Abstract The energy levels and electric dipole ( E 1 ) matrix elements of the ground state and low-lying excited states of Be atoms are calculated using the relativistic configuration interaction plus core polarization (RCICP) method. The static and dynamic E 1 , magnetic dipole ( M 1 ) and electric quadrupole ( E 2 ) polarizabilities as well as the hyperpolarizabilities of the 2 s 2 1 S 0 and 2 s 2 p 3 P 0 states are determined. Two magic wavelengths, 300.03 and 252.28 nm, of 2 s 2 1 S 0 → 2 s 2 p 3 P 0 clock transition are found. Then, the multipolar and nonlinear Stark shifts of the clock transition at the magic wavelength are discussed in detail. We find that when the laser intensity I is in the range of 14.3–15.9 kW cm−2 and the detuning δ (the frequency detuning of the lattice laser frequency relative to the magic frequency) is in the range of 40.7–40.9 MHz, the fractional Stark shifts of the clock transition are less than 1.0 × 10−18. While, when I is in the range of 15.01–15.46 kW cm−2 and δ is in the range of 40.73–40.76 MHz, the fractional Stark shifts are lower than 1.0 × 10−19.
Abstract The energy levels and electric dipole ( E 1 ) matrix elements of the ground state and low-lying excited states of Be atoms are calculated using the relativistic configuration interaction plus core polarization (RCICP) method. The static and dynamic E 1 , magnetic dipole ( M 1 ) and electric quadrupole ( E 2 ) polarizabilities as well as the hyperpolarizabilities of the 2 s 2 1 S 0 and 2 s 2 p 3 P 0 states are determined. Two magic wavelengths, 300.03 and 252.28 nm, of 2 s 2 1 S 0 → 2 s 2 p 3 P 0 clock transition are found. Then, the multipolar and nonlinear Stark shifts of the clock transition at the magic wavelength are discussed in detail. We find that when the laser intensity I is in the range of 14.3–15.9 kW cm−2 and the detuning δ (the frequency detuning of the lattice laser frequency relative to the magic frequency) is in the range of 40.7–40.9 MHz, the fractional Stark shifts of the clock transition are less than 1.0 × 10−18. While, when I is in the range of 15.01–15.46 kW cm−2 and δ is in the range of 40.73–40.76 MHz, the fractional Stark shifts are lower than 1.0 × 10−19.
We report a hitherto undiscovered frequency shift for forbidden J = 0-->J = 0 clock transitions excited in atoms confined to an optical lattice. These shifts result from magnetic-dipole and electric-quadrupole transitions, which have a spatial dependence in an optical lattice that differs from that of the stronger electric-dipole transitions. In combination with the residual translational motion of atoms in an optical lattice, this spatial mismatch leads to a frequency shift via differential energy level spacing in the lattice wells for ground state and excited state atoms. We estimate that this effect could lead to fractional frequency shifts as large as 10(-16), which might prevent lattice-based optical clocks from reaching their predicted performance levels. Moreover, these effects could shift the magic wavelength in lattice clocks in three dimensions by as much as 100 MHz, depending on the lattice configuration.
State-of-the-art optical clocks based on neutral atoms have already surpassed the world's best microwave atomic clocks in both accuracy and stability, paving the way to high-precision measurements of elementary aspects of nature, like the search for the drift of fundamental constants or the application of relativistic geodesy.These clocks operate with ensembles trapped in an optical lattice.Normally, high-power trapping laser fields induce a large AC Stark shift on the clock's transition frequency, as the clock states have different polarizabilities.A key feature of lattice clocks is the magic wavelength, for which the atomic polarizabilities of the clock states are equal, therefore compensating a differential frequency shift on the clock transition.In the periodic potential of the optical lattice, the energy spectrum of motional states features a band structure, analogous to electrons in a crystalline structure well-known from solid state physics.Within this band structure, atoms can tunnel between adjacent lattice sites.As a consequence from tunneling motion, the clock transition experiences a frequency shift and a broadening, which becomes pronounced in the shallow lattice regime.This work presents the first demonstration of a 24 Mg optical lattice clock including the precise determination of the frequency of the clock transition, its magic wavelength and its second-order Zeeman shift.Furthermore, the effect of tunneling in an optical lattice for magnesium has been extensively studied via optical spectroscopy of Bloch bands and substantiated with theoretical models, that were initially developed for a strontium optical lattice clock.Measuring lattice light induced frequency shifts of the clock transition for various lattice wavelengths and intensities, yielded a value of the magic wavelength of the 1 S 0 → 3 P 0 clock transition in 24 Mg of 468.463(207) nm.This value is in well agreement with a recent theoretical calculation of 468.45(23) nm, using a refined atomic structure model.The experimental value for the magic wavelength was independently verified with optical spectroscopy of Bloch bands carried out on the clock transition.Here, the AC Stark modified band structure is directly probed by a single frequency scan of the carrier resonance, featuring an asymmetric line shape, that is strongly dependent on the lattice wavelength.This technique demonstrates a novel method for the determination of the magic wavelength and could also be applied to other atomic species.In bosonic isotopes of alkaline earth(-like) elements, like 24 Mg, optical dipole excitation of the spin-forbidden 1 S 0 → 3 P 0 clock transition can be enhanced, e.g. by applying an external magnetic field.While these bosons are insensitive to first-order Zeeman shifts, the second-order Zeeman effect displays a dominant contribution to the clock's error budget.In this thesis, the magnetic polarizability β is measured to be -206.6(2.0)MHz/T 2 , consistent with a theoretical calculation, and the clock transition frequency was found to be 655 058 646 691(101) kHz.
