# Josephson Junctions as Single Microwave Photon Counters: Simulation and Characterization

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Detection of Axions

#### 1.2. Single Photon Counters as Detectors

## 2. A CBJJ as a Photon Detector

## 3. Device Simulation

#### 3.1. Isolated CBJJ

#### 3.2. CBJJ Coupled to an RC Circuit

#### 3.3. CBJJ Coupled to a Transmission Line

## 4. Measurements

#### 4.1. Fabrication Parameters from Simulations

#### 4.2. Setup

## 5. Results and Discussion

#### 5.1. Results

#### 5.2. Interpretations of Results

#### 5.3. Dark-Count Rate

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Peccei, R.D.; Quinn, H.R. CP Conservation in the Presence of Pseudoparticles. Phys. Rev. Lett.
**1977**, 38, 1440–1443. [Google Scholar] [CrossRef] [Green Version] - Peccei, R.D.; Quinn, H.R. Constraints imposed by CP conservation in the presence of pseudoparticles. Phys. Rev. D
**1977**, 16, 1791–1797. [Google Scholar] [CrossRef] - Weinberg, S. A New Light Boson? Phys. Rev. Lett.
**1978**, 40, 223–226. [Google Scholar] [CrossRef] - Wilczek, F. Problem of Strong P and T Invariance in the Presence of Instantons. Phys. Rev. Lett.
**1978**, 40, 279–282. [Google Scholar] [CrossRef] - Sikivie, P. Axions. In Particle Dark Matter: Observations, Models and Searches; Bertone, G., Ed.; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Preskill, J.; Wise, M.B.; Wilczek, F. Cosmology of the invisible axion. Phys. Lett. B
**1983**, 120, 127–132. [Google Scholar] [CrossRef] [Green Version] - Abbott, L.; Sikivie, P. A cosmological bound on the invisible axion. Phys. Lett. B
**1983**, 120, 133–136. [Google Scholar] [CrossRef] - Dine, M.; Fischler, W. The not-so-harmless axion. Phys. Lett. B
**1983**, 120, 137–141. [Google Scholar] [CrossRef] - Kim, J.E. Weak-Interaction Singlet and Strong CP Invariance. Phys. Rev. Lett.
**1979**, 43, 103–107. [Google Scholar] [CrossRef] - Shifman, M.A.; Vainshtein, A.I.; Zakharov, V.I. Can confinement ensure natural CP invariance of strong interactions? Nucl. Phys. B
**1980**, 166, 493–506. [Google Scholar] [CrossRef] - Dine, M.; Fischler, W.; Srednicki, M. A simple solution to the strong CP problem with a harmless axion. Phys. Lett. B
**1981**, 104, 199–202. [Google Scholar] [CrossRef] - Zhitnitsky, A.R. On Possible Suppression of the Axion Hadron Interactions. (In Russian). Sov. J. Nucl. Phys.
**1980**, 31, 260. [Google Scholar] - O’Hare, C. cajohare/AxionLimits: AxionLimits. 2020. Available online: https://zenodo.org/record/3932430#.YO-ZTPkzZPY (accessed on 15 July 2021).
- Sikivie, P. Experimental Tests of the “Invisible” Axion. Phys. Rev. Lett.
**1983**, 51, 1415–1417. [Google Scholar] [CrossRef] - Sikivie, P. Detection rates for “invisible”-axion searches. Phys. Rev. D
**1985**, 32, 2988–2991. [Google Scholar] [CrossRef] - Irastorza, I.G.; Redondo, J. New experimental approaches in the search for axion-like particles. Prog. Part. Nucl. Phys.
**2018**, 102, 89–159. [Google Scholar] [CrossRef] [Green Version] - Du, N.; Force, N.; Khatiwada, R.; Lentz, E.; Ottens, R.; Rosenberg, L.J.; Rybka, G.; Carosi, G.; Woollett, N.; Bowring, D.; et al. Search for Invisible Axion Dark Matter with the Axion Dark Matter Experiment. Phys. Rev. Lett.
