Anomalous Frequency Shift in Low-Loss Superconducting Granular Aluminum Resonators

AI summary

Overview

Research area: Superconducting circuit quantum electrodynamics (cQED) — specifically the microwave loss and frequency response of high-kinetic-inductance granular aluminum (grAl) resonators.

Technical level: Advanced. The paper combines precision microwave spectroscopy at millikelvin temperatures with a dedicated theoretical treatment (Mattis–Bardeen theory, two-level-system ensembles, and a non-uniform Josephson-junction network model).

Scope (one sentence): The paper reports an anomalous positive shift of the resonance frequency with temperature and readout power in seven low-loss grAl lumped-element resonators, attempts to explain it with quasiparticles in a medium with a spatially non-uniform superconducting gap, and shows that the accompanying internal loss is far smaller than theory predicts.

What This Paper Is About

Granular aluminum is attractive for quantum circuits because it provides large kinetic inductance, but its residual microwave loss mechanisms are poorly understood compared with pure aluminum. The authors measure how the resonance frequency and internal quality factor of grAl resonators respond to temperature, readout power, and high-energy particle impacts, expecting the standard behavior in which frequency drops as temperature rises. Instead, at the lowest temperatures the frequency increases with both temperature and power, and they set out to identify which physical mechanism could produce this sign reversal without simultaneously producing the large loss that standard models would require.

Key Contributions

1. Observation of a dual anomaly in the same devices. The authors report both an anomalous temperature dependence and an anomalous power dependence of the resonance frequency in the same low-loss grAl resonators, with the effect resolved in detail because of the devices' narrow linewidths.
2. A granularity-based quasiparticle model. They develop a theory in which grAl is treated as an array of superconducting grains with a distribution of gap values, connected by Josephson junctions, and show that quasiparticles tunneling between grains with unequal gaps produce a positive frequency shift at low temperature.
3. A quasiparticle heating model for the power dependence. They attribute the initial positive frequency shift with increasing photon number to microwave-drive-induced heating of quasiparticles, mapping the power dependence onto the temperature dependence, with a Kerr nonlinearity taking over and reversing the trend at higher powers.
4. Quantified failure of standard models. They show that Mattis–Bardeen theory fits only the high-temperature data, that the two-level-system (TLS) model predicts both the wrong sign and wrong amplitude, and that the quasiparticle models overestimate the measured internal loss by roughly two orders of magnitude at low temperature.

Main Findings

• Frequency rises before it falls. In seven devices with resonance frequencies from 5.31 to 7.53 GHz, measured in the single-photon regime (mean photon number ≈ 1) at base temperatures of 23–27 mK, the resonance frequency increases with temperature to a maximum between 100 mK and 200 mK, then decreases monotonically.
• Loss stays flat while frequency moves. The internal loss rate, expressed as 1/Q_i, remains approximately constant over the same temperature range, becoming temperature dependent only above 250 mK when thermal quasiparticles dominate.
• A loss–frequency mismatch. Below 200 mK, the internal loss contributes only about f₀/Q_i ~ 3 kHz to the resonator linewidth, while the positive frequency shift reaches 50 kHz in some devices — a disparity the authors emphasize as a major unresolved puzzle.
• Turn-around temperature tracks resistivity. The temperature T_max at which the frequency shift peaks increases as the normal-state resistivity decreases, over the range 500 μΩ cm to 4000 μΩ cm, paralleling the roughly 10% decrease of grAl film critical temperature over the same resistivity range.
• High-temperature behavior is conventional. Above roughly 350 mK, both frequency and loss follow Mattis–Bardeen theory with critical temperatures in the range 1.7–2.4 K; the fit shown uses Δ/h = 88 GHz above about 360 mK.
• A gap distribution explains the low-temperature sign. The Josephson-junction array model with a box-distributed gap — average Δ/h = 87 GHz, width δΔ₀/h = 1.4 f₀ = 10.5 GHz, at quasiparticle density x_qp = 2 × 10⁻⁵ — fits the data across the full temperature range, requiring the gap spread to be comparable to the resonator frequency.
• Power dependence is non-monotonic too. In the 7.53 GHz device at 27 mK, frequency first rises with photon number and then falls; the fall is consistent with a Kerr nonlinearity of −0.54 Hz/photon, within the theoretically predicted 0.5–5 Hz/photon range.
• The power anomaly vanishes at the frequency maximum. The positive power-induced shift shrinks as base temperature rises and disappears near T_max ≈ 200 mK for the 7.53 GHz device, where the low-power frequency is at its peak. The heat-balance model reproduces the frequency dependence only below −90 dBm.
• The TLS model is ruled out quantitatively. To produce a shift of about 15 kHz at roughly 150 mK, the TLS prefactor Γ^TLS_i,0 = 0.5 × 10⁻⁴ gives the wrong sign, and any TLS shift of the observed ~50 kHz magnitude would imply a comparably large linewidth, contradicting the measured 3 kHz.
• High-energy events produce multi-timescale, sign-changing responses. The 7.53 GHz device shows only a positive shift at the lowest temperatures, with a fast initial response followed by non-exponential relaxation on the order of a few milliseconds. The 5.31 GHz device instead shows a negative shift that vanishes within a few milliseconds, followed by a positive shift relaxing over tens of milliseconds. In both devices the positive shift disappears at elevated base temperature.

