Low-loss millimeter-wave resonators with an improved coupling structure

Extending superconducting quantum device functionality to millimeter-wave frequencies (near $100~$GHz) offers new opportunities for detection and transduction [1] as well as access to large coupling strengths for hybrid quantum experiments [2, 3]. Most importantly, the reduced sensitivity to thermal noise of higher-energy millimeter-wave photons could enable quantum experiments at 1 K, which significantly reduces cooling complexity and power dissipation constraints [4], enabling new pathways for scaling up quantum computing platforms, and could facilitate direct integration with high-speed superconducting digital logic [5, 6]. High-frequency superconducting devices have already found uses in sensitive detectors [7–9] with applications in high-frequency radio astronomy and potentially molecular spectroscopy [10–12]. In particular, superconducting resonators are well suited for millimeter-wave filter applications [13–15] making narrower line-width devices desirable.

To improve detector selectivity and sensitivity, and to establish more robust high-frequency quantum information systems, it is vital to understand and minimize decoherence in superconducting devices. Significant effort has gone towards investigating and reducing sources of loss at microwave frequencies [16], establishing that a significant remaining contribution to decoherence at the single photon level comes from two-level-systems (TLSs) found in amorphous dielectric materials. At temperatures near 1 K, these limiting loss factors can be overshadowed by dissipation from quasiparticles [17, 18] in commonly used superconductors such as aluminum or tantalum. Using a higher critical temperature (T_c ) superconductor such as niobium avoids generating quasiparticles with millimeter-wave photons and mitigates loss from thermal quasiparticles. However, since probing individual decoherence mechanisms requires reducing all other sources of loss as well, the nature and limits of TLS loss contributions and their frequency dependence when scaled to millimeter-wave frequencies remain relatively unexplored.

When measured at single-photon energies, the internal quality factor (Q_i ) of an on-chip resonant circuit provides insight into the maximum coherence of a quantum system formed by adding a source of nonlinearity (such as kinetic inductance [19–21] or a high-frequency Josephson junction [22, 23]). Significant progress has been made in improving millimeter-wave resonators [13–15, 19, 21, 24], but their single-photon quality factors remain below 2–$4\times10^4$: significantly lower than those measured in microwave circuits [25]. This can largely be attributed to two primary factors: first, millimeter-wave resonators frequently use materials with low dielectric constants (such as SiO₂) to simplify high-frequency circuit design [13, 26, 27], despite their poor dielectric loss characteristics [28] compared to crystalline silicon or sapphire. Second, many existing millimeter-wave resonators rely on coplanar stripline or microstrip transmission line components, both of which have relatively high radiation profiles resulting in increased radiative losses, particularly at high frequencies [29].

Taking inspiration from low-loss microwave devices, a ground-shielded circuit design minimizing radiative loss offers an attractive method for increasing millimeter-wave quality factors, and better control over on-chip signal propagation and coupling. Whereas low-frequency signals can be routed to an on-chip waveguide directly through wire bonds, these introduce substantial parasitic inductance [30]. Grounded circuits also present additional design challenges in millimeter-wave bands, where signals are primarily transmitted by hollow waveguides, requiring a method to efficiently direct the waveguide electromagnetic fields onto the chip. This proves to be a challenging problem, and as a result, a variety of transition structures coupling waveguides with on-chip transmission lines have been developed [31]; however, with no universal solution, transitions for specific applications are still actively studied and improved.

In this work, we use a transition specifically designed to measure ground-shielded millimeter-wave resonant circuits with improved control. We characterize a tapered coupling structure that efficiently confines the signal fields to an on-chip slotline waveguide, finding an insertion loss better than 0.5 dB over 14 GHz of bandwidth. We study niobium resonators patterned near the slotline and show that resonator coupling can be controlled independently without increasing radiation loss. With this novel resonator design combined with fabrication procedure, we improve on existing planar millimeter-wave devices and achieve high internal quality factors (Q_i ) consistently exceeding 10⁵ at single photon powers. This allows us to study the effects of oxide growth and removal on remaining loss contributions to Q_i from millimeter-wave TLSs, and show that limits from TLS can be increased as high as 10⁶.

