  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 $\omega_\text{min}$ 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 $\omega_\text{min}$. 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 $\omega \sim
\omega_\text{min}$. 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 $\omega$ regime will significantly
disentangle the pair modes of the gapless excitations. The detailed
time-dependent PHS criterion will be discussed.
