Quantized Fields in a Nonlinear Dielectric Medium: A Microscopic Approach

Theories which have been used to describe the quantized electromagnetic field interacting with a nonlinear dielectric medium are either phenomenological or derived by quantizing the macroscopic Maxwell equations. Here we take a different approach and derive a Hamiltonian describing interacting fields from from one which contains both field and matter degrees of freedom. The medium is modelled as a collection of two-level atoms, and these interact with the electromagnetic field. The atoms are grouped into effective spins and the the Holstein-Primakoff representation of the spin operators is used to expand them in inverse powers of the total spin. When the lowest order term of the interaction is combined with the free atomic and field Hamiltonians, a Hamiltonian describing a theory of noninteracting polaritons results. When higher order terms are expressed in terms of polariton operators standard nonlinear optical interactions emerge. These are then compared to the results of phenomenological and macroscopic theories. The theory is also used to derive an effective Hamiltonian describing counterpropagating modes in a nonlinear 2