Fock-space relativistic coupled-cluster study of systematic shifts and fine-structure constant sensitivity in {\rm Cf${}^{17+}$}, {\rm Cm${}^{15+}$} and {\rm Bk${}^{16+}$} highly charged ions

Highly charged ions (HCIs) are among the most promising candidates proposed for the development of next-generation optical clocks and the precision tests of fundamental physics. In this work, we investigate the clock-relevant properties of $\mathrm{C}{\mathrm{f}}^{17+}$, $\mathrm{C}{\mathrm{m}}^{15+}$ and $\mathrm{B}{\mathrm{k}}^{16+}$ HCIs using Fock-space relativistic coupled-cluster (FSRCC) theory. The corrections from Breit interaction, QED effects and perturbative triples are incorporated to improve the accuracy of results further. We compute excitation energies, transition amplitudes, lifetimes, hyperfine structure constants, electric quadrupole moments, and electric dipole polarizabilities for low-lying states of these ions. These properties are further used to characterize important clock transitions and estimate key systematic frequency shifts. Our calculations reveal that the proposed clock transitions exhibit strongly suppressed systematic frequency shifts. We also compute the sensitivity coefficients, $q$, and enhancement factors, $K_{\alpha }$, for the proposed clock transition as a probe to variation in the fine-structure constant, $\alpha$. Our calculations predict exceptionally large $K_{\alpha }$ of $-55.3$ and $57.5$ for $5f_{5\mathrm{/}2}\to 6p_{1\mathrm{/}2}$ transition in $\mathrm{C}{\mathrm{f}}^{17+}$ and $6p_{1\mathrm{/}2}\to 5f_{5\mathrm{/}2}$ transition in $\mathrm{B}{\mathrm{k}}^{16+}$, respectively. These giant $K_{\alpha }$s place the proposed transitions among the most sensitive optical probes for the laboratory searches for $\alpha$ variation. Further, a comparative analysis of proposed transitions in $\mathrm{C}{\mathrm{m}}^{15+}$ and $\mathrm{B}{\mathrm{k}}^{16+}$ shows that, $5f_{5\mathrm{/}2}\to 5f_{7\mathrm{/}2}$ transition possess smaller systematic shifts and is therefore attractive for precision frequency metrology, whereas $6p_{1\mathrm{/}2}\to 5f_{5\mathrm{/}2}$ transition exhibit significantly enhanced sensitivity to $\alpha$ variation.