Adomian Decomposition for a Bloch-Type Model: Convergence, Numerical Performance and Limitations

Splitting schemes are widely used for the numerical approximation of Bloch-type models, in particular because they preserve the qualitative properties of the model, such as the trace, hermiticity, and positivity of the density matrix over time. Their implementation, however, relies on the evaluation of matrix exponentials, whose computation may become increasingly expensive as the dimension of the system increases. In order to avoid these computationally expensive matrix exponential > evaluations, we have investigated in this work an approach based on the Adomian decomposition method. This method provides an analytical representation of the solution in the form of a series that converges uniformly to the exact solution, with a convergence rate depending on the parameters of the quantum system and the final simulation time. The numerical experiments show that the second-order truncated expansion provides a satisfactory approximation for short times and in the absence of wave-matter interaction. By contrast, as the integration time increases or after the passage of an electric pulse, this approximation gradually loses accuracy.