Background: Quadratic Bosonic Hamiltonians

We present the foundational aspects of time-independent, quadratic bosonic systems. Following a brief description of the multimode bosonic landscape we center ourselves in, we discuss the theoretical frameworks necessary for describing dynamics generated by a QBH. Of particular relevance is the intrinsic non-Hermiticity built into the Heisenberg dynamics of those systems described by QBHs without total number symmetry. Two distinct notions of stability emerge for QBHs: thermodynamical and dynamical stability. We explicitly define these notions and present equivalent characterizations in terms of the relevant dynamical matrix. Following this, we develop a Bogoliubov-de Gennes (BdG) framework that, in particular, helps reveal the underlying indefinite inner-product structure intrinsic to multimode bosonic Hilbert spaces. This will be specialized to closed (and later in Chap. 6 , open) dynamics. We conclude the chapter with a discussion of translation invariance, the key implications of Bloch’s theorem in this setting, and the basics of its generalization for diagonalization in the presence of arbitrary BCs.