Deconstructing Effective Non-Hermitian Dynamics in Quadratic Bosonic Hamiltonians

Unlike their fermionic counterparts, the dynamics of Hermitian, bosonic, quadratic Hamiltonians are governed by a generally non-Hermitian Bogoliubov de-Gennes Hamiltonian. This effective non-Hermiticity gives rise to two distinct dynamical phases: one with bounded evolution of observables in time, and one without. We elucidate the physical manifestations of the transitions between these two dynamical phases. We show how a generalized notion of PT symmetry may be used to classify the mechanisms by which this transition can occur. By combining this understanding with tools from Krein stability theory, we derive an indicator of dynamical phase boundaries inspired by the notion of phase rigidity in non-Hermitian quantum systems. As an example, we fully characterize the dynamical phase diagram of a bosonic analogue to the Kitaev-Majorana chain under a wide class of boundary conditions, and further establish a connection between phase-dependent transport properties and the onset of instability. We discuss potential applications of our techniques to quadratic Lindblad dynamics.