**2018**, 120, 151301. [Google Scholar] [CrossRef] [Green Version] - Braine, T.; Cervantes, R.; Crisosto, N.; Du, N.; Kimes, S.; Rosenberg, L.J.; Rybka, G.; Yang, J.; Bowring, D.; Chou, A.S.; et al. Extended Search for the Invisible Axion with the Axion Dark Matter Experiment. Phys. Rev. Lett.
**2020**, 124, 101303. [Google Scholar] [CrossRef] [Green Version] - Backes, K.M.; Palken, D.A.; Al Kenany, S.; Brubaker, B.M.; Cahn, S.B.; Droster, A.; Hilton, G.C.; Ghosh, S.; Jackson, H.; Lamoreaux, S.K.; et al. A quantum enhanced search for dark matter axions. Nature
**2021**, 590, 238. [Google Scholar] [CrossRef] - Choi, J.; Ahn, S.; Ko, B.; Lee, S.; Semertzidis, Y. CAPP-8TB: Axion Dark Matter Search Experiment around 6.7 μeV. arXiv
**2020**, arXiv:2007.07468. [Google Scholar] - Jeong, J.; Youn, S.; Bae, S.; Kim, J.; Seong, T.; Kim, J.E.; Semertzidis, Y.K. Search for Invisible Axion Dark Matter with a Multiple-Cell Haloscope. Phys. Rev. Lett.
**2020**, 125, 221302. [Google Scholar] [CrossRef] [PubMed] - McAllister, B.T.; Flower, G.; Ivanov, E.N.; Goryachev, M.; Bourhill, J.; Tobar, M.E. The ORGAN experiment: An axion haloscope above 15 GHz. Phys. Dark Universe
**2017**, 18, 67–72. [Google Scholar] [CrossRef] [Green Version] - Crescini, N.; Alesini, D.; Braggio, C.; Carugno, G.; Di Gioacchino, D.; Gallo, C.S.; Gambardella, U.; Gatti, C.; Iannone, G.; Lamanna, G.; et al. Operation A Ferromagn. Axion Haloscope m
_{a}=58μeV. Eur. Phys. J. C**2018**, 78, 703. [Google Scholar] [CrossRef] [Green Version] - Crescini, N.; Alesini, D.; Braggio, C.; Carugno, G.; D’Agostino, D.; Di Gioacchino, D.; Falferi, P.; Gambardella, U.; Gatti, C.; Iannone, G.; et al. Axion Search with a Quantum-Limited Ferromagnetic Haloscope. Phys. Rev. Lett.
**2020**, 124, 171801. [Google Scholar] [CrossRef] - Alesini, D.; Braggio, C.; Carugno, G.; Crescini, N.; D’Agostino, D.; Di Gioacchino, D.; Di Vora, R.; Falferi, P.; Gallo, S.; Gambardella, U.; et al. Galactic axions search with a superconducting resonant cavity. Phys. Rev. D
**2019**, 99, 101101. [Google Scholar] [CrossRef] [Green Version] - Alesini, D.; Braggio, C.; Carugno, G.; Crescini, N.; D’Agostino, D.; Di Gioacchino, D.; Di Vora, R.; Falferi, P.; Gambardella, U.; Gatti, C.; et al. Search for invisible axion dark matter of mass m
_{a}= 43 μeV with the QUAX–aγ experiment. Phys. Rev. D**2021**, 103, 102004. [Google Scholar] [CrossRef] - Melcón, A.Á.; Cuendis, S.A.; Cogollos, C.; Díaz-Morcillo, A.; Döbrich, B.; Gallego, J.D.; Gimeno, B.; Irastorza, I.G.; Lozano-Guerrero, A.J.; Malbrunot, C.; et al. Axion searches with microwave filters: The RADES project. J. Cosmol. Astropart. Phys.
**2018**, 2018, 040. [Google Scholar] [CrossRef] [Green Version] - Arguedas, C.S.; Álvarez, M.A.; Cogollos, C.; Díaz-Morcillo, A.; Döbrich, B.; Gallego, J.D.; Gimeno, B.; Irastorza, I.G.; Lozano-Guerrero, A.J.; Malbrunot, C.; et al. The 3 cavity prototypes of RADES, an axion detector using microwave filters at CAST. arXiv
**2019**, arXiv:1903.04323. [Google Scholar] - Caldwell, A.; Dvali, G.; Majorovits, B.; Millar, A.; Raffelt, G.; Redondo, J.; Reimann, O.; Simon, F.; Steffen, F. Dielectric Haloscopes: A New Way to Detect Axion Dark Matter. Phys. Rev. Lett.
**2017**, 118, 091801. [Google Scholar] [CrossRef] [Green Version] - BRASS: Broadband Radiometric Axion SearcheS. Available online: http://www.iexp.uni-hamburg.de/groups/astroparticle/brass/brassweb.htm (accessed on 15 July 2021).
- Alesini, D.; Babusci, D.; Di Gioacchino, D.; Gatti, C.; Lamanna, G.; Ligi, C. The KLASH Proposal. arXiv
