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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hopfield dielectric ： ウィキペディア英語版 Hopfield dielectric Hopfield dielectric – in quantum mechanics a model of dielectric consisting of quantum harmonic oscillators interacting with the modes of the quantum electromagnetic field. The collective interaction of the charge polarization modes with the vacuum excitations, photons leads to the perturbation of both the linear dispersion relation of photons and constant dispersion of charge waves by the avoided crossing between the two dispersion lines of polaritons. Similarly to the acoustic and the optical phonons and far from the resonance one branch is photon-like while the other charge wave-like. Mathematically the Hopfield dielectric for the one mode of excitation is equivalent to the Trojan wave packet in the harmonic approximation. The Hopfield model of the dielectric predicts the existence of eternal trapped frozen photons similar to the Hawking radiation inside the matter with the density proportional to the strength of the matter-field coupling. ==Theory== The Hamiltonian of the quantized Lorentz dielectric consisting of $N$ harmonic oscillators interacting with the quantum electromagnetic field can be written in the dipole approximation as: :$H=\backslash sum\backslash limits\_^N(a\_A+^+)\backslash cdot\; E(r\_A)+\backslash sum\_\backslash lambda\backslash sum\_k$ a_^+a_\hbar c k Assuming oscillators to be on some kind of the regular solid lattice and applying the polaritonic Fourier transform :$B\_k^+=\backslash sum\backslash limits\_^N\backslash exp(ikr\_A)a\_A^+,$ :$B\_k=\backslash sum\backslash limits\_^N\backslash exp(-ikr\_A)a\_A$ and defining projections of oscillator charge waves onto the electromagnetic field polarization directions :$B\_^+=e\_(k)\backslash cdot\; B\_k^+$ :$B\_=e\_(k)\backslash cdot\; B\_k,$ after dropping the longitudinal contributions not interacting with the electromagnetic field one may obtain the Hopfield Hamiltonian :$H=\backslash sum\_\backslash sum\_k(B\_^+B\_+)\backslash hbar\; \backslash omega\; +\backslash hbar\; cka\_^+a\_$ +\sqrt a_ +B_^+a_-B_^+a_^+ -B_a_^+ ) Because the interaction is not mixing polarizations this can be transformed to the normal form with the eigen-frequencies of two polaritonic branches: :$H=\backslash sum\_\backslash sum\_k\; \backslash left(\backslash Omega\_(k)C\_^+C\_+\backslash Omega\_(k)C\_^+C\_\backslash right)+const$ with the eigenvalue equation : One may notice that unlike in the vacuum of the electromagnetic field without matter the expectation value of the average photon number is non zero in the ground state of the polaritonic Hamiltonian $C\_|\backslash mathbf\; 0>=0$ similarly to the Hawking radiation in the neighbourhood of the black hole because of the Unruh-Davies effect. One may readily notice that the lower eigenfrequency $\backslash Omega\_$ becomes imaginary when the coupling constant becomes critical at $g>1$ which suggests that Hopfield dielectric will undergo the superradiant phase transition. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hopfield dielectric」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.