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arXiv:1607.04416 · Physical Review A

Normal modes of a superconducting transmission-line resonator with embedded lumped element circuit components

Henrik Lund Mortensen, Klaus Mølmer, Christian Kraglund Andersen

Published Nov 10, 2016• 11 citations
arXivHTMLPDF

Abstract

We present a method to identify the coupled, normal modes of a superconducting transmission line with an embedded lumped element circuit. We evaluate the effective transmission-line nonlinearities in the case of Kerr-like Josephson interactions in the circuit and in the case where the embedded circuit constitutes a qubit degree of freedom, which is Rabi coupled to the field in the transmission line. Our theory quantitatively accounts for the very high and positive Kerr nonlinearities observed in a recent experiment [M. Reh\'ak, P. Neilinger, M. Grajcar, G. Oelsner, U. H\"ubner, E. Il'ichev, and H.-G. Meyer, Appl. Phys. Lett. 104, 162604 (2014)], and we can evaluate the accomplishments of modified versions of the experimental circuit.

DOI
10.1103/physreva.94.053817

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We present a method to identify the coupled, normal modes of a superconducting transmission line with an embedded lumped element circuit. We evaluate the effective transmission-line nonlinearities in the case of Kerr-like Josephson interactions in the circuit and in the case where the embedded circuit constitutes a qubit degree of freedom, which is Rabi coupled to the field in the transmission line. Our theory quantitatively accounts for the very high and positive Kerr nonlinearities observed in a recent experiment [M. Reh'ak, P. Neilinger, M. Grajcar, G. Oelsner, U. H"ubner, E. Il'ichev, and H.-G. Meyer, Appl. Phys. Lett. 104, 162604 (2014)], and we can evaluate the accomplishments of modified versions of the experimental circuit.
We provide a general method to find the Hamiltonian of a linear circuit in the presence of a nonlinearity. Focusing on the case of a Josephson junction embedded in a transmission-line resonator, we solve for the normal modes of the system by taking into account exactly the effect of the quadratic (i.e., inductive) part of the Josephson potential. The nonlinearity is then found to lead to self and cross-Kerr effects, as well as beam-splitter-type interactions between modes. By adjusting the parameters of the circuit, the Kerr coefficient KKK can be made to reach values that are weak ($K<\ensuremath{\kappa}$), strong ($K>\ensuremath{\kappa}$), or even very strong ($K\ensuremath{\gg}\ensuremath{\kappa}$) with respect to the photon-loss rate $\ensuremath{\kappa}$. In the latter case, the resonator+junction\mathrm{resonator}+\mathrm{junction}resonator+junction circuit corresponds to an in-line version of the transmon. By replacing the single junction by a SQUID, the Kerr coefficient can be tuned in situ, allowing, for example, the fast generation of Schr\"odinger cat states of microwave light. Finally, we explore the maximal strength of qubit-resonator coupling that can be reached in this setting.
As a typical nonlinear device, Josephson junction (JJ) can be either served as a microwave device to scatter the traveling microwave or embedded in a transmission line (TL) resonator to modify the mode structure. However, two mutually exclusive circuit models have been proposed to describe the JJ embedded in a TL: one is the series embed model [see, e.g., Zueco et al., Phys. Rev. B 86, 024503 (2012)] and another is the parallel embedded one [see, e.g., Yamamoto et al., Appl. Phys. Lett. 93, 042510 (2008)]. Here, beginning with the comparison of the theoretical predictions of these two models, we fabricate the JJ-embedded TL device and measure its microwave scattering parameters at 50-mK low temperature. The experimental results support the parallel embedded model, thus rule out the series embedded one. The present work might lay the physical foundation for the designs of various microwave JJ devices, such as the microwave photonic crystals, superconducting qubits, also the Josephson parameter amplifiers, etc., by embedding the JJs into the superconducting TLs.
We present a method for the numerical analysis of superconducting quantum circuits combining lumped elements, either linear or nonlinear (i.e., Josephson junctions), and distributed coplanar waveguide (CPW) structures. CPW transmission lines and multiline couplers are directly modeled without discretizing them into lumped-element equivalents, and the circuit Hamiltonian parameters are extracted by using the energy-participation ratio (EPR) method. This approach enables fast and accurate extraction of mode frequencies, anharmonicities, cross-Kerr interactions, and Purcell decay rates without relying on full electromagnetic simulations, while naturally accounting for higher-order modes of distributed components. We have implemented the proposed method in a Python framework, QuLTRA (Quantum hybrid Lumped and TRansmission lines circuits Analyzer), which we have used to validate the approach against full electromagnetic simulations (Ansys HFSS, pyEPR), existing circuit-analysis tools (QuCAT), and designs reported in the literature. The comparisons show excellent agreement with orders-of-magnitude reductions in computational time relative to full-wave solvers. We demonstrate applications including Purcell-protected readout, multimode ultrastrong coupling, and multiplexed qubit readout, illustrating how the method can support fast and reliable early-stage circuit design.
We present the Josephson junction intersected superconducting transmission line resonator. In contrast to the Josephson parametric amplifier, Josephson bifurcation amplifier and Josephson parametric converter we consider the regime of few microwave photons. We review the derivation of eigenmode frequencies and zero point fluctuations of the nonlinear transmission line resonator and the derivation of the eigenmode Kerr nonlinearities. Remarkably these nonlinearities can reach values comparable to Transmon qubits rendering the device ideal for accessing the strongly correlated regime. This is particularly interesting for investigation of quantum many-body dynamics of interacting particles under the influence of drive and dissipation. We provide current profiles for the device modes and investigate the coupling between resonators in a network of nonlinear transmission line resonators.
In this article, we present an experimental study of microwave resonators made out of Josephson junctions. The junctions are embedded in a transmission line geometry so that they increase the inductance per length for the line. By comparing two devices with different input/output coupling strengths, we show that the coupling capacitors, however, add a significant amount to the total capacitance of the resonator. This makes the resonators with high coupling capacitance to act rather as lumped element resonators with inductance from the junctions and capacitance from the end sections. Based on a circuit analysis, we show that the input and output couplings of the resonator are limited to a maximum value of ωrZ0/4Zr, where ωr is the resonance frequency and Z0 and Zr are the characteristic impedances of the input/output lines and the resonator, respectively.
Superconducting metamaterial transmission lines implemented with lumped circuit elements can exhibit left-handed dispersion, where the group and phase velocity have opposite sign, in a frequency range relevant for superconducting artificial atoms. Forming such a metamaterial transmission line into a ring and coupling it to qubits at different points around the ring results in a multimode bus resonator with a compact footprint. Using flux-tunable qubits, we characterize and theoretically model the variation in the coupling strength between the two qubits and each of the ring-resonator modes. Although the qubits have negligible direct coupling between them, their interactions with the multimode ring resonator result in both a transverse exchange coupling and a higher-order ZZ interaction between the qubits. As we vary the detuning between the qubits and their frequency relative to the ring-resonator modes, we observe significant variations in both of these interqubit interactions, including zero crossings and changes of sign. The ability to modulate interaction terms such as the ZZ scale between zero and large values for small changes in qubit frequency provides a promising pathway for implementing entangling gates in a system capable of hosting many qubits. Published by the American Physical Society 2024