Martin Teichmann Atomes de lithium-6 ultra froids dans la ... - TEL

CHAPTER 2. THEORY
2.4 BCS theory
When Bardeen, Cooper and Schrieffer (BCS) [9] presented their theory
of superconductivity back in 1957, it was an important breakthrough
for the microscopic description of superconductors. As a core of their
theory they proposed a many-body ground state that incorporates pairing
between electrons. They showed that it is this pairing which is the origin
of superconductivity. Later, Eagles [59] and Leggett [60] realized that the
BCS ground state is more general and can be used to describe more than
just superconductivity. They realized that at large interaction strengths
(compared to the Fermi energy), the diameter of a pair will become small
and the pairs can be seen as independent bosons which may condensate
into a Bose-Einstein condensate. As an important detail, this means that
the condensation temperature and the pair binding temperature will not
be the same as is the case in normal BCS theory. This also means that
there is an intermediate region, a crossover between BCS-type pairing
and Bose-Einstein condensation.
Large attention has been drawn on this subject after Randeria et
al. suggested that such a crossover could be used to describe high
critical temperature Tc superconductors [61]. They extended an older
theory by Nozières and Schmitt-Rink, which treated the crossover for
non-zero temperatures [62]. Shortly after Bose-Einstein condensation in
cold atomic gases had been achieved, experimenters began working on
obtaining a superfluid state also in fermionic gases, bearing in mind that
the only ingredient necessary for BCS theory is an attractive interaction,
which is easily achievable by means of a Feshbach resonance. Reaching
low temperatures, however, turned out to be a bigger obstacle. Later,
Holland et al. [10], following the theory by Randeria [61], predicted that
the transition temperature in the crossover may become sufficiently
high to be achieved in cold atoms, which created excitement in the
community and hence the interest in this area.
2.4.1 The BCS ground state
The breakthrough of BCS was the realization that superconductivity can
only be described using a theory that takes pair correlations between
particles into account. Now we want to show the Hamiltonian they used
to describe such a system and the ground state wavefunction they found.
We have fermions of two different spin states, which we call ↑ and ↓.
They do not need to be spin-up and spin-down, any two different spin
states will do. Atoms in these spin states interact via a potential U. In
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