Lectures on superconductivity and noise

I forgot to write a blog entry after my previous lecture last Thursday, so today I write about two lectures, concentrating mostly on superconductivity and a bit on fluctuations. The idea of the superconductivity chapter in the book is to introduce the topic enough so that I can describe the properties of superconducting junctions. There is so much physics in them, and especially of present interest: quantum mechanics of circuits, superconducting qubits, coolers, Andreev bound states, SQUIDs and so on. Unfortunately, the fact that Chapter 5 tries to be the minimal introduction to the topic of superconductivity shows up in the treatment of the BCS theory: I should probably have written a bit more about the BCS ground state, and I should definitely have discussed the self-consistency relation in more detail. I did not have such a strong motivation for this, because the topic has been presented in many other books as well, but eventually it would not have taken very much space.

Initially I added the concept of Andreev reflection in order to be able to describe phenomena one needs to understand when dealing with small superconducting systems. Then I realized that it is a good way to provide a microscopic derivation of the Ambegaokar-Baratoff relation for the critical current via the Andreev bound states. Today in the lecture I noticed that the concept is very topical, because people try to realize Majorana fermions with their help. I am not sure if mentioning it affected the interest in the topic, but perhaps it did.

On today’s lecture I reached the discussion of the fluctuation-dissipation theorem. I will not derive it in the lectures, but I will try to make a video derivation so that I don’t have to get stuck in details in the lecture. Let’s see how that will work out.

FD theorem has two limits: that of thermal noise at low frequencies, and the vacuum fluctuations at high frequencies. There was an interesting question about the latter in the lecture: Can they be directly connected with the description of physics of vacuum fluctuations, such as the Casimir effect? I am not quite sure about the static version (Casimir force) – this is typically described merely via the energy of confinement, and the force related to the dependence of this energy on the size of the confinement. Perhaps one could do this by looking at the vacuum fluctuations in a model LC circuit. However, the dynamical Casimir effect (creation of photons from vacuum via the time-dependent confinement) can certainly be described via the vacuum fluctuations present in the second-order correlator. I think it roughly goes so that one inserts the ac driving field with frequency larger than the temperature and looks what happens to the second-order correlator at half that frequency. This should reveal the presence of the photon pairs (if I remember correctly, the photons do not have to come at half the frequency, but only the sum of their frequencies must equal that of the driving).