Design and Calibration of the 34 GHz Yale Microwave Cavity Experiment
P. L. Slocum, O. K. Baker, J. L. Hirshfield, Y. Jiang, A. T. Malagon, A. J. Martin, S. Shchelkunov, A. Szymkowiak
DDesign and Calibration of the 34 GHz Yale MicrowaveCavity Experiment
P. L. Slocum a , O. K. Baker a , J. L. Hirshfield a,b , Y. Jiang a , A. T. Malagon a ,A. J. Martin a , S. Shchelkunov a , A. Szymkowiak a,c a Department of Physics, Yale University, PO Box 208120, New Haven, CT USA 06520 b Omega-P, Inc., 291 Whitney Ave.,Suite 401, New Haven, CT 06511 c Department of Astronomy, Yale University, PO Box 208101, New Haven CT 06520
Abstract
Several proposed models of the cold dark matter in the universe includelight neutral bosons with sub–eV masses. In many cases their detectionhinges on their infrequent interactions with Standard Model photons at sub–eV energies. We describe the design and performance of an experiment tosearch for aberrations from the broadband noise power associated with a5 K copper resonant cavity in the vicinity of 34 GHz (0.1 meV). The cavity,microwave receiver, and data reduction are described. Several configurationsof the experiment are discussed in terms of their impact on the sensitivity ofthe search for axion–like particles and hidden sector photons.
Keywords: resonant cavity, microwave receiver, axion, cold dark matter,hidden sector photon
1. Introduction
Direct detection of cold dark matter (CDM) is required before many ques-tions can be addressed about its origin and implications in the universe. Cos-mological evidence for the existence of dark matter [1, 2, 3, 4] has consistentlydriven the need for measurements that can impose additional constraints ontheory.Searches for CDM with masses of GeV/c are ongoing but have thus faryielded either inconclusive [5, 6, 7, 8, 9] or negative results [10, 11, 12, 13, 14,15]. Similarly, collider searches for signals from CDM have been negative [16].These measurements have led to constructive limits on models as well as aheightened interest in light CDM candidates with masses less than 1 eV. Preprint submitted to Journal of L A TEX Templates November 13, 2018 a r X i v : . [ phy s i c s . i n s - d e t ] O c t hus far searches for light CDM in the laboratory have focused primarily onparticles that should interact infrequently, but predictably, with StandardModel (SM) photons. The pseudoscalar axion [17, 18, 19] is required tocouple to 2 SM photons by way of the Primakoff effect. Other pseudoscalarand scalar axion–like particles (ALPs) that arise in string theory [20, 21] areallowed, but not required, to couple to 2 photons. The vector hidden sectorphoton [20, 21] can also be the CDM [22, 21] and interacts with SM photonsthrough mass–dependent kinetic mixing.The nearly monoenergetic distribution of light CDM, as well as its pos-sible interaction with sub–eV ( (cid:46) GHz) photons, make it susceptible to dis-covery in searches that utilize radio frequency (RF) detection techniques. Inparticular, the approach pioneered by Sikivie [23, 24] and the ADMX collabo-ration [25, 26, 27] in which a resonant cavity sitting in a strong magnetic fieldis coupled to a microwave receiver can be an extremely sensitive probe of thesub–eV mass regime [27]. In this paper we discuss the design and calibrationof a similar experiment. A low–noise cryogenic amplifier and microwave re-ceiver are employed to search for deviations from the nominal spectrum of 34GHz power associated with a 5 K Cu resonant cavity. Several configurationsof the experiment are discussed in terms of their sensitivity to sub–eV massCDM models.
2. Experiment
The experiment sits inside an Oxford Instruments V22460 superconduct-ing magnet with a peak field of 7 T inside the warm bore of diameter 89 mm.The magnetic field is strongest in a 10 cm long region located 36 cm abovethe bottom edge of the bore. The intended purpose of the magnet is for mea-surements of nuclear magnetic resonance (NMR) with a field that is specifiedto be uniform across the peak region to a few parts per million. In the presentexperiment, however, the field uniformity is only required to be of order 10%.The temporal stability of the field has been found to be better than a fewpercent over a span of 5 years. An outline of the magnet is included inFigure 1.Also shown in Figure 1 is the liquid He cryostat system that extends intothe bore of the magnet. Its central sample tube houses the signal cavityand cryogenic amplifiers, and is cooled by He gas that has been vaporizedfrom a reservoir of liquid He. Surrounding the sample tube is a liquid N igure 1: Magnet, built by Oxford Instruments, and cold gas cryostat built by CryoIndustries of America, Inc. The experiment has two main running configurations: an experiment thatis driven with 34 GHz RF power, and alternatively, a listening mode. Thedriven experiment consists of two adjacent 34 GHz oxygen–free high thermalconductivity (OFHC) copper microwave cavities each of which supports atranverse electric (TE) mode. The “signal” cavity sits near the bottom ofthe cryostat in the region where the external B-field is maximal. The “drive”cavity sits adjacent to the signal cavity, inside the bore of the magnet butoutside the cryostat. In the second configuration, or listening mode, onlythe signal cavity is employed. One of two types of signal cavities is utilized:Either a cavity that supports a TE mode, or one that supports the transversemagnetic (TM) mode. The locations of the signal and drive cavities are shownin the schematic of Figure 2.
