Large-scale microwave cavity search for dark-matter axions

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PHYSICAL REVIEW D, VOLUME 64, 092003

Large-scale microwave cavity search for dark-matter axions

S. Asztalos, E. Daw, H. Peng, and L. J RosenbergDepartment of Physics and Laboratory for Nuclear Science, Massachusetts Institute of Technology, 77 Massachusetts Aven

Cambridge, Massachusetts 02139

C. Hagmann, D. Kinion, W. Stoeffl, and K. van BibberLawrence Livermore National Laboratory, Physics and Advanced Technology Directorate, 7000 East Avenue, Livermore, Californ

P. Sikivie, N. S. Sullivan, and D. B. TannerDepartment of Physics, University of Florida, Gainesville, Florida 32611

F. NezrickFermi National Accelerator Laboratory, Batavia, Illinois 60510-0500

M. S. TurnerNASA/Fermilab Center for Astrophysics, Fermi National Accelerator Laboratory, Batavia, Illinois 60510-0500and Departments of Astronomy & Astrophysics and Physics, Enrico Fermi Institute, The University of Chicago,

Chicago, Illinois 60637-1433

D. M. Moltz and J. PowellNuclear Science Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720

M.-O. Andre, J. Clarke, and M. Mu¨ckDepartment of Physics, University of California and Lawrence Berkeley National Laboratory, Berkeley, California 94720

Richard F. BradleyNational Radio Astronomy Observatory, Charlottesville, Virginia 22903

~Received 11 April 2001; published 5 October 2001!

We have built and operated a large-scale axion detector, based on a method originally proposed by Sikivie,to search for halo axions. The apparatus consists of a cylindrical tunable high-Q microwave cavity threadedaxially by a static high magnetic field. This field stimulates axions that enter the cavity to convert into singlemicrowave photons. The conversion is resonantly enhanced when the cavity resonant frequency is near theaxion rest mass energy. The experiment is cooled to 1.5 K and the electromagnetic power spectrum emitted bythe cavity is measured by an ultra-low-noise microwave receiver. The axion would be detected as excess powerin a narrow line within the cavity resonance. The apparatus has achieved a power sensitivity better than10223 W in the mass range 2.9–3.3meV. For the first time the rf cavity technique has explored plausibleaxion models, assuming axions make up a significant fraction of the local halo density. The experimentcontinues to operate and will explore a large part of the mass in the range of 1 –10meV in the near future. Anupgrade of the experiment is planned with dc superconducting quantum interference device microwave am-plifiers operating at a lower physical temperature. This next generation detector would be sensitive to evenmore weakly coupled axions contributing only fractionally to the local halo density.

DOI: 10.1103/PhysRevD.64.092003 PACS number~s!: 14.80.Mz, 95.35.1d, 98.35.Gi

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I. AXIONS IN QCD, ASTROPHYSICS, AND COSMOLOGY

Introduction

Many experiments, notably the negative searches foneutron electric dipole moment@1#, imply thatCP symmetryis conserved to a remarkable degree in quantum chromnamics~QCD!. This fact is a puzzle in the standard modelelementary particles becauseCP symmetry is observed to bviolated in the weak sector of the theory, e.g., in the mixof the KL and KS mesons. As discussed below, one wouexpect CP violation in the weak sector to feed into thstrong sector through the intermediary of the QCDu angle.

A solution to this ‘‘strongCP problem’’ was proposed byPeccei and Quinn@2#, which involves the spontaneous brea

0556-2821/2001/64~9!/092003~28!/$20.00 64 0920

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ing of aUPQ(1) symmetry and a concomitant quasi-NambGoldstone particle@3#, called the axion. The axion solutioto the strongCP problem is rich in experimental, observational, and cosmological implications. If its mass is of ord1025 eV, the axion is a good candidate for the dark matof the Universe. We have carried out an experiment tsearches for Milky Way halo axions by their conversion inmicrowave photons in an electromagnetic cavity permeaby a strong magnetic field. The hardware, data analysis,results of the experiment are described in Sec. II. A futupgrade is briefly discussed in Sec. III. The remainder of tIntroduction describes the strongCP problem and its solu-tion through the existence of an axion, the mechanisms

©2001 The American Physical Society03-1

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which cold dark-matter axions are produced in the early uverse, the cavity detector method, and the structure wepect to find in an axion signal.

Several detailed reviews of the theory of the axion andcosmological and astrophysical implications are foundRef. @4#. The experimental situation is reviewed in Ref.@5#,which discusses methods and constraints on axions flaboratory, astrophysical, and cosmological searches.

1. The strong CP problem and the axion

Consider the Lagrangian of QCD@6#:

LQCD521

4Gmn

a Gamn1(j 51

n

@ q jgmiD mqj

2~mjqL j1 qR j1H.c.!#1

ug2

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The last term, the product of the gluon field strength tensoGwith its dual, is a four-divergence and hence does not ctribute in perturbation theory. That term does, however, ctribute through nonperturbative effects@7# associated withQCD instantons@8#. Using the Adler-Bell-Jackiw anomaly@9#, one can show that QCD must beu dependent if none othe current quark masses vanish. If theu dependency wereabsent, QCD would have aUA(1) symmetry and would predict the mass of theh8 pseudoscalar meson to be less thA3mp'240 MeV @10#, contrary to observation. One cafurther show that QCD depends uponu only through thedifference ofu and the argument of the quark mass matr

u5u2arg~m1 ,m2 , . . . ,mn!. ~2!

If uÞ0, QCD violatesP and CP. The absence of aP andCP violation in strong interactions, therefore places an uplimit upon u. The best constraint follows from the expermental bound@1# on the neutron electric dipole momenwhich yieldsu,1029.

The question then is: why isu so small? In the standarmodel of particle interactions, the quark masses originatthe electroweak sector of the theory, which violatesP andCP. There is no reason why the overall phase of the qumass matrix should match the value ofu from the QCDsector to yield one part in 109 or better. In particular, ifCPviolation is introduced in the manner of Kobayashi aMaskawa@11#, the Yukawa couplings that give masses to tquarks are arbitrary complex numbers and hence arg demq

and u are expected to be of order one. The puzzle of wu,1029 is called the ‘‘strongCP problem.’’

Peccei and Quinn@2# proposed a solution that postulatthe existence of a globalUPQ(1) quasisymmetry.UPQ(1)must be a symmetry of the theory at the classical~i.e., at theLagrangian! level, it must be broken explicitly by those nonperturbative effects that make the physics of QCD depupon u, and finally it must be spontaneously broken. T

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axion @3# is the associated quasi-Nambu-Goldstone bosOne can show that if aUPQ(1) quasi-symmetry is presenthen

u5u2arg~m1 , . . . ,mn!2a~x!

f a, ~3!

wherea(x) is the axion field andf a , called the axion decayconstant, is of order the vacuum expectation value~VEV!which spontaneously breaksUPQ(1). It canfurther be shown@12# that the nonperturbative effects that make QCD depuponu produce an effective potentialV( u) whose minimumis at u50. Thus u is allowed to relax to zero dynamicalland the strongCP problem is solved.

The existence of an axion is the signature of the PQlution to the strongCP problem. Its properties can be derived using the methods of current algebra@13#. The axionmass is given in terms off a by

ma.6meV1012 GeV

f a. ~4!

All the axion couplings are inversely proportional tof a . Ofparticular interest here is the axion coupling to two photo

Lagg5gg

a

p

a~x!

f aEW •BW , ~5!

whereEW andBW are the electric and magnetic fields,a is thefine structure constant, andgg is a model-dependent coefficient of order one.gg50.36 in the Dine-Fischler-SrednickiZhitnitskii ~DFSZ! model @14# whereasgg520.97 in theKim-Shifman-Vainshtein-Zakharov~KSVZ! model @15#. Apriori , the value off a , and hence that ofma , is arbitrary.However, negative searches for the axion in high energynuclear physics experiments@4#, combined with astrophysi-cal constraints@16#, rule outma*1023 eV. In addition, as isdiscussed below, cosmology places a lower limit onma oforder 1026 eV from the requirement that axions do noverclose the universe.

2. Production of cold relic axions

The implications of an axion for the history of the earuniverse may be briefly described as follows. At a tempeture of order f a , a phase transition occurs in which thUPQ(1) symmetry becomes spontaneously broken. Thiscalled the PQ phase transition. At these temperatures,nonperturbative QCD effects, which produce the effectpotential V( u) are suppressed@17#, the axion is masslessand all values of a(x)& are equally likely. Axion stringsappear as topological defects. Subsequently, one must diguish two cases:~1! inflation occurings with reheat temperature less than the PQ transition temperature or~2! inflationoccuring with reheat temperature higher than the PQ tration temperature~equivalently, for our purposes, inflatiodoes not occur at all!. In case~1! the axion field gets homogenized by inflation and the axion strings are diluted awwhereas in case~2! axion strings are present from the Ptransition to the QCD epoch.

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LARGE-SCALE MICROWAVE CAVITY SEARCH FOR . . . PHYSICAL REVIEW D64 092003

When the temperature approaches the QCD scale, thetentialV( u) turns on and the axion acquires mass. Therecritical time, defined byma(t1)t151, when the axion fieldstarts to oscillate in response to the axion mass turn on@18#.The corresponding temperatureT1.1 GeV. In case~1!,where the axion field has been homogenized by inflation,initial amplitude of this oscillation depends on how far frozero the axion field is att1. The axion field oscillations donot dissipate into other forms of energy and hence contribto the cosmological energy density today@18#. This contri-bution, called ‘‘vacuum realignment,’’ is the only contribution in case ~1!. In terms of the critical densityrc

5(3H02)/(8pG), the energy density in axions is

Va5ra~ t0!

rc.

1

6a2~ t1!S f a

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~6!

where h parametrizes the present Hubble rateH05h3100 km/s Mpc anda(t1)5a(t1)/ f a is the initial misalign-ment angle. Note thatVa may be accidentally suppressedcase~1!, if the homogenized axion field happens to lie cloto zero.

In case~2! the axion strings radiate axions@19,20# fromthe time of the PQ transition untilt1. At t1, each string be-comes the boundary ofN domain walls. IfN51, the networkof walls bounded by strings is unstable@21,22# and decaysaway. If N.1, there is a domain wall problem@23# becauseaxion domain walls end up dominating the energy densresulting in a universe very different from the one observtoday. Henceforth, we assumeN51.

There are three contributions to the axion cosmologenergy density in case~2!. One contribution is from vacuumrealignment@18#. The vacuum realignment contribution is,case~2!,

Vavac.

1

3S f a

1012 GeVD 7/6S 0.7

h D 2

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In case~2!, the vacuum realignment contribution cannotaccidentally suppressed because it is an average over mhorizon volumes at QCD time, each with a causally indepdent value of the initial misalignment anglea(t1). Note alsothat the vacuum realignment contribution is larger, by aproximately a factor of 2, in case~2! than it is in case~1!with a(t1).1. This is because only the zero momentumode contributes in case~1!, whereas in case~2! there arecontributions from the zero momentum mode and frohigher modes@30#. A second contribution is from axions thawere produced in the decay of walls bounded by strings at1 @24,28–30#. The contribution from wall decay is@30#

Vad.w..

2

gS f a

1012 GeVD 7/6S 0.7

h D 2

, ~8!

whereg[^v&/ma is the average Lorentz factor of the axioproduced in the decay of walls bounded by strings. In simlations, it was found@30# thatg;7 for ln(fa /ma).6, and that

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g increases approximately linearly with ln(fa /ma). Extrapo-lation of this behavior to the parameter range of intereln(fa /ma).60, yieldsg;60, suggesting that the wall decacontribution is less than the vacuum realignment contrition.

A third contribution @19,20,24–27# is from axions thatwere radiated by axion strings beforet1.

The string decay contribution has been the most conversial, with three estimates in the literature. The resultBattye and Shellard@25# is

Vastr,BS.10.7S j

13D @~11a/k!3/221#1

h2S f a

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~9!

where we have put in the dependence onj explicitly. j pa-rametrizes the average distance between axion strings att asjt. The ratio of parametersa/k is expected@25# to be inthe range 0.1,a/k,1.0. Battye and Shellard obtain the etimatej.13 from their simulations of local string networkin an expanding universe. However, axion strings are glostrings, which are different from local strings. Global strinhave most of their energy in the Nambu-Goldstone field srounding the string core, whereas local strings have mostheir energy inside the string core. For this reason, glostring networks may behave differently from local string nworks ~see below!. Regardless, for the expected rangea/k,

Vastr,BS5~3.4 to 40!S j

13D S f a

1012 GeVD 7/6S 0.7

h D 2

. ~10!

Yamaguchiet al. @26# carried out simulations ofglobal stringnetworks in an expanding universe and findj51.0060.08.Their result for the string decay contribution to the cosmlogical axion energy density is

Vastr,YKY5~1.460.94!S f a

1012 GeVD 7/6S 0.7

h D 2

. ~11!

