  A systematic numerical approach to approximate high dimensional Lindblad
equations is described. It is based on a deterministic rank m approximation of
the density operator, the rank m being the only parameter to adjust. From a
known initial value, this rank m approximation gives at each time-step an
estimate of the largest m eigen-values with their eigen-vectors of the density
operator. A numerical scheme is proposed. Its numerical efficiency in the case
of a rank 12 approximation is demonstrated for oscillation revivals of 50 atoms
interacting resonantly with a slightly damped coherent quantized field of 200
photons. The approach may be employed for other similar equations. We in
particularly show how to adapt such low-rank approximation for Riccati
differential equations appearing in Kalman filtering.
