Microwave Cavities and Detectors for Axion Research: Proceedings of the 3rd International Workshop [1st ed.] 9783030437602, 9783030437619

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Springer Proceedings in Physics 245

Gianpaolo Carosi Gray Rybka  Editors

Microwave Cavities and Detectors for Axion Research Proceedings of the 3rd International Workshop

Springer Proceedings in Physics Volume 245

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Gianpaolo Carosi • Gray Rybka Editors

Microwave Cavities and Detectors for Axion Research Proceedings of the 3rd International Workshop

Editors Gianpaolo Carosi Lawrence Livermore National Laboratory Livermore, CA, USA

Gray Rybka Physics and Astronomy Department University of Washington Seattle, WA, USA

ISSN 0930-8989 ISSN 1867-4941 (electronic) Springer Proceedings in Physics ISBN 978-3-030-43760-2 ISBN 978-3-030-43761-9 (eBook) https://doi.org/10.1007/978-3-030-43761-9 © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The mystery of dark matter remains as one of the most pressing questions in physics today and one well-motivated candidate is the QCD axion. Although axions are predicted to have extraordinarily weak couplings to ordinary matter, it was pointed out decades ago by Pierre Sikivie that they can be detected through their resonant conversion to photons in microwave cavities threaded by a strong magnetic field. Only recently have experiments utilizing this technique, such as the Axion Dark Matter Experiment (ADMX), come online with the sensitivity required to probe large regions of parameter space. At their heart, these experiments contain large, highly tunable microwave cavities that can be cooled to dilution refrigerator temperatures in the presence of a large magnetic field. Detecting the feeble (of order 10−24 W) axion-to-photon conversion signals also requires highly specialized superconducting detectors that can operate near the quantum limit (or beyond). Designing and optimizing such systems is a nontrivial task. As a result, a series of workshops have been organized to bring together subject matter experts in axion dark matter detection, cryogenic microwave cavity design, and quantum sensor technology in order to explore new ideas and train new researchers. These proceedings are from the “3rd Workshop on Microwave Cavities and Detectors for Axion Research,” which took place at Lawrence Livermore National Laboratory (LLNL) from August 21–24, 2018. Over 40 people from around the world attended, including subject matter experts from Lawrence Livermore National Laboratory (LLNL), Stanford Linear Accelerator Center (SLAC), Lawrence Berkeley National Laboratory (LBNL), Fermi National Accelerator Laboratory (FNAL), University of Washington, University of Florida, University of California, Berkeley, University of Western Australia, CERN, and the Institute for Basic Science (IBS), South Korea. The proceedings are based on a series of lectures on topics ranging from the fundamentals of microwave simulations to new concepts for cavity systems and superconducting detectors. In addition, new axion detection techniques that are complementary to the standard microwave cavity search are presented. This workshop, along with its predecessor workshops which took place on August 25–27, 2015 and January 10–13, 2017, provided useful in sharing ideas, v

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experiences, and technologies as well as training a new generation of researchers. As a result, it is anticipated that these workshops will continue on an approximately yearly basis. All of these workshops were supported by generous contributions from the Heising-Simons Foundation and were hosted at LLNL. We thank them both for their support and would also like to thank the many participants for putting this workshop and proceedings together. Finally, we would like to thank the reader who we hope will be able to use these proceedings to learn more about the exciting field of axion dark matter detection. LLNL-PROC-784944 Livermore, CA, USA

Gianpaolo Carosi

Contents

Microwave Cavity Simulation Using Ansys HFSS. . . . . . . . . . . . . . . . . . . . . . . . . . . . Mark Jones

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Ultra-High Field Solenoids and Axion Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mark D. Bird

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Recent Results with the ADMX Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Du

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The Microstrip SQUID Amplifier in ADMX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sean R. O’Kelley, Gene Hilton, and John Clarke

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The ORGAN Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ben T. McAllister and Michael E. Tobar

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The 3 Cavity Prototypes of RADES: An Axion Detector Using Microwave Filters at CAST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sergio Arguedas Cuendis, A. Álvarez Melcón, C. Cogollos, A. Díaz-Morcillo, B. Döbrich, J.D. Gallego, B. Gimeno, I.G. Irastorza, A.J. Lozano-Guerrero, C. Malbrunot, P. Navarro, C. Peña Garay, J. Redondo, T. Vafeiadis, and W. Wünsch

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Search for 5–9 µeV Axions with ADMX Four-Cavity Array . . . . . . . . . . . . . . . . Jihee Yang, Joseph R. Gleason, Shriram Jois, Ian Stern, Pierre Sikivie, Neil S. Sullivan, and David B. Tanner

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Tunable High-Q Photonic Bandgap Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ankur Agrawal, Akash V. Dixit, David I. Schuster and Aaron Chou

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Source Mass Characterization in the ARIADNE Axion Experiment . . . . . . Chloe Lohmeyer, N. Aggarwal, A. Arvanitaki, A. Brown A. Fang, A.A. Geraci, A. Kapitulnik, D. Kim, Y. Kim, I. Lee, Y.H. Lee, E. Levenson-Falk, C.Y. Liu, J.C. Long, S. Mumford, A. Reid, Y. Semertzidis, Y. Shin, J. Shortino, E. Smith, W.M. Snow, E. Weisman, A. Schnabel, L. Trahms, and J. Voigt

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CAPP-PACE Experiment with a Target Mass Range Around 10 µeV . . . . . Doyu Lee,Woohyun Chung, Ohjoon Kwon, Jinsu Kim, Danho Ahn, Caglar Kutlu, and Yannis K. Semertzidis

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High Resolution Data Analysis: Plans and Prospects . . . . . . . . . . . . . . . . . . . . . . . . Shriram Jois, Leanne Duffy, Neil Sullivan, David Tanner, and William Wester

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Operation of a Ferromagnetic Axion Haloscope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Alesini, C. Braggio, G. Carugno, N. Crescini, D. Di Gioacchino, P. Falferi, S. Gallo, U. Gambardella, C. Gatti, G. Iannone, G. Lamanna, C. Ligi, A. Lombardi, R. Mezzena, A. Ortolan, S. Pagano, R. Pengo, A. Rettaroli, G. Ruoso, C. C. Speake, L. Taffarello, and S. Tocci

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Overview of the Cosmic Axion Spin Precession Experiment (CASPEr) . . . 105 Derek F. Jackson Kimball, S. Afach, D. Aybas, J. W. Blanchard, D. Budker, G. Centers, M. Engler, N. L. Figueroa, A. Garcon, P. W. Graham, H. Luo, S. Rajendran, M. G. Sendra, A. O. Sushkov, T. Wang, A. Wickenbrock, A. Wilzewski, and T. Wu Axion Dark Matter Search at IBS/CAPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 SungWoo Youn Multiple-Cell Cavity for High Mass Axion Dark Matter Search . . . . . . . . . . . 131 Junu Jeong, SungWoo Youn, and Yannis K. Semertzidis Exclusion Limits on Hidden-Photon Dark Matter Near 2 neV from a Fixed-Frequency Superconducting Lumped-Element Resonator . . . . . . . . 139 A. Phipps, S. E. Kuenstner, S. Chaudhuri, C. S. Dawson, B. A. Young, C. T. FitzGerald, H. Froland, K. Wells, D. Li, H. M. Cho, S. Rajendran, P. W. Graham, and K. D. Irwin Employing Precision Frequency Metrology for Axion Detection . . . . . . . . . . . 147 Maxim Goryachev, Ben T. McAllister, and Michael Tobar Bayesian Searches and Quantum Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 George Chapline and Matt Otten Status of the MADMAX Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Chang Lee Orpheus: Extending the ADMX QCD Dark-Matter Axion Search to Higher Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Gianpalo Carosi, Raphael Cervantes, Seth Kimes, Parashar Mohapatra, Rich Ottens, and Gray Rybka

Microwave Cavity Simulation Using Ansys HFSS Mark Jones

Abstract The design of microwave cavity detectors for axion dark matter research is often accomplished using advanced full-wave electromagnetic simulation software tools. These tools provide a cost-effective approach to evaluate a wide variety of cavity configurations, frequency tuning mechanisms, and conductive or dielectric materials and coatings. One simulation software package used for this application is Ansys High Frequency Structure Simulator (HFSS), which is based upon the well-established finite element method. HFSS includes numerous features useful for microwave cavity design such as parametric geometry modeling, adaptive meshing algorithm, curvilinear mesh elements, driven modal and eigenmode matrix solvers, and optimization algorithms. This paper describes the use of the HFSS software to simulate microwave cavities for axion haloscope detectors, with an example tutorial for a cylindrical cavity. Excellent agreement between the simulated and analytical results is shown for the resonant frequency, quality factor, and form factor. Keywords Ansys HFSS · Cavity simulation · Finite element modeling

1 Overview of HFSS 1.1 General Capabilities Ansys HFSS [1] is a full-wave frequency-domain three-dimensional electromagnetic field solver which uses the finite element method to solve Maxwell’s equations. It offers industry-standard accuracy, adaptive meshing of arbitrary geometries, fully parametric modeling, multiple optimization engines, high-performance computing capabilities, and multi-physics integration via the Ansys Workbench environment.

M. Jones () Pacific Northwest National Laboratory, Richland, WA, USA e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_1

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HFSS has been commercially available for approximately 30 years and is widely used to design antennas, filters, waveguides, connectors, transitions, and electronic packages. Multiple numerical solvers are available within the HFSS software, each targeted for different applications. A license allows the use of the frequency-domain finite element solver, frequency-domain finite element eigenmode solver, time-domain finite element solver, frequency-domain integral equation solver, frequency-domain finite element boundary integral hybrid solver, frequency-domain planar integral equation solver, or a linear circuit solver. A license allows the solver to use a maximum of four processor cores. Additional high-performance computing (HPC), optimization, and distributed solver licenses are available to increase computing capabilities.

1.2 User Interface The HFSS user interface is integrated into the Electronics Desktop environment which is part of the Electromagnetics Suite. Figure 1 shows the user interface of HFSS Release 19 within the Electronics Desktop. The user interface consists of multiple window panes with a menu and ribbon toolbar along the top, project manager and properties windows on the left, 3D model editor tree window in the center, graphics window on the right, and message manager and progress windows along the bottom.

1.3 Solution Types for Cavity Simulation Two solution types applicable to microwave cavity design are the frequency-domain eigenmode and frequency-domain driven modal solvers. The eigenmode solver calculates the natural resonances of the cavity based upon the geometry, materials, and boundary conditions. It calculates modal frequencies, unloaded quality factors, and electromagnetic field solutions for up to 20 modes simultaneously. This solver can be used to study the modal behavior of a resonant structure, generate mode maps for tuned cavities, and calculate field-based quantities such as the cavity form factor. The driven modal solution uses one or more ports to excite the cavity structure and provides network parameters (SYZ parameters) and electromagnetic field solutions. A wave port is a cross-section of a transmission line used to calculate the characteristic impedance and complex propagation constant of the excitation mode. The driven modal solver can be used to include antenna feed probes to predict transmission and reflection coefficients and study how to achieve critical coupling to a cavity mode. Multiple frequency sweep types are available to obtain SYZ matrix and electromagnetic field results over a user-specified bandwidth.

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Fig. 1 HFSS Release 19 user interface within the Electronics Desktop environment

1.4 Meshing Technologies for Cavity Simulation The creation of a robust, efficient mesh which accurately represents the model geometry is a critical step required to obtain accurate results from a finite element method solver. HFSS generates a tetrahedral element mesh using an iterative algorithm which refines the element size and distribution until a user-defined convergence criteria is reached. This method produces a graded mesh with fine discretization in locations required to accurately represent the field behavior, effectively tuning the mesh to the electromagnetic performance of the structure. The convergence parameter used in mesh generation is typically the modal frequency for an eigenmode solution and S-parameters for a driven modal solution. In addition to the automatic meshing algorithm, the user can manually influence the initial mesh density. These optional controls can further focus mesh elements in critical areas to reduce the number of adaptive passes needed to converge to the specified criteria, but are not required to achieve accurate results. Both rectilinear and curvilinear tetrahedral mesh elements are available to optimally represent the model geometry. Curvilinear mesh elements are recommended for the simulation of cylindrical or curved cavity structures.

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1.5 Boundary Conditions for Cavity Simulation The electromagnetic solver requires boundary conditions to define material properties for geometry surfaces. These surface definitions also can be used to simplify geometries or make the meshing process more efficient. By default, any object surface that contacts the background is automatically defined as a perfectly conducting boundary. Cavity models often include finite conductivity boundaries to model good conductors such as copper. Other applicable boundary conditions include absorbing or perfectly matched layer boundaries to model open regions, impedance boundaries for resistive materials, layered impedance boundaries for thin coatings, periodic boundaries for repeating structures, and symmetry plane boundaries to reduce model size. Frequencydependent properties can be included for the driven modal solver. Effects such as surface roughness and anisotropy can also be included as appropriate.

1.6 Electromagnetic Fields Calculator HFSS includes a calculator which can access field data to perform a wide variety of mathematical operations. The calculator can use geometric, complex, vector, and scalar data to create numerical, graphical, or exportable results. Additionally, frequently used expressions can be created and loaded into any project. The fields calculator is used to obtain the cavity form factor which is a value between 0 and 1 representing the axion coupling to a cavity mode in a haloscope detector. Although it is provided directly with the eigenmode solver results, the quality factor can also be calculated using the magnetic field data if desired.

2 Resonant Cavity Example 2.1 Creating the Model The example model described here is a cylindrical copper cavity with a radius of 21 cm and height of 100 cm. The TM010 mode is expected to resonate at 546.42 MHz with an unloaded quality factor of 61,391 [2] and form factor of 0.692 [3]. Create HFSS Project Insert project into Electronics Desktop using File > New. Set Eigenmode Solution Type Select HFSS > Solution Type and select Eigenmode. Set Model Units Select Modeler > Units and select cm.

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Set Dialog Data Entry Mode Select Tools > Options > General Options and then 3D Modeler > Drawing > Dialog. Set Default Transparency Value Select Tools > Options > General Options and then 3D Modeler > Display > Rendering. Enter a value of 0.7. Create Parameterized Cavity Select Draw > Cylinder. Enter “cavity_rad” in radius value and then 21 cm. Enter “cavity_height” in height value and then 100 cm. Assign Cavity Wall Conductivity Select the cavity in the 3D Modeler Editor tree. Select Edit > Extend Selection > All Object Faces. Select HFSS > Boundaries > Assign > Finite Conductivity. Enter “cavity_walls” in the name field. Apply Curvilinear Mesh Elements Select the cavity in the 3D Modeler Editor tree. Select HFSS > Mesh Operations > Assign > Apply Curvilinear Meshing and enter “Cavity” in the name field.

2.2 Solving the Model Add Solution Setup Select HFSS > Analysis Setup > Add Solution Setup. Enter minimum frequency = 540 MHz, number of modes = 3, maximum number of passes = 12, maximum delta frequency per pass = 2%, and minimum passes = 4. Save Project Select File > Save and enter “cavity.aedt” as the file name. Perform Validation Check Select HFSS > Validation Check and confirm a check mark appears beside each step. Solve Model Select HFSS > Analyze All in the menu bar.

2.3 Viewing the Results View Solution Data Select HFSS > Results > Solution Data. Select Eigenmode Data tab to view modal frequencies and quality factors for each requested mode. The first mode is TM010 , the second mode is TM011 , and the third mode is TE113 . The Mode 1 frequency should be 546.42 MHz and quality factor should be 61,378. The simulated frequency exactly agrees with the analytical value and the unloaded quality factor agrees within 0.02%. Select the Convergence tab to view adaptive pass information including the number of mesh elements and frequency convergence value for each pass. Select Profile tab to view the log file for the simulation. View E-Field Phase Animation Select XZ and YZ planes in the 3D Modeler Editor tree. Select HFSS > Fields > Plot Fields > E > Mag_E. Right-click on Mag_E1 plot in the Field Overlays section of the Project Manager tree. Select Phase

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Fig. 2 Plot of TM010 mode electric field magnitude in two vertical planes

as the swept variable with start value of 0◦ , stop value of 170◦ , and step value of 17 (Fig. 2). View E-Field Vector Animation Select XZ and YZ planes in the 3D Modeler Editor tree. Select HFSS > Fields > Plot Fields > E > Vector_E. Active Mode of Interest Select HFSS > Fields > Edit Sources. Enter a magnitude of 1 J for the desired mode and magnitude of 0 J for the other two modes. This will activate the desired mode for all field plots and post-processing calculations. Calculate Form Factor Select HFSS > Fields > Calculator. Due to space limitations, the detailed steps used to calculate form factor are not included here. The simulated value should be 0.692 which agrees exactly with the analytical calculation.

3 Summary This paper has described several key features of the Ansys HFSS software which is used to design microwave cavities for axion dark matter research. Effective use of advanced simulation software allows researchers to efficiently investigate

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concepts for a wide variety of cavity designs and obtain detailed insights into the electromagnetic behavior of the structure. A step-by-step procedure for an example cylindrical cavity was also given showing excellent agreement with analytical calculations for the resonant frequency, quality factor, and form factor.

References 1. Ansys HFSS Homepage, https://www.ansys.com/products/electronics/ansys-hfss/. Accessed 15 Jan 2019 2. X. Li, Y. Jiang, Design of a cylindrical cavity resonator for measurements of electrical properties of dielectric materials, University of Gavle Master Thesis, 2010 3. I. Stern, A.A. Chisholm, J. Hoskins, P. Sikivie, N.S. Sullivan, D.B. Tanner, G. Carosi, K. van Bibber, Cavity design for high-frequency axion dark matter detectors. Rev. Sci. Instrum. 86, 123305 (2015)

Ultra-High Field Solenoids and Axion Detection Mark D. Bird

Abstract High Temperature Superconducting (HTS) materials are now becoming incorporated into magnets that are being used for a variety of physics applications. Axion detection is a particularly attractive application for these conductors and there is significant promise that reliable systems can be built. However, there are still many challenges that are presently unresolved when it comes to building magnets of this scale from these materials. In particular, when a superconducting magnet quenches the energy stored in the magnetic field is converted into heat. If not controlled properly, the energy can be deposited in a non-uniform manner that results in excessive heating in some regions and damage to the magnet. For magnets using traditional Low Temperature Superconductors (LTS) methods of protecting the magnet during quench have been relatively well developed. For the HTS materials this development is presently underway, but no demonstrations protecting coils of the size needed for axion detection have yet been published. Keywords Superconducting magnet · Axion detection · Quench protection

1 Introduction The Axion Dark Matter eXperiment (ADMX) has been trying to detect axions for many years using an 8 T magnet with a 500 mm bore inside of which a radiofrequency resonant cavity has been installed along with various electronics. A primary parameter of interest in such searches is the square of the magnetic field integrated over the volume of the rf cavity, or approximately the square of the central field of the magnet multiplied by the volume of the detector, B0 2 V. Hence, large bore, high field magnets are vital to the search for axions. For ADMX, B0 2 V is approximately 12 T2 m3 .

M. D. Bird () National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL, USA e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_2

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HTS materials were first discovered in 1986 and are also known to superconduct at higher fields than the LTS materials. NbTi is the most commonly used superconductor for magnets, being used in most magnets for Magnetic Resonance Imaging (MRI) as well as most superconducting dipole and quadrupole magnets for synchrotrons. It is also the conductor used in the present AMDX magnet. However, NbTi is limited to applications below approximately 10 T. Nb3 Sn is the other commonly used superconductor for magnets. It is frequently used in magnets for Nuclear Magnetic Resonance (NMR) or Condensed Matter Physics (CMP) having attained fields as high as 23.5 T. The cheapest way to attain higher B0 2 V than in the present ADMX would be to build a large bore, modest field magnet using NbTi. For example, a commercial human whole body MRI magnet has a bore of ~90 cm and can provide field up to 7 T routinely with some examples having been delivered up to 9.4 T and one at 10.5 T. A new MRI magnet in France has been delivered and energized to 11.74 T with B0 2 V ~ 430 T2 m3 but is not yet fully operational. Another extreme example is the Compact Muon Solenoid detector installed on the Large Hadron Collider at CERN which provides 4T in a bore of 6 m, for B0 2 V ~ 5300 T2 m3 . However, given the expected energy of the axion, the rf cavities must be of modest size, which would require slaving many of them together to build a nextgeneration axion detector based on NbTi magnets. Based on these constraints in cavity design, the bore of the magnet should be ~16 cm and the length of the rf cavity should be no more than 2.5 times the diameter. This leads us to need extremely intense magnetic fields for which HTS materials are uniquely well suited.

2 Present HTS Magnet State of the Art There are presently three HTS conductors to be considered for this application: BiSCCO-2212, BiSCCO-2223, and REBCO. All three superconduct above 100 T, all have adequate current-density at 20–40 T for construction of an ultra-high-field (UHF) magnet. Rare Earth Barium Copper Oxide (Y or Gd being the Rare Earth component) was the first to become available in a high strength form suitable for UHF magnets. In 2007 SuperPower provided a conductor consisting of ~40 µm of Hastelloy with some buffer layers, ~1 µm of YBCO, Ag, and Cu cladding. The National High Magnetic Field Laboratory (MagLab) proceeded to build some test coils and secured funding to develop a 32 T superconducting magnet for CMP. This magnet has now reached field and is expected to start to serve the user community in the coming months [1]. In 2010 Seungyong Hahn, Yuki Iwasa, and others at MIT presented a new concept for UHF magnets: No-Insulation (NI-) REBCO. In this approach there is no insulation on the composite superconducting tape [2]. LTS magnets require insulation on the conductor. When an insulated conductor quenches (converts from

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superconducting to normal state), the current moves from the superconducting material to the Cu and Ag within the composite conductor. If the current-density is too high, the power density will be too high and the conductor will start to melt before the magnet is de-energized. To prevent this, a significant fraction (50–80%) of the conductor cross section is usually Cu. With HTS conductors it becomes possible to leave out the inter-turn insulation. In this case, when the conductor quenches, the current can move into Cu and Ag in adjacent turns of conductor. Less Cu is needed in individual tapes, the cross section of the conductor can become smaller, the current-density of the magnet becomes larger, and the size of the magnet becomes much smaller. The dominant stress in a solenoid is proportional to the product of magnetic field, current-density, and radius of the turn of conductor. When the size drops, the stresses reduce and less reinforcement materials are required. This results in still more reduction in size. The highest field attained purely with this technology is 26 T that was reached by a coil of only 17 cm outer diameter designed by Hahn, built by SuNAM, and tested at the MagLab in 2015 [3]. A smaller coil reached 14.4 T while operating inside a 31.1 T resistive magnet for a total field of 45.5 T at the MagLab in 2017 (see https://nationalmaglab.org/news-events/news/ mini-magnet-packs-world-record-punch). In 2013 a new ultra-high strength version of Bi-2223 tape became available from Sumitomo. It has been used by Satoshi Awaji and others at the Tohoku Magnet Lab to complete a 24 T magnet that was commissioned in early 2017 and is presently serving the CMP community [4]. Bi-2212 has been transformed in recent years into a very high current-density conductor and concepts are being developed for high-strength reinforcement to enable UHF magnets [5].

3 Proposed Magnet Concepts for Next-Generation ADMX A number of conceptual designs have been created for magnets with 16 cm bore and fields ranging between 24 T and 30 T. One approach would be to build a system that is nearly a copy of the MagLab’s 32 T magnet leaving out the innermost REBCO coil. With this approach, 24 T in 16 cm should be achieved with low risk. It should also be possible to build a 30 T, 16 cm bore magnet using this approach but using more HTS material and less LTS material than in the existing 32 T magnet. These approaches would benefit from the extensive development effort that enabled the 32 T magnet system to be completed, including extensive quench analysis and testing [6]. Another approach would be to use the newer NI-REBCO technology. This might result in much more compact coils that provide similar field in a similar bore size. However, NI-REBCO has not yet had a comprehensive analysis of behavior during quench and development of a means to prevent damage during quench.

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4 Managing High Energy Quenches During quench the energy stored in the magnetic field (½ LI2 , where L = inductance and I = current) is converted into heat. As mentioned above, if this heating is not uniform, it can destroy a coil. To reduce heating, most commercial magnets include resistors and back-to-back diodes across the coils or sections of coils. If a quench occurs in a coil, the voltage builds up until it reaches the diode breakdown voltage at which point the diode allows current to flow through a bypass around the coil. This allows the current in the normal zone to drop and avoid overheating. The energy is dissipated in the resistors. (Back-to-back diodes are used so the magnet can operate at either positive or negative field.) However, the drop in current in one coil induces voltage on the adjacent coils (transformer effect) which, coupled with the diodes, results in the current in the second coil to rise. When the current in the second coil gets too high, this coil in turn will quench and decay, and current will be induced in the next coil. During this process, a coil might operate at higher current than during normal steady-state operation and be more highly stressed than intended. To avoid either overheating or overstressing the coils, the designer must consider the interaction of all the inductances, resistors, diodes, and the evolution of the resistance of the coils during the quench process [7]. Many magnets have been destroyed over the years due to insufficient attention to quench protection. For NI-REBCO coils, the same phenomenon occurs, except on a much larger, or finer, scale. Each turn of conductor is an independent inductor and the contact area between each pair of turns is a resistive element. The turn itself has variable resistance depending on the temperature, field, and current. Instead of a few or dozens of coil sections, there are thousands or tens of thousands of turns interacting with each other. Figure 1 shows the current distribution in an NI-REBCO coil during quench as computed by Markiewicz in 2015 [8]. All turns in the coil were originally at 200 amps. When a quench was introduced at the top of the coil (disk 1, left in the figure) current started redistributing around the resistive section. Mutual inductance between the turns caused current spikes in various turns as a quench wave passes from the top of the coil to the bottom. Computed current spikes are >3 times the normal operating current of 200 A. These current spikes can result in high hoop stresses within a coil as well as high forces between multiple nested coils. The 32 T magnet at the MagLab stores 8 MJ of energy, ~0.3 MJ of it in the HTS coils. For comparison, 20 T LTS magnets sold by Oxford Instruments installed at the MagLab store 1–2 MJ depending on when they were built. A stick of dynamite also stores ~1 MJ of energy. Table 1 lists several HTS magnets in development worldwide over recent years with goals of reaching 24 T or greater as well as the amount of energy stored by the HTS parts of those magnets. The table does not include lower field magnets because such fields are attainable by LTS magnets and it does not include small test coils that stored 600 A which might lead to excessive hoop stress

magnets, so they might have destroyed coils that are not reported here. We see that only eight coils worldwide of this field and energy have been built to date and that four of those were destroyed by quench and are not expected to be put into service. The exceptions are a 24 T magnet in Sendai, Japan completed in 2017, the 32 T magnet completed in 2017 at the MagLab in Tallahassee, and the 25.8 T and 28.2 T NMR magnets by Bruker that reached performance specifications in 2019. We see that the largest amount of energy stored by a successful HTS magnet to date was 0.4 MJ. (Many smaller HTS test coils have been destroyed that stored much less than this.) Conceptual designs for axion detectors store 1.7–10 MJ.

5 Proposed Development Route Reliable quench detection systems have been developed for LTS magnets (all commercial firms and government labs have them) and they are being developed for HTS magnets also. Doing so requires meaningful numerical modeling of quench in these magnets and benchmarking of the computational results with experimental ones. Then approaches can be proposed and modeled for protection systems which will also need to be tested. The MagLab has been performing quench analysis for

Field 25 Ta 28 Ta 25 Ta 24 T 32 T 35 T 40 T 25.8 T 28.2 T 30.5 Ta 30.5 T 24 Ta 25 T 30 T 30 T

6 cm 9 cm 3 cm 10 cm 16 cm 16 cm

Cold Bore 4 cm 4 cm 5 cm 5 cm 3 cm 4 cm 3 cm

12 km ~30 km 9.4 km ~36 km [2] 30 km 40–60 km

Amount of HTS 2.1 km 2.1 km 14 km 11.4 km 9.4 km 45 km >15 km

0.4 MJ 1.7 MJ 2.5 MJ 6–10 MJ

0.3 MJ

HTS stored energy 0.1 MJ 0.1 MJ 0.4 MJ 0.4 MJ 0.3 MJ 1.8 MJ >2.8 MJ

Bold font indicates coils that have been tested to date Plain font indicates coils under construction presently Italics font indicates coils that have been proposed a Indicates coil that has been damaged and removed from service

Axions

NMR

Use Condensed matter, other

Technology I-REBCO + Bi-2223 + LTS I-REBCO + Bi-2223 + LTS I-REBCO + LTS Bi2223 + LTS Ins-REBCO + LTS NI-REBCO REBCO I-REBCO + LTS I-REBCO + LTS NI-REBCO + LTS Bi-2223 + LTS + misc. NI-REBCO NI-REBCO NI-REBCO + LTS NI-REBCO

Table 1 Partial list of various HTS coils built or under development to date that were intended to reach fields >23 T Organization Riken Riken Tohoku Tohoku MagLab KBSI, MagLab MagLab Bruker Bruker MIT RIKEN KBSI, SuNAM, CAPP Brookhaven, CAPP MagLab, ADMX MagLab, ADMX

2019 [14] 2019 [14] [15] 2023 [16] 2015 [17] 2018 [18]

Year 2015 [9] 2015 [10] 2016 [11] 2017 [12] 2017 [1] [13]

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Ultra-High Field Solenoids and Axion Detection

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I-REBCO coils for many years as part of our 32 T magnet project and has developed a reliable protection system. A particular test coil was intentionally quenched >100 times without damage Analysis of NI-REBCO coils has been published by a few groups (some results shown above). Improvement of modeling is underway and the development of protection systems is starting at the MagLab and elsewhere. However, there remains a significant amount of work in this field to be completed prior to having reliable magnets using NI-REBCO. Another challenge is materials cost. Presently HTS materials are extremely expensive. However, most manufacturers claim costs can and will be reduced as volume of production increases.

6 Conclusions While the HTS materials show tremendous potential to enable UHF magnets, there is presently only one magnet operating routinely at higher field than is available from LTS magnets. Several challenges remain to be overcome prior to reliable HTS magnets becoming widespread. Leading among these are quench protection and cost. Great progress is presently being made on quench protection. In 2017 the highest field attained by an all-superconducting magnet jumped from 27 to 32.1 T. This tremendous increase is the result of a 9-year development effort at the MagLab. While the route to still higher fields or larger bores seems much clearer than it was a few years ago, there remains a great deal of work to complete. Modeling of quenches in HTS coils is advancing quickly and reliable magnets should become routine in the coming years. Acknowledgement The author is greatly indebted to Hongyu Bai and Denis Markiewicz who created design of magnets suitable for axion detection that are listed in Table 1. Their contributions are greatly appreciated. David Tanner and Neil Sullivan brought this potential application for ultrahigh field magnets to the author’s attention and had many discussions about the requirements, goals, etc. The author also had various fruitful discussions with Hubertus Weijers and Seungyong Hahn about the possibility of designing REBCO double-pancake magnets for axion detection. The author is indebted to all of them.

References 1. H. Weijers, Characteristics of the 32 T superconducting magnet. Presented at the Low Temperature Superconductor Workshop, Jacksonville, FL, Feb. 12-14, 2018 2. S. Hahn, D.K. Park, J. Bascunan, Y. Iwasa, HTS pancake coils without turn-to-turn insulation. IEEE Trans. Appl. Supercond. 21, 1592 (2011). https://doi.org/10.1109/TASC.2010.2093492 3. S. Yoon, J. Kim, K. Cheon, H. Lee, S. Hahn, S.-H. Moon, 26 T 35 mm all-GdBa2 Cu3 O7-x multi-width no-insulation superconducting magnet. Supercond. Sci. Technol. 29, 04LT04 (2016). https://doi.org/10.1088/0953-2048/29/4/04LT04

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4. S. Awaji, K. Watanabe, H. Oguro, H. Miyazaki, S. Hanai, T. Tosaka, S. Ioka, First performance test of a 25 T cryogen-free superconducting magnet. Supercond. Sci. Technol. 30, 065001 (2017). https://doi.org/10.1088/1361-6668/aa6676 5. K. Zhang, H. Higley, L. Ye, S. Gourlay, S. Prestemon, T. Shen, E. Bosque, C. English, J. Jiang, Y. Kim, J. Lu, U. Trociewitz, E. Hellstrom, D. Larbalestier, Tripled critical current in racetrack coils made of Bi-2212 Rutherford cables with overpressure processing and leakage control. Supercond. Sci. Technol. 31, 105009 (2018) 6. M. Breschi, L. Cavallucci, P.L. Ribani, A.V. Gavrilin, H.W. Weijers, Analysis of quench in th NHMFL REBCO prototype coils for the 32 T magnet project. Supercond. Sci. Technol. 29, 055002 (2016) 7. A.A. Konjukhov et al., Quenching of multisection superconducting magnet and internal and external shunt resistors. IEEE Trans. Magn. 25(2), 1538–1540 (1989) 8. W.D. Markiewicz, J.J. Jaroszynski, D.V. Abraimov, R.E. Joyner, A. Khan, Quench analysis of pancake wound REBCO coils with low resistance between turns. Supercond. Sci. Technol. 29, 025001 (2016). https://doi.org/10.1088/0953-2048/29/2/025001 9. K. Kajita et al., Degradation of a REBCO coil due to cleavage and peeling originating from an electromagnetic force. IEEE Trans. Appl. SC 26(4), 4301106 (2016) 10. Y. Yanagisawa et al., 27.6 T Generation using Bi-2223/REBCO superconducting coils, in IEEE/CSC & ESAS Superconductivity News Forum (global edition), July 2016 11. S. Awaji et al., Learning from R&D and operation of HTS insert coil for high field magnet. Presented at 13th EuCAS, Geneva, 17-21 September 2017 12. S. Awaji et al., First performance test of a 25 T cryogen-free superconducting magnet. Supercond. Sci. Technol. 30, 065001 (2017) 13. K. Kim et al., Design and performance estimation of a 35 T 40 mm no-insulation all-REBCO user magnet. Supercond. Sci. Technol. 30, 065008 (2017) 14. Bruker representative, private communication, EuroMAR, Nantes, France, July 2018 15. P. Micheal et al., Assembly and test of a 3-nested-coil 800-MHz REBCO insert (H800) for the MIT 1.3 GHz LTS/HTS NMR magnet. Presented at 2018 applied superconductivity conference, Seattle, Oct. 29–Nov. 2, 2018 16. H. Maeda, Development of a persistent mode 1.3 GHz NMR magnet by using superconducting joints. Presented at the NHMFL, April 30, 2018 17. Y. Semertzidis, Private communication, 2018 applied superconductivity conference, Seattle, Oct. 29 – Nov. 2, 2018 18. R. Gupta et al., 25 T, 100 mm bore HTS solenoid for axion dark matter search. Presented at 2018 applied superconductivity conference, Seattle, Oct. 29 – Nov. 2, 2018

Recent Results with the ADMX Experiment N. Du on Behalf of the ADMX Collaboration

Abstract The ADMX experiment is an axion haloscope using a microwave cavity in a strong magnetic field to search for dark matter axions. I will present on results from our run 1A in which we excluded the range of axion–photon couplings predicted by QCD axions for axion masses between 2.66 and 2.81 µeV. These results marked the first time a haloscope experiment achieved sensitivity to the wellmotivated DFSZ axion. In addition, I will provide updates on the search for axions in the 2.82–3.31 μeV madss range with the ADMX experiment. Keywords Axion · Haloscope · Cavity resonator

1 Introduction Axions are a hypothetical particle that emerged as a solution to the strong-CP problem in QCD physics [12, 17, 18]. The properties of axions make them a viable dark matter candidate. For axions to be a dark matter candidate, current cosmological constraints suggest an axion mass between 1–100 µeV [1–4, 6, 13]. The coupling between axions and photons is model dependent. Two generic models for the axion are the KSVZ [8, 14] and DFSZ model [5, 19] axions, with the DFSZ coupling to axion coupling to photons being 2.7 times larger for KSVZ axions. The ADMX experiment uses an axion haloscope to search for dark matter axions [15]. Axion haloscopes are resonant cavities placed in a strong magnetic field. Axion particles inside the magnetic field couple off the field, producing microwave photons.

