Background: Quadratic Bosonic Lindbladians

This chapter provides an extension of Chap. 2 to the Markovian setting. We first introduce the basic properties of quantum Markovian dynamical semigroups and, in particular, cover the basic notions of conservation laws, symmetries, and the breakdown of Noether’s theorem. Then, we introduce the specific class of Markovian systems we are interested in, i.e., those systems whose Lindblad generator is a QBL. Such generators are defined by a QBH, paired with Lindblad dissipators that are linear in creation and annihilation operators. In particular, we highlight the key differences with the Hamiltonian formalism used up until this point. To this end, we emphasize the changes to the relevant equations of motion for linear and quadratic observables, and discuss the appropriate extensions of the stability notions in QBHs and their criteria. Of particular interest are the steady states, which we characterize in detail. The chapter concludes with a simple single-mode example that exemplifies the key parts of the formalism, in addition to an account of the implications of bulk-translation invariance for these systems.