When designing a millimeter-wave circuit to minimize loss, additional constraints apply to its coupling structure. Crystalline sapphire is an ideal substrate choice as it yields much lower dissipation than most dielectrics, but is more difficult to machine which imposes design limitations. As a result, transition designs requiring abnormally cut non-rectangular substrates [32–34], micromachining [35] or drilled holes [36] are impractical. Sapphire also presents additional challenges for high frequency circuit design due to its relatively high dielectric constant, which leads to more pronounced impedance mismatches caused by the presence of substrate in the waveguide [37]. Ensuring currents are carried by superconducting materials minimizes conduction loss. To achieve this, on-chip superconducting layers should be well separated from both waveguide and housing metal [38]: this consideration makes transitions with stripline geometries, which needs an external ground plane [39], less ideal. While potentially offering low radiation loss, coplanar waveguide transitions [35, 40] are typically more complex, requiring multiple stages and more physical space. Finline transitions, on the other hand, consist of a single taper [33, 34, 41], making them more compact, and transform the signal into a differential mode localized on the chip surface.

Our transition is defined by a superconducting niobium film patterned on the top surface of a rectangular crystalline sapphire substrate centered in a rectangular waveguide. Inspired by references [33, 36, 41], the geometry consists of a differential unilateral finline that tapers from the waveguide width down to a narrow slotline. A cutaway diagram of the complete structure is shown in figure 1, with two transitions coupled back-to-back by a length of on-chip slotline useful for coupling to resonators. Where the transition terminates, the waveguide height is reduced to increase the cutoff frequency of unwanted higher-order propagating modes, while the waveguide width is broadened to increase the usable area of the chip: this allows the device area to remain suspended away from metal surfaces, which minimizes conduction loss [38]. The only direct contact with the copper enclosure occurs where the chip is clamped at its corners.

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Due to the pronounced impedance discontinuity between the sapphire chip and the waveguide [37], a broadband matching structure is difficult to achieve with an exponential or cosine Vivaldi taper contour as is typically used in finline transitions [36, 41]. Instead, we find that a curved taper shape with a nearly linear waveguide entrance can compensate for the mismatch and can be optimized to give good performance over a section of waveguide bandwidth. The optimized contour is described by the function:

$$\begin{equation} y\left(x\right) = \left(\frac{W_0-S_1}{2}\right)\frac{x}{A_1}\sqrt{2-\left(\frac{x}{A_1}\right)^2} \end{equation} \tag{ 1 }$$

where W₀ is the smaller waveguide dimension, S₁ is the slotline width, and A₁ is the transition length. The geometry of the transition is detailed in figure 2. Using finite element method simulation software ⁶ , the above contour function and parameterized geometry dimensions are optimized for maximal transmission in the 90–$100~$GHz band. The resulting optimal dimensions are listed in table 1. Notably, by relaxing the bandwidth optimization constraint, we achieve a taper structure less than $0.9~$mm long: much smaller than the $\lambda/2$ value predicted with analytical functions [37] and more compact than many other transition structures in literature [33, 34, 36, 41, 42].

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The benefit of coupling to resonant devices through an intermediate on-chip slotline transmission line is apparent when examining the magnitude of the electric field of a wave propagating through the structure, shown in figure 3(a). Whereas some previous implementations of millimeter-wave resonators [19, 21] interact with a uniform waveguide electric field (visible on the left and right ends of figure 3(a) and rely on varying resonator dipole moments to adjust coupling, our method compresses the signal to a $40~\mu$m slotline resulting in over $50~$dB of dynamic range in electric field strength across the usable area of the chip. Consequently, the dipole coupling strength for each resonator can be set by its location on the chip, without needing to adjust the resonator's dipole moment, leaving more freedom to optimize the resonator performance.

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The taper geometry described above is defined by chlorine reactive ion etching a $100~$nm thick film of high-purity electron-beam-deposited niobium grown on a $100~\mu$m thick crystalline C-plane sapphire substrate (see appendix A for detailed procedure). Short sections of superconducting indium wire which form robust cryogenic seals compatible with low-loss superconducting devices [43], are used to secure and thermalize the chip to the copper waveguide enclosure, visible in a photograph of the mounted chip shown in figure 3(b). The edges of the on-chip taper structure match the waveguide dimensions, which enables visual alignment during mounting (necessary to maximize coupling).

The assembled structure is then measured at $0.86~$K in a helium-4 cryostat: using a vector network analyzer with millimeter-wave extension modules and a cryogenic low noise amplifier, we measure the complex response in transmission and reflection. Input attenuation and cryogenic isolators reduce thermal noise reaching the sample, enabling measurements in the single photon limit. A manual calibration removes the effects of additional hardware on the input and output lines, recovering de-embedded scattering parameters of the sample. Cryogenic calibrations are a complex problem [44–46] however due to mechanical shifts from thermal cycling and the large amounts of attenuation and gain present in the measurement network (see appendix B). This yields higher levels of uncertainty in our measured scattering parameters.