N. Du () University of Washington, Seattle, WA, USA e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_3

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2 Hardware The ADMX experiment is a 136 L cylindrical copper cavity placed inside a 6.7 T solenoid magnet. The cavity is tuned with two copper-plated rods that extend the length of the cavity and can be positioned between near the center of the cavity and the walls of the cavity. During operation, the cavity frequency is tuned by actuating a room temperature stepper motor which is connected to the tuning rods by a cryogenic gearbox. When the cavity frequency is tuned to the same frequency as the photon produced from the axion, the expected power deposited into the cavity is Paxion

      ρa B 2 C  gγ 2 V = 1.9 ∗ 10 W 136l 6.8T 0.4 0.97 0.45 GeV cm−3    f Q × , (1) 650 MHz 5000 −22



where V is the cavity volume, B is the magnetic field, C is the form factor which is defined as the amount of overlap between the electric field of the cavity mode and the external magnetic field, gγ is the model-dependent coupling to the photon, ρa is the axion dark matter density around the Earth, f is the frequency of the photons from the conversion, and Q is the loaded quality factor of the cavity. In the case of the ADMX experiment, the TM010 mode has the most alignment with the field produced by the solenoid magnet, so it was the mode used for searching for axions. An antenna that is coupled to the cavity extracts the power from the cavity and transfers it into a radio frequency amplifier chain. During the run in 2017, the amplifier chain contained a tunable Michelson SQUID amplifier (MSA) [10, 11] followed by a broadband cryogenic heterostructure field-effect transistor (HFET) amplifier. The amplified signal is transmitted to a receiver where it is mixed down with a local oscillator to a 10.7 MHz intermediate frequency for further processing and analysis. To reduce the system noise temperature within the experiment, the cavity and Michelson SQUID amplifier are cooled with a dilution refrigerator system. ADMX uses a wet dilution refrigerator developed by Janis Cryogenics which has a cooling power of 800 µW at 100 mK. During the 2017 run, ruthenium oxide thermometers measured that the dilution refrigerator cooled the cavity down to 150 mK and the MSA to 300 mK. Following the run, improvements were made to the capacity of the 1 K pot and the heat sinking of the experiment which improved the thermal performance. New measurements made during the most recent run indicate the cavity is now at 150 mK and the amplifier package was 230 mK.

ADMX Results

19

3 Data Taking During standard data taking operations, small steps are made in the tuning rods to adjust the resonant frequency of the cavity by less than one cavity line-width. Afterwards, the cavity resonant frequency and Q were measured with an S21 transmission measurement and then a power spectrum was taken by digitizing the power from the cavity in a 25 kHz span about the cavity resonant frequency. For every several scans, an S11 reflection measurement was taken to check the antenna coupling to the cavity mode. If the coupling was too low, the antenna position was adjusted accordingly. The power in the receiver was calibrated by comparing the power on and off the cavity resonance with thermometers within the experiment. Off resonance, the power came from Johnson noise from an attenuator being reflected off the cavity. On resonance, the power came from Johnson noise from the cavity itself. Using thermometry, the attenuator was measured to be 300 mK and the cavity was measured to be 150 mK. By fitting the power spectrum with a model of the radio frequency chain and using the thermometry with the experiment, we were able to measure the total system noise temperature. An example of one such scan used for calibrating the system noise power is shown in Fig. 1.

Fig. 1 A power spectrum used to calibrate the system noise in the experiment. The on resonance power comes from the receiver and a 150 mK cavity. The off resonance power comes from the receiver and a 300 mK attenuator. Comparing the off resonance and on resonance power allows for a calibration of the receiver noise temperature. The asymmetry is the result of interactions between various components within the experiment [7]

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640

650

660

Frequency (MHz) 670

680

690

700

|gaγ γ | (10

-16

16

DFSZ

14

log |ga γ γ / g

GeV-1), 100% Dark Matter

|

18

12

2 1

KSVZ

0 −1

DFSZ

This work (N-Body) 2

3

HAYSTAC

RBF

UF

ADMX 2013

4

5

6 7 8 Axion Mass ( μeV)

9

10

20

10

KSVZ 8 6

Axion Lineshape 4

Maxwellian

DFSZ

N-Body

2 0

2.65

2.7

2.75 Axion Mass (μeV)

2.8

2.85

2.9

Fig. 2 A 90% upper confidence limit on the axion–photon coupling gaγ γ and the axion mass. The red line is the limit set using a Maxwell–Boltzmann lineshape for axion lineshape derived from the isothermal halo model[16] , and the blue line is the limit set using the axion lineshape from N -body simulations [9]. We were unable to set limits between 660.16 and 660.27 MHz due to external radio frequency interference, as indicated by the gray bar. The results of this search in comparison with others are shown in the inset [7]

4 Results In 2017, ADMX searched for axions in the frequency range 645–680 MHz and found no signals consistent with axions. Two persistent signals were observed. However, a measurement of the radio interference at the experimental site determined that the signals were due to external radio interference. In the absence of axion signals in the explored frequency range, a 90% upper confidence limit was placed on axion–photon couplings using data for Maxwellian [16] and N-body [9] astrophysical models shown in Fig. 2. The upper limit excluded DFSZ axions for models that make up 100% of dark matter. This marks the first haloscope experiment that has achieved sensitivity to the well-motivated DFSZ model for axion coupling.

5 Conclusions In 2017, ADMX became the first haloscope experiment to achieve sensitivity to DFSZ axions. Since then, ADMX has begun another run in 2018 searching for axions over a different mass range. The upgrades made to the refrigeration and an increased magnetic field have increased the sensitivity of the experiment to dark matter axions. As seen in Fig. 3, the current ongoing run has already collected significantly more axion search data.

ADMX Results

21

Fig. 3 A comparison of the data collected by ADMX during our 2017 run and our ongoing 2018 run. The figure of merit is taken to be the square of the signal to noise expected from a DFSZ axion multiplied by the total band explored. During our current run, ADMX has already collected five times more data for axion searches

ADMX anticipates searching for axion masses up to 40 ueV (10 GHz) with multicavity resonators. ADMX is in a position to make a potential axion discovery at any time during the data taking process.

References 1. G. Ballesteros, J. Redondo, A. Ringwald, C. Tamarit, Unifying inflation with the axion, dark matter, baryogenesis, and the seesaw mechanism. Phys. Rev. Lett. 118, 071802 (2017) 2. E. Berkowitz, M.I. Buchoff, E. Rinaldi, Lattice QCD input for axion cosmology. Phys. Rev. D 92, 034507 (2015) 3. C. Bonati, M. D’Elia, M. Mariti, G. Martinelli, M. Mesiti, F. Negro, F. Sanfilippo, G. Villadoro, Axion phenomenology and θ-dependence from Nf = 2 + 1 lattice QCD. J. High Energy Phys. 2016(3), 155 (2016) 4. S. Borsanyi, Z. Fodor, J. Guenther, K.-H. Kampert, S.D. Katz, T. Kawanai, T.G. Kovacs, S.W. Mages, A. Pasztor, F. Pittler, J. Redondo, A. Ringwald, K.K. Szabo, Calculation of the axion mass based on high-temperature lattice quantum chromodynamics. Nature 539(7627), 69–71 (2016) 5. M. Dine, W. Fischler, M. Srednicki, A simple solution to the strong CP problem with a harmless axion. Phys. Lett. B104, 199 (1981) 6. M. Dine, P. Draper, L. Stephenson-Haskins, D. Xu, Axions, instantons, and the lattice. Phys. Rev. D 96, 095001 (2017)

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7. N. Du, N. Force, R. Khatiwada, E. Lentz, R. Ottens, L.J. Rosenberg, G. Rybka, G. Carosi, N. Woollett, D. Bowring, A.S. Chou, A. Sonnenschein, W. Wester, C. Boutan, N.S. Oblath, R. Bradley, E.J. Daw, A.V. Dixit, J. Clarke, S.R. O’Kelley, N. Crisosto, J.R. Gleason, S. Jois, P. Sikivie, I. Stern, N.S. Sullivan, D.B Tanner, G.C. Hilton, Search for invisible axion dark matter with the axion dark matter experiment. Phys. Rev. Lett. 120, 151301 (2018) 8. J.E. Kim, Weak interaction singlet and strong CP invariance. Phys. Rev. Lett. 43, 103 (1979) 9. E.W. Lentz, T.R. Quinn, L.J. Rosenberg, M.J. Tremmel, A new signal model for axion cavity searches from N -body simulations. Astrophys. J. 845(2), 121 (2017) 10. M. Mück, M.-O. André, J. Clarke, J. Gail, C. Heiden, Radio-frequency amplifier based on a niobium DC superconducting quantum interference device with microstrip input coupling. Appl. Phys. Lett. 72(22), 2885–2887 (1998) 11. M. Mück, M.-O. André, J. Clarke, J. Gail, C. Heiden, Microstrip superconducting quantum interference device radio-frequency amplifier: tuning and cascading. Appl. Phys. Lett. 75(22), 3545–3547 (1999) 12. R.D. Peccei, H.R. Quinn, CP Conservation in the presence of instantons. Phys. Rev. Lett. 38, 1440–1443 (1977) 13. P. Petreczky, H.-P. Schadler, S. Sharma, The topological susceptibility in finite temperature QCD and axion cosmology. Phys. Lett. B 762, 498–505 (2016) 14. M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Can confinement ensure natural CP invariance of strong interactions? Nucl. Phys. B166, 493 (1980) 15. P. Sikivie, Experimental tests of the invisible axion. Phys. Rev. Lett. 51, 1415–1417 (1983) 16. M.S. Turner, Periodic signatures for the detection of cosmic axions. Phys. Rev. D 42, 3572– 3575 (1990) 17. S. Weinberg, A new light Boson? Phys. Rev. Lett. 40, 223–226 (1978) 18. F. Wilczek, Problem of strong p and t invariance in the presence of instantons. Phys. Rev. Lett. 40, 279–282 (1978) 19. A.R. Zhitnitsky, On possible suppression of the axion Hadron interactions. Sov. J. Nucl. Phys. 31, 260 (1980) (in Russian)

The Microstrip SQUID Amplifier in ADMX Sean R. O’Kelley, Gene Hilton, and John Clarke

Abstract The Axion Dark Matter eXperiment (ADMX) relies equally on a low physical temperature T and low amplifier noise temperature TN to achieve a high Signal-to-Noise Ratio (SNR). The low noise amplifier is a Microstrip SQUID Amplifier (MSA) with TN of 200 mK, which compares favorably with the best available cryogenic transistor-based amplifiers with a TN of 1.5 K, allowing a search rate up to 56 times faster, depending on T . Here we present the operating principles of the MSA and some practical considerations of MSA design in a way we hope is accessible to those not otherwise familiar with superconducting electronics or amplifier design. Keywords Dark matter · Axion · Microwave · SQUID · MSA

1 The Need for a Unique Amplifier in ADMX Although “everyone knows” that the mass-energy of the universe is only about 4% ordinary matter (particles of the Standard Model, mostly existing as hydrogen), 21% dark matter, and 75% dark energy, no one knows what dark matter or dark energy is, because all of our evidence for them, though compelling, is indirect. Any attempt at direct detection of dark matter must presuppose some of its properties to guide the design of the detector. Many direct detection experiments assume a new kind of

S. R. O’Kelley () University of California, Berkeley, CA, USA Lawrence Livermore National Lab, Livermore, CA, USA e-mail: [email protected] G. Hilton NIST, Boulder, CO, USA J. Clarke University of California, Berkeley, CA, USA © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_4

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Fig. 1 The axion detection concept and example signal. (Left) About 1028 galactic halo axions pass through the ADMX detector each second. Due to the strong magnetic field and cavity Q an occasional axion converts all of its mass-energy into a single real photon. (Right) The axiogenic photons appear as a narrowband signal with the width set by the velocity distribution of galactic axions. The noise floor is set by the sum of the physical temperature and the electronics noise temperature

massive (eV to TeV range) particle, often based on supersymmetric extensions of the standard model, motivating detectors based on scintillation, ionization, phonon excitations, or other interactions between a dark matter particle and a condensed matter substrate. In contrast, ADMX assumes dark matter is composed of axions and uses a detector more akin to a radio receiver. The axion was first imagined as a solution to the extremely high symmetry observed in the strong force [1–3] and also happens to be an excellent dark matter candidate. Axions seem to have all the key properties to match our models of dark matter, namely they would be generated in great quantities in the big bang (enough to account for all the dark matter), they would be cold (non-relativistic), and they would interact with light and ordinary matter only very weakly. Because axions interact very weakly with ordinary matter we must take the best advantage we can of the small but extant interactions they have with ordinary matter. In a strong magnetic field, an axion may convert all of its mass-energy into an ordinary photon. Inside a resonant microwave cavity, the conversion rate will be further enhanced by a factor of the cavity Q. By detecting those few extra photons and reducing the background counts by minimizing the physical temperature and instrumentation noise, a direct detection of axions is possible. This concept is illustrated in Fig. 1. We do not know the axion mass (and by extension the photon frequency) a priori, so the cavity and detection electronics must be tunable over a range of frequencies. Figure 2 shows the theoretical search space and sensitivities of various axion searches. The rising diagonal lines labeled “KSVZ” [4, 5] and “DFSZ” [6, 7] indicate the two canonical axion-photon coupling strengths, which despite assuming very different axion-to-photon coupling models differ in calculated

The MSA in ADMX

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Fig. 2 The axion search space. Between a lower bound of about 10−6 eV (240 MHz) set by cosmological considerations and an upper bound of about 10−3 eV (240 GHz) set by observations of supernova 1987A, the axion mass search space spans three orders of magnitude. “Light Shining through Walls” (LSW) experiments such as the “Any Light Particle Search” (ALPS) (top orange region) depend on both generation and re-conversion of axions in the lab, and thus are sensitive only to particles with a strong interaction. Helioscopes such as the CERN Axion Solar Telescope (CAST) rely on the sun to generate axions, and are sensitive to particles with a weaker interaction. Haloscopes like ADMX tune into the pre-existing cosmological axion background and to date have the highest sensitivity. Only ADMX has approached the theoretically important KSVZ and DFSZ coupling strength limits. Adapted from [8]

coupling strength by less than a factor of three. Note that between the limit at low mass set by cosmological considerations (Too Much DM) and the limit at high mass set by observations from the 1987 supernova event (SN1987A), the search space spans three orders of magnitude in axion mass! Given the cavity size, magnetic field strength, and other experimental parameters of ADMX, the time to scan any particular frequency range to the DFSZ limit as a function of temperature is [9]  2 τ (f1 , f2 ) ≈ 4 × 1017 × (1/f1 − 1/f2 ) × Tsys /1K .

(1)

Note that the scan time scales as the square of the total effective system noise temperature Tsys . This effective temperature is composed of the physical temperature of the resonant cavity and the noise temperature of each amplifier in the detection chain according to

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Tsys = Tphys + TN = Tphys + T1 +

T2 T3 T4 + + + ···, G1 G1 G2 G1 G2 G3

(2)

where Tn and Gn are the noise temperature and power gain of the nth amplifier. With a typical gain of 20 dB or more per stage, the most important terms are the physical temperature and TN of the first amplifier.1 To illustrate the importance of Tsys in Eq. (1), suppose a physical temperature Tphys attainable by a pumped 4 He system and a noise temperature TN typical of the best transistor-based amplifiers (1.5 K and 1.7 K, respectively). The time to scan from 240 to 480 MHz at DFSZ sensitivity is about 270 years. By contrast, given a physical temperature typical of a 3 He/4 He dilution unit and the lowest noise temperature demonstrated by an MSA (50 mK and 50 mK), the time to scan the same frequency range is just 97 days. Though the scan rate of Eq. (1) could in principle also be improved by increasing the signal strength (for example, by increasing the magnetic field), reducing both the physical temperature and amplifier noise temperature is always of paramount importance for any haloscope axion search.

2 Principles of SQUIDs as Microwave Amplifiers In a sentence, the Microstrip SQUID Amplifier (MSA) is a Superconducting QUantum Interference Device, which is a remarkable flux-to-voltage transducer, with an integrated resonant microwave microstrip that couples the magnetic component of the RF input signal to the SQUID for amplification. We will expand on this sentence presently.

2.1 Superconductivity Superconductivity is a key property of the MSA. In a superconducting metal, the spin-1/2 conduction electrons (fermions) bind into spin-0 Cooper pairs (bosons), almost all of which condense into the ground state at sufficiently low temperatures to form a Bose–Einstein condensate. This gives us a macroscopic coherent quantum state of charge-carriers, described by ψ = |ψ(r)| eiθ(r)

(3)

with ψ normalized to the charge carrier density

1 Of

course, if the gain of the first amplifier G1 is not very large, the noise temperature of the later amplifiers can make a significant contribution to Tsys as well.

The MSA in ADMX

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ψ ∗ ψ = ns (r).

(4)

The charge carrier density is nearly constant throughout the device, and θ can remain coherent over the indefinite length of a superconductor. This long-range quantum coherence of the phase is the “magic sauce” that makes SQUIDs and MSAs possible. One important consequence of this macroscopic quantum coherence is flux quantization. Given a coherent quantum state, it is clear that the value of θ can have no discontinuities in a superconductor. In a ring of superconducting material, θ is allowed to have a gradient (∇θ = 0) only if it advances by an integer multiple of 2π in one trip around the ring. The current I is proportional to ∇θ , the magnetic field B is a function of the current and ring radius R (B ∝ I /R), and the flux Φ through the ring is a function of the field and ring area (Φ ∝ B ×R 2 ). Consequently, the flux threading a superconducting ring of any geometry is constrained to integer multiples of the flux quanta: Φ = nΦ0 , with Φ0 ≡ h/2e,

(5)

where h is Planck’s constant and e is the elementary charge. A second consequence of the macroscopic quantum coherent state is Josephson tunneling. At the edge of any superconductor, the amplitude of the wave function ψ does not fall to zero in zero distance, but has an exponential tail that exists outside the “classical edge” of the sample. If two superconductors are separated by an insulating gap that is not much wider than the length of this “evanescent tail,” a supercurrent can flow through the barrier with zero dissipation whereas a classical current would be forbidden. The current through the barrier and voltage between the two superconductors are given by the Josephson relations I = I0 sin δ ˙ 0 /2π , V = δΦ

(6)

where δ ≡ θ1 − θ2 is the difference in phase between the two superconductors, and I0 is a maximum dissipationless critical current that depends on specific properties of the junction (thickness, area, etc.).

2.2 The RCSJ Model The Josephson junction is a key element of the SQUID. The electrical model we use is the “Resistively Capacitively Shunted Junction” or RCSJ. Starting with the Josephson relations of Eq. (6), we recognize that the gap between conductors naturally forms a capacitor, and there may also be an ohmic conduction channel

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Fig. 3 RCSJ model and effective washboard potential. (Left) J represents the “pure” Josephson junction, R represents the parallel sum of any conduction channels in the junction and external added resistance, and C represents the sum of the junction capacitance and any explicit added capacitance. (Right) The ratio I /I0 determines the tilt of the effective potential function U (δ). For I < I0 there exist potential wells that can bind the phase particle (δ˙ = V = 0). For I > I0 , the phase particle will roll downhill (δ˙ and V > 0)

between the superconductors, either due to flaws in the insulator or explicitly added parallel resistance. The resultant equivalent circuit is shown in Fig. 3. The dynamics of the RCSJ can be understood by analogy to a particle on a tilted washboard potential where the position coordinate is the phase across the junction, the velocity is the voltage, and the tilt is set by the ratio I /I0 . For small applied dc current (low tilt) there will be no dc voltage (no net rightward velocity), but if the current I is increased to greater than I0 (washboard tilted to a slope >1) a voltage will develop (particle rolls to the right). If the current is then decreased to I < I0 (the potential again has “uphill” portions) and if there is sufficient damping (low enough R) the particle will come to rest. However if the damping is relatively small, the particle will have sufficient momentum (charge on the capacitor) to roll uphill, and regain energy lost to damping on the next downhill fall, resulting in a persistent rightward velocity (voltage) despite the existence of stable equilibrium points (I < I0 ). The result of insufficient damping is hysteresis in the I –V characteristic of the RCSJ. The need to eliminate hysteretic operation motivates the addition of parallel resistors to the junctions in practical SQUIDs and MSAs.

2.3 The DC SQUID: A Quantum Flux-to-Voltage Transducer The dc SQUID is a superconducting loop interrupted by two Josephson junctions, as shown in Fig. 4. Any magnetic flux (Φa ) threading the loop modifies the gradient ∇θ in each branch of the loop in an opposite sense, altering the relative phase difference across the two junctions. Interference between the re-combining currents results in a device

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Fig. 4 Schematic of a DC SQUID and characteristic I –V . (Left) A SQUID consists of a superconducting loop interrupted by two RSCJs. The loop has inductance L, and flux Φa is applied by an external coil. (Right) The applied flux modulates the critical current, illustrated by the I – V characteristics of the SQUID plotted for Φa /Φ0 = 0.2525 (solid blue) and Φa /Φ0 = 0.2475 (dashed red). If the bias current is held constant at an appropriate level (dashed black), a small change in applied flux will cause a large swing in output voltage

that has a critical current and an I –V trace much like a single junction, but now with the critical current Ic modulated by the external flux. The critical current Ic varies periodically from a maximum of 2I0 at Φa = nΦ0 to a minimum at Φa = (n + 1/2)Φ0 , where n is an integer. The minimum critical current depends on the inductance L of the SQUID, but is typically IcMIN ≈ IcMAX /2 for an optimized device. The maximum sensitivity to applied flux occurs when Φa ≈ (n ± 1/4)Φ0 . To use a SQUID as a high-sensitivity flux-to-voltage transducer, a constant flux bias of about Φ0 /4 and a constant bias current Ib just a little larger than Ic are applied so that a voltage appears across the terminals. A small flux variation applied to the SQUID modulates Ic and results in a large swing in the output voltage. Figure 4 (right) shows a SQUID I –V (in reduced units) for Φa = (0.25 ± .0025)Φ0 and the resulting voltage swing, given a good choice of Ib . A practical device has a flux-to-voltage sensitivity of about 600 µV/Φ0 . A dc SQUID is usually fabricated from a thin film of superconducting metal, often Nb, patterned as a square washer interrupted by a slit running between the inner and outer edges. The slit is bridged by a second superconducting layer, the “counter electrode,” connecting to each side of the slit via a Josephson junction. The external coil is integral, deposited directly on the SQUID over a thin insulating layer, often SiOx . DC SQUIDs are among the most sensitive flux transducers available to science. They operate down to arbitrarily low frequencies and up to about 200 MHz, where parasitic capacitance between the input coil and washer tends to degrade performance. The intrinsic dynamics of the junctions do not impose a limit to the frequency of operation until 10 GHz or so. Extensive analysis of dc SQUID operation can be found in [10].

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Fig. 5 Fixed frequency MSA with low TN and frequency relevant to ADMX. Lowest TN is 45 mK at a frequency 612 MHz and gain of 18 dB. Dashed-dotted red line represents the quantum limited noise temperature TQ = hf/kB . Reproduced from [11]

2.4 The DC SQUID as an RF Amplifier The key to using a dc SQUID as an rf amplifier is to design the input coil as an rf element, rather than a dc current path. To this end, the input coil over the broad SQUID washer can be seen as a microstrip waveguide, and by coupling a signal to one end of the input coil and leaving the opposite end floating (or grounded), the input coil becomes a λ/2 (or λ/4) resonator. Figure 5 (left) shows an example of rf coupling to a dc SQUID. The frequency of such a device is fixed by the geometry of the input coil and SQUID, and very low noise operation had been demonstrated for a device with a fixed resonance around 600 MHz, as shown in Fig. 5 (right). This is a very promising amplifier concept for ADMX, but is unfortunately rather narrowband—a practical device is required to work over a wide range of frequencies, as ADMX must explore a wide range of possible axion mass.

2.5 Varactor Tuning an MSA The solution to the fixed frequency problem is conceptually simple, and shown in Fig. 6 (left). By adding a varactor diode (voltage-controlled capacitance, shown in red) to the end of the input coil, the microstrip standing wave can be altered from a λ/2 to a λ/4 resonance, and the MSA thus tuned by up to a factor of two. Figure 6 (right) shows the gain profile of an MSA tuned with a single varactor.

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Fig. 6 Single-varactor tuning of an MSA

Fig. 7 MSA on its electronics support board. (Left) The signal enters from the left and travels to the MSA through a DC-blocking (rf-short) capacitor and varactor used for adjusting the input coupling strength. The amplified signal is transmitted from the MSA to the right-hand microstrip. The bias current is applied through this same output strip via an rf-blocking inductor. Various resistors (blue) and capacitors (beige) provide filtering of the DC biasing lines. (Right) The MSA chip has four bonding pads. Left pads connect the input coil to the adjustment varactors. Right pads are the SQUID terminals—one connects to ground and the other to the bias current and rf output

3 Practical MSA Design and Performance A tunable MSA has been developed and used in ADMX. Here we will present some of the practical considerations that went into its realization. Figure 7 shows an MSA installed on its electronics support board. This board consists of a coplanar microstrip waveguide for the signal input and output, with SMA connectors used for coupling to external cables. Various RC networks are used to filter the dc bias and monitoring lines, and for the voltage biasing of the tuning varactors. The varactors are used in two places: between the input line and MSA input coil to achieve a variable coupling strength, and between the far end of the MSA input coil and ground to achieve frequency tuning.

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Fig. 8 DC performance landscape of the SQUID. (Left) Map of the SQUID voltage (color) as a function of Ibias (Y) and Φa (X). The black regions indicate the superconducting zero-voltage state (Ibias < Ic ). Note that the critical current is at a minimum near Φa = Φ0 /2 (center of plot) and maximum near Φa = 0 or 1 (left and right edges). (Right) Partial derivative with respect to Φa showing the flux sensitivity ∂V /∂Φa . Black regions indicate zero flux sensitivity. Maximum gain as an rf amplifier is at bias values indicated by the red or violet areas, with a 180◦ phase difference between the two due to the sign of ∂V /∂Φa

3.1 DC Performance Landscape The performance of the MSA is determined by the dc characteristics of the SQUID. The current bias Ib must be chosen to be slightly above the critical current Ic , and the flux bias Φa must be chosen such that Ic is strongly modulated by flux perturbations. Figure 8 shows the SQUID voltage and flux sensitivity ∂V /∂Φa as a function of the current and flux biases. A high flux sensitivity is necessary to achieve high gain as an rf amplifier.

3.2 RF Performance Landscape and Feedback We can measure the rf gain of the MSA as a function of the SQUID bias parameters. Figure 9 (left) shows a color plot of rf gain as a function of bias parameters Ib and Φa . Note that although the gain has the same general outline as Fig. 8 (right), there is a marked asymmetry between the “+” and “−” lobes. The gain at the peak of the “−” lobe is about 18 dB, while the highest gain in the “+” lobe is only about 2 dB. This asymmetry arises from feedback between the rf output and input coil. Note in Figs. 6 (left) and 7 (right) that the SQUID washer (which acts as a nominal ground plane to the integrated microstrip input coil) is not connected to ground, but is rather the active output of the SQUID. The active voltage on the “ground plane” of the microstrip results in a capacitive coupling to the input coil. Figure 10 shows a diagram of a standing wave on the input coil over the active SQUID washer. In the case of a λ/2 mode (both ends weakly coupled), the anti-nodes of the voltage

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Fig. 9 RF gain of an MSA as a function of flux and current bias parameters. (Left) The colormap xaxis spans about one Φ0 of applied flux. Gain is zero at Φa = (0, 12 , 1)×Φ0 (where ∂Ic /∂Φa = 0), and maximum at intermediate flux values with Ib ≈ 1.1Ic . Note the lack of symmetry relative to Fig. 8. This arises from feedback effects—the sign of the feedback is reversed between the left and right lobes. (Right) Gain spectrum of the MSA at the bias parameters marked by the cursor on the left colormap

Fig. 10 Standing wave modes on the MSA microstrip. The spiral input coil microstrip is here imagined “straightened out” for clarity. The λ/2 resonant mode has equal and opposite voltage anti-nodes, so capacitive coupling to a signal on the SQUID washer is negligible. The λ/4 resonant mode has a non-cancelling voltage profile, so capacitive coupling to the washer is maximized, and capacitive feedback (whether positive or negative) is strong

component of the wave have equal and opposite signs, so that capacitive coupling to the washer signal is zero. In the case of a λ/4 mode (input weakly coupled, end grounded), the signal on the washer couples strongly to the voltage component of the input coil standing wave. Depending on the sign of ∂V /∂Φa (selected by the flux bias Φa ), this may result in positive or negative feedback. Thus tuning the MSA by varying the coupling to ground at the end of the microstrip with a single varactor both tunes the resonant frequency and alters the degree or even the sign of feedback. Controlling the sign and degree of feedback is necessary for maintaining optimal low TN operation, and this can be achieved by the addition of a second varactor diode.

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Fig. 11 Schematic of an MSA with two tuning varactors. By independently adjusting the coupling of the resonant microstrip both to the 50- input line and to the end termination to ground, both the wavelength and phase of the standing wave can be adjusted, maintaining optimal feedback while the frequency is tuned as needed

3.3 Two-Varactor Tuning Two-varactor tuning allows for independent control of the MSA resonant frequency and the degree of feedback, so high gain and low TN can be maintained across a wide range of frequencies. Figure 11 shows a simplified schematic of the twovaractor tuning scheme. The first varactor (controlled by “Vin ”) sets the input coupling strength, while the second varactor (controlled by “Vend ”) sets the coupling to ground. In this configuration, neither varactor is purely a “frequency tuning” or “feedback tuning” component, but rather both contribute to both parameters. Joint operation of the varactors allows us to span the space of frequency and optimal feedback. Figure 12 shows the frequency (left) and SNR (center) of an MSA as a function of the two-varactor voltages. Note that there is a “ridge” of highest SNR in Fig. 12 (center), and that the voltage pairs (Vin , Vend ) that lie along this ridge of best performance also span the space of available frequencies. From this performance map, it is simple to create an empirical function that takes desired operating frequency as input and generates two-varactor voltages as output. This is the control scheme developed at Berkeley to date, and has resulted in excellent TN performance.

4 Proven MSA Performance The lowest TN measured in a practical MSA for use in ADMX is about 200 mK (with a gain of about 20 dB). This TN measurement was made with a hot/cold load. Rather than use a simple two-temperature measurement, we stepped the load

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35

Fig. 12 Frequency and noise map of an MSA as a function of two-varactor tuning. (Left) Colormap of MSA frequency vs tuning voltages applied to varactors at the input (y-axis) and termination (x-axis) of the microstrip. Frequency ranges from 550 MHz (black) to 850 MHz (red). (Center) Colormap of SNR over the same domain of tuning voltages. SNR is a convenient empirical analog to TN . Note that the “ridge” of best SNR spans the space of frequencies. (Right) System noise temperature Tsys of an MSA optimally tuned to about 600 MHz, measured by the “gold standard” hot/cold load method. Tsys (red) is about 300 mK. The following HEMT amplifier contributes 100 mK, with the remaining 200 mK attributed to the MSA

temperature from about 100 mK to 1 K in four steps. Observing a straight line in total noise power vs load temperature assures us that the fitted value of TN is reliable. Figure 12 (right) shows results of the heated load test for an optimally tuned MSA. The hot/cold load test gives a total system noise Tsys of 300 mK. In this test, the noise contribution from the post-MSA electronics (as also measured by a hot/cold load in another test) is about 100 mK, thus the data in Fig. 12 indicates an MSA TN of about 200 mK. This low noise temperature amplifier installed in ADMX has enabled unprecedented sensitivity, unlocking the search for weakly coupled axions and tremendously speeding the search rate. In April 2018 we reported an exclusion of axions with a mass from about 2.66–2.81 µeV at a coupling equal to or greater than the DFSZ model [12]. This exclusion would have taken over 10 years using pre-MSA technology, but was obtained in a few months of operation with an MSA. Further DFSZ-sensitivity exclusions spanning a mass range from 2.81 to 3.31 µeV were published in 2020 [13], and ADMX currently continues the search to mass ranges beyond this.

References 1. R.D. Peccei, H.R. Quinn, CP conservation in the presence of pseudoparticles. Phys. Rev. Lett. 38, 1440–1443 (1977). https://doi.org/10.1103/PhysRevLett.38.1440 2. S. Weinberg, A new light boson? Phys. Rev. Lett. 40, 223–226 (1978). https://doi.org/10.1103/ PhysRevLett.40.223 3. F. Wilczek, Problem of strong P and T invariance in the presence of instantons. Phys. Rev. Lett. 40, 279–282 (1978). https://doi.org/10.1103/PhysRevLett.40.279 4. J.E. Kim, Weak-interaction singlet and strong CP invariance. Phys. Rev. Lett. 43, 103–107 (1979). https://doi.org/10.1103/PhysRevLett.43.103

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5. M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Can confinement ensure natural CP invariance of strong interactions? Nucl. Phys. B 166(3), 493–506 (1980). https://doi.org/10.1016/05503213(80)90209-6 6. M. Dine, W. Fischler, M. Srednicki, A simple solution to the strong CP problem with a harmless axion. Phys. Lett. B 104(3), 199–202 (1981). https://doi.org/10.1016/0370-2693(81)90590-6 7. A.R. Zhitnitsky, On possible suppression of the axion hadron interactions (In Russian). Sov. J. Nucl. Phys. 31, 260 (1980) 8. K.A. Olive, Review of particle physics. Chin. Phys. C 40(10), 100001 (2016). https://doi.org/ 10.1088/1674-1137/40/10/100001 9. S.J. Asztalos et al., SQUID-based microwave cavity search for dark-matter axions. Phys. Rev. Lett. 104, 041301 (2010). https://doi.org/10.1103/PhysRevLett.104.041301 10. J. Clarke, A.I. Braginski (eds.), The SQUID Handbook, vol. 1 (Wiley, Weinheim, 2004) 11. D. Kinion, J. Clarke, Superconducting quantum interference device as a near-quantum-limited amplifier for the axion dark-matter experiment. Appl. Phys. Lett. 98(20), 202503 (2011). https://doi.org/10.1063/1.3583380 12. N. Du et al., Search for invisible axion dark matter with the axion dark matter experiment. Phys. Rev. Lett. 120, 151301 (2018). https://doi.org/10.1103/PhysRevLett.120.151301 13. T. Braine et al., Extended search for the invisible axion with the axion dark matter experiment. Phys. Rev. Lett. 124, 101303 (2020). https://doi.org/10.1103/PhysRevLett.124.101303

The ORGAN Experiment Ben T. McAllister and Michael E. Tobar

Abstract In the push towards higher frequencies for axion dark matter haloscopes, new cavity designs are required to overcome a host of technical considerations associated with the standard TM010 -like modes typically employed in such searches. Through careful and novel design (such as implementation of dielectric materials) higher order modes present solutions to some of these issues. We discuss a few schemes for implementing higher order modes in haloscope searches, with a particular focus on cavity design for the ORGAN Experiment. Keywords Axion · Dark matter · Haloscope · Cavity design · Dielectric resonators

1 Introduction Many axion and axion-like particle searches have been conducted to date, and so far no evidence for axions, either as dark matter or as an exotic particle, has been found. Part of the issue is that the axion mass and axion–photon coupling strength, the two most commonly probed axion parameters, are largely unconstrained by theory and observation, leaving a large parameter space to search. As such, there is a high value placed on experiments with the capability to probe regions of this wide parameter space that are untested or difficult to access with existing techniques. An increasing number of searches for both dark matter QCD axions and axionlike particles are focusing on the so-called high mass axion regime (see ORGAN [1], HAYSTAC [2], CULTASK [3], MADMAX [4], ORPHEUS [5], QUAX [6]), above the range currently probed or projected to be probed by ADMX. This is due partially to the fact that ADMX has the capability and plans to cover the lower axion mass

B. T. McAllister () · M. E. Tobar ARC Centre of Excellence for Engineered Quantum Systems, Department of Physics, The University of Western Australia, Crawley, WA, Australia e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_5

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ranges [7], and partially due to a number of theoretical [8], computational [9], and cosmological observations [10], which lend this mass range increased interest. Despite this, a number of technical factors conspire to increase the difficulty of performing common axion detection experiments in this high mass range. This work will outline some efforts to improve the range and sensitivity of such high mass axion experiments (in particular the ORGAN Experiment), with a focus on resonator designs for high mass axion haloscopes utilizing higher order modes than the typical TM010 -like modes in cylindrical resonators.

2 ORGAN The ORGAN Experiment is a high mass (>50 µeV) axion haloscope hosted at the University of Western Australia, as a part of the ARC Centre of Excellence for Engineered Quantum Systems. The experiment has completed its pathfinding run and expects to begin long-term data collection in 2020. Figure 1 shows the pathfinder limits, as well as various projected future sensitivity limits.

