  Ionic clocks exhibit as the most promising candidates for the frequency
standards. Recent investigations show the profound advantages of interrogating
two laser beams with different frequencies in developing the frequency
standards. Here we present a scheme of a two-photon mechanism to calculate the
dynamic polarizabilities for the clock states, 6$^2$S$_{\frac{1}{2}}$ and
5$^2$D$_{\frac{3}{2}, \frac{5}{2}}$, of Ba$^+$ by employing relativistic
coupled-cluster method. We illustrate the Stark-shift cancellation between
these clock states at the two-photon magic wavelengths. These magic wavelengths
can be essential inputs to achieve better accuracy in the ionic clock
experiments. We also calculate the magic wavelengths under the single-photon
interaction to serve as the reference and for a comparative study. The
calculated single- and two-photon magic wavelengths lie in the optical region
and thus are significant for future state-of-the-art experiments. Moreover, as
an application of the two-photon polarizabilities, we investigate the impact of
these polarizabilities on the spin-mixing processes, $|0,0\rangle$
$\leftrightarrow$ $|+1,-1\rangle$ and $|0,0\rangle$ $\leftrightarrow$
$|-1,+1\rangle$, of an ultra-cold spin-1 mixture of the $^{137}$Ba$^+$ and
$^{87}$Rb atoms. We determine the protocols of selecting these spin-mixing
oscillations by changing the strength of an externally applied magnetic field
and the frequencies of the interrogating laser beams.