Hagmannet al. @24,27# carried out simulations of the motioand decay of single string loops and of oscillating bestrings, focusing their effort on obtaining a reliable estimaof the spectrum of axions radiated by strings. They assuj.1. Their result is

Vastr,HCS.0.27 261S f a

1012 GeVD 7/6S 0.7

h D 2

. ~12!

The last two results are consistent with one another. Tresult of Battye and Shellard is compatible with the othtwo if one setsj.1 instead ofj.13. Since axion strings areglobal rather than local, the global string network simutions of Yamaguchiet al. could be considered more reliabin determiningj. With j51, all estimates are consistent ansuggest

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Va5Vvac1Vd.w.1Vstr5~0.5 to 3.0!

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Equations~4!, ~7!, and~13! indicate that the mass range wplan to cover in our experiment, 1.3meV,ma,13 meV, isprime hunting ground for dark-matter axions, with the hiend favored in case~2!, and the low end in case~1!.

From the point of view of experimental design, becauwe do not know which cosmological scenario pertains,must be prepared to search from the lowest estimated oclosure bound (;1026 eV) to the upper bound set by S1987a (;1023 eV). We note that in any cosmological scnario, Va increases asma decreases, which motivatessearch beginning from the lowest possible mass and procing upwards.

The axions produced when the axion mass turns on aQCD phase transition, whether from string decay, vacurealignment, or wall decay, have momentapa;1/t1;1028 eV/c when the surrounding plasma has temperatT1.1 GeV. They are nonrelativistic from the momenttheir first appearance, and because of their negligible cplings to normal matter and radiation are therefore a formcold dark matter~CDM!. Studies of large-scale structure fomation support the view that the dominant fraction of damatter is CDM. Moreover, any form of CDM necessarcontributes to galactic halos by falling into the gravitationwells of galaxies. Hence, there is a special opportunitysearch for axions by direct detection on Earth.

3. The Sikivie microwave cavity dark-matter detector

Axions can be detected by stimulating their conversionphotons in a strong magnetic field@31#. The relevant cou-pling is given in Eq.~5!. In particular, an electromagneticavity permeated by a strong static magnetic field can degalactic halo axions. These halo axions have velocitiesb oforder 1023 and hence their energiesEa5ma1 1

2 mab2 have aspread of order 1026 above the axion mass. When the frquencyv52p f of a cavity mode equalsma , galactic haloaxions convert resonantly into quanta of excitation~photons!of that cavity mode. The power from axion-to-photon coversion on resonance is found to be@31,32#

P5S a

p

gg

f aD 2

VB02raC

1

mamin~QL ,Qa!

50.5310226 WS V

500 lD S B0

7 TD 2

CS gg

0.36D2

3S ra

1/2310224 g/cm3D3S ma

2p~GHz! Dmin~QL ,Qa!, ~14!

whereV is the volume of the cavity,B0 is the magnetic-fieldstrength,QL is its loaded quality factor,Qa5106 is the

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‘‘quality factor’’ of the galactic halo axion signal~i.e., theratio of their energy to their energy spread!, ra is the densityof galactic halo axions on Earth, andC is a mode dependenform factor given by

C5

U EVd3x EW v•BW 0U2

B02VE

Vd3x euEW vu2

, ~15!

where BW 0(xW ) is the static magnetic field,EW v(xW )eivt is theoscillating electric field of the mode in question, ande is thedielectric constant. For a cylindrical cavity and a homogneous longitudinal magnetic field,C50.69 for the TM010mode. The form factors of the other TM0n0 modes are muchsmaller, and form factors for pure transverse electric~TE!,TEM, and the remaining transverse magnetic~TM! modesare zero.

Because the axion mass is only known in order of mnitude at best, the cavity must be tunable and a large rangfrequencies must be explored in seeking a signal. The cacan be tuned by moving a dielectric rod or metal post insit. Using Eq. ~14!, one finds that to perform a search wisignal-to-noise ratio~SNR!, the maximum scanning rate is

d f

dt5

12 GHz

year S 4

SNRD 2S V

500 lD2S B0

7 TD 4

C2S gg

0.36D4

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1/2310224 g/cm3D 2S 3K

TnD 2S f

GHzD2 QL

Qa,

~16!

whereTn is the sum of the physical temperature of the cavplus the noise temperature of the microwave receiverdetects the photons from axion-to-photon conversion.

Two earlier experiments@33#, based on rf cavity axiondetection, reported limits on halo axion couplings in the aion mass window. However, these earlier experiments lacthe sensitivity, by factors of 100 or more, required to detaxions with benchmark couplings. A recent effort@34#, usingRydberg atom single-photon detection@35#, is still in devel-opment, but holds the potential for greatly improved sentivity should the Rydberg atom technique be sufficiently avanced.

4. The phase space structure of galactic halos

If a signal is found in our detector, it will be possible tmeasure the energy spectrum of cold dark-matter particleEarth with great precision and resolution. Note that thisperiment measures the total energy of the axion, e.g., itsmass plus kinetic energy. The spread (D f / f 5DEa /ma.1026) of the axion signal is due to the kinetic energymotion of the axions through the galactic halo. Our expement is equipped with a high-resolution spectrometer tresolvesD f / f ;10211 and hence divides the axion signwidth into 105 bins. We argue below that the spectrum whave narrow lines associated with the late infall of dark m

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ter onto our galaxy. Even a few percent of the axion signaany one of these lines would greatly increase the signanoise ratio and thus the experiment’s discovery potenFurther, the lines are rich in information on the history of tformation of the galaxy.

In many past discussions of dark-matter detectionEarth, it has been assumed that the dark-matter particlesan isothermal distribution. Thermalization has been arguebe the result of a period of ‘‘violent relaxation’’ following thcollapse of the protogalaxy@36#. However, substantial deviations from an isothermal distribution are expected@37# due tolate infall. If the dark matter is collisionless, as axions aour galaxy must be surrounded by a sea of dark-matterticles. The particles fall into the gravitational well of thgalaxy, forming a continual ‘‘rain.’’ The flows of lateinfalling dark matter do not get thermalized, even on timscales of the order of the age of the universe@37#. Hencethere is discrete set of dark-matter flows with well-definvelocity vectors at any location in the halo, in addition tothermalized component. One flow is comprised of particfalling onto the galaxy for the first time, a second flowcomprised of particles falling out of the galaxy for the firtime, a third flow is of particles falling into the galaxy for thsecond time, etc. Each of the flows could contain a few pcent of the local halo density@38#.

The intrinsic width of the peaks due to the most recinfall, is estimated@37# to be of order 10217, although eachpeak may be fragmented into subpeaks of that approximwidth. Because of the Earth’s rotation (v rot.1026 c) and itsrevolution around the Sun (v rev.1024 c), the peaks fre-quenciesvn are modulated by the relative amount

dvn

vn5

vW n•~vW rot1vW rev!

c21OS v rot

2 ,v rev2

c2 D , ~17!

FIG. 1. Schematic of the apparatus.

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where vn.1023 c is the velocity of thenth phase-spacesheet relative to the Sun. Observation of the diurnal fquency modulation of a peak would allow us to determthe corresponding velocity vectorvW n completely.

II. THE U.S. rf CAVITY AXION SEARCH: DESCRIPTIONAND PHYSICS RESULTS

A. Hardware

1. Hardware overview

Figure 1 is a diagram of the apparatus. The heart ofaxion detector is a cylindrical rf cavity containing two movable tuning rods positioned parallel to a strong magnetic fiproduced by a solenoid surrounding the cavity@39#. The cav-ity electromagnetic field is coupled through a small adjuable electric-field probe to ultra-low-noise receiver electroics. These components are sketched in Fig. 2.

The cavity is cooled to 1.3 K to reduce its blackbodnoise. A 7.6 T static magnetic field, parallel to the cylindaxis, threads the cavity volume. We use the TM010 cavitymode since it has the largest form factor for axion-to-phocoupling. The unloaded quality factorQ of the TM010 reso-nance is typically 7 –203104. The frequency step betweeadjacent power spectra in the data stream is typically 2 kabout 1/15th of the width of the cavity resonance widv0 /QL . Hence, many power spectra overlap each cavity

FIG. 2. Sketch of the experiment insert within the magnet stem. The experiment insert can be withdrawn without warmingmagnet.

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quency in the search range. The frequency range coverethe initial data set is 701–800 MHz, corresponding toaxion mass range 2.9–3.3meV.

The cavity electromagnetic fields are coupled to theceiver chain by an electric dipole field probe with adjustainsertion depth~i.e., with adjustable cavity loading!. Duringnormal running, the insertion depth is occasionally adjusto maintain near-critical coupling of the cavity to the crygenic amplifier input. A directional coupler between the caity and cryogenic amplifier allows for application of test acalibration signals.

The voltage across the electric-field probe is appliedcryogenic RG-402 coaxial cable@40#, then amplified by twocryogenic heterojunction field-effect transistor~HFET! am-plifiers in series, built by the National Radio Astronomy Oservatory~NRAO! @41#. The overall power gain for the twocryogenic amplifiers is approximately 34 dB. Each amplifiis thermally tied to the cavity. The first set of amplifiers usin data taking had a noise temperature of about 4.3 K.more recent amplifiers have noise temperatures better thK. Figure 3 ~upper! shows the noise temperature and gaversus frequency of one of these first amplifiers. Figure~lower! shows noise and gain for one of the later amplifieThe output of the cryogenic amplifiers is passed out ofcryostat along cryogenic RG-401 coaxial cable@40#, thenfurther amplified by a low-noise room-temperature amplifiwith about 38 dB of gain mounted in an rf shielded enclosdirectly on the room-temperature top flange of the detecThe amplified microwave signal is applied to 25 m of R213 flexible coaxial cable leading to the mixing, i.f., anaudio stages of the receiver@42,43#.

The axion receiver has a double-heterodyne desstrongly influenced by experience gained in the earFlorida and Rochester-Brookhaven-Fermilab experime@33#. The first downconversion stage, by image rejectmixing, generates a 10.7-MHz i.f. An eight-pole crysbandpass filter in the 10.7 MHz i.f. rejects power outsid35-kHz frequency window centered on the cavity resonfrequency. A second conventional mixing stage downcverts the 10.7-MHz i.f. to near audio frequency~AF! cen-tered at 35 kHz. The audio signal is applied to commercfast Fourier transform~FFT! electronics@44# which com-putes a 50-kHz bandwidth power spectrum centered akHz.

The normal data-taking sequence starts by movingtuning rods incrementally to establish a new cavity resonTM010 frequency. Then, the electromagnetic power spectrabout the cavity TM010 mode is determined by the receivelectronics and commercial FFT instrumentation. ThekHz wide FFT power spectrum consists of 400 bins, ea125 Hz wide. A uniformly weighted average of 10 000 powspectra is saved together with other parameters of the exment including the measured cavityQ and resonant fre-quency. We refer to this 10 000 spectra average as a ‘‘traA total of about 4.53105 traces were recorded over th2.9–3.3 meV mass range.

The averaged power spectrum at each cavity setting,gether with other experimental parameters~e.g., cavityQ andresonant frequency! constitutes the raw data set. The the

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malized axion signature is excess power above the baground power spectrum concentrated in a peak of bandwof the order of 1 kHz~about 6 frequency bins wide!. Thedominant background sources are relatively broadbandity blackbody noise and broadband electronic noise infirst cryogenic amplification stage.

2. The resonant cavity

The resonant cavity consists of a copper-plated stainlsteel right-circular cylinder 1 m long and 50 cm diametewith two endplates. The cavity volume is;200 l. ~The firstsuch cavity had the lower endplate welded in place andupper plate removable. This configuration was difficultelectroplate, so the design was changed to employ twomovable endplates.! Each removable endplate is firmlseated on a knife-edged lip to ensure low resistance toTM010 wall currents. The cavity components are plated onmajor surfaces with high-purity oxygen-free copper, thennealed. Both metal and dielectric tuning rods were desig

FIG. 3. ~Upper! The noise temperature and gain vs frequencyone of the first amplifiers used in the experiment.~Lower! The noiseand gain of one of our best cryogenic amplifiers. The data showere taken at a bath temperature of about 12 K. The noise tempture improves slightly~5%–10%! by lowering the bath temperaturto 4.2 K. These latter amplifiers have noise temperatures of a1.5 K.

