Quantizing the propagated field through a dielectric including general class of permutation symmetries for nonlinear susceptibility tensors

In this paper, we present a quantum theory for field propagation through a three-dimensional dielectric, when the frequency permutations of nonlinear susceptibility tensors should not be ignored. This treatment is important for the processes in which multiple frequencies are generated. The Hamiltonian of our system is obtained by introducing a unique Lagrangian and making use of the dual potential. This system is quantized by imposing the Dirac commutation relations. Finally, appropriate creation and annihilation operators are defined to derive the nonlinear Hamiltonian operator under frequency permutations of the nonlinear susceptibility tensors. As an example, our simulation results show the impact of frequency permutations on pulse propagation along a one-dimensional simple waveguide.