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This question is motivated by the question Has BCS Cooper pair condensate been observed in experiment? , and by our recent research on anyon superfluidity where anyons are emergent from a fermion system.

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Would the question be better stated "What observables are indicative of BCS Cooper pair condensation?" (which is how the title reads to me) or is it "Has BCS Cooper pair condensation been observed?" (which is closer to how I read the body)? –  dmckee Mar 24 '13 at 18:33
    
Can be this article be a possible answer? –  DaniH Mar 24 '13 at 19:49
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The question is "What observables are indicative of BCS Cooper pair condensation?" The body just contains the motivation of the question. –  Xiao-Gang Wen Mar 25 '13 at 0:35
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Isn't the proximity effect a delocalization of the condensate outside the superconductor ? Then, one can probe this effect via tunnelling (density of state probe).

Vortex are also an inhomogeneity of the condensate that one can easily visualise (STM, X-ray, ...).

Well, any kind of inhomogeneity can be seen as I believe. But I do not know of an experiment probing the stable, constant condensate (each time, one needs phase gradient in what I know).

It may also be possible to probe the edge currents proposed by London long ago (I'm not aware of such a detection, nor of an actual experiment).

EDIT: Ok, an other way of answering, I may have misunderstood the question. After reading this topic, maybe some better answers would be:

1) The coupling of two electrons to form a bound state, mediated by a phonon (à la Cooper / Bardeen and Schrieffer). So in principle one could generate it by phonon excitations (already done in the 70's if I remember correctly)

2) The emergence of a macroscopic quantum state from interacting electrons, and the creation of a quantum macroscopic state with all electrons sharing the same phase. So in principle one could observe the growing of the phase rigidity.

3) The emergence of a gapped excitation at the Fermi level.

But I still believe the question is not clear ... :-( Well, as it must at the beginning of organising minds :-)

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I think most of the above proposals measure the off diagonal long range order $<c_x c_{x+\delta}c^\dagger_0 c^\dagger_{\delta}>$. The real issue is that the appearance of fermion-pair off diagonal long range order may not imply the “BCS Cooper pair condensation". The state may be an exotic superconducting states. How to rule that out? –  Xiao-Gang Wen Mar 25 '13 at 0:43
    
@Xiao-GangWen Ok, good point. I was thinking that the exotic states were not real problem. They indeed fall into the given experiments detecting scheme. I was thinking they are also "Cooper pair condensation" plus extra features (higher crystal-like symmetries for instance). So you want to discard them... but why ? –  FraSchelle Mar 25 '13 at 5:19
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It is a matter of definition. I thought “BCS Cooper pair condensation” does not contain all the possible exotic SC states, which may contain all kind of emergent fractional statistics (ie with non-trivial topological orders). Certainly, if one define “BCS Cooper pair condensation” as off diagonal long range order in $< c_xc_{x+\delta}c^\dagger_0c^\dagger_\delta >$, then your proposals are valid. –  Xiao-Gang Wen Mar 26 '13 at 1:31
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Thanks for the comments. It reveals one important point. By definition, "BCS Cooper pair condensation" only describe those SC states that are describable by quadratic effective Hamiltonians $H_{eff}=\sum c_i^\dagger c_j + c_i c_j +h.c.$. Both "old-fashionned" BCS states and new "topological superconductors" are "BCS Cooper pair condensation" in this sense. But there are strongly interacting superconductors which may contain more exotic topological orders that can never be described by quadratic effective Hamiltonians. Do we have an experimental way to seperate the two kinds of SC states? –  Xiao-Gang Wen Mar 26 '13 at 13:31
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Here we only concern about the kinds of SC states. We do not concern about the phase transitions, which is a totally different issue. "What observables are indicative of BCS Cooper pair condensation?" –  Xiao-Gang Wen Mar 26 '13 at 18:29
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This answer, which in essence is not really mine, is intended to understand a bit better what the actual question is really about. I was opening the Feynman's book a few days ago and I remembered this question. Let's see if Feynman can help us :-)

Feynman, in his book Statistical physics - A set of lectures wrote a section entitled 10.8 - Real test of existence of pair states and energy gap which might be of interest for you.

To give you the idea developed there, let me copy a few sentences:

Any phenomenon in which scattering of electrons is involved will serve as a test for the existence of the pair states. Attenuation of phonons and paramagnetic relaxation are examples. […]

When the pair states proposed in the BCS theory exist, a scattering of an electron $k\uparrow$ induces an interference with the paired electron at $-k\downarrow$ […]

Let us now discuss gap experiments. [… then Feynman describes the tunnelling experiment to measure the DOS ...]

My feeling is that Feynman captures the essence of the BCS Cooper pair condensate. But it also seems to me that this is precisely this notion which is unclear.

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