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

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