arXiv:2109.07766 · Physical review. B./Physical review. B
Scattering coefficients of superconducting microwave resonators. II. System-bath approach
Qiming Chen, Matti Partanen, Florian Fesquet, Kedar Honasoge, Fabian Kronowetter, Yuki Nojiri, Michael Renger, Kirill G. Fedorov, Achim Marx, Frank Deppe, Rudolf Groß
We describe a unified quantum approach for analyzing the scattering coefficients of superconducting microwave resonators with a variety of geometries, and demonstrate its consistency with the classical approach [Q.-M. Chen et al., Phys. Rev. B 106, 214505 (2022)]. We also generalize the result to a chain of resonators with time delays, and reveal several transport properties similar to a photonic crystal and can be used to design high-quality resonators. These results form a firm theoretical ground for analyzing the scattering coefficients of an arbitrary resonator network. They set a step forward to designing and characterizing superconducting microwave resonators in a complex superconducting quantum circuit.
We describe a unified quantum approach for analyzing the scattering coefficients of superconducting microwave resonators with a variety of geometries, and demonstrate its consistency with the classical approach [Q.-M. Chen et al., Phys. Rev. B 106, 214505 (2022)]. We also generalize the result to a chain of resonators with time delays, and reveal several transport properties similar to a photonic crystal and can be used to design high-quality resonators. These results form a firm theoretical ground for analyzing the scattering coefficients of an arbitrary resonator network. They set a step forward to designing and characterizing superconducting microwave resonators in a complex superconducting quantum circuit.
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 their influences in typical experiments. We show that, similar to the transmission coefficient of a hanger-type resonator, the reflection coefficient of a necklace- or cross-type resonator also contains a so-called reference point that can be used to characterize the internal quality factor of the resonator. Our results provide a comprehensive understanding of superconducting microwave resonators from the design concepts to the characterization details.
The precise engineering of electromagnetic couplings is paramount for constructing scalable and high-fidelity superconducting quantum processors. While essential for orchestrating qubit operations, these couplings also present significant design challenges, including the mitigation of crosstalk and the management of environmental decoherence. A clear and unified theoretical framework is therefore crucial for the design, simulation, and analysis of these complex quantum circuits. This paper presents a comprehensive theoretical treatment of the fundamental electromagnetic coupling mechanisms in superconducting devices. Starting from first principles, we formulate the equations of motion and derive the input-output relations for canonical systems, including a single resonator coupled to a multi-port microwave network, interacting resonators, and coupled transmission lines. We review rigorous definitions for key parameters such as the energy decay rate (κ) and the dimensionless coupling coefficient (ζ) and connect these formalisms to practical methods of parameter extraction from electromagnetic simulations. This work provides a rigorous and pedagogical foundation for understanding and modeling linear electromagnetic interactions, serving as a vital resource for the development of advanced superconducting quantum hardware.
Elastic scattering is one of the useful methods to control the transmission behaviors of microwave photons transporting in microwave quantum networks without energy consumption. Therefore, it is of practical significance for developing microwave quantum devices and constructing multi-node microwave quantum networks. The transmission line embedded by a single Josephson junction can be described by different circuit models (series and parallel). In this work, we first theoretically analyze the transmission characteristics of microwave photons scattered by different elastic scattering models described by series or parallel embedding models, generated by a single LC loop or a nonlinear Josephson junction device, respectively. The classical microwave transport theory predicts that the series LC loop and the parallel LC loop lead to different elastic scattering behaviors of microwave photons, i.e. the series LC circuit yields the resonant reflection and the parallel LC circuit leads alternatively to the resonant transmission. Recently, the transport properties of microwave photons scattered by a Josephson junction embedded in a transmission line have been discussed, and the results suggested that the Josephson junction embedded in the transmission line can be described by a series embedding circuit, which implies the resonant reflection. We argue here that if the Josephson junction is embedded in parallel in the transmission line, the elastically scattered microwave photons should be transmitted by resonant transmission. In order to test which of the above two different embedding circuit models yielding the completely different elastic scattering behaviors, is physically correct, we then fabricate such a device, i.e. a single Joseph junction device embedded in a transmission line, and measure its elastic scattering transmission coefficient at an extremely low temperature. The results are consistent with the expected effect of the parallel embedding circuit model, but inconsistent with the behaviors predicted by the series embedding circuit model in the literature. According to the above theoretical and experimental analyses of the elastic scattering of a single Josephson junction device, we further propose a scheme to control the elastic scattering behavior of microwave photons by modulating a DC superconducting quantum interference device with a bypass current, which can be applied to the construction of a microwave quantum network based on elastic scattering node controls.
Superconducting qubits, though promising for both near-term problem-solving as well as the development of large-scale quantum computing systems, are limited in performance largely by materials-induced decoherence channels which are maximized at millikelvin temperatures, single photon powers, and microwave frequencies. Superconducting microwave resonators are ideal tools for measuring these materials losses due to their simplicity, well-understood behavior, and sensitivity to these same losses. In order to better understand and mitigate dominant qubit decoherence channels, accurate and precise characterization of superconducting microwave resonator performance is critical. In this paper, current measurement techniques for superconducting microwave resonators are discussed, including experimental set-up, error analysis, and cryogenic calibration of a vector network analyzer.
Instantons, semiclassical trajectories of quantum tunneling in imaginary time, have long been used to study thermodynamic and transport properties in a myriad of condensed matter and high-energy systems. A recent experiment in superconducting circuits [R. Kuzmin et al., Phys. Rev. Lett. 126, 197701 (2021)] provided first evidence for direct dynamical signatures of instantons (phase slips), manifested by order-unity inelastic decay probabilities for photons with which they interact, motivating the development of a scattering theory of instantons [A. Burshtein et al., Phys. Rev. Lett. 126, 137701 (2021)]. While this framework successfully predicted the measured inelastic decay rates of the photons for several experimental devices, it is valid only if the tunneling time of the instantons is much shorter than the relaxation time due to the environment in which they are embedded, and requires a closed analytical expression for the instanton trajectory. Here, we alleviate these restrictions by incorporating numerical methods that eliminate some of the previously applied approximations. Our results improve the agreement with the experimental measurements, also for devices with lower impedances and thus shorter relaxation times, without fitting parameters. This framework should be useful in many other quantum field theoretical contexts.
Superconducting microwave resonators are reliable circuits widely used for detection and as test devices for material research. A reliable determination of their external and internal quality factors is crucial for many modern applications, which either require fast measurements or operate in the single photon regime with small signal to noise ratios. Here, we use the circle fit technique with diameter correction and provide a step by step guide for implementing an algorithm for robust fitting and calibration of complex resonator scattering data in the presence of noise. The speedup and robustness of the analysis are achieved by employing an algebraic rather than an iterative fit technique for the resonance circle.