The cylindrical TE signal cavity supports the TE mode and is 11 mmin diameter by 17 mm in height. Its central resonant frequency is 34.29 GHz.The resonant frequency is tunable across 500 MHz using a plunger that movesvertically. There is a 1 mm gap between the plunger and the side wall whosepurpose is to break the degeneracy between the TE and the TM modes.The position of the plunger is adjusted by a lever attached to a threadedfitting that moves freely at temperatures between 7 K and 300 K. The fittingattaches to a meter–long fiberglass G10 rod that is turned by hand fromoutside the top flange of the cryostat. Figure 3 shows a drawing of theTE cavity and its tuning mechanism. Also shown are the thin OFHC finsthat increase the surface area of the assembly, helping to optimize its rate ofcooling inside the cryostat. The fins are asymmetric due to constraints onavailable space.The TE signal cavity is designed to be critically coupled at cryogenictemperatures to a few inches of copper rectangular WR28 waveguide. Thewaveguide has inside dimensions 0.280 by 0.140 inches and cutoff frequency4 g il e n t P X A N s i g n a l a n a l y z e r ~ A g il e n t P N A E C n e t w o r k a n a l y z e r D , D , C T , R M H z r e f e r e n c e MM A n r i t s u M G C s w ee p o s c ill a t o r L O = . + / - . G H z L O = . G H z L O = M H z I F = . G H z ( . G H z ) I F = M H z ( M H z ) I Q M Q M Q M Q M Q M Q S M S M S M M C M C M Q M Q S R S S R S ~ ~ S M D : D a t a t a k i n g w i t h o n e c a v i t y D : D a t a t a k i n g w i t h t w o c a v i t i e s C : C a li b r a t i o n T : T r a n s m i ss i o n R : R e f l e c t i o n L O : l o c a l o s c ill a t o r I F : i n t e r m e d i a t e f r e qu e n c y R F : c a v i t y f r e qu e n c y () : i m a g e f r e qu e n c y C , D O x f o r d I n s t r u m e n t s V T m a g n e t T A n r i t s u M G C s w ee p o s c ill a t o r m e c h a n i c a l pu m p MM MM MM H H F M F M S M R F = . + / - . G H z ( . + / - . G H z ) MM WR28 c r y o s t a t WR28 WR28 s i g n a l c a v i t y d r i v e c a v i t y N I N I d i s k D , C P a r t s li s t : S M : S p e c t r u m M i c r o w a v e w a v e g u i d e b a ndp a ss f il t e r W . G - . G - FF S M : S p e c t r u m M i c r o w a v e c a v i t y b a ndp a ss f il t e r B M - - SS S M : S p e c t r u m M i c r o w a v e t ubu l a r b a ndp a ss f il t e r B C - - SS M C : M i n i c i r c u i t s c o a x i a l l o w p a ss f il t e r S L P - + M Q : M i t e q m i x e r M W M Q : M i t e q a m p li f i e r A FS - - - P - G W - M Q : M i t e q m i x e r D M X L M Q : M i t e q a m p li f i e r A FS - - - P - M Q : M i t e q m i x e r I R M L C Q M Q : M i t e q o s c ill a t o r D L C R O - - - - P M Q : M i t e q o s c ill a t o r P L D - C - - - - P N I : N a t i o n a l I n s t r u m e n t s d i g i t i z e r U S B - o r P C I - . H : J P L C r y o g e n i c H E M T L o w N o i s e A m p li f i e r C R Y O S R S : S t a n f o r d R e s e a r c h S y s t e m s v o l t a g e a m p li f i e r S R A F M : F l a nn M i c r o w a v e w a v e g u i d e d i r e c t i o n a l c o up l e r s - MM : M a u r y M i c r o w a v e w a v e g u i d e t o c o a x i a l a d a p t e r U B MM : M a u r y M i c r o w a v e w a v e g u i d e t o c o a x i a l a d a p t e r U C W R : c o pp e r r e c t a n g u l a r w a v e g u i d e w i t h i nn e r d i m e n s i o n s . x . i n c h e s F i g u r e : S c h e m a t i c o f t h ee x p e r i m e n t . igure 3: Cross section showing the signal cavity, the plunger with bellows for tuning,and the 2 WR28 waveguides oriented vertically. The circle inside the outline of the cavitydenotes the weakly coupled port. Also shown are the threaded fittings and the connectorthat interfaces with the fiberglass G10 rod. y ( m ) -0.004-0.002 0 0.002 0.004 x ( m ) -0.004-0.00200.0020.004 ) m ( / V D | B | S / z B -60-40-20020406080100120 Axial B Field r ( m ) z ( m ) ) m ( / V D | E | S / q E Azimuthal E Field
Figure 4: Two important components of the fields in the TE cavity. The left panelshows B z , the critical component in Equation 3. The right panel depicts E θ which is thefield that enters into the numerator of Equation 1. z direction, which in the case of the TE modeis theoretically B z = k r /kJ ( k r r ) sin ( k z z ) while E z = 0. In practice B z isperturbed by its coupling to the waveguide. Its magnitude in the cavity issimulated in accordance with the actual geometry using the HFSS simulationsoftware in ANSYS (cid:114) Academic Research Release 12.1.2 and is plotted in theleft panel of Figure 4. In the second signal cavity [28] supporting the TM mode, B z = 0; the diameter is 15 mm and the height is 9 mm. The axially–aligned electric field takes the analytic form E z = k r /kJ ( k r r ) and is notplotted.The second important field component to consider is the azimuthal elec-tric field, or E θ = J (cid:48) ( k r r ) sin ( k z z ). It is the only non–zero component of theelectric field in the cylindrical TE mode. The right panel of Figure 4 showsits analytical form as a function of height and radius in an ideal cavity.7 .2.2. Drive Cavity The drive cavity runs in the TE mode at the same resonant frequencyas the TE signal cavity, at room temperature. Like the signal cavityits inner volume is on the order of 1 cm , with diameter 12 mm and height17 mm. It is typically driven by a narrow monoenergetic signal with (cid:46) The sensitivity of the experiment to new physics is governed partly bythe orientation of the electric and magnetic fields inside the resonant cavity.Each of the two running modes, and each of the two signal cavities, has afield overlap integral that defines its particular ability to detect signals drivenby interactions with new light particles.The running mode with two TE cavities side by side is optimized tofind oscillations between hidden sector photons and Standard Model photons.The sensitivity of the result depends on the overlap integral in [20] G ≡ ω o (cid:90) V (cid:48) (cid:90) V d x d y exp( ik | x − y | ) A ( y ) A (cid:48) ( x )4 π | x − y | (1)where ω is the drive frequency and k is the wavenumber of the hidden sectorphoton. Following the steps in [20] E = − dA/dt . Taking the spatial partof E θ ( x , t ) in the TE cavities from the right panel of Figure 4, | G | iscomputed numerically for the two side-by-side cavities in this experiment.Figure 5 shows the result.Alternatively, in the case of the listening mode that utilizes just one signalcavity, the overlap integral C lmn is defined with respect to the direction of thestrong external magnetic field ˆB . It is first described in [23, 24] to quantifythe response of a resonant cavity to a pseudoscalar axion in a strong magneticfield: C lmn ≡ (cid:12)(cid:12)(cid:12)(cid:82) V d x E · ˆB (cid:12)(cid:12)(cid:12) V (cid:82) V d x(cid:15) r | E | , (2)where E is the electric field in the cavity and V is the volume. In the case8 w / HSP k | G | HSP Geometry Factor
Figure 5: Plot of the field overlap integral for hidden sector photons (HSP), | G | [20],against the wavenumber k HSP . of the TM cavity employed in this experiment, C lmn is on the order of0.1 [28].For the signal cavity that runs in the TE mode, C lmn is identically zerobecause E z =0. However there is a small overlap C (cid:48) lmn in the case of scalaraxion–like particles, adapting the calculation from Eq. 2 as C (cid:48) lmn ≡ (cid:12)(cid:12)(cid:12)(cid:82) V d x B · ˆB (cid:12)(cid:12)(cid:12) V (cid:82) V d x µ r | B | . (3)The overlap is small at O (10 − ) because of the behavior of the Bessel function J ( k r r ) in the axial magnetic field, plotted on the left in Figure 4. The fieldis positive near the center of the cavity, and negative toward the outer rim.When integrated over the cylindrical volume, B · ˆB decreases. The first and most critical component in the receiver chain is a low–noise, broadband, high–electron–mobility transistor (HEMT) amplifier [29].It typically operates at frequencies of 11–26 GHz but is routinely testedfor use up to 40 GHz. In this experiment the HEMT is cooled to 7 Kand measures broadband noise near 34 GHz. Its specified noise temperaturemeasured at 22 K is approximately 35 K. In Section 3.1 the noise temperatureof the HEMT is found to be near 20 K when it is cooled to 7 K. While thereare actually two HEMT amplifiers connected in series inside the cryostat for9dequate gain, it is the first HEMT that dictates the noise temperature ofthe system. The latter remains true as long as its output power is largecompared to the noise temperature of the next amplifier.
The room–temperature receiver uses a triple heterodyne technique to mixthe RF signal down from 34 GHz to baseband. The block diagram is includedin Figure 2. The three–stage design has been chosen to avoid possible prob-lems with crosstalk related to high amplification at 34 GHz. Broadbandnoise is limited at each stage by bandpass filters placed before and after theamplifiers, thereby avoiding saturation.Power at the image frequencies is suppressed by more than 100 dB withthe bandpass filters (BPFs) that sit before the first two mixers. For exam-ple, if RF frequency f mixes with LO frequency f , then the intermediatefrequency (IF) is f -f and the image lies at 2f -f . It is this image frequencythat must lie outside the passband of the BPF that precedes each mixer. Ifit were not suppressed then the noise power would increase by a factor of2 after the mixer. In the case of the third mixer, a different technique isused to suppress the image power. The outputs of the mixer in the basebandare separated into the in–phase ( I ) and quadrature ( Q ) voltages. The imagepower in the baseband is suppressed by selecting the sign of the complexphase φ = tan − ( Q/I ).The first six harmonics of each LO are also excluded from the passband ofthe receiver. For example, the second LO in the chain oscillates at 3.5 GHz,so its first 6 harmonics are 7.0 GHz, 10.5 GHz, 14.0 GHz, 17.5 GHz, 21.0GHz, and 24.5 GHz. The frequency plan is chosen so that these harmonics donot propagate through the electronics chain into the baseband. If a harmonicdid pass into the baseband, it could mimic a real signal and cause difficulty inthe offline analysis. In spite of the above it should be noted that harmonicswith even higher orders are still expected to be present in the data; care isrequired, as always, in discriminating between a real signal and a systematicor environmental feature.As a preliminary check of the receiver’s performance, the output of theroom temperature chain has been checked against its specifications at the firsttwo stages with a 50 Ω termination at the input. The specified noise figureand gain of the components in the room–temperature chain are cascaded10omponent measured predicted(dBm/Hz) (dBm/Hz)300K 50Ω term. < -150 -174mixer 1 < -150 -173IF1 amp -112 -112mixer 2 -116 -118IF2 amp -85 -85 Table 1: Table of noise power measured after the first two stages of the microwave receiver,compared with values expected from the Friis formula for cascaded noise power. according to the Friis formula for total noise factor F = F + F − G + F − G G + · · · , (4)where F represents the total noise factor, F i is the specified noise factor forcomponent i , and G i is the gain of component i . The expected noise powerin the chain at any given point i is N S + G + 10 log ( F ) dBm/Hz where N S is the noise power of the source at the input to the chain and G is the totalgain. The predicted noise power is then compared with the measured noisepower in Table 1, with reasonable agreement.As shown in Figure 2, a low–pass filter typically precedes the digitizerin each of the two paths I and Q . Figure 6 shows the baseband powerin I and Q measured with an Agilent PXA N9030 spectrum analyzer (leftpanel) and with the digitizer (right panel). The absolute magnitudes derivedfrom each instrument are not expected to be identical due to losses that arecharacteristic of each digitizer. However the relative magnitudes of the powerin I and Q are expected to match.
