  The Exceptional Points (EPs) of non-Hermitian Hamiltonians (NHHs) are
spectral degeneracies associated with coalescing eigenvalues and eigenvectors
which are associated with remarkable dynamical properties. These EPs can be
generated experimentally in open quantum systems, evolving under a Lindblad
equation, by postselecting on trajectories that present no quantum jumps, such
that the dynamics is ruled by a NHH. Interestingly, changing the way the
information used for postselection is collected leads to different unravelings,
i.e., different set of trajectories which average to the same Lindblad
equation, but are associated with a different NHH. Here, we exploit this
mechanism to create and control EPs solely by changing the measurement we
postselect on. Our scheme is based on a realistic homodyne reading of the
emitted leaking photons with a weak-intensity laser (a process which we call
$\beta$-dyne), which we show generates a tunable NHH, that can exhibit EPs even
though the system does have any in the absence of the laser. We consider a few
illustrative examples pointing the dramatic effects that different
postselections can have on the spectral features of the NHH, paving the road
towards engineering of EPs in simple quantum systems.
