Abstract
Granular aluminum (grAl) is a promising high kinetic inductance material for detectors, amplifiers, and qubits. Here we model the grAl structure, consisting of pure aluminum grains separated by thin aluminum oxide barriers, as a network of Josephson junctions, and we calculate the dispersion relation and nonlinearity (selfKerr and crossKerr coefficients). To experimentally study the electrodynamics of grAl thin films, we measure microwave resonators with openboundary conditions and test the theoretical predictions in two limits. For low frequencies, we use standard microwave reflection measurements in a lowloss environment. The measured lowfrequency modes are in agreement with our dispersion relation model, and we observe selfKerr coefficients within an order of magnitude from our calculation starting from the grAl microstructure. Using a highfrequency setup, we measure the plasma frequency of the film around 70 GHz, in agreement with the analytical prediction.
Introduction
The introduction of crystalline defects or dopants can give rise to socalled dirty superconductors^{1}, characterized by reduced coherence length and quasiparticle mean free path. In particular, granular superconductors^{2} such as grAl^{3,4}, consisting of remarkably uniform grains connected by Josephson contacts^{5} have attracted interest since the 60s, thanks to their rich phase diagram^{6,7} and practical advantages, like increased critical temperature^{4,8}, critical field^{9,10}, and kinetic inductance^{11}. Here we report the measurement and modeling of circuit quantum electrodynamics^{12} properties of grAl microwave resonators in a wide frequency range, up to the spectral superconducting gap. Interestingly, we observe selfKerr coefficients ranging from 10^{−2} Hz to 10^{5} Hz, within an order of magnitude from analytic calculations based on grAl microstructure. This amenable nonlinearity, combined with the relatively highquality factors in the 10^{5} range, open new avenues for applications in quantum information processing^{13} and kinetic inductance detectors^{14}.
Increasing the level of disorder in a superconducting material usually decreases the superfluid density and can induce a superconducting to insulating transition. Superconductors with low superconducting carrier density can exhibit rich physical properties, arising from a variety of phenomena such as quantum phase transitions^{15} and localization^{2}. Granular aluminum is a typical example preferred by experimentalists, thanks to its relatively straightforward fabrication by aluminum evaporation in an oxygen atmosphere^{3}, which can tune the film resistivity ρ over five orders of magnitude. The phase diagram of grAl thin films, with an initial increase of the critical temperature versus resistivity^{16}, followed by a decrease and transition to an insulating state, has been extensively studied over the last 50 years, with notable recent developments in both theory^{17} and experiment^{18,19}. These studies, mostly performed by direct current measurements, or broadband THz spectroscopy, offer a solid basis to start addressing the electrodynamics of grAl in the quantum regime, defined as the limit of singlephoton excitations.
In the context of emerging quantum information platforms based on aluminum^{13}, grAl provides precious ingredients such as lowloss and highimpedance environments, tolerance to high magnetic fields, or a robust source of nonlinearity. The prospect of implementing ultrahigh impedance environments, at the level of the impedance quantum R_{Q} = h/(2e)^{2} ≃ 6.5 kΩ, for the design of qubits^{20,21,22} and parametric amplifiers^{23} or for the engineering of quantum states of light^{24} is very appealing. However, the electromagnetic properties of granular superconductors in the quantum regime are currently virtually unexplored.
Here we present a theoretical model and the corresponding experimental investigation of the dispersion relation and nonlinear Kerr coefficients for grAl resonators in the microwave regime. We will use the formalism of circuit quantum electrodynamics^{12} (cQED) and show that in a firstorder approximation, the Hamiltonian of grAl, taking into account the interaction between the resonant modes, can be written in the familiar quantum optics form^{25}
The frequencies ω_{n} form the dispersion relation, the selfKerr coefficients K_{nn} quantify the frequency shift of mode n for each added photon, and the crossKerr coefficients K_{nm}, quantify the frequency shift of mode n for an added photon in mode m. The operators a_{n} and \(a_n^\dagger\) are bosonic lowering and raising operators, and \(a_n^\dagger a_n = N\) gives the photon number.
