Frequency Light Shifts Caused by the Effects of Quantization of Atomic Motion in an Optical Lattice

Frequency light shifts resulting from the localization effects and effects of the quantization of translational atomic motion in an optical lattice is studied for a forbidden optical transition $J$=0$\to$$J$=0. In the Lamb-Dicke regime this shift is proportional to the square root from the lattice field intensity. With allowance made for magneto-dipole and quadrupole transitions, the shift does not vanish at the magic wavelength, at which the linear in intensity shift is absent. Preliminary estimates show that this shift can have a principal significance for the lattice-based atomic clocks with accuracy of order of 10$^{-16}-10^{-18}$. Apart from this, we find that the numerical value of the magic frequency depends on the concrete configuration of the lattice field and it can vary within the limits 1-100 MHz (depending on element) as one passes from one field configuration to another. Thus, theoretical and experimental investigations of contributions originated from magneto-dipole and quadrupole transitions are of principal self-dependent interest.