The effectiveness of the taper transitions is tested by using transitions to convert a waveguide signal to a slotline and back. The measurement results are summarized in figures 4 and 5. We find the transition performs best between 87.6 and 102.4 GHz, exceeding the designed range, and define this $14.8~$GHz wide frequency range as the useful operating band. Within the operating band, we find a maximum insertion loss of approximately $0.46\pm{0.35}$ dB (or 94.8% transmission), corresponding to a coupling efficiency of ${\sim}0.23$ dB for a single taper structure. However these values are likely dominated by errors introduced by calibration methods (which do not enforce passivity).

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Across the W band, we find that the de-embedded transmission and reflection of the structure are in fairly good agreement with simulations, with the exception of the region near 110 GHz: in this region unwanted resonances occur in the substrate and indium mounting regions, which are difficult to predict. For better performance in this range, our geometry could be adjusted to shift these resonances even higher in frequency, or reduce the energy participation in the indium regions. In the operating band, we find the return loss exceeds $13.1\pm8.6~$dB, which slightly deviates from simulation, but could be attributed to the significantly increased calibration uncertainty due to the high transparency of the structure and increased sensitivity to error terms. Combined, these measurements demonstrate a transition structure in good agreement with simulation, which in the operating band couples a signal on and off a chip with high efficiency.

Having demonstrated a coupling structure capable of efficiently transforming a rectangular waveguide signal to and from a localized on-chip slotline, we can now design resonators decoupled from their environment while only considering local interactions. Away from the centered slotline, the chip surface is entirely covered by superconducting material acting as a ground plane, so a resonant structure patterned in this region will have its long-range electric dipole interactions reduced. Our millimeter-wave resonator design is composed of a discrete capacitor island connected to the ground plane by a $2~\mu$m-wide meandered inductor, shown in figure 6. Every resonator is designed with an identical capacitor island, and its resonant frequency ω₀ is adjusted by changing the inductor length while keeping its width constant. The entire resonator has a rectangular footprint around 100–200 µm per side, which is not insignificant compared to signal wavelength in the slotline (${\sim}1~$mm): in this limit, the ground plane edges contribute significant reactive corrections. On each chip, five to six resonators are placed near the central slotline to allow interaction with the propagating signal. This differential geometry can be modeled by the single-ended LC circuit shown in figure 6 including asymmetric coupling (C_C and M) [47]. This accounts for reactive contributions and impedance mismatches induced by the resonator presence.

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For this resonator geometry, coupling strength can be controlled by adjusting its separation from the slotline. The interaction decreases exponentially with distance, which we demonstrate by simulating the coupling quality factor Q_e as a function of resonator separation, plotted in red in figure 7. The coupling quality factor can also be approximated empirically as a function of only the separation d by $\log_{10} Q_e = 0.06491+0.0390 d/\mu$m, shown as a dashed line in figure 7. We find that experimental measurements of Q_e (described in the next section) follow this approximation reasonably well.

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Characterizing the complex transmission spectra of these resonators at low temperatures ($0.86~$ K) allows us to explore losses at millimeter-wave frequencies. Typical normalized measurements at low average photon number ($\bar{n}\approx10$) are shown in figure 8(a). On resonance, we observe a dip in transmission as the resonance sweeps through a circle in the complex plane. This behavior is captured well by [47]:

$$\begin{equation} S_{21} = 1-\frac{Q}{Q_e^*}\frac{e^{i\phi}}{1 + 2i Q \frac{\omega-\omega_0}{\omega_0}} \end{equation} \tag{ 2 }$$

where $Q^{-1} = Q_i^{-1}+\text{Re}[Q_e^{-1}]$ [47] and the coupling quality factor Q_e is rotated in the complex plane by $Q_e = Q_e^* e^{-i\phi}$ due to asymmetric coupling to the slotline as described in the previous section. Measuring both quadratures of the transmission spectrum to capture this asymmetry is particularly important for extracting an accurate estimate of Q_i , which is sensitive to φ [48].

