Circuit quantum electrodynamics of granular aluminum resonators
Nataliya Maleeva, Lukas Grünhaupt, T. Klein, F. Lévy-Bertrand, O. Dupre, Martino Calvo, Francesco Valenti, Patrick Winkel, F. Friedrich, Wolfgang Wernsdorfer, Alexey Vladimirovich Ustinov, Hannes Rotzinger, A. Monfardini, M. V. Fistul, Ioan M. Pop
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.
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 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 100kΩ, corresponding to total inductances of several microhenries, while maintaining internal quality factors on the order of 105 in the single-photon regime. In the second part, we investigate DC-tunable resonators based on a 3λ/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.
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.
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.
High kinetic inductance materials constitute a valuable resource for superconducting quantum circuits and hybrid architectures. Superconducting granular aluminum (grAl) reaches kinetic sheet inductances in the nH/□ range, with proven applicability in superconducting quantum bits and microwave detectors. Here, we show that the single photon internal quality factor Qi of grAl microwave resonators exceeds 105 in magnetic fields up to 1 T, aligned in-plane to the grAl films. Small perpendicular magnetic fields, in the range of 0.5 mT, enhance Qi by approximately 15%, possibly due to the introduction of quasiparticle traps in the form of fluxons. Further increasing the perpendicular field deteriorates the resonators' quality factor. These results open the door for the use of high kinetic inductance grAl structures in circuit quantum electrodynamics and hybrid architectures with magnetic field requirements.
Lumped-element inductors are an integral component in the circuit QED toolbox. However, it is challenging to build inductors that are simultaneously compact, linear, and low-loss with standard approaches that either rely on the geometric inductance of superconducting thin films or on the kinetic inductance of Josephson-junction arrays. In this work, we overcome this challenge by utilizing the high kinetic inductance offered by superconducting granular aluminum (grAl). We demonstrate lumped-element inductors with a few nH of inductance that are up to 100 times shorter than inductors built from pure aluminum (Al). To characterize the properties of these linear inductors, we first report on the performance of lumped-element resonators built entirely out of grAl with sheet inductances varying from 30 – 320 pH/sq and self-Kerr nonlinearities of 0.2 – 20 Hz / photon . Further, we demonstrate integration of these grAl inductors into hybrid resonators with Al or tantalum ( Ta ) capacitor electrodes without increasing total internal losses. Interestingly, the measured internal quality factors systematically decrease with increasing room-temperature resistivity of the grAl film for all devices, indicating a trade-off between compactness and internal loss. For our lowest resistivity grAl films, we measure quality factors reaching 3.5 × 10 6 for the all-grAl devices and 4.5 × 10 6 for the hybrid grAl/ Ta devices, similar to state-of-the-art quantum circuits. Our loss analysis suggests that the surface-loss factor of grAl is similar to that of pure Al for our lowest resistivity films, while the increasing losses with resistivity could be explained by increasing inductor loss in the grAl film.
Superconducting resonators with high-kinetic inductance play a central role in hybrid quantum circuits, enabling strong coupling with quantum systems with small electric dipole moment and improved parametric amplification. However, optimizing these resonators simultaneously for high internal quality factors (Qi) and resilience to strong magnetic fields remains challenging. In this study, we systematically compare superconducting resonators fabricated from niobium nitride (NbN) and granular aluminum (grAl) thin films, each having similar kinetic inductance values (Lk∼100 pH/sq). At zero magnetic field, resonators made from grAl exhibit higher Qi compared to their NbN counterparts. However, under applied magnetic fields, NbN resonators demonstrate significantly better resilience. Moreover, NbN resonators exhibit an unexpected increase in Qi at intermediate in-plane magnetic fields (B∥∼1 T), which we attribute to an enhanced frequency detuning that reduce coupling to two-level system defects. In contrast, grAl resonators show a distinct critical field above which Qi rapidly decreases, strongly depending on resonator cross-section respect to the applied field direction. Characterization of the nonlinear properties at zero magnetic field reveals that the self-Kerr coefficient in grAl resonators is more than an order of magnitude higher than in NbN resonators, making grAl particularly attractive for applications requiring pronounced nonlinear interactions. Our findings illustrate a clear trade-off between the two materials: NbN offers superior magnetic-field resilience beneficial for hybrid circuit quantum electrodynamics applications, while grAl is more advantageous in low-field regimes demanding high impedance and strong nonlinearity.