Axion- photon coupling, gaγγ (GeV -1)

10- 10

CAST

ES DIDAT M CAN ALPs D

10- 11

10- 12 ORGAN 14 T

10- 13

RBF UF

HAYSTAC

10- 14 KSVZ DFSZ

ADMX

10- 15 QCD

A

and DM C xions

idate

8T Noise AN 2 ow Q ORG T+L Noise 4 1 Q w AN + Lo ORG T 8 AN 2 ORG

s

10-5

10- 4 Axion Mass (eV)

Fig. 1 The axion–photon coupling parameter space. Axion mass is on the horizontal axis, whilst the strength of the axion–photon coupling is on the vertical axis. The axion cold dark matter model band is outlined and highlighted in yellow, with the popular KSVZ (gold, dashed) and DFSZ (blue, dashed) models overlaid. The region in which ALPs constitute promising dark matter candidates is indicated. Exclusion limits and future projected limits from past experiments such as CAST (orange) ADMX (dark green, solid and dashed), RBF (dark blue), UF (blue-grey), and HAYSTAC (aqua) are included. The current and projected exclusion limits for ORGAN (discussed in detail in [1]) are indicated in various shades of light blue on the right-hand side

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In order to reach the desired high mass range, ORGAN must make use of novel resonant cavity designs, to overcome technical challenges associated with using ADMX-style TM010 -like modes in the millimetre-wave regime.

2.1 Haloscope Cavity Design When considering the design of a haloscope experiment, a critical parameter is the allowable rate of frequency scanning, df dt . This is due to the fact that there is a large parameter space to search for the axion, and an experiment must be able to efficiently cover a large region of this space. The frequency scanning rate is given by McAllister et al. [1] g4aγ γ B 4 C 2 V 2 ρa2 QL Qa df 1 ∝ , dt m2a (kB Tn )2 SNR2goal

(1)

Here B is the field strength of the external magnetic field, C is a mode dependent form factor of order 1, which represents the degree of overlap between the cavity mode electromagnetic field and the electromagnetic field induced due to axion photon conversion (it is an integral of the dot product of these two fields), V is the volume of the detecting cavity, QL is the loaded cavity quality factor (provided it is lower than the expected axion signal quality factor, Qa ∼ 106 ), and ρa is the local axion dark matter density. SNRgoal is the desired signal-to-noise ratio of the search, and Tn is the effective noise temperature of the first stage amplifier, with later amplifier contributions suppressed by the gain of this amplifier. This is the quantity that must be maximized in design of an experiment, for which C 2 V 2 G can be viewed as a figure of merit for resonator design as all other parameters either depend on the properties of the axion, the external magnetic field, or the first stage readout amplifier. G is the mode geometry factor which is directly proportional to the mode quality factor. From this expression we can see why axion searches become more challenging at higher frequencies. The volume, V scales inversely with frequency to the third power, surface resistances of conductors increase with frequency (lowering quality factors), the noise temperature, Tn of amplifiers increases at higher frequency, and the scan rate explicitly scales by m12 . a

3 Higher Order Mode Dielectric Resonators An obvious, naive solution to the problem of volume scaling would be to employ higher order modes than the typically employed TM010 -like mode, allowing for a larger cavity at a given frequency. However this does not provide any benefit, since the form factor, C scales down with higher order TM0n0 modes at the same rate that

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Fig. 2 Visualization of a finite element model of a practical realization of the dielectric ring cavity for a haloscope. The Ez field component is plotted. Bluer colours represent higher positive field values. As shown, the TM030 -like mode is contained in the resonator even when the ring is removed from the top of the cavity

V scales up. This is due to the fact that higher order modes have lower electric field uniformity due to an increased number of field variations. However, with careful design, this problem can be circumvented. When considering the expression for the haloscope form factor [11]  C=

V



Ez dV

2

r |E|2 dV

,

(2)

where E is the cavity mode electric field, and noting the presence of the relative dielectric constant, r , in the bottom integral, one can devise schemes where careful placement of dielectric materials (with high values of r ) inside higher order resonant modes suppresses out of phase electric field contributions to C. This boosts the form factor of these higher order modes and allows for haloscope resonances with larger volumes at higher frequencies [12]. We will now discuss a few schemes for implementing dielectrics in higher order modes of cylindrical resonators, in order to boost their haloscope form factors.

3.1 Dielectric Ring Resonators The ORGAN Experiment plans to utilize a dielectric ring resonator, as detailed in [12]. This resonator consists of a ring of sapphire embedded in a TM030 -like resonance, in order to suppress the out of phase field lobes. The ring is split in half, and the top half slides axially out of the cavity in order to tune the resonant frequency. The proposal for this kind of resonator [12] considered the theoretical implementation of a ring which exited the cavity without breaking the conducting boundary at the lid. Of course, in reality, gaps in the lid must be considered. Figure 2 shows a finite element simulation of a practical implementation of the dielectric ring resonator. In this case, the lid of the cavity is in two pieces, such

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Fig. 3 (a) Scan rate figure of merit and (b) visualization of a finite element model for the ORGAN “backup” design. In (b) the Ez field component is plotted. Redder colours represent higher positive field values. As shown in (a), the sensitivity only deviates by a factor of ∼2 over the tuning range, with the large dip around 0.008 m gap size due to a static mode interacting with the tunable TM030 like mode

that the top half of the ring can be removed from the cavity. The entire assembly is then modelled as though it is placed inside the conducting shielding of the future ORGAN Experiment. As shown, the mode can be contained within the resonant cavity, even with a sapphire ring protruding from the lid. A prototype cavity based on this design is currently undergoing testing at UWA. However, if this design proves untenable due to field leakage or other complications associated with a pseudo-open resonator, a “backup” design has been considered. In this case, the cavity is completely enclosed, the sapphire ring is still split in the middle, and the two sections are attached to the lid and base of the cavity. Then, the height of the cavity is tuned, such that the two pieces are separated from one another. This scheme presents two advantages: firstly, the structure is always symmetrical, which enhances sensitivity, and secondly, the structure is completely enclosed. Figure 3 shows a finite element model of this design, including a plot of the figure of merit for cavity design. The sensitivity of this backup design is lower than the “ideal” design (when compared with [12], but it is still an improvement over a traditionally tuned TM010 -like mode.

3.2 Dielectric Puck Resonators Another separate design was considered based on a TM021 -like mode, with a sapphire “puck” placed centrally in the bottom section of the cavity, to suppress part of the out of phase field contribution to the form factor. The key advantage of this resonant structure is that it is easily tunable, as it is a length dependent mode. As such, it can be tuned by simply adjusting the position of the lid of the cavity. This is in contrast with typical haloscope resonances which do not exhibit length dependence and require complex tuning mechanisms. The optimization of such a cavity is a subject of ongoing research, but Fig. 4 shows a finite element model of a prototype that was constructed at UWA, with the

Frequency (GHz)

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18.5 18 17.5 17 17

18

19

20

Cavity Height (mm) (a)

(b)

Fig. 4 (a) Predicted and measured resonance frequencies and (b) visualization of a finite element model for the dielectric “puck” TM021 -like resonator. In (b) the Ez field component is plotted. Redder colours represent higher positive field values. In (a) the blue dots represent the predicted frequencies, whilst the orange dots represent the measured frequencies in the prototype

predicted and measured mode frequencies presented. Although the discrepancies between prediction and experiment appear large for some cavity heights, they are of the order of a few percent of the cavity frequency. Research into this structure is ongoing.

4 Conclusion We present schemes for implementing higher order modes in haloscope searches, employing dielectric materials to boost axion sensitivity whilst still allowing larger volumes to be employed. In particular we focus on some practical considerations associated with a dielectric ring resonator, such as will be employed in the ORGAN Experiment, and a new design based on a dielectric puck resonator, which is a subject of ongoing research.

References 1. B.T. McAllister, G. Flower, E.N. Ivanov, M. Goryachev, J. Bourhill, M.E. Tobar, Phys. Dark Univ. 18, 67 (2017). https://doi.org/10.1016/j.dark.2017.09.010. [arXiv:1706.00209 [physics.ins-det]] 2. B.M. Brubaker, arXiv:1801.00835 [astro-ph.CO] 3. W. Chung, PoS CORFU 2015, 047 (2016). https://doi.org/10.22323/1.263.0047 4. A. Caldwell et al., Phys. Rev. Lett. 118(9), 091801 (2017). https://doi.org/10.1103/ PhysRevLett.118.091801. [arXiv:1611.05865 [physics.ins-det]] 5. G. Rybka, A. Wagner, A. Brill, K. Ramos, R. Percival, K. Patel, Phys. Rev.D 91(1), 011701 (2015). https://doi.org/10.1103/PhysRevD.91.011701. [arXiv:1403.3121 [physics.ins-det]] 6. N. Crescini et al., Eur. Phys. J. C 78(9), 703 (2018). Erratum: [Eur. Phys. J. C 78(9), 813 (2018)]. https://doi.org/10.1140/epjc/s10052-018-6262-6, https://doi.org/10.1140/epjc/ s10052-018-6163-8. [arXiv:1806.00310 [hep-ex]]

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7. N. Du et al., ADMX collaboration. Phys. Rev. Lett. 120(15), 151301 (2018). https://doi.org/ 10.1103/PhysRevLett.120.151301. [arXiv:1804.05750 [hep-ex]] 8. G. Ballesteros, J. Redondo, A. Ringwald, C. Tamarit, Phys. Rev. Lett. 118(7), 071802 (2017). https://doi.org/10.1103/PhysRevLett.118.071802. [arXiv:1608.05414 [hep-ph]] 9. S. Borsanyi et al., Nature 539(7627), 69 (2016). https://doi.org/10.1038/nature20115. [arXiv:1606.07494 [hep-lat]] 10. L.D. Duffy, K. van Bibber, New J. Phys. 11, 105008 (2009). https://doi.org/10.1088/13672630/11/10/105008. [arXiv:0904.3346 [hep-ph]] 11. B.T. McAllister, S.R. Parker, M.E. Tobar, Phys. Rev. Lett. 116(16), 161804 (2016). Erratum: [Phys. Rev. Lett. 117(15), 159901 (2016)]. https://doi.org/10.1103/PhysRevLett. 117.159901, https://doi.org/10.1103/PhysRevLett.116.161804. [arXiv:1607.01928 [hep-ph], arXiv:1512.05547 [hep-ph]] 12. B.T. McAllister, G. Flower, L.E.Tobar, M.E. Tobar, Phys. Rev. Appl. 9(1), 014028 (2018). [Phys. Rev. Appl. 9, 014028 (2018)]. https://doi.org/10.1103/PhysRevApplied.9.014028. [arXiv:1705.06028 [physics.ins-det]]

The 3 Cavity Prototypes of RADES: An Axion Detector Using Microwave Filters at CAST Sergio Arguedas Cuendis, A. Álvarez Melcón, C. Cogollos, A. Díaz-Morcillo, B. Döbrich, J. D. Gallego, B. Gimeno, I. G. Irastorza, A. J. Lozano-Guerrero, C. Malbrunot, P. Navarro, C. PeñaGaray, J. Redondo, T. Vafeiadis, and W. Wünsch

Abstract The Relic Axion Detector Experimental Setup (RADES) is an axion search project that uses a microwave filter as resonator for Dark Matter conversion. The main focus of this publication is the description of the three different cavity prototypes of RADES. The result of the first tests of one of the prototypes is also presented. The filters consist of 5 or 6 stainless steel sub-cavities joined by rectangular irises. The size of the sub-cavities determines the working frequency, the amount of sub-cavities determine the working volume. The first cavity prototype

S. Arguedas Cuendis () · B. Döbrich · C. Malbrunot · T. Vafeiadis · W. Wünsch European Organization for Nuclear Research (CERN), Geneva, Switzerland e-mail: [email protected] A. Álvarez Melcón · A. Díaz-Morcillo · A. J. Lozano-Guerrero · P. Navarro Department of Information and Communication Technologies, Universidad Politécnica de Cartagena, Murcia, Spain I. G. Irastorza Departamento de Física Teórica, Universidad de Zaragoza, Zaragoza, Spain J. Redondo Departamento de Física Teórica, Universidad de Zaragoza, Zaragoza, Spain Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Munchen, Germany J. D. Gallego Yebes Observatory, National Centre for Radioastronomical Technologies and Geospace Applications, Guadalajara, Spain B. Gimeno Department of Applied Physics and Electromagnetism-ICMUV, University of Valencia, Valencia, Spain C. Peña Garay I2SysBio, CSIC-UVEG, Valencia, Spain Laboratorio Subterráneo de Canfranc, Estación de Canfranc, Huesca, Spain C. Cogollos ICCUB, Universitat de Barcelona, Barcelona, Spain © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_6

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was built in 2017 to work at a frequency of ∼8.4 GHz and it was placed at the 9 T CAST dipole magnet at CERN. Two more prototypes were designed and built in 2018. The aim of the new designs is to find and test the best cavity geometry in order to scale up in volume and to introduce an effective tuning mechanism. Our results demonstrate the promising potential of this type of filter to reach QCD axion sensitivity at X-Band frequencies. Keywords Dark matter experiments · Axions · Microwave filters

1 Introduction One of the most important endeavors in physics these days is the search for dark matter. There are a plethora of theories that bring into consideration several possible dark matter candidates. One of these candidates is the axion. Axions, as well as more generic axion-like particles (ALPS), are currently considered one of the most promising fields for new physics beyond the Standard Model (SM). Axions arise in extensions of the SM through the Peccei–Quinn (PQ) mechanism [1, 2], currently the most compelling solution to the strong-CP problem [3, 4]. The exact mass of the axion is not known dependent on its cosmological history. For axion models with PQ transition happening after inflation, some lattice QCD calculations suggest a lower bound to the axion mass of around ∼ m2a > 25 µeV [5]. There is thus theoretical motivation to pursue the search for axions at these masses for which new cavity geometries can be constructed to use as detectors following the conventional axion haloscope technique [6]. This technique consists of a high-Q microwave cavity inside a magnetic field to induce the conversion of axions from our galactic dark matter halo into photons. For a cavity whose resonant frequency matches mA , the conversion is enhanced by a factor proportional to the quality factor of the cavity Q. So, these types of experiments try to detect axions by the extra power that the converted photon leaves in the cavity. The figure of merit for such an experiment is given by 4 −2 4 F ∼ gAγ m2A B 4 V 2 Tsys G Q,

(1)

where gAγ is the axion–photon coupling, B is the magnetic field, V is the volume, Tsys is the system temperature, and G is the geometrical form factor of the cavity. In order to go to higher frequencies (higher masses), one would have to reduce the size of the typical cylindrical cavity used in most haloscope experiments. This will reduce the volume of the detector and at the same time the figure of merit. To tackle this problem, RADES proposed a microwave filter composed of N cavities connected through irises. The working frequency will be determined by the size of the sub-cavity while the volume can be increased with no problems (in principle) by adding more sub-cavities to the filter. Section 2 will briefly introduce the theoretical

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model in which the design and construction were based on. Section 3 will describe the three different prototype cavities and their characterization.

2 Theoretical Model Consider a number of N cavities connected through irises, the cavities have very similar geometries and thus a similar fundamental mode at a common frequency. When excited by a monochromatic axion DM field, the system of coupled equations for the amplitudes of the fundamental mode is given by Melcon et al. [7]   ω2 1 − M E = −gAγ Be A0 ω2 G ,

(2)

where E is a vector containing the amplitudes and relative phase of the electric field in each cavity and the matrix M contains the natural frequencies, damping factors, and coupling between the cavities. It is important to emphasize that the value of the natural frequencies and coupling between the cavities can be arbitrarily chosen by altering the dimensions of the cavities and irises. Equation 2 was solved after the eigenvalue ω2 was fixed to a desired characteristic frequency. The simplest solution is to take all coupling coefficients to be equal to a fixed value K. The solution showed that all sub-cavities should resonate at the same frequency except for the first and last one. Finally, a filter that maximizes the geometric factor for that frequency was designed. The full derivation and computations mentioned above can be found in [7].

3 Cavity Prototypes RADES has produced so far a total of three cavity prototypes. The first one was built in 2017, while the other two were finished in 2018. Figure 1 shows these three microwave filters, described in the following. The length of the cavities is around 15 cm each.

3.1 Inductive Irises Prototype The first prototype consists of five sub-cavities connected with inductive irises. The cavity was first designed and modeled using CST Microwave Studio using the predictions of the theoretical model to set the dimensions of the sub-cavities. The frequency of operation was fixed to ω1 = 8.4 GHz and the geometric factor was optimized for this frequency. In 2017, this prototype was successfully installed at

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Fig. 1 The 3 RADES cavities. Left: Inductive irises prototype. Middle: Alternating irises prototype. Right: Vertical cut prototype

Fig. 2 Transmission parameter: measurement (green) at 2 K in the CAST magnet and theoretical model (gray). Red is cavity-mode to axion coupling for the five modes. Taken from [7]

the CAST magnet at CERN. Figure 2 compares the prediction of the theoretical model versus the cryogenic measurement at CAST. Also the strength of the axion coupling is plotted for the different resonances. One can see from this figure that the resonance structure behaved as expected and the first resonance peak is the one that couples to the axion. Around 350 h of data has been taken at CAST with this prototype in 2017 and 2018 and the data analysis is ongoing.

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The next step for the collaboration is to increase the volume of the cavity. However, the first simulations with 20 sub-cavities showed that there is a high chance of mode mixing for the first resonance peak due to large amount of resonances. This will reduce drastically the value of the geometric factor. Nevertheless, this can be solved if the working resonance peak is a higher mode. For higher modes, the spacing between adjacent modes is larger and the mode mixing can be avoided. Section 3.2 explains how this can be achieved.

3.2 Alternating Irises Prototype The second prototype consists of 6 sub-cavities with inductive and capacitive irises. By introducing capacitive irises, the value of coupling between two cavities can be negative. Solving Eq. (2) for alternating irises shows that the axion couples to a higher mode. CST Microwave Studio was used afterwards to design this new prototype. As expected from the theory, the simulations showed that the axion couples to the fourth mode. This can be seen in Fig. 3: in the fourth mode the electric field in all the sub-cavities is pointing to the same direction. This structure was tested at room temperature exhibiting good agreement with expectations derived from the CST model. In the future, tests will be made at 2 K to see the performance of the cavity. It is foreseen to install this cavity (or a 30 sub-cavities structure with alternating irises) at the CAST magnet in the summer of 2019 and take data with it.

Fig. 3 Electric field distribution of the six modes in the “alternating irises prototype”

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Fig. 4 Frequency of the first resonance for different gap sizes. Simulations and measurements

3.3 Vertical Cut Prototype The final important step to increase the sensitivity range of our detectors would be to tune the resonance frequency. For this reason, a five sub-cavities prototype with inductive irises was built. However, this time it was manufactured with a vertical cut, leaving two halves that can be separated from each other (see Fig. 1). By introducing a small gap between the two structures, the width of the structure changes and so does the resonance frequency. Proof of principle tests to tune the cavity were done at CERN, where spacers were manually introduced between the two halves and the resonance frequency was measured using a vector network analyzer. Figure 4 compares the results of these measurements with the results of the simulations. First results show a tuning of approximately 800 MHz. At the moment this prototype is sitting at CERN’s cryogenic laboratory and is undergoing performance tests at low temperatures (down to 9 K so far).

4 Conclusions and Future Plans There is convincing theoretical motivation to look for axions with a mass above 25 µeV. The conventional cylindrical cavity geometry can be challenging when going to higher frequencies. RADES presented a new geometry that allows searches at an axion mass of around 34 µeV. The theoretical model was developed and then tested using simulations. Afterwards, the first prototype was successfully built, installed, and used for data taking at the CAST magnet. New geometries to upgrade

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the detector were built and are under study at the moment. The first tests of the new structures show promising results in order to avoid mode mixing and eventually develop tuning for the cavity. In 2019, the vertical cut cavity was fully characterized at 2 K. The construction of an alternating cavity with 30 sub-cavities is planned. This cavity would be installed at the CAST magnet for data taking. A mechanical system to tune the vertical cut cavity will be developed and tested at room temperature. Studies of the possibilities of electrical tuning are also ongoing. Acknowledgments This work has been funded by the ERC-STG 802836 (AxScale), the Spanish Agencia Estatal de Investigacion (AEI), and Fondo Europeo de Desarrollo REgional (FEDER) under project FPA-2016-76978.

References 1. R.D. Peccei, H.R. Quinn, CP conservation in the presence of instantons. Phys. Rev. Lett. 38, 1440–1443 (1977). [328(1977)] 2. R.D. Peccei, H.R. Quinn, Constraints imposed by CP conservation in the presence of instantons. Phys. Rev. D16, 1791–1797 (1977) 3. S. Weinberg, A new light boson? Phys. Rev. Lett. 40, 223–226 (1978) 4. F. Wilczek, Problem of strong p and t invariance in the presence of instantons. Phys. Rev. Lett. 40, 279–282 (1978) 5. V.B. Klaer G.D. Moore, The dark-matter axion mass. J. Cosmol. Astropart. Phys. 2017(11), 049–049 (2017) 6. P. Sikivie, Experimental tests of the “invisible” axion. Phys. Rev. Lett. 51, 1415–1417 (1983) 7. A.A. Melcon et al., Axion searches with microwave filters: the RADES project. JCAP 1805(05), 040 (2018)

Search for 5–9 µeV Axions with ADMX Four-Cavity Array Jihee Yang, Joseph R. Gleason, Shriram Jois, Ian Stern, Pierre Sikivie, Neil S. Sullivan, and David B. Tanner

Abstract An axion is a particle that could resolve some issues with strong interaction CP symmetry and dark matter. A tunable microwave cavity permeated by a strong external magnetic field can be used to detect axions. The Axion Dark Matter eXperiment (ADMX) has developed a four-cavity array design to explore the axion mass range of 5–9 µeV. Tuning of each cavity frequency is done primarily by a coarse tuning rod, with a fine tuning rod used to lock frequencies together. Aluminum prototypes at 1:3 scale have been built and tested. At 300 K, the TM010 frequency varied from 4.4 to 6.4 GHz and crossed three TE modes. Electroplating pure aluminum on the cavity improved the Q factor by 27%. Preliminary results from 4.2 K test showed a 2% reduction of resonant frequency and an improvement in Q by a factor of 2.7 compared to the 300 K values. Keywords Microwave · Tunable · Cavity · Axion · Dark matter · ADMX

1 Introduction The axion is a particle related to two key issues in particle physics and astrophysics: the origin of CP symmetry in the strong interactions [1–3] and the composition of the dark matter of the universe [4–8]. The present laboratory, astrophysical, and cosmological constraints suggest axions have a mass in the 1 µeV–1 meV range. Axions are especially well-motivated candidates for dark matter if their mass is in the range 1–10 µeV [4–7]. As shown 35 years ago [9], these dark matter axions may be detected by their coupling to photons through the E · B0 interaction in a

J. Yang () University of Washington, Seattle, WA, USA e-mail: [email protected] J. R. Gleason · S. Jois · I. Stern · P. Sikivie · N. S. Sullivan · D. B. Tanner Department of Physics, University of Florida, Gainesville, FL, USA e-mail: [email protected]; [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_7

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tunable high-Q microwave cavity permeated by a strong external magnetic field B0 . ADMX is one of many past, existing, and proposed experiments that exploit this coupling [10–20]. ADMX is by far the most sensitive of these experiments [13, 14]. Benefiting from the addition of a high capacity dilution refrigerator, ADMX has in the last year reached the sensitivity to detect even the most weakly coupled QCD axions [14]. If axions are the dominant constituent of our galactic halo, their local energy density is of order 0.45 GeV/cc. Using 12 µeV as an estimate of the axion mass, the axion number density at the Earth is n ≈ 2.3 × 1013 axions/cm3 . Galactic halo axions have velocities v/c of order 10−3 and hence their energies, a = ma c2 + ma v 2 /2, have a spread of order 10−6 above the axion mass. These axions convert to microwave photons in a cavity permeated by a strong static magnetic field. The power on resonance from axion → photon conversion into the TMn0 mode is [9]  2   Cnl  gγ 2 B0 V P = 130 yW 200  8 Tesla 0.5 0.36     fa QL ρa , · 1 GH z 100, 000 0.5 yg/cm3 

(1)

where V is the cavity volume, B0 is the strength of the static field, Cnl is a form factor representing the overlap of the cavity resonance with the static field (only large for n = 0 and  = 1), gγ is the coupling of the axion to two photons, ρa is the axion mass density at the Earth, fa is the photon frequency (equal to ma c2 / h), and QL is the loaded quality factor of the cavity. The quantity gγ is a model-dependent parameter of the theory. For example, in all grand-unified models, such as the Dine–Fischler–Srednicki–Zhitnitskii model (DFSZ) [21], gγ ≈ 0.36. However, the Kim–Shifman–Vainshtein–Zakharov (KSVZ) model [22] has gγ ≈ −0.97. A definitive axion search must be sensitive to the weaker DFSZ coupling.

2 Multiple Cavity Array Design As the search moves from 2–4 µeV to higher frequencies, the cavity volume is reduced, because the radius r of the cavity is related to the TM010 frequency f by 11.5 GHz r = . 1 cm f

(2)

At 2.2 GHz (around 9 µeV), the cavity radius is 5.2 cm. This size is considerably smaller than the ∼25 cm radius of the ADMX magnet bore. ADMX plans to use multiple cavities in order to maintain an efficient use of the magnet volume. The

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cavities will be tuned to identical resonant frequencies and their outputs will be added in phase, combining amplitudes, so that they act as a single detector. This approach, with cavities constructed with high precision, maintains a better form factor than a complex multi-cell design according to simulations and experiments [23, 24]. The Florida ADMX group is designing and building a four-cavity array for detecting axions that have masses in the 5–9 µeV range. The design of this array and the results from the 1:3 scale prototypes are described here. We model the cavity by assuming it to be made from a perfectly conducting metal in the shape of a cylinder of radius R and length L. The Helmholtz equation can be solved using variable separation method both inside and outside the cylinder and should satisfy the boundary conditions Ez ∂Bz ∂ρ

= 0.

(3)

ρ=R

The resonance frequencies depend on the eigenvalues of Jm , kmn , as well as the length of the cavity [25]

fmnp =

c 2π

2 kmn p2 π 2 + . 2 R L2

(4)

For prototype one, the radius of the cylinder was taken to be 2.79 cm and the length to be 17.6 cm. The frequency of the TM and TE modes was obtained by solving the transcendental equations analytically. The empty cylinder had the TM010 mode at f010 = 4.1067 GHz. As the cavity is tuned, the TM010 mode would cross the TE114,115,116 and TE211,212,213,214 , a total of 7 mode crossings. When a metal tuning rod is added to the cavity, in addition to the original boundary, the solution should satisfy the boundary conditions on the surface of the tuning rod. The TM modes for the case when the rod is at the center can be obtained by solving the equation with outer radius a and inner radius b  Jm

       kmn kmn kmn kmn b Nm a = Jm a Nm b . a+b a+b a+b a+b

(5)

The TE modes can be obtained by solving

Jm



       kmn kmn kmn kmn

b Nm a = Jm a Nm b . a+b a+b a+b a+b

(6)

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3 The ADMX Four-Cavity Array The design of the ADMX run-2A cavity array aims to achieve a tuning range of 1.5–2.2 GHz using an array of four cavities of approximately 8 cm radius and 97 cm length (Fig. 1). A major design challenge of this multi-cavity system is to maintain a frequency lock between the cavities in order to use the array as a single higher frequency cavity that exploits the maximum volume available within the ADMX magnet bore. Like the current single ADMX cavity, each cavity of the array contains a large coarse tuning rod (2.9 cm diameter copper in this design) to set each cavity frequency equal to the target within the margin of accuracy of the coarse tuning system. Additionally, in the four-cavity array, each cavity uses a smaller fine tuning rod to compensate for frequency differences generated by mechanical deviations in the coarse tuning system. These fine tuning elements are 2.16 cm diameter sapphire rods actuated by piezoelectric linear motors that insert or retract the rods through the top of each cavity. Because the frequency lock between the cavities must be re-acquired after each successive step in coarse tuning, and the fine tuning range is limited to approximately 3.3 MHz, it is critical that the mechanical motion of the coarse tuning rods does not differ too greatly between cavities. Consistency in coarse tuning rod motion is achieved using a single rotary assembly to which all four coarse tuning rods are mounted. This assembly consists of upper and lower wheels

Antenna Actuator

Fine Tuning Rod Actuator

Upper Coarse Tuning Wheel

Coarse Tuning Rotary Actuator Coarse Tuning Rod Cavity Axle Lower Bearing

Fig. 1 Drawings of cavity 2A assembly design. (Left) Whole cavity 2A assembly. (Middle) Cross section view of cavity 2A assembly. (Right) Top view of cavity 2A assembly

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joined by a central axle that rotates on high precision ceramic bearings mounted in the cavity end plates. The coarse tuning rods are held in place by sapphire pins that mount in the upper and lower wheels and pass through a slot in each cavity end. A single rotary piezo motor actuates the entire rotary mechanism. After assembly, initial frequency offsets between the cavities due to mechanical tolerances can be compensated by manual adjustments to flexures in the coarse tuning wheels.

4 First Prototype (v1) The four-cavity array prototype provides a test bed for the first multi-cavity array for the ADMX experiment. Designed for a frequency range of 4–6 GHz, the prototype array uses a cavity cross section that is roughly 1:3 scale to that of the planned fullsized experiment. The first prototype array (v1) was built using 6061 aluminum and with a shortened aspect ratio for ease of manufacturing. The fundamentals of the design are identical to that of the full size array. The prototype v1 was tested at room temperature (300 K) and atmospheric pressure first. A network analyzer (Keysight E5063A) was used to measure the S parameters for each cavity shown in (Fig. 2a). On the end plates, two fixed length stub antenna pins were placed to measure the S12 parameter. From those S12 parameter measurements, modes were identified throughout the coarse tuning range and mapped out on a mode map (Fig. 2b). The lowest mode that varies with the tuning angle is the TM010 mode (black dots). The frequency of the TM010 mode varies from 4.37 GHz (coarse rod at the wall of cavity; 0◦ ) to 6.39 GHz (coarse rod at the center of cavity; ∼27◦ ). The TM010 mode encounters 3 major TE mode

Fig. 2 Left: (a) Representation of the S parameter measurement setup. S parameters are taken from the setup to identify resonances (modes) in each cavity to obtain a mode map. Right: (b) Mode map plot for cavity C at 300 K. Black dots represent the TM010 mode and gray dots represent all other modes. The TM010 mode has three major crossings throughout the entire coarse angle range

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Flexure

Fine Tuning Linear Stage Coarse Tuning Rotary Stage Upper Coarse Tuning Wheel Upper Bearing

Cavity Coarse Tuning Rod Coarse Tuning Offset Adjustment +/- ~100 MHz

Axle

Antenna Actuator

Fig. 3 Drawings of prototype (v2) assembly design. (Left) Coarse tuning assembly including tuning wheel with flexures and tuning rods. (Right) Cross section of prototype assembly

crossings in the tuning range. The largest mode crossing (avoided crossing at 14◦ ) has a frequency separation gap of 80 MHz. Over the coarse angle range of 0◦ –27◦ , the average unloaded quality factor (Q) was 6700. The average frequency spread between the four cavities was 9 MHz with a maximum range (difference between maximum and minimum values) of 13 MHz. Because there was an offset frequency spread that cannot be corrected for solely by fine tuning rods, a flexure design was introduced to adjust the offset in the prototype v2 (Fig. 3).

5 Second Prototype (v2) The four-cavity array prototype v2 builds on lessons learned from the first prototype. The gap between the coarse tuning rod ends and cavity end plates was reduced from 0.50 to 0.18 mm to improve the capacitance effect. A more complex coarse tuning wheel was designed to allow independent manual adjustment of the coarse tuning rod positions to compensate for initial offsets caused by machining and assembling process. The length of the cavity was changed to alter the frequencies of the TE

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Fig. 4 Plots of Q factor (upper plot) and coupling (lower plot) vs antenna depth. The antenna depth of an adjustable antenna is varied in the cavity A to measure Q factor and antenna coupling changes. When the antenna is critically coupled (−35 dB coupling), the Q value decreases to about half of the unloaded value

modes. As an additional test, the v2 array was plated with 99.99% aluminum to improve the conductivity, perhaps permitting the achievement of the anomalous skin effect regime at low temperatures. This second prototype was tested under the same conditions as the first prototype to study its RF properties. As a result of the different cavity lengths, the overall TM010 frequency tuning region was shifted down by 25 MHz and mode crossings appeared at frequencies that were an average of 150 MHz lower than the first prototype crossings. The average Q factor at room temperature (300 K) was 8500, which is a 27% improvement over the first (non-electroplated) prototype value of 6700. Moreover, flexures were used to bring cavity frequencies together and the maximum range of the frequency spread becomes just 4 MHz instead of 13 MHz. This reduced frequency spread is well within the tolerance of the fine tuning range, 13.4 ± 4.5 MHz, in all frequency regions. A fixed length antenna (weakly coupled) and an adjustable antenna (RG402) were used to inspect Q and antenna coupling changes in the cavity. For each cavity, S12 and S11 parameters were measured while the antenna depth was varied. Figure 4

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shows Q factor and coupling vs antenna depth plots in cavity A at a coarse angle of 5◦ . When the antenna is critically coupled (coupling of −35 dB), the Q value becomes half of the unloaded Q value as expected. The average antenna depth for critical coupling across all cavities and coarse tuning angles was 3.4 mm. At 4.2 K in a liquid 4 He environment, the TM010 frequency and Q value were measured with the coarse tuning rod at the wall position. The frequency at this temperature is predicted to shift down due to changes in the cavity radius and the dielectric constant of helium filling the cavity. Considering the relationship √ f ∝ 1/(r 1 ), where f is the resonant frequency, r is the cavity radius, and

1 is the dielectric constant, the TM010 frequency should be about 98% of the 300 K value [26, 27]. The observed frequency reduction was within 0.05% of that predicted value. With a given antenna depth, the coupling of the antenna became 2.6 times stronger and the Q value increased to Q = 16,000, a value about 2.4 times larger than the 300 K values. If we compare this Q value to the one at 300 K with equivalent antenna coupling, the Q improvement by cooling down the cavity is a factor of 2.7.

6 Summary The four-cavity array layout presented here is designed to achieve efficient tuning and locking frequencies through coarse and fine tuning rods. Having a central control system for all coarse rods is beneficial in terms of simplicity of the mechanical and electrical motion systems. Flexures on the coarse tuning wheel and fine tuning rod are intended to lock the frequencies of all cavities together. The aluminum prototype cavity results showed three major TE mode crossings within the tuning range of 4.4–6.4 GHz. These mode crossings were located an average of 150 MHz lower due to the longer cavity length of second prototype. It is shown that the frequency spread offset between cavities is successfully reduced by 69% within the tolerance of fine tuning by applying flexure design. The quality factor of the TM010 mode in the 6061 aluminum prototype (v1) was 6700 and has been improved by 27% by electroplating the cavity with pure aluminum and by a factor of 2.7 by cooling down to 4.2 K. Acknowledgments This work was supported by the Department of Energy Grant No. DESC0010296 and by Fermi National Accelerator Laboratory contract 640844 at the University of Florida.

