Okay, so welcome back. Today is the first time we really want to start discussing a physical system, namely quantum optics and superconducting circuits.

But before I come to that, let me just remind you again of what we said last time about the two-diver system.

So we came to the conclusion that many physical systems are well described as a two-level system if we concentrate on the ground state and the first excited state.

And actually when we discuss superconducting circuits today we will find one example which is called the Cooper pair box.

And then we just asked ourselves what if we just have a two-dimensional Hilbert space, what is the most general Hamiltonian.

We found that we could write down the most general Hamiltonian in this two-dimensional Hilbert space and we found a very nice format for doing this.

Namely we could write it as a scalar product between a vector epsilon, which is something like a magnetic field, and a vector containing as entries the so-called polymatrices.

And the polymatrices, I just repeat here, are a useful basis to describe any operator in the space of two-by-two matrices.

So sigma x is 0, 1, 1, 0, sigma y is the minus in the top, and sigma z is 1, minus 1, 0.

And then of course if you really want to have any operator you should add the identity.

So then any two-by-two operator can be with meso-linear superposition of those.

Okay, and one of the things we found was that the energy levels of this are very simply connected to this vector epsilon.

Namely they are just, there might be two of them since we are in a two-dimensional Hilbert space, plus or minus the magnitude of epsilon.

Now the fact that they are symmetric around 0 is simply due to the fact that I omitted a constant which you could add and which would simply multiply the identity, but this would be a trivial energy shift so I left it away.

Okay, so these are the energy levels.

And so you can ask yourself what if I change, say, one of the three parameters, one of the three components of epsilon, by that whole constant, the other ones.

So this is a situation which arises very often, plot the energy levels E as given here, versus say epsilon z, while you hold constant epsilon x and epsilon y.

And then in this case this E would be square root of epsilon z squared plus epsilon x squared plus potentially epsilon y squared, if that is not 0.

And so what you see is these are just hyperbolas.

So typically when you draw it you want to insert the lines first that you would get if epsilon x and epsilon y were equal to 0.

In particular this includes the so-called degeneracy point where in the absence of epsilon x and epsilon y again the energy levels would be degenerate.

And then since there are these terms what you get are these hyperbolas.

And this splitting would be related to, in this case, in the general case, square root of epsilon x squared plus epsilon y squared.

And we also reminded ourselves that this feature is called an avoided crossing in the sense that in the absence of these extra terms epsilon x and epsilon y you would get a crossing of energy levels.

But because the two levels are still coupled by these terms you get an avoided crossing.

So much for the statics, we also started looking at the dynamics.

And in order to discuss the dynamics it's very useful to introduce what we call, what everyone calls the Bloch vector.

That is simply the expectation value of the vector containing the polymatrices.

So that's the expectation value of sigma x, sigma y and sigma z.

And the point is that since you can express any operator in this two-dimensional Hilbert space as a linear superposition of the polymatrices plus the identity,

knowing the Bloch vector gives you the expectation value of any arbitrary operator.

So this is everything. This fully defines the quantum state of the system.

And we found that it obeys a very simple equation of motion, namely it precesses around a vector given by epsilon if epsilon is constant in time.

And if epsilon is not constant in time then it just precesses around a time dependent vector.

And so in the end we asked ourselves what happens if we drive the system with some external time dependent field.

And so we chose the simplest possible situation which would be say having only the z component

which just means that we take the static Hamiltonian to be already diagonal which we can always do by switching into the correct basis.

Plus some time dependent term and we chose it to be as simple as possible.

First of all, in order to get something non-trivial, the operator involved in this time dependent term should not just be equal to sigma z,

nor should it be equal to the identity because then it would be entirely trivial.

So we chose sigma x and as a homework problem you look into sine omega t multiplying this term.

We chose cosine omega t. The end result is only trivial and different.

And the pre-factor then defines the amplitude say of the oscillating electric field or the oscillating magnetic field.

And we gave it the name h bar omega so that capital omega here is the frequency and it will turn out to be what is called the lobby frequency.

So little omega is the external driving frequency.

Capital omega is related to the external driving amplitude.

And in order to solve this problem we found that it is very convenient to switch into what is called a rotating frame.

That is to factor out the precession that takes place with this frequency level omega of the driving.

And then you can again within some approximation, namely the circle rotating wave approximation,

come to a Schrödinger equation that only invokes a time independent Hamiltonian written down in the rotating frame.