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to be used in the experiment. The metal rods consist of cper cylinders, capped at both ends, plated with oxygen-high-purity copper and then annealed. The dielectric rconsist of low-microwave-loss alumina. The rods wemounted at the end of alumina swing arms, which pivabout alumina axles that penetrate the top and bottomplates of the cavity. By rotating these axles, the tuning rmay be swung in circular arcs from close to the cavity swall to close to the center. Metal rods increase the frequeof the cavity, whereas dielectric rods decrease it. Figurea photograph of two tuning rods within the cavity. The axand arms used to move the tuning rods are visible. Inpicture, the upper-right tuning rod is made of alumina. Ofirst data-taking run used copper tubes for tuning rods, esimilar to the lower-left rod in the picture, 8.25 cm in diameter. The tuning rods are driven by stepper motors mounon top of the cryostat. The stepper motors drive G10 rinto a two-stage antibacklash gear reduction mounted oncavity endplate. The reduction, besides allowing small tunsteps, also reduces the effects of play in the G10 rods. Tare 8.4 million motor steps per full revolution, with a singstep corresponding to a change in the angle of the tuningby 0.15 arcseconds. The tuning precision is about 1 kHz100 MHz.

a. Quality factor measurement.TheQ of the TM010 modeis determined from the transmission response. A swepsignal is applied through an electric dipole probe to the vweakly coupled rf calibration port on the top plate of tcavity, and the transmitted signal from the major port isrected to a scalar network analyzer. Figure 5 is the swtransmission response across the TM010 resonance for oneparticular tuning-rod configuration. The width of the resnance is around 7 or 8 kHz~at resonant frequencies near 7MHz! with the cavity critically coupled to the receiver chaiimplying an unloadedQ of ;23105.

b. Obtaining critical coupling.The scan rate at constanSNR depends on the coupling of the external amplifier tocavity; the rate has a broad maximum at slightly over criticoupling. Hence, during normal data taking, the insertdepth of the major port electric-field probe was periodicaadjusted to maintain critical coupling between the cav

FIG. 4. The resonant cavity viewed from above with the tflange removed. The cavity is a right circular cylinder of diame50 cm and depth 1 m. An alumina tuning rod is at the upper riga copper tuning rod is at the lower left.

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TM010 mode and the receiver chain. At critical coupling, tpower incident on the cavity from the major port is absorbwithout reflection. To set the coupling, a swept rf signalinjected into the field probe through the weak-coupled sof the directional coupler~see Fig. 1!. Almost all the signalpower is absorbed in the cold 50V terminator. The remain-ing power ~around 1 part in 1000! is directed towards thecavity. A network analyzer and cryogenic directional coupare used to measure the power reflected from the cavwhen the probe is critically coupled essentially no powerreflected at the cavity resonant frequency. Figure 6 showsreflected power in the neighborhood of the TM010 mode afterthe critical coupling procedure. The reflected power is msured after the rf amplifier chain using a scalar network alyzer. The bottom of the absorption dip is the noise floorthe electronics, the actual reflected power is consideraless. We considered a;30 dB absorption dip critical coupling.

c. The cavity mode structure and form factor.There is acomplicated mode structure within the cavity. There areand TM modes associated with the smooth-wall cylindrigeometry. TEM modes are introduced by the metallic tun

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FIG. 5. The cavity transmission resonance taken from theline data acquisition display. The vertical axis is relative transmitpower after amplification. The horizontal axis is the swept inpfrequency.

FIG. 6. The relative reflected power off the cavity major portthe neighborhood of the TM010 mode after the critical couplingprocedure. The resonant frequency is 805.457 MHz.

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rods. There are, in addition, hybrid modes associated wdeviations from a uniform cross section. The cavity itselfnot a perfect circular cylinder, and the insertions and tunrods also distort the cylindrical geometry, thus there is csiderable mixing among modes. Consequently, there issufficiently accurate closed analytic form for the cavity mostructure, so the form factorC is therefore estimated by numerical simulation.

With a cavity containing two metal tuning rods, the TM010mode is lowest in frequency when both copper rods are nest the wall. As one or both rods approach the cavity centhe frequency increases. Figure 7~upper! shows the frequencies of various cavity modes versus tuning rod positionsthe case of one rod fixed near the wall and the other movtowards the center. The vertical axis is cavity frequencyf 0and the horizontal axis is number of stepper motor stfrom a rod position near the wall. These mode maps wobtained by feeding a swept rf signal into the cavity throuthe weak-coupled rf port and measuring the transmitpower. The TE and TEM modes of the cavity are onweakly excited by the rf probes oriented perpendicular toTE and TEM electric fields.

FIG. 7. ~Upper! The cavity mode structure with one rod fixenear the wall and the other rotating towards the cavity center.horizontal axis is the rod position in units of stepper motor ste~Lower! Sketch of an avoided mode crossing.

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Notice that there are mode crossings~mixings! in the vari-ous TM modes. These crossings occur when the frequenof two modes are nearly degenerate and the modes mix.ure 7~lower! is a sketch of mode crossing of a TM with a Tor TEM mode. The mode crossings introduce frequency gthat cannot be covered with a pure TM010 mode. These gapswere scanned by filling the cavity with liquid helium at 1K. Liquid helium has a relative dielectric permittivity o1.055 and hence alters the microwave index of refractiona factor of 1.027. The frequencies of all the modes ofcavity and the mode crossings are decreased by 2.7%MHz at 700 MHz. By combining data taken when the cavis filled with low-pressure gas, as is normally the case wsome data taken when the cavity is filled with helium, wcover the entire frequency range without gaps. There islight penalty in filling the cavity with helium as the reduceelectric field gives a lower axion conversion rate.

The form factorC010 of the TM010 mode, defined in Eq.~15!, is calculated from a numerical solution of the waequation for the axial component of the electric fieldEz inthe cavity, including tuning rods. A relaxation code using tGauss-Seidel method@45# is used on a 1103110 lattice forthe simplified case of a constant two-dimensional cavcross section and tuning rods.~The effects of the approxi-mately 1-cm gap at each end between the tuning rods andcavity end plates were neglected; we estimate this simpcation results in an error on the form factor of the order1%.! The resultant axial electric field is combined with thcalculated magnetic-field shape~calculated from the actuamagnet winding current density! to yield Fig. 8, a plot of theform factor versus TM010 resonant frequency for copper rodwith dimensions the same as in our experiment. The tlines represent the form factor with the cavity filled wilow-density gas and the cavity filled with liquid4He. Theform factor for different tuning rod materials~metallic anddielectric! and rod diameters is shown in Fig. 9; here thorizontal axis is the cavity resonant frequency, and the vtical axis is the form factorC010.

3. The receiver electronics

a. The cryogenic amplifiers.The characteristics of thecryogenic amplifiers are particularly important because th

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FIG. 8. The calculated cavity form factorC010 as a function ofthe cavity resonant frequency. One curve is for the cavity containa small amount of helium vapor, the other for the cavity filled wsuperfluid helium.

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noise dominates the system noise. Two cryogenic amplifibuilt by NRAO @41#, are cascaded so as to have sufficiegain to render negligible further noise contributions. Eacryogenic amplifier is a balanced design; the input signasplit into two paths by a 90° hybrid~one path phase shifteby 90° with respect to the other!. Each path is then amplifiedand recombined by another 90° hybrid~reversing the initial90° phase shift!. Any reflection from the input from one patis canceled by the phase-shifted reflection from the otpath. The reflection instead is directed to a termination retor at the phase-shifted input of the 90° hybrid@46#. Such abalanced amplifier provides good matching without isolator other nonreciprocal devices that are difficult to use inlarge ambient magnetic field. This balanced design is rized by a pair of single stage HFET amplifiers, the twomatched pair with nearly identical characteristics, wsingle-pole matching networks at the input. The amplifipackage is cooled to near liquid-helium temperatures.

The input impedance of our balanced amplifiers are tycally well matched to 50V over 500-MHz bandwidths; theinput power reflection coefficient is smaller than218 dBeverywhere between 600 and 1300 MHz, limited by the chacteristics of the hybrids.

b. Cryogenic amplifier characteristics in high ambiemagnetic fields.We were careful to ensure that the ganoise, and bandwidth of the cryogenic amplifiers are notgraded by the experiment’s ambient magnetic field.In situmeasurements of the gain of the cryogenic amplifier wcompared with bench-test data taken by NRAO.

FIG. 9. The frequency dependence of the form factor. The hzontal axis is the cavity resonant frequency, the vertical axis isform factor C010. The curves show the form factor for differentuning rod materials and rod geometries:~1! two alumina rods (e59.5), ratio of rod diameter~r! to cavity diameter~R! r /R50.13,one rod fixed at cavity center;~2! two alumina rods,r /R50.13, onerod fixed at cavity center;~3! one metal rod,r /R50.16; ~4! twometal rods,r /R50.16, one rod fixed at wall~for the 500–700 MHzrange! or fixed at the cavity center~for the 700–800 MHz range!;~5! two metal rods, ther /R50.16 rod fixed at the cavity center, thr /R50.3 rod moving.

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For the in situ gain measurements, a swept rf signalapplied through the cryogenic directional coupler to tcryogenic amplifier input. With the experiment insert at rootemperature and the magnetic field off, a scalar network alyzer at the output of the two cryogenic amplifiers allows fdetermining the cable and connector attenuation resposince the room temperature gain and other amplifier chateristics are known. The insert is then cooled and the nommagnetic field restored~the magnetic field in the neighborhood of the amplifier is around 4 T!. Keeping the sweptpower the same, another power measurement at the outpthe two amplifiers yields the cryogenic amplifier gain in higambient magnetic fields. Studies of the passive componshowed that temperature-dependent losses are negligblein situ gain is shown in Fig. 10 and is in good agreemewith independent measurements by NRAO.

The noise temperature of the cryogenic amplifier is msured with the amplifier installed for data taking. Ultimateour calibration of amplifier noise temperature is referencedthe temperature of the cavity as a Johnson noise source

We first critically coupled the amplifier to the cavity vithe reflection minimization method. On resonance, thepedance looking towards the cavity from the amplifier50 V; the cavity is then effectively a 50-V resistor whoseNyquist noise temperature is the physical temperature ofcavity. The measurements begin with the cavity temperaat 1.3 K, and the liquid-helium reservoir below the cavempty. A small amount of helium liquid is then released inthe cavity space, forming a vapor of a few Torr and ensurgood thermal contact between the tuning rods and the rethe cavity. Heaters on the top and bottom of the cavity thwarm the cavity to about 10 K over 1 h. During warming,swept signal is injected through the weak port and the tramitted spectral density determined with a scalar netwanalyzer~the swept signal is used to correct for any sligthermal drift, of order 1 dB or less, of the amplifier gain ov

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FIG. 10. The power gain of a typical cryogenic amplifier inpmeasuredin situ. The error bars are due to variations in cable losdue to temperature changes and flexing. Our results are in gagreement with independent measurements of the same devicNRAO, which are also shown.

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the measurement temperature swing!. Between sweeps,90-s power spectrum of 30-KHz bandwidth about the cavresonance is recorded by the receiver. Figure 11 showsgain-corrected power in a 125 Hz wide bandwidth atcenter of the power spectrum~on resonance! of an earlycryogenic amplifier as a function of the physical temperatof the cavity tuned to 700 MHz. The straight line is a leasquares fit. The noise temperature is the absolute value ointercept of the fitted line with the cavity temperature axhere the inferred value of the noise temperature is 4.6 Kvalue consistent with the noise temperature of that amplmeasured independently by NRAO. We ascribe a 10% msurement uncertainty to ourin situ noise temperature resulThe amplifiers currently taking data are considerably betwith measured noise temperatures below 2 K, as determindependently by NRAO and us. Figure 3~lower! shows thenoise and gain for one of these recent amplifiers at a btemperature of around 12 K; the noise is slightly reduced~by5–10%! at the operating temperature of the amplifier in tdetector.

The orientation of the amplifier during the production dataking is such that the magnetic field is parallel the HFchannel electron flow. During commissioning we observthat if the cryogenic amplifier was rotated so the magnefield is perpendicular to the electron flow in the HFET juntions, the amplifier noise temperature rises from 4.6 Karound 8 K. A simple model is that the Lorentz force on telectrons in the HFET junctions distorts the electron pathsthat their trajectory across the gate region lengthens. Insimplest version of the model, the HFET amplifier noise teperature is proportional to the length of the electron p

FIG. 11. Results of an earlyin situ noise measurement ontypical cryogenic amplifier. The vertical axis is rf spectral densrecorded by the receiver chain, the horizontal axis is the catemperature. The straight line is a least-squares fit.

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through the gate. The model predicts an increase in ntemperature consistent with that seen in@47#.

c. The room temperature receiver electronics.The outputof the cryogenic amplifiers is carried by a cryogenic RG-4coaxial cable@40# to a low noise room-temperature postamplifier mounted in a room-temperature rf-shielded encloson top of the cryostat. The power gain of the postamplifieabout 35 dB in the frequency range 300 MHz–1 GHz wnoise temperature around 90 K. The power gain of the tcryogenic amplifiers in series is 34 dB, hence the contrition of the postamplifier to the system noise temperaturenegligible, less than 0.03 K. The signal lines terminate inshielded enclosure and mate with type-N bulkhead connec-tors. The connectors then mate with flexible RG-213 cableading to the room-temperature receiver electronics inanalysis hut. Overall, there are three rf lines from the dettor to the analysis hut: one for the cryogenic amplifier outpone for the weakly coupled port, and one for the directiocoupler. The total power gain from cavity probe to the poamplifier output is 69 dB (83106); with this gain, the noisepower from the cavity and amplifier at the postamplifier oput in a 125-Hz bandwidth is 8.3310214 W.

