## 3 Dynamics of Self-Localization  
Let us now proceed to the non-adiabatic quantum dynamics of self-localization, including the kinetic energy term of the phonon in Equation (1). The wavelength of visible light is quite a lot longer than the lattice constant of the crystal. This means that the wave vector of the visible photon is almost zero, because it is extremely smaller than the other wave vectors of an exciton in the first Brillouin zone of this crystal. Consequently, as already mentioned in the previous section, the initial FC type excited state (\(\equiv\mathrm{i}\,\mathrm{FC}>\)) is the Bloch wave whose total wave vector (\(\equiv\mathrm{k}\)) is almost zero, having the same translational symmetry as that of the original crystal. It is given by  
\[\left|\mathrm{FC}>\ =\ \mathrm{N}^{-\frac{1}{2}}\ \sum_{l}\mathrm{e}^{- \mathrm{i}\,\mathrm{k}\cdot l}\ \mathrm{F}_{l}^{+}\ \left|0>,\ \mathrm{k}\to 0,\right. \tag{6}\]  
\(10>\equiv\mathrm{Exciton\cdot phonon}\) true vacuum. Thus, the probability density of the exciton at each lattice site of the crystal is inversely proportional to N (volume of the crystal),  
\[<\mathrm{FC}|\mathrm{F}_{l}^{+}\mathrm{F}_{l}\ |\mathrm{FC}>\ =\ 1/\mathrm{N} \tag{7}\]  
(the unit of length is the lattice contant). Meanwhile, the self-localization mentioned above, is often misunderstood to be a sudden shrinkage of the excitation energy or the excitation wave function from the infinitely extended Bloch state \(\mathrm{i}\,\mathrm{FC}>\) to a localized one within a lattice site, say, only at the central lattice site of the crystal. This picture of sudden shrinkage, however, is completely wrong. Even if it will shrink, it will do so, not only to the central site, but also to all the other sites simultaneously and equally, with a certain transient quantum coherence among them. This is not the shrinkage any more. Before, during, and even after the self-localization, the wave function never shrinks, as shown by [PERSON] and [PERSON] [13]. They have proposed the following simple but Bloch type self-localized state (\(\equiv\)\(|\)STE \(>\)),  
\[|\text{STE}>\ =\ \text{N}^{-\frac{1}{2}}\ \sum_{I}\ \text{e}^{-\text{i}\ \text{k} \cdot I-\text{S}(\text{F}_{I}^{+}\text{F}_{I})\ \frac{\partial}{\partial\text{Q}_{I}}}\ \text{F}_{I}^{+}|\text{O}>\,\ \text{k}\to 0. \tag{8}\]  
In this Bloch type STE state, the density at each lattice site of the crystal is unchanged from Equation (7), and is still inversely proportional to N,  
\[<\text{STE}|\text{F}_{I}^{+}\text{F}_{I}\ |\text{STE}>\ =\ 1/\text{N}. \tag{9}\]  
As described in Equation (8), however, through the following displacement operator for phonons,  
\[\text{e}^{-\text{S}(\text{F}_{I}^{+}\text{F}_{I})\frac{\partial}{\partial\text {Q}_{I}}}, \tag{10}\]  
the self-localized state induces a large (S \(\gg\) 1) lattice distortion only in its site. This phonon displacement will appear and disappear according to the presence or the absence of exciton, since it is just proportional to F\({}_{I}^{+}\)F\({}_{I}\). In other words, once this large local lattice distortion occurs, the exciton is heavily dressed in phonons. Hence, even if it tries to move only to a neighboring lattice site from its original one, it has to annihilate all these phonons (larger distortion) and has to make them again at the neighboring site, newly. This phonon dressing picture was also developed by [PERSON] and his co-workers [14].