  Quantum trajectories describe the stochastic evolution of an open quantum
system conditioned on continuous monitoring of its output, such as by an ideal
photodetector. In practice an experimenter has access to an output filtered
through various electronic devices, rather than the microscopic states of the
detector. This introduces several imperfections into the measurement process,
of which only inefficiency has previously been incorporated into quantum
trajectory theory. However, all electronic devices have finite bandwidths, and
the consequent delay in conveying the output signal to the observer implies
that the evolution of the conditional state of the quantum system must be
non-Markovian. We present a general method of describing this evolution and
apply it to avalanche photodiodes (APDs) and to photoreceivers. We include the
effects of efficiency, dead time, bandwidth, electronic noise, and dark counts.
The essential idea is to treat the quantum system and classical detector
jointly, and to average over the latter to obtain the conditional quantum
state.
  The significance of our theory is that quantum trajectories for realistic
detection are necessary for sophisticated approaches to quantum feedback, and
our approach could be applied in many areas of physics.
