The inclusion of a double excitation subspace can make our δ E very large, forcing us to choose a very small ∆t. Note that the double-excitation subspace has a diagonal energy that is about twice that of the single-excitation subspace (when it has zero light-matter detuning). Further, the Hamiltonian has a block-diagonal structure of ĤQ , illustrated in Fig. , where the single and double excitation manifolds are not coupled to each other through Hamiltonian operators (they are connected through the dipole operator and will have transitions upon laser excitation). This observation leads us to separate the first and second excitation subspaces with their individual spectral radius. The operation Ô |Ψ⟩, via Eq. 28 is thus performed separately for the different subspaces. We separate the system Hilbert space into different excitation manifolds,