Nonstandard finite difference schemes for the two-level Bloch model

Overview & Synopsis

In this paper, we present splitting schemes for the two-level Bloch model. After proposing two ways to split the Bloch equation, we show that it is possible in each case to generate exact numerical solutions of the obtained sub-equations. These exact solutions involve matrix exponentials which can be expensive to compute. Here, for $2\times2$ matrices we develop equivalent formulations which reduce the computational cost. These splitting schemes are nonstandard ones and conserve all the physical properties (Hermicity, positiveness and trace) of Bloch equations. In addition, they are explicit, making effective their implementation when coupled with the Maxwell equations.