References 1. R.D. Peccei, H. Quinn, CP conservation in the presence of pseudoparticles. Phys. Rev. Lett. 38, 1440 (1977); R.D. Peccei, H. Quinn, Constraints imposed by CP conservation in the presence of pseudoparticles. Phys. Rev. D 16, 1791 (1977)

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2. S. Weinberg, A new light boson?. Phys. Rev. Lett. 40, 223 (1978) 3. F. Wilczek, Problem of strong P and T invariance in the presence of instantons. Phys. Rev. Lett. 40, 279 (1978) 4. L. Abbott, P. Sikivie, A cosmological bound on the invisible axion. Phys. Lett. B 120, 133 (1983) 5. J. Preskill, M. Wise, F. Wilczek, Cosmology of the invisible axion. Phys. Lett. B 120, 127 (1983) 6. M. Dine, W. Fischler, The not-so-harmless axion. Phys. Lett. B 120, 137 (1983) 7. M.S. Turner, Windows on the axion. Phys. Rep. 197, 67 (1990); G.G. Raffelt, Astrophysical methods to constrain axions and other novel particle phenomena. Phys. Rep. 198, 1 (1990) 8. C. Amsler et al. (Particle Data Group), Review of particle physics. Phys. Lett. B 667, 1 (2008) 9. P. Sikivie, Experimental tests of the ‘invisible’ axion. Phys. Rev. Lett. 51, 1415 (1983) 10. S. DePanfilis, A.C. Melissinos, B.E. Moskowitz, J.T. Rogers Y.K. Semertzidis, W.U. Wuensch, H.J. Halama, A.G. Prodell, W.B. Fowler, F.A. Nesrick, Limits on the abundance and coupling of cosmic axions at 4.5 < ma < 5.0 µeV. Phys. Rev. Lett. 59, 839 (1987); W.U. Wuensch, S. DePanfilis-Wuensch, Y.K. Semertzidis, J.T. Rogers, A.C. Melissinos, H.J. Halama, B.E. Moskowitz, A.G. Prodell, W.B. Fowler, F.A. Nezrick, Results of a laboratory search for cosmic axions and other weakly coupled light particles. Phys. Rev. D 40, 3153 (1989) 11. C. Hagmann, P. Sikivie, N.S. Sullivan, D.B. Tanner, Results from a search for cosmic axions. Phys. Rev. D 42, 1297 (1990) 12. R. Bradley, J. Clarke, D. Kinion, L.J. Rosenberg, K. van Bibber, S. Matsuki, M. Mück, P. Sikivie, Microwave cavity searches for dark-matter axions. Rev. Mod. Phys. 75, 777 (2003) 13. S.J. Asztalos, G. Carosi, C. Hagmann, D. Kinion, K. van Bibber, M. Hotz, L.J Rosenberg, G. Rybka, J. Hoskins, J. Hwang, P. Sikivie, D.B. Tanner, R. Bradley, J. Clarke, SQUID-based microwave cavity search for dark-matter axions. Phys. Rev. Lett. 104, 041301/1–041301/4 (2010) 14. N. Du, N. Force, R. Khatiwada, E. Lentz, R. Ottens, L.J Rosenberg, G. Rybka, G. Carosi, N. Woollett, D. Bowring, A.S. Chou, A. Sonnenschein, W. Wester, C. Boutan, N.S. Oblath, R. Bradley, E.J. Daw, A.V. Dixit, J. Clarke, S.R. O’Kelley, N. Crisosto, J.R. Gleason, S. Jois, P. Sikivie, I. Stern, N.S. Sullivan, D.B. Tanner, G.C. Hilton, Search for invisible axion dark matter with the axion dark matter experiment. Phys. Rev. Lett. 120, 151301/1–151301/5 (2018) 15. K. Zioutas et al. (CAST Collab.), First results from the CERN axion solar telescope. Phys. Rev. Lett. 94, 121301 (2005); E. Arik et al. (CAST Collab.), Search for Sub-eV mass solar axions by the CERN Axion solar telescope with 3 He Buffer Gas. Phys. Rev. Lett. 107, 261302 (2011) 16. K. Ehret et al. (ALPS Collab.), New ALPS results on hidden-sector lightweights. Phys. Lett. B 689, 149 (2010) 17. B.M. Brubaker, L. Zhong, Y.V. Gurevich, S.B. Cahn, S.K. Lamoreaux, M. Simanovskaia, J.R. Root, S.M. Lewis, S. Al Kenany, K.M. Backes, I. Urdinaran, N.M. Rapidis, T.M. Shokair, K.A. van Bibber, D.A. Palken, M. Malnou, W.F. Kindel, M.A. Anil, K.W. Lehnert, G. Carosi, First results from a microwave cavity axion search at 24 µeV. Phys. Rev. Lett. 118, 061302 (2017) 18. R. Baehre et al. (ALPS Collab.), The ALPS II technical design report. JINST 8, T09001 (2013) 19. J.E. Kim, Y.K. Semertzidis, and S. Tsujikawa, Bosonic coherent motions in the Universe. Front. Phys. 2, 60 (2014) 20. E. Armengaud (IAXO Collab.), Conceptual design of the International Axion Observatory (IAXO). JINST 9, T05002 (2014) 21. M. Dine, W. Fischler, M. Srednicki, A simple solution to the strong CP problem with a harmless axion. Phys. Lett. 104B, 199 (1981); A.P. Zhitnitskii, Sov. J. Nucl. Phys. 31, 260 (1980) 22. J. Kim, Weak-interaction singlet and strong CP invariance. Phys. Rev. Lett. 43, 103 (1979); M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B166, 493 (1980) 23. I. Stern, A.A. Chisholm, J. Hoskins, P. Sikvie, N.S. Sullivan, D.B. Tanner, G. Carosi, K. van Bibber, Cavity design for high-frequency axion dark matter detectors. Rev. Sci. Instrum. 86, 123305 (2015)

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Tunable High-Q Photonic Bandgap Cavity Ankur Agrawal, Akash V. Dixit, David I. Schuster, and Aaron Chou

Abstract We present the design of a tunable high-Q photonic bandgap (PBG) cavity, made from a periodic arrangement of dielectric rods in a woodpile like structure. We know that the axion signal rate diminishes as we move to higher axion mass searches. High quality factor achieved with this type of cavity will enable higher mass axion search with improved signal sensitivity and faster scanning rate. Quality factor in excess of the axion-Q will enable us to design interesting experiments such as the quantum non-demolition (QND) measurement of itinerant microwave photons using superconducting qubits and stimulated emission of photons. Keywords Photonic bandgap · Dark matter axion · Microwave cavity

1 Introduction A woodpile structure made out of dielectric rods exhibits an omnidirectional photonic bandgap (PBG) which forbids the propagation of electromagnetic wave with energy within a certain range in all directions. We designed an electromagnetic cavity by creating a defect inside the crystal, such that its frequency lies within the forbidden bandgap. Very high Q-factors can be achieved since the light has no way to escape because of the bandgap and is only limited by the dielectric loss in the material. We predict the quality factor of such a cavity to be close to 108 near 10 GHz. The cavity frequency is tuned by sliding the rods in and out. One of the potential applications is in the axion dark matter search, which is currently limited by the use of low Q-factor copper cavities due to the presence of a strong magnetic

A. Agrawal () · A. V. Dixit · D. I. Schuster The University of Chicago, Chicago, IL, USA e-mail: [email protected] A. Chou Fermilab National Laboratory, Chicago, IL, USA © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_8

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field. We predict the Q-factor of a PBG cavity to increase in the presence of a large magnetic field due to the shift in the two-level system energies to a higher level.

2 Photonic Bandgap The control of electric currents achieved with the semiconductors such as silicon is dependent on a phenomenon called the bandgap: a range of energies in which the electron is blocked from traveling through the semiconductor. Scientists have produced material with a photonic bandgap—a range of wavelengths of light which is blocked by the material. The analogy between electron-wave propagation in real crystals and the electromagnetic-wave propagation in a multidimensionally periodic structure has proven to be a fruitful one. It allows us to manipulate light in addition to electric currents. Photonic crystals occur in nature in the form of structural coloration and animal reflectors as shown below (Figs. 1 and 2). The existence and properties of an electronic bandgap depend crucially on the type of atoms in the material and their crystal structure—spacing between the atoms and the shape of the crystals they form. In the optical analogue, atoms or molecules are replaced by macroscopic media with differing dielectric constants, and the periodic potential is replaced by the periodic dielectric function.

Fig. 1 Natural periodic micro-structures responsible for the iridescent color in stones and feathers (Source: Wikipedia). (a) Opal Armband. (b) Peacock Feathers

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Fig. 2 (Left) The multilayer films, a one-dimensional photonic crystal. The system consists of alternating layers of materials with different dielectric constants, with a spatial period of a. (Right) The photonic band structure of a multilayer film with lattice constant a and alternating layers of different widths. The width of the = 13 layer is 0.2a and the width of the = 1 layer is 0.8a [3]

3 Photonic Bandgap Crystal The simplest photonic bandgap material is the periodic arrangement of alternating layers of materials with different dielectric constants. In a one-dimensional photonic crystal, the band-stop edges are determined by the frequency of the waves propagatc/n ing in the dielectric band and the air band, given by fair = λc0 and fdiel = λ0diel , respectively. It has been shown that the size of the bandgap increases with the increase in dielectric contrast, and it reaches a maximum at an optimal volume filling fraction [1]. The presence of an omnidirectional bandgap in the 3D woodpile structure was demonstrated early [1], and the defect cavity modes within the bandgap were well studied. In recent experiments, nanocavities created inside the woodpile crystals made out of semiconducting material have achieved Q’s as high as 38,500 at 4 K [2]. These Q-values are already high compared to the copper cavities which can achieve Q ∼104 at 10 GHz at 20 mK. Inspired by these results, we constructed the woodpile structure using commercially available alumina bars. The woodpile structure is assembled by stacking together layers of dielectric rods, with each layer consisting of parallel rods with a center-to-center separation of d. The rods are rotated by 90◦ in each successive layer. Next, the rods in third and fourth layers are displaced by a distance of 0.5d perpendicular to the first and the second layer, respectively, as shown in Fig. 3. This results in a face-centered-tetragonal (fct) symmetry where each unit cell repeats √ after four layers. We kept the separation, d = a/ 2 and the width and the height of bars to be 0.25a (w = h = 0.25a). We simulated a 3D woodpile structure using the MIT mpb simulation package [4] to determine the optimal lattice parameters. The simulation predicts a complete bandgap of 14.24 % at the center frequency as shown in the band-structure diagram in Fig. 4. The band-stop edges are given by 0.490 ac and 0.562 ac (9.5–

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Fig. 3 (Left) Isometric view of the unit cell. (Right) Front view of the unit cell depicting all the lattice parameters

Fig. 4 (Left) The photonic band structure for the lower bands of a woodpile structure simulated using mpb. (Right) Measured transmission S21 through the woodpile photonic crystal; bandgap of 1.6 GHz centered around 10.3 GHz

11.1 GHz), where a is the periodicity along the stacking direction. We measured the transmission spectra using a vector network analyzer. The cables were calibrated up to the ends of the dipole antennae. The measured transmission spectra are shown in Fig. 4 where the bandgap lies between 9.5 GHz and 11.1 GHz, which agrees well with the simulation result. We varied the power incident on the crystal to check if the lowest level of the transmission in the bandgap is limited by the noise floor of the VNA and we did not observe any change in the lowest level, implying the lowest

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Fig. 5 (Left column) The assembled woodpile PBG crystal. Defect layer shown in the front and top view. (Right) The experimental apparatus showing the input–output ports connected to a VNA. (Right-bottom) The exponential attenuation of the electric field as it enters the crystal [3]

level is indeed the suppression in transmission. Further, the entire structure was covered with ECCOSORB sheets to provide absorbing boundary condition at the surface and suppress any direct coupling between the antennae. The experimental setup is presented in Fig. 5.

4 Photonic Bandgap Cavity The defect state within the bandgap is created by either removing material or by adding material to the crystal. The measured spectrum of a defect cavity in the woodpile crystal is shown in (Fig. 6). We studied the variation of Qi as a function of the cavity frequency within the bandgap. We observed that the Qi reaches a maximum value at the center of the bandgap as seen in Fig. 7. It is reasonable to expect this hump in the center since the field will be attenuated maximally.

4.1 Estimation of Field Attenuation To maximally utilize the space inside a magnet bore, we need to estimate the number of periods required to attenuate the field such that the quality factor is not

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Fig. 6 (Left) Transmission (S21 ) spectra of the defect cavity. (Right) Reflection (S11 ) spectra of the defect cavity

Fig. 7 Variation of the internal quality factor, Qi as a function of the defect frequency in the bandgap. Blue dashed line represents the quality factor of a 10 GHz copper cavity at 20 mK

limited by the boundary condition. We know that the loss rate due to multiple decay mechanisms is given by κ = κ1 + κ2 + · · ·

(1)

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where κ is the total loss rate. The internal quality factor, Qi is related to the dielectric (Qd ), conductor (Qc ), and radiation losses (Qr ) by κ=

1 1 1 1 = + + Q Qd Qc Qr

(2)

The power loss in the conductor is proportional to the square of the surface current [3] such that the loss is proportional to the square of the magnitude of the field at the boundary Pc ∝ J 2 ∝ H (r)2

(3)

and the power loss in a dielectric material is given by its loss tangent which would maximally be limited by the fraction of field residing inside the lossy dielectric material Qd =

1 tan(δ)

(4)

In summary, the total loss rate is given by κ = wc κc + wd κd + wr κr

(5)

where wc , wd , and wr are the participation ratios of the mode in the dielectric, conductor, and radiation.

5 Conclusion We have demonstrated a complete bandgap in the woodpile like photonic crystal. The defect-cavity Q-factor measured at room temperature is higher than the copper cavity at 20 mK, which is promising. At cryogenic temperature, we expect to see an improvement in the Q-factor by at least an order of magnitude. Ultimately, we would like to construct this PBG cavity using high purity Sapphire (tan(δ)∼10−8 − 10−9 at 27 mK at 13 GHz) [5] to achieve Q-factors higher than 108 . It will hugely impact Qi the axion search by improving the scanning rate to a factor of 10 4 . In other words, for a Qi = 106 we can scan the axion mass range 100 times faster than the current technology. Acknowledgments This research was supported by the Heising-Simons Foundation. We would like to thank Steven G. Johnson for simulation scripts and helpful discussion. This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics.

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References 1. K.M. Ho, C.T. Chan, C.M. Soukoulis, Existence of a photonic gap in periodic dielectric structures. Phys. Rev. Lett. 65, 3152–3155 (1990) 2. A. Tandaechanurat, S. Ishida, D. Guimard, M. Nomura, S. Iwamoto, Y. Arakawa, Lasing oscillation in a three-dimensional photonic crystal nanocavity with a complete bandgap. Nat. Photonics 5, 91 (2011) 3. J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade, Photonic Crystals Molding the Flow of light (Princeton University Press, Princeton 2017) 4. S. Johnson, J.D. Joannopoulos, Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis. Opt. Express 8(3), 173–190 (2001) 5. D.L. Creedon, Y. Reshitnyk, W. Farr, J.M. Martinis, T.L. Duty, M.E. Tobar, High q-factor sapphire whispering gallery mode microwave resonator at single photon energies and millikelvin temperatures. Appl. Phys. Lett. 98, 222903 (2011)

Source Mass Characterization in the ARIADNE Axion Experiment Chloe Lohmeyer for the ARIADNE collaboration, N. Aggarwal, A. Arvanitaki, A. Brown, A. Fang, A. A. Geraci, A. Kapitulnik, D. Kim, Y. Kim, I. Lee, Y. H. Lee, E. Levenson-Falk, C. Y. Liu, J. C. Long, S. Mumford, A. Reid, Y. Semertzidis, Y. Shin, J. Shortino, E. Smith, W. M. Snow, E. Weisman, A. Schnabel, L. Trahms, and J. Voigt

Abstract The Axion Resonant InterAction Detection Experiment (ARIADNE) is a collaborative effort to search for the QCD axion using nuclear magnetic resonance (NMR), where the axion acts as a mediator of spin-dependent forces between an unpolarized tungsten source mass and a sample of polarized helium-3 gas. Since the experiment involves precision measurement of a small magnetization, it relies on limiting ordinary magnetic noise with superconducting magnetic shielding. In C. Lohmeyer () · N. Aggarwal · A. A. Geraci Center for Fundamental Physics, Department of Physics and Astronomy, Northwestern University, Evanston, IL, USA e-mail: [email protected] A. Arvanitaki Perimeter Institute, Waterloo, ON, Canada A. Brown · I. Lee · C. Y. Liu · J. C. Long · A. Reid · J. Shortino · W. M. Snow · E. Weisman Department of Physics, Indiana University, Bloomington, IN, USA A. Fang · A. Kapitulnik Department of Physics and Applied Physics, Stanford University, Stanford, CA, USA D. Kim · Y. Kim Center for Axion and Precision Physics Research, IBS, Daejeon, South Korea Department of Physics, KAIST, Daejeon, South Korea Y. H. Lee KRISS, Daejeon, Republic of Korea E. Levenson-Falk · S. Mumford Department of Physics, Stanford University, Stanford, CA, USA Y. Shin · Y. Semertzidis Center for Axion and Precision Physics Research, IBS, Daejeon, South Korea E. Smith Los Alamos National Laboratory, Los Alamos, NM, USA A. Schnabel · L. Trahms · J. Voigt Physikalisch-Technische Bundesanstalt, Berlin, Germany © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_9

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addition to the shielding, proper characterization of the noise level from other sources is crucial. We investigate one such noise source in detail: the magnetic noise due to impurities and Johnson noise in the tungsten source mass. Keywords Axion · ARIADNE · Nuclear magnetic resonance · 3He · SQUID · Interaction · Fifth-force · Superconductor · Magnetic impurity

1 Introduction Axions or axion-like particles arise in many theories beyond the standard model [1–3]. One such particle, the Peccei–Quinn (PQ) axion, was postulated in the 1970s as a solution to the strong CP problem in quantum chromodynamics (QCD) [2]. The strong CP problem arises from experimental limits on the electric dipole moment (EDM) of the neutron being magnitudes smaller than would be expected theoretically. This discrepancy is usually parameterized by an angle θQCD , which is generally expected to be of order unity under QCD, but is observed to be less than 10−10 by searches for the neutron EDM [4, 5]. In addition to its usefulness in explaining the smallness of the θQCD angle [2, 6], the axion is also considered to be a promising dark matter candidate [7, 8]. The parameter space for the QCD axion in particular can be parameterized by mass, which is inversely proportional to the PQ symmetry breaking scale fa [2]. The mass range has an upper bound of mA < 10 meV which is implied by astrophysical limits [9] and a lower bound dependent upon cosmological theories of inflation where mA > 1 µeV [7, 8] for models with the energy scale of inflation above the PQ phase transition. The mass of the QCD axion can be significantly lighter in models with low energy-scale inflation [7, 8]. ARIADNE aims to search for an axion field in the mass range 1 µeV < mA < 6 meV that couples to nucleons via their spin [1]. Other axion experiments and proposals for new experiments probe for axions in different mass ranges, using a variety of couplings (i.e., to photons, electrons, nucleons), and unlike our approach, search for interactions directly with the hypothetical cosmic dark matter axion field [10–18]. A comparison can be found in reference [19], showing ARIADNE’s potential to fill in an uncharted and complementary region of the QCD axion parameter space. ARIADNE searches for the QCD axion from the perspective that it mediates short-range spin-dependent forces between two nucleons. The interaction energy can be written as [1] Usp (r) =

h¯ 2 gsN gpN 8π mf



1 1 + 2 rλa r



 r  e− λa σˆ · rˆ ,

(1)

where gs and gp are the monopole and dipole coupling constants, respectively, λa is the Compton wavelength of the axion, mf is the fermion mass, σˆ is the fermion spin, and r is the vector between the source mass and spin. This coupling to the fermion

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Fig. 1 (a) Illustration of the experiment concept: the unpolarized tungsten source mass is placed near a quartz block with a thin film of niobium shielding. Inside the block is the spheroidal vessel housing 3 He and the SQUID magnetometer. (b) Detailed schematic of the bottom region of the cryostat with three 3 He sample vessels surrounding the tungsten sprocket. The sample cells will be connected to a copper stage at 4 K

spin allows us to define an effective or fictitious magnetic field. In order to detect this field, we plan to use an NMR technique. An ensemble of nuclear spins in gaseous 3 He will be hyperpolarized by metastability-exchange optical pumping (MEOP). An external magnetic field Bext will be applied to the nuclear spins to set their Larmor precession frequency at approximately 100 Hz. A dense, nominally nonmagnetic source mass is modulated in close proximity to the 3 He sample resulting in a modulation of the effective transverse magnetic field. When the frequency of the field modulation matches the Larmor frequency, the fermion spins will be resonantly driven into precession [1]. This results in a transverse magnetization which can be detected using a SQUID magnetometer. A full description of the experiment can be found in Ref. [19]. The source mass will be a tungsten sprocket with 11-fold symmetry, as shown in Fig. 1a. The source mass will be rotated such that the “teeth” transit the 3 He cell at the Larmor frequency. The receptacle which houses the 3 He sample will be a spheroidal shaped cavity in a piece of fused quartz. Three 3 He sample chambers will be positioned around the sprocket so that cross-correlation among them can be used to suppress backgrounds (see Fig. 1b). The average gap between the tungsten source mass and the 3 He sample is kept below 200 µm, (within the Compton wavelength of the axion, as determined by the targeted mass range) in order to detect the pseudomagnetic field. Since the axion signal manifests as a magnetization of the nuclear spins, we must shield the spin sample from electromagnetic backgrounds. In order to achieve this, a superconducting magnetic shielding will be deployed. This shield is formed by sputtering a ∼1 µm layer of niobium onto the surfaces of the quartz chambers. The shield is anticipated to suppress magnetic backgrounds for sufficiently low

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magnetization and contamination of the source mass. A thorough characterization of a source mass prototype is the main focus of this paper. Here we present a description of the source mass fabrication and specifications as well as residual magnetization measurements. The application of a large external magnetic field to the prototype resulted in a residual permanent magnetization, indicating the presence of magnetic impurities. However, this magnetization could be reduced by an order of magnitude by degaussing. The remaining magnetization as well as Johnson noise ultimately determines the magnetic shielding requirements for the thin film Nb shield. The results obtained thus far are sufficiently low for the anticipated shielding factor.

2 Source Mass Fabrication and Characterization 2.1 Prototype Source Mass The raw tungsten for the source mass is quoted to be greater than 99.95% pure [20]. This material was chosen for its high nucleon density as well as its low magnetic susceptibility. The sprocket was machined using wire electrical discharging machining [21]. The sprocket contains 11 teeth equally spaced around its circumference. The teeth ensure that the source drive frequency differs from the frequency of modulation in order to isolate vibrational backgrounds at the rotation frequency. The outer radius is 19 mm at the teeth and 18.8 mm between teeth. Thus, the source mass modulation amplitude is 200 microns as determined by the target axion mass range. The sprocket radii were machined with a dimensional tolerance of ±10 microns in order for the modulation of the axion signal to be uniform. The sprocket thickness is 1 cm in order to fully subtend the 3 He sample cell. The sprocket will need a high level of radial stability as it will be placed 50 µm from the surface of the quartz vessel that houses the helium sample. A fiber optic interferometer system will be employed to monitor a laser reflected off the bottom surface of the sprocket and assess both the speed stability and “wobble” of the sprocket. The surface finish of the sprocket will need to be highly reflective in this respect. The top and bottom flat surfaces were finished with 220 and 440 grit abrasive sheet, giving an RMS surface roughness of 1.4–2.1 µm as observed via profilometry measurements [22]. In the experiment, the sprocket’s rotational velocity needs to remain stable such that the modulation frequency stays matched to the Larmor precession. In order to monitor rotational velocity, a higher surface quality will be needed to facilitate the optical interferometry. The periodic velocity pattern will therefore be applied to an optically flat silicon disk epoxied to the bottom surface of the sprocket (Fig. 2a).

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Fig. 2 (a) Schematic detailing the pattern that will be applied to the surface of the sprocket in order to monitor the rotational speed. (b) A photograph of the prototype tungsten sprocket

2.2 Relevant Magnetic Backgrounds The sensitivity of the experiment to an effective magnetic field is expected to be limited by the transverse projection noise in the 3 He sample, to the level of Bmin = √ −19 3 × 10 T (1000 s/T2 ) T / H z, where the T2 is the 3 He relaxation time. With a T2 of 1000 s and sufficient integration time 106 s, this sensitivity would be sufficient to probe the axion coupling (Eq. (1)) at the level below the current constraint set by the EDM limits. Thus it is important to keep other magnetic backgrounds below Bmin . We anticipate the following to be the dominant sources of magnetic background: thermal Johnson noise, the Barnett effect, magnetic impurities in the source mass, and the magnetic susceptibility of the source mass. Johnson Noise All electrical conductors have thermal motion of electrons which generate electronic noise, known as Johnson noise [23]. As a conductor, the tungsten sprocket will have an anticipated level of 10−12 √TH z of Johnson noise at its expected temperature of ∼170 K. A magnetic shielding factor f of 108 lowers this background to 10−20 √TH z (Table 1).

Barnett Effect When an uncharged object is rotated on its axis, it will acquire a magnetization dependent on its magnetic susceptibility χ and frequency of rotation ω [24]: M=

χω γ

(2)

where γ is the gyromagnetic ratio. For tungsten, χ = 6.8 × 10−5 [25] and the frequencies of rotation relevant for this experiment will be about 5–9 Hz. We expect

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Table 1 Estimated source mass related systematic error and noise sources, as discussed in the text Systematic effect/ Noise source Johnson noise Barnett effect Magnetic impurities in mass Mass magnetic Susceptibility

Background level √ 8 10−20 ( 10f )T/ Hz 8 10−22 ( 10f ) T 10−25 − 10−17 ( 8

10−22 ( 10f ) T

Notes f is SC shield factor (100 Hz) Can be used for calibration above 10 K

η 108 1ppm )( f )

T

η is impurity fraction Assuming background field is 10−10 T

Background field can be larger if f > 108 √ 1/2 T/ Hz. Further discussion of The projected sensitivity of the device is 3 × 10−19 ( 1000s T2 ) systematic effects and noise levels can be found in Ref. [19]

a background magnetic field of 10−14 T to be produced by the Barnett effect. With a magnetic shielding factor of 108 the background noise is reduced to 10−22 T. Magnetic Susceptibility We anticipate a background level of 10−22 T due to the magnetic susceptibility of the mass, assuming a background field of 10−10 T and a shielding factor of 108 . Magnetic Impurities Iron contaminants can be present in “pure” samples of tungsten metal. Assuming Fe impurities at a concentration of 1 ppm, we expect the residual magnetic field to be between 10−17 T and 10−9 T, depending on whether the moments are oriented randomly or perfectly aligned. Again, assuming a shielding factor of 108 , this background is reduced to 10−25 T–10−17 T which is above the specification but only for the worst-case scenarios.

2.3 Characterization Measurement Procedure In this section, we present results on the magnetization and magnetic impurity tests of the tungsten source mass and describe a future method for obtaining higherprecision results. Measurements were carried out at the Physikalisch-Technische Bundesanstalt (PTB) laboratory in Berlin. To enhance the sensitivity of these measurements to potential contaminants, the source mass was magnetized with a 30 mT neodymium magnet. The resulting magnetization was then measured in a magnetically shielded room with a multi-channel superconducting quantum interference device (SQUID) system with the sprocket at room temperature. This SQUID system contains 19 modules where the relevant plane to this measurement is the plane shown in Fig. 3c. The blue dots and arrows indicate the orientation of each magnetometer pickup loop. Three of them are oriented to measure in the Z directions (Z1, Z2, Z3) and three oriented to measure in the X and Y directions (X1, V1, A1). These six signals can be used to assess the direction and localization

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Fig. 3 (a) Photograph of the tungsten sprocket in a wooden holder used for rotation positioned below the bottom surface of a dewar containing the SQUID measurement system. (b) Photograph showing the wooden rotation assembly and string for manual rotation. (c) A top view schematic of one of the 19 modules of the multi-channel SQUID system. SQUIDs Z1, Z2, Z3 are in the positive Z direction (out of the plane of the page, and thus vertical in photo (a)), and the SQUIDS X, V, and A are oriented to measure in the X and Y directions. The radius of the dotted black line is 16.667 mm. The brown dotted rectangle is indicating the location and orientation of the rotor

of magnetic contaminants. The brown dotted rectangle indicates the location and orientation of the wheel. Static and rotational tests were performed. Using the measured field values from the SQUID sensor array, the net magnetization of the sample and inferred surface field was determined. A hard disk eraser generating a 500 mT magnetic field amplitude at 50 Hz was then used to demagnetize the sprocket via a usual decreasing AC-amplitude degaussing process. The magnetization was measured before and after degaussing. The rotational tests were performed by placing the source mass in a wooden structure (Fig. 3a) and placed below the dewar containing the SQUID system (Fig. 3c) where the brown dotted rectangle depicts the orientation of the rotor. The rotation was done manually by string (Fig. 3b) where the frequency of rotation was inconsistent. This rotational test was done before and after degaussing for overall magnetic field signal comparison. Further tests can be performed to analyze more closely the magnetic impurity orientations and locations. These tests will include measurements with QuSPIN [26] optical magnetometers allowing for more control of the distance between sensor and sprocket, and a servo motor setup to provide rotation speed control. This allows a more direct measurement of the near-surface fields, which are most relevant for the ARIADNE experiment.

3 Results The static test found that after magnetization the tungsten source mass exhibited a net magnetic dipole moment of about 20 nAm2 corresponding to an approximate surface magnetic field of about 300 pT, thus indicating that the source mass or

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Fig. 4 Output from the six SQUID sensors (see Fig. 3c for the orientation) for the rotational test done before degaussing (in blue) and output after degaussing (in orange)

its impurities can be slightly magnetized. The value of this dipole moment was deduced from the data collected from the 19-module squid array. After degaussing, the residual magnetic moment in the sample was reduced to a level which precluded a good fit to a magnetic dipole. The corresponding field signal at the sensors was reduced to approximately 2 pT at about a 3 cm distance. For the rotational tests, a string was pulled to rotate the wheel by approximately 14 full rotations over a time period of 120 s. Figure 4 shows the output from the 6 SQUID sensors in the lowest level module. The largest signals before degaussing have an amplitude around 8 pTpp (Fig. 4). Here the separation between the sensors and the edge of the wheel is 30 mm ± 5 mm. After degaussing, the signals were reduced to approximately 2 pTpp (Fig. 4). Additionally, the wheel was found to generate Johnson noise on the level of 1–1.5 pTpp for a bandwidth of about DC100 Hz (250 Hz sampling rate). Note that in Fig. 4 the 14 peaks present in the data (e.g., from sensor Z3 or Z2) correspond to the 14 manual rotations of the wheel. As the string was pulled manually, the time between revolutions of the wheel is not uniform. Future tests will employ a rotation mechanism with constant speed which will allow us to analyze the Fourier harmonics of the signal from the rotation and separate out any underlying signal which occurs at the 11-fold periodic pattern of the teeth in the wheel. Considering the data, we can roughly estimate that the residual field near the surface of the wheel after degaussing will be at or below the level of

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∼30 pT. The data are consistent with a single, or possibly double, localized region of contaminant, as opposed to uniformly distributed contaminants in the sprocket, since the modulated signal does not appear to be varying 11-fold upon each sprocket rotation. The residual signal at this level with 11-fold response per rotation can be bounded approximately at the few pT level, which would require a shielding factor in ARIADNE of approximately 108 to achieve full design sensitivity. We find these preliminary results to be encouraging; however, further tests using the SQUID system and QuSPIN optical magnetometer system will be performed to obtain more quantitative results on the localization and orientation of any magnetic contaminants. Also a clearer measurement of the residual field from Johnson noise should be obtainable. These results will be compared with the background specifications for ARIADNE.

4 Discussion and Outlook To summarize, we have tested a procedure to fabricate the source mass for the ARIADNE experiment and we have characterized its residual magnetism. Preliminary tests have shown that the impurities can be magnetized for detailed characterization, as well as demagnetized for eventual use in the experiment. Currently, we are developing methods to probe the field at a shorter distance from the source mass. This will also permit better control of its rotation speed as well as its position relative to the probe. This is an important step to ascertain if the current metal purity level and the fabrication procedure are acceptable for use in the ARIADNE experiment, or if there are any unexpected background sources. Besides the source mass, work is underway to construct the main cryostat that contains the liquid 4 He cooling system, the quartz holder for the sample 3 He, and superconducting niobium shield. Niobium coatings are being characterized for the critical temperature, shielding factor, and adhesion to a quartz surface. Acknowledgments We acknowledge the work of Alex Brown, who as an undergraduate at Indiana University produced the mechanical drawings for the tungsten sprocket described in this work, and conducted measurements to confirm that the raw tungsten material met the necessary magnetic specifications before the sprocket was cut. Alex passed away in the fall of 2018 while beginning the graduate program in Physics at Florida State University. We acknowledge the support from the U.S. National Science Foundation, grant numbers NSF PHY-1509176, NSF PHY-1510484, NSF PHY-1506508, NSF PHY-1806671, NSF PHY-1806395, NSF PHY-1806757.

References 1. A. Arvanitaki, A.A. Geraci, Resonantly detecting axion-mediated forces with nuclear magnetic resonance. Phys. Rev. Lett. 113, 161801 (2014). https://link.aps.org/doi/10.1103/PhysRevLett. 113.161801

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2. R.D. Peccei, H.R. Quinn, CP conservation in the presence of pseudoparticles. Phys. Rev. Lett. 38, 1440–1443 (1977). https://link.aps.org/doi/10.1103/PhysRevLett.38.1440 3. R. Bähre, et al., Any light particle search II—Technical Design Report. JINST 8, T09001 (2013) 4. B. Graner, Y. Chen, E.G. Lindahl, B.R. Heckel, Reduced limit on the permanent electric dipole moment of 199 Hg. Phys. Rev. Lett. 116, 161601 (2016). https://link.aps.org/doi/10.1103/ PhysRevLett.116.161601 5. J.M. Pendlebury, et al., Revised experimental upper limit on the electric dipole moment of the neutron. Phys. Rev. D92(9), 092003 (2015) 6. J.E. Moody, F. Wilczek, Phys. Rev. D 30(1), 130 (1984) 7. Gondolo, P., Visinelli, L.: Axion cold dark matter in view of bicep2 results. Phys. Rev. Lett. 113, 011802 (2014). https://link.aps.org/doi/10.1103/PhysRevLett.113.011802 8. D.J.E. Marsh, D. Grin, R. Hlozek, P.G. Ferreira, Tensor interpretation of BICEP2 results severely constrains axion dark matter. Phys. Rev. Lett. 113, 011801 (2014). https://link.aps. org/doi/10.1103/PhysRevLett.113.011801 9. Beringer, J. et al., Review of particle physics. Phys. Rev. D 86, 010001 (2012). https://link.aps. org/doi/10.1103/PhysRevD.86.010001 10. S.J. Asztalos, G. Carosi, C. Hagmann, D. Kinion, K. van Bibber, M. Hotz, L.J. Rosenberg, G. Rybka, J. Hoskins, J. Hwang, P. Sikivie, D.B. Tanner, R. Bradley, J. Clarke, Squid-based microwave cavity search for dark-matter axions. Phys. Rev. Lett. 104, 041301 (2010). https:// link.aps.org/doi/10.1103/PhysRevLett.104.041301 11. D. Budker, P.W. Graham, M. Ledbetter, S. Rajendran, A.O. Sushkov, Proposal for a cosmic axion spin precession experiment (casper). Phys. Rev. X 4, 021030 (2014). https://link.aps.org/ doi/10.1103/PhysRevX.4.021030 12. G. Rybka, A. Wagner, K. Patel, R. Percival, K. Ramos, A. Brill, Search for dark matter axions with the orpheus experiment. Phys. Rev. D 91, 011701 (2015). https://link.aps.org/doi/10.1103/ PhysRevD.91.011701 13. A. Caldwell, G. Dvali, B. Majorovits, A. Millar, G. Raffelt, J. Redondo, O. Reimann, F. Simon, F. Steffen, Dielectric haloscopes: a new way to detect axion dark matter. Phys. Rev. Lett. 118(9), 091801 (2017) 14. A. Phipps, S.E. Kuenstner, S. Chaudhuri, C.S. Dawson, B.A. Young, C.T. FitzGerald, H. Froland, K. Wells, D. Li, H.M. Cho, S. Rajendran, P.W. Graham, K.D. Irwin, Exclusion limits on hidden-photon dark matter near 2 neV from a fixed-frequency superconducting lumpedelement resonator (2019) 15. L. Zhong, et al., Results from phase 1 of the HAYSTAC microwave cavity axion experiment. Phys. Rev. D97(9), 092001 (2018) 16. Y. Kahn, B.R. Safdi, J. Thaler, Broadband and resonant approaches to axion dark matter detection. Phys. Rev. Lett. 117, 141801 (2016). https://link.aps.org/doi/10.1103/PhysRevLett. 117.141801 17. R. Barbieri, C. Braggio, G. Carugno, C.S. Gallo, A. Lombardi, A. Ortolan, R. Pengo, G. Ruoso, C.C. Speake, Searching for galactic axions through magnetized media: the QUAX proposal. Phys. Dark Univ. 15, 135–141 (2017) 18. P. Sikivie, N. Sullivan, D.B. Tanner, Proposal for axion dark matter detection using an LC circuit. Phys. Rev. Lett. 112(13), 131301 (2014) 19. H. Fosbinder-Elkins, C. Lohmeyer, J. Dargert, Cunningham, M., M., H., E. Levenson-Falk, S. Mumford, A. Kapitulnik, A. Arvanitaki, I. Lee, E. Smith, E. Weisman, J. Shortino, J.C. Long, W.M. Snow, C.Y. Liu, Y. Shin, Y. Semertzidis, Y.H. Lee, Progress on the ARIADNE axion experiment, in Microwave Cavities and Detectors for Axion Research: Proceedings of the 2nd International Workshop (2018), pp. 151–161 20. Source: Midwest Services, 544 IL-185 (Vandalia IL 62471) 21. Wire EDM performed by Stacy Machine and Tooling, Inc., 2810 industrial ln, Broomfield Co 80020 22. A. Brown, Development of a tungsten rotor for the Axion resonant interaction detection experiment (ARIADNE), Masters thesis, Indiana University, Bloomington (2018)

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23. T. Varpula, T. Poutanen, Magnetic field fluctuations arising from thermal motion of electric charge in conductors. J. Appl. Phys. 55(11), 4015–4021 (1984). https://doi.org/10.1063/1. 332990 24. S.J. Barnett, Magnetization by rotation. Phys. Rev. 6, 239–270 (1915). https://link.aps.org/doi/ 10.1103/PhysRev.6.239 25. R. Weast, Handbook of Chemistry and Physics (CRC, New York, 1984), p. E110 26. Quspin Inc., 331 south 104th street, suite 130, Louisville, Co 80027

CAPP-PACE Experiment with a Target Mass Range Around 10 µeV Doyu Lee, Woohyun Chung, Ohjoon Kwon, Jinsu Kim, Danho Ahn, Caglar Kutlu, and Yannis K. Semertzidis

Abstract CAPP-PACE is a pilot experiment of IBS/CAPP for direct detection of axions with a mass around 10 µeV based on Sikivie’s microwave cavity scheme. The detector is equipped with an 8 T superconducting magnet and employs a dilution refrigerator that lowers the physical temperature of a high Q-factor split-type cavity to less than 45 mK. The frequency tuning system utilizes piezoelectric actuators with interchangeable sapphire and copper rods. The feeble signal ( 10−24 W) from the cavity is amplified and transmitted through the RF receiver chain with a HEMT amplifier whose noise temperature is around 1 K. I will present the results of CAPP’s first physics data runs in the axion mass range from 2.45 to 2.75 GHz and discuss our future plans and R&D projects. Keywords Axion · RF cavity · CAPP · Dilution refrigerator · Frequency tuning

1 Introduction The axion is a hypothetical particle which was theorized in the early 1970 as a solution of strong CP problem in the standard model [1–4]. The axion search has been accelerated since it is announced that the light axion is an excellent candidate D. Lee () · J. Kim · D. Ahn · Y. K. Semertzidis Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea Center for Axion and Precision Physics Research, Institute for Basic Science, Daejeon, Republic of Korea e-mail: [email protected] W. Chung · O. Kwon Center for Axion and Precision Physics Research, Institute for Basic Science, Daejeon, Republic of Korea C. Kutlu Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_10

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for the cold dark matter and since P. Sikkive suggested that the microwave resonant cavity in the strong magnetic field should detect the axion mass [5]. In Korea, the Center for Axion and Precision Physics Research (CAPP) at Institute for Basic Science (IBS) was established in 2013 and we have been built the state-of-the-art axion detector based on the P. Sikkivie’s method. We named the haloscope experiments in CAPP as CAPP’s Ultra-Low Temperature Axion Search experiment in Korea (CULTASK) and the first experimental project of the CULTASK, i.e., CAPP-PACE (Pilot Axion Cavity Experiment) has started taking data since Jan 2018 aiming to scan 10.13∼11.37 µeV (2.45 ∼ 2.75 GHz). In this proceeding, we briefly show the experimental aspects of CAPP-PACE.