3. Measurements
The system noise temperature T sys is the figure of merit that drives thestatistical uncertainty σ T in the measurements, according to the Dicke ra-diometer equation [30] σ T = T sys √ ∆ ντ (5)11 Baseband Frequency (MHz) R e l a t i ve P o w e r ( d B ) -76-74-72-70-68-66-64-62-60-58-56 I Q Receiver VSA Output
Baseband Frequency (MHz) P o w e r ( d B m / H z ) -124-122-120-118-116-114-112-110-108-106 +Q I I Q Receiver Digitizer Output
Figure 6: Plots comparing the output of the receiver chain in the baseband. The leftpanel shows the outputs in I and Q measured with the spectrum analyzer. The rightpanel shows the outputs after the digitizer and offline analysis. where ∆ ν is the resolution bandwidth and τ is the integration time. Fromthe equation, σ T decreases with longer integration times and increases withnarrower bandwidths. In this experiment, T sys = T HEMT + T th , where T HEMT is the noise temperature of the HEMT and T th is the physical temperatureof the HEMT’s chassis and connectors.The total uncertainty σ tot in the measurements is driven by both thestatistical uncertainty σ T and the systematic uncertainties σ sys . Effectivelythe two sources of error are combined in quadrature as σ tot = (cid:113) σ T + σ sys . (6)From Equation 6, σ tot improves with integration time only until σ T < σ sys .This means that in a system with high T sys and small ∆ ν , long integrationtimes ( ∼ hours) can be beneficial. Conversely, for low T sys and large ∆ ν , thecondition σ T < σ sys happens quickly and σ tot may be optimal after relativelyshort integration times ( ∼ seconds). The system noise temperature is measured using several approaches, andthe results are compared. First is a measurement of the total output noisepower with a matched 50 Ω terminator at the input to the cold HEMT. Inthis approach, the mean output power divided by the gain of the electronicschain is defined as T sys . For this purpose the gain of the electronics chainis measured with a test signal that is sent through the electronics chain by12ay of the calibration waveguide. However, a problem with this techniqueis that the power contained in the test signal is reduced by the line loss, orthe loss in the waveguide and through both ports in the cavity at cryogenictemperatures. Because these behaviors are hard to characterize in the pres-ence of reflections and standing waves, the gain inferred from a test signal isreported only as a function of the line loss. Figure 7 shows an example of atest signal sitting on a background of amplified thermal and electronic noise. Baseband Frequency (Hz) · P o w e r D e n s i t y d B m / H z -105-104-103-102-101-100-99-98-97-96-95 Test Signal
Figure 7: Broadband noise measured in the baseband without the low pass filters in place.The spike at 4 MHz is a test signal that was sent into the cavity through the weaklycoupled calibration port.
With the gain defined as above, T sys can be derived as a function of theline loss. With a 50 Ω termination at the input to the 7 K HEMT, the outputpower is corrected for the hardware transfer function and divided by an arrayof values for the gain. An example for one value of the gain is shown on theleft in Figure 8, and the results for the other gains are included on the rightin Figure 9.The noise factor of the electronics, driven by T HEMT , can also be deducedwith the “twice–power” method. In this approach a narrow test signal withpower P in is injected into the electronics until it sits 2 × higher than thanthe baseline noise at the output of the electronics. Then, the noise factor F ≡ P in /k B T ∆ ν where k B T ∆ ν is the baseline noise at the input. Both P in and k B T ∆ ν are extracted using the gain of the electronics chain, which asstated above, is still a theoretical function of the line loss. On the right inFigure 8 is a plot of P out against P in for a set of test signals with a range ofmagnitudes. The condition P in /P baseline ≡ F . The noise factor and noise temperature T of theHEMT are related as T = ( F − T where T is the temperature of theHEMT’s chassis. T sys is then derived by adding the chassis temperature T th to T . The right panel of Figure 9 includes T sys derived from F as a functionof the line loss. As shown on the plot, the results agree well with T sys takenfrom the measurements of the mean broadband noise from the matched 50 Ωtermination. Frequency (GHz) O upu t P o w e r / G a i n ( K ) Termination W Cold 50 (mW) * 10 in P -3 · b ase li n e / P ou t P Twice Power
Figure 8: The left panel shows broadband noise from the receiver, corrected for the hard-ware transfer function, with a 7 K matched 50 Ω termination at the input. The spike at34.423 GHz is related to the 10 MHz reference oscillator. The circular points in the rightpanel show the magnitudes of a series of test signals relative to the surrounding noise,plotted against the presumed input power P in . The input power for which the outputpower is doubled is inferred by interpolation, shown as the diamond marker on the plot.The magnitudes of P in have been temporarily scaled by a factor of 10 to accommodatethe precision of the fitting algorithm on the 32-bit computer. The uncertainty caused by the line loss described in the previous sectionis removed with a Y-factor measurement. In this approach, the frequency–dependent noise temperature of the amplifier is derived from the ratio oftwo spectra with different input temperatures while the temperature of theHEMT is held constant. The first spectrum is taken using a cold source( ∼ ∼