**2017**, arXiv:1707.06010. [Google Scholar] - Gatti, C.; Alesini, D.; Babusci, D.; Braggio, C.; Carugno, G.; Crescini, N.; Di Gioacchino, D.; Falferi, P.; Lamanna, G.; Ligi, C.; et al. The Klash Proposal: Status and Perspectives. In Proceedings of the 14th Patras Workshop on Axions, WIMPs and WISPs (AXION-WIMP 2018) (PATRAS 2018), Hamburg, Germany, 18–22 June 2018. [Google Scholar]
- Alesini, D.; Babusci, D.; Björkeroth, F.; Bossi, F.; Ciambrone, P.; Monache, G.D.; Di Gioacchino, D.; Falferi, P.; Gallo, A.; Gatti, C.; et al. KLASH Conceptual Design Report. arXiv
**2019**, arXiv:1911.02427. [Google Scholar] - Gleyzes, S.; Kuhr, S.; Guerlin, C.; Bernu, J.; Deleglise, S.; Hoff, U.B.; Brune, M.; Raimond, J.-M.; Haroche, S. Quantum jumps of light recording the birth and death of a photon in a cavity. Nature
**2007**, 446, 297–300. [Google Scholar] [CrossRef] [Green Version] - Schuster, D.I.; Houck, A.A.; Schreiner, J.A.; Wallraff, A.; Gambetta, J.M.; Blais, A.; Frunzion, A.; Majer, J.; Johnson, B.; Devoret, H.; et al. Resolving photon number states in a superconducting circuit. Nature
**2007**, 445, 515–518. [Google Scholar] [CrossRef] [Green Version] - Johnson, B.R.; Reed, M.D.; Houck, A.A.; Schuster, D.I.; Bishop, L.S.; Ginossar, E.; Gambetta, J.M.; DiCarlo, L.; Frunzio, L.; Girvin, S.M.; et al. Quantum Non-Demolition Detection Single Microwave Photons A Circuit. Nat. Phys.
**2010**, 6, 663–667. [Google Scholar] [CrossRef] [Green Version] - Besse, J.C.; Gasparinetti, S.; Collodo, M.C.; Walter, T.; Kurpiers, P.; Pechal, M.; Eichler, C.; Wallraff, A. Single-Shot Quantum Nondemolition Detection of Individual Itinerant Microwave Photons. Phys. Rev. X
**2018**, 8, 021003. [Google Scholar] [CrossRef] [Green Version] - Kono, S.; Koshino, K.; Tabuchi, Y.; Noguchi, A.; Nakamura, Y. Quantum non-demolition detection of an itinerant microwave photon. Nat. Phys.
**2018**, 14, 546–549. [Google Scholar] [CrossRef] - Lescanne, R.; Deléglise, S.; Albertinale, E.; Réglade, U.; Capelle, T.; Ivanov, E.; Jacqmin, T.; Leghtas, Z.; Flurin, E. Irreversible Qubit-Photon Coupling for the Detection of Itinerant Microwave Photons. Phys. Rev. X
**2020**, 10, 021038. [Google Scholar] [CrossRef] - Inomata, K.; Lin, Z.; Koshino, K.; Oliver, W.D.; Tsai, J.-S.; Yamamoto, T.; Nakamura, Y. Single microwave-photon detector using an artificial Λ-type three-level system. Nat. Commun.
**2016**, 7, 1–7. [Google Scholar] [CrossRef] [PubMed] - Chen, Y.F.; Hover, D.; Sendelbach, S.; Maurer, L.; Merkel, S.T.; Pritchett, E.J.; Wilhelm, F.K.; McDermott, R. Microwave Photon Counter Based on Josephson Junctions. Phys. Rev. Lett.
**2011**, 107, 217401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kuzmin, L.S.; Sobolev, A.S.; Gatti, C.; Di Gioacchino, D.; Crescini, N.; Gordeeva, A.; Il’ichev, E. Single Photon Counter Based on a Josephson Junction at 14 GHz for Searching Galactic Axions. IEEE Trans. Appl. Supercond.
