  The dynamics of Markovian open quantum systems are described by Lindblad
master equations. For fermionic and bosonic systems that are quasi-free, i.e.,
with Hamiltonians that are quadratic in the ladder operators and Lindblad
operators that are linear in the ladder operators, we derive the equation of
motion for the covariance matrix. This determines the evolution of Gaussian
initial states and the steady states, which are also Gaussian. Using ladder
super-operators (a.k.a. third quantization), we show how the Liouvillian can be
transformed to a many-body Jordan normal form which also reveals the full
many-body spectrum. Extending previous work by Prosen and Seligman, we treat
fermionic and bosonic systems on equal footing with Majorana operators, shorten
and complete some derivations, also address the odd-parity sector for fermions,
give a criterion for the existence of bosonic steady states, cover
non-diagonalizable Liouvillians also for bosons, and include quadratic systems.
In extension of the quasi-free open systems, quadratic open systems comprise
additional Hermitian Lindblad operators that are quadratic in the ladder
operators. While Gaussian states may then evolve into non-Gaussian states, the
Liouvillian can still be transformed to a useful block-triangular form, and the
equations of motion for $k$-point Green's functions form a closed hierarchy.
Based on this formalism, results on criticality and dissipative phase
transitions in such models are discussed in a companion paper
[arXiv:2204.05346].
