Semiclassical propagation of coherent states using complex and real trajectories

We study the semiclassical propagation of coherent states in $d$ dimensions, which in general involves complex classical dynamics. Several simple approximations are derived that depend only on real classical trajectories, among them the thawed Gaussian approximation (TGA). Apart from the TGA, all other possibilities are able to reproduce interference and tunnelling effects, and involve propagating a set of classical initial conditions compatible with the quantum uncertainties. The accuracy of the results is verified in two dimensions for the scattering by an attractive potential, for a bound nonlinear system, for motion inside a circular billiard and for a system involving tunnelling.