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We describe a unified quantum approach for analyzing the scattering coefficients of superconducting microwave resonators with a variety of geometries. We also generalize the method to a chain of resonators with time delays, and reveal several transport properties similar to a photonic crystal. It is shown that both the quantum and classical analyses provide consistent results, and they together reveal different decay and decoherence mechanisms in a general microwave resonator. These results form a solid basis for understanding the scattering spectrums of networks of microwave resonators, and pave the way for applying superconducting microwave resonators in complex circuits.

We describe a unified classical approach for analyzing the scattering coefficients of superconducting microwave resonators with a variety of geometries. To fill the gap between experiment and theory, we also consider the influences of small circuit asymmetry and the finite length of the feedlines, and describe a procedure to correct them in typical measurement results. We show that, similar to the transmission coefficient of a hanger-type resonator, the reflection coefficient of a necklace- or bridge-type resonator does also contain a reference point which can be used to characterize the electrical properties of a microwave resonator in a single step. Our results provide a comprehensive understanding of superconducting microwave resonators from the design concepts to the characterization details.

Funding Information: The authors thank R. Assouly, R. Dassonneville, and B. Huard for useful discussions. All authors acknowledge support from QMiCS (Grant No. 820505) of the EU Flagship on Quantum Technologies. T.G.-R. and M.S. acknowledge financial support from the Basque Government QUANTEK project under the ELKARTEK program (KK-2021/00070), the Spanish Ramón y Cajal Grant No. RYC-2020-030503-I, project Grant No. PID2021-125823NA-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” and “ERDF Invest in your Future”, OpenSuperQ (820363) of the EU Flagship on Quantum Technologies, the EU FET-Open projects Quromorphic (828826) and EPIQUS (899368), and IQM Quantum Computers under the project “Generating quantum algorithms and quantum processor optimization”. M.C. and Y.O. thank the support from FCT – Fundação para a Ciência e a Tecnologia (Portugal), namely through project UIDB/04540/2020, as well as from project TheBlinQC supported by the EU H2020 QuantERA ERA-NET Cofund in QuantumTechnologies and by FCT (QuantERA/0001/2017). M.C. acknowledges support from the DP-PMI and FCT through scholarship PD/BD/135186/2017. M.R., F.F., F.D., and K.F. acknowledge support from the German Research Foundation via Germany’s Excellence Strategy (EXC-2111-390814868), Elite Network of Bavaria through program ExQM, and the German Federal Ministry of Education and Research via project QUARATE (Grant No. 13N15380) and project QuaMToMe (Grant No. 16KISQ036). This research is part of the Munich Quantum Valley, which is supported by the Bavarian state government with funds from the Hightech Agenda Bayern Plus. The research of V.S. is supported by the Basque Government through the BERC 2022–2025 program and by the Ministry of Science, Innovation, and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718. M.M. acknowledges funding from the European Research Council under Consolidator Grant No. 681311 (QUESS), from the Jane and Aatos Erkko Foundation and the Technology Industries of Finland Centennial Foundation through their Future Makers program, and from the Academy of Finland through its Centers of Excellence Program (Projects No. 312300 and No. 336810). | openaire: EC/H2020/820505/EU//QMiCS | openaire: EC/H2020/681311/EU//QUESS Microwave technology plays a central role in current wireless communications, including mobile communication and local area networks. The microwave range shows relevant advantages with respect to other frequencies in open-air transmission, such as low absorption losses and low-energy consumption, and in addition, it is the natural working frequency in superconducting quantum technologies. Entanglement distribution between separate parties is at the core of secure quantum communications. Therefore, understanding its limitations in realistic open-air settings, especially in the rather unexplored microwave regime, is crucial for transforming microwave quantum communications into a mainstream technology. Here, we investigate the feasibility of an open-air entanglement distribution scheme with microwave two-mode squeezed states. First, we study the reach of direct entanglement transmission in open air, obtaining a maximum distance of approximately 500 m with parameters feasible for state-of-the-art experiments. Subsequently, we adapt entanglement distillation and entanglement swapping protocols to microwave technology in order to reduce the environment-induced entanglement degradation. The employed entanglement distillation helps to increase quantum correlations in the short-distance low-squeezing regime by up to 46%, and the reach of entanglement increases by 14% with entanglement swapping. Importantly, we compute the fidelity of a continuous-variable quantum teleportation protocol using open-air-distributed entanglement as a resource. Finally, we adapt this machinery to explore the limitations of quantum communication between satellites, where the impact of thermal noise is substantially reduced and diffraction losses are dominant.

International audience; Narrow-band-gap IV-VI semiconductors represent a historically important class of thermoelectric materials. As one of the representative compound of this class, SnTe has been reinvestigated over the last years demonstrating its potential as a high-temperature p-type thermoelectric material. Here, we present a detailed study of the influence of very low doping levels of In, from 0.05% up to 2%, on the high-temperature transport properties of the selfcompensated Sn1.03Te compound. Our results evidence a strong impact of In on the transport properties, consistent with the presence of an In-induced resonant level (RL) in the valence bands of Sn1.03Te. This peculiar behavior is confirmed by electronic band structure calculations performed using the Korringa-Kohn-Rostoker method with the coherent potential approximation (KKR-CPA) revealing a narrow and sharp peak in the density of states (DOS) induced by the hybridization of the In s-states with the electronic states of Sn1.03Te. This distortion in the DOS results in a spectacular increase in both the thermopower and electrical resistivity at 300 K. Although the influence of the RL is somewhat lessened at higher temperatures, a significant enhancement in theZTvalues is nevertheless achieved with a peak ZTof 0.75 at 800 K which represents an increase of 35% over the values measured in Sn1.03Te. Of relevance for practical applications, the weak dependence of the RL on temperature leads to enhanced averageZTvalue.