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Niobium quarterwave resonator with the optimized shape for quantum information systems
EPJ Quantum Technology volume 7, Article number: 7 (2020)
Abstract
Quantum computers (QC), if realized, could disrupt many computationally intense fields of science. The building block element of a QC is a quantum bit (qubit). Qubits enable the use of quantum superposition and multistate entanglement in QC calculations, allowing a QC to simultaneously perform millions of computations at once. However, quantum states stored in a qubit degrade with decreased quality factors and interactions with the environment. One technical solution to improve qubit lifetimes and network interactions is a circuit comprised of a Josephson junctionbased qubit located inside of a high Qfactor superconducting 3D cavity.
It is known that niobium resonators can reach \(Q_{0}>10^{11}\). However, existing cavity geometries are optimized for particle acceleration rather than hosting qubits. RadiaBeam Technologies, in collaboration with Argonne National Laboratory and The University of Chicago, has developed a niobium superconducting radio frequency quarterwave resonant cavity (QWR) for quantum computation. A 6 GHz QWR was optimized to include tapering of the inner and outer conductors, a toroidal shape for the resonator shorting plane, and an inner conductor tip to reduce parasitic capacitance. In this paper, we present the results of the resonator design optimization, fabrication, processing, and testing.
Introduction
Nearly all areas of modern life are influenced by the incredible impact of computational capabilities. Quantum computers may make many computationally intense fields of science, such as cosmology, quantum field theory, particle interactions, and nuclear physics, tractable. The building block element of a QC is a quantum bit, which is a twolevel quantum system. Qubits enable the use of quantum superposition and multistate entanglement in QC calculations, allowing a QC to perform millions of quantum mechanical computations at once [1]. Entanglement lets a QC change the state of multiple qubits simultaneously via adjusting the state stored in a single bit, enabling computational power scalability unachievable with traditional computers [2]. These advantages are not just theoretical: it was recently reported that quantum supremacy was experimentally demonstrated for the first time using Google’s superconducting Sycamore processor [3].
One of the greatest challenges in building QCs is controlling quantum decoherence [4], rapid degradation of the qubit’s quantum state due to interactions with the environment and the integrated control channels. Because of decoherence, the relevant quantum behavior is lost, and timeconsuming tasks may render some quantum algorithms inoperable [5].
One approach to building a quantum computer is using superconducting RF oscillator circuits with Josephson junctions for anharmonic dynamics [6]. Planar chips allow integrating a large number of qubits in topologies that increase the lifetimes of quantum states by applying errorcorrection techniques [7]. However, the low intrinsic Qfactors of microstrip resonators limits the coherence times of the qubits [8] while planar geometries require nontrivial solutions for coherencepreserving coupling to the chip [9, 10]. Another solution to improve qubit lifetimes and allow for network interactions is to couple the Josephson junction through a high Qfactor superconducting 3D cavity [11]. The quantum state excited in the Josephson junction is protected from environmental noise and loss via the encoding qubit states in the highQ resonant cavity modes [12–15].
The coherence time is closely related to the Qfactor of the resonator and its energy dissipation. Current qubit 3D resonators can achieve \(Q\sim 10^{8}\) with coherence times of several milliseconds [16]. On the other hand, niobium resonators used in particle accelerators reach quality factors of ∼10^{11} [17], potentially enabling storage times approaching seconds [18] if adopted for operation in a QC.
To pursue this opportunity, we developed a 3D superconducting RF (SRF) quarterwave resonator (QWR) with a shape optimized for highQ operation in the quantum regime [19] (see Fig. 1). The QWR is an attractive choice, due to the simplicity of its integration with the Josephson junction, which can be placed near the central conductor providing a high coupling strength, and for its feasibility for scaling to multiqubit systems. The optimization of the cavity includes the introduction of inner and outer conductor tapering, toroidal shaping of the resonator ends and optimization of the inner conductor tip to reduce parasitic capacitances. These geometrical features provide a better Qfactor and thus a longer lifetime for quantum memory. In addition, we performed a series of numerical simulations and optimizations to decrease energy dissipation in the cavity due to both surface currents and dielectric losses in the niobium oxide layer.
