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arXiv:2205.06291 · APL Quantum

Power and temperature dependent model for high Q superconductor resonators

Ashish Alexander, Christopher Weddle, Christopher J. K. Richardson

Published Jul 23, 2025• 4 citations
arXivHTMLPDF

Abstract

Measuring the internal quality factor of coplanar waveguide superconducting resonators is an established method of determining small losses in superconducting devices. Often, resonator losses are attributed to two-level system (TLS) defects using a power dependent model for the quality factor. However, excess non-equilibrium quasiparticles can also limit the quality factor of superconducting resonators. Here, a two-temperature, power and temperature dependent model is used to evaluate resonator losses for titanium nitride and aluminum coplanar waveguide resonators with loss tangents of 10−5–10−6. This model enables a set of parameters to describe both the internal quality factor and fractional frequency shift of the resonator across a broad range of average cavity photon numbers, from 1 to 106, and temperatures between those limited by the experimental apparatus near 50 mK and thermal quasiparticle dominated. The presented model also offers potential insight into observing the temperature and power dependence of TLS–TLS interactions in these resonators.

DOI
10.1063/5.0273506

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Measuring the internal quality factor of coplanar waveguide superconducting resonators is an established method of determining small losses in superconducting devices. Often, resonator losses are attributed to two-level system (TLS) defects using a power dependent model for the quality factor. However, excess non-equilibrium quasiparticles can also limit the quality factor of superconducting resonators. Here, a two-temperature, power and temperature dependent model is used to evaluate resonator losses for titanium nitride and aluminum coplanar waveguide resonators with loss tangents of 10−5–10−6. This model enables a set of parameters to describe both the internal quality factor and fractional frequency shift of the resonator across a broad range of average cavity photon numbers, from 1 to 106, and temperatures between those limited by the experimental apparatus near 50 mK and thermal quasiparticle dominated. The presented model also offers potential insight into observing the temperature and power dependence of TLS–TLS interactions in these resonators.
High quality factor, i.e. low loss, microwave resonators are important for quantum information storage and addressing. In this work we study the resonance frequency and loss in superconducting coplanar waveguide resonators as a function of power and temperature. We find that there is increased loss at low power and low temperature. The increased loss is attributed to the existence of two-level systems (TLS) at the surfaces, interfaces, and in the bulk of insulators deposited on the structures. We show that both the temperature dependence of the resonant frequency and the power dependence of the loss can be used to find the TLS contribution to the loss. The TLS intrinsic loss tangent derived from the frequency shift data at high power is shown to agree well with the direct loss measurement at low power. The former allows for a relatively fast measurement of the TLS loss. As an example, we measure the properties of amorphous AlOXdeposited on the resonators and find a TLS loss tangent of 1 × 10-3.
The loss in superconducting microwave resonators at low-photon numbers and low temperatures is not well understood but has implications for achievable coherence times in superconducting qubits. We have fabricated single-layer resonators with a high quality factor by patterning a superconducting aluminum film on a sapphire substrate. Four resonator geometries were studied with resonant frequencies ranging from 5 to 7 GHz: a quasi-lumped element resonator, a coplanar strip waveguide resonator, and two hybrid designs that contain both a coplanar strip and a quasi-lumped element. Transmitted power measurements were taken at 30 mK as a function of frequency and probe power. We find that the resonator loss, expressed as the inverse of the internal quality factor, decreases slowly over four decades of photon number in a manner not merely explained by loss from a conventional uniform spatial distribution of two-level systems in an oxide layer on the superconducting surfaces of the resonator.
