  In order to understand the nature of the two-dimensional Bose-Einstein
condensed (BEC) phase in BaCuSi2O6, we performed detailed 63Cu and 29Si NMR
above the critical magnetic field, Hc1= 23.4 T. The two different alternating
layers present in the system have very different local magnetizations close to
Hc1; one is very weak, and its size and field dependence are highly sensitive
to the nature of inter-layer coupling. Its precise value could only be
determined by "on-site" 63Cu NMR, and the data are fully reproduced by a model
of interacting hard-core bosons in which the perfect frustration associated to
tetragonal symmetry is slightly lifted, leading to the conclusion that the
population of the less populated layers is not fully incoherent but must be
partially condensed.
