Characterizing the attenuation of coaxial and rectangular microwave-frequency waveguides at cryogenic temperatures
- Philipp Kurpiers^{1}Email authorView ORCID ID profile,
- Theodore Walter^{1},
- Paul Magnard^{1},
- Yves Salathe^{1} and
- Andreas Wallraff^{1}
Received: 2 March 2017
Accepted: 4 April 2017
Published: 4 May 2017
Abstract
Low-loss waveguides are required for quantum communication at distances beyond the chip-scale for any low-temperature solid-state implementation of quantum information processors. We measure and analyze the attenuation constant of commercially available microwave-frequency waveguides down to millikelvin temperatures and single photon levels. More specifically, we characterize the frequency-dependent loss of a range of coaxial and rectangular microwave waveguides down to \(0.005\ \mbox{dB}/\mbox{m}\) using a resonant-cavity technique. We study the loss tangent and relative permittivity of commonly used dielectric waveguide materials by measurements of the internal quality factors and their comparison with established loss models. The results of our characterization are relevant for accurately predicting the signal levels at the input of cryogenic devices, for reducing the loss in any detection chain, and for estimating the heat load induced by signal dissipation in cryogenic systems.
Keywords
1 Introduction
Interconverting the quantum information stored in stationary qubits to photons and faithfully transmitting them are two basic requirements of any physical implementation of quantum computation [1]. Coherent interaction of solid-state and atomic quantum devices with microwave photons has been experimentally demonstrated for quantum dot systems [2–5], individual electron spin qubits [6], ensembles of electronic spins [7], superconducting circuits [8–10] and Rydberg atoms [11–14].
In the field of circuit quantum electrodynamics, experiments show the ability to use single itinerant microwave photons [15, 16] or joint measurements [17] to generate entanglement between distant superconducting qubits [18]. In these probabilistic entanglement schemes the entanglement generation rate is inversely proportional to the signal loss between the two sites. Furthermore, entanglement can be generated deterministically by transmitting single microwave photons with symmetric temporal shape [19] which can be emitted [20, 21] and reabsorbed with high fidelity [22]. However, the fidelity of the entangled state is dependent on the signal loss for most protocols. Therefore, the ability to transmit microwave photons with low loss, which we address in this manuscript, is essential for the realization of quantum computation with solid-state and atomic quantum systems.
In addition, studying the reduction of loss of superconducting waveguides has the potential to contribute to improving the fidelity of qubit state measurements [1] by minimizing the loss of the signal between the read-out circuit and the first amplifier [23]. Knowing the loss of microwave waveguides also enables more accurate estimates of the signal levels at the input of cryogenic devices and could be used to better evaluate the heat load induced by signal dissipation.
Previous studies of the attenuation constant were performed for different types of superconducting coaxial cables down to \(4\ \mbox{K}\) by impedance matched measurements [24–31]. In those works, the attenuation constant is typically evaluated from measurements of the transmission spectrum of the waveguide, which is subsequently corrected for the attenuation in the interconnecting cables from room temperature to the cold stage in a reference measurement. In these studies lengths of the low-loss superconducting waveguides between \(20\ \mbox{m}\) and \(400\ \mbox{m}\) were used for the measurements to be dominated by the device under test.
In this paper, we study the loss of coaxial cables and rectangular waveguides using a resonant-cavity technique from which we extract attenuation constants down to \(0.005\ \mbox{dB}/\mbox{m}\) accurately between room and cryogenic temperatures at the tens of millikelvin level. By utilizing higher-order modes of these resonators we measure the frequency dependence of the attenuation for a frequency range between 3.5 and \(12.8\ \mbox{GHz}\) at cryogenic temperatures only limited by the bandwidth of our detection chain. By comparing our data to loss models capturing this frequency range we extract the loss tangent and relative permittivity of the dielectric and an effective parameter characterizing the conductor loss.
