  We consider the numerical solution of a nonlocal partial differential
equation which models the process of collective spontaneous emission in a
two-level atomic system containing a single photon. We reformulate the problem
as an integro-differential equation for the atomic degrees of freedom, and
describe an efficient solver for the case of a Gaussian atomic density. The
problem of history dependence arising from the integral formulation is
addressed using sum-of-exponentials history compression. We demonstrate the
solver on two systems of physical interest: in the first, an initially-excited
atom decays into a photon by spontaneous emission, and in the second, a photon
pulse is used to an excite an atom, which then decays.
