Search for Invisible Axion Dark Matter of mass ma = 43 μeV with the QUAX–aγ Experiment

D. Alesini, C. Braggio, 3 G. Carugno, 3 N. Crescini, 3, ∗ D. D’Agostino, 6 D. Di Gioacchino, R. Di Vora, 7, † P. Falferi, 9 U. Gambardella, 6 C. Gatti, G. Iannone, 6 C. Ligi, A. Lombardi, G. Maccarrone, A. Ortolan, R. Pengo, A. Rettaroli, 10, ‡ G. Ruoso, L. Taffarello, and S. Tocci INFN, Laboratori Nazionali di Frascati, Frascati, Roma, Italy INFN, Sezione di Padova, Padova, Italy Dipartimento di Fisica e Astronomia, Padova, Italy INFN, Laboratori Nazionali di Legnaro, Legnaro, Padova, Italy Dipartimento di Fisica E.R. Caianiello, Fisciano, Salerno, Italy INFN, Sez. di Napoli, Napoli, Italy Dipartimento di Scienze Fisiche, della Terra e dell’Ambiente, Università di Siena, via Roma 56, 53100 Siena, Italy Istituto di Fotonica e Nanotecnologie, CNR Fondazione Bruno Kessler, I-38123 Povo, Trento, Italy INFN, TIFPA, Povo, Trento, Italy Dipartimento di Matematica e Fisica, Università di Roma Tre, Roma, Italy (Dated: December 18, 2020)

The classical haloscope detection scheme consists of a resonant cavity immersed in a static magnetic field to stimulate the axion conversion into photons through the Primakoff effect. When the cavity resonant frequency ν c is tuned to the axion mass m a c 2 /h, the expected power * Present address: IBM Research-Zürich, Säumerstrasse 4, CH-8803 Rüschlikon, Switzerland † divora@pd.infn.it ‡ alessio.rettaroli@lnf.infn.it deposited by DM axions is given by [20] where ρ a ∼ 0.4 − 0.45 GeV/cm 3 [21] is the local DM density, α is the fine-structure constant, Λ = 78 MeV is a scale parameter related to hadronic physics, and g γ is a model dependent parameter equal to −0.97 (0.36) in the KSVZ (DFSZ) axion model [22,23]. The constant g γ is related to the coupling g aγγ = (g γ α/πΛ 2 )m a appearing in the Lagrangian describing the axion-photon interaction. The second parentheses contain the vacuum permeability µ 0 , the magnetic field strenght B 0 , the cavity volume V , its angular frequency ω c = 2πν c , the coupling between cavity and receiver β and the unloaded quality factor Q 0 . C 010 is a geometrical factor equal to about 0.69 for the T M 010 mode of a cylindrical cavity.
Presently, different solutions are being devised to improve the signal-to-noise ratio. The resonant cavity design is moving towards the multiple-cell concept [11] and the employment of different materials, like superconductors (Refs. [16,24] and Ref. [25]) or dielectrics [26,27]. On the amplification side, state-of-art experiments operate at the SQL − with SQUIDs [6,28] or Josephson Parametric Amplifiers (JPAs) [7] − while there is an attempt to circumvent it using squeezed-state receivers [8]. Yet, it's clear that the turning point in future experiments will be the introduction of single microwave photon counters in the amplification chain [29,30].
In this work we describe the operation of a classical haloscope of the QUAX-aγ experiment using a copper cavity coupled to a JPA and immersed in a static magnetic field of 8.1 T, all cooled down with a dilution refrigerator at a working temperature T ∼ 150 mK. These features improve the precedent work of Ref. [16], allowing us to exclude values of g aγγ > 0.639 · 10 −13 GeV −1 at 90% C.L.
In Sec. II we describe the experimental setup along with its calibration, while in Sec. III we present the results and data analysis, and prospects for QUAX-aγ in Sec. IV. The haloscope, assembled at Laboratori Nazionali di Legnaro (LNL), is composed by a cylindrical OFHC-Cu cavity ( Fig. 1), with inner radius of 11.05 mm and length 210 mm, inserted inside the 150 mm diameter bore of an 8.1 T superconducting (SC) magnet of length 500 mm. The total volume of the cavity is V = 80.56 cm 3 . The whole system is hosted in a dilution refrigerator with base temperature of 90 mK. Each cavity endplate hosts a dipole antenna in the holes drilled on the cavity axis. The cavity was treated with electrochemical polishing to minimize surface losses. We measured the resonant peak of the T M 010 mode at 150 mK and magnet on with a Vector Network Analyzer obtaining the frequency ν c = 10.4018 GHz and an unloaded quality-factor Q 0 = 76,000 in agreement with expectations from simulation performed with the ANSYS HFSS suite [31]. During data-taking runs, the cavity was critically coupled to the output radiofrequency (RF) line and the loaded qualityfactor was measured to be about Q L = 36,000.

