  We consider the transfer functions describing the input-output relation for a
class of linear open quantum systems involving feedback with nonzero time
delays. We show how such transfer functions can be factorized into a product of
terms which are transfer functions of canonical physically realizable
components. We prove under certain conditions that this product converges, and
can be approximated on compact sets. Thus our factorization can be interpreted
as a (possibly infinite) cascade. Our result extends past work where linear
open quantum systems with a state-space realization have been shown to have a
pure cascade realization [Nurdin, H. I., Grivopoulos, S., & Petersen, I. R.
(2016). The transfer function of generic linear quantum stochastic systems has
a pure cascade realization. Automatica, 69, 324-333.]. The functions we
consider are inherently non-Markovian, which is why in our case the resulting
product may have infinitely many terms.
