  A network of optical parametric oscillators is used to simulate classical
Ising and XY spin chains. The collective nonlinear dynamics of this network,
driven by quantum noise rather than thermal fluctuations, seeks out the Ising /
XY ground state as the system transitions from below to above the lasing
threshold. We study the behavior of this "Ising machine" for three canonical
problems: a 1D ferromagnetic spin chain, a 2D square lattice, and problems
where next-nearest-neighbor couplings give rise to frustration. If the pump
turn-on time is finite, topological defects form (domain walls for the Ising
model, winding number and vortices for XY) and their density can be predicted
from a numerical model involving a linear "growth stage" and a nonlinear
"saturation stage". These predictions are compared against recent data for a
10,000-spin 1D Ising machine.