Results
Electrodynamic model
The microstructure of grAl consists of pure aluminum grains, with the average diameter a, separated by thin aluminum oxide barriers, as schematically illustrated in Fig. 1a. For films fabricated at room temperature with ρ > 10 μΩ cm, the grain size is homogeneous and independent of resistivity, a = 3 ± 1 nm^{4}. We use grAl films with a resistivity between 40 μΩ cm and 4000 μΩ cm, below the superconducting to insulating transition at ρ ≃ 10^{4} μΩ cm^{7}, and for which the kinetic inductance dominates over the geometric inductance^{11}. We model this medium as a network of effective Josephson junctions (JJ), which provides a handle to calculate its dispersion relation^{26} and the Kerr coefficients^{27,28}.
For elongated structures, such as stripline resonators (Fig. 1a), the calculation of the lowfrequency dispersion relation and nonlinearity can be performed in the limit of onedimensional (1D) current distributions along the stripline (see Supplementary Discussion), resulting in an effective JJ chain model (see Fig. 1b). The current is homogeneously distributed through the sample crosssection due to the fact that the thickness d ≃ 20 nm is much smaller than the magnetic field penetration depth, λ_{L} > 0.4 μm, depending on the film resistivity ρ, and the width b is smaller than the screening distance, \(\lambda _ \bot = \lambda _{\mathrm{L}}^2{\mathrm{/}}d > 8\) μm^{3}. The equivalent electrical schematics is shown in Fig. 1c, where each superconducting section of length a with self capacitance C_{0} is connected by effective JJs with critical current I_{c} and capacitance C_{J}.
The classical equation of motion for the phase difference φ_{n} across the nth JJ is
The resonator drive is introduced as an external current applied to the mth cell, δ_{m,n}I_{ext} cos(ωt), where δ_{m,n} is the Kronecker delta. In order to derive the eigenfrequencies, we use firstorder Taylor expansion for the Josephson currents (see Supplementary Discussion). Thus we obtain the dispersion relation:
sketched in Fig. 1d, which is approximately linear for the lowest modes, and it saturates at the effective plasma frequency \(\omega _{\mathrm{p}} = \omega _{n = \ell /a} = \sqrt {2eI_{\mathrm{c}}{\mathrm{/}}\hbar C_{\mathrm{J}}}\), as measured on mesoscopic JJ arrays^{29}. As we will show in the following, the fundamental frequency f_{1} = ω_{1}/2π, designed in the low GHz range, can provide a convenient link through the crossKerr effect to the higher modes of the dispersion relation, spanning up to ~100 GHz.
To derive the Kerr coefficients of the fundamental mode in Eq. (1), we solve the equation of motion, expanding the nonlinear terms up to third order. This method is similar to the one recently used to derive the nonlinearity of mesoscopic arrays of JJ^{27,30}. By relating the phase response amplitude to the circulating photon number \(\bar N\) (see Supplementary Discussion), we obtain the selfKerr and crossKerr coefficients for the fundamental mode:
Here, e is the electron charge, a is the grain size, j_{c} = I_{c}/bd is the critical current density, ω_{n} are the eigenfrequencies given by Eq. (3), and \(V_{{\mathrm{grAl}}} = bd\ell\) is the volume of grAl threaded by the current, see Fig. 1a. \({\cal C}\) is a numerical constant of order one, which for a sinusoidal current distribution is \({\cal C} = 3{\mathrm{/}}16\) for n = 1 and \({\cal C} = 1{\mathrm{/}}4\) for n > 1. Using the expression for the singlephoton current as a function of frequency and total inductance, \(I_{\bar N = 1}^2 = 2fh{\mathrm{/}}L\) and L ∝ 1/j_{c}, Eq. (4), can be rewritten in a qualitatively similar form to the K_{11} coefficient estimated from Mattis–Bardeen theory for dirty superconductors^{11,23}, \(K_{11} \propto \left( {I_{\bar N = 1}{\mathrm{/}}I_ \ast } \right)^2\). The depairing current \(I_ \ast\) is of the same order of magnitude as the critical current of the strip I_{c}. In contrast, Eq. (4) offers a quantitative model for the nonlinearity of grAl, starting from the film properties. Remarkably, this analytic result agrees within an order of magnitude with the K_{11} coefficients measured on 14 grAl samples, spanning from K_{11} = 2 × 10^{−2} Hz to K_{11} = 3 × 10^{4} Hz.
