ac Stark shift of the Cs microwave atomic clock transitions

The present definition of the unit of time, the second, is based on the frequency of the microwave transition between two hyperfine levels of the Cs atom. Recently, it has been realized that the accuracy and stability of atomic clocks can be substantially improved by trapping atoms in optical lattices operated at a certain “magic” wavelength 1,2 . At this magic wavelength, both clock levels experience the same ac Stark shift; the clock frequency becomes essentially independent on trapping laser intensity. This effect was demonstrated 3–5 for optical clocks using strontium atoms. An extension of this idea to microwave clocks with alkali-metal atoms Rb and Cs was considered in Ref. 6 . A multitude of magic wavelengths for the hyperfine transition was identified. Unfortunately, detailed analysis presented below shows the conclusions of that paper to be erroneous: there is no magic wavelength for Cs, at least for clock levels with zero projections of the total angular momentum MF on the quantizing magnetic field. In a separate paper 7 we analyze the case of circular light polarization and MF 0 levels and demonstrate that the ac shift can be eliminated by an appropriate “magic angle” choice of the direction of the magnetic field with respect to the light propagation. This paper presents a detailed theoretical analysis of the frequency shift of a microwave clock involving a hyperfine transition. The analysis requires the calculation of the differential polarizability involving third-order expressions, quadratic in the field strength and linear in the hyperfine interaction. Evaluation of the resulting expressions is carried out using relativistic many-body theory. The second part of the paper reports measurements of the clock shift at two laser wavelengths. The results of the calculations are in good agreement with the experimental measurements. II. THEORY