We carry out relativistic many-body calculations of the static and dynamic dipole polarizabilities of the ground 6s 2 1 S 0 and the first excited 6s6p 3 P o 0 states of Yb. With these polarizabilities, we compute several properties of Yb relevant to optical lattice clocks operating on the 6s 2 1 S 0 –6s6p 3 P o 0 transition. We determine (i) the first four magic wavelengths of the laser field for which the frequency of the clock transition is insensitive to the laser intensity. While the first magic wavelength is known, we predict the second, the third and the fourth magic wavelengths to be 551 nm, 465 nm and 413 nm. (ii) We re-evaluate the effect of black-body radiation on the frequency of the clock transition, the resulting clock shift at T = 300 K being −1.41(17) Hz. (iii) We compute long-range interatomic van der Waals coefficients (in a.u.) C 6 (6s 2 1 S 0 + 6s 2 1 S 0 ) = 1909(160), C 6 (6s 2 1 S 0 + 6s6p 3 P 0 ) = 2709(338) and C 6 (6s6p 3 P 0 + 6s6p 3 P 0 ) = 3886(360). Finally, we determine the atom-wall interaction coefficients (in a.u.), C 3 (6s 2 1 S 0 ) = 3.34 and C 3 (6s6p 3 P 0 ) = 3.68. We also address and resolve a disagreement between previous calculations of the static polarizability of the ground state.
After being pumped from the 5 s 1/2 ground state to the 6 p 1/2 state, the population inversion between 6 s 1/2 and 5 p 1/2,3/2 can be established for a rubidium four-level active optical clock. We calculate the ac Stark shift due to lattice trapping laser which dominates the frequency shift of clock transition in a lattice trapped rubidium four-level active optical clock. Several magic wavelengths are found, which can form desired optical lattice trapping potential. By choosing a proper intensity and linewidth of the trapping laser, the fractional frequency uncertainty of clock transition due to the ac Stark shift of the trapping laser, is estimated to be below 10 −18 .
Magnetic-dipole, electric-quadrupole, and hyperpolarizability effects on clock energy levels are analyzed in detail for Sr atoms in Stark potentials of red-detuned and blue-detuned magic-wavelength optical lattices. A difference between ac Stark shifts in traveling and standing waves is determined numerically. Differences between magic wavelengths for atoms in traveling and standing waves are presented and strategies for minimizing uncertainties of the clock frequency are indicated explicitly. Significant suppression of hyperpolarizability effect is demonstrated analytically for a blue-detuned in comparison with a red-detuned lattice, thus enabling essential deepening of trap potentials, reducing tunneling between lattice wells and collision effects.
Higher-order perturbation theory is used to study frequency shifts of the spin-forbidden 3 P J - 1 S 0 transitions for Mg, Sr and Yb atoms in an external laser field. Calculations of the higher-order dynamic Stark effect have been performed in connection with precise measurements on cooled and trapped atoms and the design of ultra stable optical clock in an engineered light shift trap. The form of the model potential of the valence electrons suggested by Simons is used for the calculation of atomic dynamic characteristics given by second- and fourth-order perturbation theory for the interaction of the atoms with the electromagnetic field. Dispersion properties of the polarizability and hyperpolarizability components and their behavior in the vicinity of both resonances and so-called magic wavelengths for the upper and lower atomic states, are analyzed. Calculations of ac multipole polarizabilities and dipole hyperpolarizabilities for the 3 P 0 - 1 S 0 clock transition in 87 Sr indicate that the contribution of the higher- order light shifts can be reduced to less than 1 mHz, allowing for a projected accuracy of better than 10 -17 .