Summary of waveguide and measurement parameters
ID | CC085NbTi | CC085Nb | CC141Al | CC085Cu | WR90Alc | WR90Al | WR90CuSn |
---|---|---|---|---|---|---|---|
dim. [mm (in)] | 2.2 (0.085) | 2.2 (0.085) | 3.6 (0.141) | 2.2 (0.085) | WR90 | WR90 | WR90 |
conductor | NbTi/NbTi | Nb/Nb | Al/SPC | Cu/SPCW | coated Al | Al | Cu-Sn |
dielectric | ldPTFE | ldPTFE | ldPTFE | sPTFE | vacuum | vacuum | vacuum |
length [mm (in)] | 110 | 110 | 900 | 120 | 304.8 (12) | 304.8 (12) | 304.8 (12) |
T(BT) [mK] | 120 | 50 | 60 | 15 | 60 | 25 | 50 |
T(4K) [K] | 4.0 | 4.0 | 4.0 | 4.1 | 4.3 | 4.0 | 4.0 |
ν range [GHz] | 4.2-12.7 | 4.1-12.5 | 3.5-12.7 | 3.6-12.7 | 7.9-12.3 | 7.7-12.8 | 7.7-12.8 |
n̅(BT) | 0.1-2 | 0.3-10 | 1-2 | 1-3 | 0.2-1 | 1-4 | 1-3 |
n̅(4K) | 0.4-5 | 0.2-4 | 8-16 | 4-9 | 2-10 | 3-12 | 5-20 |
2 Experimental setup
At room temperature (RT) we use a vector network analyzer (VNA) and a through-open-short-match (TOSM) calibration to set the measurement reference plane to the input of the coupling ports of the waveguide according to the schematic presented in Figure 1(c) and adjust the input and output coupling to be approximately equal. For measurements at cryogenic temperatures the microwave signal propagates through a chain of attenuators of \(20\ \mbox{dB}\) each at the 4 K, the cold plate and the base temperature stages before entering the waveguide (Figure 1(d)). The output signal is routed through an isolator with a frequency range of 4-\(12\ \mbox{GHz}\) and an isolation larger than \(20\ \mbox{dB}\), a high-electron-mobility transistor (HEMT) amplifier with a bandwidth of 1-\(12\ \mbox{GHz}\), a gain of \(40\ \mbox{dB}\) and a noise temperature of \(5\ \mbox{K}\), as specified by the manufacturer. After room temperature amplification and demodulation, the signal is digitized and the amplitude is averaged using a field programmable gate array (FPGA) with a custom firmware.
The waveguides are characterized at a nominal temperature of \(4\ \mbox{K}\) (4K) using the pulse tube cooler of a cryogen-free dilution refrigerator system in which also the millikelvin temperature (BT) measurements are performed. We thermally anchor the waveguides to the base plate of the cryostat using OFHC copper braids and clamps. The actual waveguide temperatures are extracted in a measurement of the resistance of a calibrated ruthenium oxide (RuO) sensor mounted at the center of the coaxial cables or at the end of the rectangular waveguides and are listed in Table 1.
For the measurements at base temperature BT (\({\sim}10\ \mbox{mK}\)) it proved essential to carefully anchor all superconducting waveguide elements at multiple points to assure best possible thermalization. The measured temperatures listed in Table 1 are found to be significantly higher than the BT specified above. We attribute the incomplete thermalization of the superconducting waveguides to the small thermal conductivity of the employed materials below their critical temperature \(T_{\mathrm{c}}\) [36]. We note that when using only a minimal set of anchoring points, we observed even higher temperatures.
3 Measurements of the attenuation constant
3.1 Illustration of the measurement technique
3.2 Analysis of coaxial lines
The surface resistance of a normal conductor \(R_{\mathrm{s}}^{\mathrm{nc}}(\nu )\) is proportional to \(\sqrt{\nu}\) and to the direct current (dc) conductivity \(1/\sqrt{\sigma}\) [38]. The theory of the high-frequency dissipation in superconductors [39–42] shows a quadratic dependence of \(R_{\mathrm {s}}^{\mathrm{sc}}(\nu)\).