2 CAPP-PACE (CAPP’s Pilot Axion Cavity Experiment) The axion haloscope model can be summarized by the following equation which describes the axion search scan speed [5] df −2 ∼ B 4 V 2 C 2 QL Tsys dt

(1)

As seen in the equation, the magnetic field is the most powerful parameter to decide scanning speed. Moreover, it gives constraints for the cavity relevant parameters such as cavity volume, axion searching mass range, and so on because the bore of the solenoid magnet has limited space. At CAPP-PACE, we have used 12 cm bore of AMI magnet, the center field of which is 8T [6]. Fortunately this is an adequate bore size for the frequency range aimed at CAPP-PACE. Figure 1 shows the three-dimensional rendering image as well as the realistic photo image of the CAPP-PACE setup. The key features are: Cryogen-free dilution refrigerator from BlueFors can reach below 10 mK and cool down the cavity below 45 mK during the experimental runs. [7] 8T Low Temperature Superconductor (LTS) magnet with 12 cm bore. High quality-factor (Q-factor) (unloaded Q-factor >100,000) of resonance cavity with a vertical split design minimizes the Q-factor degradation of TM010 mode at cavity assembling joints. The cavity volume is 0.59L for the first run and 1.12L for other runs. The frequency tuning system with piezoelectric is capable of kilohertz level adjustment at sub-Kelvin temperature. Best commercial High-Electron-Mobility Transistor (HEMT) amplifier has ∼1 K noise temperature [8]. The data acquisition system (DAQ) with online monitoring and safety alarm minimizes the human fatigue during the experiment with magnet ON. Typical but sure configuration of RF layout. Figure 2 shows the RF readout chain of PACE which is typical but sure configuration. In the cryostat, we amplify the signal by 80 dB using two low noise amplifiers, and we amplify it by 23 dB in the room temperature. We analyze the data using Rohde and Schwarz FSV 7 spectrum analyzer that has more than 90% efficiency.

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Fig. 1 Three-dimensional rendering image of CAPP-PACE detector setup with BlueFors LD400 model (left) and the photo of the inside cryostat (right). The gold disk structures are the still plate, the 100 mK plate, and the MXC plate, respectively, from top to bottom, and the lowest part is the microwave cavity that is thermally linked to the MXC plate through oxygen free high thermal conductivity copper (OFHC) fixture

Our first engineering run was done in Jan 2018. Its frequency range was 2.45∼2.5 GHz, and target sensitivity was 10 × KSVZ. Frequency tuning system with a sapphire rod and piezoelectric actuator was flawlessly operating. All the DAQ controls and monitoring were functional stably even for more than a month of operation. In the second run, we upgraded the cavity by doubling the cavity volume and we changed the tuning range by inserting an additional copper rod. We started our second run aiming for 2.5∼2.6 GHz with 10 × KSVZ sensitivity, and we could achieve about four times faster-scanning speed than the first run. In the next run, we targeted the KSVZ sensitivity with narrower frequency range, i.e., 2.5905∼2.5916 GHz. Figure 3 schematically shows the entire schedules of each experimental run of CAPP-PACE in 2018.

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Fig. 2 RF receiver chain of CAPP-PACE detector. Block diagram of the cryogenic RF chain (left) and room temperature RF chain (lower right), DAQ monitor and structure (upper right)

10 KSVZ 1st run

26 days

10 KSVZ 2nd run

Jan 19 - Feb 13, Target sensivity is 10 KSVZ, scan frequency is from 2.45GHz to 2.50GHz.(50MHz range) 10 days

10 KSVZ 3rd run

Jul 23 - Aug 01, Target sensivity is 10 KSVZ, scan frequency is from 2.5GHz to 2.55GHz.(50MHz range)

18 days GD\V

KSVZ 1st run KSVZ 2nd run

30 days

Sep 27 - Oct 26, Target sensivity is KSVZ, scan frequency is from 2.5909GHz to 2.5916GHz.(0.7MHz range)

KSVZ 2nd run

30 days

Sep 27 - Oct 26, Target sensivity is KSVZ, scan frequency is from 2.5909GHz to 2.5916GHz.(0.7MHz range)

10 KSVZ 2nd run 10 KSVZ 3rd run

10 KSVZ 1st run Jul 23

2018

Aug 14 - Aug 23, Target sensivity is 10 KSVZ, scan frequency is from 2.55GHz to 2.61GHz.(60MHz range)

Sep 01 - Sep 21, Target sensivity is KSVZ, scan frequency is from 2.5905GHz to 2.5909GHz.(0.4MHz range) GD\V 24 days

Jan

Jul 23

Feb 13

Feb

Jul

Aug 01

Aug

KSVZ 1st run

Aug 14 Aug 23 Sep 01

Sep

KSVZ 2nd run

Sep 21 Sep 27

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Oct 26

Nov

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Fig. 3 2018 experiment schedule of CAPP-PACE. There were three runs targeting to 10 KSVZ sensitivity (total ∼160 MHz), and there were two runs targeting to KSVZ sensitivity (total ∼1.1 MHz)

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3 Conclusions and Prospects In the coldest circumstance ever of the axion search, CAPP’s first physics run, CAPP-PACE has been started since Jan 2018. We divide the 2.45∼2.75 GHz region into three regions and took data with 10×KSVZ sensitivity. Moreover, we took data with KSVZ sensitivity at 1 MHz region near 2.59 GHz. In 2019, we expect that we can use a superconducting quantum interface device (SQUID) based amplifiers such as Microstrip SQUID Amplifier (MSA) and Josephson Parametric Amplifier (JPA) that we have been testing [9]. Thus we can speed up the axion search at least 20 times faster than 2018. After 2020, CAPP will have much stronger and bigger magnets. 25 T, 10 cm bore HTS magnet will be delivered in 2020 by Brookhaven National Laboratory (BNL) [10], and also we will get 12 T, 32 cm bore LTS magnet from Oxford Instruments [11] in 2021. Therefore, CAPP will have the ability to scan broad candidate areas that are likely to have axions not just with KSVZ but also with the DFSZ sensitivity. Acknowledgments This work was supported by IBS-R017-D1-2018-a00 in the Republic of Korea.

References 1. R.D. Peccei et al., CP conservation in the presence of pseudoparticles. Phys. Rev. Lett. 38, 1440 (1997). https://doi.org/10.1103/PhysRevLett.38.1440 2. R.D. Peccei et al., Constraints imposed by CP conservation in the presence of pseudoparticles. Phys. Rev. D 16, 1791 (1997). https://doi.org/10.1103/PhysRevD.16.1791 3. S. Weinberg, A new light boson?. Phys. Rev. Lett. 40, 223 (1978). https://doi.org/10.1103/ PhysRevLett.40.223 4. F. Wilczek, Problem of strong P and T invariance in the presence of instantons. Phys. Rev. Lett. 40, 279 (1978). https://doi.org/10.1103/PhysRevLett.40.279 5. P. Sikivie, Phys. Rev. Lett. 51, 1415 (1983). Erratum: [Phys. Rev. Lett. 52, 695 (1984)]. https:// doi.org/10.1103/PhysRevLett.51.1415, https://doi.org/10.1103/PhysRevLett.52.695.2 6. American Magnetics. http://www.americanmagnetics.com 7. Bluefors LD400 Dilution Refrigerator. https://www.bluefors.com 8. Low Noise Factory. https://www.lownoisefactory.com/ 9. A. Matlashov, M. Schmelz, V. Zakosarenko, R. Stolz, Y.K. Semertzidis, Squid amplifiers for axion search experiments. Cryogenics 91, 125–127 (2018) 10. Brookhaven National Laboratory. https://www.bnl.gov 11. Oxford Instruments Nanoscience. https://nanoscience.oxinst.com/

High Resolution Data Analysis: Plans and Prospects Shriram Jois, Leanne Duffy, Neil Sullivan, David Tanner, and William Wester

Abstract A report on the progress on the high resolution data analysis of the ADMX experimental results is presented. In this paper, tools are developed and tested on a blind injection mimicking a Maxwellian like signal in the frequency domain. This blind injection will be used as a test bed which can be later implemented on all the high resolution data. The high resolution data is stored in the Fermilab server. In this analysis a PostgreSQL query was made to ensure the blind injection is in the middle of the frequency spectrum and 19 such files were found. The time series data is read using a c++ program. An apodization function is applied on the time series data and zero filled to reduce the frequency spacing in order to achieve a better interpolation. A FFTW header is used to compute the Fourier transform of the time series data. A Savitzky–Golay filter is applied on the unnormalized power which then can be used to remove the spectral shape. Each frequency spectrum has a bandwidth of 50 kHz. Keywords ADMX · Axion · Dark matter · Sine wave injection · Synthetic axion injection · High resolution data analysis · Apodization functions · Spectral shape · Savitzky–Golay filter · Exponential distribution

S. Jois () · N. Sullivan · D. Tanner Department of Physics, University of Florida, Gainesville, FL, USA e-mail: [email protected]; [email protected]; [email protected] L. Duffy Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: [email protected] W. Wester Fermi National Accelerator Laboratory, Batavia, IL, USA e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_11

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1 Introduction The Axion Dark Matter eXperiment (ADMX) uses a cylindrical RF cavity in a high magnetic field and a superconducting amplifier to search for a hypothetical particle called the axion, which if it exists is a solution to the strong CP problem in QCD [1] and a possible cold dark matter candidate [2]. During the experiment the axion decays into photons in the presence of a high magnetic field in a process called the Primakoff conversion. Since the mass of axion is unknown, to detect these photons the RF cavity is tuned to a frequency that matches the total energy of the axion 2 and resonantly enhances the conversion process (fa = mahc ). Since the galactic halo axions are non-relativistic, the total energy of axion is given by Ea = ma c2 + 1 2 2 ma v . The signal undergoes a diurnal and annual modulation due to the Earth’s rotation and revolution around the Sun. The frequency modulation is given by δf fa = va |δv| . The frequency shift has been calculated by Ling et al. in 2004 [3]. The shift in c2 a 100s time interval due to the Earth’s orbital motion Δfo is O(10−12 f0 ) and for the Earth’s rotation Δfr is O(10−11 f0 ). The high resolution channel has a frequency

resolution of 0.01 s and will be able to detect these frequency modulation when the axions are found, whereas the velocity dispersion of axions in the medium resolution channel is O(10−4 c) [4]. To test and develop a search pattern, a sine wave injection or blind injection can be used. This has been demonstrated by the advanced LIGO group to validate the discovery of gravitational waves [5].

2 Fast Fourier Transform Techniques The time series gets truncated due to the finite number of points. The instrument function I (ω) of the time series data f (t) is defined by the exponential Fourier transformation 1 I (ω) = 2π



T 2 T 2

dt f (t)a(t)e−iωt .

(1)

If the time series data, f (t) contains a periodic signal, e.g. a sine wave, the Fourier transform of f (t) a(t) will be a convolution of F (ω) and A(ω). In the absence of an apodization function, a(t)  = 1. The Fourier transform of the rectangular apodization is, A(ω) =

1 2π

sin ω T2 ω T2

. Because the F (ω) is a delta function, the

convolution of F (ω) and A(ω) will be a sinc function. To minimize the sidelobes in the sinc function, an apodization function is often used in the spectral analysis. In a high resolution data analysis, the spectrum is dominated by noise, hence minimizing the side lobes is not required. The frequency resolution of the spectrum is set by the measurement time which for this case is 0.01 Hz. However a better interpolation is

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achieved by adding zeros to the time series [6]. If f(n) is the time series and f(k) its Fourier transform f (k) =

N −1 n=0

2π kn

f (n)e−i N+M +

−1 : 0 N kn 2π kn 2π −i N+M = f (n)e f (n)e−i N+M .  

M+N −1

n=N

(2)

n=0

Here N is the number of points in the time series and M is the zeros added to the time series. The frequency resolution of the original time series was Ω = 2π N . This 2π got reduced to Ωe = N +M . The data analysis can proceed in two steps. The first is to find the candidates. Once the candidates are found, it can be zero filled to reduce the frequency spacing which results in a better interpolation. A peak detection algorithm is likely to find the center frequency of the axion signal more accurately in a zero filled spectrum than the original frequency spectrum due to smaller frequency spacing. A compromise needs to made since a higher order zero will fill up the disk space faster than a lower order zero fill (Fig. 1).

Fig. 1 This figure shows the Fourier transform of a sine wave with a 25 kHz frequency with various degrees of zero fill. The peak of the signal is closer to 25 kHz on a zero filled spectrum than the original spectrum

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3 Power Spectrum and Distribution The time series data has a time spacing of Δt = 10.24 µs. Each data file contains a total of 9 million data points. The zero fill is done as a multiple of the total number of data points. Making the total number of points a power of two will make the program run faster. Once the apodization function is chosen and the amount of zero fill is decided, a Fourier transform of the time series is computed (Fig. 2). In this paper, a FFTW package was used on a c++ program to compute the Fourier transform 1 [7]. The Fourier frequency is computed as fF = kΔt and k can have values from n−1 0 to 2 , where n is the total number of data points in the time series. The start frequency of the spectrum is stored in the time series file which can be read and added to the Fourier frequency to get the actual frequency of the spectrum. The complex Fourier transform can be expressed as sine and cosine transforms. The sine and cosine transform of the noise in the time series follows a Gaussian distribution with two random variables in the frequency domain. The distribution function is given by −x 2 −y 2 dP 1 2σ 2 e 2σ 2 , = e dA 2π σ 2

(3)

Fig. 2 Comparison of a Hann apodization and a no-apodization frequency spectrum with background noise and a blind axion injection

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converting x and y to polar coordinates r and θ . The θ integral can be integrated to get 2π . This gives us the Rayleigh distribution 1 P = 2π σ 2





∞

r dr 0

2π

dθ e

−r 2 2σ 2

0

1 = 2 σ



∞

−r 2

r dr e 2σ 2 .

(4)

0

Since we are interested in the power due to Primakoff conversion, the amplitude can be squared to find the rms power. The resulting probability distribution function is an exponential distribution 1 −p dP = 2 e σ2 . dp σ

(5)

Taking the logarithm of the above equation, we find log(P (x)) = −

p + c. σ2

(6)

This is of the form y = mx + c, a straight line with a negative slope (Fig. 3). A generalized distribution function for n-bin data has been derived by L. Duffy et. al [8].

Fig. 3 Histogram of unfiltered noise spectrum on a semilog scale which follows Eq. (7). The scattering at the end of the line is due to the blind axion injection. A spectrum with a real axion signal is expected to have a similar distribution

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4 Spectral Shape of the Noise The spectral shape of the noise (Fig. 2) refers to the amplitude dependence of the noise as a function of the frequency and is due to the systematic effects that are introduced in the receiver chain. These systematic errors should be corrected before establishing a noise baseline. In the previous high resolution searches [4, 8], the spectral shape was removed by dividing the frequency data by a polynomial of order n as a function of frequency offset using the equivalent circuit model which estimates the rms value of noise power at the NRAO output. The spectral shape can be removed even without the equivalent circuit model by a textbook implementation of Savitzky–Golay digital filter [9] (Fig. 4). Due to the number of points in the time series, the window size of such a filter should be large to achieve a smaller standard deviation in the filtered data. With the use of high performance computing techniques, the Savitzky–Golay filters can be employed for the removal of spectral shape. The Savitzky–Golay filter has two variables, window size and order. The filtered data is a convolution of Savitzky–Golay coefficients and the unnormalized data and can be written as [10]

Fig. 4 Plot of the frequency spectrum after the application of a Savitzky–Golay filter on the noise spectrum. The Savitzky–Golay has preserved the spectral shape of the noise distribution which can be used to flatten the noise spectrum

HiRes

95 M−1

PjSG (f ) =

2

Ci Pi+j (f ).

(7)

i=− M−1 2

Here, the window size of the filter is M and Ci are the normalized Savitzky–Golay convolution coefficients. The coefficients can be calculated by forming a least square polynomial of order n and using the Moore–Penrose inverse to solve a set of M+1 linear equations. The sum of all coefficients is one. The sharp dip at the ends of the Savitzky–Golay filtered spectrum is due to the Gibbs phenomenon. Since the Savitzky–Golay filter requires convolution, the data needs to be extended at the start and the end. This determines the length of the dip at the ends of the spectrum. One way to remove the dip is to use Gegenbauer polynomials [11]. However, in this paper it has been removed by chopping off the ends. The data that has been lost can be covered in the next spectrum.

Fig. 5 Plot of the flattened frequency spectrum after the removal of the spectral shape. The Savitzky–Golay has preserved the spectral shape of the noise distribution which was used to flatten the frequency spectrum. The mean is plotted in red and is equal to 1

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5 Future Work Since the window size of the Savitzky–Golay filter is chosen such that the mean of the filtered data is equal to the unnormalized power, the mean of the normalized spectrum is equal to 1. This can be seen in Fig. 5. To improve the S/N, the frequency data can be averaged by sub-dividing the time series data and zero filling the subdivided time series and averaging all the frequency spectra. This is similar to what the spectrum analyzer does. Once the apodization function is chosen and the time series is subdivided and zero filled, the exclusion limits can be set. The search pattern that is developed on the blind injection can be used in the analysis of the data obtained in the experimental search for axions. Acknowledgments This work was supported by the Department of Energy Grant No. DE181SC0010296 at the University of Florida.

References 1. R.D. Peccei, H.R. Quinn, CP conservation in the presence of pseudoparticles. Phys. Rev. Lett. 38(25), 1440 (1977) 2. J.R. Primack, B. Seckel, B. Sadoulet, Detection of cosmic dark matter. Annu. Rev. Nucl. Part. Sci. 38(1), 757–807 (1988) 3. F.-S. Ling, P. Sikivie, S. Wick, Diurnal and annual modulation of cold dark matter signals. Phys. Rev. D 70, 123503 (2004) 4. J. Hoskins, N. Crisosto, J. Gleason, P. Sikivie, I. Stern, N.S. Sullivan, D.B. Tanner, C. Boutan, M. Hotz, R. Khatiwada, D. Lyapustin, Modulation sensitive search for nonvirialized darkmatter axions. Phys. Rev. D 94(8), 082001 (2016) 5. C. Biwer, D. Barker, J.C. Batch, J. Betzwieser, R.P. Fisher, E. Goetz, S. Kandhasamy, S. Karki, J.S. Kissel, A.P. Lundgren, D.M. Macleod, Validating gravitational-wave detections: the advanced LIGO hardware injection system. Phys. Rev. D 95(6), 062002 (2017) 6. M.L. Forman, Spectral interpolation: zero fill or convolution. Appl. Opt. 16(11), 2801–2801 (1977) 7. M. Frigo, S.G. Johnson, The design and implementation of FFTW3. Proceedings of the IEEE 93.2 216–231 (2005). 8. L.D. Duffy, P. Sikivie, D.B. Tanner, S.J. Asztalos, C. Hagmann, D. Kinion, L.J. Rosenberg, K. Van Bibber, D.B. Yu, R.F. Bradley, High resolution search for dark-matter axions. Phys. Rev. D 74(1), 012006 (2006) 9. A. Savitzky, M.J. Golay, Smoothing and differentiation of data by simplified least squares procedures. Anal. Chem. 36(8), 1627–1639 (1964). Vancouver 10. S.J. Orfanidis, Introduction to signal processing (Prentice-Hall, Inc, New York, 1995) 11. D. Gottlieb, C.W. Shu, On the Gibbs phenomenon and its resolution. SIAM Rev. 39(4), 644– 668 (1997)

Operation of a Ferromagnetic Axion Haloscope D. Alesini, C. Braggio, G. Carugno, N. Crescini, D. Di Gioacchino, P. Falferi, S. Gallo, U. Gambardella, C. Gatti, G. Iannone, G. Lamanna, C. Ligi, A. Lombardi, R. Mezzena, A. Ortolan, S. Pagano, R. Pengo, A. Rettaroli, G. Ruoso, C. C. Speake, L. Taffarello, and S. Tocci

Abstract The QUAX proposal studies the interaction of dark matter axions with electrons. As the effect of axions can be described as an effective magnetic field, the signal is a magnetization oscillation of a sample. To detect it, the material is placed

D. Alesini · D. Di Gioacchino · C. Gatti · C. Ligi · A. Rettaroli · S. Tocci INFN, Laboratori Nazionali di Frascati, Frascati, RO, Italy C. Braggio · G. Carugno · S. Gallo INFN, Sezione di Padova, Padova, Italy Dip. di Fisica e Astronomia, Padova, Italy N. Crescini () Dip. di Fisica e Astronomia, Padova, Italy INFN, Laboratori Nazionali di Legnaro, Legnaro, PD, Italy e-mail: [email protected] P. Falferi Istituto di Fotonica e Nanotecnologie, CNR-Fondazione Bruno Kessler, Povo, TN, Italy INFN, TIFPA, Povo, TN, Italy U. Gambardella · G. Iannone · S. Pagano Dip. di Fisica E.R. Caianiello, Fisciano, SA, Italy INFN, Sez. di Napoli, Napoli, Italy G. Lamanna Dip. di Fisica and INFN, Sez. di Pisa, Pisa, Italy A. Lombardi · A. Ortolan · R. Pengo · G. Ruoso INFN, Laboratori Nazionali di Legnaro, Legnaro, PD, Italy R. Mezzena INFN, TIFPA, Povo, TN, Italy Dip. di Fisica, Povo, TN, Italy C. C. Speake School of Physics and Astronomy, Univ. of Birmingham, Birmingham, UK L. Taffarello INFN, Sezione di Padova, Padova, Italy © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_12

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in a resonant cavity and polarized with a static magnetic field. Here we describe the operation of such a device, with a resonance frequency of 14 GHz, corresponding to an axion mass ma = 58 µeV. Eventually some recent developments on the resonant cavities design and test are detailed. Keywords Axion · Dark matter · DFSZ · Haloscope · Spin

1 Introduction The QUAX (QUaerere AXion) program explores the feasibility of an apparatus to detect axions as a dark matter (DM) component by exploiting its interaction with the spin of electrons (see [1] and the references therein). Thanks to the relative motion of Solar System with respect to the Milky Way’s dark matter halo, an observer on Earth is subjected to a wind of axionic DM. The effect of such wind on a sample’s electrons is similar to the one of a radiofrequency magnetic field, as it effectively modulates its magnetization. The frequency of the field is fixed by the axion mass ma , its amplitude is proportional to the axion–electron coupling constant gaee , and depends on the Earth-DM relative speed and on the local DM density. The axion field has its own coherence time τa , so the optimal detection scheme is Larmor resonance at the frequency of the axion mass and with a T2 as long as τa . This is achieved by immersing the sample in a static magnetic field, which tunes the precession frequency of the electrons to ma . For example, a 0.5 T field produces a Larmor frequency of 14 GHz, corresponding to an axion mass of 58 µeV. The interaction of the axionic wind with the sample’s polarized electrons modulates its total magnetization, and in principle these oscillations can be detected by a suitable spin-magnetometer. To optimize the detection scheme, the sample is placed inside a microwave cavity and cooled down at cryogenic temperature.

2 The QUAX Prototype The TM110 mode magnetic field of a cylindrical cavity is uniform along the cavity axis, and it is thus suitable to be coupled to an arbitrary large material quantity. The cavity diameter is 26.1 mm and the length is 50.0 mm, resulting in a resonance frequency fc  13.98 GHz and linewidth of 400 kHz at liquid helium temperature. Scalability is an important feature of the apparatus, as by increasing the quantity of material also the axionic signal increases. To this aim, five 1 mm GaYIG (Gallium doped Yttrium Iron Garnet) spheres are placed on the cavity axis held by a PTFE support where the material is free to rotate. Once the static magnetic field B0 is turned on, the ferrite is going to align with it. For B0 = 0.5 T the Larmor and cavity mode, singular frequencies coincide, resulting in two hybrid cavity-material modes with a quality factor Qh  104

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Low frequency I

A3I ADC

Fig. 1 Left—Scheme (not to scale) of the experimental setup, HTE and LTE stand for high temperature and low temperature electronics, respectively, while SO is the source oscillator. Right—Electronics layout. Figure taken from [2]

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Q

A3Q

output LO

SO

input A2

+ −

-20

heater

A1 T

T

RJ

50 Ω switch

GaYIG T

MW cavity Magnetic field T = 4K

which is the average of the previous two. The field B0 has to be as uniform as 1/Qh over all the spheres, to avoid the inhomogeneous broadening of the linewidth. The apparatus employs a main cylindrical magnet and a uniformity corrector, both made of NbTi. The current is supplied by a high-precision, high-stability current generator, injecting 15.416 A into the magnet with a precision better than 1 mA, while a stable current generator provides 26.0 A for the correction magnet. The cavity-material hybrid system is an axionic-to-rf signal transducer. This signal must then be collected with a suitable detection electronics, reported in Fig. 1 (right). The input of the amplification chain can be switched between two channels called Input Channel 1 and 2. Channel 1 measures the signal power, while Channel 2 has calibration and characterization purposes. A cryogenic switch is used to select the desired channel: 1. the hybrid system output power is collected by a dipole antenna with variable coupling. To extract the maximum possible signal the coupling is set to critical, and the doubling of the sub-critical linewidth of the selected mode is checked; 2. a load attached to a heater emits a power, proportional to the load temperature, which is collected by the chain. Its output is thus used to calibrate the noise temperature and the total gain of the system.

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Before every measurement the setup is calibrated with Channel 2, typical noise temperatures and gains are Tn  10 K and Gtot  106 dB.

3 Results The signal of Channel 1 is heterodyned, and its in-phase and quadrature components are separately acquired by an ADC. A complex FFT yields the signal power spectrum sω2 with positive frequencies for f > fLO and negative frequencies for f < fLO . Figure 2 reports an example run of ∼2.3 h, consisting in 2,048,000 FFTs of 8192 244 Hz-bins each, which were square averaged and rebinned to the bandwidth Δf = 7.8 kHz (256 bins), close to the axion linewidth. A degree 5 polynomial is fitted to the averaged spectrum to get the residuals and thus the power sensitivity. The averaged spectrum is reported in Fig. 2 together with the fitting function. Figure 2 shows a plot of the residuals and their histogram. The average value is −4.6 × 10−23 W with standard deviation σP = 2.2 × 10−22 W. The result is compatible with Dicke radiometer equation

 Δf  8280 s  Δf −22 = 2.1 × 10 W, σD = kB (Tc + Tn ) t 7.8 kHz t

(1)

7

8

7

Fig. 2 Averaged spectra collected by the ADC in an example run. The error of the black datapoints is within the symbol dimensions, and the red line is a polynomial fit. The residuals are in blue, the shaded region is the 1σ level, and their histogram is shown in the inset. Figure taken from [2]

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where t is the total integration time, Tn system noise temperature, and Tc cavity √ temperature. This means that the standard deviation of the noise decreases as 1/ t trend at least within this time span. The measured rf power is compatible with the modeled noise for every bin and we found no statistically significant signal consistent with axions. The upper limit at the 95% C.L. is 2σP = 1.1 × 10−21 W. This value can be converted to an equivalent axion field [2], obtaining Bm < 2.6 × 10−17

 14 GHz  f+

×

 2.13 · 1028 /m3  0.11 μs  2.6 mm3 1/2 nS

τ+

Vs

T, (2)

where f+ , τ+ are the working mode frequency and relaxation time, Vs the GaYIG volume: all the reported parameters have been explicitly measured within the run. The measurement has been repeated several times, with small changes of f+ from run to run. This is probably due to mechanical instabilities and to the low resolution of the correction magnet power supply and results in a limited frequency scan over a ∼3 MHz range. The maximum integration time for a 1 MHz band was 6 h, and no √ deviations from the 1/ t scaling of σP were found. Our results are a limit on the axion–electron coupling constant, since the relation between Bm and gaee is known once the axion dark matter density is fixed. This result is far from an actual axion search since the sensitivity is not enough to reach the axion–electron coupling. However, these results can be used to limit the presence of DM in the form of axion-like particles, which may also compose it. Data are acquired when the axion signal is on the daily maximum, thus we can use all of them to produce the upper limit on the ALP-electron coupling with the best possible sensitivity of the setup. By repeating the analysis procedure for seven measurement runs and averaging together overlapping bandwidths, we produce the plot in Fig. 3. The minimum measured value of gaee is 4.9×10−10 , corresponding to an equivalent axion field limit of 1.6 × 10−17 T.

4 Cavity R&D To improve the coherence time to get closer to the axion one, which is ∼ 1 µs, it is necessary to improve the quality factor of the hybrid mode. To achieve such result, both the linewidths of cavity and material have to be narrowed; hereafter, we report the recent developments on the cavity side, where two prototypes were tested. The two considered cavities are a cylindrical copper cavity with conic end-caps and a NbTi thin film sputtered on the walls parallels to its axis, and a photonic-bandgap cavity made with sapphire rods in a copper frame also with conic end-caps. The photos of the two cavities are reported in Fig. 4. The conic end-caps, common to both the cavities, create an evanescent field at the two ends improving the overall quality factor which results practically limited only by the walls. The simulation of

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Fig. 3 First exclusion plot of the coupling between dark matter axions and electrons (blue), obtained assuming that all dark matter is composed of axions. The dashed black line is a limit obtained with the axio-electric effect of solar axions, while the green region is an astrophysical limit. The orange line is the predicted value of the DFSZ axion–electron coupling. Figure taken from [2]

Fig. 4 Left—NbTi superconducting cavity halves. Right—Photonic-bandgap cavity

the TM110 mode for both the cavities is of the order of the million; thus, they were produced and assembled. As for the NbTi sputtered cavity, we introduced a layer of copper between the two halves, to block the supercurrents and avoid the Meissner effect, which otherwise forbids the magnetic field to enter in the cavity. The resonant mode was measured at liquid helium temperature, yielding a zero-field quality factor of 3 × 105 which becomes 2 × 105 with a 2 T field. By inserting a YIG sample inside the cavity we verified that the field is entering in the cavity volume, since the hybridization was obtained as expected. We are confident that also the field uniformity is preserved since the linewidth of the hybrid mode is the average of the cavity and material ones, excluding inhomogeneous broadening [3]. The photonic cavity yielded a quality factor of around 105 at room temperature, which improves more than a factor 2 at

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77 K; liquid helium tests have not yet been performed. Both these cavities are also suitable for operating the haloscope to search for the axion-to-photon conversion: a narrow bandwidth measurement is on the way. Acknowledgments The authors wish to thank Mario Tessaro, Fulvio Calaon, Massimo Rebeschini, Enrico Berto, Mario Zago, and Andrea Benato for the help with cryogenics and for the mechanical and electronic work on the experimental setup. We acknowledge the support of Giampaolo Galet and Lorenzo Castellani for the building of the magnet power supply, and Riccardo Barbieri for the stimulating theoretical discussions. We eventually acknowledge Gianpaolo Carosi and Gray Rybka for all the time spent working for this proceedings.

References 1. R. Barbieri et al., Phys. Dark Univ. 15, 135 (2017) 2. N. Crescini et al., Eur. Phys. J. C 78, 703 (2018) 3. D. Alesini et al., Phys. Rev. D 99, 101101(R) (2019)

Overview of the Cosmic Axion Spin Precession Experiment (CASPEr) Derek F. Jackson Kimball, S. Afach, D. Aybas, J. W. Blanchard, D. Budker, G. Centers, M. Engler, N. L. Figueroa, A. Garcon, P. W. Graham, H. Luo, S. Rajendran, M. G. Sendra, A. O. Sushkov, T. Wang, A. Wickenbrock, A. Wilzewski, and T. Wu

Abstract An overview of our experimental program to search for axion and axion-like-particle (ALP) dark matter using nuclear magnetic resonance (NMR) techniques is presented. An oscillating axion field can exert a time-varying torque on nuclear spins either directly or via generation of an oscillating nuclear electric dipole moment (EDM). Magnetic resonance techniques can be used to detect such an effect. The first-generation experiments explore many decades of ALP parameter space beyond the current astrophysical and laboratory bounds. It is anticipated that future versions of the experiments will be sensitive to the axions associated with quantum chromodynamics (QCD) having masses 10−9 eV/c2 . D. F. Jackson Kimball () California State University – East Bay, Hayward, CA, USA e-mail: [email protected] S. Afach · J. W. Blanchard · T. Wu Helmholtz Institute, Mainz, Germany D. Aybas · A. O. Sushkov Boston University, Boston, MA, USA G. Centers · M. Engler · N. L. Figueroa · A. Garcon · M. G. Sendra · A. Wickenbrock · A. Wilzewski Johannes Gutenberg-Universität and Helmholtz Institute, Mainz, Germany D. Budker Johannes Gutenberg-Universität and Helmholtz Institute, Mainz, Germany University of California, Berkeley, CA, USA P. W. Graham Department of Physics, Stanford University, Stanford, CA, USA H. Luo Shanghai Institute of Ceramics, Chinese Academy of Sciences, Beijing, China S. Rajendran Johns Hopkins University, Baltimore, MD, USA T. Wang Princeton University, Princeton, NJ, USA © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_13

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Keywords Axion · Dark matter · NMR · ALPs

1 Coupling of Dark-Matter Axions and Axion-Like-Particles (ALPs) to Spins The original motivation for introducing axions (pseudoscalar particles) emerged from an elegant solution to the strong-CP problem1 by Peccei and Quinn [1, 2]. The strong-CP problem is the observation that the theory of quantum chromodynamics (QCD) requires extreme fine-tuning of parameters in order to be reconciled with the observation that the strong interaction is found experimentally to respect CP symmetry to a high degree, as evidenced by experimental constraints on the neutron permanent electric dipole moment (EDM) [3]. The so-called QCD axion emerges when the strong-CP problem is resolved by introducing a new symmetry that is broken at a high energy scale fa , possibly as high as the Planck scale [4, 5]. Since the original proposal of Peccei and Quinn [1, 2], the axion concept has had a broad and significant impact on theoretical physics. Axion-like-particles (ALPs) emerge naturally whenever a global symmetry is broken [4–9]: such global symmetry breaking is a ubiquitous feature of beyond-Standard-Model physics, including grand unified theories (GUTs), models with extra dimensions, and string theory [10, 11]. In the following, we use the term axion to refer to both the QCD axion and ALPs as long as the discussion pertains to both and make distinctions when necessary. Interactions of the axion field a, via QCD or other mechanisms in the case of ALPs [12, 13], generate a potential energy density ∼m2a c2 a 2 /(2h¯ 2 ), where ma is the axion mass. Initial displacement of the axion field from the minimum of this potential results in oscillations of the axion field at the Compton frequency ωa =

ma c 2 . h¯

(1)

The energy density in these oscillations can constitute the mass-energy associated with dark matter. The temporal coherence of oscillations of the dark-matter axion field a(r, t) observed in a terrestrial experiment is limited by relative motion through random spatial fluctuations of the field. The size of such fluctuations corresponds to the axion de Broglie wavelength λdB , thus the coherence time is τa ≈

2π h¯ λdB ≈ , v ma v 2

(2)

where v ∼ 10−3 c is the galactic virial velocity of the dark-matter axion field. Therefore the “quality factor” Q corresponding to a dark-matter axion field is given 1 CP

refers to the combined symmetry with respect to charge-conjugation (C), transformation between matter and anti-matter, and spatial inversion, i.e., parity transformation (P ).