The amplified rf signal from the detector then entersdouble-heterodyne section of the receiver. Figure 12 ischematic of the receiver i.f. and af processing componeThe first component in this section is an image rejectmixer; it shifts rf power from the TM010 resonant frequencyto an i.f. frequency range centered at 10.7 MHz, whilejecting image noise power from the i.f. The image rejectimixer is a MITEQ IRM045-070-10.7@42# with insertion lossof 6 dB, and image rejection better than 20 dB. During nmal running, the local oscillator frequency is maintained10.7 MHz below the measured cavity resonant frequencythat the cavity resonance is centered at 10.7 MHz in thestrip.

After the first mixer is an adjustable attenuator~maximum63 dB attenuation!. Attenuation is needed to avoid saturatinthe receiver during tests with the cryogenic componentsroom temperature. The i.f. signal is then amplified by ab20 dB and passes through weakly coupled signal sampused for monitoring the internal receiver power levels.

The i.f. section has a bandpass filter to suppress noutside a 30-kHz bandwidth centered on the 10.7 MHzOut-of-band power would otherwise have been aliased inaf output by the next mixing stage. On account of the strodependence of the receiver transfer function to the filteresponse, the filter is enclosed in a temperature controoven maintained at 4060.5° C. The filter has eight poleand ripple from this pole structure appears across theceiver passband. After another 20 dB of amplification, the

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FIG. 12. Simplified schematic of the intermediate~IF! and near-audio frequency~AF! compo-nents in the receiver chain.
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signal is mixed down again to near af in the range 10–kHz. The output of the second mixer leads to a commerFFT spectrum analyzer@44#, and the resulting power spectrare saved to disk as our raw data. The frequency of the loscillators is stabilized to drift less than60.005 Hz by acesium frequency reference. The frequency stability ofreceiver chain is establishedin situ by noting that occasionalarge external radio peaks drift in frequency by less th60.02 Hz.

The receiver response, including the filter, is calibratedmeasuring the receiver output spectrum with a white-nosource at the input. Noise generated by a Noise/Com 3broadband noise source@43#, amplified by 35 dB, is injectedat the rf input of the image rejection mixer. At the outputthe receiver, the FFT spectrum analyzer takes and aver

FIG. 13. The spectral output density at the output of thesection with a spectrally flat noise source at the i.f. input. The strture shows mainly the response of the crystal i.f. filter.

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400-point~125 Hz per point! power spectra of the filter passband response. After a total integration time of 2.2 h,have the calibration of the receiver response shown in F13. The vertical axis is power per 125 Hz, the horizontal ais frequency at the af output of the receiver.

The search for axionlike signals in our data is affectedeven very small drifts in the receiver response. A typicintegration time for the acquisition of a single averaged Fpower spectrum~containing 104 individual spectra, describedin the next section! is 80 s. Hence fluctuations in the powespectrum are expected to be 1/A10000 of the raw noisepower. A receiver response change of only 1% or less notably affects the averaged spectrum. Changes of apprmately 0.2% in the receiver response over the first 1.5 yeof operation were seen; the treatment of these drifts iscussed in Sec. II B.

d. Signal and noise power through the receiver chaTable I shows the approximate power at the output ofmajor components of the receiver electronics. We list powover two different bandwidths: the first over a 125-Hz banwidth ~the width of the frequency bins in the FFT spectruanalyzer!, the second over the full bandwidth of the specicomponent~this to ensure the power output of a componeover its full bandwidth is not so large that the next compnent in the receiver chain is saturated!. The input power overa 125-Hz bandwidth incident on the first cryogenic amplifiis around 10220 W (2170 dBm). Notice the i.f. crystal fil-ter significantly reduces the broadband power in decreathe bandwidth from about 1 GHz to 35 kHz.

e. The FFT spectrum analyzer and a typical power sptrum. The FFT spectrum analyzer operates here at an eftive sampling rate of 100 kHz. Every 8 ms, an individusingle-sided spectrum~one in which the negative and pos

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TABLE I. Estimated power levels at selected points in the receiver electronics chain. We takbandwidth of the cryogenic amplifiers as 1 GHz and the cavity plus amplifier noise temperature of 6

Component Gain~dB! Output Power Output powerper 125 Hz over full bandwidth

~dBm! ~dBm!

Cavity 2170 2101Cryogenicamplifiers 34 2136 267 ~over 1 GHz!Room temperaturepostamplifier 35 2101 232 ~over 1 GHz!Flexible cableto analysis hut 26 2107 238 ~over 1 GHz!Image rejectmixer 27 2114 245 ~over 1 GHz!First i.f.amplifier 30 284 215 ~over 1 GHz!Crystalfilter 23 287 260 ~over 30 kHz!Second i.f.amplifier 30 257 230 ~over 30 kHz!i.f.-afmixer 27 264 237 ~over 30 kHz!

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tive frequency components are folded on top of each oth!is taken with uniform~flat! windowing consisting of 400bins, each 125 Hz wide, spanning the frequency range 10kHz. Over 80 s of integration time per frequency settin10 000 such individual spectra are taken and averaged ais this averaged spectrum that is saved as raw data. Figushows a typical averaged power spectrum taken duringduction running.

The most obvious feature of this spectrum is the rafalloff in power at the edges of the 20–50 kHz frequenrange at the skirts of the i.f. crystal filter response. Thereslight differences between Figs. 13 and 14 due to additiostructure in the latter introduced by standing waves incavity, transmission line, and amplifier system. There is aa slight difference in noise power levels on- and oresonance; the Johnson noise power on resonance~where thecavity noise source dominates! is less than the Johnson noispower off resonance~where internal noise sources in the amplifier dominate!. Understanding the underlying structurethe raw data spectra is important as the expected axion sappears as additional structure above noise backgroacross six adjacent bins in the power spectrum.

f. Extremely narrow bin electronics.There is, in addition,a special electronics channel optimized for the detectionextremely narrow lines from late-infall axions. For this cathe af cavity signal~centered at 35 kHz! is fed into a passiveLC filter @48# ~passband.6.5 kHz), amplified, and mixeddown to a 5-kHz center frequency. The signal is applied toADC/DSP board@49# in a PC host computer, where it ioversampled at 20 kHz by a 16-bit ADC and stored in Dmemory. The PC receives the start command along with city Q and frequency data from the main DAQ computer imediately after the main FFT channel begins collecting dA single spectrum is obtained by acquiring 220 points inabout 53 s, performing the FFT calculation in about 8 s, auploading the 219-point power spectrum to the PC host fanalysis. The entire process is finished before the medresolution spectrum is done and the cavity frequencytuned.

About 3.43105 points lie in the 6.5 kHz passband withfrequency spacing of 19 mHz. In order to get higher sen

FIG. 14. A typical trace from the raw data. The vertical axisspectral density at the af output. The mixed-down cavity resonfrequency is near the center of the frequency range.

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tivity for lines of intermediate width, we averaged nonovelapping segments of 8 and 64 points, resulting in two adtional power spectra of resolution 152 mHz and 1.2 Hrespectively.

4. The cryogenic hardware

a. The magnet.The magnetic field is produced by a sperconducting solenoid@50#. The magnet consists oniobium-titanium windings immersed in a liquid4He cry-ostat. The helium consumption of the magnet is appromately 55 l per day. The axial field at the center of the sonoid is 7.6 T, falling to about 70% of this value at the cavend plates.

b. The insert and helium system.Figure 2 shows a sketchof the insert and the magnet showing the cryogenic systThe cavity is at the bottom of the insert. The entire insert cbe withdrawn from the magnet bore without warming tmagnet.

The cavity is surrounded by a stainless-steel can. Liqhelium is released into the bottom of this can from a resvoir directly above the cavity space through a thin capillaline and an adjustable inlet valve. During normal operatthe liquid helium forms a pool several cm’s deep in the spunder the cavity. A level gauge monitors the depth of thelium pool and controls the release of liquid helium frothe valve.

A 4-in diameter stainless-steel column rises from theof the can enclosing the resonant cavity through the centethe liquid-helium reservoir, through radiation baffles, acontinues to a manifold protruding from the top of the detetor. Through this column, a large Roots blower pumps onhelium pool below the cavity, thus cooling the cavity tophysical temperature of 1.3 K. The pressure of the heligas in the insert is typically less than 1 Torr. While scannnear mode crossing regions, the liquid-helium pool islowed to rise until the entire cavity is filled.~This liquidhelium is eventually boiled off by heaters attached tobottom of the cavity.! The reservoir supplying liquid heliumto the insert is refilled once every two weeks. Betweeninsert and the inner wall of the cylindrical magnet dewaran insulating space at 1026 Torr vacuum. Heat losses fromthe insert are slight as it is surrounded on three sides bymagnet dewar and on the fourth side by layers of vacooled baffles.

B. Analysis method

Our analysis is a search for the signature of axions inspectra from the detector@51#. The analysis has two pathsThe first path~the ‘‘single-bin’’ search! is motivated by thepossibility that at least some halo axions may not have thmalized and therefore have a negligible velocity dispersithese axions would deposit their energy into a single pospectrum bin. The second path~the ‘‘six-bin’’ search! makesno assumption about the halo axion energy distribution otthan its velocity dispersion isO(1023c) or less. ~Axionswith velocity larger than about 231023 escape from thehalo.! The results from the six-bin search are the most cservative since they are valid whether or not the halo axi

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have thermalized, completely or in part. Should the halononthermal because of late infall, the single-bin analywould be more sensitive. The axion search using antremely narrow line channel is outlined at the end of tsection. Both search paths incorporate a simple power exin the search statistic.

1. Overview of the single- and six-bin search methodology

Both 1- and 6-bin searches use the same raw data, wcovers without gaps 701–800 MHz. These raw data conof a set of overlapping set of averaged power spectra.refer to this data set as the ‘‘run 1 raw data.’’ These speare combined into a single power spectrum over the en701–800 MHz range with a bin width of 125 Hz.

a. The single-bin search.For the single-bin search, individual frequency bins exceeding a power level thresholdselected from the combined power spectrum. The threshis chosen relatively low so as to select a considerable numof candidates. For just the selected candidate frequenciesecond, independent, set of raw data are taken with the sintegration time. This new data set is combined with the fiproducing a single combined power spectrum with higsignal-to-noise ratio at the selected candidate frequencThe selection process is repeated; individual frequencyexceeding a power level threshold are selected from thecombined power spectrum, and so on. The few survivcandidates are carefully checked as to whether they are itified with known sources of external interference. If all cadidates were to be identified with external interference, thno candidates would survive and the excluded axion cplings are computed from the near-Gaussian statistics ofsingle-bin data.

b. The 6-bin search.For the 6-bin search, all six adjacefrequency bins exceeding a power level threshold arelected from the combined power spectrum. The thresholagain chosen relatively low so as to accept a considernumber of candidates.~These candidates are correlated wcandidates from the 1-bin search.! For just the selected candidate frequencies, a second, independent, set of raw dataken with the same integration time. These new datacombined with the first, producing a single combined powspectrum with higher signal-to-noise ratio at the seleccandidate frequencies. Again, repeating the selection procany six adjacent frequency bins exceeding a power lethreshold are selected from the new combined power strum. For just these remaining candidate frequencies, ythird, independent set of raw data are taken with the saintegration time as the initial data set, then combined wthe previously taken data. Six adjacent frequency binsceeding a certain power level threshold are selected fromnew combined power spectrum. After these three stepcandidate selection there are few surviving candidates,again we carefully check whether they are identified wknown sources of external interference. Should all candates be identified with external interference, then no cadates survive and the excluded axion couplings are compby Monte Carlo techniques.

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a. Preliminary treatment of raw traces.Figure 14 is atypical ‘‘trace’’ from raw data.~This trace is an 80-s singlesided power spectrum centered on the cavity resonantquency. It contains 400 frequency bins, each bin 125wide.! The first stage in the processing of a raw trace isremove frequency bins outside the filter passband. From400 bins in each trace, the first 100 and the last 125 binsdiscarded. Slightly more bins at the higher-frequency sidethe passband are removed due to a narrow pole in cryfilter response at this side of the filter passband with a wiof about 10 bins, which is somewhat close to the expecaxion signal width~approximately 6 bins!. We refer to theportion of each trace remaining after the end bins arecarded as the ‘‘cropped trace.’’

Figure 13 shows the receiver passband response caltion. Comparing this with Fig. 14, much of the slowly varying structure in the power spectrum is due to the receipassband response. The next operation in data analysis tfore divides the cropped trace bins by the corresponding bin the receiver passband response. The result of performthis normalization operation on the trace in Fig. 14 is shoin Fig. 15.

