  One of the most intriguing phenomena in nature is the fractional quantum Hall
effect (FQHE) observed in the half-filled second Landau level which, arising in
even-denominator filling factors, $\nu=5/2$ and $7/2$, is completely different
from other FQHEs in its origin, all of which, except for those two filling
factors, occur in odd-denominator fractions. Usually formulated in terms of a
trial wave function called the Moore-Read Pfaffian wave function, current
leading theories attribute the origin of the 5/2 FQHE to the formation of
Cooper pairs, not of electron, but of the true quasi-particle of the system
known as composite fermion. The nature of superconductivity resulting from such
Cooper pairing is particularly puzzling in the sense that it apparently
coexists with strong magnetic fields, which poses an interesting dilemma since
the Meissner effect is {\it the} most important defining property of
superconductivity. This apparent dilemma is resolved by the fact that composite
fermions do not respond to external magnetic field at even-denominator filling
factors. To provide direct evidence that it is composite fermions that actually
form the superconducting condensate, here, we develop a numerically exact
method of creating a Cooper pair of composite fermions and explicitly compute
the superconducting order parameter as a function of real space coordinates. As
results, in addition to direct evidence for superconductivity, we obtain
quantitative predictions for superconducting coherence length. Obtaining such
theoretical predictions can serve as an important step toward fault-tolerant
topological quantum computation.