28 K). Thetemperature of the HEMT is reasonably constant, ranging between 4.5 and8 K. The temperatures are determined with the thin–film resistance cryogenictemperature sensors. The noise temperature of the amplifier is defined as theX–intercept on a graph of output power plotted against source temperature.The data are collected using an Agilent 9030A spectrum analyzer.14uring the measurement, a Cu block is in thermal contact with a 50 Ωtermination that is connected to the input of the HEMT using a 5 cm long0.085 (cid:48)(cid:48) diameter cable with stainless steel jacket, PTFE dielectric interior, andstainless steel inner conductor. The cable provides some thermal isolationbetween the HEMT and the termination. Additional isolation is achievedby wrapping the block loosely with mylar lined on the inside with a layer ofDacron mesh. For temperature stability, the HEMT is thermally coupled toa pool of liquid He at the bottom of the cryostat with an OFHC Cu coldfinger. The temperature of the source is increased by applying current to aresistor that is in thermal contact with the Cu block.Figure 9 summarizes the results of the Y-factor measurement. The leftpanel contains the extracted noise temperature of the electronics, driven bythe HEMT. The largest sources of uncertainty come from imperfect thermalcontact between the temperature sensors and the warm and cold termina-tions, and from unwanted heating of the amplifier during the measurement.Additional errors arise from a lack of thermal equilibrium between the Cublock and the 50 Ω termination, time lag between heating of the source andthe data collection, and uncertainty in the amount of RF power lost in thestainless steel cable. The loss in the stainless steel cable is estimated to be0.6 dB, adjusted downward from 0.9 dB at room temperature according tothe expected lowering of electrical resistivity at 5 K [31]. Considering allof the above, T HEMT is probably near 20 ± T sys = T HEMT + T th ∼ =27 ± ± ± hf /k B =1.6 K. While this isnot atypical for a HEMT amplifier whose semiconductor structures are builtfrom InP [32], it is perhaps notable that the noise performance does notappear to be significantly degraded by the strong 7 T magnetic field. Dawand Bradley [33] found that the noise performance of a HEMT built withGaAs/AlGaAs and operating at 683 MHz was degraded by the presence of anambient 3.6 T magnetic field. The effect was found to be highly dependent onthe orientation of the amplifier relative to the field. A quantitative accountfor the behaviors was given in terms of the electrons’ trajectories across thetwo-dimensional electron gas between semiconductor layers [33].For the case of the InP HEMT in the present experiment, there is no15 frequency (GHz)
32 32.5 33 33.5 34 34.5 35 35.5 36 N o i se T e m p e r a t u r e ( K ) -5051015202530 =32K h =4K, T c T =27K h =4K, T c T =32K h =6K, T c T =27K h =6K, T c T Amplifier Noise Temperature
Input Line Loss (dB) -52 -50 -48 -46 -44 -42 -40 N o i se T e m p e r a t u r e ( K ) termination W th + T (F-1)T System Noise Temperature
Figure 9: The left panel shows the amplifier noise temperature derived from the Y–factormeasurement. The legend contains several sets of plausible cold and warm source temper-atures T c and T h . In the right panel, the shaded region shows T sys inferred from the leftpanel assuming a cryostat temperature T th of 7 K. The right panel also depicts T sys ascalculated from measurements with a 50 Ω termination and from the twice-power method. obvious reason why the electrons in the two-dimensional gas should not beaffected by the magnetic field similarly to [33]. Furthermore, through en-gineering constraints the amplifier is oriented relative to the magnetic fieldsuch that if there is such an effect, it should be maximal. Additional inves-tigation is therefore required to fully characterize the noise performance ofthe InP HEMT in the magnetic field, including observations of the currentdrawn, the gain, and the inferred noise temperature.As a complement to the above measurement of the mean T sys , it is impor-tant to examine the distribution of individual power samples. Their valuesare expected to have a predictable behavior when sampled with an ideal totalpower radiometer. The raw sampled voltages V should follow a Gaussian dis-tribution in number density centered around 0 V. The power measurementsshould fall inside the same Gaussian, squared [34] P ( V ) ∼ V e ( − V / (2 σ )) , (7)where σ is the standard deviation of the Gaussian and σ becomes the mean ofthe Gaussian squared [34]. Applying the envelope P ( V ) to a random numbergenerator simulates the output of an ideal radiometer. Figure 10 shows a plotof the simulated data compared with real data from this experiment. Themeans in each distribution are the same. From the plot it is apparent thatthe qualitative behavior of the two data sets are similar. Differences are also16 Output Power (mW/Hz) -9 · ( nu m b e r o f sa m p l es ) N Log measuredideal
Power Distribution Before Averaging
Figure 10: Distribution of power measured with the receiver, plotted simultaneously withthe ideal distribution having the same mean. present: The ideal data contain samples at the low and high energies thatare not present in the real data. These are attributed to the finite noise floorof the digitizer on the low end, and to infrequent saturation of the receiveron the high end. The slight discrepancy near 0.05 mW/Hz is caused by thelow energy tail combined with the arbitrary requirement that the mean beidentical in both data sets.
In the previous section the system noise temperature T sys was discussed.In this section T sys will be considered only as it applies to the system witha resonant cavity at the input. Following the approach discussed in [35, 36]for a noisy two–port device coupled to a noise source, a model of the presentexperiment is constructed. Figure 11 shows a block diagram of the cavity,WR28–to–coaxial adapter, and the HEMT amplifier with the noise fields andtheir directions. As in [35] A n and B n are the complex ingoing and outgoingnoise waves of the amplifier. A th , A ad , and A cav are the waves associated withthe physical temperatures of the 3 components, where | A th | > | A ad | > | A cav | .The reflection and transmission coefficients Γ and τ are determined from thereturn loss (RL) such that RL(dB) = -20log ( | Γ | ) and | Γ | + | τ | ≡ L and L are electrical lengths.Summing the fields at the input to the HEMT gives A n + A th + Γ B n + A ad + τ ( A cav + Γ ( B n τ + A ad )) , which reduces to A n + A th + Γ B n + τ Γ B n because the HEMT is warmerthan the adapter and the cavity. The power at the input to the HEMT17 EMT B n A th A n cavity A cav G adapter A ad A ad G ,t T=7K T=4K L L Figure 11: Block diagram as the basis for a wave model of the noise properties of theexperiment, after [35]. L and L are electrical lengths, and Γ and τ are reflection andtransmission coefficients. The fields A n and B n are characteristic of the amplifier. Allother fields are derived from physical temperatures.
