  We propose a trial wave function for the quantum Hall bilayer system of total
filling factor $\nu_T=1$ at a layer distance $d$ to magnetic length $\ell$
ratio $d/\ell=\kappa_{c1}\approx1.1$, where the lowest charged excitation is
known to have a level crossing. The wave function has two-particle correlations
which fit well with those in previous numerical studies, and can be viewed as a
Bose-Einstein condensate of free excitons formed by composite bosons and
anti-composite bosons in different layers. We show the free nature of these
excitons indicating an emergent SU(2) symmetry for the composite bosons at
$d/\ell=\kappa_{c1}$, which leads to the level crossing in low-lying charged
excitations. We further show the overlap between the trial wave function and
the ground state of a small size exact diagonalization is peaked near
$d/\ell=\kappa_{c1}$, which supports our theory.