Summary of the extracted relative permittivities \(\pmb{\epsilon _{\mathrm{r}}}\) and loss tangents tan δ of the tested dielectric materials
T/parameter | Micro-Coax LD PTFE | Keycom ldPTFE | Micro-Coax sPTFE |
---|---|---|---|
\(\epsilon_{\mathrm{r}}\) | |||
RT | 1.70 ± 0.01 | 1.72 ± 0.06 | 1.98 ± 0.07 |
4K | 1.70 ± 0.01 | 1.72 ± 0.06 | 2.01 ± 0.07 |
BT | 1.70 ± 0.01 | 1.72 ± 0.06 | 2.01 ± 0.07 |
tanδ [×10^{−5}] | |||
RT | 9 ± 1 | 25 ± 4 | |
4K | 8.5 ± 0.2 | 0.8 ± 0.2 | 22 ± 2 |
BT | 6.6 ± 0.2 | 0.7 ± 0.1 | 19 ± 4 |
We extract the relative permittivities \(\epsilon_{\mathrm{r}}\) from Eq. (2) and the loss tangent tanδ from fitting Eq. (5) to the measured \(Q_{\mathrm{i}}(\nu)\) for each coaxial cable (Table 2). The values for the Micro-Coax ldPTFE are determined from CC141Al measurements, for the Micro-Coax sPTFE from CC085Cu and for the Keycom ldPTFE from CC085NbTi and CC085Nb measurements. We extract tanδ of the Keycom ldPTFE from the fit assuming \(R_{\mathrm{s}}^{\mathrm{sc}}(\nu) \propto\nu^{\mathrm{p}}\) at 4K and BT. Due to the low internal quality factors \(Q_{\mathrm{i}}<100\) limited by the low RT conductivity (measured \(\sigma\approx8\times 10^{5}\ \mbox{S}/\mbox{m}\) for NbTi and Nb) we are unable to extract these quantities at RT. At cryogenic temperature we observe that tanδ of the ldPTFE of Micro-Coax and Keycom differ by a factor of ∼10. \(\epsilon_{\mathrm{r}}\) is found to be ∼1.7 for ldPTFE and ∼2 for sPTFE and is nearly temperature independent.
3.3 Analysis of rectangular waveguides
4 Conclusions
We have presented measurements of the attenuation constant of commonly used, commercially available low-loss coaxial cables and rectangular waveguides down to millikelvin temperatures in a frequency range between 3.5 and \(12.8\ \mbox{GHz}\). We have performed measurements of attenuations constants down to \(0.005\ \mbox{dB}/\mbox{m}\) using a resonant-cavity technique at cryogenic temperatures. In this method, we employ weak couplings to the waveguides resulting in resonant standing waves and measure their quality factors. We have extracted the loss tangent and relative permittivity of different dielectric materials by comparing our measurement results to existing loss models. The frequency dependence of the internal quality factors of the normal conducting waveguides are well described by the loss model, while the tested CC085NbTi and CC085Nb show small deviations from the predictions for the high-frequency dissipation in superconductors [39, 40]. We have also studied the power dependence of the attenuation constant which we find to be independent of the input power in a range from −140 to \(-80\ \mbox{dBm}\) (see Appendix 5).
Our results indicate that transmitting signals on a single photon level is feasible within laboratory distances, e.g. 95% of the signal can be transmitted over distances of \(28\ \mbox{m}\) using commercial rectangular waveguides or \(8\ \mbox{m}\) using coaxial cables. Furthermore, we find no significant dependence of the attenuation constants on the ambient residual magnetic fields in measurements performed with and without cryoperm magnetic shielding (see Appendix 6).
Comparing our results to recent measurements of high quality 3D cavities [45] with quality factors up to \(7\times 10^{7}\) indicate that improving the surface treatment of rectangular waveguides may lead to a even lower attenuation constant of rectangular waveguides down to \({\sim}10^{-4}\ \mbox{dB}/\mbox{m}\). Furthermore, our measurements show that the loss tangent tanδ strongly dependents on the PTFE composite where \(\operatorname{tan}\ \delta{\sim} 2\times10^{-6}\) of PTFE have been reported at cryogenic temperatures [46, 47] about a factor of 4 lower than those measured here. This suggests that the loss of superconducting coaxial cables may also be further reduced.
Declarations
Acknowledgements
The authors thank Tobias Frey, Silvia Ruffieux and Maud Barthélemy for their contributions to the measurements, Oscar Akerlund for his support with the numerical integration, Christopher Eichler for discussing the manuscript and Keycom Corporation for providing superconducting coaxial cables. This work is supported by the European Research Council (ERC) through the “Superconducting Quantum Networks” (SuperQuNet) project, by National Centre of Competence in Research “Quantum Science and Technology” (NCCR QSIT), a research instrument of the Swiss National Science Foundation (SNSF), by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office grant W911NF-16-1-0071 and by ETH Zurich. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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