Wavelike bosonic dark matter candidates like axions and dark photons can be detected using microwave cavities known as haloscopes. Traditionally haloscopes consist of tunable copper cavities operating in the inlineformulammlmath displayinlinemmlmrowmmlmsubmmlmrowmmlmiTMmmlmimmlmrowmmlmrowmmlmn010mmlmnmmlmrowmmlmsubmmlmrowmmlmathinlineformula mode but ohmic losses have limited their performance. In contrast superconducting radio frequency SRF cavities can achieve quality factors of inlineformulammlmath displayinlinemmlmrowmmlmommlmommlmsupmmlmrowmmlmn10mmlmnmmlmrowmmlmrowmmlmn10mmlmnmmlmrowmmlmsupmmlmrowmmlmathinlineformula perhaps 5 orders of magnitude better than copper cavities leading to more sensitive dark matter detectors. In this paper we first derive that the scan rate of a haloscope experiment is proportional to the loaded quality factor inlineformulammlmath displayinlinemmlmrowmmlmsubmmlmrowmmlmiQmmlmimmlmrowmmlmrowmmlmi mathvariantnormalLmmlmimmlmrowmmlmsubmmlmrowmmlmathinlineformula even if the cavity bandwidth is much narrower than the dark matter halo line shape. We then present a proofofconcept search for dark photon dark matter using a nontunable ultrahighquality SRF cavity. We exclude dark photon dark matter with kinetic mixing strengths of inlineformulammlmath displayinlinemmlmrowmmlmimmlmimmlmogtmmlmommlmn1.5mmlmnmmlmommlmommlmsupmmlmrowmmlmn10mmlmnmmlmrowmmlmrowmmlmommlmommlmn16mmlmnmmlmrowmmlmsupmmlmrowmmlmathinlineformula for a dark photon mass of inlineformulammlmath displayinlinemmlmrowmmlmsubmmlmrowmmlmimmmlmimmlmrowmmlmrowmmlmsupmmlmrowmmlmiAmmlmimmlmrowmmlmrowmmlmommlmommlmrowmmlmsupmmlmrowmmlmsubmmlmommlmommlmn5.35mmlmnmmlmtext mmlmtextmmlmtext mmlmtextmmlmrowmmlmi mathvariantnormalmmlmimmlmieVmmlmimmlmrowmmlmrowmmlmathinlineformula achieving the deepest exclusion to wavelike dark photons by almost an order of magnitude.
['Astrophysics and astroparticle physics', 'High Energy Physics - Experiment', 'Astrophysics - Instrumentation and Methods for Astrophysics', 'High Energy Physics - Phenomenology']
Deepest sensitivity to wavelike dark photon dark matter with superconducting radio frequency cavities
{'Deepest sensitivity to wavelike dark photon dark matter with superconducting radio frequency cavities': "R. Cervantes, 1, ∗ J. Aumentado, 2 C. Braggio, 3, 4 B. Giaccone, 1 D. Frolov, 1 A. Grassellino, R. Harnik, 1 F. Lecocq, 2 O. Melnychuk, 1 R. Pilipenko, 1 S. Posen, 1 and A. Romanenko 1  \n1  \nFermi National Accelerator Laboratory, Batavia IL 60510 2 National Institute of Standards and Technology, Boulder, CO 80305 3 Dip. di Fisica e Astronomia, Universit`a di Padova, 35100 Padova, Italy 4 INFN - Sezione di Padova, 35100 Padova, Italy (Dated: September 10, 2024)  \n1  \nWavelike, bosonic dark matter candidates like axions and dark photons can be detected using microwave cavities known as haloscopes. Traditionally, haloscopes consist of tunable copper cavities operating in the TM 010 mode, but ohmic losses have limited their performance. In contrast, superconducting radio frequency (SRF) cavities can achieve quality factors of ∼ 10 10 , perhaps five orders of magnitude better than copper cavities, leading to more sensitive dark matter detectors. In this paper, we first derive that the scan rate of a haloscope experiment is proportional to the loaded quality factor Q L , even if the cavity bandwidth is much narrower than the dark matter halo line shape. We then present a proof-of-concept search for dark photon dark matter using a nontunable ultrahigh quality SRF cavity. We exclude dark photon dark matter with kinetic mixing strengths of χ > 1 . 5 × 10 -16 for a dark photon mass of m A ' = 5 . 35 µ eV, achieving the deepest exclusion to wavelike dark photons by almost an order of magnitude.", 'I. INTRODUCTION': "There is overwhelming evidence that 84.4% of the matter in the Universe is made out of dark matter (DM) [17]. The ΛCDM model describes dark matter as feebly interacting, nonrelativistic, and stable on cosmological timescales. Not much else is known about the nature of dark matter, particularly what particles beyond the standard model it is made of.  \nThe dark photon (DP) is a compelling dark matter candidate. It is a spin-1 gauge boson associated with a new Abelian U(1) symmetry and is one of the simplest possible extensions to the Standard Model (SM) [8-10]. The dark photon, having the same quantum numbers as the SM photon, generically interacts with the SM photon through kinetic mixing [11, 12] described by the Lagrangian  \nL = -1 4 ( F µν 1 F 1 µν + F µν 2 F 2 µν -2 χF µν 1 F 2 µν -2 m 2 A ' A ' 2 ) , (1)  \nwhere F µν 1 is the electromagnetic field tensor, F µν 2 is the dark photon field tensor, χ is the kinetic mixing strength, m A ' is the DP mass, and A ' is the DP gauge field. If both m A ' and χ are sufficiently small, then it is stable on cosmological timescales [13]. The dark photon is then an attractive dark matter candidate. If its mass is less than an eV, the dark photon dark matter (DPDM) is in the wavelike regime, where it is best described as a coherent wave oscillating at the frequency of its rest mass rather than a collection of particles. The dark matter  \nkinetic energy distribution sets the degree of coherence of wavelike dark matter to be of order v 2 DM ∼ 10 -6 [14, 15].  \nSeveral mechanisms could produce a relic of cosmic dark photons. One simple example is the displacement of the DP field through quantum fluctuations during inflation [16]. These fluctuations in the DP field serve as the initial displacement for dark photon field oscillations, which commence once the universe's expansion rate H falls below the DP mass, i.e., H ℏ < mc 2 . Other mechanisms are possible and are described in [10, 17].  \nDPDM can be detected through its mixing with the SM photon. If dark photons convert into SM photons inside a microwave cavity with a large quality factor, then a feeble EM signal accumulates inside the cavity, which can be read by ultralow noise electronics. This type of detector is called a haloscope and is often deployed to search for axionic DM [18]. The SM photon frequency f is related to the dark photon energy E A ' by hf = E A ' ≈ m A ' c 2 .  \nIn natural units, the dark photon signal power inside the cavity is [9, 19-21]  \nP S = P 0 β β +1 L (∆) , (2)  \nwhere  \nP 0 = { ηχ 2 m A ' ρ A ' V eff Q L , if Q L << Q DM ηχ 2 m A ' ρ A ' V eff Q DM , if Q L >> Q DM (3)  \nV eff = (∫ dV E ( ⃗x ) · A ' ( ⃗x ) ) 2 ∫ dV ϵ r | E ( ⃗x ) | 2 | A ' ( ⃗x ) | 2 , (4)  \nη is a signal attenuation factor, ρ A ' is the local density of dark matter, V eff is the effective volume of the cavity, Q L is the cavity's loaded quality factor, Q DM is the  \ndark matter 'quality factor' related to the dark matter coherence time, and β is the cavity coupling coefficient. The Lorentzian term is L ( f, f 0 , Q L ) = 1 / (1+4∆ 2 ), where ∆ ≡ Q L ( f -f 0 ) /f 0 is a detuning factor that depends on the SM photon frequency f , cavity resonant frequency f 0 , and Q L . V eff is the overlap between the dark photon field A ' ( ⃗x ) and the dark photon-induced electric field E ( ⃗x ). Equations 3, and 4 assume that the cavity size is much smaller than the DP de Broglie wavelength ( ∼ 300 m).  \nThe dark photon mass is unknown, so haloscopes must be tunable to search through the χ vs. m A ' parameter space. Thus, the scan rate is a crucial figure of merit for haloscope experiments. Most haloscope literature has focused on the case where Q L << Q DM because copper cavities have been traditionally used. However, superconducting niobium cavities with Q L ∼ 10 10 [22] are readily available for DPDM haloscope searches, and superconducting cavities resistant to multi-Tesla magnetic fields with Q L > Q DM , consisting of dielectrics, Nb 3 Sn, or HTS tapes, will soon be readily available [23-26].  \nThis Article first derives that the haloscope scan rate is proportional to Q L , even in the case where Q L >> Q DM and the DP signal power saturates (Eq. 3). This conclusion strongly motivates the pursuit of ultrahigh Q haloscopes. This Article then reports a DPDM search using a nontunable 1 . 3 GHz cavity with Q L ∼ 10 10 . The search demonstrates superior sensitivity enabled by the ultrahigh quality factor and achieves the deepest exclusion to wavelike DPDM to date by almost an order of magnitude.", 'II. THE SCAN RATE FOR AN ULTRAHIGH Q HALOSCOPE': "The haloscope scan rate is strongly dependent on the signal-to-noise ratio (SNR), where SNR = ( P S /P n ) √ b ∆ t [27, 28]. P n is the noise power, b is the frequency bin width, and ∆ t is the integration time. P n is the combination of the cavity's blackbody radiation and the receiver's Johnson noise. The noise power can be expressed as P n = k b bT n , where k b is the Boltzmann constant, and T n is the system noise temperature referenced to the cavity output.  \nIt is common for a microwave haloscope experiment to implement a circulator at the same temperature as the cavity. For such a system, T n is constant and independent of the cavity detuning ∆ and cavity coupling β [29]. In other words, the noise temperature is expected to be the same inside and outside the cavity bandwidth.  \nIf Q L >> Q DM , the cavity width is smaller than the dark matter halo line shape width ∆ f DM . The resulting dark matter signal will follow the Lorentzian cavity response with bandwidth ∆ f c = f 0 /Q L . Fortunately, a haloscope is sensitive to a distribution of possible dark photon rest masses corresponding to the cavity resonant frequency f c . In other words, a single cavity tuning step can probe the entire dark matter line shape bandwidth, and the tuning steps need only to be comparable  \nto ∆ f DM . However, this scanning strategy would be sensitive to the virialized portion of dark matter. Any component of dark matter with a sharper dispersion could remain undetected by scanning in steps of ∆ f DM .  \nThe frequency bin width b is typically chosen to be comparable to the dark matter signal bandwidth. Typical haloscope experiments use copper cavities with Q L << Q DM , so b ∼ ∆ f DM = f 0 /Q DM . However, if Q L >> Q DM , the signal bandwidth is the same as the cavity bandwidth and b ∼ f 0 /Q L . Thus, the noise power is inversely proportional to the Q L , i.e., P n ∼ k b ( f 0 /Q L ) T n . The higher Q L is, the lower the noise power.  \nAn estimate of the integration time can be obtained by rearranging the SNR equation ∆ t = 1 /b (SNR × P n /P S ) 2 . The tuning steps are ∆ f ∼ f 0 /Q DM . Putting all this together, the instantaneous scan rate for a dark photon haloscope consisting of an ultrahigh Q microwave cavity, i.e., Q L >> Q DM , is  \nd f d t = ∆ f ∆ t ∼ Q L Q DM ( ηχ 2 m A ' ρ A ' V eff β T n ( β +1)SNR ) 2 . (5)  \nNote that Equation 5 does not include the dead time of the experiment associated with tuning the cavity and other operations like rebiasing quantum amplifiers.  \nThe scan rate equation, Equation 5, happens to be the same whether Q L >> Q DM or Q L << Q DM [30]. In both cases, the scan rate is directly proportional to Q L [31].  \nAs a comparison, ADMX uses copper cavities with Q L ∼ 8 × 10 4 [32], whereas niobium SRF cavities can achieve Q L ∼ 10 9 -10 11 depending on the temperature and cavity treatment [22]. This suggests that SRF cavities can increase the instantaneous scan rate of haloscope experiments by as much as a factor of 10 5 .", 'III. DARK PHOTON DARK MATTER SEARCH WITH AN SRF CAVITY': "The Superconducting Quantum Materials and Systems (SQMS) Center, hosted by Fermilab, performs a wide range of multidisciplinary experiments with SRF cavities for quantum computing and quantum sensing. One of these efforts includes a family of SRF haloscope experiments known as SERAPH (SupERconducting Axion and Paraphoton Haloscope). The first phase of SERAPH is a proof-of-principle dark photon search using existing accelerator SRF cavities. Using an SRF cavity with Q L ≈ 4 . 7 × 10 9 , a HEMT amplifier, and the standard haloscope analysis is enough to demonstrate superior sensitivity to wavelike dark photons compared to previous searches.  \nThe haloscope consists of a TESLA-shaped singlecell niobium cavity [33] with TM 010 resonant frequency f 0 ≈ 1 . 3 GHz. The cavity is made of fine-grain bulk niobium with a high residual resistivity ratio of ≃ 300. The  \nFIG. 1. A conceptual diagram of the dark photon dark matter search using an SRF cavity. Power from the cavity is amplified by a cryogenic HEMT amplifier before reaching the room-temperature receiver. A series of circulators prevent the cavity from being excited by the amplifier's noise. The variable temperature stage (VTS) is used to perform the noise calibration. A technical schematic of the experiment is shown in Fig. 4.  \n<!-- image -->  \ncavity volume is 3875 mL, and the effective volume calculated from Equation 4 is V eff = 669mL, assuming a randomly polarized DP field. Electromagnetic coupling to the cavities is performed using axial pin couplers at both ends of the beam tubes.  \nThe cavity underwent heat treatments in a customdesigned oven to remove the niobium pentoxide (Nb 2 O 5 ) and to mitigate the two-level system (TLS) dissipation [34, 35]. The central cell 1 . 3 GHz cavity is heat treated at ∼ 450 · C in vacuum for one hour.  \nThe cavity is cooled to ≈ 33 mK using a BlueFors XLD 1000 dilution refrigerator. A double-layer magnetic shielding around the entire cryostat is used, and magnetometers placed directly on the outside cavity surfaces indicate that the DC ambient magnetic field level is shielded to below 2 mG.  \nA conceptual diagram of the microwave electronics is shown in Fig. 1, and a technical drawing is shown in Fig. 4. A series of attenuators on the cavity input line attenuates the room-temperature noise. The power from the cavity is first amplified by a cryogenic HEMT amplifier. Four circulators are installed between the HEMT and SRF cavity. The circulators prevent the HEMT amplifiers from injecting noise into the cavity. The signal is further amplified at room temperature and then injected into the appropriate measurement device (spectrum analyzer, network analyzer, or phase noise analyzer).  \nThe system noise temperature referenced to the cavity output, T n , is measured using the Y-factor method [36, 37]. For this measurement, the cryogenic switch in Fig. 1 is connected to a Variable Temperature Stage (VTS) instead of the SRF cavity. The VTS consists of a 50 Ω load, a resistive heater, and a Cernox thermometer mounted on a copper plate. The VTS is anchored to the mixing chamber plate via stainless steel standoffs. In this configuration, the system noise power, P 0 , depends linearly on the VTS temperature, T VTS , via P = k b Gb ( T VTS + T add ), where G is the system gain and T add is the added noise from the electronics and includes the insertion loss from the transmission cables and circulators. T VTS is heated from 0 . 16 K to 7 . 6 K, and the power out of the HEMT is monitored with a spectrum analyzer. From the Y-factor  \nmeasurement, it is determined that T add = 7 . 1(4) K. This number is consistent with a HEMT noise temperature of 4 . 6 K and a 2 dB insertion loss from the circulators at the mixing chamber temperature. When the switch is connected to the cavity, T n = T cav + T add . During the experiment, T cav = 33mK, leading to T n = 7 . 1(4) K (Boson statistics need not be considered in the Rayleigh-Jeans limit, k b T cav >> hf ).  \nThe cavity's resonant frequency is identified using a self-excited loop (SEL) [38, 39]. The thermal noise from the output of the cavity is amplified, phase shifted, and fed back into the input of the cavity. A power splitter feeds the cavity's output power to the spectrum analyzer to monitor the response to the SEL. The peak of the power spectrum corresponds to the cavity resonance.  \nThe SEL was also used to characterize cavity microphonics, i.e., detuning induced by vibrations. Microphonics causes the cavity's instantaneous frequency to fluctuate over time, which causes the dark matter signal to spread beyond the cavity bandwidth. This frequency modulation causes the dark matter signal power to leak into different sidebands. Two prominent sidebands were identified at 14 . 3 Hz and 57 . 2 Hz. Their combined effect caused the dark matter signal amplitude to be reduced by 0 . 54 dB. The signal attenuation factor from Equation 3 is thus η = 0 . 88. The SEL implementation, measurement, and characterization of microphonics are described in more detail in Appendix C.  \nThe cavity's loaded quality factor Q L is measured using a decay measurement [40, p.155], where the cavity is first energized and the time constant τ L in which the energy decays is extracted. The decay measurements consistently demonstrated Q L = 2 πf 0 τ L = 4 . 7 × 10 9 . The antenna external quality factors Q e 1 and Q e 2 are measured beforehand in a separate liquid helium bath test stand following the procedure outlined in Reference [41]. The external quality factors were determined to be Q e 1 = 8 . 4 × 10 9 and Q e 2 = 2 . 3 × 10 12 with 10% uncertainty. This results in an unloaded Q, Q 0 = 1 . 1 × 10 10 . The cavity coupling coefficient of the output port is determined to be β = ( Q L /Q e ) / (1 -Q L /Q e ) = 1 . 3(3).  \nFor this proof-of-principle measurement, there is no tuning mechanism. A single power spectrum is measured. In the absence of a discovery, an exclusion on the kinetic mixing strength χ is determined from the measured power spectrum, the system noise temperature, and cavity properties. The relevant properties for determining the dark photon signal power and system noise temperature are shown in Table I.  \nη β V eff m A ' ∆ t Q L T n b 0 . 88 1 . 3(3) 669 mL 5 . 4 µ eV 1000s 4 . 7 × 10 9 7 . 1(4) K 0 . 1 Hz  \nTABLE I. Operating parameters for the dark photon dark matter search with the SQMS SRF cavity.  \nThe detector sensitivity is estimated from the operating parameters shown in Table I. The detector sensitivity can be estimated by rearranging the SNR equation to  \nFIG. 2. Raw spectrum as measured by the spectrum analyzer. The spectrum bin size is 100 mHz, and 100 averages were taken for a total integration time of 1000 s. The cavity frequency is f 0 = 1 . 294 605 478 GHz and the cavity bandwidth is 277 mHz. Assuming the spectrum is just noise, the raw spectrum is the noise power of the system.  \n<!-- image -->  \nsolve for the kinetic mixing parameter χ :  \nχ = √ β +1 β SNR × T n ηm A ' ρ A ' V eff Q DM ( ∆ f c ∆ t ) 1 / 4 . (6)  \nFor this estimate, the bandwidth b is set to the cavity bandwidth, ρ A ' = 0 . 45 GeV / cm 3 , and Q DM ≈ 10 6 . For this sensitivity estimate, setting SNR = 2 approximates a 90% exclusion limit. The parameters in Table I are converted to natural units, and the projected detector sensitivity χ proj is estimated to be χ proj = 1 . 9 × 10 -16 .  \nFor the dark photon search, power from the cavity is measured using the Rohde & Schwarz FSW-26 Signal and Spectrum Analyzer. The cavity Q L = 4 . 7 × 10 9 , so the frequency resolution needs to be b ∼ 100 mHz. This sub-Hertz resolution is achieved using the spectrum analyzer's I/Q-analyzer mode. These measurements use a 1 kHz sample rate and a 10 s sweep time. The power spectrum consists of 100 averages, resulting in a total integration time of 1000s. The spectrum's center frequency is set to the cavity's resonant frequency of 1 . 294 605 478 GHz. The frequency bin size is 100 mHz. The cavity bandwidth is 277mHz, so the cavity spans about 2 . 77 bins. After the data is recorded, the spectrum is truncated further so the span is about 100 Hz centered around the cavity resonance. Most of the power spectrum is outside of the cavity bandwidth, allowing for a convenient verification that the noise is Gaussian. On this frequency scale, the power fluctuations are Gaussian without the need for the application of a low pass filter (the Savitzky-Golay filter is typical of haloscope analysis [42, 43]). The resulting power spectrum is shown in Fig. 2.  \nOnce the power spectrum is measured, the standard haloscope analysis is applied to either find a spectrally narrow power excess consistent with a dark photon signal or to exclude parameter space. The procedure for  \nderiving the exclusion limits follows the procedure developed by ADMX and HAYSTAC [42-44], and is adapted for dark photon searches [9, 10, 19].  \nThere are a few important deviations from the standard haloscope analysis for this search. First, only one spectrum was measured, so combining many spectra at different RF frequencies is unnecessary. Second, the frequency range of interest is narrow enough such that the measured power is unaffected by the frequencydependent gain variation of the electronics. So, the Savitzky-Golay filter is not needed to remove this gain variation. This is advantageous because the SavitzkyGolay filter is known to attenuate the dark matter signal by as much as 10%-20% [42, 43].  \nThird, most of the data points in Fig. 2 used to verify the noise's Gaussianity and determine the statistical parameters are well outside the cavity bandwidth. Fortunately, in the absence of a dark matter signal, the statistical distribution of the power fluctuations is the same inside and outside of the cavity bandwidth.  \nFourth, past haloscope experiments with Q L << Q DM typically convolved the spectra with the dark matter halo line shape to account for the signal being spread across multiple bins. For this search, Q L >> Q DM , so the signal will be Lorentzian from the cavity response. So the spectrum is convolved with the cavity line shape L ( f, f 0 , Q L ) = 1 / (1 + 4∆ 2 ).  \nFifth, a single measurement is sensitive to a range of dark photon masses. Thus, the excluded power on resonance is convolved with the dark matter halo line shape. This convolution was also performed in other dark photon searches with Q L > Q DM [45]. When performing this convolution, it should be noted that the photon frequency (which corresponds to the cavity frequency) is fixed, and it is the dark photon mass that varies.  \nNo spectrally narrow power excess with an SNR > 5 is found in the measured power spectrum. The excluded parameter space using a 90% confidence limit is shown in Fig. 3. The derived limit assumes dark photon dark matter is randomly polarized, and the dark photon energy distribution follows the standard halo model. The excluded kinetic mixing strength is χ 90% = 1 . 