Direct Experimental Evidence of the Statistical Nature of the Electron Gas in Superconducting Films

In an Nb film an alternate electrical current is partitioned at a Y-shaped obstacle into two splitted beams. The intensity-fluctuation correlation of the two beams (cross-correlation) and the intensity- fluctuation correlation of one beam (auto-correlation) are measured within a low-frequency bandwidth as a function of the incident beam intensity, at temperatures T above or below the temperature Tc of the superconductive transition. The results of these measurements reveal the statistical nature of the electron gas in the normal film and in the superconducting film. The conceptual scheme of the present experiment is a version of the Hanbury Brown and Twiss (HBT) experiment, here adopted for a gas of particles in a solid.


Introduction
The Cooper pairs, which play the fundamental role in the BCS theory of low-temperature superconductivity, are generally described, although with explicit caution, as bound pairs of electrons in a highly coherent state; the formation of few pairs encourages the formation of others in a cooperative way, a tendency quite analogous to what happens in a Bose condensation. For instance, with Feynman's wording [1]: "The two electrons which form the pair are not like a point particle… They are really spread over a considerable distance… The mean distance between pairs is relatively smaller than the size of a single pair. Several pairs are occupying the same space at the same time… We will accept, however, the idea that electrons do, in some manner or other, work in pairs… behaving more or less like particles. Since electron pairs are bosons, when there are a lot of them in a given state there is an especially large amplitude for other pairs to go to the same state. So nearly all of the pairs will be locked down at the lowest energy in exactly the same state…". Now, such a general and qualitative description of the physical nature of the Cooper pairs being universally taken as true, it would be nice to verify it by a direct experimental test. Such a test is feasible, as we discuss in the present letter describing the experiment we have performed.
First we have argued that the physical nature (bosonic, fermionic, or classical) of a beam of particles can effectively be distinguished by measuring its temporal intensity-correlation function. In this regard we may recall that Hanbury Brown and Twiss (HBT) [2] long ago measured the bunching effect of a beam of optical photons, so showing their bosonic nature; and more recently the antibunching effect of fermions was observed in a beam of free non-interacting neutrons [3]. Other fermionic and bosonic systems have also been studied [4] [5]. These results have convinced us that a HBT experiment might bring directly to light the statistical properties of a beam of Cooper pairs, and have motivated the measurements presented below in this Letter.

Method
Let ( ) n n ∆ ∆ of the occupation numbers, n a and n b , of the two partitioned beams, as it is shown in [6] for beams emitted from an equilibrium reservoir ( of bosons or fermions ).
In the presence of a steady current, provided that eV , the cross correlation for transport fluctuations of the two beams I a and I b , in small conductors and in the limit of low frequencies, has been derived theoretically as [6]: is the transmission probability of the portioned beam-i, I is the total current intensity (for an alternate current we adopt the notation eff I I = ), q is the carrier charge (q = e for fermions and q = 2e for Cooper pairs), and ∆ν is a flat frequency band-width within the low-frequency spectrum of the current fluctuations. Here a small electrical conductor is a conductor of small resistance, such that the wavelike transport of carriers preserves phase coherence over distances larger than the distance between the two contacts of the beam partition. With a = b (auto-correlation), the sign of the correlation is always positive, both for fermions and bosons, and formula (1) reduces to the standard expression for shot noise ( ) for fermions the correlation is always negative. For bosons, even though as a rule the correlation would be negative because flux conservation requires that an increase at one partition must be compensated by a decrease at the other partition, yet there are many circumstances under which the sign of the cross-correlations is positive, among them, the conductors with one of the two terminals partitioned into two leads, like our Nb samples [6]. Experimental evidence of the anticorrelation predicted by formula (1) for a gas of fermions was obtained on a beam of electrons in the quantum Hall regime [7].

Experiment and Results
Let us consider now superconducting films. Obviously, expression (1) with negative sign applies also to such films at temperatures above the critical temperature T c (Fermi gas); differently, at temperatures below T c the Cooper pairs should behave like bosons with consequent positive cross-correlation at a Y-shaped partition. Yet, the fluctuation correlations cannot be discussed without addressing the cooperative effect of such pairs analogous to the Bose condensation transition: therefore, now the incident state (i.e. the state arriving at the partition obstacle ) is a single state of energy and momentum containing a precise number of many particles (nearly all of the pairs), and the fluctuations ΔI a and ΔI b , which are a consequence of the probabilistic transmission either in partition a or b, will have opposite signs if measured at the same time. With n I as the occupation number of the incident state containing a precise number of particles, and n a , n b as the occupation numbers of the transmitted states, the cross-correlation is n n n t t ∆ ∆ = − , and the auto-correlation is ( ) ( ) n n n t t ∆ = ∆ = . This yields: Consequently, the expectation value of the correlation is still given by expression (1), with q = e. An experiment aimed at enlightening the statistical nature of the Cooper pairs will therefore measure the cross correlation a b I I ∆ ∆ above and below T c , testing the validity of such predictions. We have performed an experiment whose general scheme is represented in Figure 1. The correlator outcome is a voltage V C proportional to a b I I ∆ ∆ . The measurements have been carried out, at different temperatures above and below T c , on various specimens of Nb films of small resistance, R ≤ 25Ω at T = 293 K. The particle  source was a current source, and the two exit terminals of the Y-partition, virtually grounded, were connected to a current-voltage converter which allowed to measure separately I a and I b . The width of the pass-band filters was ∆ν = 300 Hz -600 KHz for the all measurements.
All the specimens showed up the same behaviour; and Figure 2 shows the results recorded with the 20Ωspecimen at T = 15.0 K (>T c = 9.2 K) and T = 6.5 K (<T c ). At T = 15.0 K, the negative slope and value of the cross-correlation as well as the positive slope and value of the auto-correlation, expected for Fermi particles according to formulae (1), are clearly recorded. At T = 6.5 K, the sign and the value of the measured slop are the same as those above T c , as predicted by formulae (2) for condensation of the Cooper pairs. At these low temperatures, below or little above T c , the measured slope of the correlations yields a value of 0.25 a b t t  , in good agreement with the reasonable prediction of 50% a b t t =  . At much higher temperatures, the thermal noise reduces both the cross-correlation and the auto-correlation, as it is shown in Figure 3 1 .