So let's continue with the James Cummings model.

I just want to remind you of the energy level diagram that we have for the James Cummings

model that is the interaction between an atom and a field.

So the point is that there are just two quantum numbers, which is the excitation state of

the atom and the number of photons inside the field.

And then you can draw the energy levels at least for the uncoupled system.

So we had a level sitting at zero energy in our notation, which had the atom in the ground

state and zero photons.

And then you would have the state that still has the atom in the ground state but one photon

and obviously this is removed by the photon energy.

And then you have a full ladder of such states above.

Now the other thing that can happen is that the atom is excited and so you get another

level, which just to not confuse the picture we draw to the right, which has the atom excited

and still zero photons.

Since the atomic transition frequency in general can be different from the cavity frequency,

we will have an offset here.

And this offset we call the detuning delta.

And then on top of this level there is still again the same ladder of states

with a spacing which is given by H bar omega.

So we have two harmonic oscillator ladders and the interaction now couples these levels.

In particular, if we only keep the most important couplings, then these are those where we have

the atom in the ground state with a number n of photons and a transition an excited atom

and n minus one photons.

And we learned that the coupling matrix element would be H bar g for the lowest two levels

that are coupled and H bar g square root of two for the next one and H bar g square root

of three for the one after that.

And in general H bar g times square root of n where n is the number of expectations.

So we already discussed the situation what happens if delta equals zero, so that was

the resonant case.

And then we can simply diagonalize in these two dimensional subspaces and find that the

new eigenstates are symmetric and antisymmetric superpositions.

And then we can look at the time evolution for example.

And so the specific example we looked at was what happens if we placed n photons inside

the cavity but the atom is in its ground state.

And so what happens is that we cycle between two states namely between atom and the ground

state and n photons and the other state that has atom in the excited state and n minus

one photons.

So this gives rise to a coherent oscillation as a function of time and one quantity which

you can choose to plot would be say the probability to find the atom in the ground state that

is in other words the probability to find the total system at this state.

And say if you start with this state then you will start at one and you will observe

cosine type oscillations.

And the important thing is that these oscillations have a frequency or a period that depends

on the coupling strength g.

And since the coupling is given by h by g times square root of n the coupling frequency

is that then g times square root of n or rather multiplied with a factor of two if you consider

the difference between energy levels.

And so the period here is two pi divided by two g times square root of n.

So the more photons you have the faster are these oscillations.

And then we identified them as a kind of quantized form of Rabi oscillations because when you