In SRF cavities the unloaded Qfactor can be defined as \(Q_{0} = \frac{G}{R_{s}}\). Here, G is a geometry factor, which ranks the cavity’s effectiveness in providing “useful” electric field due to the influence of its shape alone and excludes specific material wall loss. The Gfactor can be formally defined as \(G= \frac{\omega \mu _{0} \int \vert {\vec{H}} \vert ^{2} \,dV}{\int \vert {\vec{H}} \vert ^{2} \,dA}\), where ω is the resonant frequency, \(\mu _{0}\) is permeability of free space, the top integral is for volumetric RF magnetic field and the bottom is for surface RF magnetic field [20]. Increasing the value of G results in higher \(Q_{0}\) and thus indicates a better cavity design. \(R_{s}\) is the surface resistance, defined by the material and operating conditions. The surface resistance \(R_{s}\) can be expanded into two terms: BCSresistance and residual resistance, \(R_{s} = R_{\mathrm{BCS}} + R_{\mathrm{res}}\). The BCSresistance is given by the Bardeen–Cooper–Schrieffer theory [21], in which the superconducting Cooper pairs, which have zero resistance for DC current, have finite mass and their momentum alternates sinusoidally due to the AC currents of RF fields, giving rise to energy loss. BCS resistance for niobium depends on frequency and temperature, \(R_{\mathrm{BCS}} \sim \frac{A}{T} f^{2} e^{ \frac{\Delta T}{kT}}\) [22], and thus one should keep the frequency and temperature as low as possible to maximize the unloaded Qfactor. The residual resistance arises from several sources, such as material defects, oxides and hydrides that can form on the surface due to hot chemistry and slow cooldown, and other sources related to cavity processing and surface treatment.
Methods/experimental
Geometrical optimization
As a reference model for comparison of our optimization efforts, we used the straight QWR shape used by Yale [23]. As an initial step for optimization, we adopted the shape of the 72.5 MHz QWR cavity developed at ANL [24]. This geometry was designed for use with highpower accelerating fields with limitations on the maximum surface electric field and surface magnetic fields of 35 MV/m and 50 mT respectively [25]. These limitations are introduced by electric (E) field stimulated emission and thermal superconductivity breakdown caused by surface currents induced by the surface magnetic (H) field. While these surface fields limitations are not relevant for lowpower quantum applications, the optimization approach used for these cavities helps to reach better Gfactor by reducing peak and integral surface Hfields. We performed further shape optimization by adjusting the geometrical parameters shown in Fig. 2. The simulations were performed in CST Microwave Studio [26].
The machining methods used to produce the cavity put limits on the cavity dimensions. Here we used a milling center to hollow out the cavity resonator from a solid block of niobium. The cutting tool width is constrained by the gap between the inner and outer conductors \(w_{b}\) and vibrations experienced during cutting. By analyzing the cutting tool and cavity dimensions (Fig. 3), a simple relation was inferred: \(w_{b} = l_{1} + l_{3} + w_{c}\), where \(l_{1}\), \(l_{3}\) are the outer and inner conductor tapering parameters, and the minimum width of the cutting tool \(w_{c}\) is limited by vibrations during machining. This relation generally means that for bigger tapering we will need to increase the gap width. Therefore, we performed simulations to find the dependence of the RF parameters on these dimensions and then optimized them.
First, we adjusted the geometry of the top part of the resonator. For the QWR geometry, this is the region where the magnetic field energy density is highest (see Fig. 4). By increasing the volume of this part of the resonator, the magnetic energy is distributed over a larger volume and the peak energy density is decreased. Decreasing the magnetic energy density reduces the magnetic surface field and improves the Gfactor. We started by simulating geometries with different blending radius of the top part of the cavity. After we explored the outer conductor blending, we optimized the shape of inner and outer conductors by adding tapering. We optimized the taper shape by adjusting the tapering start point for the inner and outer conductors. To maximize the coupling strength with the transmon (Josephson junction), we optimized the inner and outer conductor dimensions, as well as the inner conductor radius. Finally, we performed simulations to find the optimal width of the gap between the inner conductor tip and outer conductor wall. The simulated Gfactor dependences on the geometric parameters are presented in Fig. 5.
Turning to geometrical optimization to reduce \(R_{s}\), the primary source of residual resistance is the thin dielectric layer that is usually present on the surface of superconducting niobium. As shown in recent work at Fermi National Accelerator Laboratory [27, 28], removing this layer can improve the Qfactor by an order of magnitude. Losses in the dielectric layers can be calculated by integrating the Efield over the thin surface dielectric volume:
where ω is the working frequency and \(\varepsilon ^{\prime \prime }\) is the imaginary part of complex dielectric permittivity. By creating a more uniform Efield distribution, we can potentially lower the surface layer integral and thus decrease the dielectric loss. The Efield is concentrated at the end of the quarterwavelength loading element and concentrated by the sharp edges of the straight geometry of the central pin as shown in Fig. 4. We blended this feature to achieve a more uniform Efield distribution. Full blending of the inner conductor tip helped to reduce the Efield integral by 22%, thus significantly reducing this source of dielectric loss.
