By M. Szymanska
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Additional resources for Bose Condensation and Lasing in Optical Microstructures - Part I [thesis]
14) We then have a self-consistency equation for the off-diagonal part of the Green’s function G which is equal to the average coherent polarisation of the media 1 N ˜ −g ψ a†j bj = P = β −1 j ωn ,j ˜ ω˜n 2 + (ǫ˜j − µ)2 + g 2 ψ 2. 15) Both order parameters, the coherent polarisation and the coherent photon field, are connected through equation ψ = −g P . 4). A number of electronic excitations refered to later as inversion can be obtained from the diagonal elements of the Green’s function and is equal to 1 2 j (b†j bj − a†j aj ) = β −1 −2(˜ ǫj − µ) ωn ,j ˜ ω˜n 2 + (ǫ˜j − µ)2 + g 2 ψ 2.
We will show in the Chapter 3 that the decoherence processes drive this crossover. The model studied by Eastham and Littlewood is a closed system model with the dipole interaction between excitons and photons only. However, in real microcavities other interactions are also present. There are elastic collisions between excitons, exciton interactions with phonons and impurities and radiative decay to modes different that the cavity mode. Excitons are also subject to pumping and the photon field decay from the cavity.
Thus this model: • gives a very good description of tightly bound, Frenkel - type of excitons localised by disorder or bound on impurities, molecular excitons in organic materials or atoms in the solid state, 3. Self-consistent Green’s-Functions Approach 37 • gives a qualitative description within a mean-field approximation for other types of excitons like Wannier excitons or excitons propagating in a sample. The second application follows from the fact that the dipole interaction between excitons and photons is a dominant interaction at high excitation densities (large photon fields).