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Repeating these measurements at varying powers shows that Q_i increases with power, as shown in figure 8(b). This increase can be explained by a power-dependent loss mechanism from saturating TLSs [49–52] described by:

$$\begin{equation} Q_\text{TLS}\left(\bar{n},T\right) = \frac{Q_{\text{TLS},0}}{\tanh{\frac{\hbar\omega}{kT}}}\sqrt{1 + \left(\frac{\bar{n}}{n_c}\right)^\beta \tanh{\frac{\hbar\omega}{kT}}}. \end{equation} \tag{ 3 }$$

Here $Q_{\text{TLS},0}$ is the inverse linear absorption from TLSs, ω is the resonant frequency, and β and n_c are parameters characterizing TLS saturation [49, 52]. Nonlinear effects (from kinetic inductance) [21] limit the power range where linear measurements can be performed, however at high powers we observe that Q_i begins to saturate, indicating the presence of other loss mechanisms.

By examining the power and temperature dependence of Q_i , we can further distinguish between sources of loss. The full behavior is captured with a model that includes TLS loss (Q_TLS) [49–52], equilibrium quasiparticle loss (Q_σ ) [17, 53, 54] and other loss mechanisms that are power and temperature-independent (Q_other):

$$\begin{equation} \frac{1}{Q_i\left(T,\bar{n}\right)} = \frac{1}{Q_\text{TLS}\left(\bar{n},T\right)} + \frac{1}{Q_\sigma\left(T\right)} + \frac{1}{Q_\text{other}}. \end{equation} \tag{ 4 }$$

The quasiparticle loss term is parameterized by:

$$\begin{equation} Q_\sigma\left(T\right) = Q_{\sigma_0}\frac{\sigma_2\left(T,T_c\right)}{\sigma_1\left(T,Tc\right)} \end{equation} \tag{ 5 }$$

where σ₁ and σ₂ respectively are the real and imaginary parts of the complex surface conductance, calculated by numerically integrating the Mattis–Bardeen equations for $\sigma_1/\sigma_n$ and $\sigma_2/\sigma_n$, and σ_n is the normal conductance. [17, 53, 54].

To investigate the effects of quasiparticle loss, we measure Q_i with TLS loss mostly saturated (at $\bar{n}\sim10^5$) for several representative resonators as a function of temperature. We show the results in figure 8(c) along with $Q_\sigma(T)$ and fits to the full model described above. Below approximately $1.5~$K we find that Q_i is almost unaffected by Q_σ and is nearly temperature independent for most devices. As the temperature approaches a significant fraction of niobium's critical temperature ($9.2~$K), Q_σ rapidly becomes dominant. However, at the low temperatures relevant for quantum experiments, Q_σ exceeds measured values of Q_i by several orders of magnitude, suggesting that quasiparticle loss contributions are negligible in this regime.

Having determined that the dominant loss contributions come from power-independent loss Q_other and Q_TLS, we can neglect thermal contributions to Q_i at low temperatures. Upon inspection of the power dependence of a typical resonator in figure 8(b), we find the increase of Q_i from TLS saturation is relatively small, unlike what is seen in many microwave loss studies [55, 56]. Using the model above, we find $Q_{\text{TLS},0} = 0.953\times10^6$ while $Q_{\text{other}} = 1.17\times10^5$, indicating that TLSs are not the dominant source of loss.

For one resonator in particular we observe quality factors significantly higher than average, providing additional insight into loss origins. This device falls very close in frequency to a nearby higher-linewidth resonance, which results in more asymmetric coupling as shown in figure 8(d), and likely modifies its radiation properties [57]. We find equation (2) is able to accurately capture Q_i from the response, but unlike the other resonators, the power dependence shown in figure 8(e) is much more pronounced. For this device, we find single-photon $Q_i = 0.827\times10^6$, which is similar to state-of-the-art microwave resonators [55, 56], and the loss sources can be disentangled into $Q_{\text{TLS},0} = 1.03\times10^6$ and $Q_{\text{other}} = 4.18\times10^6$. This independent loss limit is significantly higher than in most of our resonators, and Q_i is instead primarily dominated by Q_TLS. The latter is consistent with the other resonators on chip, giving us a better insight into millimeter-wave TLS loss.

5.1. Reducing losses with surface oxide etch

When exposed to air, niobium is known to slowly evolve a lossy amorphous surface oxide layer containing dissipative sub-oxides [58] and TLSs [56]. Surface treatments are commonly used to remove this surface layer in microwave resonators to reduce loss [56]. Although TLS density has not yet been investigated in the W-band, losses from niobium sub-oxides are believed to be more pronounced at higher frequencies [58]. To study the effect of surface processing on Q_i , we repeat the measurements summarized in figures 8(a) and (b) for devices that underwent different aging times and etch conditions. In figure 8(f), we plot the low and high power limits of measured Q_i as well as the fitted value of $Q_{\text{TLS},0}$ for devices from five separate chips.

