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Multipolar Black Body Radiation Shifts for the Single Ion Clocks

Appraising the projected $10^{-18}$ fractional uncertainty in the optical frequency standards using singly ionized ions, we estimate the black-body radiation (BBR) shifts due to the magnetic dipole (M1) and electric quadrupole (E2) multipoles of the magnetic and electric fields, respectively. Multipolar scalar polarizabilities are determined for the singly ionized calcium (Ca$^+$) and strontium (Sr$^+$) ions using the relativistic coupled-cluster method; though the theory can be exercised for any single ion clock proposal. The expected energy shifts for the respective clock transitions are estimated to be $4.38(3) \times 10^{-4}$ Hz for Ca$^+$ and $9.50(7) \times 10^{-5}$ Hz for Sr$^+$. These shifts are large enough and may be prerequisite for the frequency standards to achieve the foreseen $10^{-18}$ precision goal.