We refer to the traces normalized by the receiver passbresponse as ‘‘corrected traces.’’ Notice there is additiosmooth structure in the corrected traces. Furthermore,structure is not symmetric about the center bin~correspond-ing to the resonant frequency of the cavity!.

b. Origin of asymmetry in the receiver corrected tracesIfall the structure in the raw traces were attributable tofrequency-dependent effects in components downstreamthe receiver rf amplifier, the corrected traces would consisnearly Gaussian fluctuations on a background, symmeabout the cavity resonant frequency. It is clear from Fig.that, in fact the power spectra are not symmetric. Furthmore, the structure of the corrected traces depends oncavity TM010 resonant frequency. Figure 16 shows correctraces at two different resonant frequencies. Such frequedependence implies that the structure of the corrected trcomes from an interaction between the rf cavity and the fi

FIG. 15. A raw trace after the first 100 and last 125 bins~out of400 bins! have been removed, then divided bin-by-bin by theceiver response. The cavity resonant frequency was mixed-downear 34 kHz af.

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nsee,


cryogenic amplifier. This interaction is complicated; therenoise sources both in the cavity and in the amplifier, a tramission line connecting the two, and complex impedanlooking into the cavity and amplifier. In order to understathe asymmetric structure, we developed an equivalent cirmodel of the amplifier coupled to the cavity.

c. Equivalent circuit model for the resonant cavicoupled to the first cryogenic amplifier.Our equivalent cir-cuit for the resonant cavity coupled to the amplifier is shoin Fig. 17. This equivalent circuit is approximate in that whave assumed that the amplifier~which is a relatively com-plicated balanced design! can be represented as a perfecmatched 50-V input impedance containing current and voage noise sources. WithP the power at the amplifier outpuand D the displacement from the center frequency ofpower spectrum~in units of 125 Hz! the power spectrumfrom the equivalent circuit is

dP

d f~D!5

a118a3~D2a5 /a2!214a4~D2a5 /a2!

114~D2a5 /a2!2 ,

~18!

FIG. 16. Typical corrected traces at the af output for two diffent cavity resonant frequencies. Points are measured relative pThe line is from an equivalent circuit model of the amplifier, tranmission line, and cavity interaction.

FIG. 17. An equivalent circuit model of the amplifier, transmsion line, and cavity interaction.TC is the noise contributed fromcavity Johnson noise.TV and TI are the voltage and current noiscontributed by the amplifier.

09200

es-s

it

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e

where a15Tc1TI1TV , a25 f 0 /Q, a35TI1TV1(TI2TV)cos 2kL, and a45(TI2TV)sin 2kL, with k the wavenumber corrresponding to frequencyf 5 f 01125 Hz D, andthe remaining parameters~the electrical lengthL from cavityto amplifier, and the amplifier voltage and current noise ctributions TI and TV) are as shown in Fig. 17. The last coefficient a5 is actually not an equivalent circuit parametand requires some explanation: The receiver downconvpower from about the measured cavity resonant frequenca band centered at 35 kHz. Therefore, in principle the 10bin of the corrected spectrum should be the cavity resonfrequency. Actually, the cavity resonant frequency is cetered by the DAQ system to about6500 Hz. The final pa-rametera5 is therefore the displacement of the cavity resnant frequency~in 125 Hz units! from the 100th bin in thecorrected trace.

The lines overlaid on the corrected traces of Fig. 16 shthe result of the equivalent circuit model. The equivalecircuit description describes the smooth background w@52#.

d. Fit parameters and the radiometer equation.Theequivalent circuit description of the corrected traces givesestimate of the average power for each. The residual binbin fluctuations about this average power level shouldapproximately Gaussian-distributed with rms deviations csistent with the radiometer equation@53#. We shall nowcheck that this holds.

Recall that each trace is divided by its correspondingted average power; the resulting trace consists of fluctuatabout one. To simplify later processing, we subtract one freach bin, leaving Gaussian-distributed fluctuations abzero. Figure 18 is the corrected trace from the top left corof Fig. 16 normalized again from the five parameter equilent circuit model.

We refer to the dimensionless fluctuations about zerothese traces as ‘‘deltas.’’An important quantity for our anasis is the rms of these deltas for a single trace

s5A(i

n d i2

n, ~19!

-er.

-

FIG. 18. A single trace after correcting for the receiver respoby the equivalent circuit model of the amplifier, transmission linand cavity interaction.

3-14

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em

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f-fre--cetc.

is,he

onthe

nallisce.

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isthe

c-

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ee

id


wheren is the number of bins andd i is the delta from thei thbin. There is a simple relationship betweens and N, thenumber of spectra in the linear average taken by the FFform the trace. This relationship comes from the radiomeequation@53#: s51/AN, providing a simple relationship between the rms deviations in a normalized trace and the nber of spectra averaged together to form the trace. LetNeff51/s2 be the estimate of the number of averages fromone trace. Directly from thex2 probability distribution fors2, we have the probability distribution forNeff given by

P~Neff!5S n

2D n/2Neff2(11n/2)e2n/2Neffs

2

snG~n/2!, ~20!

where n5175 for the raw traces. Figure 19 is the measudistribution of Neff from the rms deviations for all the normalized traces~approximately 250 000 in number! in the firstrun data. The measured distribution is centered near 10averages, which is also the number of spectra averagedeach trace in the first run data. The curve is the expectedNeffdistribution. The reasonable agreement suggests we dohave a large contribution from spurious non-Poisson or nstationary noise sources.

Figure 20 is the same normalized trace shown in Fig.except that, in software, we have simulated an axion cadate at 32.5 kHz. The corresponding power from this candate peak in the normalized trace of heightHs is Ps5HskbTNB. Notice the heightHs of a signal seen in thenormalized trace is inversely proportional to the systnoise temperatureTN . Software peak injection of knownpower into the analysis chain enables us to establish codence limits.

e. Factors affecting the relative weights of data from dferent raw traces.The change in TM010 mode frequency between adjacent traces is typically 1/15 of the receiver paband. As a result, any one 125-Hz frequency bin appearmany traces, and these data points are combined so amaximize sensitivity. Four factors determine the weightsigned to a bin in combining the data. First, there issystem noise temperatureTN for the trace. Second, the num

FIG. 19. Histogram of the inferred number of effective averagNeff from the rms spread in the 175 bins in each trace. The linthe expectation for Gaussian noise.

09200

tor

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00for

ot-

8i-i-

fi-

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ber of spectraN averaged to form the trace. Third, the diference between the bin frequency and the resonantquency of the TM010 mode. Finally, the power from axionto-photon conversion, which may vary from trace to tradue to changes in the magnetic field, cavity form factor, e

We now consider the effect of the position of the TM010mode on the weighting applied to a bin from a trace, thatproperly accounting for the frequency offset in bins from tpeak of the cavity resonance curve. We defineh as the ratioof the signal height in a bin from axion-to-photon conversito the signal height seen if the bin were at the peak ofTM010 mode resonance. We haveh51 for a bin exactly onresonance,h50 for a bin far off resonance, and betweethese extremesh is described by a Lorentzian shape. Recthat in operation of the experiment, the first local oscillatorset to center the cavity resonance in the middle of the traIn terms ofn, the number of 125 Hz wide bins to the centof the resonance, andG, the Lorentzian resonance widthh isgiven by

h~n!51

114~125!2n2/G2 . ~21!

Our weighting strategy in combining overlapping tracesas follows: First, consider just two overlapping traces; letbin from the first ~second! have power excessd1 (d2),Lorentzian factorh1 (h2), rms power fluctuationss1

W (s2W),

and hypothetical signal powerP1 (P2). The weightingsw1andw2 that maximize signal-to-noise ratio~SNR! are @54#

dWS5w1d11w2d25h1P1

~s1W!2 d11

h2P2

~s2W!2 d2 . ~22!

The signal-to-noise ratio in any one trace isS5hP/s. Thesignal-to-noise ratio for a single bin in two overlapping spetra is

SWS5H h1P1

s11

h2P2

s2J 1/2

5AS121S2

2. ~23!

The extension to more than two overlapping tracesstraightforward.

sis

FIG. 20. A single trace from the raw data with an overlaartificial single bin axion peak.

3-15

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3. Useful quantities from the combined data

In the previous section we described the weightscheme. The weighted sum ofdWS in a single bin of thecombined data, is given by

dWS5(i

hi Pid i

~s iW!2 , ~24!

where the indexi refers to thei th spectrum contributing tothe single bin in the combined data. The standard deviafor this weighted sumsWS is

sWS5H(i

hi Pi

~s iW!2J 1/2

. ~25!

The data array in which we search for peaks is the ratio

dWS

sWS5

(i

hi Pid i /~s iW!2

H(i

hi Pi /~s iW!2J 1/2. ~26!

Notice that neitherdWS nor sWS has units of watts; they ardimensionless as a result of multiplying the excess powed icontributing to a bin by a weighting factorwi , having di-mensions of 1/W. To express the weighted-sum deviationunits of power, divide by the summed weights

(i

wi5(i

hi Pi

~s iW!2

. ~27!

Using the summed weights to normalize the weighted sumthe deltas, we obtain the following expression for fluctuatabout the mean power in watts,d j (W), in the j th bin of thecombined data

d j~W!5

(i

hi Pid i /~s iW!2

(i

hi Pi /~s iW!2

. ~28!

Similarly, the standard deviation in watts,s j (W), in the j thbin of the combined data, is

s j~W!5

H(i

hi2Pi

2/~s iW!2J 1/2

(i

hi Pi /~s iW!2

. ~29!

The expected signal power in wattsPSW in the combined data

is similarly

PSW5

(i

hi2Pi

2/~s iW!2

(i

hi Pi /~s iW!2

. ~30!

09200

n

in

ofn

Finally, the SNR in the combined data is obtained by addin quadrature the SNR’s from the contributing bins in traw traces

SNR5H(i

S hi Pi

s iW D 2J 1/2

. ~31!

4. The run 1 raw data

Production data were taken over the frequency ra701–800 MHz corresponding to a range of axion mas2.90–3.31meV. This data consists of 4.23105 raw tracessaved to disk.

a. Cuts on the raw data.Three different cuts were usethat together removed 1.4% of the raw traces. The firstremoved traces taken when the pressure in the resonantity exceeded 0.8 Torr. These relatively high pressures almalways developed as sudden rises, resulting in rapid vations in the cavity resonant frequency.

The second cut removed data that occurred whenchange in cavity resonant frequency between successpectra exceeded 7 kHz. Failing this cut was correlated wfailing the first cut, as sudden changes in cavity resonfrequency were often caused by sudden changes in presin the cavity. Data taken during rapid frequency changwould suffer from poorly measured cavityQ and resonantfrequency.

The third cut on the raw data eliminated traces writtwhen the cavity temperature exceeded 2 K. Such a relativhigh cavity temperature indicated a failure in the vacuusystem or the detector warming because the liquid-helsupply system had failed.

b. The run 1 data.Rather than having a single continuousweep over the frequency range, we scanned the frequinterval with several sweeps. This allows us to identify bacground from transient rf peaks~e.g., if a high-power peakappears at some frequency, but fails to reoccur at the sfrequency weeks or months later, it is unlikely to be duean axion!.

Run 1 occurred in three distinct stages. First, three sweof the entire frequency span were made. Second, a fouless continuous sweep of parts of the span was made tothe SNR across the frequency search range to a nomvalue. Third, gaps left in the first four sweeps due to mocrossings were filled in with the cavity flooded with liqui4He.

c. Signal-to-noise ratio in the run 1 combined data.Fig-ure 21 shows the expected power from KSVZ axions verfrequency and the corresponding rms noise. Figure 22 shthe signal-to-noise ratio versus frequency assuming an aenergy density 0.45 GeV/cm3.

This signal-to-noise ratio assumes that all the sigpower appears in a single 125-Hz bin. In the case wheresignal power is thermalized over 6 bins, the rms noise leincreases by a factor ofA6. This implies that the run 1 ratioof the KSVZ signal power to the rms noise in a 750-Hz~6bin! bandwidth is at least 4.7 everywhere in our frequenrange.

3-16

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d. The statistics of the run 1 combined data.The raw datafrom run 1 was combined according to the algorithm dscribed above. In this combined spectrum, the bin contedeviate slightly from a mean-zero Gaussian distribution.though the shape is Gaussian, the mean of the deviationot zero, so we have not entirely removed the non-Gauscomponents of the power spectra. This result is to bepected, because knowledge of the underlying physicalrameters governing the shape of the power spectra is limFor instance, there are many noise sources in the HFETplifier, yet our equivalent circuit model uses only two. Futhermore, the model does not include the effects of the innal 90° hybrids and the directional coupler betweencavity and the amplifier input. Introducing more fit parameters is undesirable because axionlike signals in the spewould be then unacceptably diluted in the fit. A secondample of our imperfect knowledge of the contributionspower spectra shapes is the crystal filter. We have assuthat the transfer function of the receiver electronics is wknown, and can be applied as an errorless correction to eraw trace. The temperature of the crystal filter was regulato better than 0.5° C, yet it is possible that some crysaging caused the transfer function to change over time.ternatively, the gain of one of the other components in

FIG. 21. Signal~upper line! and noise~lower line! power spec-tral density vs frequency, assuming that all the signal power appin a single 125 Hz bin. If, as expected, the signal power is sprover 6 bins the noise power increases by a factor ofA6.

