Static and dynamic polarizabilities of Yb-ion]{Accurate determination of black-body radiation shift, magic and tune-out wavelengths for the $6{\mathrm{S}}_{1\mathrm{/}2}\to 5{\mathrm{D}}_{3\mathrm{/}2}$ clock transition in Yb${}^{+}$

We present precise values of the dipole polarizabilities ($\alpha$) of the ground $[4{\mathrm{f}}^{14}6\mathrm{s}]{}^2{\mathrm{S}}_{1\mathrm{/}2}$ and metastable $\rm [4f^{14} 5d] ~ ^2D_{3/2}$statesofYb$^+$, that are %vital {\bf important} in reducing systematics in the clock frequency of the $\rm[4f^{14}6s] ~ ^2S_{1/2} \rightarrow [4f^{14}5d] ~ ^2D_{3/2}$transition\mathrm{.}Thestaticvaluesof$\alpha$forthegroundand$\rm [4f^{14} 5d] ~ ^2D_{3/2}$statesareestimatedtobe$9.8(1) \times 10^{-40} ,,\rm Jm^2V^{-2}$and$17.6(5) \times 10^{-40},, \rm Jm^2V^{-2}$,respectively,whilethetensorcontributiontothe$\rm [4f^{14} 5d] ~ ^2D_{3/2}$stateas$- 12.3(3) \times 10^{-40},, \rm Jm^2V^{-2}$comparedtotheexperimentalvalue$-13.6(2.2) \times 10^{-40},,\rm Jm^2V^{-2}$. This corresponds to the differential scalar polarizability value of the above transition as $-7.8$(5)$,\times, 10^{-40},\rm Jm^2 V^{-2}$incontrasttotheavailableexperimentalvalue$-6.9$(1.4)$,\times, 10^{-40}$$\rm Jm^2V^{-2}$. This results in the black-body radiation (BBR) shift of the clock transition as $-0.44(3)$ Hz at the room temperature, which is large as compared to the previously estimated values. Using the dynamic $\alpha$ values, we report the tune-out and magic wavelengths that could be of interest to subdue %major systematics due to the Stark shifts and for constructing lattice optical clock using Yb${}^{+}$.