Furthermore, the crossKerr coefficients, K_{1n}, follow the functional dependence of the dispersion relation, ω_{n}, given by Eq. (3) and reach a maximum at the effective plasma frequency ω_{p} (see Fig. 1d). Due to the high crossKerr interaction and highmode density around ω_{p}, we expect a strong response of the fundamental mode for drive frequencies in the vicinity of ω_{p}/2π. As discussed in detail in the next section, for highly resistive samples (grAl#3) with ρ = 3000 μΩ cm, for which ω_{p} is low enough to be in the measurable range, we observe the expected plasma frequency response in the vicinity of 70 GHz.
Measurements
To measure the dispersion relation, microwave losses, and the nonlinearity of grAl structures, we use three types of resonators of various shapes and sizes (see Methods), optimized for two complementary measurement setups (see Fig. 2), covering a broad frequency range up to 200 GHz.
In Fig. 3a, b, we plot a typical amplitude and phase response measured for stripline resonators in the singlephoton regime, \(\bar N \approx 1\), which is relevant for quantum information applications. We extract an internal quality factor Q_{i} = 10^{5}, comparable to values obtained for JJ array superinductances^{29}. We obtain similar results for Q_{i} measurements on Hilbertshaped (Fig. 2c) and aluminumshunted stripline resonators, for tens of resonators, with grAl resistivities up to 4000 μΩ cm, corresponding to ~kΩ characteristic impedance. As discussed in ref.^{31}, we estimate that Q_{i} is dominated by nonequilibrium quasiparticle dissipation, which could be suppressed by phonon and quasiparticle traps.
Using a twotone spectroscopy, similar to a superconducting qubit readout procedure^{12}, we measure higher modes of the dispersion relation for stripline resonators. Due to the symmetry of the electric field, the next mode, above the fundamental, coupled to the waveguide is the third. For sample grAl#1, we measure f_{1} = 6.287 GHz and f_{3} = 18.255 GHz. Notice that the dispersion relation already shows a measurable deviation from linear behavior, 3 × f_{1} − f_{3} = 606 ± 1 MHz, which, using Eq. (3), allows us to estimate an effective plasma frequency ω_{p} = 68 ± 0.1 GHz (see Supplementary Discussion), as shown in Fig. 4a).
Indeed, using a Martin–Puplett Interferometer (MPI) as a broadband illumination source up to 200 GHz and a Hilbertshaped set of resonators (grAl#3) with similar sheet resistivity as grAl#1 mounted in an optical access cryostat, we observe a shift of the fundamental mode for illumination frequencies in the range 60–80 GHz (red curve in Fig. 4b). This shift is comparable to the pairbreaking response at twice the gap, and significantly above the noise floor. We interpret this response to be the cumulated crossKerr shift due to the population of the highmode density region of the effective plasma frequency (see Fig. 4a).
As expected, for resonators with 50 times higher critical current densities j_{c}, the effective plasma frequency can no longer be measured (green line in Fig. 4b), as it is above the spectroscopic gap frequency. To confirm the correct calibration of the MPI setup, we measured the response of a standard 25nm aluminum film using an additional 180GHz lowpass filter. The MPI measurements (blue line in Fig. 4b) indicate the expected Al spectral gap value of 100 GHz, above which the illumination can break the Cooper pairs, inducing a shift of the fundamental mode and a Q_{i} decrease^{14}. Finally, notice that the spectroscopic gap of samples grAl#2 and grAl#3 increases with resistivity, as expected^{7}.
To measure the selfKerr coefficient, K_{11}, we monitor the fundamental frequency as a function of photon population \(\bar N\) using the lowfrequency setup (Fig. 2b). Typical measurement results are shown in Fig. 3c, d in linear and logarithmic scale, respectively. In Fig. 5, we report the measured K_{11} for 14 types of grAl resonators, grouped in three different geometries: KID (in blue), striplines (in green), and Alshunted striplines (in red); details on the resonators' geometry are given in the Supplementary Discussion. For a direct comparison with Eq. (4) represented by the black line, we plot the measured selfKerr coefficients versus\(f_1^2{\mathrm{/}}j_{\mathrm{c}}V_{{\mathrm{grAl}}}\) using a measured j_{c} = 1.1 mA/μm^{2} for ρ = 1600 μΩ cm (see Supplementary Discussion) and scaling it according to j_{c} ∝ 1/ρ for all resistivities^{7}. We would like to emphasize that there are no fitting parameters. We estimate the main source of error, responsible for the scatter of the points and for the deviation compared to Eq. (4) to be the photon number calibration. We can only perform this calibration by estimating the total attenuation of the input lines at various frequencies. We estimate this method to be accurate only within a factor of 10. Remarkably, the selfKerr coefficient of grAl can be tuned over six orders of magnitude by varying the room temperature resistivity ρ ∝ 1/j_{c} and the resonator volume V_{grAl}, without compromising the internal quality factor.
