Chapter 10 deals with the exact solution to the BCS problem, as obtained by analytically solving Richardson-Gaudin equations. The chapter first provides a simple derivation of these equations and shows how to solve them for arbitrary N. The explicit N dependence of the N–Cooper pair energy obtained from this resolution provides invaluable insights into the role of the Pauli exclusion principle in paired fermion systems because the reduced BCS potential does not directly act between Cooper pairs. Revelation of this subtle point makes it possible to understand (i) why Cooper pairs can strongly overlap without breaking, whereas, at large density, an exciton system dissociates into an electron-hole plasma, and (ii) why the gap energy is far larger than the average pair energy. This understanding is further supported by the Cooper pair many-body parameter, which has the same form as the ones for Wannier or Frenkel excitons do.
In a device with a superconductor coupled to two parallel quantum dots (QDs) the electrical tunability of the QD levels can be used to exploit nonclassical current correlations due to the splitting of Cooper pairs. We experimentally investigate the effect of a finite potential difference across one quantum dot on the conductance through the other completely grounded QD in a Cooper pair splitter fabricated on an InAs nanowire. We demonstrate that the nonlocal electrical transport through the device can be tuned by electrical means and that the energy dependence of the effective density of states in the QDs is relevant for the rates of Cooper pair splitting (CPS) and elastic cotunneling. Such experimental tools are necessary to understand and develop CPS-based sources of entangled electrons in solid-state devices.
We use Richardson-Gaudin exact equations to derive the ground-state energy of
N composite bosons (cobosons) interacting via a potential which acts between fermion pairs having zero center-of-mass momentum, that is, a potential similar to the reduced BCS potential used in conventional superconductivity. Through a density expansion, we show that while for two-dimensional (2D) systems, the
N-coboson correlation energy undergoes a surprising cancellation which leaves the interaction part with an $N(N\ensuremath{-}1)$ dependence only, such a cancellation does not exist in 1D, 3D, or 4D systems (which correspond to 2D parabolic traps) nor when the cobosons interact via a similar short-range potential but between pairs having an arbitrary center-of-mass momentum. This shows that the previously found cancellation which exists for the Cooper-pair correlation energy results not only from the very peculiar form of the reduced BCS potential, but also from a quite mysterious dimensionality effect, the density of states for Cooper pairs feeling the BCS potential being essentially constant, as for 2D systems.