  Nuclear magnetic resonance (NMR) study of the high magnetic field (H) part of
the Bose-Einstein condensed (BEC) phase of the quasi-onedimensional (quasi-1D)
antiferromagnetic quantum spin-chain compound NiCl2-4SC(NH2)2 (DTN) was
performed. We precisely determined the phase boundary, Tc(H), down to 40 mK;
the critical boson density, n_c(Tc); and the absolute value of the BEC order
parameter S_perp at very low temperature (T = 0.12 K). All results are
accurately reproduced by numerical quantum Monte Carlo simulations of a
realistic three-dimensional (3D) model Hamiltonian. Approximate analytical
predictions based on the 1D Tomonaga-Luttinger liquid description are found to
be precise for Tc(H), but less so for S_perp(H), which is more sensitive to the
strength of 3D couplings, in particular close to the critical field. A
mean-field treatment, based on the Hartree-Fock-Popov description, is found to
be valid only up to n_c = 4% (T < 0.3 K), while for higher n_c boson
interactions appear to modify the density of states.