**2018**, 28, 1–5. [Google Scholar] [CrossRef] - Semenov, A.D.; Gol’tsman, G.N.; Sobolewski, R. Hot-electron effect in superconductors and its applications for radiation sensors. Supercond. Sci. Technol.
**2002**, 15. [Google Scholar] [CrossRef] - Paolucci, F.; Buccheri, V.; Germanese, G.; Ligato, N.; Paoletti, R.; Signorelli, G.; Bitossi, M.; Spagnolo, P.; Falferi, P.; Rajteri, M.; et al. Development of highly sensitive nanoscale transition edge sensors for gigahertz astronomy and dark matter search. J. Appl. Phys.
**2020**, 128, 194502. [Google Scholar] [CrossRef] - Komiyama, S. Single-Photon Detectors in the Terahertz Range. IEEE J. Sel. Top. Quantum Electron.
**2011**, 17, 54–66. [Google Scholar] [CrossRef] - Dixit, A.V.; Chakram, S.; He, K.; Agrawal, A.; Naik, R.K.; Schuster, D.I.; Chou, A. Searching for Dark Matter with a Superconducting Qubit. Phys. Rev. Lett.
**2021**, 126, 141302. [Google Scholar] [CrossRef] [PubMed] - Kutlu, Ç.; van Loo, A.F.; Uchaikin, S.V.; Matlashov, A.N.; Lee, D.; Oh, S.; Kim, J.; Chung, W.; Nakamura, Y.; Semertzidis, Y.K. Characterization of a flux-driven Josephson parametric amplifier with near quantum-limited added noise for axion search experiments. Supercond. Sci. Technol.
**2021**. [Google Scholar] [CrossRef] - Revin, L.S.; Pankratov, A.L.; Gordeeva, A.V.; Yablokov, A.A.; Rakut, I.V.; Zbrozhek, V.O.; Kuzmin, L.S. Microwave photon detection by an Al Josephson junction. Beilstein J. Nanotechnol.
**2020**, 11, 960–965. [Google Scholar] [CrossRef] [PubMed] - Alesini, D.; Babusci, D.; Barone, C.; Buonomo, B.; Beretta, M.M.; Bianchini, L.; Castellano, G.; Chiarello, F.; Di Gioacchino, D.; Falferi, P.; et al. Development of a Josephson junction based single photon microwave detector for axion detection experiments. J. Phys. Conf. Ser.
**2020**, 1559, 012020. [Google Scholar] [CrossRef] - Lamoreaux, S.K.; van Bibber, K.A.; Lehnert, K.W.; Carosi, G. Analysis of single-photon and linear amplifier detectors for microwave cavity dark matter axion searches. Phys. Rev. D
**2013**, 88, 035020. [Google Scholar] [CrossRef] [Green Version] - Alesini, D.; Babusci, D.; Barone, C.; Buonomo, B.; Beretta, M.M.; Bianchini, L.; Castellano, G.; Chiarello, F.; Di Gioacchino, D.; Falferi, P.; et al. Status of the SIMP Project: Toward the Single Microwave Photon Detection. J. Low Temp. Phys.
**2020**, 199, 348–354. [Google Scholar] [CrossRef] - Barone, A.; Paternò, G. Physics and Applications of the Josephson Effect; Wiley: New York, NY, USA, 1982. [Google Scholar]
- Kogan, S. Electronic Noise and Fluctuations in Solids; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
- Martinis, J.M.; Devoret, M.H.; Clarke, J. Energy-Level Quantization in the Zero-Voltage State of a Current-Biased Josephson Junction. Phys. Rev. Lett.