5 × 10 -16 for a dark photon mass of m A ' = 5 . 354 µ eV. This χ 90% is consistent with the expected sensitivity estimated from Equation 6. It is also the deepest exclusion to wavelike dark photon dark matter by almost an order of magnitude.", 'IV. OUTLOOK AND THE POTENTIAL OF SRF CAVITIES FOR AXION DARK MATTER SEARCHES': 'The exclusion in Fig. 3 is an impressive demonstration of how SRF cavities can benefit dark matter searches. To extend the range of frequencies in a broader dark matter search, SQMS is currently developing experiments using widely-tunable SRF cavities.  \n<!-- image -->  \nFIG. 3. Top: A 90% exclusion on the kinetic mixing strength parameter space. Bottom: SQMS limits in the context of other microwave cavity haloscopes. Figure adapted from [46].  \n<!-- image -->  \nIn addition to dark photons, there is a growing interest in dark matter axions. Axions are particularly well motivated because they solve the strong CP problem [47]. Axion haloscope searches require multi-Tesla magnetic fields to be sensitive enough to the QCD axion. The scan rate for axion haloscope searches is still directly proportional to Q L . Unfortunately, the performance of superconductors degrades under an external magnetic field. Achieving high quality factors is thus a very active area of research [23-26], and it seems likely that axion haloscopes with Q L > 10 7 are achievable in the near future.  \nThis experiment also demonstrates that axion haloscopes with Q L ∼ 10 10 are worth striving for. Applying a hypothetical 8 T magnetic field to the dark photon data in Fig. 2 would have led to an exclusion on the axion-photon coupling constant at g aγγ ∼ 3 × 10 -16 , well below DFSZ coupling ( g aγγ = 8 × 10 -16 ).  \nDespite decades of searching for the axion, only a small fraction of the QCD axion parameter space has been explored. Perhaps a combination of ultrahigh Q cavities, subSQL metrology [45, 48], multiwavelength detector designs [49-56], and innovations in multi-Tesla continuous magnets will enable experiments to probe most of the post-inflation QCD axion parameter space within the next few decades.  \nAcknowledgements -The authors thank Asher Berlin, Yonatan Kahn, Akash Dixit, and Benjamin Brubaker for fruitful discussions regarding scanning strategies and data analysis with ultrahigh Q cavities. The authors thank Andrew Penhollow and Theodore C. Ill for the cavity assembly. This material is based upon work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Superconducting Quantum Materials and Sys-  \ntems Center (SQMS) under contract number DE-AC0207CH11359', 'Appendix A: Detailed Schematic': 'The conceptual diagram in Fig. 1 is simplified to increase the readability of the main text. The complete schematic of the electronics in the dilution refrigerator is shown in Fig. 4. The cryogenic switch is a Radiall R583423141. The cryogenic HEMT amplifier is from Low Noise Factory (LNF-LNC0.3 14A [57]). At 1 . 3 GHz, the manufacturer has measured the amplifier noise temperature at 4 . 9(5) K and the gain at 36 dB. The four circulators are QuinStar QCY-G0110151AS. Each circulator is expected to have a 0 . 5 dB insertion loss and 18 dB isolation. The bias tee is a Marki DPXN-M50. A room-temperature low-noise amplifier (Fairview Microwave FMAM1028, not shown) is used to amplify the signal further before reaching the spectrum analyzer.  \nA Josephson Parametric Amplifier (JPA) was installed but not used for the dark matter search. The JPA  \n0 d  \nFIG. 4. A more detailed schematic of the wiring and microwave electronics in the dilution refrigerator. The blue dashed lines represent superconducting NbTi cables.  \n<!-- image -->  \nFIG. 5. The decay measurement of the 1 . 3 GHz cavity used to derive the loaded quality factor.  \n<!-- image -->  \nwas fabricated by the National Institute of Standards and Technology (NIST). The JPA gain is maximized at 1 . 9 GHz and had a tuning range between 1 . 2 GHz and 2 . 5 GHz, but otherwise this JPA was never fully characterized. During this cooldown, the JPA was measured to have an overall gain of 3 dB at 1 . 