We present a numerical method to approximate the long-time asymptotic solution $rho_infty(t)$ to the Lindblad master equation for an open quantum system under the influence of an external drive. The proposed scheme uses perturbation theory to rank individual drive terms according to their dynamical relevance, and adaptively determines an effective Hamiltonian. In the constructed rotating frame, $rho_infty$ is approximated by a time-independent, nonequilibrium steady-state. This steady-state can be computed with much better numerical efficiency than asymptotic long-time evolution of the system in the lab frame. We illustrate the use of this method by simulating recent transmission measurements of the heavy-fluxonium device, for which ordinary time-dependent simulations are severely challenging due to the presence of metastable states with lifetimes of the order of milliseconds.
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