The condensates of cold atoms at zero temperature in the tunable binary Bose-Einstein condensate system are studied with the Rabi transition between atomic hyperfine states where the system can be represented by a coupled two-field model of gapless excitations and gapped excitations. We set up the configuration of the supersonic and subsonic regimes with the acoustic horizon between them in the elongated two-component Bose-Einstein condensates, trying to mimic Hawking radiations, in particular due to the gapped excitations. The simplified step-like sound speed change is adopted for the subsonic-supersonic transition so that the model can be analytically treatable. The effective energy gap term in the dispersion relation of the gapped excitations introduces the threshold frequency $ω_\text{min}$ in the subsonic regime, below which the propagating modes do not exist. Thus, the particle spectrum of the Hawking modes significantly deviates from that of the gapless cases near the threshold frequency due to the modified grey-body factor, which vanishes as the mode frequency is below $ω_\text{min}$. The influence from the gapped excitations to the quantum entanglement of the Hawking mode and its partner of the gapless excitations is also studied according to the Peres-Horodecki-Simon (PHS) criterion. It is found that the presence of the gapped excitations will deteriorate the quantumness of the pair modes of the gapless excitations when the frequency of the pair modes in particular is around $ω\sim ω_\text{min}$. On top of that, when the coupling constant between the gapless and gapped excitations becomes large enough, the huge particle density of the gapped excitations in the small $ω$ regime will significantly disentangle the pair modes of the gapless excitations. The detailed time-dependent PHS criterion will be discussed.

38 pages, 11 figures