FIG. 22. Signal-to-noise ratio vs frequency, assuming that allsignal power appears in a single 125 Hz bin.

09200

-ts-is

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edllchdll-e

receiver chain could change in time. Our receiver normalition corrections would then be inaccurate.

We found that the non-Gaussian component of the pospectra in the run 1 production data originates in the corrtion made to each power spectrum for the receiver respolikely due to a sudden aging of the filter crystals betwecalibrating the response during commissioning and rundata. The response function used throughout run 1 is shin Fig. 13. Fortunately, throughout run 1 we found this effehad stabilized. We can therefore generate an effectiveceiver calibration by taking a linear average of all the tracfrom run 1~power spectra that have been normalized boththe receiver response and the five parameter fit!. That is, binj of this average ofN spectra is (1/N)( i

NPi , j . We refer tothis average as the ‘‘run 1 fit residual.’’

We hypothesize that for any single receiver normalizpower spectrum, the distribution of power in 125-Hz frquency bins about the run 1 fit residual is Gaussian, andtest this hypothesis by simulation. We take each raw tracthe run 1 data and normalize the trace with the receiversponse of Fig. 13 and fit the normalized trace to the fiparameter cavity-amplifier model. We then keep the fitcurve and discard the real trace. We then build an artifictrace to replace the real one as follows: We start with the1 fit residual curve described above. We add to it compugenerated Gaussian distributed noise of rms deviation 0~as would be expected for a trace from an average of 10power spectra!. We then multiply this curve by the fit curveto the real trace. The resulting artificial trace is then analyexactly like a real trace from the data. It is fitted and normized with the five parameter model, and the resulting normized trace is added to the fake combined traces exactly asreal trace would have been.

This procedure is repeated for every raw trace unticomplete set of artificial combined traces has been produTo improve the statistics of the resulting data set, the produre is repeated ten times, so the artificial set of combitraces, resulting from the simulation, has roughly ten timthe statistics of the combined data. Figure 23 shows thetogram of the real combined data~points! overlaid by thesimulation~line! adjusted for the different normalizations.

rsd

e

FIG. 23. Dispersion of single 125 Hz bin power about the mefor the run 1 data. The line is the prediction from Monte Casimulation.

3-17

te fre-an. The


TABLE II. Persistent candidates in the 1-bin search channel. The first column gives the candidaquency. The second and third columns give the heights of the candidate signals in run 1 and the rescfinal column gives the results of investigation of the persistent candidate by manual scanning.

Frequency Run 1 Run R Comment and~MHz! Pk height Pk height ID method

729.910 750 3.5655 17.958 T, A748.944 125 3.7367 3.4041 N750.014 000 3.5827 3.5450 N771.218 500 28.893 3.6456 N771.218 625 28.893 3.6456 N771.234 375 23.576 5.3310 N771.265 750 27.365 9.5139 N780.299 625 3.5043 3.3814 N

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en a

5. The single-bin peak search

The single-bin peak search is most sensitive to signwith a bandwidth less than 125 Hz. The search statistipower excess in any single 125-Hz frequency bin of the cobined data. We assume that the distribution of single-power excesses in single bins of the combined data is puGaussian, so that the sensitivity to axions in this channelbe calculated analytically. The search proceeds in sevsteps. There is first a single-bin search in the run 1 dfollowed by the single-bin rescan~‘‘run R’’ ! of candidates.Finally we carefully study any candidate frequency passthe cut in both scans to see whether it was due to axionphoton conversion.

a. Single-bin peak search in the run 1 data.The datasample is the array of power excess in units of standdeviation in each of the 792 000 125-Hz-wide frequency bbetween 701 and 800 MHz. Single-bin candidates arelected by requiring the single-bin power exceed a fixed nuber of standard deviations above the mean power. Frequbins passing this cut are then rescanned to see if the poexcess is persistent. In the single-bin channel the samedidate selection criterion is applied to run 1 and run R.

In the single-bin search channel we use a threshold3.2s power excess. As discussed above, the effect of driftthe receiver passband response on the single-bin data dbution is to shift the mean of the Gaussian distribution uwards by 0.1s. Therefore for a 3.2s cut on the Gaussian datcentered at10.1s, the threshold must be at13.3s in thecombined data set. This resulted in 538 single-bin candates. The expected number of candidatesnc in a sample ofN bins from a Gaussian distribution of unit width witthreshold atp sigma is

nc5N

2 S 12erfp

A2D ~32!

or 544 candidates above13.2s.The frequency of each of the 538 candidates in the 1-

channel was rescanned, with enough traces acquired atfrequency to equal or exceed the run 1 SNR. The rescanwas combined according to the same algorithm as that uin run 1, and the same 3.2s candidate threshold applied i

09200

lsis-

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the combined run R data. Of the 538 candidates from the1 search, 8 reappeared as rescan candidates.

b. Investigation of persistent 1-bin candidates.Table II isa list of candidates surviving both the run 1 and the runselection. The fourth column contains one or more codeters indicating how the persistent candidate was eliminaas a possible axion signal. Below is an explanation of ecode letter:

A: Elimination by detection with an antenna. Here, tcable connecting the output of the room-temperature pamplifier to the input of the axion receiver is disconnecteand a stub antenna connected to the receiver input inplace. A trace is taken to see if the candidate corresponda background signal in the laboratory. If it does, this isstrong indication, but not proof, that the peak in the detecis due to rf contamination from outside the detector.

T: To establish that a peak is due to rf contamination,two main sources of rf leakage into the cavity, the weacoupled port and the cable connected to the directional cpler weak port, are terminated at the rf feedthroughs onof the cryostat. The candidate is finally discarded if the pedisappears. More specifically, a candidate is discardeddoes not exceed the 3.2s candidate threshold at 99.5% confidence following termination of test ports.

N: Nonrecurring peak. On-line software accumulatesaverage of at least 100 traces centered at the frequencypersistent candidate. If the Gaussian probability is less t0.5% that the bin height we measured is a power greater3.2s in run 1 plus a large negative statistical fluctuation cosistent with no peak observed in the 100 averaged trathen we conclude that the peak seen in run 1 was duetransient rf source and the candidate is rejected.

As can be seen from Table II, all of the candidates surving run 1 and run R were rejected. Figure 24 showsresulting axion-to-two-photon couplings excluded as a fution of axion mass at 90% confidence assuming the axline is significantly narrower than 125 Hz and a halo densof 0.45 GeV/cm3.

6. The 6-bin peak search

The 6-bin ~750 Hz! search is well matched to the linshape of thermalized galactic halo axions. It is based o

3-18

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power excess in the sum of any 6 adjacent frequency binthe combined data~the 6 bins covering a total bandwidth o750 Hz!. There are three factors which, while not affectithe single-bin search, are important for the 6-bin search.

First, because the tail of the assumed Maxwellian haxion velocity distribution extends the axion line shapehigh frequency, we do not expect all axions would convinto photons within a 750-Hz search bandwidth; 20% ofpower is in the tail of the distribution above a frequenma1750 Hz. At most, 80% of power from axion-to-photoconversion is detected in any adjacent 6 bins. Second,750-Hz bandwidth axion signal is present in a raw trace,five parameter fit to the trace baseline will deviate fromshape it would have in the absence of the signal. For KSaxion signals, the resultant dilution of the signal strenbetween a raw trace and the resulting normalized tracbetween 10% and 15%. The third effect concerns the statused to search for the 6-bin power excess. The simplestcedure is to separaten 125-Hz-wide bins inton/6 nonover-lapping 750-Hz-wide bins, each of which is the sum ofunderlying 125-Hz-wide bins. Unfortunately, an axion signwould be split between two adjacent 750-Hz bins. Depeing on the position of the peak with respect to the chobins, the power excess in a single 750-Hz bin is degradedup to 50%.

a. Use of overlapping frequency bins.To avoid the thirdproblem above, we take overlapping 6-bin segments of cbined data. Thei th 6-bin segment is the sum of thei th to the( i 15)th 125-Hz bins, the (i 11)th 6-bin segment is the sumof the (i 11)th to the (i 16)th 6-bin segment, and so onHence a set ofn 125-Hz bins yields (n25) 6-bin segmentsThis algorithm ensures that the bulk of the power in taxion line will be deposited in at least one of the 6-bin sements. The penalty of this gain in SNR over the previomethod is that there are now;n750 Hz wide bins com-pared to then/6 from the above nonoverlapping method. Othe other hand, adjacent bins in the array are highly colated, so that a thermalized Maxwellian axion signal wyield power excesses in several neighboring 6-bin sThese adjacent candidates are rescanned together, wiincurring extra rescan time. We refer to this procedure

FIG. 24. Axion-photon couplings excluded at 90% confidenassuming the axion line is significantly narrower than 125 Hz.

09200

in

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constructing overlapping 6-bin segments as ‘‘coadding’’ tdata.

The search starts by setting a candidate threshold at avalue (6A6s was used in our search!. If any bin exceeds thethreshold, then the frequency of that bin is added to a reslist. This bin and the surrounding 16 bins are flagged so tnone of the nearby correlated bins exceeding the thresare duplicated on the rescan list. The candidate threshlevel is then reduced by a small amount (0.1A6s) and theprocess repeated until some target threshold is reachedrefer to this technique as a ‘‘running sigma cut.’’

One might guess that if the same threshold level werein the n/6 uncorrelated 6-bin segments generated usinsimple 6-bin cut, and in then correlated 6-bin segments generated using a running sigma cut, there would be the sanumber of candidates in the two cases since the runnsigma cut prevents two correlated bins from both exceedthe threshold. However, in simulation studies, a search oncorrelated bins with a running sigma cut consistently yield2–3 times as many candidates as a search onn/6 uncorre-lated bins at the same threshold level.

Owing to its greater sensitivity, the correlated bin seais adopted in our 6-bin search channel. The data after pasthrough the running sigma cut filter is non-Gaussian, andsensitivity to axion signals using this search method musdetermined from simulation.

b. Simulation of the 6-bin peak search in the run 1 daThe actual candidate threshold level required in the rudata was determined from Monte Carlo simulation. Tsimulation generates a set of combined traces with artificiinjected KSVZ axionlike signals injected at a random setfrequencies. The array of power excesses from the combtraces is coadded to make correlated simulated 6-bin dand a running sigma cut peak search is implemented ondata set.

Figure 25 is a plot of the fraction of the injected pealocated by the peak search on the Monte Carlo genercombined traces versus cut level. From this simulationcandidate threshold in the 6-bin peak search on the run 1

,FIG. 25. ~Right axis! The fraction of injected artificial detecte

peaks vs the candidate threshold, and~left axis! the time needed torescan noise candidates passing the candidate threshold. The cdate threshold applied in this analysis yields a confidence grethan 90%.

3-19

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was set at 2.25s. There were 6535 candidate frequencexceeding the threshold.

c. Re-scanning the 6-bin candidates.Each candidate wasubsequently rescanned in run R. The run R data was cbined using the same data combining algorithm as in runNext, the run 1 and run R data were combined with eaother into a data set called the ‘‘11R combined data.’’

The combined data from run R differs from that in runin one important aspect: the effect of the receiver drift onrun R data is more severe than on the run 1 data. Thibecause in run R, the cavity resonant frequency is heldfixed value while a single candidate frequency is rescannhence, the same systematic shift is applied to every new tthat is added to the combined data. If the systematic shidSYS and the rms spread in Gaussian fluctuations in a sintrace iss, then the ratioRs of the amplitude of the systematic shift to the rms spread in the Gaussian noise in theof n traces weighted equally isRs5(dSYSAn)/s.

In order to correct the run R combined data for theceiver drift, we exploit the fact that in these data the resfrequency is always close to the center of the raw traces,that the systematic shift in spectra taken within a few weof each other is nearly the same. We take the set of rucombined data points near rescan frequencies, consistin6535 data points.

Associated with each of these data points is the avertime when the underlying traces contributing to each powere taken. These data points were then rearranged intcreasing average time order. For each of these points, aceiver drift correction was computed from the average of 3nearby points.

With Rs a good approximation and the raw traces contruting to each candidate point taken within a short enoutime so that the evolution of the systematic shift is small,statistics of the corrected combined data will be GaussFigure 26 is the histogram of the dispersion at candidfrequencies in the combined data from run R; it agrees wthe Gaussian expectation.

d. The algorithm for combining the run 1 with the rundata.The run 1 combined data and the run R combined dnear each rescan frequency are combined using a weighscheme similar to that used in combining the raw traces.

FIG. 26. Dispersion of single 125 Hz bin power about the me~points! for the (11R) combined data. The curve shows the exptation from Monte Carlo simulation.

09200

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41 nearest 125-Hz frequency bins surrounding each resfrequency are taken from each of the two data sets. Matchbins from the two data sets are combined as follows. Wd1 (dR) the contents of one of the 41 run 1~run R! bins, theweighted sumd11R of a bin from run 1 with the corresponding bin from run R is given by

d11R5d1S11dRSR

AS121SR

2. ~33!

The SNR in the combined data set is the quadrature sumthe component SNR’s.

