminated from the output sig-
nalbythesubsequentappropriatedataprocessing. Itcouldbementionedthatingeneralcase,this
filtering can not be done by in real time by using a causal filter. However, for the GW detection
thisisnotanissuebecauseoff-linecancellationwithanacausalfiltercanbeusedinthiscase.
Theremaining“ ”channelquantumnoisespectraldensitycanbecalculatedusingEqs.(1,18,
−
22b,24). Assumingthattheoutputofthischannelismeasuredbythehomodynedetectorwiththe
homodyneangleζ ,weobtain:
−
(cid:18) (cid:19)
S 1
SQL
S = 2cotζ +K , (25)
EPR 2 K sin2ζ − − −
− −
where
(γ γ )2J
K =S G 2 = 1− 2 (26)
− SQL| −| (γ1+γ2)(γ12+Ω2)(γ22+Ω2)
is the Kimble factor [13] for the “ ” channel. The three terms in parentheses in (25) originate,
−
respectively,fromtheshotnoise,thecross-correlationbetweenthetheshotnoiseandtheradiation
pressurenoise,andtheradiationpressurenoise. Theydirectlyrelatetothecorrespondingtermsin
Eqs.(1)and(9).
It is instructive to compare K with the corresponding factors for the position meter (Michel-
−
son)interferometers[13]:
2γJ
K =S G 2 = (27)
PM SQL| PM| Ω2(γ2+Ω2)
andforthespeedmeterSagnacinterferometer[19]:
4γJ
K =S G 2 = . (28)
SM SQL| SM| (γ2+Ω2)2
Note that in order to fulfill the speed meter conditions (10, 12), the optomechanical coupling has
tobesufficientlystrong,K 1,whichrequiressufficientlyhighcirculatingopticalpower:
SM
≥
γ3
J . (29)
≥ 4
8
More sophisticated frequency dependence of K allows to alleviate this problem. Indeed K
− −
has a flat speed meter-like behaviour only up to the smaller of the two bandwidths γ ; to be
1,2
specific,wesupposethatγ <γ . AthigherfrequenciesΩ γ ,itcanbeapproximatedas
2 1 2
(cid:29)
(γ γ )2J
1 2
K − (30)
− ≈ (γ1+γ2)(γ12+Ω2)Ω2
and has the position meter type frequency dependence with the bandwidth equal to γ , compare
1
with(27).
Consider therefore the strongly asymmetric case with γ γ and with γ is equal to the re-
2 1 1
(cid:28)
quired bandwidth of the interferometer. In this case the speed meter conditions (10, 12) give the
followingrequirementforthecirculatingopticalpower:
(γ +γ )γ2γ2
J 1 2 1 2 γ γ2 (31)
≥ (γ γ )2 ≈ 1 2
1 2
−
which is a factor of 1/4 (γ /γ )2 weaker than (29). With the account for this moderate optical
1 2
×
power, the quantum noise at frequencies above the γ will be dominated by the shot noise, with
2
the position meter-like K providing 1/4 (γ /Ω)2 lower shot noise spectral density within the
1
interferometerbandwidth−(Ω(cid:46)γ )thanin×theSagnacspeedmetercase.
1
The price to pay, however, is the approximately two times lower high-frequency asymptotic
of K (30) compared to that of the ordinary position meter (27) (see also Fig.3). The reason
−
for this is evident: only the part of the total circulating optical power which corresponds to the
“ ” channel, that is about half of the total power, participates in the measurement. Whether this
−
disadvantage outweighs the advantage of more effective K discussed above or not, depends on
−
thespecificparametersoftheinterferometer.
IV. PROSPECTSOFUSEINGWDETECTORS
A possible method of implementation of two independent position meters within the standard
Michelson interferometer topology is to use the two orthogonal polarisations of light. Note that
polarisation-based schemes of speed meter (the Sagnac interferometer or the Michelson inter-
ferometer with the additional sloshing cavity) were discussed already in literature several times
[20,30–34].