We further investigated the dielectric loss using simulations of a thin Nb oxide layer on the cavity surface. The prevailing surface oxide in niobium RF cavities is \(\mathrm{Nb} _{2}\mathrm{O}_{5}\) [29]. RF measurements in thin \(\mathrm{Nb} _{2}\mathrm{O}_{5}\) films [30] have found a dielectric constant of about 50 with a loss tangent of 0.01 for temperatures below \(\sim 100\mbox{ K}\). The results presented in [31] show that the loss tangent remains constant for frequencies larger than 1 MHz. Thus we used these parameters in simulations to estimate the upper limit of the Qfactor.
The typical thickness of the oxide dielectric film in niobium RF cavities is around 50 Å [32]. This is much smaller than the cavity dimensions, which are on the order of millimeters. Thus such thin films cannot be simulated directly. Instead, we simulated losses in a thicker 1 μm layer, and divided the simulated by a factor of 200 to approximate the expected actual film thicknesses. By doing this, we assumed that the Efield density does not change in this thin oxide film. Inferred dielectric losses thus should depend linearly on the surface thickness h:
In these simulations, as we are interested in the Qfactor as limited by dielectric losses, we assumed infinite electric conductivity for the cavity walls. The results of these simulations showed that a cavity with a 50 Å thick \(\mathrm{Nb} _{2}\mathrm{O}_{5}\) layer in the optimized geometry almost two times better than the simulated dielectric Qfactor the nonoptimized QWR geometry.
We then studied how to reduce the RF losses in the joints between the different components of the cavity assembly. Our fabrication method introduces limits on the dimensions of the cavities. One of the most substantial restrictions is the cavity length. From a manufacturing point of view, it is desirable to make the cavity as short as possible. The shorter cavity allows using a shorter cutter (Fig. 3), reducing vibrations and improving the quality of the machined surface.
We thus introduced the concept of machining a short cavity elongated by attaching an additional part (Fig. 6) made of a different material. An obvious material choice is aluminum, as this becomes superconducting at 1.2 K, well above our 10 mK operating temperature.
Our new design added a number of parameters for optimization. We performed simulations to define the minimum distance from the niobium inner conductor to the niobium to aluminum connection (\(L_{\mathrm{g}}\)) shown in Fig. 7. The next step was to simulate the joint losses. The surface magnetic field in the region of the QWR joints drives currents across the seam. Lower seam conductivity introduces additional losses; therefore we tried to minimize the Hfield on the seam.
One design solution to reduce the joint loss is an RF choke (Fig. 8(e)). The addition of the choke, however, introduces difficulties with the cavity coupling, where the deep choke groove pushes the RF coupling pin (see Fig. 6) too far from the cavity, making the coupling between the cavity and RF transmission line unacceptably weak. On the other hand, a big concern was that coupling through the choke may introduce parasitic modes. This is inacceptable for qubit cavities, which require a clean spectrum for single mode operation. Moreover, eigenmode simulations showed a lowerorder mode with Efield concentrated inside the choke. The appearance of a lowerorder mode was a significant drawback, and the choke could not be used.
The evolution of the design during the optimization process is illustrated in Fig. 8 and compared quantitatively in Table 1; we managed to increase the Gfactor by 65%. Further optimization seems to be possible; however, those designs are not feasible with machining fabrication.
Engineering and fabrication
In order to test this approach, we engineered and built a proof of principle prototype. The engineering design of the prototype includes system integration, thermal management, magnetic shielding, vacuum considerations, and signal integration. The complete system is shown in Fig. 9. Starting from the inside out, the original fulllength cavity was truncated to have a separate lower field region made from aluminum that would also provide connection points for the RF signals through nonmagnetic SMA connectors. Then considerations were made for sufficient thermal contact and choice of fastener. This was accomplished both through analytical calculation and computational stress analysis in ANSYS Multiphysics [34], as seen in Fig. 10, shown with magnified deformation scales.
Next, a simple magnetic shield was designed in Cryoperm 10 [35], a unique cryogenically friendly steel alloy that retains functional permeability at cryogenic temperatures more effectively than conventional MuMetal [36]. This magnetic shield required two separate shielding cans, each manufactured from 1 mm thick Cryoperm 10, that are separable for assembly of the device under test while permitting small penetrations for SMA cables. Finally, the full system was reanalyzed to ensure that the additional material did not affect the original assembly thermal calculations and provisions were made to ensure the full system could be mounted onto the available test plate within a dilution refrigerator operated at 10 mK. At this stage, spring washers were incorporated into the design, and a torque value and target compression were calculated and provided to technicians for assembly.
