Finite Element Time Domain Discretization of a Semiclassical Maxwell-Schrödinger Model of a Transmon Qubit

In superconducting circuit quantum computers, classical microwave pulses are used to control and measure the state of the qubits. The design of microwave pulses for these purposes typically use simple theoretical or numerical models that do not account for the self-consistent interactions of how the presence of the transmon modifies the applied microwave field. In this work, we present the formulation and finite element time domain discretization of a semiclassical Maxwell-Schrödinger model that can self-consistently characterize the interactions between a transmon and classical microwave pulses. Our numerical results demonstrate the validity of this approach for describing the central effects related to the control and measurement of the state of a transmon. In the future, our numerical method can be used to efficiently investigate a wider range of operating regimes than existing approaches are valid for to search for more effective control and measurement protocols.