18s taken from the square of the summed fields. Following the steps in [35]which are partially summarized in Equations 8–11, correlated cross termsretain a phase shift and uncorrelated cross terms collapse to zero. The noisetemperature of the amplifier is proportional to | A n | . The waves A n and B n going in and out of the amplifier are correlated by a phase shift φ c . Theelectrical lengths L and L + L behave as phase shifts in the reflectioncoefficients Γ and Γ . | A n | = k B T a ∆ f (8) | B n | = k B T b ∆ f (9) A n ∗ B n = k B T c ∆ f e iφ c (10)Γ = | Γ | e iφ s where φ s = 2 L /λ and φ s = 2( L + L ) /λ. (11)Squaring the fields and collecting the cross terms, the power at the inputto the HEMT is expected to be T a + T th + | Γ | T b + | Γ | T b | τ | + 2 | Γ | T c cos ( φ s + φ c ) +2 | Γ | T c τ cos ( φ s + φ c ) + 2 | Γ || Γ | T b τ cos ( φ s + φ s ) . (12)The first two terms T a + T th are equivalent to T sys , estimated to be 27 Kin Section 3.1. The parameters | Γ | and | Γ | are measured at cryogenictemperatures in terms of return loss with the Agilent PNA E8364C net-work analyzer and are labeled S11 and S22 in Figure 12. Oscillations withfrequency 50 MHz can also be seen on the plots. These features indicatethe presence of well-behaved reflections inside the 6 m round-trip length ofwaveguide. The loaded Q of the TE cavity is measured at 5 K and islabeled S21 in Figure 12. From the S22 of the cavity on resonance (-12 dB)and the measurement of Q at 10 , the frequency–dependent Γ behaves as aLorenztian | Γ | = 10 . − ./ . (cid:16) Q ( f − f ) f (cid:17) . Remaining parameters that are unmeasured in the model are the three phaseshifts φ , φ , and φ c , as well as T b and T c . The bottom right panel ofFigure 12 shows an overlay of the model with the data from the receiverwith the 5 K resonant cavity at the input. The free parameters have beenadjusted by hand in the plot to be T c =3.0 K, T b =0.5 K, φ c =0, φ s =0, and φ s =0. Although the solution is not a unique one, it is still encouraging thatthe model can account for the observations made in this experiment.19 Frequency (GHz) M a gn i t ud e S ( d B ) -90-80-70-60-50 Signal Cavity S21
Frequency (GHz) M a gn i t ud e S ( d B ) -11-10-9-8-7-6-5-4-3 Signal Cavity S22
Frequency (GHz) M a gn i t ud e S ( d B ) -14-13.5-13-12.5-12-11.5-11-10.5 Adapter S11
Baseband Frequency (Hz) -5000 -2500 0 2500 5000 · O u t pu t P o w e r ( m W / H z ) -12 · Cavity
Figure 12: Upper left panel: Transmitted power from which the Q of the TE cavity isderived. Upper right: Power reflected from the cold TE cavity tuned to 34.294 GHz.Lower left: Power reflected from the cold waveguide to coaxial adapter. Lower right: Wavemodel (solid line) from equation 12 incorporating the measurements from the other threepanels in this figure. .3. Data Reduction The aim of the data processing is to optimize the signal to noise ratio SN = P sig √ τk B T sys √ ∆ ν (13)where P sig is the signal power, k B σ T ∆ ν is the noise power, and σ T is definedin Equation 5. The bandwidth of P sig in Equation 13 is assumed to benarrower than ∆ ν . From the expression it is clear that P sig increases as √ τ and as 1 / √ ∆ ν . It is therefore beneficial to choose the narrowest possiblevalue of ∆ ν while still requiring that it encapsulate the width of P sig .After selecting the resolution bandwidth, the data are converted from araw time series of voltages in I and Q to a power spectrum in frequency.This is done using a complex Fast Fourier Transform (FFT) taken from thefftw3 [37] libraries. After the FFTs there are N = τ ∆ ν frequency spectrathat are averaged together. In the case of the two–cavity experiment, the data reduction is a straight-forward search for a monoenergetic signal at the same frequency as the drivesignal. Also, since the drive signal is narrow ( <
10 mHz) and is frequency–locked to the receiver chain, ∆ ν should be narrow to optimize the signal tonoise ratio (e.g. [38, 39]).Figure 13 shows an example of a data run with an RF leak (left panel),and with the RF leak suppressed (right panel). The expected frequency ofthe drive signal in the baseband is noted with an arrow. In looking at thefigure one might notice that the width of the RF leak appears to span morethan one bin. At first glance this may be surprising given that the source isnarrow, ∆ ν is 6.7 mHz, and the system is frequency–locked. However in thisparticular measurement the RF leak was found to be related to a ground loopthat included the two cryogenic amplifiers and their power supplies, whichprobably means that the power in this RF leak was not well locked to therest of the system. This can account for the excess width in the signal.The shielding between the two cavities is demonstrated, also in Figure 13.The power contained in the RF leak is 10 − Watts. The signal appears at theexpected frequency in the baseband which demonstrates that the detectionand analysis are working correctly. With the RF leak suppressed, as in theright panel, the shielding is estimated from the drive power and σ tot to be P sig / ( k B σ tot ∆ ν ) ≥
223 dB. 21 aseband Frequency (Hz) · O u t pu t P o w e r ( m W / H z ) -140-130-120-110-100-90 RF Leak at 6.7 mHz Resolution
Baseband Frequency (Hz) · O u t pu t P o w e r ( m W / H z ) -140-130-120-110-100-90 Figure 13: The left panel shows the power from the signal cavity in the presence of anRF leak. The right panel shows the power spectrum without the RF leak. The arrowin each plot marks the frequency in the baseband where the drive signal should be afterdownmixing.
In the swept search for halo ALPs with one cavity, the data analysisbegins with the FFT as above in the driven experiment. Following the FFTare several additional steps that are needed to account for the interaction ofthe ALP–driven photons with the apparatus. These steps are more criticalhere than they are in the driven experiment largely because the frequencyof the signal is unknown. Thus it becomes important to characterize eachbin for every cavity frequency before averaging the overlapping scans. Thesteps outlined here are similar to those described in [25] and [40]. To datethe analysis has been focused on a search for statistical aberrations thatare one bin wide, using a resolution bandwidth of 34 kHz across a tuningbandwidth of 600 MHz. Wider scan ranges are possible but are probablybest approached with an automated experiment.
