  The time evolution of entanglement tracks how information propagates in
interacting quantum systems. We study entanglement entropy in CFT$_2$ with a
time-dependent Hamiltonian. We perturb by operators with time-dependent source
functions and use the replica trick to calculate higher order corrections to
entanglement entropy. At first order, we compute the correction due to a metric
perturbation in AdS$_3$/CFT$_2$ and find agreement on both sides of the
duality. Past first order, we find evidence of a universal structure of
entanglement propagation to all orders. The central feature is that
interactions entangle unentangled excitations. Entanglement propagates
according to "entanglement diagrams," proposed structures that are motivated by
accessory spacetime diagrams for real-time perturbation theory. To illustrate
the mechanisms involved, we compute higher-order corrections to free fermion
entanglement entropy. We identify an unentangled operator, one which does not
change the entanglement entropy to any order. Then, we introduce an interaction
and find it changes entanglement entropy by entangling the unentangled
excitations. The entanglement propagates in line with our conjecture. We
compute several entanglement diagrams. We provide tools to simplify the
computation of loop entanglement diagrams, which probe UV effects in
entanglement propagation in CFT and holography.
