Bloch equations Model Numerical issues ## Relaxation terms
All the models in the literature agree on the form of transverse relaxation terms (those who apply to coherences): On the contrary many models can be found for the longitudinal relaxation applying to populations without any comparison in the physics literature. We comapre theme from the point of view of the good modelling of spontaneous emission and trace conservation [Bid01, Bid02, BBR01]. This study shows that the best model is the Pauli master equation
which equilibrium states (
We want to preserve positiveness properties through the time evolution such as - ρ
_{nn}∈ [0,1]; - |ρ
_{nm}|^{2}≤ ρ_{nn}ρ_{mm}; - ρ is a positive (linear) operator.
To this aim we have to impose conditions on the coefficients. For the first two properties, we find necessary and sufficient conditions that are valid in all the physical contexts:
γ For the last property, we can make the (very restrictive but always verified in physical models) assumption that
γ
in which case a sufficient condition is γ |