Discussion
According to the strength of the nonlinearity, we can divide the possible grAl applications into three categories. First, for superinductors^{29,32,33,34} and microwave kinetic inductance detectors^{14,35,36}, the nonlinearity should be as low as possible. The devices plotted in green in Fig. 5 could be used as superinductors with a selfKerr coefficient of only tens of Hz, which would be at least three orders of magnitude lower than the stateoftheart^{27}. Second, for parametric devices, such as amplifiers^{23} or frequency converters^{37,38}, the selfKerr coefficient should be in the kHz range, as shown by the devices plotted in red in Fig. 5. Fabricating them using grAl instead of mesoscopic JJ arrays offers the advantages of compactness and singlestep fabrication. Finally, in the case of transmon qubits^{39}, the selfKerr nonlinearity should be even higher, in the tens of MHz range, which could be achieved by reducing the grAl volume and increasing the resistivity of the film.
In conclusion, granular aluminum is a superconductor with high characteristic impedance, low microwave losses, and amenable nonlinearity, which recommend it as a material of choice for quantum information processing. Using a highfrequency setup, including a Martin–Pupplet interferometer, we observe the effective plasma frequency of highly inductive grAl devices in the range of 70 GHz, which is in agreement with estimates based on a 1D JJ array model and the measured lowfrequency spectrum. The measured selfKerr coefficients agree within an order of magnitude with our analytic model, and they are in the range of applications for parametrically pumped devices, such as quantum amplifiers. Highly inductive grAl films could implement lowloss superinductors for quantum circuits or ultrasensitive kinetic inductance detectors.
Methods
Experimental apparatus
The dispersion relation for grAl resonators spans up to ~100 GHz. To cover this wide frequency range, we employ two complementary measurement setups, and we use the first mode as a link between them, via the crossKerr effect.
Lowfrequency setup
The lowfrequency part of the spectrum (n = 1–3), up to 20 GHz, is measured using microwave transmission and reflection measurements in a standard cQED setup^{12} (Fig. 2b). The grAl stripline resonators (Fig. 2a) are mounted in a 3D waveguide (WG) sample holder, housed inside a hermetic copper shield coated with infraredabsorbing material. In this lownoise setup, all microwave lines are filtered above 8 GHz using commercial lowpass filters, circulators, and infrared absorbers identical to the setup in ref. ^{31} in order to reduce stray radiation. Even though the Hilbertshaped grAl resonators and their aluminum sample holder (Fig. 2c) are designed to operate as kinetic inductance detectors (KIDs), which are required for the measurement of their highfrequency spectrum by means of direct optical spectroscopy (Fig. 2d), they were also measured by standard microwave transmission in the lownoise, shielded setup of Fig. 2b. The high level of filtering and superior shielding, offered by the measurement setup optimized for low frequencies, is required for the protection of the fundamental mode against stray excitations, which is essential for the measurement of its coherence and nonlinear properties (selfKerr and crossKerr).
Highfrequency setup
For the measurement of the effective plasma frequency, we use the widefrequency band setup of Fig. 2d, consisting of an optical access cryostat coupled to a Martin–Puplett Interferometer (see Supplementary Discussion). The fundamental mode is continuously measured via microwave transmission measurements, while its frequency is shifted by crossKerr interactions with optically populated higher modes of the dispersion relation.
Data availability
All relevant data are available from the authors.
References
 1.
Anderson, P. Theory of dirty superconductors. J. Phys. Chem. Solids 11, 26–30 (1959).
 2.
Beloborodov, I. S., Lopatin, A. V., Vinokur, V. M. & Efetov, K. B. Granular electronic systems. Rev. Mod. Phys. 79, 469–518 (2007).
 3.
Cohen, R. W. & Abeles, B. Superconductivity in granular aluminum films. Phys. Rev. 168, 444–450 (1968).
 4.