3 GHz. It was decided that this additional gain was not worth the added complexity and systematic uncertainties in the noise calibration.', 'Appendix B: Decay measurement': 'The decay measurement is implemented with a Rohde & Schwarz ZNA in the pulse mode. At the resonant frequency, the network analyzer injects a 15 dBm signal with a bandwidth of 200 Hz into the input transmission line. Port 2 of the VNA measures the absolute power from the cavity output line. The network analyzer source is then turned off, and the output power is observed to decay over several seconds until it reaches an equilibrium. The measured power is fitted using P t = A exp( -t/τ L ), and Q L = 2 πf 0 τ L . The result is shown in Fig 5.', 'Appendix C: Self-excited Loop and Microphonics': "A conceptual diagram of the self-excited loop (SEL) is shown in Fig. 6. The phase shifting is performed with an ATM PNR P2214. A power splitter feeds the cavity's output power to the spectrum analyzer to monitor the response to the SEL. In the real implementation, three amplifiers were used: one cryogenic HEMT, the Fairview Microwave FMAM1028 LNA, and a Minicircuits 15542 Model No. ZHL-42. A directional coupler was used along with the power splitter so that the spectrum analyzer and phase noise analyzer could monitor the SEL simul-  \nFIG. 6. A schematic for the self-excitation loop used to find the cavity's resonant frequency and characterize microphonics.  \n<!-- image -->  \nFIG. 7. The power spectrum coming out of cavity from selfexcitation loop. Microphonics, mostly from the dilution refrigerator pulse tubes, introduces a modulating effect. The sideband amplitude was greatly diminished by turning off the pulse tubes. The 14.3 Hz and 57.2 Hz sidebands discussed extensively in the text are marked with diamonds.  \n<!-- image -->  \ntaneously. A bandpass filter (Lorch 6BC-1300/75-S) was used to reduce broadband noise, and an isolator was used between the Fairview LNA and Minicircuits amplifier to mitigate unwanted self-oscillations.  \nThe resulting power is shown in Fig. 7. The central peak corresponds to the resonant frequency. There are prominent sidebands spaced 14 . 3 Hz and 57 . 2 Hz apart from the central peak and a forest of smaller peaks; both sidebands are roughly 15 dB smaller than the carrier frequency. The power within the carrier peak and sidebands are confined within the cavity bandwidth. These sidebands are caused by the modulation of the resonant frequency due to microphonics. These vibrations originate primarily from the dilution refrigerator's pulse tubes. Fig. 7 demonstrates that these sidebands are mitigated when the pulse tubes are turned off.  \nMicrophonics affects the dark matter search sensitivity by spreading the potential dark matter signal from the central cavity frequency into sidebands. The frequency modulation framework is useful for understanding the microphonics and the effects on the potential dark matter signal. The Rohde & Schwarz FSWP Phase Noise Analyzer is used to measure the cavity frequency as a function  \nFIG. 8. The instantaneous cavity frequency as a function of time, as measured by a phase noise analyzer.  \n<!-- image -->  \nof time (Fig. 8). The root-mean-square of the frequency deviation is 24 . 7 Hz.  \nAs demonstrated in Fig. 9, taking the amplitude of the Fourier Transform of the cavity frequency reveals the modulation frequencies f m = (14 . 3 Hz , 57 . 2 Hz). The corresponding cavity detuning amplitude, i.e., frequency deviation, is f ∆ = (5 . 5 Hz , 18 . 2 Hz). The modulation indices are h = f ∆ /f m = (0 . 4 , 0 . 3), which correspond to sideband amplitudes of ( -14 . 5 dBc, -16 . 1 dBc) and is consistent with what is observed in Fig. 7. There are other notable modulation frequencies in Fig. 9, but they have smaller modulation indices, so their corresponding sideband amplitudes are subdominant to the 14 . 3 Hz and 57 . 2 Hz sidebands. One can see from Fig. 7 that the 37 Hz modulation frequency corresponding to sideband amplitude 10 dB smaller than the 14 . 3 Hz and 57 . 2 Hz sidebands. Other sidebands are even more subdominant.  \nThe modulation frequencies (14 . 3 Hz, 57 . 2 Hz) with detuning amplitudes (5 . 5 Hz, 18 . 2 Hz) correspond to a reduction in the carrier amplitude by (0 . 22 dBc, 0 . 32 dBc). This leads to a total reduction of the DM signal by 0 . 54 dB, corresponding to a dark matter signal attenuation factor of η = 0 . 88. This reduction in the carrier amplitude is confirmed numerically by taking the FFT of  \ny ( t ) = cos ( 2 πf c t + f ∆1 f m 1 sin(2 πf m 1 t ) + f ∆2 f m 2 sin(2 πf m 2 t ) ) ,  \nwhere f c = 100Hz, f ∆1 = 5 . 5 Hz, f m 1 = 14 . 