Methodology in Plain English

The team fabricated lumped-element microwave resonators from granular aluminum films of constant thickness 91 ± 1 nm on sapphire, using a millimeter-scale two-pad capacitor shunted by a central inductive strip between 30 μm and 500 μm long, read out in a hanger configuration through a coupling capacitance. For two devices, the capacitor electrodes were made from pure aluminum. They measured the complex transmission coefficient as a function of drive frequency and power at millikelvin temperatures, extracting resonance frequency and internal and external quality factors — the latter on the order of 10⁶ for all devices. Photon numbers were converted from incident power using a standard formula involving internal and external quality factors; at higher powers, where the resonance line shape distorts, they fitted to a Duffing resonator response. They then compared the data against three theoretical frameworks: Mattis–Bardeen theory for a uniform superconductor, the phenomenological TLS ensemble model, and a custom model treating the grAl strip as a network of Josephson junctions between grains whose superconducting gaps are drawn from a box distribution. Finally, they built a heat-balance model in which the microwave drive heats the quasiparticle population, and they monitored the resonator response over time to characterize relaxation after high-energy events.

Why This Matters

Impact on research. The work sharpens the benchmark for what "low loss" in grAl actually implies: the authors note that internal quality factors as high as four million have recently been demonstrated, and it is precisely in this high-Q regime that the standard loss models break down most visibly. The paper reframes the problem — the puzzle is not just why the frequency shifts, but why the dissipation does not follow it — and supplies a concrete candidate mechanism (gap non-uniformity across grains) that other groups can test. It also cautions against interpreting frequency shifts as loss proxies in high-kinetic-inductance materials.

Real-world applications (as motivated by the paper's subject matter):

• Compact inductors for superconducting quantum information circuits, enabled by high sheet inductance.
• Devices requiring resilience to magnetic fields, a property the authors attribute to high-kinetic-inductance materials.
• Quantum circuits needing low nonlinearity alongside high inductance.
• Radiation and high-energy-particle sensing, since the devices show characteristic multi-timescale responses to ionizing events.

Industry relevance. grAl fabrication is compatible with standard Josephson junction processes, so improvements in understanding its loss mechanisms translate directly into integration with existing superconducting circuit foundries rather than requiring a new process line.

Future Directions

• Close the loss discrepancy. The quasiparticle model reproduces the frequency shift but overestimates internal loss by about two orders of magnitude at T ≲ 100 mK (and by a factor of 2–3 at high temperature); finding the mechanism that suppresses dissipation while preserving the dispersive response is the central open problem.
• Extend the heating model. The heat-balance description accounts for the power dependence only below −90 dBm, leaving the intermediate-power regime unexplained.
• Pin down microscopic gap statistics. The model assumes a box distribution of gap values; independent structural or spectroscopic measurement of grain-to-grain gap variation would test whether a width comparable to hf₀ is realistic.
• Explain the device-to-device spread in radiation response. The differing high-energy-event signatures between the 7.53 GHz and 5.31 GHz devices — which differ in grAl normal-state resistivity and sapphire substrate type — remain speculative and warrant systematic study.

Target Audience

This paper is aimed at researchers and graduate students in superconducting quantum circuits and materials for quantum devices — particularly groups working with high-kinetic-inductance superconductors, granular aluminum, resonator loss characterization, or quasiparticle dynamics. It is also relevant to theorists interested in disordered superconductors and Josephson-junction networks, and to device engineers who need to know where standard TLS-based loss models stop being predictive. Readers without a background in Mattis–Bardeen theory, kinetic inductance, and quality-factor extraction will find the theory sections demanding.

Authors’ 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.