II. EXPERIMENTAL SETUP
FIG. 2. Schematics of the experimental apparatus. The microwave cavity (orange) is immersed in the uniform magnetic field (blue shaded region) generated by the magnet (crossed boxes). A1 and A2 are the cryogenic and room-temperature amplifiers, respectively. The JPA amplifier has three ports: signal (s), idler (i), and pump (p). Superconducting cables (red) are used as transmission lines for RF signals from 4 K stage to 150 mK stage. Thermometers (red circled T) are in thermal contact with the resonant cavity and the signal port on the JPA. Attenuators are shown with their reduction factor in decibels. The horizontal lines (blue) identify the boundaries of the cryogenic stages of the apparatus, with the cavity enclosed within the 150 mK radiation shield. The magnet is immersed in liquid helium.
The RF setup is the same as our previous measurement [15] and is shown in Fig. 2. It consists of four RF lines used to characterize and measure the cavity sig-nal and to determine attenuations and gains. Starting from the left of Fig. 2, the "SO" line connects the source oscillator to the fixed, weakly coupled antenna D2 and is used to inject calibration and probe signals into the cavity. The "Pump" line connects the pump-signal generator to the corresponding port (p) of a JPA amplifier. The cavity is critically coupled to the "Readout" line through the antenna D1, tunable via a micrometric screw. The emitted power enters the JPA on the "s" port and is reflected, amplified, toward the HEMT cryogenic (A1) and HEMT room-temperature (A2) amplifiers. The signal is then downconverted with an I-Q mixer with a 100 MHz IF-band, the phase and quadrature components of the heterodyne signal are post-amplified in a 10 MHz band and finally sampled via an analog-to-digital converter (ADC) with a 2 MHz bandwidth. The "Aux" line is an auxiliary line introduced for calibration purposes. To minimize the Johnson noise contribution at the coldest stage we inserted attenuators and circulators in the RF lines. A non-optimal attenuation of the "Aux" line with 10 dB attenuation at 1 K and 10 dB at 150 mK causes an excess Johnson noise of about 95 mK on the circulator and on the cavity, corresponding to an effective temperature of the circulator of 273 mK at 10 GHz. We monitored the temperatures with RuO 2 thermometers, one in thermal contact with the cavity and the other with the mixing chamber. Due to some unexpected behaviour of the termometers we only estimated a temperature between 100 mK and 150 mK in the mixing chamber and between 200 mK and 250 mK on the cavity.
The JPA in our setup, first realized in [32], has noise temperature expected at the quantum-limit of about 0.5 K (including 0.25 K from vacuum fluctuations) and a resonance frequency tunable between 10 and 10.5 GHz by varying the pump amplitude and frequency and by applying a small magnetic field for fine regulation. After tuning the resonance frequency of the JPA to that of the cavity we measured a gain of 18 dB in a 10 MHz bandwidth.

