Equation of motion approach to black-box quantization: taming the multi-mode Jaynes-Cummings model

An accurate modeling of a Josephson junction that is embedded in an arbitrary environment is of crucial importance for qubit design. We present a formalism to obtain a Lindblad master equation that describes the evolution of the system. As the qubit degrees of freedom oscillate with a well-defined frequency $\omega_q$, the environment only has to be modeled close to this frequency. Different from alternative approaches, we show that this goal can be achieved by modeling the environment with only few degrees of freedom. We treat the example of a transmon qubit coupled to a stripline resonator. We derive the parameters of a dissipative single-mode Jaynes-Cummings model starting from first principles. We show that the leading contribution of the off-resonant modes is a correlated decay process involving both the qubit and the resonator mode. In particular, our results show that the effect of the off-resonant modes in the multi-mode Jaynes-Cummings model is perturbative in $1/ \omega_q$.