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by Q=

 c 2 ωa τa ≈ ≈ 106 . 2π v

(3)

The Q of the axion field can also be understood by noting that due to the secondorder Doppler shift that arises from the contribution of the kinetic energy of the axions to their total energy, axions moving at v relative to a detector are measured to have a frequency   v2 ω ≈ ωa 1 + 2 . 2c

(4)

Thus the spread in axion velocities leads to a spread in observed frequencies, giving the axion field the Q shown in Eq. (3).   The axion field seen by a detector on the Earth is a(t) = a0 cos ω t , a field oscillating at ≈ωa , and the amplitude of the field a0 can be estimated by assuming the energy of the axion field comprises the totality of the local dark matter energy density ρ DM ≈ 0.4 GeV/cm3 [14–16]: ρ DM ≈

c2 2h¯ 2

m2a a02 .

(5)

In order to interpret experiments searching for axion dark matter fields, it is important to take into account the fact that the amplitude, velocity, and phase of the axion field vary stochastically on scales ∼λdB , as discussed in detail in Ref. [17]. To detect such an oscillating axion field with a terrestrial sensor, one searches for (non-gravitational) interactions of a(r, t) with Standard Model fields and particles. Axion/ALP fields a(r, t) possess three such interactions, in general, that can be described by the Lagrangians (given in natural units where h¯ = c = 1) [18] L EM ≈ gaγ γ a(r, t)Fμν F˜ μν , L EDM ≈ L spin

− 2i gd a(r, t)Ψ n σμν γ5 Ψn F μν

,   ≈ gaN N ∂μ a(r, t) Ψ n γ μ γ5 Ψn ,

(6) (7) (8)

where gaγ γ parameterizes the axion–photon coupling, gd parameterizes the axiongluon coupling that generates nuclear EDMs, gaN N parameterizes the coupling to nuclear spins,2 Fμν is the electromagnetic field tensor, Ψn is the nucleon wave function, and σ and γ are the standard Dirac matrices. Note that the coupling constants gaγ γ , gd , and gaN N are proportional to 1/fa [18]. The axion–photon

2 There

can be a similar coupling to electron spins.

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interaction described by Eq. (6) is used in a variety of “haloscope” experiments3 to search for axion dark matter as discussed elsewhere in this volume, such as the Axion Dark Matter eXperiment (ADMX) [24, 25], the Haloscope At Yale Sensitive To Axion Cold dark matter (HAYSTAC) [26], and CAPP’s (Center for Axion and Precision Physics Research) Ultra Low Temperature Axion Search in Korea (CULTASK) [27]. In contrast to other haloscope experiments searching for axion– photon interactions, the Cosmic Axion Spin Precession Experiment (CASPEr, see Ref. [28] and an earlier online version of this paper [29]) exploits the axion couplings to nuclear spins described by Eqs. (7) and (8). The Lagrangian L EDM describes an oscillating nuclear EDM dn (t), generated by a(t) along the direction of the nuclear spin σˆ n , given in natural units by the expression dn (t) = gd a0 cos(ωa t)σˆ n

(9)

that interacts with an external electric field E. The oscillating EDM amplitude can be estimated from Eq. (5):   gd GeV−2 gd  −25 , 2ρ DM ≈ 6 × 10 e · cm × dn = gd a0 ≈ ma ma [eV]

(10)

where [· · · ] indicates the units of the respective quantity. The non-relativistic Hamiltonian describing this interaction is H EDM = −dn (t) · E ,

(11)

and there is a corresponding spin-torque τ EDM τ EDM = dn (t) × E .

(12)

The Lagrangian L spin results in a non-relativistic Hamiltonian (in natural units) H spin = gaN N ∇a(r, t) · σˆ n ,

(13)

which describes the interaction of nuclear spins with an oscillating “pseudomagnetic field” generated by the gradient of the axion field. The magnitude of the

3 Haloscope

experiments directly detect the dark matter from the galactic halo [19, 20]. Complementary approaches include (1) “helioscope” experiments that search for axions emitted by the Sun; (2) “light-shining-through-walls” experiments where axions are created from an intense laser light field passing through a strong magnetic field (which facilitates mixing between photons and axions) and then detected by converting them back to photons after they cross a wall that is transparent to them but completely blocks the light; and (3) indirect experiments that search for modifications of known interactions due to exchange of virtual axions. See Refs. [21–23] for reviews of these alternative approaches.

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gradient ∇a(r, t) can be estimated by noting that since p = −i h∇ ¯ |∇a| ≈

ma v a0 . h¯

(14)

This is the so-called axion wind which acts as a pseudo-magnetic field directed along v [18]. The resultant Hamiltonian is H wind ≈ gaN N ma a0 cos(ωa t)v · σˆ n ,  ≈ gaN N 2h¯ 3 cρ DM cos(ωa t)v · σˆ n ,

(15) (16)

where the amplitude of the axion field assumed in Eq. (5) was used in Eq. (16), in which H wind is expressed in Gaussian units. Given a nucleus with a particular gyromagnetic ratio γn = gn μN /h, ¯ where gn is the nuclear Landé factor and μN is the nuclear magneton, the amplitude Ba of the oscillating pseudo-magnetic field produced by the axion field can be estimated to be Ba ≈ 10−3 ×

gaN N hγ ¯ n

Ba [T] ≈ 10−7 ×

 2h¯ 3 c3 ρ DM ,

  gaN N GeV−1 , gn

(17) (18)

where we have assumed that v ≈ 10−3 c, the virial velocity of the axions. Analogous to the case of the axion-induced oscillating nuclear EDM dn (t) discussed above, here we have an oscillating pseudo-magnetic field Ba (t) that interacts with the nuclear magnetic dipole moment μn : H wind = −μn · Ba (t) ,

(19)

and a corresponding spin-torque τ wind τ wind = μn × Ba (t) .

(20)

On the one hand, the interactions described by Eqs. (11 and 12) and (19 and 20) have similar signatures and thus suggest similar experimental approaches: in both cases the goal of an axion dark matter experiment would be to detect an oscillating nuclear-spin-dependent energy shift (or, analogously, an oscillating torque on nuclear spins). The well-developed techniques of nuclear magnetic resonance (NMR) are ideally suited to this task. On the other hand, there is an essential difference between the signatures of L EDM and L spin , namely in the case of L EDM an electric field E is required for observation of the effect. For this reason, the CASPEr experimental program is divided into two branches: CASPEr Electric,

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which searches for an oscillating EDM dn (t), and CASPEr Wind, which searches for an oscillating pseudo-magnetic field Ba (t) [28].

2 General Principles of the CASPEr Nuclear Magnetic Resonance (NMR) Approach Both CASPEr Electric and CASPEr Wind use NMR techniques to search for axion dark matter. The experimental geometries are shown in Fig. 1. The leading field B0 determines the Larmor frequency ΩL = γn B0 ,

(21)

for the nuclear spins, which are initially oriented along B0 . If ΩL = ωa , then the time-dependent torques given by Eqs. (12) and (20) average out and there is no appreciable effect on the spins. However, when ΩL ≈ ωa , a resonance occurs and the spins are tilted away from the direction of B0 . Viewed from the frame rotating with the spins at ΩL , the effective leading field in the rotating frame goes to zero Beff (rot) = B0 −

ΩL =0. γn

(22)

The axion-induced torque oscillating in the laboratory frame has a static component in the rotating frame when ΩL ≈ ωa , and thus is able to tilt σˆ n away from the direction of B0 (see, for example, Problem 2.6 of Ref. [30] for a tutorial discussion of this well-known effect). In the laboratory frame, the tilted nuclear spins are observed to precess in B0 . This axion-induced nuclear spin precession at ΩL ≈ ωa is the signature of the axion dark matter detected in both CASPEr Electric and Fig. 1 Experimental geometries for CASPEr Electric (top) and CASPEr Wind (bottom). In both cases, the nuclear spins σˆ n are oriented along a leading magnetic field B0 . An oscillating torque, τ EDM = dn (t) × E in the case of CASPEr Electric and τ wind = μn × Ba (t) in the case of CASPEr Wind, tips the nuclear spins away from B0 if the Larmor frequency ΩL matches ωa

CASPEr Electric

CASPEr Wind

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sample magnetometer (e.g., SQUID)

Fig. 2 Schematic diagram of the CASPEr experiment. When ΩL ≈ ωa , the nuclear spins in the sample are tipped away from their initial orientation along B0 due to the axion-induced torque. The precessing magnetization at ΩL can be detected with a magnetometer such as a SQUID placed near the sample

CASPEr Wind. The precessing magnetization can be measured, for example, by induction through a pick-up loop or with a Superconducting QUantum Interference Device (SQUID), see Fig. 2. The amplitude of the NMR signal is proportional to the tilt angle ϕ of the spins (since under our conditions ϕ  1), which for CASPEr Electric is given by

ϕ EDM

ET2 dn ET2 ≈ ≈ gd h¯ hm ¯ a

2h¯ 3 ρ DM , c

(23)

and for CASPEr Wind is given by ϕ wind ≈

 μn T2 μn Ba T2 2h¯ 3 c3 ρ DM , ≈ gaN N h¯ 1000h¯ 2 γn

(24)

where T2 is the spin-precession (transverse) coherence time. The ultimate limit on T2 is the axion coherence time τa [Eq. (2)], but in many cases T2 is determined by the material properties of the nuclear spin sample. A key to CASPEr’s sensitivity is the coherent “amplification” of the effects of the axion dark matter field through a large number of polarized nuclear spins. For example, in the case of CASPEr Wind, the spins tip by an angle ϕ wind during the coherence time [Eq. (24)]. Consider a spherical sample with a polarized nuclear spin density of n. The amplitude of the oscillating field B1 generated by the precessing magnetization that is detected by the magnetometer as shown in Fig. 2 is  B1 ≈

   8π 8π 2 μn n ϕ wind ≈ μ nT2 Ba = ηBa , 3 3h¯ n

(25)

where the factor η is the effective enhancement of the measured oscillating field B1 over the pseudo-magnetic field Ba generated by the axion dark matter field. The

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enhancement factor η can be considerable: for example, a fully polarized sample of liquid 129 Xe has n ≈ 1022 spins/cm3 and can have T2  1000 s [31], which gives η  106 . The axion mass ma to which an NMR experiment of this type is sensitive is set by the resonance frequency ΩL = ωa (although note that Refs. [32–34] discuss alternative broadband methods suitable for very low ma ). Thus there is an upper limit to the possible range of axion masses explored by CASPEr, given by the largest dc magnetic field achievable in laboratories (∼30 T) which corresponds to ωa ∼ 400 MHz and ma c2 ∼ 10−6 eV. The procedure to search for axions of different masses (with an exception for very small ωa as discussed in Refs. [32–34]) is to scan B0 in increments proportional to the width of the NMR resonance (≈ 1/T2 ). The integration time at each point in the magnetic field scan should be T2 in order to take full advantage of the coherent build-up of tilted magnetization.

3 Low Mass (1 µeV) Axions and Cosmology As noted above, CASPEr searches for axions with masses ma c2  10−6 eV. It is worth mentioning that cosmological constraints have been considered for QCD axions with masses below 10−6 eV [12, 13]. However, these constraints are highly dependent upon assumptions about unknown initial conditions of the universe. Such lighter mass QCD axions are not ruled out either by experimental or astrophysical observations. Cosmological arguments suggesting that ma c2  10−6 eV are based on a particular scenario for the earliest epochs in the universe, a time about which we know little. The original argument posited that QCD axions with masses below 10−6 eV produced too much dark matter and thereby “over-closed” the universe [12, 13]. In the absence of additional physics, this statement is true if inflation precedes axion production. On the other hand, if inflation occurs after axion production, the axion abundance in the universe depends sensitively upon the unknown initial “misalignment” angle of the axion field [35], the displacement of the axion field from the local minimum of its potential. If the misalignment angle is sufficiently small overclosure can be avoided. An initial misalignment angle 1 can be naturally explained by anthropic considerations: namely the idea that there is an initial random distribution of misalignment angles in the universe so that, of course, humans exist in regions of the universe where the misalignment angle is such that life is possible [35, 36]. Note that such anthropic arguments are a generic explanation for the observed coincidence in the energy density of dark matter and the baryon content of the universe [37]. This coincidence is particularly suggestive for axion dark matter since the axion and the baryon abundances arise from completely different physics. It should also be noted that such “tuned” values of the axion misalignment angle also naturally emerge in models such as the relaxion scenario where the axion solves the hierarchy problem [38].

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It is also crucial to note that the possible cosmological constraints discussed above do not apply to ALPs. In summary, not only do axions with masses 10−6 eV fit well within the landscape of theoretical particle physics, but they also probe particularly interesting physics as they arise from symmetry breaking at the GUT and Planck scales [10, 11, 18, 21, 22, 28].

4 CASPEr Electric The sample to be employed in the CASPEr Electric experiment is a ferroelectric crystal such as those considered for static nuclear EDM experiments [39–42]; recently experiments were carried out with paramagnetic ferroelectrics to search for the electron EDM [43–45]. A ferroelectric crystal possesses permanent electric polarization that creates a strong effective internal electric field E∗ (on the order of 100 MV/cm)4 with which an axion-induced nuclear EDM can interact. Searching for an axion-induced EDM requires a heavy atom in order to minimize the Schiff screening, which arises because the positions of charged particles in the sample tend to adjust themselves to cancel internal electric fields [47, 48]. In ferroelectric crystals, the effective energy shift produced by a nuclear EDM is given by S dn E ∗ , where S is the Schiff suppression factor [39–41]. It is advantageous for NMR detection that the heavy atom has a large nuclear magnetic moment, and for minimizing spin relaxation it is desirable that the nucleus has spin I = 1/2 [39–41]. The sample material must also have noncentrosymmetric crystal structure in order to provide the effective electric field E∗ . Based on these requirements, we have identified several potential samples to investigate: lead germanate (PGO, Pb5 Ge3 O11 ), lead titanate (PT, PbTiO3 ), lead zirconate titanate (PZT, PbZry Ti1−y O3 ), lanthanum-doped lead zirconium titanate (PLZT, Pbx La1−x Zry Ti1−y O3 ), lead magnesium niobate-lead titanate [PMN-PT, (1 − x)PbMg1/3 Nb2/3 O3 − (x)PbTiO3 ], cadmium titanate (CdTiO3 ), and paraelectric materials such as SrTiO3 , LiTaO3 , and KTaO3 . The different samples have different potential advantages, such as tunable ferroelectric properties and optical transparency (which may enable optical control of nuclear spin relaxation or even optical pumping of nuclear spins [49, 50]). Phase I of CASPEr Electric will use thermally polarized 207 Pb nuclear spins (I = 1/2, 22% abundance) in ferroelectric PMN-PT crystals at cryogenic temperatures. The ferroelectric displacement of the Pb ion with respect to its surrounding oxygen cage gives rise to a large effective electric field E ∗ ≈ 3 × 108 V/cm [41]. The transverse coherence time for the 207 Pb nuclear spins in single-crystal PMN-PT samples is expected to be T2 ≈ 10−3 s, similar to that observed for PbTiO3 [51]. Thus over the entire range of ωa values to be searched, T2  τa . The

4 The mechanism generating E∗

is similar to that generating the effective electric fields experienced by EDMs in polar molecules [46].

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longitudinal nuclear spin polarization time is expected to be T1 ≈ 1000 s at cryogenic temperatures [51], and so the strategy is to start at a relatively high magnetic field B0 ∼ 10 T and then ramp down B0 over a time scale ∼T1 while there remains significant thermal spin polarization. Note that in this scheme there is a lower bound on B0 due to relaxation caused by paramagnetic impurities in the PMN-PT sample, i.e., when B0 becomes too small local magnetic fields due to paramagnetic impurities cause spins to dephase with respect to one another. The amplitude of the transverse precessing component of magnetization M that arises due to interaction with the dark-matter axion field, for a spin density n and polarization P , is given by ∗

M ≈ nP μn ϕ EDM ≈ nP μn S dnh¯E T2  ∗ gd 2h¯ 3 ≈ nP μn S Eh¯ T2 m c ρ DM , a

(26) (27)

where Eqs. (23) and (10) were used to derive Eqs. (26) and (27), respectively. The thermal polarization of 207 Pb nuclei at a temperature of T ≈ 4 K and B0 ≈ 10 T is 

μn B0 P = tanh kB T

 ≈

μn B0 ≈ 3 × 10−4 , kB T

(28)

where μn = h¯ γn μN /2 and γn ≈ 0.58 for 207 Pb. The spin density n ≈ 1022 cm−3 and we assume S ≈ 10−2 is the Schiff suppression factor for 207 Pb in PMN-PT, similar to PbTiO3 [41]. A major limiting factor in the measurement sensitivity is the magnetometer detecting M(t) as shown in Fig. 2. For frequencies 10 Hz  ωa √ /(2π )  106 Hz, SQUID magnetometers offer the best sensitivity (≈10−15 T/ Hz), while for frequencies 106 Hz an inductive pick-up coil connected to an RF amplifier offers superior sensitivity, although potentially advantageous alternative atomic magnetometry schemes were considered in Ref. [52]. To reduce noise and systematic errors, multiple samples with different orientations of their internal electric fields E∗ will be used. This allows rejection of many types of common-mode noise that all samples would share, such as vibrations or uniform magnetic field noise. Samples with opposite orientations should exhibit oscillating axion-induced transverse magnetizations 180◦ out-of-phase. An important effect to consider in CASPEr Electric is chemical shift anisotropy (CSA). Chemical shift refers to the shift of the NMR frequency of a nucleus from a reference NMR frequency for that particular nucleus (measured under standard conditions). This shift is primarily produced by the local distribution of electron currents near the nucleus in question and therefore varies as the chemical structure/environment varies. There is both isotropic chemical shift, an overall offset of the NMR frequency independent of the orientation of B0 , and anisotropic chemical shift that depends on the relative orientation of B0 with respect to the local environment (crystal axes). For a single-crystal sample, the CSA can be

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Fig. 3 Experimental reach of CASPEr Electric. The green band is excluded by astrophysical observations [22, 23]. The blue region shows the axion mass range covered by ADMX and HAYSTAC. Orange, red, and maroon regions show sensitivity projections explained in text. Phases II and III reach the QCD axion coupling strength. The fundamental quantum sensitivity limit is given by magnetization noise, shown by the dashed red line. The vertical dashed gray line indicates the mass ma and frequency ωa corresponding to axions generated by symmetry breaking at the Planck scale. See Ref. [28] for details of these estimates

fully characterized and, in principle, relatively narrow NMR lines (widths ∼1/T2 , where T2 is the transverse spin relaxation time) can be obtained by various NMR techniques. The orange-shaded region in Fig. 3 shows the parameter space to which CASPEr Electric Phase I will be sensitive based on the above estimates and an overall integration time of 106 s. Phases II and III of CASPEr Electric rely on improving several important experimental parameters: 1. increasing the degree of nuclear polarization P by using optical pumping and other hyperpolarization techniques; 2. increasing the nuclear spin relaxation time T2 by using decoupling protocols; 3. implementing resonant circuit detection schemes; 4. increasing the sample size; 5. longer integration time. The risk factors for CASPEr Electric Phases II and III are mainly technical in nature, representing uncertainties on how well the sample material can be fabricated free of paramagnetic impurities and how effectively vibrations can be controlled. The main scientific uncertainty is the achievable degree of nuclear spin

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hyperpolarization. The ultimate sensitivity limit is given by the nuclear spin noise of the sample (Fig. 3), which has been experimentally observed in the past using similar tools [53] and appears to be a feasible long-term sensitivity goal.

5 CASPEr Wind In CASPEr Wind, no electric field is required and so different possibilities for the choice of the sample are opened. Otherwise, the procedure is similar to that for CASPEr Electric: the sample is placed within a magnetic field B0 ; as B0 is scanned, the corresponding Larmor frequency changes; and if ΩL is tuned to resonance with the axion oscillation frequency, an oscillating magnetization M(t) will build up in the sample. The oscillating magnetization is detected with a pick-up loop connected to a SQUID or RF amplifier. The sample of choice for CASPEr Wind is liquid 129 Xe, a high-density sample that can be hyperpolarized through spin-exchange with optically pumped Rb [31]. Because 129 Xe is in the liquid phase, the environment is isotropic on average which removes CSA. The transverse spin relaxation times T2 for liquid 129 Xe can be on the order of 1000 s and over essentially the entire range of axion masses to be investigated by CASPEr, the factor limiting the integration time will be τa . As discussed in Sect. 4, different detectors are optimal for different frequency (and therefore, magnetic field) ranges. Above ≈1 MHz (B0 ≈ 0.1 T for 129 Xe), standard inductive detection using an LC circuit gives optimal signal-to-noise. However, below ≈1 MHz, SQUID magnetometers perform better. At near-zero fields corresponding to ωa /(2π )  10 Hz, other experimental strategies become viable. Thus the CASPEr Wind experiment is being realized with three distinct setups: CASPEr Wind High Field (magnetic fields 0.1 T  B0  14 T), CASPEr Wind Low Field (10−4 T  B0  10−1 T), and CASPEr Wind ZULF (zero-toultralow field) that probes B0  10−4 T.

5.1 CASPEr Wind: High and Low Field For the CASPEr Wind High and Low Field experiments the preparation of the hyperpolarized liquid Xe sample is identical (and in fact both experiments use the same Xe polarizer). To prepare the spin-polarized liquid Xe sample, first Xe gas is mixed with other gases (N2 , He, and Rb) and then hyperpolarized via spinexchange optical pumping (SEOP, see Refs. [54, 55]). The hyperpolarized Xe is condensed into solid form inside a region cooled by a liquid-nitrogen bath in the presence of a leading magnetic field. He and N2 used in SEOP are then vented out. Subsequently the frozen Xe is sublimated to become a gas and the valve to the experimental cell is opened and a piston compresses the Xe into liquid form. The liquid Xe is then flowed into the experimental cell. After the Xe polarization

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decays, the Xe is pumped out and recycled. This procedure has been demonstrated to achieve near unity polarization and spin densities on the order of 1022 per cm3 [31, 56]. The design of the experiment should allow nearly continuous cycling between the Xe polarizer and the experimental cell, enabling a high duty cycle for the measurements. For CASPEr Wind High Field, a tunable magnetic field of up to ≈14 T will be applied to the liquid Xe with a cryogen-free superconducting magnet, thus enabling access to axion frequencies of up to ≈60 MHz. One of the technical challenges to be addressed is maintaining the required magnetic-field homogeneity over the course of a scan. This challenge can be addressed by dynamic shimming techniques aided by in-situ-magnetic-field gradient measurements with additional magnetometers. Another possible approach is to take advantage of magnetic field gradients to multiplex the measurement and operate the experiment in analogy with a magnetic resonance imaging (MRI) measurement where different regions of the sample access different ωa . For CASPEr Wind Low Field, the magnet requirements are less challenging, thus allowing for more rapid development of the experiment. CASPEr Wind Low Field will employ a sweepable superconducting magnet assembly inside a superconducting magnetic shield. The projected sensitivities of Phase I CASPEr Wind High Field and Low Field experiments are shown in Fig. 4. The sensitivity of CASPEr Wind Phase II is improved by significantly increasing the sample size: for Phase I the sample volume is ≈1 cm3 .

10-8

10-10

10

10 4

10 6

phase I-HF CASPEr ZULF

phase I-LF

-12

10-14

10 8

SN1987A

phase II

fa ~ Planck

ALP nucleon coupling gaNN (GeV -1)

ALP Compton frequency (Hz) 10 2

spin nois

10-16

ADMX

e

QCD Axion 10 -12

10 -10

10 -8

10 -6

ALP mass (eV) Fig. 4 Experimental reach of CASPEr Wind. Red, orange, and gray regions show sensitivity projections for high field (HF), low field (LF), and ZULF experiments

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5.2 CASPEr Wind: Zero-to-Ultralow Field (ZULF) An intriguing new possibility for a CASPEr Wind experiment at ultralow magnetic fields has recently emerged based on ZULF-NMR (see Refs. [57, 58] for reviews). The basic idea of ZULF-NMR is that the usual requirement of large magnetic fields for nuclear polarization, information encoding, and read-out is alleviated by the use of hyperpolarization techniques, encoding based on intrinsic spin–spin interactions, and detection methods not based on Faraday induction (e.g., SQUIDs and atomic 15 magnetometers). 15 N,13 C2 -acetonitrile (13 CH13 3 C N) appears to be a suitable candidate for a CASPEr Wind sample, since non-hydrogenative parahydrogeninduced polarization (NH-PHIP) has been shown [59] to produce nuclear spin polarizations on the order of 10%. Additional benefits of this experimental approach are that it is relatively low-cost and non-destructive (the isotopically enriched acetonitrile can be reused indefinitely), and it enables parallel observation of three different frequencies simultaneously  [the  probed   gyromagnetic ratios for the  nuclei are γn 1 H = 42.57 MHz/T, γn 13 C = 10.71 MHz/T, and γn 15 N = −4.32 MHz/T], which reduces the scanning time and may be advantageous for understanding and mitigating systematic effects. Low-field NH-PHIP experiments take advantage of matching conditions that occur in μT magnetic fields between hydride 1 H spins and other nuclear spins in transient iridium “polarization-transfer” catalysts. Essentially, parahydrogen is bubbled into a solution containing an iridium catalyst and the analyte molecule. The Ir forms a complex wherein the parahydrogen, which is in a singlet state, transfers its spin order to other nuclear spins via electron-mediated indirect dipole–dipole coupling (J-coupling). Because the binding of parahydrogen and the analyte ligands is reversible (residence times on the order of a ms), many parahydrogen molecules are brought into contact with many analyte molecules over the bubbling timescale, allowing for a buildup of substantial non-thermal spin populations over several seconds. Which spin states are populated is dependent on the specific spin-topology of the parahydrogen-iridium-analyte complex, but it has been consistently shown that polarization enhancements of 4–5 orders of magnitude can be achieved [59]. If a version of this experiment is performed at zero or ultralow field (the ZULF regime, B  1 µT), two-spin order having no net magnetization will be produced. This spin order may be converted to magnetization via sequences of magnetic field pulses. Alternatively, it may be possible to search for oscillating magnetization produced by selective population of nuclear spin states by the oscillating axion field. Because the singlet spin order has no net magnetization, it is immune to decoherence due to dipole–dipole interactions. Such long-lived nuclear spin states (demonstrated in other systems, such as 13 C-formic acid at zero field [60]) may allow for longer integration times and longer experimental runs between repolarization. Another intriguing aspect of the CASPEr ZULF experiment is that one can use a nonresonant measurement consisting of searching for sidebands induced by modulation of ΩL by the axion wind interaction, removing the need to scan for the resonance. This is discussed in detail in Ref. [32] and has been experimentally

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realized as described in Ref. [34]. The potential sensitivity of the CASPEr ZULF experiments using NH-PHIP is shown in Fig. 4. Results of ZULF-NMR CASPEr searches for axion dark matter using thermally polarized samples have been recently reported in Refs. [33, 34].

6 Conclusion The CASPEr program involves a multi-pronged experimental strategy employing NMR techniques to search for the coupling of axion dark matter to nuclear spins. First-generation experiments have the potential to probe a vast range of unexplored ALP parameter space. Future generations of CASPEr experiments should achieve sufficient sensitivity to search for the QCD axion. Acknowledgments The authors are grateful for the generous support of the Simons and HeisingSimons Foundations, the European Research Council, DFG Koselleck, and the National Science Foundation.

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Axion Dark Matter Search at IBS/CAPP SungWoo Youn On behalf of IBS/CAPP

Abstract The axion is a consequence of the PQ mechanism to solve the strong-CP problem and has been considered as an attractive candidate for cold dark matter. The Center for Axion and Precision Physics Research (CAPP) of the Institute for Basic Science (IBS) in South Korea has completed the construction of the infrastructure for axion dark matter search experiments. Multiple experiments are currently under preparation for parallel operation targeting at different mass ranges. The ultimate goal of our center is to be sensitive to the QCD axion models over a wide range of axion mass. The current approaches to achieve this goal are three folds: commissioning high-field magnets, designing high frequency cavities, and developing low noise amplifiers. We present the status of the experiments and discuss future prospects. Keywords Axion · Dark matter · Microwave cavity · CAPP

1 Introduction As a dynamic solution to the strong-CP problem in particle physics, a new global symmetry introduced by Peccei and Quinn is spontaneously broken at a particular energy scale. It was immediately noticed that, as a consequence, a pseudoscalar Nambu–Goldstone boson appears with its mass depending on the symmetry breaking energy scale. This hyperthetical particle is called axion. Experimental disproval of its existence at the electroweak scale suggested the invisible axion extending the mass range to sub-eV region. Later on, due to its properties, the axion has been considered as a charming candidate for cold dark matter with its mass falling in a certain range. The detection methodology of axion dark matter employs the inverse S.W. Youn () CAPP of IBS, Daejeon, Republic of Korea e-mail: [email protected]

© This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_14

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Primakoff effect, by which axions are converted into photons in a strong magnetic field and resonated in a microwave cavity. The axion-to-photon conversion power is given by 2 Pa→γ γ = gaγ γ

ρa 2 B V C min(QL , Qa ), ma 0

(1)

where gaγ γ is the axion-to-photon coupling, ρa is the local halo density, ma is the axion mass, B0 is the magnetic field, V is the cavity volume, C is the mode dependent form factor, and QL and Qa are the cavity and axion quality factors, respectively. As the axion mass is a priori unknown, all possible mass ranges need to be scanned and the scan rate is expressed as df = dt



1 SNR

2 

Pa→γ γ kB Tsys

2

Qa −2 ∼ B04 V 2 C 2 QL Tsys , QL

(2)

where SNR is the signal-to-noise ratio, kB is the Boltzmann constant, and Tsys is the total noise temperature of the system. The scan rate is the figure of merit in designing an axion haloscope. The Center for Axion and Precision Physics Research (CAPP) of the Institute for Basic Science (IBS) was established in 2013 to build state-of-the-art experiments for axion dark matter search in Korea. A tremendous effort has been put to complete the construction of the infrastructures including low vibration pads, cryogenic systems, superconducting magnets, etc. Currently several experiments are being set up, with one in DAQ mode. The CAPP’s strategy is to operate multiple experiments in parallel targeting at different mass ranges to cover as large parameter space as possible. Table 1 summarizes the equipment available at IBS/CAPP and experiments under preparation. Table 1 List of cryogenic systems and superconducting magnets at IBS/CAPP and their major specifications. The experiments are named after the magnet that they utilize. The last two rows are for upcoming equipment and planned experiments in the near future Refrigerator Manufacturer BlueFors BlueFors BlueFors BlueFors Janis Oxford Leiden

Model LD400 LD400 LD400 LD400 HE-3-SSV Kelvinox DRS1000

TB [mK] 10 10 10 10 300 30 100

Magnet Manufacturer

Experiment Bmax [T] Bore [mm] Name

AMI AMI Cryomagnetics SuNAM Oxford IBS/BNL

8 8 9 18 12 25

125 165 125 70 320 100

CAPP-PACE CAPP-8TB CAPP-9T MC CAPP-18T CAPP-12TB CAPP-25T

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2 Axion Research at CAPP The detection principle of axion dark matter research at IBS/CAPP relies on the haloscope technique and the major efforts focus on improvement of the experimental sensitivity by optimizing a combined parameter, B 2 V T −1 , in three fold. The following subsections will cover in detail the research activities at IBS/CAPP to enhance this experimental parameter.

2.1 High-Field and Large-Volume Magnet Since B0 contributes to the sensitivity in the same power as gaγ γ , any improvement in magnetic field strength is linearly reflected to improvement in axion-to-photon coupling. CAPP is currently commissioning and procuring two high-field largeaperture superconducting (SC) magnets. A 25 T/100 mm SC magnet is under development as a commission project with the Brookhaven Nation Laboratory. The magnet consists of a single layer of 14 double pancakes of 12 mm high temperature superconducting (HTS) REBCO tapes. It employs the no-insulation winding technique to take advantage of the selfprotecting feature against quenches. This magnet is scheduled to be delivered in early 2020 to be installed at the CAPP-25T experiment as a strong booster of axionto-photon conversion. A 12 T/320 mm magnet, designed and manufactured by Oxford Instruments, is under procurement process. Conventional low temperature superconducting cables (Nb3 Sn) are used to generate a 12 T magnetic field. The large aperture and large volume will enable us to be sensitive down to DFSZ physics in a low-frequency region between 0.7 and 3 GHz. This magnet will also be delivered in early 2020 for CAPP-12TB, one of the ultimate experiments at CAPP.

2.2 Quantum-Limited Noise Amplifier A Superconducting Quantum Interference Device (SQUID) is the most sensitive magnetic sensor utilizing two quantum phenomena: Josephson effect and flux quantization. The standard quantum-limited noise is given as TSQL = 48 mK × f , where f is the frequency in GHz. Two types of SQUID-based amplifiers are considered at CAPP: microstrip SQUID amplifier (MSA) and Josephson parametric amplifier (JPA). CAPP has been developing units from various resources for their application to axion search experiments. The Korea Research Institute of Standards and Science has developed MSAs based on octagonal washers targeting at maximal amplifications at around 2.5 GHz. The typical gain is about 15 dB with a large bandwidth, ∼100 MHz. CAPP also has

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a unit of MSA provided by UC Berkeley, a modified version of the original unit designed for the ADMX experiment. This unit has a series of varactor diodes in the input line providing a frequency tuning capability from 1.2 to 1.5 GHz with gains above 20 dB. For suitable amplification at high frequencies, a SQUID should be designed with the smallest possible tunnel junction capacitance, reasonably low SQUID loop inductance, and maximal steepness of the transfer function. Sub-micron size Josephson junctions with capacitance 0.04 pF initially were used for low-frequency SQUID current sensors designed at IPHT. Such slightly modified sensors were tested as microwave amplifiers in a frequency range of 0.5−5 GHz at CAPP. The high-gain performance over a wide frequency bandwidth indicates that this new SQUID design is applicable to wideband axion searches [1].

2.3 High Frequency Approach In haloscopes, exploring high frequency regions requires a smaller cavity size which gives rise to decrease in detection volume. A conventional way to overcome this defect is to bundle an array of identical cavities and combine the individual signals coherently. An in-depth study of phase-matching for multiple-cavity systems has been performed at CAPP [2]. However, since this approach is still inefficient in the usage of a given magnet volume, CAPP introduces a new concept of cavity design, as dubbed pizza cavity [3]. A pizza cavity consists of a single large cavity fitting into a magnet bore with multiple identical cells split by metal partitions placed with equidistant intervals. Benefiting from the large volume usage, this multiple-cell design yields an improved sensivity by a factor of 2 comparing to the conventional multiple-cavity design. In addition, an innovative idea of introducing a small hollow gap in the middle of the cavity provides various critical advantages: (1) breaking the frequency degeneracy; (2) enabling a single pick-up antenna to extract the signal from the entire cavity volume and thereby simplifying the readout chain; and (3) facilitating the phasematching mechanism by dissipating the coupling strength to higher modes. An experimental demonstration using a double-cell cavity shows the stability of the tuning mechanism and a negligible DAQ dead-time owing to phase-matching, which verifies that this unique design is promising for high mass axion searches.

3 Axion Experiments at CAPP The strategy of CAPP for axion dark matter search is to run several experiments simultaneously targeting at different frequencies in order to cover a large parameter space as quickly as possible. Currently, one experiment is in DAQ mode and three

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experiments are under construction. The following subsections will describe the status of the individual experiments at CAPP.