**1985**, 55, 1543–1546. [Google Scholar] [CrossRef] [Green Version] - Devoret, M.H.; Martinis, J.M.; Clarke, J. Measurement of Macroscopic Quantum Tunneling out of the Zero-Voltage State of a Current-Biased Josephson Junction. Phys. Rev. Lett.
**1985**, 55, 1908. [Google Scholar] [CrossRef] [PubMed] - Piedjou Komnang, A.; Guarcello, C.; Barone, C.; Gatti, C.; Pagano, S.; Pierro, V.; Rettaroli, A.; Filatrella, G. Analysis of Josephson junctions switching time distributions for the detection of single microwave photons. Chaos Solitons Fractals
**2020**, 142, 110496. [Google Scholar] [CrossRef] - Ben-Jacob, E.; Bergman, D. Thermal noise effects on the microwave-induced steps of a current-driven josephson junction. Phys. Rev. A
**1984**, 29, 2021–2028. [Google Scholar] [CrossRef] - Yurke, B.; Denker, J. Quantum Network Theory. Phys. Rev. A
**1984**, 29, 1419. [Google Scholar] [CrossRef] - Schondorf, M.; Govia, L.; Vavilov, M.; McDermott, R.; Wilhelm, F. Optimizing microwave photodetection: Input—Output theory. Quantum Sci. Technol.
**2018**, 3, 024009. [Google Scholar] [CrossRef] [Green Version] - Kivioja, J.M.; Nieminen, T.E.; Claudon, J.; Buisson, O.; Hekking, F.W.J.; Pekola, J.P. Observation of Transition from Escape Dynamics to Underdamped Phase Diffusion in a Josephson Junction. Phys. Rev. Lett.
**2005**, 94, 247002. [Google Scholar] [CrossRef] [Green Version] - Kivioja, J.M.; Nieminen, T.E.; Claudon, J.; Buisson, O.; Hekking, F.W.J.; Pekola, J.P. Weak coupling Josephson junction as a current probe: Effect of dissipation on escape dynamics. New J. Phys.
**2005**, 7, 179. [Google Scholar] [CrossRef] - Longobardi, L.; Massarotti, D.; Stornaiuolo, D.; Galletti, L.; Rotoli, G.; Lombardi, F.; Tafuri, F. Direct Transition from Quantum Escape to a Phase Diffusion Regime in YBaCuO Biepitaxial Josephson Junctions. Phys. Rev. Lett.
**2012**, 109, 050601. [Google Scholar] [CrossRef] [Green Version] - Bae, M.H.; Sahu, M.; Lee, H.J.; Bezryadin, A. Multiple-retrapping processes in the phase-diffusion regime of high-T
_{c}intrinsic Josephson junctions. Phys. Rev. B**2009**, 79, 104509. [Google Scholar] [CrossRef] [Green Version] - Yu, H.F.; Zhu, X.B.; Peng, Z.H.; Tian, Y.; Cui, D.J.; Chen, G.H.; Zheng, D.N.; Jing, X.N.; Lu, L.; Zhao, S.P.; et al. Quantum Phase Diffusion in a Small Underdamped Josephson Junction. Phys. Rev. Lett.
**2011**, 107, 067004. [Google Scholar] [CrossRef] [Green Version] - Massarotti, D.; Longobardi, L.; Galletti, L.; Stornaiuolo, D.; Montemurro, D.; Pepe, G.; Rotoli, G.; Barone, A.; Tafuri, F. Escape dynamics in moderately damped Josephson junctions. Low Temp. Phys. Fiz. Nizk. Temp.
**2012**, 38, 336. [Google Scholar] - Castellano, G.; Torrioli, G.; Chiarello, F.; Cosmelli, C.; Carelli, P. Return current in hysteretic Josephson junctions: Experimental distribution in the thermal activation regime. J. Appl. Phys.
**1999**, 86, 6405. [Google Scholar] [CrossRef] - Fenton, J.C.; Warburton, P.A. Monte Carlo simulations of thermal fluctuations in moderately damped Josephson junctions: Multiple escape and retrapping, switching- and return-current distributions, and hysteresis. Phys. Rev. B
**2008**, 78, 054526. [Google Scholar] [CrossRef] [Green Version] - Yoon, Y.; Gasparinetti, S.; Pekola, M.M.J.P. Capacitively Enhanced Thermal Escape in Underdamped Josephson Junctions. J. Low Temp. Phys.
**2011**, 163, 164. [Google Scholar] [CrossRef] - Martinis, J.M.; Devoret, M.H.; Clarke, J. Experimental tests for the quantum behavior of a macroscopic degree of freedom: The phase difference across a Josephson junction. Phys. Rev. B
**1987**, 35, 4682–4698. [Google Scholar] [CrossRef]

