Entanglement dynamics across dynamical stability transitions in monitored free bosons

Understanding the interplay between unitary dynamics, which tends to generate entanglement, and measurement, which tends to suppress it, has recently garnered considerable attention in both the quantum information and condensed-matter communities. Most studies to date have focused on fermionic or spin systems with finite-dimensional Hilbert spaces. Bosonic Hamiltonians, which feature an infinite-dimensional Hilbert space, have typically been explored only within parameter regimes that do not fully exploit the richness of their unbounded nature -- by assuming they are thermodynamically and dynamically stable. If local observables are continuously monitored, the entanglement entropy is then conjectured to obey an area law. By considering arbitrary quadratic bosonic Hamiltonians, we uncover a qualitative difference in the spatial correlations and entanglement production depending on whether the Hamiltonian is dynamically stable, metastable, or unstable -- despite the existence of a unique, stabilizing, steady-state solution to the full non-linear evolution of the covariance matrix in each case. In particular, we identify a sharp transition in the convergence rate to the area law, as a function of system size, between these dynamical stability phases. Crucially, we also observe that in the dynamically unstable and metastable regimes, the strength of the measurement governs the rate of saturation to the area law.