The scheme which we propose here is shown in Fig.2. In this scheme, in order to excite two
orthogonal linear polarisations of the carrier light in the interferometer, the polarisation of the
pumpinglaserhastobetiltedbytheangledefinedbythecondition(21):
E2 γ
arctan 2 =arctan 2 (32)
E2 γ
1 1
Two strongly different values of the bandwidths γ γ can be created by using the signal re-
1 2
(cid:29)
cycledconfigurationoftheinterferometer[35,36]withabirefringentplateinsertedintothesignal
recycling cavity. This plate has to introduce the phase shift π/2 between the two polarisations
passing through it (the quarterwave plate). In this case, for one polarisation we obtain a resonant
sideband extraction scheme (large bandwidth), while for the other polarisation we obtain a signal
9
FIG. 2. Possible implementation of the EPR speed meter based on the standard signal recycled interfer-
ometertopologyofthemodernGWdetectors. SRM:signalrecyclingmirror; BP:birefringentplate; PBS:
polarisingbeamsplitter.
recyclingscheme(lowbandwidth). Thebandwidthsofthetworeadoutsaregivenby:
c T T
SRM ITM
γ = , (33a)
1
4L 1 √R R
SRM ITM
−
c T T
SRM ITM
γ = , (33b)
2
4L 1+√R R
SRM ITM
where R , T are the power reflectivity and transmissivity of the arm cavities input mirrors,
ITM ITM
andR ,T arethepowerreflectivityandtransmissivityofthesignalrecyclingmirror.
SRM SRM
Finally,inordertocreatetheoutputchannels“+”and“ ”,thepolarisationbeamsplittertilted
−
by45 relativetothelinearpolarisationscirculatingintheinterferometercanbeused.
◦
Forthesensitivityestimatesforthisscheme,weusetheparametersplannedforthenextgener-
ationGWdetectorLIGOVoyager[25],seeTableI.Notethatduetotheincreasedmirrormassand
thechangedlaserwavelength(comparedtoAdvancedLIGO)anddespitetheincreasedcirculating
optical power, this detector will be even more underpowered than the Advanced LIGO, with the
normalizedopticalpowerJ γ3. Therefore,weassumehereinmostcasesthatthephasequadra-
(cid:28)
tureoftheoutputchannel“ ”ismeasured,whichcorrespondstohomodyneangleζ =π/2,see
− −
Eq.(25). Inthisregime,thespeedmetercannotovercometheSQL,butstillhasmuchbetterlow-
frequency sensitivity than the position meter, see Sec.II. At the same time, this regime requires
smalleropticalcirculatingpowerthanthemoreadvancedSQL-beatingonewithζ =π/2.
(cid:54)
Figures 3 and 4 show the optomechanical transfer function and the (single-sided) linear quan-
tumnoisespectraldensity,normalisedtotheequivalentGWsignal:
(cid:112) 2
Sh = √2S, (34)
L
respectively. For comparison, the corresponding plots for the Sagnac speed meter interferometer,
alsoforthecaseofζ =π/2,andforaMichelson/Fabry-Perot(positionmeter)interferometerwith
thesamevalueofthehomodyneanglearealsoprovided.
10
m
/
2
/
1
z
H
,
|
G 1020
|
n
o
i
ct “+”channel
n
u
f “ ”channel
r −
e
f MichelsonPM
s
n
a SagnacSM
Tr 1019
101 102 103
f, Hz
FIG. 3. Optomechanical transfer functions for: the “+” and the “ ” channels of the EPR speedmeter
(cid:112) −
noise with γ = 2π 500Hz and γ = J/γ 2π 31Hz; the Michelson/Fabry-Perot position meter
1 2 1
× ≈ ×
interferometer with γ = 2π 500Hz; the Sagnac speed meter interferometer with γ = (4J)1/3 2π
× ≈ ×
125Hz. AllotherparameterscorrespondtotheonesoftheplannedLIGOVoyagerGWdetector,seeTable
I.
EPRspeedmeter
/2 10 22 MichelsonPM
1 −
− SagnacSM
z
H
SQL
,
h
S
√
10 23
n −
i
a
r
t
s
W
G
10 24
−
101 102 103
f, Hz
(cid:112)
FIG. 4. Quantum noise of: the EPR speedmeter with γ =2π 500Hz and γ = J/γ 2π 31Hz;
1 2 1
× ≈ ×
the Michelson/Fabry-Perot position meter interferometer with γ =2π 500Hz; the Sagnac speed meter
×
interferometer with γ = (4J)1/3 2π 125Hz. In all cases, the phase quadrature of the output light is
≈ ×
supposedtomeasured(ζ =π/2). AllotherparameterscorrespondtotheonesoftheplannedLIGOVoyager
GWdetector,seeTableI.