Intensity nonlinearity of the error-signal frequency shift in the modulation spectroscopy of dark resonances and approaches to its reduction

We have found that the error-signal frequency corresponding to the coherent population trapping resonance can be displaced from that of $0-0$ transition unperturbed by the optical field, although the frequency is not sensitive to changes in its intensity. We consider the double $Lambda$-system of levels interacting with the asymmetric polychromatic optical field to demonstrate that this effect arises due to intensity nonlinearity of the error-signal frequency shift. The experiment with $^{87}$Rb atoms in Ar-N$_2$ buffer gas atomic cell shows how the displacement value depends on different parameters. The possible influence of the effect on the clocks frequency stability and reproducibility are discussed.