Derivation of a Semiclassical Model for a Transmon Capacitively Coupled to a Transmission Line

Accurately controlling the state of a transmon qubit via applied microwave pulses is essential in successfully running quantum computers that use these qubits. Currently, control pulses are designed using classical simulations of time-dependent quantum systems that neglect any back-action of the transmon on the transmission lines that deliver the microwaves. In this work, we develop a semiclassical model for a transmon capacitively coupled to a transmission line that treats the transmon quantum mechanically and the transmission line classically. This model can be used in the future to study the detailed interactions of microwave control pulses with transmon qubits to better optimize the control pulses. We show results from a classical model of this system to highlight that the back-action of the transmon may not be negligible for common experimental parameters, emphasizing the need for more rigorous models as experimental parameters are changed to accommodate scaling in device density.