Solving quasi-free and quadratic Lindblad master equations for open fermionic and bosonic systems

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