Liouville Operator Theory Of Nonlinear Optical Susceptibilities

Expressions for the n-th order nonlinear susceptibility, x
⁽ⁿ⁾
, are given using Liouville operators. When damping is included, the n-th order susceptibility operator is expressed as a product of propagators, the output polarization dipole operator, and n input dipole Liouville operators operating on the unperturbed density operator with a sum performed over the input frequency permutations. When damping can be neglected, the susceptibility can be reexpressed as a product of propagators, the n+1 dipole operators, and the unperturbed density operator with the sum performed over the permutations of the input frequencies and the output frequency.