Explanation for few things in this paper about photonic crystals In this paper two weakly coupled cavities are excited with light. Their frequencies are modulated by a mechanical pulse. I have the following questions:

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*What is the Hamiltonian of this system ? I don't find it in this paper.


*The figure 1 (d), at page 3 we have the time evolution of the frequencies ("normal modes"). The thick solid lines are colored and the color value is labled $\frac{|a_1|^2}{|a_1|^2 + |a_2|^2}$. What do $a_1$ and $a_2$ on the color bar on right represent ?


*The eigenvalues of the matrix of the system (3 & 4) are the normal modes in eq.(1), page 2. The functions $a_j$ here might be those above and are called here amplitudes. What are these amplitudes exactly, definition as an operator ?


*Most importantly, How do we get these equations (3 & 4) ?
 A: This is a little bit of stretch for me, but let me give it a try.
The Hamiltonian in detail would be complex to describe. If you had no defects in the photonic crystal you could write out a Hamiltonian that would describe the band structure of the crystal. By introducing defects into the crystal, they have essentially created an energy well or defect level in the band structure analogy where the photon is confined. In that area of confinement there are specific energy levels for the trapped mode (only certain frequencies or wavelengths are confined) the Q or sharpness of the energy level would depend on the details of the surrounding crystal, but basically for most systems you can think of it being a well like a harmonic oscillator or a finite square well or infinite well. The details of the potential would matter as to the exact energy level value, or spacing to higher energy levels. But experimentally you could observe the levels without having to exactly know all of that. Or in the case of finite element analysis be able to solve numerically without writing and explicit hamiltonian out.
In the paper they create two defects separated by a short distance so there are two well with energy levels since the defects are slightly different they might have slightly different frequencies.  The goal is examine the coupling between the two defects with the surface acoustic wave being part of the coupling.
If the two defects were identical and were close enough you could think of the of an evanescent wave extending from one defect to the other, and then the energy would slosh around between the two defects.
(If it were propagating waves in a waveguide coupler and the wave guides were close together you would use coupled mode theory and see the intensity of the light go from one waveguide to another in the interaction region.)
In this case they had the idea that they could more efficiently exchange energy from defect to another, by increasing the coupling using a surface acoustic wave. Probably the best coupling is achieved at a certain surface acoustic have energy.
So they did the experiment and could probably observe the position of the light emitted (which defect it was coming from), or the frequency of the light emitted and how that varied with the frequency and phase of the surface acoustic wave. I think their $a_1$ and $a_2$ are probably just simple harmonic functions. So the ratio would correspond to how much intensity is in one well vs the other one as a function of time since they are taking the ratios of the magnitudes.
To derive 3 and 4 take a look at coupled mode theory, for generalized coupled mode equations. Probably easier to find for optical or waveguides with propagating modes, but it is general to any system like two springs that are coupled. Maybe the spring or mechanical resonator system is more applicable since they are not propagating waves. With equation 3 and 4 they are just saying there is weak coupling, and that the solution a1 for one defect and solution a2 for the other defect will exchange energy according to the coupling constant J that they measured.