The first step in the analysis after the FFT is to extract the raw fluc-tuations from around the mean power. To do this, the mean is calculatedempirically and is subtracted from the power spectrum. Typically the meanis derived from the data in groups of ∼ .5.2. Mismatch The impedance mismatch between the cavity and the waveguide acts asa filter during the measurement. For the case of a cavity critically coupledto the waveguide on resonance, half of the power in the cavity propagates tothe waveguide. Off resonance, less than half of the power propagates. Thefunction describing this behavior in frequency is a Lorentzian curve withthe Q of the cavity, peaking at 0.5 . It is applied to the fluctuations bydivision, which worsens the resolution everywhere by at least a factor of 2.The coupling of the TM cavity to the waveguide has been measured to beclose to critical in [28]. For the case of a cavity with some other degree ofcoupling, the mismatch filter is derived similarly but with a different peakvalue that corresponds to the fraction of power transmitted out of the cavityon resonance. The narrow signal that is expected from ALP couplings coexists with therandom thermal and electronic noise characterized by T sys . Therefore thesignal to noise ratio is improved by an algorithm that reduces the randomnoise power. The Wiener filter [41] is one example of a useful and accessibletool for this purpose, implemented as in [42] b i = µ + ( σ − ν ) σ ( a i − µ )where a i ( b i ) is the unfiltered (filtered) fluctuation in bin i , σ is the standarddeviation in the neighborhood of bin i , ν is the average value of σ near bin i , and µ is the mean of a i near bin i . The typical granularity of the filter inthis analysis is about 10 bins. The last step is to consider the power spectrum of photons that shouldcome from axion couplings within the volume of the resonant cavity. Thepower P for the case where the axion mass is equal to the resonant frequencyof the cavity is calculated from the Lagrangian in [23, 24]. Where the massis off resonance, the power spectrum is also derived from the Lagrangianand follows the Lorentzian shape of the cavity resonance [23, 24, 43]. Inpractice this behavior is applied to the data in frequency as a division by aLorentzian function that peaks at unity. Figure 14 shows a typical data setmeasured in the listening mode with the TM cavity, before and after thedata reduction. 23 RF frequency (GHz) po w e r ( m W / H z ) -2-1.5-1-0.500.511.52 -12 · Raw Fluctuations
RF frequency (GHz) po w e r ( m W / H z ) -2-1.5-1-0.500.511.52 -12 · All Filters
Figure 14: Two plots showing a sample of raw fluctuations in the data before (top panel)and after (bottom panel) the data reduction in the listening experiment, using with onecavity in the TM mode. The resolution bandwidth is 34 kHz. . Summary The design and nominal behavior of the Yale 34 GHz resonant cavityexperiment has been described. The noise temperature of the electronicshas been measured. The observations made with the apparatus have beencompared with expected values, and the data reduction has been outlined.The measurements and their derivation are in accordance both with idealmodels and with techniques used in other experiments [23, 27]. It is thereforereasonable to conclude that if an unexpected signal were to be found in thisexperiment it should be investigated as a possible sign of new physics.
5. Acknowledgments
The authors are grateful to the United States Office of Naval ResearchDirected Energy Program and to Yale University for their generous financialsupport. The authors also wish to thank Professor Kurt Zilm for the use ofthe NMR magnet throughout this experiment.
References [1] F. Zwicky, On the Masses of Nebulae and of Clusters of Nebulae, As-trophys.J. 86 (1937) 217–246. doi:10.1086/143864 .[2] Y. Sofue, M. Honma, T. Omodaka, Unified Rotation Curve of theGalaxy – Decomposition into de Vaucouleurs Bulge, Disk, Dark Halo,and the 9-kpc Rotation Dip –, Publ.Astron.Soc.Japan 61 (2009) 227–236. arXiv:0811.0859 .[3] N. Jarosik, C. Bennett, J. Dunkley, B. Gold, M. Greason, et al., Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations:Sky Maps, Systematic Errors, and Basic Results, Astrophys.J.Suppl.192 (2011) 14. arXiv:1001.4744 , doi:10.1088/0067-0049/192/2/14 .[4] Tauber, Jan, Planck 2013 results. i. overview of products and scientificresults, A&A doi:10.1051/0004-6361/201321529 .URL http://dx.doi.org/10.1051/0004-6361/201321529 [5] C. Aalseth, et al., CoGeNT: A Search for Low-Mass Dark Matter usingp-type Point Contact Germanium Detectors, Phys.Rev. D88 (1) (2013)012002. arXiv:1208.5737 , doi:10.1103/PhysRevD.88.012002 .256] G. Angloher, M. Bauer, I. Bavykina, A. Bento, C. Bucci, et al.,Results from 730 kg days of the CRESST-II Dark Matter Search,Eur.Phys.J. C72 (2012) 1971. arXiv:1109.0702 , doi:10.1140/epjc/s10052-012-1971-8 .[7] R. Bernabei, et al., First results from DAMA/LIBRA and the combinedresults with DAMA/NaI, Eur.Phys.J. C56 (2008) 333–355. arXiv:0804.2741 , doi:10.1140/epjc/s10052-008-0662-y .[8] C. Savage, G. Gelmini, P. Gondolo, K. Freese, Compatibility ofDAMA/LIBRA dark matter detection with other searches, JCAP 0904(2009) 010. arXiv:0808.3607 , doi:10.1088/1475-7516/2009/04/010 .