Regarding the fabrication of the cavity, we encountered challenges during niobium machining that were amplified by overannealing of the niobium by the vendor. This resulted in a ‘gummier’ metal consistency, which leads to surface burnishing rather than clean shearing of material and may have limited the Q of the prototype.
Optical measurements were made on the first fabricated prototype cavity. All dimensions of the cavity were within the design tolerances except for the inner conductor length and inner conductor tilt. The former was decreased by 270 μm which, according to simulations, could lead to a frequency increase of 140 MHz. Inner conductor tilt was 1.6 degrees and, according to simulations, could not disturb RF parameters of the cavity. Furthermore, this tilt could be due to pressure applied to the sample during the cut for inspection.
We used optical measurements to estimate surface roughness on the outer conductor tapering which was one of the most challenging surfaces from a fabrication point of view. The measured profile is presented in Fig. 11. The surface roughness estimate based on this measurement is 20 μm which, however, could be up to 30 μm due to the high error of optical measurements of surface profiles.
Two solid niobium 6.2 GHz quarter wave resonators were cleaned, etched and high pressure rinsed at Argonne National Laboratory (see Fig. 12). Figure 13 shows the cavities before and after etching. Both cavities were processed according to the following procedure: (1) a 1 hour ultrasonic cleaning in a 40^{∘}C 4% alconox solution [37]; (2) a ultrahigh purity water rinse; (3) dried with filtered and deionized nitrogen boiloff gas; (4) a 150 minute buffered chemical polished (BCP) in 1:1:2 (HF:\(\mathrm{HNO} _{3}\):\(\mbox{H}_{3}\mathrm{PO} _{4}\)); (5) a ultrahigh purity water rinse; (6) another 1 hour ultrasonic cleaning at 40^{∘}C in a 4% alconox solution; (7) a ultrahigh purity water rinse; (8) a highpressure ultrahigh purity water rinse; and finally (9) air dried in class 10 clean room and bagged in a class 100 clean room for transfer to the cryogenic laboratory.
Resonator tests
We performed a series of room temperature (RT) measurements for all fabricated cavities to keep track of their performance. First, we measured the RT quarterwave resonant frequency and Qfactor (Fig. 14, left). We used a single coupling pin (Fig. 14, middle) to perform reflectiontype measurements. The pin length was chosen to be 4.1 mm above the aluminum part edge to provide sufficient coupling. During the experiment, the cavity was tightly clamped to the aluminum part (Fig. 14, right), until a plateau in the measured Qfactor was observed.
We started from measuring the magnitude of \(S_{11}\) (Fig. 15, left), which gave us information about the resonant frequency, \(f_{0}=6160\text{ MHz}\). We used the Smith chart representation (Fig. 15, right) to determine the frequencies \(f_{1}\) and \(f_{2}\) where the real and imaginary parts of the load impedance were equal. These values then were used to calculate the internal Qfactor of the cavity
Measurements of the nonoptimized and optimized niobium and copper cavities are summarized in Table 2. The nonoptimized niobium cavity appeared to be detuned by \(+15\mbox{ MHz}\), while the optimized cavity with 400 μm shorter inner conductor was detuned by \(150\mbox{ MHz}\), which was 100 MHz less than expected. In order to implement the S11curve measurement method, we added a circulator to the excitation port to measure the reflected power (we note that the unshielded magnetic fields from the circulators may affected the measured Qfactor of the resonators).
In order to accurately calculate the internal Qfactor \(Q_{0}\) of the cavity, the external Qfactors \(Q_{\mathrm{EXT}}\) of the probes should differ (relative to \(Q_{0}\)) by no more than by an order of magnitude, and ideally by no more that a factor of two [38]. However, since the \(Q_{0}\) of the cavity was initially unknown, we had to perform a series of measurements of loaded Qfactor \(Q_{L}\) with different external Qfactors \(Q_{\mathrm{EXT}}\) of the probes, i.e. the coupling pin of various lengths (Fig. 16, left). For these measurements, a copper cavity with already measured internal Qfactor (at room temperature) was used, so we just calculated external Qfactor for each coupler length by inferring the coupling coefficient from \(S_{11}\) measurements. However, since the room temperature \(Q_{0}\sim 10^{3}\), these measurements could only be done for \(Q_{\mathrm{EXT}}\sim 10^{5}\). For shorter couplers, we used electromagnetic simulations (Fig. 16, right).
In order to measure the Qfactor of the resonator, we adopted the ringdown technique that is widely used for measurements of superconducting accelerating cavities [39]. In this method, the resonant cavity is driven by a short rf pulse into one port. Then the exponentially decaying cavity signal coming out of the second port is captured by an oscilloscope where we measure the decay time. One method of exciting the cavity is with positive feedback or a selfexcited loop (SEL) [40], which works without any external rf reference matched to the cavity resonance.
The schematic of the SEL circuit built for our tests is shown in Fig. 17. When the rf switch is in the on state, the rf signal is amplified by a lownoise amplifier (LNA) from a noise that further on gets filtered by a combination of the test cavity itself and an additional bandpass filter. As this signal fills up the cavity the switch turns off and the stored energy decay time is measured by the scope. This method has the advantage of being independent of resonant frequency due to the microphonics, which is essential for measuring highQ cavities with extremely narrow bandwidths.
