  We simulate the collective dynamics in spin lattices with long range
interactions and collective decay in one, two and three dimensions. Starting
from a dynamical mean-field approach derived by local factorization of the
density operator we improve the numerical approximation of the full master
equation by including pair correlations at any distance. This truncations
enable us to drastically increase the number of spins in our numerical
simulations from about ten spins in case of the full quantum model to several
ten-thousands in the mean-field approximation and a few hundreds if pair
correlations are included. Extensive numerical tests help us identify
interaction strengths and geometric configurations where these approximations
perform well and allow us to state fairly simple error estimates. By simulating
systems of increasing size we show that in one and two dimensions we can
include as many spins as needed to capture the properties of infinite size
systems with high accuracy, while in 3D the method does not converge to desired
accuracy within the system sizes we can currently implement. Our approach is
well suited to give error estimates of magic wavelength optical lattices for
atomic clock applications and corresponding super radiant lasers.