[9] R. Agnese, et al., Silicon Detector Dark Matter Results from the FinalExposure of CDMS II, Phys.Rev.Lett. 111 (2013) 251301. arXiv:1304.4279 , doi:10.1103/PhysRevLett.111.251301 .[10] E. Aprile, et al., Limits on spin-dependent WIMP-nucleon cross sectionsfrom 225 live days of XENON100 data, Phys.Rev.Lett. 111 (2) (2013)021301. arXiv:1301.6620 , doi:10.1103/PhysRevLett.111.021301 .[11] J. Angle, et al., A search for light dark matter in XENON10 data,Phys.Rev.Lett. 107 (2011) 051301. arXiv:1104.3088 , doi:10.1103/PhysRevLett.110.249901,10.1103/PhysRevLett.107.051301 .[12] E. Armengaud, et al., Final results of the EDELWEISS-II WIMP searchusing a 4-kg array of cryogenic germanium detectors with interleavedelectrodes, Phys.Lett. B702 (2011) 329–335. arXiv:1103.4070 , doi:10.1016/j.physletb.2011.07.034 .[13] E. Armengaud, et al., A search for low-mass WIMPs with EDELWEISS-II heat-and-ionization detectors, Phys.Rev. D86 (2012) 051701. arXiv:1207.1815 , doi:10.1103/PhysRevD.86.051701 .[14] D. Y. Akimov, H. Araujo, E. Barnes, V. Belov, A. Bewick, et al., WIMP-nucleon cross-section results from the second science run of ZEPLIN-III, Phys.Lett. B709 (2012) 14–20. arXiv:1110.4769 , doi:10.1016/j.physletb.2012.01.064 . 2615] D. Akerib, et al., First results from the LUX dark matter experiment atthe Sanford Underground Research Facility, Phys.Rev.Lett. 112 (2014)091303. arXiv:1310.8214 , doi:10.1103/PhysRevLett.112.091303 .[16] G. Aad, et al., Search for dark matter candidates and large extradimensions in events with a jet and missing transverse momentumwith the ATLAS detector, JHEP 1304 (2013) 075. arXiv:1210.4491 , doi:10.1007/JHEP04(2013)075 .[17] R. Peccei, H. R. Quinn, CP Conservation in the Presence of Instantons,Phys.Rev.Lett. 38 (1977) 1440–1443. doi:10.1103/PhysRevLett.38.1440 .[18] S. Weinberg, A New Light Boson?, Phys.Rev.Lett. 40 (1978) 223–226. doi:10.1103/PhysRevLett.40.223 .[19] F. Wilczek, Problem of Strong p and t Invariance in the Pres-ence of Instantons, Phys.Rev.Lett. 40 (1978) 279–282. doi:10.1103/PhysRevLett.40.279 .[20] J. Jaeckel, A. Ringwald, A Cavity Experiment to Search for HiddenSector Photons, Phys.Lett. B659 (2008) 509–514. arXiv:0707.2063 , doi:10.1016/j.physletb.2007.11.071 .[21] P. Arias, D. Cadamuro, M. Goodsell, J. Jaeckel, J. Redondo, et al.,WISPy Cold Dark Matter, DESY-11-226, MPP-2011-140, CERN-PH-TH-2011-323, IPPP-11-80, DCPT-11-160 arXiv:1201.5902 .[22] A. E. Nelson, J. Scholtz, Dark Light, Dark Matter and the MisalignmentMechanism, Phys.Rev. D84 (2011) 103501. arXiv:1105.2812 , doi:10.1103/PhysRevD.84.103501 .[23] P. Sikivie, Experimental Tests of the Invisible Axion, Phys.Rev.Lett. 51(1983) 1415–1417. doi:10.1103/PhysRevLett.51.1415 .[24] P. Sikivie, Detection rates for ’invisible’ axion searches, Phys. Rev. D32(1985) 2988. doi:10.1103/PhysRevD.36.974,10.1103/PhysRevD.32.2988 .[25] H. Peng, S. J. Asztalos, E. Daw, N. Golubev, C. Hagmann, et al.,Cryogenic cavity detector for a large scale cold dark-matter axion27earch, Nucl.Instrum.Meth. A444 (2000) 569–583. doi:10.1016/S0168-9002(99)00971-7 .[26] R. Bradley, J. Clarke, D. Kinion, L. Rosenberg, K. van Bibber, et al.,Microwave cavity searches for dark-matter axions, Rev. Mod. Phys. 75(2003) 777–817. doi:10.1103/RevModPhys.75.777 .[27] S. Asztalos, et al., A SQUID-based microwave cavity search for dark-matter axions, Phys.Rev.Lett. 104 (2010) 041301. arXiv:0910.5914 , doi:10.1103/PhysRevLett.104.041301 .[28] A. T. Malagon, Search for 140 µ eV Pseudoscalar and Vector Dark Mat-ter Using Microwave Cavities, Ph.D. Thesis, Yale University (2014).[29] S. Weinreb, M. W. Pospieszalski, R. Norrod, Cryogenic HEMT Low-Noise Receivers for 1.3 to 43 GHz Range, Microwave Symposium Digest,1988., IEEE MTT-S International 2 (1988) 945–948. doi:10.1109/MWSYM.1988.22187 .[30] R. H. Dicke, Rev. Sci Instrum. 17 (1946) 268.[31] J. G. Hust, P. J. Giarratano, Standard Reference Materials: ThermalConductivity and Electrical Resistivity. Standard Reference Materials:Austenitic Stainless Steel, SRMs 735 and 798, From 4 to 1200 K., Nat.Bur. Stand. (U.S.) Spec. Publ. 260-46, Washington, D.C., 1975.[32] M. Reid, Low-noise systems in the Deep Space Network, Wiley, Hobo-ken, N.J, 2008.[33] E. Daw, R. F. Bradley, Effect of High Magnetic Fields on the Noise Tem-perature of a Heterostructure Field-Effect Transistor Low-Noise Ampli-fier, J.Appl.Phys. 82 (1997) 1925. doi:10.1063/1.366000 .[34] J. J. Condon, S. M. Ransom, Essential radio astronomy.URL [35] R. P. Meys, A wave approach to the noise properties of linear microwavedevices, IEEE Trans. on Microwave Theory and Tech. 26 (1) (1978) 34–37. 2836] S. W. Wedge, D. B. Rutledge, Wave Techniques for Noise Modeling andMeasurement, IEEE Trans. on Microwave Theory and Tech. 40 (11)(1992) 2004–2013.[37] M. Frigo, S. G. Johnson, The design and implementation of FFTW3,Proc. IEEE 93 (2005) 216–231. doi:10.1109/JPROC.2004.840301 .[38] F. Caspers, J. Jaeckel, A. Ringwald, Feasibility, engineering aspectsand physics reach of microwave cavity experiments searching for hid-den photons and axions, JINST 4 (2009) P11013. arXiv:0908.0759 , doi:10.1088/1748-0221/4/11/P11013 .[39] M. Betz, F. Caspers, M. Gasior, M. Thumm, S. Rieger, First re-sults of the CERN Resonant Weakly Interacting sub-eV Particle Search(CROWS), Phys.Rev. D88 (7) (2013) 075014. arXiv:1310.8098 , doi:10.1103/PhysRevD.88.075014doi:10.1103/PhysRevD.88.075014