  A method for deriving accurate analytic approximations for Markovian open
quantum systems was recently introduced in [F. Lucas and K. Hornberger, Phys.
Rev. Lett. 110, 240401 (2013)]. Here, we present a detailed derivation of the
underlying non-perturbative jump expansion, which involves an adaptive
resummation to ensure optimal convergence. Applying this to a set of exemplary
master equations, we find that the resummation typically leads to convergence
within the lowest two to five orders. Besides facilitating analytic
approximations, the optimal jump expansion thus provides a numerical scheme for
the efficient simulation of open quantum systems.