III. ANALYSIS AND RESULTS
We first measured the transmittivity of the RF lines and the amplification gain as described in detail in [15]. Then we calibrated the power scale by injecting a known signal. Finally, we measured the system noisetemperature resulting in T n = (0.99±0.15 cal ±0.04 stab ) K where the errors result from the uncertainty in the calibration scale due to a limited tunability of the coupling of the antenna D1 and to the temperature variation during the data taking run. This value, within the error, is in agreement with our estimate of 0.83 K obtained from the single contributions reported in table I.
After setting to 8.1 T the magnetic field, we performed the axion search for a total time ∆t = 4203 s with an ADC sampling of 2 Msps with the cavity tuned at the fixed-frequency of ν c = 10.4018 GHz. We first computed  the average power-spectrum with a fine frequency-bin of 651 Hz, corresponding to 1/8th of the expected axionsignal width [33], in order to identify and remove noisy lines. We then excluded from our analysis a 200 kHz frequency region around the local oscillator frequency, ν lo = 10.4015 GHz, which was affected by 1/f and pickup noise. For the same reason we also excluded a single bin in the cavity region. Finally, we considered only the region of the Lorentzian distribution of the cavity power-spectrum with an expected power at least 10% of the peak value. The resulting spectrum is shown in Fig. 3. In order to extract the residuals, we modeled the system composed of the cavity and the "Readout" line with an equivalent electrical circuit. We derived the following expression of the power spectrum as the sum of the added noise of the JPA and HEMT (A1) amplifiers, T A,tot = 0.50 K as reported in table I, the Johnson noise from the circulator on the "AUX" line at temperature T 1 ∼ 270 mK reflected on the cavity, and that emitted by the cavity itself at the temperature T c left as a free parameter: whereT 1,c are the noise temperatures k BT = hν c /(exp (hν c /k B T ) − 1) + hν c /2 including the contribution from vacuum fluctuations, Q L is the loaded qualityfactor, δ = (ν/ν c − ν c /ν), ν c the cavity resonance frequency and G T OT (ω) is the total gain function. We fit the power spectrum expressing G T OT (ω) as 2-nd and 4th order polynomials in the left and right branch, respectively. Given the large number of unknown parameters we fix all known quantities to our best knowledge, taking into account measurement errors. The best fit is obtained for ν c = 10.40176 GeV and Q L = 35, 000 in reasonable agreement with our measurements. When fixing the "AUX" circulator temperature to T 1 = 273 mK, we obtain a cavity temperature T c = 250 mK, compatible with our expectations. The fit has χ 2 /n = 1226/1032 and is shown by the red line in Fig. 3. Changing T 1 in an interval between 150 and 273 mK does not impact on the quality of the fit and just reduces the cavity temperature down to about 100 mK in the former case. The maximum sensitivity to the axion signal is obtained by recalculating the power spectrum with a frequency spacing equal to the expected axion-signal width of about 5 kHz. We recompute the residuals and normalize it to the expected noise power P Dicke = 1.52 · 10 −23 W derived from the Dicke radiometer formula [34] using the system temperature T n = 0.99 K. The distribution of the normalized residuals is shown in Fig. 4 together with the result of a Gaussian fit showing an r.m.s. compatible with 1 within the statistical error, although some contribution to the total spread from temperature variations cannot be excluded. Interpreting our result as an exclusion test for the axion existence in this mass range, we calculate the limit to the axion-photon coupling g aγγ with a 90% confidence level. To compute the g aγγ we use our r.m.s. multiplied by the most conservative noise power P Dicke = 1.81 · 10 −23 W, obtained with the maximum temperature allowed within its error. We show in Fig. 5 the limit calculated for each frequency bin as a colored box ending with a marker, together with a solid purple line showing the expected limit in case of no signal. The dashed line represents the expected limit if the analysis were carried out assuming the axion deposits its power in two adjacent bins. Compatibly with statistics, we observe 10 bins out of 131 with a power less than 1.28σ, below the threshold for the 90% single-sided C.L. For these bins, the nominal expected 90% C.L. limit is plotted in Fig. 5.
In Fig. 6 we compare the limit we observed, g aγγ < 0.639 · 10 −13 GeV, in a mass window ∆m a = 3.7 neV centered at the mass m a = 43.0182 µeV with those obtained in previous searches.

IV. CONCLUSIONS
We report results of the search with an haloscope for galactic axions with mass about 43 µeV in a small frequency region of 3.7 neV. By cooling the system to about 150 mK in a dilution refrigerator and employing a Josephson Parametric Amplifier with noise at the Standard Quantum Limit, we set a limit on the axion-photon coupling of about a factor 1.5 from the QCD band. We showed directly that, even at frequency as large as 10 GHz, haloscopes will soon have the sensitivity to observe QCD axions. The total noise, estimated as twice the Standard Quantum Limit, can be further reduced by improving the thermalization of the resonant cavity, the line filtering, and by reducing the noise contribution from the HEMT.

ACKNOWLEDGMENTS
We are grateful to E. Berto, A. Benato and M. Rebeschini, who did the mechanical work, F. Calaon and M. Tessaro who helped with the electronics and cryogenics, and to F. Stivanello for the chemical treatments. We thank G. Galet and L. Castellani for the development of the magnet power supply, and M. Zago who realized the technical drawings of the system. We deeply acknowledge the Cryogenic Service of the Laboratori Nazionali di Legnaro, for providing us large quantities of liquid helium on demand.