  We show the emergence of a third Goldstone mode in binary condensates at the
phase-separation in quasi-1D optical lattices. We develop the coupled discrete
nonlinear Schr\"odinger equations (DNLSEs) using Hartree-Fock-Bogoliubov theory
with Popov approximation in the Bose-Hubbard model to investigate the mode
evolution at zero temperature. In particular, as the system is driven from
miscible to immiscible phase. We demonstrate that the position swapping of the
species in $^{87}$Rb-$^{85}$Rb system is accompanied by a discontinuity in the
excitation spectrum. Our results show that in quasi-1D optical lattices, the
presence of the fluctuations dramatically change the geometry of the ground
state density profile of TBEC.
