Microscopic quantum theory

In this section we develop a microscopic quantum model which accounts both for positional and orientational disorder in such a system. Using the bosonic Hamiltonian for the system, we find the structure of the eigenstates (i.e. the weights of the electronic excitations on different molecules of the disordered medium) in the intervals where the wavevector of the cavity polaritons is a good quantum number. These weights will be used in Ch. 13 in consideration of the upper polariton nonradiative decay and also for estimations of the rate transition from incoherent states to the lowest energy polariton states. 

Let at(q) and a(q) be the creation and annihilation operators of cavity photons, and B and B, be the creation and annihilation operators of an excitation on the ith molecule . We do not account for the dipole-dipole interactions between different molecules , as we consider the coupling by light to be the principal coupling mechanism. Thus the Hamiltonians of noninteracting cavity photons and molecular excitations are 

Following the usual procedure (see Ch. 4), we write the Hamiltonian of the exciton-photon interaction near the anticrossing region in the form  

To find the eigenstates of the full Hamiltonian, which is the sum of the Hamiltonians (10.32) and (10.33), we introduce new Bose operators and t for the mixed exciton-photon states as follows  

The coefficients a(q) and /% determine the fraction of light (with wavevector q) and the fraction of the excitation on the th molecule in the polaritonic states respectively. From the Hamiltonian eigenvalue problem we find that the coefficients a(q) and /% obey the following system of equations  

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THE MICROSCOPIC QUANTUM THEORY OF LOW TEMPERATURE AMORPHOUS SOLIDS... 

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