Deutscher, G., Fenichel, H., Gershenson, M., Grünbaum, E. & Ovadyahu, Z. Transition to zero dimensionality in granular aluminum superconducting films. J. Low. Temp. Phys. 10, 231–243 (1973).
 5.
Parmenter, R. H. Isospin formulation of the theory of a granular superconductor. Phys. Rev. 154, 353–368 (1967).
 6.
Dynes, R. C. & Garno, J. P. Metalinsulator transition in granular aluminum. Phys. Rev. Lett. 46, 137–140 (1981).
 7.
Pracht, U. S. et al. Enhanced cooper pairing versus suppressed phase coherence shaping the superconducting dome in coupled aluminum nanograins. Phys. Rev. B 93, 100503 (2016).
 8.
Abeles, B., Cohen, R. W. & Cullen, G. W. Enhancement of superconductivity in metal films. Phys. Rev. Lett. 17, 632–634 (1966).
 9.
Deutscher, G. & Dodds, S. A. Criticalfield anisotropy and fluctuation conductivity in granular aluminum films. Phys. Rev. B 16, 3936–3942 (1977).
 10.
Chui, T., Lindenfeld, P., McLean, W. L. & Mui, K. Coupling and isolation: critical field and transition temperature of superconducting granular aluminum. Phys. Rev. B 24, 6728–6731 (1981).
 11.
Rotzinger, H. et al. Aluminiumoxide wires for superconducting high kinetic inductance circuits. Supercond. Sci. Technol. 30, 025002 (2017).
 12.
Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162 (2004).
 13.
Gu, X., Kockum, A. F., Miranowicz, A., xi Liu, Y. & Nori, F. Microwave photonics with superconducting quantum circuits. Phys. Rep. 718–719, 1–102 (2017).
 14.
Day, P. K., LeDuc, H. G., Mazin, B. A., Vayonakis, A. & Zmuidzinas, J. A broadband superconducting detector suitable for use in large arrays. Nature 425, 817 (2003).
 15.
Emery, V. J. & Kivelson, S. A. Importance of phase fluctuations in superconductors with small superfluid density. Nature 374, 434 (1995).
 16.
Deutscher, G., Gershenson, M., Grünbaum, E. & Imry, Y. Granular superconducting films. J. Vac. Sci. Technol. 10, 697–701 (1973).
 17.
Pracht, U. S. et al. Optical signatures of the superconducting goldstone mode in granular aluminum: experiments and theory. Phys. Rev. B 96, 094514 (2017).
 18.
Bachar, N. et al. Mott transition in granular aluminum. Phys. Rev. B 91, 041123 (2015).
 19.
Bachar, N. et al. Signatures of unconventional superconductivity in granular aluminum. J. Low. Temp. Phys. 179, 83–89 (2015).
 20.
Astafiev, O. V. et al. Coherent quantum phase slip. Nature 484, 355 (2012).
 21.
Manucharyan, V., Koch, J., Glazman, L. & Devoret, M. Fluxonium: single cooperpair circuit free of charge offsets. Science 326, 113–116 (2009).
 22.
Gladchenko, S. et al. Superconducting nanocircuits for topologically protected qubits. Nat. Phys. 5, 48 (2008).
 23.
Ho Eom, B., Day, P. K., LeDuc, H. G. & Zmuidzinas, J. A wideband, lownoise superconducting amplifier with high dynamic range. Nat. Phys. 8, 623 (2012).
 24.
Puri, S., Boutin, S. & Blais, A. Engineering the quantum states of light in a kerrnonlinear resonator by twophoton driving. npj Quantum Inf. 3, 18 (2017).
 25.
Walls, D. F. & Milburn, G. J. Quantum Optics. (SpringerVerlag, Berlin Heidelberg, 2008).
 26.
Hutter, C., Tholén, E. A., Stannigel, K., Lidmar, J. & Haviland, D. B. Josephson junction transmission lines as tunable artificial crystals. Phys. Rev. B 83, 014511 (2011).
 27.
Weissl, T. et al. Kerr coefficients of plasma resonances in josephson junction chains. Phys. Rev. B 92, 104508 (2015).
 28.
Bourassa, J., Beaudoin, F., Gambetta, J. M. & Blais, A. Josephsonjunctionembedded transmissionline resonators: From kerr medium to inline transmon. Phys. Rev. A. 86, 013814 (2012).
 29.
