Volume91A,number4 PHYSICSLETFERS 6 September1982

A PROTOTYPE BACK-ACTION EVADING TRANSDUCER

SUITABLE FOR GRAVITATIONAL RADIATION ANTENNAE

David BLAIRDepartmentofPhysics,Universityof WesternAustralia,Nedlands,WA. 6009, Australia

Received14 May 1982

Thispaperdescribesa practicalconfigurationfor a back-actionevadingmotiontransducersuitablefor useon Weber-typegravitationalradiationantennae.Testson a prototypearedescribed,noisesourcesarediscussedandrealisticsensitivitylimits areestimated.

Quantumnon-demolitionmethodshavebeenpro- describea practicalrealisationof the “continuousposedasa meansof circumventingthelinearampli- back-actionevadingtransducer”proposedby Cavesetfier quantumlimit (standardquantumlimit, SQL) to al. [1], whichhasbeentestedin a roomtemperaturethesensitivity of preciseexperiments[1]. In particu- prototype.The deviceis basedona 10 GHz doublelar, Webertypegravitationalradiation(GR)antennae re-entrantmicrowavecavity.Fig. 1 showstheequiva-reachtheSQLat amplitudesensitivity ~10—2O_ lent circuit of the BAE transducer.The inputsignal10—21 m, comparableto the signalcomplitudeex- (motionof theGRantenna)is appliedacrossa splitpectedfrom extragalacticGRsources.To be confi- capacitorof total capacitanceC, to which a microwavedentof GRdetectionit is importantto surpassthis excitationsignalis appliedvia a couplingtransformerlimit, andthehighQ resonatorof which the capacitoris

The SQLcanbeovercomeby designinga trans- part.Themotionsignalis extractedthroughthetrans-ducerto measureonly oneof the symmetricaloscil- former.TheresonatorQ factordeterminesthedegreelator coordinatesor quadratureamplitudesX1 and of filtering of variableX2 (unwanted)relativeto theX2. Severalequivalentcircuits[1—3]andtwo designs observedvariableX~.[4,5] for suchdeviceshavebeenpublished.Here I Fig. 2ais a schematicdiagramof theprototype

transducer.A doublere-entrantcavity of resonant

~Jmicrowave frequency~2with amicrowavechoketo preventra-probe diationloss, shieldedinductancepolesanda narrow

______ slit to acceptthe centralcapacitorplate is placednearJ ~ theendof a Weberbar of frequencyw.-~ ,jj-J The cavity is excitedby the sidebands~l ±w of a

L T ~2 R microwavecarrier &2, obtainedusinga sidebandmod-

ulator

V~x. V0 sin ~2t coswt. (1)

______ Notethat Vex is modulatedin the cosineor X2 qua-motion drature.

Fig. 1. Equivalentcircuit of thetransducer.Thetransformer A correspondingtimevaryingelectricdipolemo-representsthecoupling probethroughwhich signalsVexand mentis inducedon the centralcapacitorplate,whichV5arecoupled. in turnmixeswith the incomingmotion signalsoas

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Volume91A, number4 PHYSICSLETTERS 6 September1982

whereI3~is thecavity couplingfactoron resonance,andI~2~is the cavity coupling at frequencies&2 ±2w.

The back-actionforceFba,which actsbackon theantennadueto theJohnsonnoisefrom the resistanceR

1at theinputof the amplifierhasthe form

cavityFba = (—~~V0130Z2/D)coswt

choke

+ termsat frequencies(2~7±w) , (3)

bar whereZ2 isthe amplitudeof themicrowavequadrat-

ure componentsat frequency~l. Eq.(3) showsthat

(a) althoughsomeof theamplifier noisepoweris downconvertedto thesignalfrequency,this downconverted

cavity mounting block component appears in the cosine X2 quadrature,and

II~i~JI is thereforein phasewith the phaseof themodula-~ tion. No back-actionforcesappearin thesine quadra-

ture.bar

bending In the prototypethe cavity Q is low (~400)so thatplate X2 is not filtered by the transducer cavity (~2 9.6