The 6-bin search is more subtle. In constructing the 6-search statistic, we need to consider that the axion peamany 125 Hz bins wide, and each bin also contains Gausnoise. A peak may pass the run 1 cut due to a power exin the i th bin; if in the rescan thei th bin happens to containa large negative noise fluctuation while thej th bin ~whereu j 2 i u, axion linewidth! exceeds the threshold, then by onre-examining the same bin where we first found the powexcess, we would miss the axion. It is most likely that tsame bin will pass the cut in the two data sets (i 5 j ), withthe probability of two bins separated by (i 2 j ) passing cutsin runs 1 and R decreasing asi 2 j increases.

We therefore developed a candidate search in the run1R) combined data as follows. First the arrays of 41 babout each rescan frequency are coadded, generatinglapping 6-bin sets. The 41 element array is then multiplby the function in Fig. 27, with the peak of the functiocentered on the rescan frequency. This function represthe probability, due to noise, that a simulated axion siginjected in a particular bin appears instead in a nearbyThe two curves show the initial bin relocation after runand the second relocation after run (11R). We refer to theproduct of the overlapping six bin sets near each rescanquency and the weighting function as the weighted (11R)distribution. Candidates are those frequencies in the up

n- FIG. 27. Neighboring bin weighting functions for~circles! the(11R) combined data and~squares! the (11R1P) combined data.These functions represent the probability, due to noise, that an asignal at a particular bin appears instead in a nearby bin. The cirshow the initial bin relocation after run 1, and the squares the rcation after run (11R).

3-20

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10% probability in the weighted (11R) distribution; the cor-responding candidate threshold level in the weighted1R) distribution is 2.06A6s. The number of surviving 6-bincandidates is 654. These candidates are scanned for atime in ‘‘run P.’’

e. 6-bin search sensitivity in the run (11R) data. Thesensitivity of the experiment when the axion signal exceboth the 2.25A6s threshold in the run 1 data and th2.06A6s threshold in the run (11R) data is determinedfrom Monte Carlo simulation. In this simulation, combinetraces are created that replicate the run 1 and the run Rexcept that axionlike signals are injected at a random suof the rescan candidate frequencies~approximately 46 per 10MHz of bandwidth! passing the run 1 cut.~We inject theaxionlike signals at this stage because in the run R data,frequencies on the run 1 candidate list have a good sigto-noise ratio.! The peak search procedure is the same asapplied to the data. First a 2.25A6s threshold is set for candidates from the fake combined traces corresponding torun 1 data. A list is made of the frequencies that exceedthreshold and are correlated with the frequencies at whartificial peaks were injected. Then the simulated combintraces, corresponding to runs 1 and R, are combined uthe same algorithm as for combining the run 1 and R data2.06A6s candidate threshold is set for these simulated cobined traces. Finally, a list is made of the frequenciesinjected signals that exceeded both thresholds.

This simulation is repeated for various power levels ofinjected signals. The full frequency range 701–800 MHzsplit up into 20 ranges. The result of the simulation is tfraction of peaks injected in each of the 20 ranges exceeboth thresholds as a function of the injected signal powInverting this result gives the signal power detectable atconfidence level as a function of frequency.

The Monte Carlo simulation was implemented in theprogramming language and ran on the MIT LaboratoryNuclear Science ‘‘Abacus’’ computer facility. Figure 28 isplot of the power level that exceeeds both thresholds inMonte Carlo simulation at 93% confidence as a functionfrequency. The power level is in units of power expectfrom KSVZ axions assuming a halo density0.45 GeV/cm3. Referring to Fig. 22, the regions where w

FIG. 28. Axion power~in units of KSVZ axion power! excludedat 93% confidence after serially applying the~1! and (11R) can-didate thresholds.

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are most sensitive corresponds to the regions of highest Sin the run 1 combined data.

f. The run P combined data.The 654 6-bin candidate frequencies passing the 2.06A6s threshold in run R werescanned again. We refer to this scan as run P. Three disphases occured in the run P data taking. First persistentdidates between 778 MHz and 800 MHz were scanned. Nthe cavity was filled with liquid helium and persistent canddates near mode crossings were scanned. Finally, the rethe run P candidates were scanned. There resulted 28 treach of 10 000 power spectra, for each candidate freque

To correct for the receiver drift in the run P data, threceiver passband response was re-measured every mand the corresponding measurement used to normalize trtaken within 62 weeks. The run P combined data setmade using a minor extension of the procedure used in ming the run 11R combined data. Withd1, the deviation frommean power in one of the 41 bins near a rescan frequencthe run 1 combined data and withdR (dP) the correspondingpower excess in the run R~P! combined data, and the signato-noise ratios at that frequency byS1 , SR , andSP , respec-tively, the weighted sumd11R1P of the three data streams

d11R1P5d1S11dRSR1dPSP

AS121SR

21SP2

. ~34!

The SNR of the combined data is again the quadratureof the component SNR’s.

The 41 bin regions about each persistent candidatequency in the run~11R1P! data were coadded and agaweighted according to a weighted sum of these bins,shown in Fig. 27 as the weighting function. The weightifunction used for the run~11R1P! data is narrower than theone used to process the run~11R! data. This is because thSNR in the run~11R1P! data is higher, hence it is leslikely that fluctuations result in a bin far away in frequenfrom the bin with the maximum signal power.

Figure 29 is the weighted sum for each rescan frequein the run ~11R1P! data. Also shown is the candidat

FIG. 29. The weighted sum of bins in the neighborhood of psistent candidate frequencies. The line is the candidate thresho

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threshold that determines which frequencies to re-examThese candidates were carefully examined individuallydetermine whether the peaks at those frequencies weresistent with axion-to-photon conversion.

The effect of the run P candidate selection procedurdetermined using a simulation similar to that of the r~11R! combined data. Axion signals were added to a seMonte Carlo generated raw traces similar to the run 1, runand run P data. The power of injected axion signal depen

FIG. 30. Power of injected artificial peaks~in units of KSVZaxion power! vs the fraction of peaks passing all three candidthresholds. The KSVZ axion would be detected at over 90% codence.

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on frequency; the injected power at a frequencyf was setequal to the signal power detectable at 93% confidenceruns 1 and R, multiplied by a scale factor. The simulatiwas run at various values of the scale factor. Figure 30plot of the fraction of signals passing the cuts in the runrun ~11R!, and the run~11R1P! combined data as asfunction of the scale factor. The 6-bin search channel is ssitive to KSVZ axions at over 90% confidence.

g. Examination of surviving 6-bin candidates. TableIII shows all the 6-bin candidate frequencies surviving tcuts in runs 1, R, and P. The second, third, and fourth cumns in the table are the candidate peak heights in the tdata sets. The fifth column encodes the method used to enate that candidate. For examining these few remaining cdidates, an on-line trace combining software is used. Mraw traces are acquired at a candidate frequency andaverage of these traces is computed in real time. Then, olapping sets of 6 bins are coadded for the 6-bin peak sea

‘‘R, T’’: The axion receiver is disconnected from the loncable connecting it to the room-temperature post amplifiestub antenna is attached and one or more traces are acqat the candidate frequency. The flag R means that a rapeak was detected in the lab at the same frequency ascandidate. This hints there may be leakage of a radio siginto the axion detector. The flag T means that for such cdidates, the weakly coupled port and the line to the cryogedirectional coupler weak port are terminated on top ofcryostat. If the peak then disappears, it was eliminated acandidate.

‘‘N’’: During this examination no peak was seen at thfrequency. To establish that there is no peak at high co

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TABLE III. A list of candidate frequencies surviving cuts in runs 1, R, and P. The last column givemethod by which the candidate frequency was eliminated. Candidates marked with an asterisk werethe run P candidate threshold (3.5s), but were examined anyway.

Frequency Run 1 Run R Run P Method of~MHz! Pk height Pk height Pk height elimination

707.792 250 2.2633 7.0965 7.8610 R, T714.362 250* 2.2589 2.2014 3.4795* N718.078 125* 2.2712 2.6417 3.4418* N722.143 875 3.2541 3.1852 3.5557 N729.489 750* 2.4225 2.1543 2.9880* N729.910 750 3.3062 2.6648 4.0381 R, T731.953 750 2.6281 2.5627 3.6342 N740.692 500* 2.7180 2.0656 3.2812* N748.426 625 6.4740 6.4740 5.5094 N748.434 250 6.9001 5.3010 5.6607 N750.427 125 6.2973 4.2480 3.5190 R, T754.498 125 3.4347 3.4347 3.5136 N759.703 000 3.2476 2.7648 3.5429 N775.749 500 8.6955 4.8389 6.1293 R, T771.249 375 15.559 9.1932 5.2231 R, T783.249 375 8.5138 5.6073 5.9795 R, T790.000 000* 2.4320 2.2307 3.3705* N796.315 000 3.1884 2.8133 12.017 N799.976 250* 5.4750 3.1005 2.5656* R, T

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dence, the coadded power spectrum is calibrated by calcing the power level corresponding to 1s rms in the Gaussiannoise background from the amplifier and cavity noise teperature. Figure 21 is used to determine the signal poThe actual height of the candidate frequency bin is msured, and if the probability is less than 0.5% that theheight seen is from a signal of the power level from Fig. 2then the candidate is eliminated.

After this examination, none of the 6-bin persistent cadidates remained.

h. The exclusion limit in the 6-bin search. The sensitivityto axions in the 6-bin search is deduced from the resultsthe Monte Carlo simulation of Fig. 28. A slight dilution icandidate selection efficiency from the run P cut at 3.sreduces the confidence level to just above 90%~see Fig. 30and the preceding discussion!. Recall that the signal power inour cavity is proportional to the square of the axion phocoupling constantgAgg

2 , and together with the relation between axion mass and PQ symmetry breaking scale, wetain the following expression for the limit ongAgg

2 imposedby this experiment:

gAgg~excluded!2

5~3.631027 GeV21!2S mA

1 eVD 2Pexcl90

PKSVZ, ~35!

where PKSVZ is the power expected in the KSVZ axiomodel andPexcl90 is the signal power excluded at 90% cofidence.

Figure 31 shows the range ofgAgg2 excluded at 90% con

fidence by the 6-bin search. On the main figure, the indepdent variable has been divided bymA

2 to flatten the exclusionline. The inset shows the results from the two previous pexperiments@33#. Also shown are the KSVZ and DFSmodel predictions. This experiment has achieved an

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provement in sensitivity of two orders of magnitude ovprevious experiments, and is the first such search to prrealistic axion models.

7. The search for extremely narrow lines

According to the radiometer equation, the sensitivityour search can be increased if strong and narrow spelines exist.

For our nominal experimental conditions, the integratitime for each tuning rod setting is around 60 s. The fractioDoppler shift due to the Earth’s rotation in this time;10212 and amounts to; mHz ~less than the inverse integration time!.

Besides the highest-resolution spectrum of bandwidthmHz, two more high-resolution spectra~of Nyquist resolu-tions 152 and 1200 mHz! are formed by combining adjacenbins of the highest-resolution spectrum. The three hiresolution spectra were examined for peaks on-line by calating the mean and standard deviation of each 100 bin wsegment. Bins with an excess of 11s in the unaveraged spectrum, and excesses of 5s and 4s in the two lower resolutionspectra~152-mHz and 1.2-Hz binwidth, respectively! wereflagged and saved for further analysis.

After completion of all frequency scans, the frequencoverage was fairly uniform with every frequency samplseveral times. Any undersampled regions were rescanseparately to achieve a minimum of four samples at efrequency.

In order to reduce the number of candidate peaks, a ccidence search was performed. Since the contributing spewere generally taken at different times, the annual frequemodulation of the narrow axion lines must be considered.required the peaks to fall within a variable frequency wdow of sizeD f / f 5431027usin(pDt/365 days)u, whereDtis the time separation.

To reduce the number of candidates for subsequentscans, the thresholds for coincidence pairs at the three rlutions was set to 15s, 8s, and 5s, respectively. This pro-

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FIG. 31. The range ofgAgg2 ex-

cluded at 90% confidence by th6-bin search. On the larger figurethe independent variable has beedivided bymA

2 to flatten the exclu-sion line. The inset shows the results from the two previous pilotexperiments. Also shown are thKSVZ and DFSZ model predic-tions.

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duced a total of several hundred candidate frequenciesthe three resolutions. We later examined each peak by tasix additional spectra. If the peak persisted, further tests wperformed, similar to the 6- and 1-bin searches, by discnecting or attenuating various diagnostic coaxial cables leing into the experiment dewar. Some spurious peaks dispeared with this procedure, while the remainder wmatched to strong narrow band signals in the environme

The minimum detectable power in a bin of bandwidthB isgiven by Pmin5kBTsBNs , whereNs is the number of standard deviations above the mean. Because the signal canbetween two bins, the detectable power is smaller thanPmin .A simulation shows that at an approximate 90% confideand with the signal frequency falling randomly between tbin frequencies, the least power in the strongest of thebins for two out of four trials is 0.73Pmin . Thus, the mini-mum detectable power in for the three different resolutioare approximatelyPmin.3310223 W, 5310223 W, and 9310223 W, respectively, assuming a system noise tempeture of Ts.5.5 K. This represents sensitivity to approxmately 10%, 17%, and 30% of the total KSVZ axion signpower.

