Exact parameterization of qubit dynamics with coupling to a resistive element in the drive line

Understanding the coupling between a quantum circuit and its environment is crucial for qubit development. In this work we derive a complete description of the qubit dynamics, where the solution is given in terms of the experimental circuit parameters. We start from the circuit model for the qubit plus environment, the latter realized by a variable temperature attenuator [1] acting as a thermal bath with Johnson-Nyquist noise. We apply methods of circuit quantum electrodynamics for the qubit-environment combination, following Ref. 2, and obtain a Hamiltonian from which we derive a master equation in the open quantum systems formalism. This approach gives insight into how each circuit element and the drive line temperature affect the qubit dynamics. As proof of concept, we solve the master equation for the specific case of a fixed-frequency, dispersive transmon, recovering the expected result of decohering qubit without dissipation.

[1] S. Simbierowicz et al., Rev. Sci. Instrum. 92, 034708 (2021). 

[2] M. Cattaneo and G. Paraoanu, Adv. Quantum Technol. 2100054 (2021).