GHzandw 1260 Hz). Backactionevasionis notachieved,sincefrequencyselectionis only obtainedby a 1260Hz bandpassfilter in the input circuit ofthe lock-in amplifier (seefig. 1).Howeverthe selec-

(b) tive sensitivityto X1 is easilyobserved,andin this in-Fig. 2. (a)Schematicdiagramof theprototypetransducer stancethe sensitivityratio X1 /X2 1 o~—1o~.Onlyanddemodulationcircuit. Thetransducercavity is cylin- at Q-levels~107— ~0

8,achievablewith superconduct-drical,with two opposingpostsmountedradially. Thecavity ing cavitieswifi BAE occur.diameteris 8 mm,and thepostsare2 mm diameterand 3 i now go on to dicussseveralpracticalaspectsofmm long. Thecentralcapacitorplatepenetratesthecavity

this transducerdesign.througha slit betweentwo semicircularshieldingplatesatthebottomof the cavity. (b) Schematicdiagramof theset-up (a) Seismicnoise.Low frequencyseismicnoisefor a cavity mountedon a barwith a bendingmodeimpe- (<10 Hz) canbea seriousproblemin GRantennaedancematchingplate. [6], sincefor earthbasedantennaethe amplitudeis

alwayslargeandvibration isolationat low frequenciesto “re-radiate” to the inputprobesignalcomponents is difficult. Thisparticularlyappliesto BAE or strobo-at frequencies&7 and ~2±2w via capacitorC

2 in fig. 1. scopicschemesusingconventionalre-entrantcavityAs shownin refs. [1,4] thefrequency&2 containsonly transducers[7], whereonefinds that theX1 signalisX1 information.Thefrequencies~2±2w containboth contaminatedby a prohibitivelevel of seismicnoise.X1 andX2information,butpowerflow at thesefre- The split capacitorschemehasthe advantagethat toquencieswill besuppressedif thecavity Q-factorQe first orderthe total seriescapacitanceis independentis such that Qe ~‘ &l/w. of distanceandtheresonantfrequencyof thecavity

Thefrequency~2component of the signalvoltage is notaffectedby low frequencyvibration. Thustheobservedat the outputof thecavity is enhancedby excitationsignalis notamplitudemodulatedby thethe Q-factorof the cavity, up to thelimit ~2/w, so seismicnoiseandcontaminationof theX1 signalisthat in thehigh Q limit prevented.

V~(t) ~V0(J30/D)(&2/w)X1~ ~ Measurementson a prototypecavity areshown infig. 3. Secondordereffectsclearly occur,as one

— ~V~J(J32L,/D)(X1sin f2t cos2wt would expect,but thedatashowsthatover abroadcentralregionvariationsof capacitancewith distance

+X2 sin &1t sin2wt), (2) areweak. Fromthis datait follows that the contam-

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Volume91A, number4 PHYSICSLETFERS 6 September1982

tion. Local oscillatorphasenoiseisindistinguishablefrom signal,andin practicehasto bereducedby fre-

9660 quencystabilisationsystems[9]. In thepresentcase

N 9650 -, - - -. -. - -.- .• theweak dependenceof resonantfrequencyon dis-- ~•- - -. placementleadsto reducedphasediscriminationso

9640 • thatphasenoisesensitivity isreduced.HoweverAM

~. 9630 noisenow becomesthedominantnoisesource.Let

9620 SA(w)be the AM noisespectraldensity of the car-rier ~2before modulation. Mter modulation this

-0 4 -0 3 -0 2 -0.1 01 02 0 3 0 4 sidebandnoisepoweris convertedto noiseat fre-X/

2Q ‘ quency~2:

