Quadrupole Shifts for the $^{171}$Yb$^+$ Ion Clocks: Experiments versus Theories

Quadrupole shifts for three prominent clock transitions, $[4f^{14} 6s] ^2S_{1/2} \rightarrow [4f^{14} 5d] ^2D_{3/2}$, $[4f^{14} 6s] ^2S_{1/2} \rightarrow [4f^{14} 5d] ^2D_{5/2}$ and $[4f^{14} 6s] ^2S_{1/2} \rightarrow [4f^{13} 6s^2] ^2F_{7/2}$, in the Yb$^+$ ion are investigated by calculating the quadrupole moments ($\Theta$s) of the $5d_{3/2,5/2}$ and $4f_{7/2}$ states using the relativistic coupled-cluster (RCC) methods. We find an order difference in the $\Theta$ value of the $4f_{7/2}$ state between our calculation and the experimental result, but our result concur with the other calculations that are carried out using different many-body methods than ours. However, our $\Theta$ value of the $5d_{3/2}$ state is in good agreement with the available experimental result and becomes more precise till date to estimate the the quadrupole shift of the $[4f^{14} 6s] ^2S_{1/2} \rightarrow [4f^{14} 5d] ^2D_{3/2}$ clock transition more accurately. To justify the accuracies in our calculations, we evaluate the hyperfine structure constants of the $6s_{1/2}$, $5d_{3/2,5/2}$ and $4f_{7/2,5/2}$ states of $^{171}$Yb$^+$ ion using the same RCC methods and compare the results with their experimental values. We also determine the lifetime of the $5d_{3/2}$ state to eradicate the scepticism on the earlier measured value as claimed by a recent experiment.