3 Hz, f ∆2 = 18 . 2 Hz, and f m 2 = 57 . 2 Hz [58]. This numericallycalculated FFT also confirms the amplitude of the sidebands.  \nThere was an initial attempt to mitigate the effects of microphonics on the dark matter signal by turning off the pulse tubes during the dark matter search. The temperatures of both the cavity temperature and dilution refrigerator mixing chamber are stable for more than  \nFIG. 9. The Fourier Transform amplitude of the microphonics measurement from Fig. 8, revealing the cavity vibration frequencies and their corresponding detuning amplitudes. The 14.3 Hz and 57.2 Hz sidebands discussed extensively in the text are marked with diamonds.  \n<!-- image -->  \n20 min immediately after the pulse tubes are off. Unfortunately, the HEMT amplifier's physical temperature rises from 2 K to 9 K during this period, making the noise calibration questionable. Thus, the dark matter search incorporating microphonics is used for this publication.  \nOne might make a more sophisticated analysis that searches for the dark matter signal lost in the sidebands to improve sensitivity, but this has yet to be incorporated into this analysis.", 'Appendix D: Cavity stability': 'A previous experiment by SQMS (a material property study of an SRF cavity not intended for dark matter searches) demonstrates that the cavity is stable for 1000 s. The previous setup was similar to Fig. 4, except that the cavity was connected directly to a HEMT (with three 0 dB attenuators in between for thermalization). There was no switch, cavity bypass line, JPA line, or series of circulators. That means that the noise from the HEMT was being directly injected into the cavity, exciting the cavity. The resulting power spectrum is shown in Fig. 10. There is a power excess that corresponds to the cavity frequency. A different cavity is used here with Q L = 3 . 07 × 10 9 , corresponding to a cavity bandwidth of 433 mHz. The power spectrum was taken with a bin size of 100 mHz. One hundred averages were taken, resulting in a total integration time of 1000 s. The bandwidth of the power excess corresponds to the bandwidth of the cavity. This measurement thus shows that the cavity central frequency is stable on this timescale.  \nA future study to determine the stability of the cavity frequency would be to measure the Allan standard deviation of the cavity frequency.  \nFIG. 10. The power spectrum of a 1 . 3 GHz cavity being excited by the noise from HEMT amplifiers. The bin size is 100 mHz and the cavity bandwidth for this measurement 433 mHz.  \n<!-- image -->  \nFIG. 11. The variable temperature stage is used to perform the noise calibration. It consists of a 50 Ω matched load and a resistive heater anchored to a copper plate.  \n<!-- image -->', 'Appendix E: Noise calibration': "The Variable Temperature Stage (VTS) is shown in Fig. 11. It consists of a 50 Ω matched load anchored to a 50 mm × 41 mm × 3 . 2 mm copper plate. A power 100Ω VPR221 power resister from Vishay Precision Group heats the copper plate and is anchored to the other side of the copper plate. Two Cernox temperature sensors (CX1010-CU-HT-0.1L) are used to measure the temperature of the copper plates. The VTS is attached to the copper brackets surrounding the 1 . 3 GHz cavity via stainless  \n- [1] V. C. Rubin, W. K. Ford, Jr., N. Thonnard, and D. Burstein, Rotational properties of 23 SB galaxies, Astrophys. J. 261 , 439 (1982).  \nsteel standoffs. Alumina fish beads surround the stainless steel standoffs. Alumina has a higher thermal conduction than stainless steel above 3 K, allowing for quicker cooldowns.  \nThe noise calibration is performed by varying the VTS temperature T VTS from base temperature to 7 . 6 K and monitoring the power out of the HEMT with a spectrum analyzer. The spectrum analyzer is centered at the cavity resonant frequency and is set to a 100 kHz span with 1000 sweep points and a 1 kHz resolution bandwidth (RBW). The noise power does not vary appreciably over this fre-  \nFIG. 12. The noise calibration is performed using the Yfactor method, where the noise power of the electronics chain is measured as a function of a varying noise source. The shaded red regions are excluded from the noise calibration fit.  \n<!-- image -->  \nquency range, so the noise power at a particular T VTS is taken to be the mean of the sweep points, and the uncertainty is taken to be the standard deviation. 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