  Open quantum systems that interact with a Markovian environment can be
described by a Lindblad master equation. The generator of time-translation is
given by a Liouvillian superoperator $\mathcal{L}$ acting on the density matrix
of the system. As the Fock space for a single bosonic mode is already
infinite-dimensional, the diagonalization of the Liouvillian has to be done on
the creation- and annihilation-superoperators, a process called `third
quantization'. We propose a method to solve the Liouvillian for quadratic
systems using a single symplectic transformation. We show that the
non-Hermitian effective Hamiltonian of the system, next to incorporating the
dynamics of the system, is a tool to analyze its symmetries. As an example, we
use the effective Hamiltonian to formulate a $\mathcal{PT}$-`symmetry' of an
open system. We describe how the inclusion of source terms allows us to obtain
the cumulant generating function for observables such as the photon current.