3.1 CAPP-PACE CAPP-PACE, a R&D machine for the CAPP-25T axion experiment, provides the necessary experience in dealing with the comprehensive detecting system. The major components of the experiment include an ultra-low temperature cryogenic system and an 8 T/125 mm SC magnet. The detecting system consists of a high Qfactor resonant cavity, a reliable frequency tuning system, highly sensitive cryo-RF electronics, and a DAQ/control system including monitors ensuring the data quality and safe environment. The CAPP-PACE detector has grown into a complete system and since the beginning of 2018 it has been taking physics data to test every aspect of the axion dark matter experiment at around 2.5 GHz. The goal of this experiment lies in filling out the unexplored frequency gap by the RBF collaboration, 2.45– 2.76 GHz, with a sensitivity down to one of the two main axion benchmark models (KSVZ axions) by adopting a quantum-limited noise amplifier.

3.2 CAPP-8TB The CAPP-8TB experiment utilizes an 8 T SC magnet with a relatively bigger bore (165 mm). Taking advantage of the large volume and using a single dielectric tuning rod, it aims to be sensitive at low-frequency regions, 1.6–1.7 GHz. A cryogenic environment below 50 mK is achieved by employing a dilution refrigerator. The cavity response and system noise are well understood through intensive studies. The experiment plans to run at two stages: the first one will use a HEMT-based amplifier for pre-amplification to touch the theoretical QCD axion band with 3 month operation, and the second stage will receive a SQUID-based amplifier to reach near the KSVZ QCD axion model.

3.3 CAPP-9T MC An experiment, named CAPP-9T MC, is currently under preparation in order to exploit the pizza cavity design described in Sect. 2.3. The experiment utilizes a He-3 cryogenic system and a 9 T superconducting (SC) magnet. By keeping the charcoal temperature at ∼40 K and the 1 K pot temperature as low as possible, the system can be operated in continuous (non-evaporating) mode with He-3 gas in condensed state, cooling the He-3 plate down to less than 2 K. Operation of the He-3 system in continuous mode enables us to maintain the cavity temperature at 2.0 K. A

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capability of operation in persistent mode reduces the heat load on the 4K cold mass and provides the stability of the magnetic field. Multiple-cell cavities with 110 mm inner diameter and 220 mm inner height are designed to maximize the sensitivity. A cryogenic readout chain consists of a series of low noise HEMT amplifiers. The CAPP-MC experiment consists of three phases depending on the cell multiplicity of the detector: double-cell, quadruple-cell, and octuple-cell. Each phase will cover a frequency range of 2.8–3.3, 3.8–4.5, and 5.8–7.0 GHz, respectively, with a target sensitivity of 10 times the KSVZ QCD axion model.

3.4 CAPP-18T CAPP-18T utilizes a 18 T HTS SC magnet manufactured by a local company, SuNAM Inc. The magnet was fabricated using GdBCO HTS tapes and featured by multi-width and no-insulation technologies to be robust against the mechanical stress and magnet quenches, respectively. The performance of the magnet shows a good uniformity (93% at z ± 100 mm) and stability (150 h. Test operations with free load confirm the desired specifications of the system.

2.2 Superconducting Magnet A superconducting (SC) magnet with 5

diameter clear bore was manufactured by Cryomagnetics. The magnet, fabricated using NbTi to generate a 9 T center field at 81 A, was designed to compromise with the Janis cryogenic system. The capability of operation in persistent mode is a feature of the system and provides a stable magnetic field with minimal heat generation. The magnet design was modeled by simulation using the finite element method based on its mechanical structure and the field distribution was verified to be consistent with the company measurement.

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Table 2 Experimental parameters for each stage of the CAPP-MC experiment Configuration Average magnetic field [T] Frequency [GHz] (fmax − fmin ) Quality factor (RRR = 9) Form factor Volume [L] DAQ efficiency Tsys [K] Scan rate for 10 × KSVZ [GHz/year]

2-cell 7.8 2.8–3.3 (0.5) 60,000 0.45 2.0 0.5 2.1 + 2.0 5.4

4-cell

8-cell

3.8–4.5 (0.7) 51,000 0.45 1.9

5.8–7.0 (1.2) 51,000 0.40 1.7

2.1 + 3.0 4.8

2.1 + 4.0 5.0

2.3 System Setup Multiple-cell cavities are designed with inner diameter of 110 mm and inner height of 220 mm to maximize the sensitivity for the given magnet field profile. Four thick copper bars support the cavity from the He-3 plate to be located at the middle of the magnet bore. A single piezo rotator located underneath the cavity rotates the alumina tuning rods inside the cavity simultaneously to tune the resonant frequency. The coupling strength to the cavity is adjusted by a linear piezo installed on the top of the cavity. A cryogenic readout chain consists of a cascade connection of two HEMT amplifiers with either a circulator or isolator attached in front.

2.4 Prospects The CAPP-MC experiment plans to proceed in three steps depending on the cell multiplicity of the cavity detector: double-cell, quadruple-cell, and octuple-cell. Each stage of the experiment covers a frequency range of 2.8–3.3, 3.8–4.5, and 5.8–7.0 GHz, respectively. Using a series of HEMT amplifiers, the total noise temperature of the system is conservatively assumed to be between 4.1 and 6.1 K. target Targeting at a sensitivity of 10×KSVZ QCD axion model, i.e., (gaγ γ = 10 × KSVZ ), the estimated scan rates are approximately 5 GHz/year and the time scales gaγ γ to scan the corresponding frequency ranges are one to 3 months. Table 2 summarizes the experimental parameters for each stage.

3 Summary A new concept of multiple-cell cavity design is introduced having benefits of detection volume in searches for high mass axions. The characteristics of the design are studied to derive a frequency tuning mechanism, and a demonstration of

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the experimental feasibility is performed at room temperature. A proof-of-concept experiment, CAPP-MC, employs a Janis He-3 system and a SC magnet to prove the multiple-cell cavities as axion haloscopes. The experiment targets at a sensitivity of 10 times the KSVZ QCD model for various high mass regions of axion dark matter.

References 1. C. Hagmann et al., Cavity design for a cosmic axion detector. Rev. Sci. Instrum. 61, 1076 (1990) 2. D.S. Kinion, First results from a multiple microwave cavity search for dark matter axions. Ph.D. Thesis, University of California, Davis (2001) 3. J. Jeong, S. Youn, S. Ahn, C. Kang, Y.K. Semetrzidis, Phase-matching of multiple-cavity detectors for dark matter axion search. Astropart. Phys. 97, 33 (2017) 4. J. Jeong, S. Youn, S. Ahn, J.E. Kim, Y.K. Semertzidis, Concept of multiple-cell cavity for axion dark matter search. Phys. Lett. B. 777, 10 (2018)

Exclusion Limits on Hidden-Photon Dark Matter Near 2 neV from a Fixed-Frequency Superconducting Lumped-Element Resonator A. Phipps, S. E. Kuenstner, S. Chaudhuri, C. S. Dawson, B. A. Young, C. T. FitzGerald, H. Froland, K. Wells, D. Li, H. M. Cho, S. Rajendran, P. W. Graham, and K. D. Irwin

Abstract We present the design and performance of a simple fixed-frequency superconducting lumped-element resonator developed for axion and hidden-photon dark matter detection. A rectangular NbTi inductor was coupled to a Nb-coated sapphire capacitor and immersed in liquid helium within a superconducting shield. The resonator was transformer-coupled to a DC SQUID for readout. We measured a quality factor of ∼40,000 at the resonant frequency of 492.027 kHz and set a simple exclusion limit on ∼2 neV hidden photons with kinetic mixing angle ε  1.5×10−9 based on 5.14 h of integrated noise. This test device informs the development of the Dark Matter Radio, a tunable superconducting lumped-element resonator which will search for axions and hidden photons over the 100 Hz to 300 MHz frequency range.

A. Phipps () · S. E. Kuenstner · S. Chaudhuri · C. S. Dawson · H. Froland · K. Wells P. W. Graham Department of Physics, Stanford University, Stanford, CA, USA e-mail: [email protected] B. A. Young · C. T. FitzGerald Department of Physics, Santa Clara University, Santa Clara, CA, USA D. Li · H. M. Cho Technology Innovation Directorate, SLAC National Accelerator Laboratory, Menlo Park, CA, USA S. Rajendran Johns Hopkins University, Baltimore, MD, USA K. D. Irwin Department of Physics, Stanford University, Stanford, CA, USA Technology Innovation Directorate, SLAC National Accelerator Laboratory, Menlo Park, CA, USA © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_16

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Keywords Lumped-element · Superconducting resonator · Hidden photon · Dark Matter Radio

1 Introduction The Dark Matter Radio (DM Radio) [1, 2] is a new experiment to search for ultralight, wavelike dark matter—particles with individual mass m0  1 eV/c2 . If these particles account for a substantial fraction of the cold dark matter, they must be bosonic with a high occupation number. As a consequence, this type of dark matter behaves like a classical, oscillating wave with a narrow bandwidth near frequency f = m0 c2 /h. Rather than search for the scattering or absorption of a single dark matter particle, we search for a persistent, narrow-band signal caused by weak coupling of the dark matter field to standard model particles. The QCD axion [3–5] is a strongly motivated candidate dark matter wave. Axion haloscopes [6] use the inverse Primakoff effect to search for the conversion of axion dark matter into photons by immersing a tunable microwave cavity in a strong DC magnetic field. If the cavity is tuned to match the frequency of the axion, the signal rings up the cavity, which leads to a power excess above the thermal noise. The sensitivity increases with resonator volume, quality factor, and magneticfield strength. ADMX [7] and HAYSTAC [8] have used this technique to place constraints on the axion–photon coupling gaγ γ for axion masses ∼1 µeV/c2 and higher. Lumped-element resonators can also be used to search for the “hidden-sector photon,” a spin-1 vector dark matter wave, which can be produced in the observed dark matter abundance by inflationary fluctuations [9] or the misalignment mechanism [10]. Kinetic mixing results in the conversion of hidden photons to ordinary photons. An external magnetic field is not required. Enhancement of the signal by a suitably tuned resonator also occurs, allowing axion haloscopes to place constraints on the kinetic mixing angle ε for ∼>1 µeV hidden photons. Cavity haloscope searches are only sensitive to axions/hidden photons whose Compton wavelength is comparable to the size of the cavity, limiting practical searches to m0  1 µeV/c2 . Lumped-element resonators (a wire-wound inductor connected to a monolithic capacitor) are free from this geometric restriction, motivating their use when the dark matter Compton wavelength is much larger than the size of the detector [1, 11–13]. An optimized single-pole, lumped-element resonator can theoretically achieve ∼70% of the fundamental quantum limit on detection of axion and hidden-photon dark matter in the sub-wavelength regime [14]. Tunable lumped-element resonators with a high quality factor will be required to cover QCD axion parameter space with m0 < 1 µ eV/c2 . DM Radio will consist of a tunable, superconducting, lumped-element resonator designed to search for axion and hidden-photon dark matter over the ∼peV to ∼µeV (100 Hz to 300 MHz) mass range. In this paper, we describe the performance of a simple fixed-frequency, superconducting lumped-element test resonator, and compute the first direct detection limits on hidden photons in a narrow range around 2.035 neV.

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2 Experimental Setup The resonator used in this experiment is shown in Fig. 1a. The inductor is a 40turn (0.05 inch pitch) Formvar-coated rectangular niobium-titanium (NbTi) coil wound on a polytetrafluoroethylene (PTFE) form, enclosing a volume of ∼100 mL. Opposite faces of a 500 µm thick, 100 mm diameter sapphire wafer were coated with 800 nm of Nb to form the capacitor. Electrical connections were created by Nb wire bonds between the capacitor electrodes and small Nb blocks. The ends of the inductor wire were also spot-welded to the blocks. A single-turn NbTi coil on the end of the PTFE form was used to inductively couple to the resonator. The ends of the transformer coil passed through a small hole into a separate Nb annex which houses a custom DC SQUID. The transformer coil is connected to two Nb screw terminal blocks using Nb screws and washers. The input coil of the SQUID was connected to the blocks using Nb wire bonds. The resonator and annex were enclosed in a 21.6 cm tall, 14.4 cm diameter cylindrical Nb shield with 2 mm wall thickness. The shield was mounted to a stainless steel probe and cooled to 4.2 Kelvin by insertion into a liquid helium dewar lined with Cryoperm-10. Wiring in the center tube of the probe connected the SQUID to room-temperature electronics. Biasing and pre-amplification were provided by Magnicon XXF-1 electronics. The SQUID was operated open-loop with 3.5 MHz bandwidth. Additional amplification was provided by two Stanford Research Systems SR560 channels before digitization. The equivalent circuit model of this resonator is shown in Fig. 1b. Dedicated calibration runs were performed to measure the inductance of the resonator (LR = 59 ± 6 µH), transformer (LT = 842 ± 29 nH), their mutual inductance (M = 1.15 ± 0.06 µH), and the SQUID input coil inductance (Lin = 2.74 ± 0.06 µH). The measured resonant frequency of 492.027 kHz was used to find the total capacitance (C = 1.78 ± 0.18 nF). The resonator had a quality factor of ∼40,000 as determined by a Lorentzian fit to the thermal noise peak (see Fig. 2a), implying an equivalent series resistance of approximately 4.5 mOhm.

a)

b)

resonator

transformer

M R C

LR

LT Lin

dark matter signal Fig. 1 (a) The lumped-element resonator. (b) The equivalent circuit model

SQUID

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Fig. 2 (a) Noise model fit to the thermal peak (see text). √ (b) Histogram of the excess power distribution. Data points are shown at the bin centers, with N error bars

A known AC current was injected into a separate inductor coil coupled to the SQUID while recording the output voltage, providing a calibration of the system gain. This measurement determines the SQUID input coil sensitivity using known values for the SQUID couplings.

3 Expected Signal-to-Noise Ratio In the sub-wavelength limit, hidden photons can be treated as isotropic, oscillating effective current density JHP which fills the superconducting shield [1]. If we assume that the effective hidden-photon current points along the longitudinal axis of the shield and ignoring their small velocity, this current takes the form  JHP = −ε 0

mc2 h¯

2 

   imc2 t h¯  2μ0 ρDM exp zˆ , mc h¯

(1)

where ε is the kinetic mixing angle, m is the hidden photon mass, and ρDM is the local dark matter density. Unlike microwave cavities, the dimensions of the shield are much smaller than the Compton wavelength of the dark matter particles. The boundary conditions result in a suppression of the electric field compared to the magnetic field and magnetoquasistatic approximations apply [1]. JHP sources an oscillating, quasistatic magnetic field inside the shield, which is sensed by the resonator inductor coil. The induced EMF was found by Faraday’s law to be VHP

iε = c



mc2 h¯

2





imc2 t 2μ0 ρDM exp h¯



 Nh  2 2 y2 − y1 , 4

(2)

where N is the number of coil turns, h is the height of the rectangular coil, and (y1 , y2 ) denote the horizontal offset of the coil edges to the center of the shield.

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The SQUID input-coil current due to a voltage V in series with the resonator inductance is given by iin =

V M , LT + Lin Zr

(3)

where Zr is the impedance of a series RLC resonator with L = Leff = LR − M 2 /(LT +Lin ) due to screening by the transformer. Equation 3 was used to calculate the input-coil current spectral density iHP due to an assumed VHP . Loss mechanisms in the resonator (represented √ by the resistance R) produce a Johnson–Nyquist noise voltage density Vth = 4kTN R in series with the resonator inductance. The effective noise temperature TN determines the height of the inputreferred thermal noise peak, while R determines the bandwidth. We denote the input-coil current noise due to this thermal noise as ith . The SQUID itself adds an additional input-referred white noise iw . We fit the input-referred average power spectral density (PSD) to the uncorrelated sum of ith √ and iw (Fig. 2a) which gave R = 4.53 mOhm, TN = 9.34 K, and iw = 1.41 pA/ Hz. Using these parameters, the input-referred expected signal-to-noise ratio, η, is η=

|iHP |2 . |ith |2 + |iw |2

(4)

4 Data Collection and Analysis The test data was recorded on the morning of July 28, 2018. The amplified SQUID output voltage was sampled at 25 MHz with a 16-bit digitizer. The average PSD was formed from 6900 independent 2.68 s time records (226 samples, Δf = 0.37 Hz), equal to a total integration time of τ = 5.14 h. A Blackman–Harris (BH) window was applied to each time trace prior to PSD computation. Analysis of the data loosely followed the procedure outlined in [15]. The spectral baseline was removed by applying a Savitzky–Golay (SG) filter (W = 50, d = 6) to the average PSD. The PSD and SG output were truncated to an analysis band of 492.027±1.5 kHz. The residual excess power in each frequency bin was determined by normalizing the PSD to the SG output and subtracting 1. In the absence of a signal, each resulting bin should be an √ independent sample drawn from a Gaussian distribution with μ = 0 and σ = 1/ τ Δf . A histogram of the excess power data is shown in Fig. 2b. While the distribution has a mean of zero, the width is narrower than expected (σM = 0.86). We can attribute this narrowing to the combined effects of the BH window and SG filter, which was confirmed by a Monte Carlo simulation in which the excess power distribution was formed from computer-generated noise traces with and without application of windowing/filtering. We conclude that the distribution of excess power over frequency bins was consistent with noise. In future analyses, a standard rectangular window will be used.

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Fig. 3 (a) The 90% C.L. exclusion line on hidden-photon dark matter (see text) with ±1σ band due to systematic error. The exclusion is calculated assuming that the effective hidden-photon current points along the longitudinal axis of the shield; field disalignment would weaken this limit. (b) Wider view of hidden-photon parameter space. The result presented here is shown in red. Projected limits from a 1-year scan of the DM Radio Pathfinder are shown in blue. Limits from axion haloscopes are shown in green. The dashed line is a model-dependent constraint above which hidden photons would not account for the total dark matter density [10]

We set an upper limit on the hidden-photon kinetic mixing angle ε (normalized to a local dark matter density of ρDM = 0.45 GeV/c2 ) using the maximum observed single-bin excess power of 3.34σ . The exclusion is calculated assuming that the effective hidden-photon current points along the longitudinal axis of the shield; field disalignment would weaken this limit. Three orthogonal experiments could be used to eliminate this assumption, or multiple rescans with one experiment could be used to set a more rigorous limit with a more general assumption about directionality. We divide this value by σM = 0.86 to correct for the effects of the BH window and SG filter. While we only performed a single-bin search for excess power, the standard halo model predicts greater than 45% of the total signal power would be contained within a single bin. We divide our single-bin excess value by this as a correction. For each frequency bin, we use Eq. 4 to determine the kinetic mixing angle which would produce an excess power above 3.34σ /(0.86 × 0.45) = 8.63σ at 90% confidence. The result is shown in Fig. 3a, excluding kinetic mixing angles of ε  1.5 × 10−9 in a band around 2.035 neV. We note that even for frequency bins detuned by several resonator bandwidths, the exclusion limit is not significantly degraded due to out-of-band sensitivity [14].

5 Discussion and Conclusion The resonator described here was designed as a test structure for DM Radio. It lacks the physical volume, frequency scanning, and maturity of modern axion haloscopes, and the exclusion limit it demonstrates is not competitive with larger experiments (although it is in a new mass regime). This work is just a first step

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towards using lumped-element resonators to search for dark matter waves in the sub-µeV regime. The resonator quality factor of ∼40,000, already comparable to copper microwave cavities, is believed to be limited by a combination of the low characteristic resonator impedance and loss in the sapphire capacitor. Operation in liquid helium with a non-optimized SQUID is responsible for the high noise temperature of 9.34 K. Even with these limitations, a 5.14-h integration was able to set an exclusion limit on the kinetic mixing angle of ∼2.035 neV hidden-photon dark matter and is the first direct detection limit in this mass range. The next step in the DM Radio program is the DM Radio Pathfinder [2], a tunable resonator with 7x larger volume and 10x higher characteristic impedance. Projected hidden-photon limits from a 1-year scan of the Pathfinder are shown in Fig. 3b. The full DM Radio will include a DC magnetic field to search for axions, with an ultimate goal of probing the QCD band in the 10 neV/c2 –1 µeV/c2 mass range, and a broader range of axion-like-particles and hidden photons across peV–µeV/c2 masses. Acknowledgments This research was sponsored by a seed grant from the Kavli Institute for Particle Astrophysics and Cosmology, by the Laboratory Directed Research and Development Program of the SLAC National Accelerator Laboratory, and by DOE HEP QuantISED award #100495.

References 1. S. Chaudhuri et al., Phys. Rev. D 92, 075012 (2015) 2. M. Silva-Feaver et al., IEEE Trans. Appl. Supercond. 27, 1400204 (2017) 3. R.D. Peccei, H.R. Quinn, Phys. Rev. Lett. 38, 1440 (1977) 4. S. Weinberg, Phys. Rev. Lett 40, 223 (1978) 5. F. Wilczek, Phys. Rev. Lett. 40, 279 (1978) 6. P. Sikivie, Phys. Rev. Lett 51, 1415 (1983) 7. N. Du et al., Phys. Rev. Lett. 120, 151301 (2018) 8. L. Zhong et al., Phys. Rev. D 97, 092001 (2018) 9. P.W. Graham, J. Mardon, S. Rajendran, Phys. Rev. D 93, 103520 (2016) 10. P. Arias et al., JCAP06 13 (2012) 11. B. Cabrera, S. Thomas, Workshop Axions 2010, U. Florida. http://www.physics.rutgers.edu/~ scthomas/talks/Axion-LC-Florida.pdf (2008) 12. P. Sikivie, N. Sullivan, D.B. Tanner, Phys. Rev. Lett. 112, 131301 (2014) 13. Y. Kahn et al., Phys. Rev. Lett. 117, 141801 (2016) 14. S. Chaudhuri, K.D. Irwin, P.W. Graham, J. Mardon, arXiv:1803.01627 (2018) 15. B.M. Brubaker et al., Phys. Rev. D 96, 123008 (2017)

Employing Precision Frequency Metrology for Axion Detection Maxim Goryachev, Ben T. McAllister, and Michael Tobar

Abstract Precision frequency metrology can overcome critical challenges in axion dark matter searches based on the power detection method. Using a generalization of the standard axion haloscope approach onto a case with two resonant modes, it is demonstrated how dark matter axion signal is converted into resonant mode phase fluctuations. The method is not only sensitive enough in principle to achieve the predicted model bands, but also remove the main complications such as the need for strong magnetic fields, large cavity volumes, and quantum sensors making the whole experiment much more cost effective. Keywords frequency metrology · dark matter · axions · oscillators · phase noise

1 Introduction In standard contemporary cosmology, 26.8% of the universe exists in the form of dark matter. Despite being such a large portion (compared to ordinary matter accounting for only about 5%), dark matter has not been observed directly using laboratory methods. So, after the recent direct detection of gravitational waves, direct observation of dark matter is the next biggest challenge in modern physics. Despite a plethora of experimental attempts, the goal seems to remain distant for a few reasons. Not only is dark matter coupling to observable quantities extremely weak, leading to requirements for extremely high sensitivities, but also the nature of the dark matter, including its mass and other relevant parameters are not know M. Goryachev () · B. T. McAllister · M. Tobar ARC Centre of Excellence for Engineered Quantum Systems, Department of Physics, University of Western Australia, Crawley, WA, Australia e-mail: [email protected] https://equs.org/

© This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_17

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and are highly model dependent. As a result most dark matter detectors are bulky, expensive, and only capable of scanning in a limited region of the large parameter space. Thus, the dark matter research community is constantly looking for new sensitive approaches. The axion is a promising candidate for galactic dark matter. This theory motivated particle is very alluring as it is not only a dark matter candidate but also solves the strong CP (charge-parity) problem in quantum chromodynamics [1, 2]. Moreover, this low mass dark matter candidate is increasingly promising as no signs of dark matter have been observed in the extensively searched MeV–TeV mass range. Although most accepted axion models, KSVZ (Kim–Shifman–Vainshtein– Zakharov) and DFSZ (Dine–Fischler–Srednicki–Zhitnitskii), do not stipulate its mass or coupling to baryonic matter, they limit these values to certain regions of the parameter space if axions constitute the bulk of galaxy dark matter. Most experimental approaches to detect cold dark matter axions, postulated to exist as a dark halo of our galaxy, employ an axion haloscope based on strong DC magnetic fields, a detector proposed by P. Sikivie [3]. Its working principle is based on the inverse Primakoff effect in which axions interact with virtual photons (from an external DC magnetic field), producing detectable real photons whose frequency corresponds to the axion mass. In all these power detection experiments, axions would appear as a “smeared out” signal (due to velocity dispersion), a weak excess of power over the detector noise floor. To achieve sensitivity to probe the theoretically predicted regions of the parameter space in such a haloscope, a typical experiment [4, 5] is required to employ highly tunable, low loss, large cavities with maximally uniform modes situated in very strong magnetic fields at temperatures of a few mK. Moreover, such cavities must be measured with quantum limited amplification over time spans of several years. The readout noise floor, cavity geometry and quality factor, and mode form factor set the scanning rate at which the detector can exclude the parameter space at a desired level of sensitivity. The challenge increases dramatically if one wishes to move into the higher axion mass/frequency range, as typical haloscope cavities get smaller and exhibit higher losses, while at the same time the noise floor of the readout typically increases. So, recently it has become obvious that in order to improve the scanning rate and relax several technological requirements, one needs to look beyond the traditional haloscope technique based on direct power detection. This task may be accomplished by a fruitful alternative based on precision frequency metrology [6].

2 Basic Principles In a low enough mass range, axions may be approximated by a scalar field weakly coupled to photons via the Lagrangian term aE · B. Equivalently, axion effects can be represented by additional terms in Maxwell’s equations constituting the so-called axion electrodynamics [7, 8]. For galactic halo axions, terms containing spatial derivatives of the field vanish on the scale of a realistic laboratory experiment,

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leaving only an addition to Ampere’s law proportional to da dt B. This equation allows one to understand the axion haloscope based on the inverse Primakoff effect: by supplying a large DC magnetic field, one is trying to observe the resulting electric field source term via power detection. In such experiment, the cavity plays the role of a resonant antenna accumulating the power over a certain region of the space inside the magnet. A natural generalization of these effects is a cavity with two photonic modes without any external DC fields. By transforming the axion Lagrangian term into the form of the Hamiltonian density, the term may be understood as an axion induced coupling term between two photonic modes. By integrating the Hamiltonian density over the cavity volume, one can transform this coupling term into the language of creation/annihilation operators (cn† /cn ), where all spatial dependence is reduced to geometric form factors ξ describing orthogonality of the two modes over the cavity volume. This form allows us to take the rotating wave approximation for two particularly interesting cases [6]. The first case is referred as axion upconversion and constitutes the situation for which axion mass/frequency is equal to the sum of the frequencies of the two cavity modes (fa = f1 + f2 ). The resulting interaction Hamiltonian is simply the linear or beam–splitter interaction of the type a ∗ c1† c2 + ac1 c2† where axion field amplitude a may be understood as a complex coefficient, i.e. a phase shifter. The second case is called downconversion describing the situation in which the axion frequency equals the difference between frequencies of the two cavity modes (fa = f1 − f2 ). Here, the mode interaction is described by the parametric amplification term (ac1† c2† + a ∗ c1 c2 ). Based on these interaction Hamiltonians, it is possible to derive a transfer function between from one mode that is externally pumped to another playing the role of an “idler” mode. This means that the two mode experiment is a further generalization of the Sikivie axion haloscope for which virtual photons of the DC magnetic field are replaced with real photons of the pumped mode. As a natural generalization, the dual mode experiment provides extra features and information, in particular, the axion phase relative to the pump phase. On the other hand, the deposited power of the axion signal is now proportional to the power driven in the pumped mode instead of the squared magnitude of the DC magnetic field [6]. This fact means that for practical implementations, power-excess based measurements in the dual mode configuration would be much less sensitive because it is impossible to deposit enough power in a cavity mode to match the strength of modern superconducting magnets. Since this gap is impossible to recover with higher Q-factors in fully superconducting cavities, such experiments have been never implemented [9]. Improvement of the sensitivity of the dual mode experiment comes from the phase sensitivity of the measurements. Indeed, instead of asking how much power is deposited by axions into modes, one may ask a question of how much phase or frequency shift is induced by the axion mediated coupling between the modes [6]. The strength of the axion mediated interaction term is proportional to its

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amplitude and causes shift of eigenfrequencies of the coupled mode system that can be effectively measured. This shift of paradigm transforms the detection problem into the domain of frequency metrology.

3 Details of the Proposed Approach Frequency metrology techniques [6] allow to detect either phase or frequency fluctuations induced by axion mediated mode coupling. To estimate the magnitude of the frequency shifts, one may calculate the eigenvalues of the coupled mode system that show that in the ideal case of axion upconversion, the real part of the frequency shift is proportional to the axion–photon coupling times the axion field magnitude in frequency units. In the case of downconversion, the effect comes through in the imaginary parts of eigenfrequencies due to parametric amplification nature of the coupling. This result relates the axion coupling and strength to sensitivity of a proposed detector through its frequency stability [6]. Although frequency shift measurements are very sensitive, their direct practical implementation could be challenging because the effect is at DC. To make this approach more practical, it is straightforward to shift from direct frequency estimations to phase noise measurements, where the axion induced signal is at some offset frequency in the Fourier spectrum of phase fluctuations. This could be done by splitting the actual axion frequency fa into an exact difference or sum between the modes fa = f1 ± f2 and a small offset f (Fourier frequency), i.e. fa = fa + f = f1 ± f2 + f and deriving a transfer function between the axion mediated slowly varying (around frequency f ) coupling parameter and output phase fluctuations. This can be done by writing the system equations in terms of slowly varying magnitudes and phases and linearize them for small fluctuations around steady state values. The resulting function in the phase space is a second-order low pass filter whose gain is set by the ratio of magnitudes stored in both modes. This means that the sensitivity no longer depends on a bare amplitude, and significant results could be reached even in the regime of modest amplitudes. In practice, the phase noise axion detection can be implemented with either a dual mode passive phase noise measurement scheme or a dual mode oscillator [6]. In the first scheme, signals from two external signal sources are split into two arms each, a dual mode cavity is put into one arm of each reference signal. The signals that pass through the cavity are then compared to the properly phase shifted original signals to measure the cavity induced fluctuations. In the second approach, a dual loop feedback oscillator is built around the four port cavity without any external signal sources giving significant improvement of the sensitivity. Calculations of phase fluctuations for such systems suggest that by implementing low noise measurement techniques for a cryogenic microwave oscillator it is possible to achieve both the KSVZ and DFSZ axion model bands [6]. As a separate experiment, it is possible to construct a broadband haloscope based on frequency metrology by matching two photonic mode frequencies f1 = f2 ,

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so that the axion frequency fa is the same as the Fourier frequency f . Here, by measuring the cavity phase noise in the Fourier spectrum, one searches for low mass axions over a few decades. Although for the best sensitivity of the setup over such large frequency range, it might be necessary to implement both cryogenic and room temperature experiments as one gives better sensitivity for close to carrier and the other for higher Fourier frequencies.

4 Advantages of Axion Frequency Metrology The proposed precision frequency metrology method compared to traditional power detection axion search schemes demonstrates certain advantages that make such experiment considerably more cost effective. These advantages are: – No DC magnetic fields required. The proposed approach does not rely on any DC magnetic fields comparing to traditional haloscope approaches; – No requirement for superconducting technology such as SQUIDs. Calculation of sensitivity of a frequency detector [6] is based on phase noise specifications of traditional semiconductor amplifiers. Superconductor based amplifiers might lead to some improvement, though their usage is not crucial comparing to power detection methods; – Independence of cavity volume. Unlike for conventional haloscopes, sensitivity of the frequency detection method is not directly proportional to this parameter, though cavity volume is still related to resonance frequencies and Q factors. This advantage solves the major problem for higher mass axion searches (fa > 10 GHz); – Operation near liquid-helium temperatures (>4 K). Moreover, only a few components such as the amplifier and the cavity should be cooled to this temperature. Indeed, all ultra-stable microwave and optical clocks and oscillators operate at bulk temperatures above 4 K. Thus, dilution refrigeration (temperatures below 100 mK), key component in traditional haloscopes, is not required making such detection principles more cost effective and accessible to a broader research community. Although in future research, milli-Kelvin temperatures might result in some improvements in axion sensitivities; – Possibility to probe otherwise almost inaccessible higher and lower frequency ranges. Because axion mass is either the sum or difference of cavity resonance mode frequencies, one can search for axions in frequency ranges where the power detection technique is limited by extra challenges. For instance, working around 20 GHz, one is able to probe axion masses in the vicinity of 40 GHz, where traditional haloscope experiments are significantly more difficult. Moreover, optical cavities might be used to probe otherwise inaccessible regions of the THz and infrared spectra, as well as millimeter-wave and microwave frequencies; – No high power levels required. Though sensitivity of the frequency technique is explicitly dependent on amplitudes of circulating signals, it is proportional to a

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ratio of deposited amplitudes rather than to their absolute values. The sensitivity calculations [6] assume very modest levels of stored signal powers; Axion phase sensitivity. Unlike power detection haloscopes based on DC magnets, the frequency metrology technique based on two resonance modes will give additional information about the axion, i.e. its phase relative to oscillator or pump signals; KSVZ/DSFZ sensitivity. The proposed cryogenic dual mode experiment [6] is able to reach the KSVZ/DSFZ axion dark matter models with optimistic but achievable parameters. Moreover, even a tabletop search may lead to competitive limits on dark matter; Possibility of a broadband low mass axion search. It is demonstrated that one can construct a wide-band experiment to search for low mass axions that does not require tuning; Fractional order sensitivity in the axion–photon coupling parameter gaγ γ . It has been demonstrated that the sensitivity of the axion induced frequency shifts Δf √ may be of a fractional order of the axion coupling constants (Δf ∼ 2 gaγ γ and √ Δf ∼ 3 gaγ γ ), by implementing an exceptional point system [10] and leading to unprecedented improvement in sensitivity.

5 Conclusions and Perspectives We demonstrate that precision frequency metrology is a very sensitive tool for axion dark matter search that requires less resources than traditional power detection techniques. Only two basic schemes based on frequency metrology have been considered so far. The approach could be further improved by implementing more sophisticated schemes, employing optics or reducing phase noise by other methods. Additional discussion can be found in subsequent publications [11]. Acknowledgments This research was supported by the Australian Research Council Centre of Excellence for Engineered Quantum Systems (project ID CE170100009) and DP160100253.