The initial tests of the ringdown circuit were carried out on a 6 GHz copper test resonator (see Fig. 18). Typical waveforms are shown in Fig. 19. The decay time τ was measured, and then the loaded Q is calculated using \(Q_{L} = \tau \pi f\), where f is the cavity resonant frequency.
Measurements of the loaded Q using Sparameter measurements in the frequency domain using a vector network analyzer (VNA) showed \(Q_{L}=1500\), which was in good agreement with the results of the ringdown measurements that showed \(Q_{L}=1550\) (\(\tau \sim 82\mbox{ ns}\)). The SELbased ringdown measurement circuit proved to work well and matched the results from the frequency domain measurements at room temperature.
Moving to the cryogenic tests, two etched resonators, one with the simple and the other with the optimized shape (shapes b and d, as shown in Fig. 8) were used for Qfactor measurements in the quantum regime (10 mK temperature and a singlephoton power level). The assembly is shown in Fig. 20.
We connected the resonators using the amplification chain installed inside the dilution refrigerator. The room temperature measurements of the installed resonators revealed that the resonant frequencies of both resonators increased by 200250 MHz after polishing. The frequency of the nonoptimized resonator became 6226 MHz, while the frequency of the optimized resonator increased to 6098 MHz (see Table 2, columns 2 and 4 for the frequencies before etching).
Both Sparameters and ringdown measurements were used to measure the loaded Qfactor (\(Q_{L}\)), which is a combination of the resonator Qfactor (\(Q_{0}\)), external Qfactor of the coupling elements (\(Q_{\mathrm{EXT}}\)) and the Qfactor of the parasitic elements such as aluminum piece, seam losses, etc. (\(Q_{\mathrm{PAR}}\)):
Therefore, in order to measure the real Qfactor, we need to operate in the undercoupled regime: \(Q_{\mathrm{EXT}}\gg Q_{0}\). However, since \(Q_{0}\) is not zero and \(Q_{\mathrm{EXT}}\) scales directly with the coupling pin length, we decided to perform a series of Qfactor measurements with different \(Q_{\mathrm{EXT}}\), starting from ∼10^{5}. One of the couplers was used to excite the signal in the resonator, and the other one was used as the field probe.
For the first measurement, SMA couplers lengths were adjusted to \(Q_{\mathrm{EXT}1}=4\cdot 10^{5}\) and \(Q_{\mathrm{EXT}2}=10^{7}\) according to the calibration curve shown in Fig. 16. Then, on each successive run we increased the \(Q_{\mathrm{EXT}}\) by about an order of magnitude by reducing the coupler length. When the \(Q_{0}\) starts to dominate in the \(Q_{L}\), we should see no further changes in the measured Qfactor. Due to the project constraints, we had to limit the number of runs to three.
Results and discussion
For each measurement, the resonators were then cooled down to 10 mK within 24 hours. It is worth mentioning that the resonators remained in the 100–150 K region for about 9 hours, long enough to be affected by Qdisease and have their Qfactor reduced [41, 42].
The results for all three runs are summarized in Fig. 21. We can see a clear trend, where the Qfactor of the resonator with the optimized shape is higher by ∼25% than the Qfactor of the nonoptimized resonators, measured by different techniques. Also, it is important to note that the Qfactor has not reached a plateau and grows exponentially with the length of the coupler, which may indicate that we are still in the overcoupled regime.
Finally, we have measured the Qfactor of the optimized resonator at single photon power levels and below and observed no change for the level of one photon or above. The Qfactor drops rapidly for levels below one photon. However, the signal to noise ratio is very low at these power levels. The results of the Qfactor measurements as a function of average input power level and temperature are shown in Fig. 22. The saturation of losses at higher power levels demonstrates that the lowpower Qfactor is limited by losses in the dielectric layer and material imperfections [33, 43, 44].
Conclusions
RadiaBeam, in collaboration with the University of Chicago and Argonne National Laboratory, has developed a 3D SRF quarterwave resonator with shape optimized for operation in the quantum regime. We used the known merits of SRF resonator design performance to demonstrate superior Qfactor performance of the optimized resonator. We have fabricated and etched two niobium resonators: one with simple and one with optimized shape. These prototype resonators were machined out of highRRR niobium, chemically polished and highpressure rinsed at Argonne National Laboratory and tested at the University of Chicago at 10 mK up to 1 photon power levels. We used several methods to measure Qfactor, which demonstrated that the Qfactors of the resonators with optimized shaped are ∼25% higher than for the nonoptimized resonators (see Fig. 23). This result demonstrates the proofofconcept of the higher Qfactor due to shape optimization. Our future work will focus on improving the fabrication and surface processing of these resonators while incorporating new methods to limit the role surface oxides and nitrides play in reducing the cavity quality factors.
Abbreviations
 AC:

