  The stochastic Schr\"odinger equation, of classical or quantum type, allows
to describe open quantum systems under measurement in continuous time. In this
paper we review the link between these two descriptions and we study the
properties of the output of the measurement. For simplicity we deal only with
the diffusive case. Firstly, we discuss the quantum stochastic Schr\"odinger
equation, which is based on quantum stochastic calculus, and we show how to
transform it into the classical stochastic Schr\"odinger equation by
diagonalization of suitable quantum observables, based on the isomorphism
between Fock space and Wiener space. Then, we give the a posteriori state, the
conditional system state at time $t$ given the output up to that time and we
link its evolution to the classical stochastic Schr\"odinger equation. Finally,
we study the output of the continuous measurement, which is a stochastic
process with probability distribution given by the rules of quantum mechanics.
When the output process is stationary, at least in the long run, the spectrum
of the process can be introduced and its properties studied. In particular we
show how the Heisenberg uncertainty relations give rise to characteristic
bounds on the possible spectra and we discuss how this is related to the
typical quantum phenomenon of squeezing. We use a simple quantum system, a
two-level atom stimulated by a laser, to discuss the differences between
homodyne and heterodyne detection and to explicitly show squeezing and
anti-squeezing and the Mollow triplet in the fluorescence spectrum.
