  At zero temperature and strong applied magnetic fields the ground sate of an
anisotropic antiferromagnet is a saturated paramagnet with fully aligned spins.
We study the quantum phase transition as the field is reduced below an upper
critical $H_{c2}$ and the system enters a XY-antiferromagnetic phase. Using a
bond operator representation we consider a model spin-1 Heisenberg
antiferromagnetic with single-ion anisotropy in hyper-cubic lattices under
strong magnetic fields. We show that the transition at $H_{c2}$ can be
interpreted as a Bose-Einstein condensation (BEC) of magnons. The theoretical
results are used to analyze our magnetization versus field data in the organic
compound $NiCl_2$-$4SC(NH_2)_2$ (DTN) at very low temperatures. This is the
ideal BEC system to study this transition since $H_{c2}$ is sufficiently low to
be reached with static magnetic fields (as opposed to pulsed fields). The
scaling of the magnetization as a function of field and temperature close to
$H_{c2}$ shows excellent agreement with the theoretical predictions. It allows
to obtain the quantum critical exponents and confirm the BEC nature of the
transition at $H_{c2}$.