Fig. 3. Dependenceof thecavity resonantfrequencyon the s o~= s ( ~ 5staticpositionof thecentralcapacitanceplate,measuredin A’ / aunitsx/D. wherea takesa valuebetween..J~and 2 depending

onthe degreeof correlationof noisein the upper andinationofX1 from low frequencyamplitudeexcur- lowersidebands.Now, ff Qe � fu/w (asis requiredforsions~O.2D,isreducedby ~8 X 10—6 comparedwith QND)the spectraldensityofnoisepowerin thecavitya conventionalre-entrantcavity transducer.Fora 1% S~(0)will be increasedby alorentzianfactorduetotuningaccuracy,seismiccontaminationisreducedto thepreferentialfiltering of thenoiseat frequency~l.~2 X 108. II oneassumes,asa worstcase,that the That isseismicnoiseoccursat the samplingperiod r5 (e.g. ~ ~ ~ ~

~AW)*(~eW/~) SA 0 . 60.1 sor 10 Hz) thenthe seismicnoiseamplitudeatthetransducermustbe~l 0 11 cm to reachthe SQL. Thisdemonstratesthat for this transducertheratioThisis a tight,but achievablecondition. By modify- Qe~&~I~

2shouldnotbe increasedwithout limit. Theingthe cavity geometryit is possiblethat theoffend- optimumvaluedependson SA(w)for theexcitationing secondordereffectscouldbereducedfurther, signalsource.In additiontechnicalproblemsariseif

(b) Coupling.It isimportantthatanytransducer Qe is muchlarger than~/w sincethecouplingof thebewell coupledor impedancematchedto the signal excitationsignalto thecavity is reducedandincon-source.Them~~tchingis characterisedby the dimen- vementlylargeexcitationsignalsare required.sionlessmatchingratio (3. Forthe transducerdescribed To detecta signalS~(0)in thepresenceof AMhere,in thehighQ limit, noiserequiresS

5(0)/S~(0)�1. This limit coincides

1 2 2 2 withtheSQL(in themicrowavedetectioncircuit)~ ~(~Z/w)CV 1mw D (4) for S~(0) 10—17 if ~2/2ir 1010 Hz. Suchlevelsofpracticallyidenticalto the couplingof conventional AM noisecanbeachievedwith superconductingcay-parametricupconvertertransducers[8]. Foralow ity stabilisedoscillators[9] for w/2ir ~ i0

3—i04.massantenna(e.g. the67 kg antennaat UWA) it is Theanalysisby JohnsonandBocko [5J of a SQUIDpossibleto achieve(3 values~‘ 0.1 [8] if D issmall basedBAE transduceravoidsthe AM noiseproblem(�lOOiim). Formoremassiveantennae(e.g. 1.5 by assuminganidealnon-resonantforcedmodulationtonneNb antennaat UWA) it isnecessaryto usean Vex.The technicaldifficulty in achievingsuchcou-impedancematchingelementto obtain(3valuesap- plingmaybeconsiderable.proachingunity. Fig. 2b showsschematicallya sim- (d) Thermalnoisein GRantennae.BAE tech-ple matchingschemewherethe endof a baris cut to niquesoffernohopeof surpassingthequantumlimitform atunedbendingplate throughwhichthe signal until highmechanicalQ-factors,short samplingtimesis coupled. r

5 andultralow temperaturescanreducefluctuations(c) Noisein the excitationsignal. Conventionalre- in occupationnumberof the fundamentalmodeMT

entrantcavity transducersextractsignalinformation suchthat (M/) ~ i in the time r~.ThisrequiresQ toby measuringthe fractionalchangesin phaseshift satisfytheapproximaterelationcausedby themodulationof capacitanceby themo-

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Volume91A, number4 PHYSICSLETTERS 6 September1982

urationdescribedhasadvantagesin seismicnoise iso-Q� l0~~r~T. lation. However,theuseof a singleresonantcircuit

