Low rank approximation for the numerical simulation of high dimensional Lindblad and Riccati equations

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.