  Models based on non-Hermitian Hamiltonians can exhibit a range of surprising
and potentially useful phenomena. Physical realizations typically involve
couplings to sources of incoherent gain and loss; this is problematic in
quantum settings, because of the unavoidable fluctuations associated with this
dissipation. Here, we present several routes for obtaining unconditional
non-Hermitian dynamics in non-dissipative quantum systems. We exploit the fact
that quadratic bosonic Hamiltonians that do not conserve particle number give
rise to non-Hermitian dynamical matrices. We discuss the nature of these
mappings from non-Hermitian to Hermitian Hamiltonians, and explore applications
to quantum sensing, entanglement dynamics and topological band theory. The
systems we discuss could be realized in a variety of photonic and phononic
platforms using the ubiquitous resource of parametric driving.