Althoughultra low temperaturerefrigerationtech- for bothexcitationandreadout,has the disavantageniques,combinedwithhighQ sapphireorsiliconcrys- thatAM noisein the excitationsignalis enhanced.tals 1101,canenablethis inequalityto be satisfied,thetechnicaldifficulties in doingsoare significant I wishto thankM.J. Buckingham,A.G. Mannandunless~ canbereducedto ~l0~ 210 ~s. In the L.D. Mannfor usefuldiscussions.Thiswork was sup-absenceof thermalnoise,matchingconditionsfor the portedby the AustralianResearchGrantsCommittee.transducerrequire r5 ([3w~1, Themost realisticantennaconfigurationmight thenbeanoptimised Referencestransducerwith (3 0.1 usedwith an Al or Nb bar ofQ~ 108 at temperatures~ I K. [1] C.M. Caves,KS. Thorne,R.W.P.Drever, U.D. Sandberg

(e) Possibilityof reachingthe SQL. Braginskyhas andM. Ziemmermann,Rev.Mod. Phys.52 (1980) 341.

shownthat in the absenceof thermalnoisetheuncer- 12] K.S. Thorne, in: Theoreticalprinciplesin astrophysicsandrelativity, eds. N.R. Leibovitz, W.H. Reid andP.O.tainty~X1 in X1 is givenby Vandervoort(Universityof Chicago,1978).

= (Ah/mw)(�2/wQe), [3JW.G. Unruh, in: Gravitationalradiationand collapsedobjects,Lecturenotesin physics,VoL 124, ed.C.

whereA is theamplifiernoisenumber[11]. Thus, to Edwards(Springer,Berlin , 1980).

achievethe SQLrequiresQe �A~2/w.In practiceat- 141 V.1. PanovandF. Ja.Khalili, 9th mt. Conf. on Gen.Rd. andGray. Abstracts2 (1980)397.tainmentof the SQLrequiresanamplifier withA [5] W.W. Johnsonand M. Bocko,Phys.Rev.Lets. 47

~ 10 (suchasa GaAsFETor parametricamplifier) (1981) 1184.[12] anddueto the difficulty in obtaininghigh Q~ [6] E. Amaldi eta!.,NuovoCimento4C (1981) 295.

valueswith a small gap D, (needed to obtain a high [71D.G. Blair et a!., J. dePhys.Lett. 40 (1979)Li 13;13)it may benecessaryto use lowervaluesof ~ SeealsoW.C. Oelfkc,W.O. HamiltonandD. Darling,

IEEE Trans.MAG 17 (1981)853.that ~2/w ~ 10~. [8J D.G. Blair eta!.,in: Proc.22ndMarcelGrossmann

(1’) if l’~is simply an unmodulatedcarrier V0 Conf. on Gen.Re!. andGray.,ed. R. Ruffini, to beX coswt the device acts as a lineartransducerwithout published.backactionevasionproperties,butstill possessingthe [9] A.G. Mannand D.G. Blair, submittedto J. Phys.D.

advantageof reducedsensitivity to seismicnoise corn- 110] Kh.S. Bagdasanor,V.S. BraginskyandV.P. Mitrofonov,Soy. Phys.Crystallogr. 19 (1975)549.

paredwith conventionaltransducers. D.F. McGuiganeta!.,J. Low Temp.Phys.30 (1978)

In conclusion,a backactionevadingtransducer 621.

with sensitivityZ~sX1~ SQLappearsfeasiblefor fre- [11] C.M. Caves,in: Quantumoptics,experimentalgravita-quenciesw/2ir l0’~and&2/2 ir 109_~10b0Hz. tion andmeasurementtheory,eds.P. MeystreandMO.

Am noisein theexcitationsignalsourceSA(w) Scully (Plenum,New York) to be published.

shouldbereducedto �10 17 andthe microwaveam- [12] R.N. Jameseta!.,9th Int. Conf. on Gen.Re!. andGray.Abstracts2 (1980) 392.plifier shouldhavenoisenumberA ~ 10. The config-

200