III. THE U.S. rf CAVITY AXION SEARCH:HIGHER SENSITIVITY UPGRADE

A. dc SQUID amplifiers in the axion search

For a given axion-photon coupling, the scan rate improwith decreasing system noise temperature asTs

22 , and for agiven scan rate, the power sensitivity, which can be reachis proportional torAgAgg

2 , and goes asTs . We are thereforeplanning to improve the scan rate and sensitivity by lowerthe system noise temperatureTs5Tn1T, i.e., the electronicnoise temperature plus the cavity blackbody temperaturethe present experiment,Tn;T;1.5 K, and thusTs;3 K atbest. Recently, however, the Berkeley group has develodc superconducting quantum interference device~SQUID!amplifiers in the 100–1000 MHz range specifically for taxion experiment. This section is a brief description of theSQUID amplifiers, and the new results.

The essence of the dc SQUID is shown in Fig. 32~a!. Twojunctions are connected in parallel, breaking a superconding loop of inductanceL. Each junction has a capacitanceCand is shunted by a resistanceR. The I –V characteristics areshown in Fig. 32~b! for F5nF0 and (n11/2)F0, whereFis the external flux applied to the loop,n is an integer, andF05h/2e'2.07310215 Wb is the flux quantum. When thSQUID is biased with a constant current~greater than 2I 0,whereI 0 is the critical current of each junction!, the voltageacross the SQUID oscillates with periodF0 as F is in-creased@Fig. 32~c!#. To maximize the performance of thamplifier, the SQUID is biased at the steep portion ofV–F curve, where the flux-to-voltage transfer coefficieVF[u]V/]Fu I , is a maximum. Thus, the SQUID producean output voltage in response to a small input fluxdF(!F0), and is effectively a flux-to-voltage transducer.

The signal and noise properties of the SQUID aretained by solving numerically the equations relating the vo

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age across the SQUID to the bias current and the applied@55#. The noise arises from Nyquist current in the shuntsistors, which has a current spectral density 4kT/R. Onefinds thatVF peaks smoothly as a function of bias currewith a shallow maximum around (2n11)F0/4. Optimumperformance is found forbL52LI 0 /F051.

For a typical SQUID at 4.2 K withbL51 and 2pkT/I 0F050.05, one finds the flux-to-voltage transfer coefficientgiven approximately byVF'R/L. The corresponding whitevoltage noise across the SQUID has a spectral denSv( f )'16kTR, approximately 8 times the Nyquist noise inresistanceR/2. The spectral density of the equivalent flunoise in the SQUID is thusSF( f )5Sv( f )/VF

2 '16kTL2/R.It is often convenient to introduce the flux noise energy:

E~ f !5SF~ f !/2L'9kTL/R. ~36!

These expressions forSv( f ), SF( f ), andE( f ) show thatthe noise power of the SQUID is expected to scale withtemperatureT. For a representative SQUID at 4.2 K withL5100 pH, I 0510 mA, and R54V, one finds VF

580 mVF021 andE( f )'1.3310232 J/Hz.

The most widely used configuration for the dc SQU@56# is shown schematically in Fig. 32~d!. The loop of theSQUID consists of a square washer of niobium with athrough one side. The two Josephson junctions are growneach side of the slit near the outer edge, and connecteda second niobium film to form a closed loop. The bias crent is passed between the SQUID washer and the couelectrode of the junctions. The magnetic flux is applied toSQUID by passing a current through a spiral coil deposiover the square washer, with an intervening insulating la

To form a tuned amplifier, one connects the voltasource in series with a capacitor and the input coil ofSQUID to form a tuned circuit. The lowest noise temperatu

FIG. 32. Left:~a! The dc SQUID;~b! I –V characteristics;~c! Vvs F/F0 at constant bias currentI B ; ~d! square washer dc SQUID

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at the resonant frequency of the tank circuitv/2p isachieved for a source resistancevMi

2VF /R, whereMi is themutual inductance between the input coil and the SQUloop. Under these conditions, one finds an optimized notemperature

Tn'7vT/VF'7T/G, ~37!

whereG'VF /v is the corresponding power gain. Expemental results in good agreement with these predictions hbeen attained at frequencies up to about 100 MHz@57#. Un-fortunately, at higher frequencies, the effects of parasiticpacitance between the input coil and the SQUID wascause a substantial roll-off in the amplifier gain@58#. Al-though attempts have been made to circumvent this probby placing the input coil inside the hole in the SQUID@59#,because of the decreased coupling between the coil andSQUID, the resultant gain is quite small.

An alternative way of achieving high gains and low noitemperatures at high frequencies is to configure the SQinput resonantly in order to tune out the parasitic loadiHere, the input coil is used as a microstrip resonator@60#, theinput signal is no longer coupled to the two ends of the incoil, but rather between one end of the coil and the SQUloop, which acts as a ground plane for the coil. The micstrip resonator is thus formed by the open-ended stripincluding the inductance of the input coil and its grouplane and the capacitance between them.

A number of such SQUID amplifiers have been fabricaand operated. These started as conventional square-wSQUID’s, with inner and outer dimensions of 0.2 m30.2 mm and 1 mm31 mm; the 31-turn input coil had awidth of 5 mm and a lengthl'71 mm. Since the conventional washer SQUID is an asymmetric device~the two Jo-sephson junctions are situated close together rather thaopposite sides of the SQUID loop!, one can either ground thwasher or ground the counterelectrode close to the Josepjunctions. Using the washer as a ground plane for the incoil suggests one should ground the washer. However,also possible to ground the counterelectrode and havewasher at the output potential. In this case, depending onsign of VF , one can obtain either a negative or positifeedback from the output to the input. For the sign of ttransfer function~denotedVF

1! the power gain was 18 dB a620 MHz and theQ of the input resonator was about 31.

The SQUID’s were mostly operated with the counter eltrode grounded and the washer at output potential. Fig33~a! shows the gain achieved for four different lengthscoil on the same SQUID. The gain is typically 18 dB.

The noise temperature of a number of SQUID amplifiwere measured referenced to a heated resistor as asource. Figure 33~b! shows the gain and noise temperatufor an 11-turn SQUID. On resonance, the gain is 22 dB athe noise temperatureTn50.9 K60.3 K, measured at abath temperature of 4.2 K. This is the overall system notemperature, which includes a 0.4 K contribution fromroom-temperature post amplifier.

In order to reduce this noise contribution, a single-staHFET amplifier was operated at a bath temperature of 1.8

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The lowest noise temperature achieved with this ampliwas about 12 K. Two SQUID amplifiers were characterizone with a resonant frequency of 250 MHz and a secowith 365 MHz. When the SQUID’s were cooled to about 1K, the gain increased slightly, with a system noise tempeture of about 0.360.1 K and 0.2560.1 K, respectively@thelatter shown in Fig. 33~c!#. To our knowledge, these are th

FIG. 33. ~a! Gain vs frequency for four coils on the samSQUID, with signal applied to the innermost turn. Data arecounterelectrode grounded andVF

1 . ~b! Gain and noise temperaturvs frequency for an 11-turn SQUID cooled to 4.2 K.~c! Noisetemperature vs frequency for a 29-turn SQUID cooled to 1.8with a HFET postamplifier at the same temperature.

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lowest noise temperatures achieved in this frequency ra~The apparent narrow bandwidth was set by the resoninput matching network of this specific post amplifier, adoes not reflect the bandwidth intrinsic to the SQUID.!

More recently, a microstrip SQUID amplifier was cooleto 0.4 K, and found to have a noise temperature of aboutK. In addition, demonstrations have been made of adjustmicrostrip tuning with the stripline loaded by a varactoSQUID post amplification, and SQUID amplification withresonant input load. Further studies are planned to charaize the properties of these SQUID’s at dilution refrigeratemperatures.

B. Cryogenic upgrade

The upgraded detector will be a considerably more coplex instrument than the present one. The upgrade ofexperiment to incorporate SQUID amplifiers poses sevunique design challenges. Since the system noise tempture Ts5T(117v/VF), the physical temperatureT will bereduced to around 200 mK using a dilution refrigerator.practice,Tn saturates at about 100 mK because of hot etron effects@61#.

The upgraded experimental insert will be compatible wthe existing 8 T magnet, i.e., the cavity arrays and refrigetor will fit into a 53-cm diameter clear bore. The cavity arrwith its vacuum and thermal shields (;200 kg) occupiesthe lower third of the tower, situating it in the middle of thsolenoidal magnet. The array is suspended from the cgenic assembly, comprised of the dilution refrigerator, ficompensation coils, LHe reservoir, thermal baffles, suppmembers, etc., occupying the upper two-thirds of the towThe entire tower is suspended from a 60-cm diameter flanmating with the top of the magnet cryostat via an O-riseal. The dilution refrigerator itself will fit within a 17.5-cmbore down the center, which in addition to serving asvacuum pump-out column, accommodates a helium pre-cline, microwave signal cables, actuator, and sensor wiresminimize the thermal load from the outside, all mechanimotion for tuning and coupling will be performed by piezelectric actuators; these have already been fitted topresent experiment.

The SQUID amplifier requires a field-free environmefor operation (&1 mT at the shields!, but is located lessthan a meter above the magnet coil where the fringe fiwould normally be on the order of 1 T. Hence, there is a ficompensation coil to reduce the field to a few tens of msurrounding the hermetic niobium can containing tSQUID. The field compensation coil is actually a pairopposing coils mechanically and electrically locked togethwith relative dipole moments such that the net force onpair is zero, but able to produce a zero field region atlocation of the SQUID. Whereas a single bucking coil wouhave an upward force of several tons, this configurationonly internal forces between the two coils for any excitatiof the main field magnet. Figure 34 exhibits key details.

IV. CONCLUSIONS

While much has happened both in particle physics acosmology over the past two decades, the motivation for

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axion remains strong. It is still the most compelling mechnism to enforce strong-CP conservation, and the sum total oall cosmological observables still point to a large CDM coponent in the Universe.

Figure 31 shows the rf cavity experimental constraintsthe axion axion mass and coupling through the axion-phocoupling, along with two prototypical axion models~KSVZand DFSZ!. It provokes several key points.

The first concerns axion models. While it is encouragithat two very different axion models have axion-photon coplings of the same order of magnitude, one should be ctious in drawing conclusions. Should the ratio of the electmagnetic and color anomalyE/N of the PQ symmetry be faaway from that corresponding to exact suppress(;1.95) there could be a pleasant surprise for experimenists; conversely a value close to it could make the axnearly impossible to see. Kim has evaluated numerous mels with very different but plausible PQ charges@62#, andnevertheless finds their correspondinggAgg

2 clustered inroughly two orders of magnitude, mostly more favorabthan the implementation of the benchmark DFSZ modSecond, the open range of axion masses (mA

51026–1023 eV) is ‘‘soft’’ on both ends, particularly onthe low-mass side.

It is also interesting that the axion-photon couplinggAgg

has lent itself to the greatest variety of imaginative and prtical search techniques. If the axion is someday found, it w

likely be through theEW •BW interaction. Purely laboratory experiments, such as the precision measurement of vacuumrefringence using lasers and magnets, are elegant andsoon provide beautiful demonstrations of higher-order QEOn the other hand, it strikes us as unlikely those techniqcould ever be sensitive to DFSZ axions, although verifyi

FIG. 34. Sketch of upgraded experiment, and field profile wcompensation coil pair.

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the astrophysical limits would be an important and signcant achievement.

The most encouraging development in the past two yehas been our demonstration that the standard axion mocould be tested with the microwave cavity experiments. Ugrades to extend the sensitivity and mass reach are alrbeing planned. In the meantime, we plan to operate theperiment indefinitely; the (1 –10meV) region of mass isarguably the most interesting range to explore.

The microwave cavity experiments are also not witholimitations. The local CDM density of our halo is not wedetermined. Furthermore, we do not see how to easily exthe mass search of the microwave cavity searches to cthe entire range, particularly upward. IfmA@1024 eV, an-other technique will need to be found as we consider itrealistic to operate an experiment with hundreds or m

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cm-scale cavities. It is impressive to consider that as smathe axion signal is expected to be—requiring radio receivorders of magnitude more sensitive than anything elseearth—power sensitivity is not likely to be a limitation of thmicrowave cavity experiments, in view of the very promiing work on dc SQUID amplifiers.

ACKNOWLEDGMENTS

This research is supported by the U.S. Department ofergy under Contract Nos. DE-FC02-94ER40818, W-74ENG-48, DE-FG02-97ER41029, DE-AC02-76CH0300, DDE-FG02-90ER40560, DE-AC-03-76SF00098, andNational Science Foundation under Award No. PH9501959. We thank the LLNL Technical Information Depament for timely and efficient production of the graphics.

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Large-scale microwave cavity search for dark-matter axions