Okay, so let me first repeat what we said about the Cooper Pair box last time, which

is the simplest interesting circuit you can build using superconducting elements, and

it just consists of two islands, two metallic islands, that have been cooled to low temperatures,

say below 1 Kelvin, in order to make them superconducting.

And then we learned that there is not very much about superconductivity, which you have

to know to understand these things.

There are actually only two things.

One of them is that the many-particle ground state of such a superconducting island, for

a fixed particle number, is separated by an energy gap with respect to all the higher

excited states.

So the energy would have the ground state sit in here, and then it would have the continuum

of excited states up there, and there would be a finite energy gap, conventionally called

two-dial time, that is independent of the size.

So then we decided it would be a good choice as a basis to just give the number of particles

sitting on this island.

And that's the other thing you have to know, which is that the number of electrons can

only change by integer multiples of two, because electrons in the superconducting state pair

into what is called Cooper Pairs that contain two electrons.

So then in order to have any interesting dynamics at all, you need two of these metallic islands,

such that Cooper Pairs can tunnel from one to the other island.

This is the idea.

So that means if I call in the number of Cooper Pairs that have tunneled, say, from the upper

island to the lower island, with respect to the neutral state, then this defines a basis

of states.

And so we set about writing the Hamiltonian given this basis of states, and we realized

that there would be two contributions to the energy, to the Hamiltonian.

One of the contributions would tell you, given the number of Cooper Pairs sitting on the

lower island, what is the total electrostatic field energy, because we found that, say,

if a Cooper Pair is tunneled, then it would be, in total, two positive elementary charges

on the upper island and two negative elementary charges on the lower island, and these charges

would be distributed smoothly on the surface of this metallic island, and typically they

would be distributed next to the other island, such as to form something like a capacitor.

And you would try to calculate the field energy stored inside this electric field configuration,

and we found that this energy should be proportional to the square of the field and therefore proportional

to the square of the total charge that has been transmitted, and then there's a single

number that contains all the details about the geometry, and that would be the capacitance

C.

And we also asked ourselves how could one tune this energy, and one could tune it, obviously,

by applying an external electric field, because then, for example, it would be, for example,

more favorable for a Cooper Pair to sit on the lower island if it then gets closer to

the positive charges of an external gate electrode that produces this external electric field.

So we found that the overall charging energy would be given as Q minus some number, which

we call the gate charge, which just tells us what would be the preferred charge on the

lower island if it were able to change its charge continuously.

But as a matter of fact, of course, it can't change the charge continuously, because the

charge is just two times the elementary charge of an electron times the number of Cooper

Pairs that have tunneled, and that is an integer.

Okay, but anyway, this is the charging energy part of the Hamiltonian, and then there is

another part to the Hamiltonian which just describes tunneling of these Cooper Pairs.

And so in order to describe the tunneling quantum mechanically, we write down the elementary