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Results from higher order mean field calculations of light interacting withatom arrays are presented for calculations of one- and two-time expectationvalues. The atoms are approximated as two-levels and are fixed in space.Calculations were performed for mean field approximations that include theexpectation value of one operator (mean field), two operators (mean field-2),and three operators (mean field-3). For the one-time expectation values, weexamined three different situations to understand the convergence withincreasing order of mean field and some limitations of higher order mean fieldapproximations. As a representation of a two-time expectation value, wecalculated the $g^{(2)}(\tau )$ for a line of atoms illuminated by aperpendicular plane wave at several emission angles and two differentintensities. For many cases, the mean field-2 will be sufficiently accurate toquantitatively predict the response of the atoms as measured by one-timeexpectation values. However, the mean field-3 approximation will often beneeded for two-time expectation values.

Beyond lowest order mean field theory for light interacting with atom arrays