Okay, so we have derived the James Cummings model for the case of the transmission line

resonator and now the obvious next step would be just to solve this model that describes

the interaction between an atom and a field model.

But I want to take this opportunity and just point out how general these simple models

turn out to be.

And so we will go through a list of examples of where the James Cummings model appears.

So we have learned this model describes the interaction of a single field mode and that

means for us a harmonic oscillator with an atom.

Again for us in this context this means a two-level system.

And the interaction, at least the one we discussed, was of the simple type sigma x, that is the

operator that induces transitions between the two energy eigenstates of the two-level

system, times a plus a dagger.

And if this is to be an interaction Hamiltonian you have to multiply it with an energy and

express the energy by a frequency that is typically done.

So g would be a frequency and we will see that this describes the frequency at which

a photon turns into an excitation of the atom and back again.

So the point I want to make now is, although we have derived this for a very specific case,

that is we have looked at a superconducting circuit, we have looked at the special case

of a transmission line, we have cut this transmission line into points and then made a resonator

out of it and this resonator for microwaves contains several discrete field modes and

one of those field modes can be in resonance with the Coupapay box in this case, which

is the artificial atom.

And so in the end we arrived at this interaction.

But the point I want to make is that this model is much more general and so this follows

a general trend.

If you have derived a simple model by taking your microscopic model and throwing away a

lot of things, so in our case this would be for example you throw away all the other field

modes, you throw away all the other levels of the atom and so you arrive at a very simplified

model.

And then it turns out that this very same simplified model could also arise in a large

number of different microscopic models.

And so the goal now is to show that indeed the James Cummings model is of this type.

So the reason why theoretical physicists like simple models is not only that sometimes they

may be easy to solve, in fact often times they are not so easy to solve.

For example the James Cummings model has no analytical exact solution, but still a simple

model that is one which has a simple structure, which has a simple Hamiltonian containing

only a few parameters is very likely to be general because it is the end result of a

few simplifications applied to many different microscopic models.

Okay so just to go through some of the examples that we had discussed things in the case of

the microwave transmission line resonator.

So graphically this would mean a piece of superconducting matter that can support standing

microwaves, and then these microwaves or rather the electric field of these microwaves

will interact with another small superconducting circuit that in this case we discussed would

be the Kuhn-Pair box.

Now that was one example.

You can have the same not with microwaves but in the optical domain.

So you would have an optical cavity and the typical picture to draw would be two mirrors,

two curved mirrors that support a standing optical field mount.

So these would be the wave fronts of the standing field mount.

And the purpose of having these mirrors curved is just to have a stable field mount in between