Diffusive entanglement growth in a monitored harmonic chain

We study entanglement growth in a harmonic oscillator chain subjected to the weak measurement of observables which have been smeared-out over a length scale R. We find that entanglement grows diffusively (S t^1/2) for a large class of initial Gaussian states provided the measurement scale R is sufficiently large. At late times t O (L^2) the entropy relaxes towards an area-law value which we compute exactly. We propose a modified quasi-particle picture which accounts for all of these main features and agrees quantitatively well with our essentially exact numerical results. The quasiparticles are associated with the modes of a non-Hermitian effective Hamiltonian. At small wave-vector k, the quasiparticles transport entropy with a finite velocity, but have a lifetime scaling as 1/k²; the concurrence of these two conditions leads directly to the observed t^1/2 growth.