We present the measurements of the temperature dependence of the internal quality factor ( Q i ) of a microwave resonator, well below the superconducting transition temperature. The device is a quarter-wavelength niobium ( T c = 9.2 K) coplanar waveguide resonator. The measured | S 21 | parameter shows typically the skewed Lorentzian distributions, from which the fitted quality factor monotonically decreases with the temperature increasing from 30 mK to 900 mK. It is observed that for the lower temperature range (i.e., at T < 700 mK) the temperature dependence of the fitted Q i deviates significantly from the predictions of the usual Mattis—Bardeen theory. The measured 3 dB internal quality factor Q ′ i also verifies such an anomalous temperature dependence. Physically, this phenomenon could be attributed dominantly to the effects of the two-level systems in the device, rather than the usual temperature-dependent complex conductance.
The high critical magnetic field and relatively high critical temperature of niobium nitride (NbN) make it a promising material for applications in superconducting quantum technology. However, NbN-based superconducting circuits are sensitive to decoherence sources such as two-level system (TLS) defects. Here, we numerically and experimentally investigate NbN superconducting microwave coplanar waveguide resonator arrays with a 100 nm thickness, capacitively coupled to a common coplanar waveguide on a silicon chip. We observe that the resonators' internal quality factor (QiQ_{i}Qi​) decreases fromQi∼1.07×106Q_{i}\sim 1.07\times 10^{6}Qi​∼1.07×106in a high power regime toQi∼1.36×105Q_{i}\sim 1.36\times 10^{5}Qi​∼1.36×105in the single photon regime. Data from this study is consistent with the TLS theory, which describes the TLS interactions in resonator substrates and interfaces. Moreover, we study the temperature (TTT) dependence ofQiQ_{i}Qi​and resonance frequencyfrf_{r}fr​of coplanar waveguide resonators to characterize the quasiparticle density. We observe that the increase in kinetic inductance at higher temperatures is the main reason for the frequency shift. Finally, we measurefrf_{r}fr​andQiQ_{i}Qi​in response to in-plane magnetic fields (B∣∣B_{||}B∣∣​). We verify thatQiQ_{i}Qi​stays well above10410^{4}104up toB∣∣B_{||}B∣∣​= 240 mT in the photon number< ⁣nph ⁣>< \! n_{ph} \! ><nph​>= 1.8 atTTT= 100 mK. Our results could enable the development of robust microwave superconducting circuits for circuit quantum electrodynamics at high magnetic fields required for hybrid superconducting-semiconducting quantum circuits.
Superconducting coplanar waveguide (SCPW) resonators have a wide range of applications due to the combination of their planar geometry and high quality factors relative to normal metals. However, their performance is sensitive to both the details of their geometry and the materials and processes that are used in their fabrication. In this paper, we study the dependence of SCPW resonator performance on materials and geometry as a function of temperature and excitation power. We measure quality factors greater than 2 × 106 at high excitation power and 6 × 105 at a power comparable to that generated by a single microwave photon circulating in the resonator. We examine the limits to the high excitation power performance of the resonators and find it to be consistent with a model of radiation loss. We further observe that while in all cases the quality factors are degraded as the temperature and power are reduced due to dielectric loss, the size of this effect is dependent on resonator materials and geometry. Finally, we demonstrate that the dielectric loss can be controlled in principle using a separate excitation near the resonance frequencies of the resonator.
The high critical magnetic field and relatively high critical temperature of niobium nitride (NbN) make it a promising material candidate for applications in superconducting quantum technology. However, NbN-based devices and circuits are sensitive to decoherence sources such as two-level system (TLS) defects. Here, we numerically and experimentally investigate NbN superconducting microwave coplanar waveguide resonator arrays, with a 100 nm thickness, capacitively coupled to a common coplanar waveguide on a silicon chip. We observe that the resonators' internal quality factor (Qi) decreases from Qi ~ 1.07*10^6 in a high power regime (< nph > = 27000) to Qi ~ 1.36 *10^5 in single photon regime at temperature T = 100 mK. Data from this study is consistent with the TLS theory, which describes the TLS interactions in resonator substrates and interfaces. Moreover, we study the temperature dependence internal quality factor and frequency tuning of the coplanar waveguide resonators to characterise the quasiparticle density of NbN. We observe that the increase in kinetic inductance at higher temperatures is the main reason for the frequency shift. Finally, we measure the resonators' resonance frequency and internal quality factor at single photon regime in response to in-plane magnetic fields B||. We verify that Qi stays well above 10^4 up to B|| = 240 mT in the photon number < nph > = 1.8 at T = 100 mK. Our results may pave the way for realising robust microwave superconducting circuits for circuit quantum electrodynamics (cQED) at high magnetic fields necessary for fault-tolerant quantum computing, and ultrasensitive quantum sensing.