  Studying entanglement growth in quantum dynamics provides both insight into
the underlying microscopic processes and information about the complexity of
the quantum states, which is related to the efficiency of simulations on
classical computers. Recently, experiments with trapped ions, polar molecules,
and Rydberg excitations have provided new opportunities to observe dynamics
with long-range interactions. We explore nonequilibrium coherent dynamics after
a quantum quench in such systems, identifying qualitatively different behavior
as the exponent of algebraically decaying spin-spin interactions in a
transverse Ising chain is varied. Computing the build-up of bipartite
entanglement as well as mutual information between distant spins, we identify
linear growth of entanglement entropy corresponding to propagation of
quasiparticles for shorter range interactions, with the maximum rate of growth
occurring when the Hamiltonian parameters match those for the quantum phase
transition. Counter-intuitively, the growth of bipartite entanglement for
long-range interactions is only logarithmic for most regimes, i.e.,
substantially slower than for shorter range interactions. Experiments with
trapped ions allow for the realization of this system with a tunable
interaction range, and we show that the different phenomena are robust for
finite system sizes and in the presence of noise. These results can act as a
direct guide for the generation of large-scale entanglement in such
experiments, towards a regime where the entanglement growth can render existing
classical simulations inefficient.
