Hamiltonian operator, nonlinear optics

Substituting the expansion (3.197) into (3.19) and (3.29) we obtained the desired expansion of the Hamiltonian in powers with respect to Bose operators, when not only the dynamic, but also the kinetic interaction is taken into account. The new anharmonicity terms do not contain kinematic corrections. The role of this anharmonicity in the theory of third order nonlinear optical effects has been discussed in the article by Ovander (92). 

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... 

Hamiltonian (6), the operator Gln = (— I )bi b " (equation (5)) highly nonlinear in the phonon-1 appears mediated by phonons 2. It introduces multiple electron oscillations between the split levels mediated by continuous virtual absorption and emission of the phonons 1. The effect is analogous to Rabi oscillations in quantum optics due to photons [9]. Let us note that Rabi oscillations assist both the interlevel onsite and intersite electron transitions mediated by the electron transfer T. 

As a result of its dependence on the density (pa), the one-electron operator H is a pseudo-Hamiltonian, and the corresponding Schrodinger equation is nonlinear, so that its solution (for a fixed pin) must be self consistently adjusted to (e.g., by iteration) [3,53], In the case of full equilibrium, when pm = pa, both optical and inertial potentials (4>RF) depend on pa. As discussed below, the eigenstates of H (i.e., the electronically adiabatic states) are distinct from the diabatic states used to characterize the ET process (see footnote 5). 

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