**Figure 1.**Axion parameter space showing the exclusion limits of experiments. The yellow band is representative of the QCD axions, together with the brown lines showing theoretical predictions of the KSVZ [9,10] and DFSZ [11,12] models. The plot is freely available from the online repository of [13].

**Figure 2.**(

**a**) An electrical model of a JJ with intrinsic and external current sources. (

**b**) An electrical model of a JJ attached to a transmission line. (

**c**) An electrical model of a JJ with a parasitic RC load.

**Figure 3.**Equivalent potential of a JJ. The phase value is represented by the green particle which can overcame the energy barrier after the absorption of a suitable stimulus.

**Figure 4.**(

**a**) Schematic image of the shadow mask evaporation technique. The copolymer/PMMA bilayer was exposed by EBL and developed, in order to obtain a self-standing bridge of the desired geometry, as shown by the scanning electron microscope (SEM) micrograph in (

**b**). A two-angle evaporation with an oxidation step in between defines the junction geometry (light and dark gray in (

**a**). (

**c**) SEM micrograph of the junction after the lift-off process used to remove the bilayer mask. (

**d**) An atomic force microscope (AFM) characterization of a typical junction.

**Figure 6.**Measured voltage-current characteristic of a Josephson junction (2 $\mathsf{\mu}\mathrm{m}\times $ 2 $\mathsf{\mu}\mathrm{m}$).

**Figure 7.**Filtered escape current distributions for $2\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}\times 2\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ junction (

**a**) and for $2\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}\times 4\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ junction (

**b**). The ranges of temperatures are slightly different and are shown in the legends. Rightmost curves are taken at the lowest temperature and leftmost curves are taken at highest temperature. The effect of the filter was a reduction of $\sigma $ up to 20% with respect to the unfiltered data (not shown); $\langle {I}_{c}\rangle $ was left unchanged.

**Figure 8.**$\langle {I}_{c}\rangle $ and $\sigma $ for $2\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}\times 2\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ junction (

**a**) and for $2\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}\times 4\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ junction (

**b**). In both plots, the red squares are relative to the left axis (widths) and the empty blue squares are relative to the right axis (mean currents), as indicated by the arrows.

**Figure 9.**Mean switching currents (

**a**) and widths (

**b**) of the simulated data for different simulation parameters. In the legend, TA refers to thermal activation processes, QT to quantum tunneling, and RC to the dissipation effect due to the external circuit.

**Figure 10.**Comparison of the mean switching currents and widths of the simulated data with ${R}_{p}=1$ k$\mathsf{\Omega}$, $C=2$ pF, ${R}_{J}=100$ k$\mathsf{\Omega}$, ${I}_{0}=143$ nA, and ${C}_{J}=0.2$ pF with respect to experimental results of the $2\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}\times 2\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ junction, shown in empty squares.

**Figure 11.**Experimental dark-count rate for the $2\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}\times 2\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ junction at 50 mK, compared with quantum tunneling escape rate (red dashed line) and quantum tunneling plus the effect of 1.1 nA of Gaussian smearing (green solid line).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rettaroli, A.; Alesini, D.; Babusci, D.; Barone, C.; Buonomo, B.; Beretta, M.M.; Castellano, G.; Chiarello, F.; Di Gioacchino, D.; Felici, G.;
et al. Josephson Junctions as Single Microwave Photon Counters: Simulation and Characterization. *Instruments* **2021**, *5*, 25.
https://doi.org/10.3390/instruments5030025

**AMA Style**

Rettaroli A, Alesini D, Babusci D, Barone C, Buonomo B, Beretta MM, Castellano G, Chiarello F, Di Gioacchino D, Felici G,
et al. Josephson Junctions as Single Microwave Photon Counters: Simulation and Characterization. *Instruments*. 2021; 5(3):25.
https://doi.org/10.3390/instruments5030025

**Chicago/Turabian Style**

Rettaroli, Alessio, David Alesini, Danilo Babusci, Carlo Barone, Bruno Buonomo, Matteo Mario Beretta, Gabriella Castellano, Fabio Chiarello, Daniele Di Gioacchino, Giulietto Felici,
and et al. 2021. "Josephson Junctions as Single Microwave Photon Counters: Simulation and Characterization" *Instruments* 5, no. 3: 25.
https://doi.org/10.3390/instruments5030025