Dynamics of a dispersively coupled transmon qubit in the presence of a noise source embedded in the control line

We describe transmon qubit dynamics in the presence of noise introduced by an impedance-matched resistor
(50 Ω) that is embedded in the qubit control line. To obtain the time evolution, we rigorously derive the circuit
Hamiltonian of the qubit, readout resonator and resistor by describing the latter as an infinite collection of
bosonic modes through the Caldeira-Leggett model. Starting from this Jaynes-Cummings Hamiltonian with
inductive coupling to the remote bath comprised of the resistor, we consistently obtain the Lindblad master
equation for the qubit and resonator in the dispersive regime. We exploit the underlying symmetries of the master
equation to transform the Liouvillian superoperator into a block diagonal matrix. The block diagonalization
method reveals that the rate of exponential decoherence of the qubit is well-captured by the slowest decaying
eigenmode of a single block of the Liouvillian superoperator, which can be easily computed. The model captures
the often used dispersive strong limit approximation of the qubit decoherence rate being linearly proportional to
the number of thermal photons in the readout resonator but predicts remarkably better decoherence rates when
the dissipation rate of the resonator is increased beyond the dispersive strong regime. Our work provides a full
quantitative description of the contribution to the qubit decoherence rate coming from the control line in chips
that are currently employed in circuit QED laboratories, and suggests different possible ways to reduce this
source of noise.