alternating current
 ANL:

Argonne National Laboratory
 BCP:

buffered chemical polishing
 BCS:

Bardeen–Cooper–Schrieffer theory
 EM:

electromagnetic
 IR:

infrared
 LNA:

lownoise amplifier
 QC:

quantum computers
 QWR:

quarterwave resonant cavity
 RF:

radio frequency
 RRR:

residualresistivity ratio
 RT:

room temperature
 SBIR:

Small Business Innovation Research
 SEL:

selfexcited loop
 SMA:

SubMiniature connector version A
 SRF:

superconducting radiofrequency
 VNA:

vector network analyzer
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Acknowledgements
The authors would like to thank Dr. Alexander Romanenko from Fermi National Accelerator Laboratory for the important discussions about SRF qubit cavities. We would also like to acknowledge our RadiaBeam colleagues Peter Phillip and Daniel Villaseñor for their work on cavity fabrication, as well as Salime Boucher for constructive criticism of the manuscript.
Availability of data and materials
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
Funding
This work was supported by the U.S. Department of Energy, Office of High Energy Physics, under SBIR grant DE SC0018753.
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Affiliations
Contributions
The resonator design and optimization were done by KT and SK. Engineering design and fabrication studies were performed by RA, AM and PC. AS developed the SEL measurement stand. ZC performed the chemical treatment of the niobium cavities and consulted the authors in SRF technology. AC provided consultation in quantum technologies, qubit integration and hosted the experiment. The experiment was performed by KT, ED, AS and SK. The data was analyzed and interpreted by KT, ED, ZC, SK and AS. The manuscript was written and edited by SK, KT, ZC, RA, AS, ED and AC. The project was led by SK. All authors commented on the manuscript. All authors have read and approved the final manuscript.
Corresponding author
Correspondence to S. V. Kutsaev.
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Kutsaev, S.V., Taletski, K., Agustsson, R. et al. Niobium quarterwave resonator with the optimized shape for quantum information systems. EPJ Quantum Technol. 7, 7 (2020). https://doi.org/10.1140/epjqt/s40507020000828
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