  The measurement-result-conditioned evolution of a system (e.g. an atom) with
spontaneous emissions of photons is well described by the quantum trajectory
(QT) theory. In this work we generalize the associated QT theory from
infinitely wide bandwidth Markovian environment to the case of finite bandwidth
non-Markovian environment. In particular, we generalize the treatment for
arbitrary spectrum, which is not restricted by the specific Lorentzian case. We
rigorously prove a general existence of a perfect scaling behavior jointly
defined by the bandwidth of environment and the time interval between
successive photon detections. For a couple of examples, we obtain analytic
results to facilitate QT simulations based on the Monte-Carlo algorithm. For
the case where analytical result is not available, numerical scheme is proposed
for practical simulations.
