Understanding the Cooper Pair in a Box: Questions & Intro

[0xDEADBEEF]

I am trying to understand the cooper pair in a box. They don't seem to behave like normal particles. Can someone point me to a good intro. Especially on that focuses on how they are not like a quantum well. Some questions:

1) Does the cooper pair in the box see anything of the geometric shape of the box?
2) Can I excite the cooper pair?
3) How does the phase behave, I suppose it's not a standing wave, since the currents prefer the surface?
4) What is the aequivalent for cooper pairs of the dressed state of a quantum dot in cavity qed? Do we get Bloch equations?

 

[f95toli]

No, you are right; it is quite different from a quantum well. There is a good introduction in Tinkham's book ("introduction to superconductivity").

However, note that the physics doesn't really change THAT much as you go from normal particles to Cooper pairs (as people frequently do in real experiment by suppressing the superconductivity using a magnetic field); there are of course differences but most of the "basics" are the same. The main point in both cases is simply that the charging energy is much larger than the thermal (and in the superconducting case Josephson) energy.

Let's see if I can answer some of your questions

1) No, the only condition is the charging energy.
2) Yes, if you by "excite" mean go from one band to another. This is how a charge qubit works (the splitting energy is just he Josephson energy).
3) There is no global phase as such; this is very much a "particle" phenomena (essentially number states).
4) You can get Block oscillations, but as far as I remember it is difficult to observe them experimentally in a "basic" CPB. There are a number of papers out there where this is described in detail; see e.g. Kuzmin and Haviland Phys. Rev. Lett. 67, 2890 - 2893 (1991) (which isn't a CPB as such since only one junction is involved, but the idea is the same).
There is also a more recent paper from Saclay which is on the arXiv,

http://arxiv.org/abs/cond-mat/0605061

But again, the quantronium is of course not a "basic" CPB.

 

[Minich]

I am trying to understand the cooper pair in a box.

Can you define: what is it Cooper pair?

If You can define "Cooper pair" may be You needn't ask such question.

 

[0xDEADBEEF]

A cooper pair is the bound state of two electrons in a superconductor, where the attractive force is provided for by lattice distortions (phonon exchange) or possibly spin waves. Since the two electron system has an overall spin that is integer it displays bosonic behavior and can do Bose Einstein condensation which is believed to take place at the superconducting phase transition. The energy of the bosons in the crystal lattice is completely different from a free particle so the standard particle in a box calculation cannot be used.
So what was your snippy comment trying to tell me?

 

[hiyok]

a few years ago i also strayed into your problem: a cooper pair in a box. i made some guesses that are however not verified. One such guess is that, as long as the size of the box approaches that of a free cooper pair (for metal superconductors usually hundreds of microns) it may be broken, namely, the binding potential is no longer possible to prevail over the kinetic energy increase due to displacing the wall of the box. So, in my opinion, a ccoper pair should see the wall of the box. On the other hand, if the box is very large relative to the pair, one may take a cooper pair as ordinary particles. Further, I hope to remark that, although a cooper bears zero spin, it can still not be taken as a perfect boson (pls see the BCS paper in 1957 for details). So, what if two cooper pairs are present?

 

[Minich]

A cooper pair is the bound state of two electrons in a superconductor, where the attractive force is provided for by lattice distortions (phonon exchange) or possibly spin waves. Since the two electron system has an overall spin that is integer it displays bosonic behavior and can do Bose Einstein condensation which is believed to take place at the superconducting phase transition. The energy of the bosons in the crystal lattice is completely different from a free particle so the standard particle in a box calculation cannot be used.
So what was your snippy comment trying to tell me?

Excuse, i tryed answer to your question with ideas from original BCS paper but used my own words, so:
Now i am waiting for permission to quote Bardeen, Cooper, Shriffer original paper on PFs. Wouldn't You wait a little bit?

 

[f95toli]

The BCS theory is more or less irrelevant when it comes to the physics of a CPB. None of theory for superconducting charge-based devices depends on the details of the theory for the bulk superconductor.
This is why the physics of normal single-electron transistors (made form e.g. semiconductors) is so similar to the physics of superconducting SETs (the main difference being the factor of 2 due to the pairs).

The basic physics of charge based devices is really very simple and most of it is just circuit theory with capacitors and voltage sources. The only "superconducting" part of the theory is the presence of Josephson coupling but that does not have anything as such to do with BCS theory.

Now, there are obviously some practical differences between superconductors and normal metals even in this case; e.g. the presence of a gap that makes superconducting devices less sensitive to noise and thermal fluctuations than a normal metal.
However, unless you are actually performing an experiment or a calculation that takes dissipation into consideration this won't affect the physics in any way.

 

[0xDEADBEEF]

The BCS theory is more or less irrelevant when it comes to the physics of a CPB. None of theory for superconducting charge-based devices depends on the details of the theory for the bulk superconductor.
[...]

That was very helpful. Thank you.

 

[M@2]

I am trying to understand the cooper pair in a box. They don't seem to behave like normal particles. Can someone point me to a good intro. Especially on that focuses on how they are not like a quantum well. Some questions:

1) Does the cooper pair in the box see anything of the geometric shape of the box?
2) Can I excite the cooper pair?
3) How does the phase behave, I suppose it's not a standing wave, since the currents prefer the surface?
4) What is the aequivalent for cooper pairs of the dressed state of a quantum dot in cavity qed? Do we get Bloch equations?

It is VERY interesting question!
I try to to prove it by example.

Consider two body problem in a box. Masses of bodies are M and m. They interact as quantum infinite well (ie the second box). It is very easy to show, that ground state of two bodies IN A BOX can be much less than we usually think (from standard two body solution). Especially when M>>m
You should consider superposition of wave functions:
psi=exp(iP1*x1+iP2*x2)+exp(-iP1*x1-iP2*x2)

Taking into account boundary conditions for A first BOX and for interparticle second box we can minimize energy and get surprising result :!)