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arXiv:2608.29282

Anomalous Frequency Shift in Low-Loss Superconducting Granular Aluminum Resonators

Patrick Winkel, Daniil S. Antonenko, Vishakha Gupta, Neel Thakur, Pavel Kurilovich, Luigi Frunzio, Leonid I. Glazman, Robert J. Schoelkopf

quant-phcond-mat.mtrl-scicond-mat.supr-conSubmitted Aug 29, 2026• Announced Sep 1, 2026• 0 citations

Abstract

Superconducting high-kinetic inductance materials like granular aluminum (grAl) are a versatile part of the circuit quantum electrodynamics (cQED) toolbox, provided their losses at microwave frequencies are low enough. To further advance the use of grAl in quantum devices, it is indispensable to identify the dominant loss mechanisms. The standard approach pairs electromagnetic simulations of resonator geometry with measured temperature and power dependence of resonance frequency and loss rate. Most materials follow the phenomenological theory of two-level systems (TLSs) at low temperatures, resulting in an initial decrease of the resonance frequency with increasing temperature. In our work, we observe an opposite behavior at the lowest temperatures: The resonance frequency initially increases with both temperature and readout power, contradicting the standard TLS model predictions. We are able to explain a part of the data by an alternative mechanism associated with the effect of superconducting quasiparticles in a granular system with spatially-nonuniform gap. Yet, the observed change in the resonance frequency with temperature is not matched by a proportional change in the loss rate. We also observe anomalous positive frequency responses following high-energy events, characterized by several time scales. While anomalous behavior has been reported previously in grAl, the disagreement with standard models is particularly visible in our devices thanks to their exceptionally low loss rates.

Comments: 10 pages, 8 figures (main text) and 11 pages, 7 figures (supplementary material)

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Superconducting high-kinetic inductance materials like granular aluminum (grAl) are a versatile part of the circuit quantum electrodynamics (cQED) toolbox, provided their losses at microwave frequencies are low enough. To further advance the use of grAl in quantum devices, it is indispensable to identify the dominant loss mechanisms. The standard approach pairs electromagnetic simulations of resonator geometry with measured temperature and power dependence of resonance frequency and loss rate. Most materials follow the phenomenological theory of two-level systems (TLSs) at low temperatures, resulting in an initial decrease of the resonance frequency with increasing temperature. In our work, we observe an opposite behavior at the lowest temperatures: The resonance frequency initially increases with both temperature and readout power, contradicting the standard TLS model predictions. We are able to explain a part of the data by an alternative mechanism associated with the effect of superconducting quasiparticles in a granular system with spatially-nonuniform gap. Yet, the observed change in the resonance frequency with temperature is not matched by a proportional change in the loss rate. We also observe anomalous positive frequency responses following high-energy events, characterized by several time scales. While anomalous behavior has been reported previously in grAl, the disagreement with standard models is particularly visible in our devices thanks to their exceptionally low loss rates.

Superconducting high kinetic inductance elements constitute a valuable resource for quantum circuit design and millimeter-wave detection. Granular aluminum (grAl) in the superconducting regime is a particularly interesting material since it has already shown a kinetic inductance in the range of nH/□ and its deposition is compatible with conventional Al/AlOx/Al Josephson junction fabrication. We characterize microwave resonators fabricated from grAl with a room temperature resistivity of 4×10^{3} μΩ cm, which is a factor of 3 below the superconductor to insulator transition, showing a kinetic inductance fraction close to unity. The measured internal quality factors are on the order of Q_{i}=10^{5} in the single photon regime, and we demonstrate that nonequilibrium quasiparticles (QPs) constitute the dominant loss mechanism. We extract QP relaxation times in the range of 1 s and we observe QP bursts every ∼20 s. The current level of coherence of grAl resonators makes them attractive for integration in quantum devices, while it also evidences the need to reduce the density of nonequilibrium QPs.

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 (self-Kerr and cross-Kerr coefficients). To experimentally study the electrodynamics of grAl thin films, we measure microwave resonators with open-boundary conditions and test the theoretical predictions in two limits. For low frequencies, we use standard microwave reflection measurements in a low-loss environment. The measured low-frequency modes are in agreement with our dispersion relation model, and we observe self-Kerr coefficients within an order of magnitude from our calculation starting from the grAl microstructure. Using a high-frequency setup, we measure the plasma frequency of the film around 70 GHz, in agreement with the analytical prediction.

Superconducting resonators are widely used in many applications such as qubit readout for quantum computing, and kinetic inductance detectors. These resonators are susceptible to numerous loss and noise mechanisms, especially the dissipation due to two-level systems (TLS) which become the dominant source of loss in the few-photon and low temperature regime. In this study, capacitively-coupled aluminum half-wavelength coplanar waveguide resonators are investigated. Surprisingly, the loss of the resonators is observed to decrease with a lowering temperature at low excitation powers and temperatures below the TLS saturation. This behavior is attributed to the reduction of the TLS resonant response bandwidth with decreasing temperature and power to below the detuning between the TLS and the resonant photon frequency in a discrete ensemble of TLS. When response bandwidths of TLS are smaller than their detunings from the resonance, the resonant response and thus the loss is reduced. At higher excitation powers, the loss follows a logarithmic power dependence, consistent with predictions from the generalized tunneling model (GTM). A model combining the discrete TLS ensemble with the GTM is proposed and matches the temperature and power dependence of the measured internal loss of the resonator with reasonable parameters.

Superconducting resonators are widely used in many applications such as qubit readout for quantum computing applications, and kinetic inductance detectors. These resonators are susceptible to numerous loss and noise mechanisms under microwave excitation, especially the dissipation due to non-equilibrium quasi-particles and two-level systems (TLS), which can result in a decrease of the superconducting intrinsic quality factor (

Q_i

) in high quality superconducting resonators. Particularly in the few-photon and low temperature (T) regime, TLS losses can become a dominant loss mechanism. In this study, novel aluminum half-wavelength resonators are investigated, focusing on the loss properties at extra-low power and low temperature. An unusual increase of

Q_i(T)

with deceasing temperature is observed. This behavior is attributed to the increase of TLS coherence time (

T_2

) at ultra-low temperatures and powers. This

T_2

increase is consistent with other work on resonant frequency noise in resonators and measurements of individual TLS, and likely arises from interacting TLS in the aluminum half-wavelength resonators.

Superconducting circuits based on high kinetic inductance materials are promising candidates for future quantum technologies, owing to their magnetic field resilience, low microwave losses, and lithographic scalability. In this thesis, we explore the properties of granular aluminum (grAl) resonators in two distinct geometries, using the nonlinear kinetic inductance of grAl for applications in superconducting quantum circuits. In the first part, we present compact meandered ring resonators fabricated using a single-step electron-beam lithography and lift-off process. By optimizing film resistivity and geometry, we achieve superinductors with characteristic impedances exceeding

100~\mathrm{k}\Omega

, corresponding to total inductances of several microhenries, while maintaining internal quality factors on the order of

10^{5}

in the single-photon regime. In the second part, we investigate DC-tunable resonators based on a

3\lambda/4

fractal geometry that allows direct current injection for in-situ tuning of the resonance frequency. We achieve frequency tunability up to

4.5\%

and analyze the nonlinear kinetic inductance using models based on Ginzburg-Landau, Mattis-Bardeen-BCS, and an effective Josephson-junction-array model. This work establishes granular aluminum as a material for realizing high-impedance and frequency-tunable superconducting circuits, offering new opportunities for advanced applications in protected qubits, hybrid quantum systems, broadband phase shifters, and three-wave mixing parametric amplifiers.