Masluk, N. A., Pop, I. M., Kamal, A., Minev, Z. K. & Devoret, M. H. Microwave characterization of josephson junction arrays: implementing a low loss superinductance. Phys. Rev. Lett. 109, 137002 (2012).
 30.
Tancredi, G., Ithier, G. & Meeson, P. J. Bifurcation, mode coupling and noise in a nonlinear multimode superconducting microwave resonator. Appl. Phys. Lett. 103, 063504 (2013).
 31.
Grünhaupt, L. et al. Loss mechanisms and quasiparticle dynamics in superconducting microwave resonators made of thinfilm granular aluminum. Phys. Rev. Lett. 121, 117001 (2018).
 32.
Bell, M. T., Sadovskyy, I. A., Ioffe, L. B., Kitaev, A. Y. & Gershenson, M. E. Quantum superinductor with tunable nonlinearity. Phys. Rev. Lett. 109, 137003 (2012).
 33.
Hazard, T. M. et al. High kinetic inductance nbn nanowire superinductors. arXiv:1802.01723 (2018).
 34.
Niepce, D., Burnett, J. & Bylander, J. Nanowire superinductance fluxonium qubit. arXiv:1805.00938 (2018).
 35.
Baselmans, J. et al. A broadband superconducting detector suitable for use in large arrays. J. Low. Temp. Phys. 151, 524–529 (2008).
 36.
Cardani, L. et al. New application of superconductors: high sensitivity cryogenic light detectors. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel., Spectrometers, Detect. Assoc. Equip. 845, 338–341 (2017). Proceedings of the Vienna Conference on Instrumentation 2016.
 37.
Sliwa, K. M. et al. Reconfigurable josephson circulator/directional amplifier. Phys. Rev. X 5, 041020 (2015).
 38.
Lecocq, F. et al. Nonreciprocal microwave signal processing with a fieldprogrammable josephson amplifier. Phys. Rev. Appl. 7, 024028 (2017).
 39.
Koch, J. et al. Chargeinsensitive qubit design derived from the cooper pair box. Phys. Rev. A. 76, 042319 (2007).
 40.
Martin, D. & Puplett, E. Polarised interferometric spectrometry for the millimetre and submillimetre spectrum. Infrared Phys. 10, 105–109 (1970).
 41.
Catalano, A. et al. Bilayer kinetic inductance detectors for space observations between 80–120 ghz. A&A 580, A15 (2015).
Acknowledgements
We are grateful to O. Buisson, D. Basko, G. Weiss, and A. Shnirman for fruitful discussions and to L. Radtke and A. Lukashenko for technical support. Facilities used waere supported by the KIT Nanostructure Service Laboratory (NSL). Funding was provided by the Alexander von Humboldt foundation in the framework of a Sofja Kovalevskaja award endowed by the German Federal Ministry of Education and Research. This work was partially supported by the Ministry of Education and Science of the Russian Federation in the framework of the Program to Increase Competitiveness of the NUST MISIS, contracts no. K22016063 and K22017081 (experiments), and K2217085 (theory).
Author information
Affiliations
Contributions
N.M. and M.V.F. performed the theoretical study. L.G., F.V., and P.W. designed, fabricated the samples, and performed the measurements in the lowfrequency setup. F.L.B., O.D., M.C., and A.M. performed the MPI measurements and analysis. P.W., F.F., and W.W. designed and performed the switching current experiments. N.M. and I.M.P. lead the paper writing, while all other authors contributed to the text. IMP supervised and coordinated the project.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Maleeva, N., Grünhaupt, L., Klein, T. et al. Circuit quantum electrodynamics of granular aluminum resonators. Nat Commun 9, 3889 (2018). https://doi.org/10.1038/s41467018063869
Received:
Accepted:
Published:
Further reading

DEMETRA: Suppression of the Relaxation Induced by Radioactivity in Superconducting Qubits
Journal of Low Temperature Physics (2020)

A new setup for probing condensed matter in the Far IR to THz ranges at subKelvin temperatures on the AILES beamline at SOLEIL
Infrared Physics & Technology (2020)

Hybrid superconductorsemiconductor systems for quantum technology
Applied Physics Letters (2020)

PhotonicCrystal Josephson TravelingWave Parametric Amplifier
Physical Review X (2020)

Kubo spins in nanoscale aluminum grains: A muon spin relaxation study
Physical Review B (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.