At this point, we have produced three of an infinite ladder of coupled equations. As can be seen from the third equation, higher order correlations are necessary. The differential equation for the time evolution of ⟨ ⟨ a † † σ σ ⟩ ⟩ will contain expectation values of higher order products of operators, thus leading to an infinite set of coupled equations. We heuristically make the approximation that the expectation value of a product of operators is equal to the product of expectation values of the individual operators. This is akin to assuming that the operators are uncorrelated, and is a good approximation in the classical limit. It turns out that the resulting equations give the correct qualitative behavior even in the single excitation regime. Additionally, to simplify the equations we make the following replacements