BEC of triplon in the complex quantum spin liquid BaCuSi$_2$O$_6$

We present a $^{63,65}$Cu and $^{29}$Si NMR study of the quasi-2D coupled spin 1/2 dimer compound BaCuSi$_2$O$_6$ in the magnetic field range 13-26 T and at temperatures as low as 50~mK. NMR data and neutron scattering experiments in the gapped phase reveal that below 90~K different intra-dimer exchange couplings and different gaps ($\Delta_{\rm{B}}/\Delta_{\rm{A}}$ = 1.16) exist in every second plane along the $c$-axis, in addition to a planar incommensurate (IC) modulation. $^{29}$Si spectra in the field induced magnetic ordered phase reveal that close to the quantum critical point at $H_{\rm{c1}}$ = 23.35 T the average boson density $\overline{n}$ of the Bose-Einstein condensate (BEC) is strongly modulated along the $c$-axis with a density ratio for every second plane $\overline{n}_{\rm{A}}/\overline{n}_{\rm{B}} \simeq 5$. An IC modulation of the local density is also present in each plane.