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Register a new userAnnexing magic and tune-out wavelengths to the clock transitions of the alkaline-earth metal ions
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We present additional magic wavelengths ($lambda_{rm{magic}}$) for the clock transitions in the alkaline-earth metal ions considering circular polarized light aside from our previously reported values in [J. Kaur et al., Phys. Rev. A {bf 92}, 031402(R) (2015)] for the linearly polarized light. Contributions from the vector component to the dynamic dipole polarizabilities ($alpha_d(omega)$) of the atomic states associated with the clock transitions play major roles in the evaluation of these $lambda_{rm{magic}}$, hence facilitating in choosing circular polarization of lasers in the experiments. Moreover, the actual clock transitions in these ions are carried out among the hyperfine levels. The $lambda_{rm{magic}}$ values in these hyperfine transitions are estimated and found to be different from $lambda_{rm{magic}}$ for the atomic transitions due to different contributions coming from the vector and tensor part of $alpha_d(omega)$. Importantly, we also present $lambda_{rm{magic}}$ values that depend only on the scalar component of $alpha_d(omega)$ for their uses in a specially designed trap geometry for these ions so that they can be used unambiguously among any hyperfine levels of the atomic states of the clock transitions. We also present $alpha_d(omega)$ values explicitly at the 1064 nm for the atomic states associated with the clock transitions which may be useful for creating high-field seeking traps for the above ions using the Nd:YAG laser. The tune out wavelengths at which the states would be free from the Stark shifts are also presented. Accurate values of the electric dipole matrix elements required for these studies are given and trends of electron correlation effects in determining them are also highlighted.
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The frequency dependent polarizabilities of the francium atom are calculated from the available data of energy levels and transition rates. Magic wavelengths for the state insensitive optical dipole trapping are identified from the calculated light shifts of the $7s~^2S_{1/2}$, $7p~^2P_{1/2, 3/2}$ and $8s~^{2}S_{1/2}$ levels of the $7s~^{2}S_{1/2}-7p~^{2}P_{1/2,3/2}$ and $7s~^{2}S_{1/2}-8s~^{2}S_{1/2}$ transitions, respectively. Wavelengths in the ultraviolet, visible and near infrared region is identified that are suitable for cooling and trapping. Magic wavelengths between 600-700~nm and 700-1000~nm region, which are blue and red detuned with the $7s-7p$ and $7s-8s$ transitions are feasible to implement as lasers with sufficient power are available. In addition, we calculated the tune-out wavelengths where the ac polarizability of the ground $7s~^{2}S_{1/2}$ state in francium is zero. These results are beneficial as laser cooled and trapped francium has been in use for fundamental symmetry investigations like searches for an electron permanent electric dipole moment in an atom and for atomic parity non-conservation.
Derivations for the higher tensor components of the quadrupole polarizabilities are given and their values for the metastable states of the Ca$^+$, Sr$^+$ and Ba$^+$ alkaline earth-metal ions are estimated. We also give the scalar quadrupole polarizabilities of the ground and metastable states of these ions to compare our results with the previously available theoretical and experimental results. Reasonably good agreement between our calculations with the previous values of scalar quadrupole polarizabilities demonstrate their correctness. The reported scalar and tensor quadrupole polarizabilities could be very useful to estimate the uncertainties due to the gradient of the electric fields in the clock frequencies of the above alkaline earth-metal ions when accuracies of these frequency measurements attain below 10$^{-19}$ precision level.
Apropos to the growing interest in the study of long-range interactions which for their applications in cold atom physics, we have performed theoretical calculation for the two-dipole $C_6$ and three-dipole $C_9$ dispersion coefficients involving alkaline-earth atoms with alkaline-earth atoms and alkaline-earth ions. The $C_6$ and $C_9$ coefficients are expressed in terms of the dynamic dipole polarizabilities, which are calculated using relativistic methods. Thereafter, the calculated $C_6$ coefficients for the considered alkaline-earth atoms among themselves are compared with the previously reported values. Due to unavailability of any other earlier theoretical or experimental results, for the $C_6$ coefficients for alkaline-earth atoms with alkaline-earth ions and the $C_9$ coefficients, we have performed separate fitting calculations and compared. Our calculations match in an excellent manner with the fitting calculations. We have also reported the oscillator strengths for the leading transitions and static dipole polarizabilities for the ground states of the alkaline-earth ions, i.e., Mg$^+$, Ca$^+$, Sr$^+$, and Ba$^+$ as well as the alkaline-earth atoms, i.e., Mg, Ca, Sr, and Ba. These, when compared with the available experimental results, show good agreement.
We present precise values of the dipole polarizabilities ($alpha$) of the ground $rm [4f^{14}6s] ~ ^2S_{1/2}$ and metastable $rm [4f^{14} 5d] ~ ^2D_{3/2}$ states of Yb$^+$, 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. The static values of $alpha$ for the ground and $rm [4f^{14} 5d] ~ ^2D_{3/2}$ states are estimated to be $9.8(1) times 10^{-40} ,,rm Jm^2V^{-2}$ and $17.6(5) times 10^{-40},, rm Jm^2V^{-2}$, respectively, while the tensor contribution to the $rm [4f^{14} 5d] ~ ^2D_{3/2}$ state as $- 12.3(3) times 10^{-40},, rm Jm^2V^{-2}$ compared to the experimental value $-13.6(2.2) times 10^{-40},,rm Jm^2V^{-2}$. This corresponds to the differential scalar polarizability value of the above transition as $-7.8$(5)$,times, 10^{-40},rm Jm^2 V^{-2}$ in contrast to the available experimental value $-6.9$(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$^+$.
We present a novel method for engineering an optical clock transition that is robust against external field fluctuations and is able to overcome limits resulting from field inhomogeneities. The technique is based on the application of continuous driving fields to form a pair of dressed states essentially free of all relevant shifts. Specifically, the clock transition is robust to magnetic shifts, quadrupole and other tensor shifts, and amplitude fluctuations of the driving fields. The scheme is applicable to either a single ion or an ensemble of ions, and is relevant for several types of ions, such as $^{40}mathrm{Ca}^{+}$, $^{88}mathrm{Sr}^{+}$, $^{138}mathrm{Ba}^{+}$ and $^{176}mathrm{Lu}^{+}$. Taking a spherically symmetric Coulomb crystal formed by 400 $^{40}mathrm{Ca}^{+}$ ions as an example, we show through numerical simulations that the inhomogeneous linewidth of tens of Hertz in such a crystal together with linear Zeeman shifts of order 10~MHz are reduced to form a linewidth of around 1~Hz. We estimate a two-order-of-magnitude reduction in averaging time compared to state-of-the art single ion frequency references, assuming a probe laser fractional instability of $10^{-15}$. Furthermore, a statistical uncertainty reaching $2.9times 10^{-16}$ in 1~s is estimated for a cascaded clock scheme in which the dynamically decoupled Coulomb crystal clock stabilizes the interrogation laser for an $^{27}mathrm{Al}^{+}$ clock.
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