Analytical Quantum Full-Wave Analysis of Photon Transport Through a 3-D Rectangular Waveguide Cavity Containing a Transmon Qubit

Superconducting quantum circuits are a leading hardware approach for implementing many emerging quantum information processing applications. Despite the promise of these devices, their complexity makes designing them particularly difficult due to the limited number of efficient and high-fidelity numerical modeling tools available for these purposes. There is growing interest in creating such full-wave modeling tools, but one challenge in accomplishing this is the inherent difficulties in validating these emerging numerical methods. To aid in addressing this gap, in our prior work we created the first-ever analytical quantum full-wave solutions for a superconducting circuit quantum device that could be used for validating future quantum full-wave numerical methods (S. Moon, D.-Y. Na, and T. E. Roth, “Analytical Quantum Full-Wave Solutions for a 3-D Circuit Quantum Electrodynamics System,” IEEE Transactions on Antennas and Propagation, vol. 72, no. 8, pp. 6702–6716, 2024). In this prior work, we specifically considered the case of photons scattering through an empty rectangular waveguide cavity fed by coaxial ports, as well as a photon in a closed rectangular waveguide cavity interacting with transmon qubits. In this current work, we build on these prior analytical quantum full-wave solutions to consider the case of photon scattering through a rectangular waveguide cavity that contains a single transmon qubit. The presence of a transmon qubit greatly alters the photon transport properties through the cavity in comparison to those of an empty cavity, which can provide useful signatures for validating future quantum full-wave numerical methods. To aid in creating our analytical solutions, we build on the input-output formalism (E. Rephaeli and S. Fan, “Few-Photon Single-Atom Cavity QED With Input-Output Formalism in Fock Space,” IEEE Journal of Selected Topics in Quantum Electronics, vol. 18, pp. 1754-1762, 2012) by incorporating the multi-level nature of trans-mon qubits that is distinct from the ideal two-level system behavior considered previously. To validate the full-wave aspects of our analytical solution, we compare to results using the finite element method to compute the eigenmodes of the 3-D system. We then use our quantum solutions to study the photon transport properties through the qubit-cavity system when the qubit and cavity coupling strengths are varied between weak and strong coupling regimes. In the future, our analytical solutions can find use in validating quantum full-wave numerical methods, helping advance this important field.