References 1. J. Jaeckel, A. Ringwald, The low-energy frontier of particle physics. Annu. Rev. Nucl. Sci. 60, 405–437 (2010). https://doi.org/10.1146/annurev.nucl.012809.104433 2. R.D. Peccei, H.R. Quinn, CP conservation in the presence of pseudoparticles. Phys. Rev. Lett. 38, 1440–1443 (1977). https://doi.org/10.1103/PhysRevLett.38.1440 3. P. Sikivie, Experimental tests of the “invisible” axion. Phys. Rev. Lett. 51, 1415–1473 (1983). https://doi.org/10.1103/PhysRevLett.51.1415 4. S.J. Asztalos, G. Carosi, C. Hagmann, D. Kinion, K. van Bibber, M. Hotz, L.J. Rosenberg, G. Rybka, J. Hoskins, J. Hwang, P. Sikivie, D.B. Tanner, R. Bradley, J. Clarke, SQUID-based microwave cavity search for dark-matter axions. Phys. Rev. Lett. 104, 041301 (2010). https:// doi.org/10.1103/PhysRevLett.104.041301

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5. B.T. McAllister, G. Flower, J. Kruger, E.N. Ivanov, M. Goryachev, J. Bourhill, M.E. Tobar, The ORGAN experiment: an axion haloscope above 15 GHz. Phys. Dark Univ. 18, 67–72 (2017). https://doi.org/10.1016/j.dark.2017.09.010 6. M. Goryachev, B.T. McAllister, M.E. Tobar, Axion detection with precision frequency metrology. Phys. Dark Univ. 26, 100345 (2019). https://doi.org/10.1016/j.dark.2019.100345 7. F. Wilczek, Two applications of axion electrodynamics. Phys. Rev. Lett. 58, 1799–1802 (1987). https://doi.org/10.1103/PhysRevLett.58.1799 8. M.E. Tobar, B.T. McAllister, M. Goryachev, Modified axion electrodynamics through oscillating vacuum polarization and magnetization and low mass detection using electric sensing. Phys. Dark Univ. 26, 100339 (2019). https://doi.org/10.1016/j.dark.2019.100339 9. P. Sikivie, Superconducting radio frequency cavities as axion dark matter detectors. arXiv:1009.0762 [hep-ph] (2013). https://arxiv.org/abs/1009.0762 10. M. Goryachev, B.T. McAllister, M.E. Tobar, Probing dark universe with exceptional points. Phys. Dark Univ. 23, 100244 (2018). https://doi.org/10.1016/j.dark.2018.11.005 11. M. Goryachev, B.T. McAllister, M.E. Tobar, Precision frequency metrology for axion searches, in 2019 Joint Conference of the IEEE International Frequency Control Symposium and European Frequency and Time Forum (EFTF/IFC), Florida (2019). https://doi.org/10.1109/ FCS.2019.8856108

Bayesian Searches and Quantum Oscillators George Chapline and Matt Otten

Abstract A new LLNL Strategic Initiative is focused on developing improved methods for Bayesian inference when the input data depends on hidden parameters. Part of this effort involves investigating the idea of using an array of quantum oscillators (viz., microwave cavities) as an analog computer for implementing Bayesian model selection. The practical motivations are twofold: (1) Bayesian model selection problems are often intractable using conventional digital computers and (2) quantum information processing may allow detection of weak analog signals below the usual quantum noise threshold. Keywords Bayesian search · Quantum oscillators · Quantum information · Quantum noise

The working hypothesis for our current LLNL quantum Bayesian effort is that quantum dynamics of coupled harmonic oscillators may alleviate some of the difficulties encountered when trying to optimize Bayesian model selection problems, e.g. finding the optimum strategy for complex Bayesian search problems. This hypothesis already has a certain plausibility by virtue of the discovery at Google Brain [1] that neural network approaches to pattern recognition can in many situations of interest be represented in terms of manipulation of Gaussian processes (GPs), i.e. variables in a large dimensional vector space where all the variables of interest are represented using iid Gaussian probability densities. Since the quantum wave functions for harmonic oscillators involve Gaussian functions, it seems natural to surmise that the quantum dynamics of oscillator arrays might G. Chapline () LLNL, Livermore, CA, USA e-mail: [email protected] M. Otten ANL, Lemont, IL, USA

© This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_18

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potentially be useful for finding weak signals in analog signals. Indeed this may be of some interest in connection with the problem of searching for axions with superconducting microwave cavities. Impetus for this effort is provided by another LLNL-LDRD project aimed at the development of a four qudit quantum computer as a user facility, where each qudit consists of the lowest quantized energy levels of a superconducting microwave cavity. If one interprets the lowest N levels of a microwave oscillator as representing N locations, and each location is paired with a qubit, then one might imagine that such a system could be used to represent a “Monty Hall” type Bayesian search. In this simple type of a Bayesian search one is searching for an object concealed in one of the N boxes, and the auxiliary qubit can be used to represent the probability that the object is present or not at the paired location. For example, in the case of the axion search one might identify each location with the axion frequency. This type of problem differs from the typical machine learning problem, where one is just trying to fit the explanations for a training set of observed data inputs with a parameterized interpolation model, in that there is an additional hidden “factor” that influences the input data—the location of the object. In this talk I describe some of the progress we have made towards emulating the probabilistic dynamics of a Bayesian search with an array of quantum oscillators. In the 1990s it began to be appreciated that the GPs representations for both input data and data interpretation models would provide a way of using Bayes’ formula for posterior probabilities to construct analytic regression models, whose parameters are conditioned with past data, and which are able to provide an explanation for new data [2]. It has also been shown [3] that GP representations can be used to facilitate the use of Bayesian methods for robotic control problems. The application of GP methods to control problems is based on the application of Bayes theorem at each stage of the search or control process to make probabilistic predictions for the future state of the system given a control decision. Actually there is a close relationship between Bayesian approaches to robotic control and Bayesian searches. However, whereas the only model parameters for GP-Bayesian control applications are the GP covariances and means, Bayesian searches involve a hidden parameter: the location of the object being sought. It seems natural to assume that the successes of GP-Bayesian techniques for solving robotic control problems can be extended to Bayesian searches, although this remains to be demonstrated.

1 Bayesian Filter Approach to Search Problems A Bayesian search is defined by a sequence of control decisions UN ≡ {xn+1 − xn , n = 1, . . . , N − 1} leading to a sequence XN ≡ {xn , n = 1, . . . , N} of locations or system states that will be interrogated for the presence of a desired object or system state x∗ . The interrogations result in a sequence of measurements YM ≡ {yk , k = 1,

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. . . , M} that are intended is to determine at each step k whether the desired object or state x∗ is present. In reality the result of each observation, yk , only determines probabilistically whether xk is the desired state x∗ . Based on the collection Yn of measurements gathered during the first n-steps of a search and the prior posterior probability density pn−1 (x), one can estimate the posterior probability density at step n pn (x) ≡ pn (x|Yn , Xn , pn−1 ) that the desired object or state x∗ the “Bayes Filter” for predicting the future becomes   P (Yn |Un , x ∗ = x, pn−1 ) pn−1 (x ∗ = x|Yn−1 , U n−1 ) pn x ∗ = x|Yn , U n , pn−1 = P (Yn |Un ) (1) Since the results of using a Bayes Filter are expressed as probabilities, it is natural to inquire how much information has been gained about the location of the desired object after N decision steps. A natural measure of the information gathered after n-steps is the Shannon entropy  H (pn ) = −

pn (x) log pn (x)dx

where the log is base 2. The optimal search strategy can now be characterized as the problem of finding a policy for choosing the control sequence {Un } so that the Shannon entropy H(pN ) is minimized after N-steps. As shown in Ref. [3] the use of Gaussian process representations for the variables XN and UN allows one to write down analytic expressions for the posterior probabilities pn (x∗ = x) given a set of training data, which in turn allows one to make incremental improvements in the cost function H(pN ). Unfortunately when many input data streams or hidden variables are involved evaluating the Bayes filter predictions can easily become intractable—even with the use of Markov chain Monte Carlo sampling techniques. As it happens there is a neural networklike architecture, the Helmholtz machine, and associated algorithm, the wake-sleep algorithm, for choosing the hidden parameters that minimize the Shannon entropy [4]. Unfortunately, in contrast with the gradient descent algorithm that can be used to find the parameters (means and variances) of a GP [3], use of the wakesleep algorithm to find optimal hidden parameters with existing formulations of the Helmholtz machine, which use the binary spin nodes usually used for neural networks, is typically an intractable problem. Whether there is a GP version of the classical Helmholtz machine would be useful is an area of active research. On the other hand, we already have evidence that the quantum dynamics of interacting qubits might provide a framework for realizing the Helmholtz machine approach to finding hidden variable models for input data.

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2 Self-Organization Approach to Model Selection Our original inspiration for the idea that the quantum dynamics of oscillators might be useful for solving Bayesian inference problems with hidden variables was provided the observation that the Durbin–Willshaw elastic net method [5] (that actually predated the Helmholtz machine) for solving the traveling salesman problem (TSP) (cf. Fig. 1) might be reformulated as quantum dynamics problem [6]. As the name suggests the elastic net approach to solving the traveling salesman problem involves replacing the path of the “salesman” by a string of points—the square points in Fig. 1—connected by springs, while the connection of these points to the cities that are to be visited—the round points in Fig. 1—reflects the likelihood that a chosen point represents the visit to a particular city as part of the minimum path. It is almost obvious way that this setup can be translated to method for solving Bayesian search problems by representing the input data {Yi } as the positions of the round points in Fig. 1, while the positions of the square points are parameters w(xn ) defining a model for the path traveled. The string running through the square nodes plays the role of MacKay’s regression model Z(x) for input data. In the limit of weak coupling one can describe the Bayesian regression of the interpolation model as the quantum motion of the string variables string acted on by a stochastic classical field arising from the stochastic nature of the distances d(i,μ) the stochastic nature of the influence of the oscillators attached to the cities provides the dissipation needed to minimize the information cost for representing the positions of the nodes. The net result is the prior distribution for the feature variables has the form         α ∂w 2 P w rn = exp − 2 ∂rα n

(2)

This “string action” plays essentially the same role as the prior distribution for the parameters for the generative model for input data used in the “sleep phase” of the Helmholtz machine [4]. In the Durbin–Willshaw method the data model representing the square points is adapted to the data by introducing in addition Fig. 1 Elastic net solution for the traveling salesman problem

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to the prior model eq. 5 a cost function for the difference between the model and observations of the form:      β 2 EC [{w (rn )}] = − exp − |w (rn ) − φm | log (3) 2 n m This cost function causes the feature vectors to be attracted to the “true” values for the model parameters and corresponds to a likelihood predictive model of the form. P (φi | {w (rn )}) =

N  n=1

  β exp − |w (rn ) − φi |2 2

(4)

It is perhaps worth noting that from the point of view of quantum mechanics the Durbin–Willshaw attraction between the string nodes representing the locations of the cities and the fixed nodes representing the cities to be visited corresponds to attracting the Gaussian wave functions for the model oscillators to the ground state wave functions for particles bound to the city locations by a harmonic potential. We can now see how a quantum version of the Durbin–Willshaw network might be able to deal with optimizing the type of hidden parameter that occurs in the Monty Hall problem. The hidden variables in the traveling salesman problem can be thought of as variables λni that link a particular point of the string with a particular city. It is fairly obvious from the structure of the Durbin–Willshaw energy function that the optimal values for these variable emerge automatically as the string relaxes to the shortest path threading the cities. In the quantum version of the Durbin–Willshaw network the string length becomes the action function for the quantum oscillators representing the string. Evidently then the Durbin–Willshaw self-organizing net method for solving the traveling salesman problem (TSP) is essentially equivalent to taking the classical limit of a Feynman path integral for a system of points linked by quantum oscillators that are acted on by a stochastic external field.

3 Quantum Dynamics for an Oscillator Coupled to Qubits The density matrix propagator for a quantum harmonic oscillator subject to an environment consisting of a classical noisy signal f (t) random classical force is  J =

  ei{S[x(t)]−S[y(t)]}/ Φ x(t) − x (t) Dx(t)Dy(t)

(5)

where S[x(t)] is the classical action for an isolated oscillator with mass m and angular frequency ω0

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 e

i {S[x(t)]−S [x (t)]}/

 =

 im  2 2 2 2 2 2 x˙ − ω0 x − y˙ + ω0 y , exp 2 

and [k(t)] is the functional Fourier transform of the probability density functional for the noisy signal f (t):  Φ [k(t)] =

ei



k(t)f (t)dt

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For a Gaussian process signal one has     −1         1

t, t f t − f t dtdt P [f (t)] = exp − f (t) − f (t) A 2 where A is the autocorrelation function for the signal. If instead of a classical noise signal f (t) the free oscillator is coupled to a quantum environment, then the real classical functional  is replaced by a complex valued influence functional F{[x(t), x (t )]. The resulting theory of quantum noise, due independently to Feynman and Keldysh, implies that the density matrix obeys a non-Markovian master equation, which in the case of a superconducting cavity coupled to an array of qubits has the form:    ∂ρ i ∂ 2 ρ ∂ 2 ρ = + ∂t 2L ∂Q2 ∂Q 2 ⎛ ⎞ ⎞ ti =t ti  C 2 ln Δ /Δ     iLω02  2 max min 2 +⎝ Q(t)−Q (t) Q(s)−Q (s) ds ⎠ ρ ⎠ Q −Q − 2 2 Δmax t =−∞ i

ti −τ

(7)

where min and max are the minimum and maximum qubit level splitting. As first showed by Caldeira and Leggett the nature of the density matrix evolution implied by (7) is that the forward and backward trajectories, Q(t) and Q (t), converge with time. It is of course encouraging that this “wake-sleep” like behavior is just what we are looking for if we wish the behavior of the quantum oscillators representing models for either input data or data models to converge as indicated in Fig. 1. There is a Hamiltonian formulation of (7) which is more easily adapted to the qudit case     y ↑ z 1 a Δ σ +igσ −a H = ωr a ↑ a+ 12 + j j j 2 j   gi  + −iΔ(t−t ) 0 − σ − e iΔ(t−t0 ) + F (t) σ e a˙ = −iω0 − i j j  F (t) = −

 j

j

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t t0

        sin t − t σjz t a + t − a t dt

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Because the “starting times” t0 for the episodes of coherent evolution for each qubit are randomly distributed the influence function F can now be regarded as a random function of time whose fluctuations can be measured by the autocorrelation function for the oscillator amplitude. ⎞ ⎛ t t  gj2 ⎜      

⎟ e−iΔj (t−t ) x t −eiΔj (t−t ) y t [x(t)−y(t)] dt dt ⎠ F (t) ≈ exp − ⎝ 2 j

tj 0 tj 0

Thus it appears that the quantum dynamics of oscillators coupled to qubits does in fact provide a kind of quantum analog of the GP representations that have been found to be useful in the classical Bayesian analyses of regression and robotic control problems. Of course, in practice, one would have to work with qudits rather than the full Hilbert space of an oscillator as is implied in Eq. (7), and this is what we are currently pursuing. One important issue is how to represent the measurements that are in the inputs for a Bayesian search. We have already carried out simulations where at each instant of time, the density matrix is either left unchanged or reset to zero at a specified rate. The result that the initial excitation of the qudit decays with time is shown in Fig. 2. One interesting aspect of this calculation is that the qudit decay occurs even if the coupling of the qubit to the qudit is so weak that it is not excited to its excited state.

Fig. 2 Decay of a qudit due to coupling to a qudit that is sporadically reset to |0>

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Acknowledgments This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and was supported by the LLNL-LDRD Program under Project No. 16-ERD-013.

References 1. Lee, J. et. al., Deep Neural Networks as Gaussian Processes. ArXiv reprints (2017) 1711.00165 2. D. Mackay, Information Theory, Inference, and Learning Algorithms (Cambridge U Press, Cambridge, 2003) 3. M. Diesenroth, D. Fox, C. Rasmussen, Gaussian processes for data efficient learning in robotics and control. IEEE Trans. Pattern Anal. Mach. Intell. 37, 408–423 (2015) 4. G. Hinton et al., Wake-sleep algorithm for unsupervised neural networks. Science 268, 1158 (1995) 5. R. Durban, D. Willshaw, An analogue approach to the traveling salesman problem using an elastic net method. Nature 326, 689 (1987) 6. G. Chapline, Quantum mechanics as self-organization. Phil. Mag. B 81, 541 (2001)

Status of the MADMAX Experiment Chang Lee On behalf of the MADMAX collaboration

Abstract Relic axions generated after cosmic inflation may have a mass of around 100 µeV. While being an excellent candidate for cold dark matter, axions of this range are hardly covered by existing detection techniques, which are based on electromagnetic resonance of 3D cavities. A dielectric haloscope is a promising alternative technique for a search in this mass range. The MADMAX collaboration is aiming to build a dielectric haloscope that is sensitive to post-inflation QCD axions. Here, we present the principle, current status, and future plans of the MADMAX experiment. Keywords Post-inflation Axion · Dielectric haloscope

1 Motivation The Peccei–Quinn symmetry [1] solves the strong CP problem in QCD by promoting the CP-violation angle θ into a dynamical field. The fluctuation of θ , the axion, can have a characteristic energy scale fP Q  fweak , which makes it light with feeble interactions with standard model particles. Nonetheless, an axion is a compelling candidate for cold dark matter because it can be generated by nonthermal mechanisms such as vacuum misalignment [2] after the Big Bang. Astronomical observations [3] bound the axion mass ma loosely, to be above a few µeV. An axion with ma ∼ 100 µeV can constitute the majority of dark matter if it was generated after the cosmological inflation. When the PQ symmetry is broken, each causally connected patches of the universe can have different θ between −π and π , and each patch has the corresponding initial axion density. Cosmic inflation

C. Lee () Max Planck Institute for Physics, München, Germany e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_19

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can enlarge one of these patches and its axion density to the current values. If the initial θ was extremely close to 0, even the lowest ma can explain the current dark matter density. Without inflation, on the other hand, the current axion density is the average density of multiple patches. The post-inflation axion scenario, combined with production mechanisms other than vacuum realignment, brings attention to the axion with ma ∼ 100 µeV, which is significantly higher than ∼2 µeV of the conventional resonant cavity experiments [4]. The “high mass” axion poses experimental challenges. The detection principles do not change from those of the resonant cavity experiments: The axion number density inside a de Broglie wavelength is still so immense that axions can be treated as a coherent classical field within typical length scale of an experiment. The field induces a spatially constant electric field via an inverse-Primakoff effect [5] inside a strong magnetic field with a frequency νa , which corresponds to the axion mass ma . For ma = 100 µeV, νa is 25 GHz. Unfortunately, fundamental mode resonant cavities rapidly lose their volume and quality factor at this high frequency, and thus become much less effective.

2 Dielectric Haloscope A dielectric haloscope [6, 7] is a compelling idea for detection of axions for ma in 40−400 µeV, whose schematic is shown in Fig. 1. It contains a flat mirror and multiple dielectric disks in a strong transverse magnetic field. Strength of the axioninduced electric field is inversely proportional to relative permittivity of materials and is discontinuous at boundaries between the materials, which include the surface of the mirror and disks. To match the E and H fields here, additional propagating electromagnetic waves oscillating at νa are emitted. However, the propagating wave, i.e. the axion signal, has negligible power. The power from a single boundary is only around 10−27 W/m2 at 10 T and is hardly detectible with existing technologies. By controlling the gaps between the disks, the axion-induced propagating waves from multiple boundaries can constructively interfere, and the total signal power can Fig. 1 Schematic of a dielectric haloscope. Image from [6]

Be

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be boosted to a detectible level. The enhanced amplitude is parametrized by boost factor (β). The boost factor and frequency can be flexibly tuned by varying the spacing between the flat mirror and disks. Sensitivity to QCD axions in 40 < ma < 400 µeV requires 80 of 1 m2 -area (A) disks aligned in a 10-T magnetic field (Be ). The system temperature (Tsys ) needs to be kept low to suppress the thermal noise and increase the scan rate. System temperature of 8 K is achievable by cooling the whole system to 4.2 K by liquid helium and using a high electron mobility transistor (HEMT). The scan rate [6] to reach |Cαγ | = 1, where Cαγ is a model-dependent proportionality constant for the axion–photon interaction, is  SNR 2  400 4  1 m2 2  m 2 Δt a ∼ 1.3 days 5 β A 100 µeV  T 2  10 T 4  0.8 2 sys −4 × Cαγ . 8K Be η Here, SNR is the signal-to-noise ratio, and η is the signal collection efficiency.

3 MADMAX

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Reflecvity Measurement Reflecvity Simulaon Group Delay Measured Group Delay Simulated

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The MADMAX collaboration aims to probe the QCD axions with a dielectric haloscope. To understand the exact behavior of the frequency dependent boost and to quantify deviations from the previously performed 1D calculations [7], 3D electromagnetic simulations are being developed [9]. The simulations provide estimation of loss, 3D-beam shape and its coupling to antenna, and the effect of tiled disks. The proof-of-principle setup shown in Fig. 2 helps to understand the optical behavior of the dielectric haloscope. This minimal setup consists of a mirror, sapphire disks, and an antenna at room temperature. Amplitude and phase of

2.5 0.15

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Fig. 2 The proof-of-principle setup (left). A receiver antenna is inside the copper tube on right. Comparison of the reflectivity measurement to simulation (right). Images from [8]

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the reflected wave are measured by a vector network analyzer and compared to simulation. The boost factor of a dedicated measurement can be calculated from simulations reproducing the electromagnetic response. With five equidistant sapphire disks, initial results show that the center of the boost curve relative to its FWHM and the boost power varies less than 5% due to inaccuracies of the disk spacings [8]. Work is ongoing to reduce the reflection of the antenna so that more disks can be included. The proof-of-principle setup is also used to study stability of the boost factor against misplacement or tilt of the disks. During axion search, a focusing reflector and an antenna collect the power emitted from the booster. For frequencies below 30 GHz, a HEMT with a noise temperature near 5 K has been set up. The pre-amplified signal is mixed-down to a frequency accessible by samplers, which digitize the signal and save its Fourier transform. Such a receiver system successfully detected a “fake axion" power of 10−23 W over a week. Above 30 GHz, quantum noise is non-negligible, and a different amplifier needs to be devised. Dielectrics other than sapphire are considered as well. High relative permittivity and low tan δ are desirable for maximum signal power. Perovskite oxides including LnAlO3 and LSAT appear promising [10]. Application of a cryogenic glue to assemble small single-crystal pieces that have a minimal tan δ into 1-meter-diameter disks is being tested. First a scaled down prototype booster will be built and commissioned. It is presently being considered to use this prototype inside the MORPURGO magnet [11] at CERN to search for ALPs. The magnet offers a 1.6 T dipole in a 160 cm warm bore. The prototype booster contains 20 disks of 30-cm diameter. The disks will be positioned by piezo motors inside the cryostat. Cryogenic, optic, and mechanical aspect of the prototype are being investigated currently. In addition to serving as a technical test bed for the full-scale MADMAX detector, the prototype will yield the first WISP search data for ma in 79.6−87.9 µeV. The first operation is scheduled during the SPS shutdown from 2022, and its 3-year projected sensitivity for the axion–photon coupling is shown in Fig. 3. Figure 4 shows a schematic of the final MADMAX experiment. The design figure of merit (FoM) for the final MADMAX dipole magnet is 100 T2 m2 in a warm bore of 1.25 m diameter. Design studies have shown that based on the “block design” such a magnet can be built using NbTi superconductor providing the required FoM with a homogeneity within 15%. The detector will be built inside the existing iron yoke of the H1 detector, which strengthens the field in the bore while reducing the stray field. HERA North Hall can meet the cryogen requirements and support the weight of the magnet. The full MADMAX is sensitive to the DFSZ model in the same mass range. The search above 100 µeV can be significantly accelerated when quantum-noise-level amplifiers that can work above 30 GHz are developed.

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Fig. 3 Projected axion sensitivity of MADMAX for axion to photon coupling vs. axion mass. The orange and yellow bands correspond to theoretical estimations for pre- and post-inflationary dark matter axions. The exclusion limits from ADMX [4] and HAYSTAC [12] are from resonant cavities. Also shown are limit from CAST [13] and the projected sensitivities of ALPS II [14] and IAXO [15] Fig. 4 Preliminary design of the full-scale Madmax detector

References 1. R.D. Peccei, H.R. Quinn, CP conservation in the presence of instantons. Phys. Rev. Lett. 38, 1440–1443 (1977) 2. J. Preskill, M.B. Wise, F. Wilczek, Cosmology of the invisible axion. Phys. Lett. B 120, 127–132 (1983) 3. G.G. Raffelt, Astrophysical axion bounds. In: M. Kuster, G. Raffelt, B. Beltrán (eds.) Axions. Lecture Notes in Physics, vol. 741 (Springer, Berlin, 2008)

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4. N. Du et al., Search for invisible axion dark matter with the axion dark matter experiment. Phys. Rev. Lett. 120, 151301 (2018) 5. D.A. Dicus et al., Astrophysical bounds on the masses of axions and Higgs particles. Phys. Rev. D 18, 1829–1834 (1978) 6. A. Caldwell et al., Dielectric haloscopes: a new way to detect axion dark matter. Phys. Rev. Lett. 118, 091801 (2017) 7. A.J. Millar et al., Dielectric haloscopes to search for axion dark matter: theoretical foundations. J. Cosmol. Astropart. Phys. 2017, 01 (2017) 8. MADMAX Collaboration, A new experimental approach to probe QCD axion dark matter in the mass range above 40 μeV. Eur. Phys. J. C 79, 3 (2019) 9. S. Knirck et al., A first Look on 3D effects in open axion haloscopes. J. Cosmol. Astropart. Phys. (2019) arXiv:1906.02677 10. C. Zuccaro et al., Materials for HTS-shielded dielectric resonators. IEEE Trans. Appl. Supercond. 7, 3715–3718 (1997) 11. M. Morpurgo, A large superconducting dipole cooled by forced circulation of two phase Helium. Cryogenics 19, 411 (1979) 12. B. Brubaker et al., First results from a microwave cavity axion search at 24 μeV. Phys. Rev. Lett. 118, 061302 (2017) 13. CAST Collaboration, New CAST limit on the axion-photon interaction. Nat. Phys. 13, 584 (2017) 14. R. Bähre et al., Any light particle search II – technical design report. J. Inst. 8(9), T09001 (2013) 15. IAXO Collaboration, Physics potential of the International Axion Observatory (IAXO) (2019). arXiv:1904.09155

Orpheus: Extending the ADMX QCD Dark-Matter Axion Search to Higher Masses Gianpalo Carosi, Raphael Cervantes, Seth Kimes, Parashar Mohapatra, Rich Ottens, and Gray Rybka

Abstract Axions in our local dark matter halo could be detected using an apparatus consisting of a resonant microwave cavity threaded by a strong magnetic field. The ADMX experiment has recently used this technique to search for the QCD axions in the few µeV/c2 mass range. However, the ADMX search technique becomes increasingly challenging with increasing axion mass. This is because higher masses require smaller-diameter cavities, and a smaller cavity volume reduces the signal strength. Thus, there is interest in developing more sophisticated resonators to overcome this problem. We present the progress of the ADMX Orpheus prototype experiment. This uses a dielectric-loaded Fabry–Perot resonator to search for axions with masses approaching 100 µeV/c2 [1]. Keywords Axion · Haloscope · Dielectric resonator · Fabry–Perot resonator · Open resonator · High-mode resonator · Numerical simulations

1 Motivation for Dielectric Haloscopes Axions from the dark matter halo can be detected using their coupling to photons. For instance, axions in a strong magnetic field can convert to photons. This conversion is enhanced in a resonator if the photon frequency matches that of the resonator. The power of the axion signal is then Pa ∝ Bo2 QVeff , where Bo is the external magnetic field, Q is the resonator’s quality factor, and Veff is the effective volume of the resonator. Veff can be thought of as the product of the resonator’s physical volume and the fraction of the axion’s electric field Ea that aligns with  the external magnetic field. It is calculated as Veff =

| dV Bo ·Ea |2  . Bo2 dV εr |Ea |2

This apparatus

G. Carosi · R. Cervantes () · S. Kimes · P. Mohapatra · R. Ottens · G. Rybka Physics and Astronomy Department, University of Washington, Seattle, WA, USA e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020 G. Carosi, G. Rybka (eds.), Microwave Cavities and Detectors for Axion Research, Springer Proceedings in Physics 245, https://doi.org/10.1007/978-3-030-43761-9_20

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consisting of a resonator inside of a strong magnetic field designed to search for dark matter axions is known as a haloscope [2]. ADMX has used haloscopes to search for QCD axions with masses of a few µeV [3]. The ADMX haloscope consists of right-circular cavity inside of a solenoid magnet. The cavity is operated in the TM010 mode, so the electric field is aligned with the axis of the cavity (like a parallel plate capacitor). Thus the electric field is aligned with the solenoid’s magnetic field, maximizing Veff . ADMX’s current haloscope has Veff ≈ 50 L, Q ∼ 50,000, Bo ≈ 7 T, giving an axion signal power of Pa ≈ 10−22 W. This haloscope design becomes increasingly difficult with increasing axion mass. Increasing mass corresponds to higher-frequency photons. Operating at the TM010 mode would require smaller-diameter cavities, and a smaller cavity volume reduces the signal strength. Furthermore, the decreased volume-to-surface ratio decreases Q, further decreasing the signal. One can think about keeping a large volume and operating at a mode higher than the TM010 . But then portions of Ea are anti aligned with Bo , and | dV Bo · Ea | ≈ 0. The effective volume approaches 0 even though the physical volume is large. Thus, there is no benefit to operating an empty cylindrical cavity at a higher-order mode. However, high-order modes can couple to the axion when dielectrics are placed inside of the resonator. Dielectrics suppress electric fields. If dielectrics can be placed where the electric field is anti-aligned with the magnetic field, then the overlap between the axion’s electric field and external magnetic field is greater than 0 (see Fig. 1). Thus the effective volume can become arbitrarily large and the axion signal power is greater than what it would have been for a cylindrical cavity operating at the TM010 mode. Overall, dielectric resonators can be operated at higher-order modes while maintaining coupling to the axion and can be made arbitrarily large, making them suited for higher-frequency searches.

2 The Orpheus Haloscope The ADMX Orpheus haloscope [1] will use dielectrics to search for axions with masses around 70 µeV. Orpheus is a dielectrically loaded Fabry–Perot open resonator placed inside of a dipole magnet (Fig. 1). Dielectrics are placed every half-wavelength to suppress the electric field where it is anti-aligned with the dipole field. Orpheus will be designed to search for axions between 15 and 18 GHz. The resonator is tuned by changing its length. There are several benefits to the open resonator design. Less metallic walls lead to less ohmic losses and a higher Q. Less metallic walls also mean less resonating modes. This leads to a sparse spectrum and less mode crossings, making it easier to maintain the axion-coupling mode. However, this experiment has many challenges. First, the optics must be designed to maintain good axion coupling from 15 to 18 GHz. This includes choosing the right radius of curvature for the Fabry–Perot mirrors and right dielectric thicknesses.

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Fig. 1 A schematic of how dielectrics allow high-order modes to couple to the axion. The red lines represent Ea , the dashed blue lines represent Bo , shaded rectangles represent dielectrics, and the black lines represent the resonator metallic walls. Ea is suppressed in the dielectrics, causing a positive overlap between the electric and magnetic field for an arbitrarily large volume. This concept of the dielectrically loaded resonator will be implemented in the Orpheus experiment

Fig. 2 The target gaγ γ sensitivity of the current cryogenic design, compared to the best limit set by CAST

This might even mean curving the dielectrics so that they act like lenses that collimate the electric field. Another challenge is the requirement of a dipole magnet instead of a solenoid. Dipoles are about 10 times more expensive than solenoids. Finally, the dielectric losses will decrease the resonator Q. A successful Orpheus experiment would be more sensitive to axions around 70 µeVthan previous experiments (Fig. 2). The sensitivity scales as gaγ γ ∝ 1 Bo Δt 1/4

Tsys SNR Veff Q ,

where Δt is the integration time, Tsys is the system noise temperature, and SNR is the signal-to-noise ratio. A previous Orpheus prototype

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had reached a coupling sensitivity of gaγ γ ≈ 10−6 [4]. Hardware improvements will increase the new experiment’s sensitivity by 30,000: the 0.85 mT wireplane magnet will be replaced by a 1.5 T racetrack dipole magnet, cooling by liquid helium will reduce Tsys from 1420 K to about 20 K, and higher-quality mirrors will improve Q from 5,000 to 10,000. The cryogenic design would be sensitive to gaγ γ = 3.3 × 10−11 , beating the 2016 CAST limit by over half an order of magnitude.

3 Orpheus Progress The ADMX collaboration has made progress on Orpheus in several ways. The preliminary cryogenic design has been made and the electrodynamics has been simulated. Resonator properties are currently being tested with table-top prototype. Figure 3 shows the preliminary cryogenic design. The dielectric material will be alumina because it has a high dielectric constant of r = 10 and low dielectric loss of tan δ = 0.0002 at 10 GHz. The scissor jacks keep the alumina plates evenly spaced between the Fabry–Perot mirrors. The resonator will be encased by a 1.5 T racetrack dipole magnet and will sit in the belly of a liquid helium dewar. We have developed a room-temperature table-top experiment for fast and cheap prototyping (Fig. 3). This allows us to test the mechanics, electronics, and measurement techniques. Delrin is used for the dielectrics because it is cheap and easily machineable. The scissors jacks are directly attached to the dielectrics and Fabry–Perot mirrors. Both mirrors are 6" in diameter. One mirror is flat, the other has a radius of curvature of 33 cm, allowing Ea to have a near-confocal configuration. Both mirrors have a coupling aperture in their centers. WR62

Fig. 3 Left: the cryogenic design for the Orpheus experiment. The dielectrics will be made of alumina and will be kept evenly spaced between the Fabry–Perot mirrors using the scissor jacks. Right: A room-temperature table-top prototype. The dielectrics are Delrin because it is cheaper and more easily machineable

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rectangular waveguides are attached to both mirrors. One of the waveguides has stubs to allow for impedance matching. Each waveguide connects to a coax-towaveguide adaptor (CMT RA62-SMA-F_1A-A). These adaptors are then connected to a coaxial cable and then to a network analyzer (Agilent FieldFox N9918A). This setup allows the measurement of the S12 transmission coefficient of the resonator. Since there are 9 dielectric plates, the Gaussian TEM00−18 mode is the mode that couples best to the axion. This mode consists of 19 half-wavelengths: 9 halfwavelengths reside within the dielectrics and 10 half-wavelengths reside between the dielectrics. This setup will demonstrate whether the axion-coupling mode can be tracked while tuning the resonator. If the electrodynamics is well understood, then Veff can be calculated. The strategy is to first study the empty resonator. Analytical predictions, simulations, and measurements will be compared. If they agree, then simulations and measurement techniques are validated. Next, the resonator is loaded with dielectrics. Since an analytical understanding is difficult, simulations are used to determine the TEM00−18 mode frequency. Transmission measurements to simulations are then compared to check for agreement. The TEM00−18 mode of the Fabry–Perot resonator was simulated using finite element analysis software (ANSYS HFSS). For a given resonator length, S12 was simulated as a function of frequency using a driven modal analysis. The peak of the S12 gives the resonant frequency, and plotting the electric field confirms that this resonance corresponds to the TEM00−18 mode. This simulation was performed for different lengths to obtain the TEM00−18 resonant frequency as a function of resonator length. Simulations were performed with both the empty and Delrinloaded resonator. Figure 4 shows the transmission through the resonator as a function of resonator length for both the empty and Delrin-loaded configurations. The strong transmission peaks correspond to resonant modes. These modes tune to lower frequencies as the resonator length increases. The empty configuration shows well-defined modes. Both the analytical predictions and the simulation results are overlaid on top of the measurement. Measurement, analytical prediction, and the simulation agree with each other to one part in 10,000. The other measured transmission peaks correspond to other harmonic modes. The Delrin-loaded transmission measurements are similar to the empty transmission measurement. However, the resonances are much less well-defined because Delrin is a very lossy plastic. The loss tangent is tan δ ∼ 0.005, so if you filled the resonator with Delrin, then the unloaded Q would be less than 200. The simulated TEM00−18 mode follows the same trend as the measured TEM00−18 mode from S21 measurements. However, both uncertainty in Delrin’s dielectric constant at 15–18 GHz and the mechanical instability may cause the difference between the simulation and measured peak. Nevertheless, it is clear that we can track the TEM00−18 mode in a dielectrically loaded resonator. With more precise machining and with well-characterized, lower-loss dielectrics, the agreement between simulation and measurement will improve.

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Fig. 4 Left: The transmission coefficient of an empty Fabry–Perot resonator. High-transmission corresponds to resonant modes. Both the analytical prediction and simulated resonant frequency for the TEM00−18 mode are overlaid on top of the measured transmission. Right: The transmission coefficient of a Delrin-loaded Fabry–Perot resonator. The resonances are much broader because of the dielectric losses from Delrin. The resonator length is shorted because the wavelength in Delrin is smaller. The simulated resonant frequency for the TEM00−18 mode is overlaid on top

4 Orpheus Outlook We are currently improving the optical and mechanical design. Table-top testing with these improvements will be done by March 2019. The DAQ and motor systems will be developed in parallel. Then the cryogenic experiment will be put together by November 2019. Data-taking and analysis will occur afterward. We will have the first results between 2 and 3 years. With more resources and R&D an Orpheus-like experiment can reach DFSZ sensitivity. Assume that a quantum-limited noise level can be achieved at 18 GHz. Then for Q = 105 , SNR = 3.5, and a scan rate of df dt = 1 GHz/yr, then our experiment needs to have Bo2 Veff = 200 LT2 . That means that if a DFSZ sensitive Orpheus had a 1 T dipole magnet, the effective volume would need to be Veff = 200 L (the physical volume would probably need to be about 1000 L). While these requirements are formidable, they are still achievable. For reference, the current ADMX experiment has a Bo2 Veff ≈ 2500 LT2 . With parallel advances in magnet technology and low-noise amplifiers, Orpheus appears to be a viable resonator design for axion searches approaching 100 µeV.

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References 1. R. Cervantes, Orpheus: Extending the ADMX QCD Dark-Matter Axion Search to Higher Masses. APS April Meeting (2019). http://meetings.aps.org/Meeting/APR19/Session/Z09.5 2. P. Sikivie, Detection rates for “invisible”-axion searches. Phys. Rev. D 32, 2988 (1985). https:// link.aps.org/doi/10.1103/PhysRevD.32.2988 3. N. Du et al., Search for invisible axion dark matter with the axion dark matter experiment. Phys. Rev. Lett. 120, 151301 (2018). https://link.aps.org/doi/10.1103/PhysRevLett.120.151301 4. G. Rybka et al., Search for dark matter axions with the Orpheus experiment. Phys. Rev. D 91, 011701 (2015). https://doi.org/10.1103/PhysRevD.91.011701