Evaluation of the ultimate performances of a Ca+ single-ion frequency standard

We numerically evaluate the expected performances of an optical frequency standard at 729 nm based on a single calcium ion. The frequency stability is studied through the Allan deviation and its dependence on the excitation method (single Rabi pulse or two Ramsey pulses schemes) and the laser linewidth are discussed. The minimum Allan deviation that can be expected is estimated to $\sigma_y(\tau) \approx 2.5\times 10^{-15}/\sqrt{\tau}$ with $\tau$ the integration time. The frequency shifts induced by the environmental conditions are evaluated to minimize the uncertainty of the proposed standard by chosing the most suited environment for the ion. If using the odd isotope $^{43}$Ca$^{+}$ and a vessel cooled to 77 K, the expected relative shift is $-2 \times 10^{-16}$ with an uncertainty of $\pm 4\times10^{-16}$, mainly due to the quadrupole shift induced by the unknown static electric field gradient .