On polariton anharmonicity in the nonlinear optical response

Nonlinear optical effects in crystals can be investigated also microscopically without using the phenomenological Maxwell equations. In the framework of this approach one has to keep, in the Hamiltonian of the crystal (formed, for example, by multilevel molecules), not only quadratic but also terms of third, fourth, etc. order with respect to the Bose amplitudes of excitons and photons. The part of the Hamiltonian which is quadratic with respect to the Bose amplitudes (see Ch. 4), can be diagonalized by making use of new Bose operators s(k) and j(k) (see eqn 4.16) so that 

In the relation (7.63) the coefficients W exhibit symmetry properties which follow from the hermiticity of the operator HU1. In particular 

The coefficients W determine the probabilities of third-order nonlinear optical processes in an unbounded crystal. An analogous expression can be derived for the coefficients determining the probabilities of fourth-order nonlinear optical processes. As already mentioned the derivation for multilevel molecules is rather complicated and has not yet been obtained. However, the simplicity of the final result, that is the simplicity of the nonlinear Hamiltonian, determines the simplicity of the calculations of nonlinear processes. Note also that a similar polariton approach can be applied for consideration of nonlinear processes in low-dimensional nanostructures (chains, quantum wells). For such structures just resonances of the pumping radiation with polaritons of low-dimensional structure and not with excitons will determine the resonances in the absorption of light as well as resonances in nonlinear processes. 

B3In the expression determining H111 cubic terms of the form and are omitted. 

These terms give only small contributions to the probability of third-order nonlinear processes as compared to (7.63). 

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