Between samples A and B we observe that 5 days of aging reduces both $Q_{\text{TLS},0}$ and Q_other, leading to lower quality factors for resonators exposed to air for several days, which is consistent with niobium oxide regrowth that has been shown to increase loss in microwave devices [56]. This can be mitigated by selectively removing the surface layer of niobium oxide after fabrication using a buffered solution of hydrogen fluoride (BHF) [56], which we achieve by immersing samples C, D and E in a 5% BHF solution for $40~$min immediately prior to mounting and measurement.

We observe that this BHF treatment can reverse the effects of air exposure in Sample C, which despite experiencing a longer air exposure of 8 days, has higher average $Q_{\text{TLS},0}$ and Q_other than Sample B after the BHF treatment. Applying this surface treatment to samples D and E, which experienced reduced initial air exposure yield consistently lower losses than samples A-C, including the highest-Q resonator in this study described above. Combining minimal air exposure with surface oxide removal using BHF in sample E, we are able to consistently obtain resonators with single photon internal quality factors above $1.4\times10^5$, and an average $Q_{\text{TLS},0} = 1.04\times10^6$: significantly higher than previously measured for planar millimeter-wave devices [13–15, 19, 21, 24].

While these values of loss are much closer to those reported for microwave devices [56], in the power-dependent measurements above we observe that our millimeter-wave resonators are on average limited by power and temperature-independent loss Q_other to a much greater extent than TLS loss, which limits microwave devices [55, 56]. This loss could come from a variety of sources, including remnants of conduction loss from the copper enclosure [38], seam loss from an imperfect seal between the halves of the enclosure [59], radiation loss [18] or additional power-independent microscopic relaxation channels such as conductive loss in the niobium sub-oxides [56]. To estimate the impact of radiation losses, we can verify that our ground-shielded resonator design protects the resonance from radiative loss induced by coupling. In figure 9 we plot Q_i for single-photon and high-power limits (Q_other) as a function of Q_e , and observe no correlation with either. Thus we can design a resonant circuit with a wide range of coupling strengths without affecting coupling to lossy or radiative channels. As a final note, an examination of all the devices in figure 8(f), shows that on average $Q_{\text{TLS},0}$ and Q_other scale similarly relative to each other when affected by aging and surface treatment. This is highly suggestive that the remaining millimeter-wave source of loss is still tied to materials-induced decoherence in the superconductor surface, and warrants further studies.

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We have demonstrated an on-chip millimeter-wave resonator design with a ten-fold improvement in loss over previous work, and leveraged this platform to investigate sources of single-photon decoherence in the W band, finding that scaling up superconducting device frequency does not significantly change the nature of decoherence. Using a specifically-designed waveguide to slotline transition based on a finline taper, we demonstrate a packaging technique for probing on-chip devices at high frequencies which improves on dipole coupling techniques previously used to address millimeter-wave resonators. The increased control afforded by this coupling method enables more complex device designs with high-frequency signal routing, and we show that our coupling strengths can be adjusted over a wide range without introducing circuit losses.

Having shown that planar millimeter-wave resonators compatible with planar fabrication techniques can achieve performance comparable to microwave quantum circuits [55, 56], we now have suitable infrastructure to develop more advanced superconducting devices for astronomical detection and studying complex spin systems in this frequency regime. This platform could be further enhanced by incorporating materials with improved surface oxide characteristics [55], and the operating range could be extended to even greater temperatures with even higher-T_c materials such as niobium nitride or niobium titanium nitride. Furthermore, introducing a high-frequency nonlinear element [21–23] into our design would enable a millimeter-wave artificial atom operating at liquid-helium-4 temperatures, paving the way for a new generation of high-frequency higher-temperature quantum tools.

The authors thank P Duda for assistance with fabrication process development, and M W, A Kumar and A Suleymanzade for useful discussions. This work is supported by the U.S. Department of Energy Office of Science National Quantum Information Science Research Centers as part of the Q-NEXT center, and partially supported by the University of Chicago Materials Research Science and Engineering Center, which is funded by the National Science Foundation under Grant No. DMR-1420709. This work made use of the Pritzker Nanofabrication Facility of the Institute for Molecular Engineering at the University of Chicago, which receives support from Soft and Hybrid Nanotechnology Experimental (SHyNE) Resource (NSF ECCS-2025633).

All data that support the findings of this study are included within the article (and any supplementary files).