So we are trying to finish the chapter on superconducting circuits and I promised to

tell you about the general theory of superconducting circuits so that can be used to write down

the Hamiltonian for any network of Josephson junctions, capacitors, inductances, transmission

lines and transmission line resonators and it's far more general than in the context

that we used it so far.

And last time we discussed the three basic elements, the three basic non-dissipative

elements that you can have in such a circuit.

These are just capacitors, inductances and Josephson junctions.

And so for each of the elements you can write down a relation that is characteristic of

this element, for example, giving the current in terms of another variable.

So this was for example the capacitor or the capacitance C and then you know that the voltage

and the charge are related.

So we wrote down the voltage of this branch that is such an element is called a branch

would be equal to a charge that has accumulated divided by the capacitance.

And then you would have the inductance L and here rather than writing down something for

the voltage and the charge you could write down a relation between the current and the

flux.

And the flux simply was given as the time integral over the voltage.

So that is a relation which we already encountered when discussing the transmission line because

it proved to be very useful to describe the transmission line classically in terms of

these variables and then to quantize it.

And then finally, and this is new in our description, we want to describe a single Josephson turnar

junction and here again there is a parameter which turns out to be the dimensions of the

current and it is actually the critical current and we will discuss its meaning later.

But for now we just write down again a relation that looks like the one for an inductance

maybe relating the current with the corresponding branch flux only that in this case the relation

is no longer linear but it is a nonlinear relation involving the sine function of the

flux.

And the pre-factor in front of the flux to make everything dimensionless of course has

to have the dimensions of an inverse flux and it is a fundamental constant.

It is a flux quantum.

So the last thing we discussed was that you could view the Josephson turnar junction as

a nonlinear inductance and that is simply because you see that whereas here the inductance

is always 1 over L, in that case if you were to take the derivative with respect to the

flux then that derivative would define an effective inductance of the inverse of an

effective inductance but it would depend on the current value of the flux.

So now the question is how to write down say the diagram of such a circuit and the trouble

is just as for the transmission line, just as in the example of the transmission line

that you don't want to over count the degrees of freedom.

So in fact if you were to look only at the branches then if you have a network of n nodes

in the most extreme case you could try to connect each node to every other node and

so you would have on the order of n squared branches if you have n nodes.

But then again the number of degrees of freedom is really provided by the nodes not by the

branches that you can draw.

So you would want to go from these branch variables to node variables such as the voltage

on a node or as it will turn out to be more convenient the flux associated with a node

and in order to do so you have to remind yourself that there are constraints between the branch

variables that result in the fact that there are many more, that there are many less degrees

of freedom than you would think at first sight.

And these constraints are very well known, these constraints are simply Kirchhoff's