An effective model for describing coherent population trapping resonances, which correctly takes into account the off-resonant frequency components in periodically modulated laser field

TL;DR

This paper introduces an effective mathematical model for describing coherent population trapping resonances in periodically modulated laser fields, accurately accounting for off-resonant components and revealing new insights into light shifts.

Contribution

The authors develop a simplified yet accurate model that includes off-resonant effects via new operators, providing a clearer physical understanding of CPT resonances under modulation.

Findings

Model accurately matches exact calculations for three-level systems.

Off-resonant beat contributions can dominate traditional ac Stark shifts.

Standard spectrum analysis is insufficient to determine CPT resonance light shifts.

Abstract

We have developed an effective mathematical model to calculate the coherent population trapping (CPT) resonance in periodically modulated light, when the modulation frequency $f$ varies near the fractional part of hyperfine splitting in the ground state $\Delta_{\rm hfs}/N$ (where $N=1,2,...$). In such polychromatic field, only two frequency components that are most resonant with atomic optical transitions are taken into account accurately, while all other off-resonant components are taken into account using the second-order perturbation theory in the field. Within the presented concept, equation for atomic density matrix is obtained, in which the contribution of all off-resonant components is reduced to the appearance of two new operators (non-diagonal, in general case): the shift operator and relaxation operators. In the case of three-level $\Lambda$-system, the adequacy of presented effective model was verified by numerical calculations of various dependencies, in which we did not find visual differences from the exact calculations. In addition to a significant mathematical simplification, our model provides a clear physical picture of various features of CPT spectroscopy in a periodically modulated laser field, including effects that have not been discussed in the scientific literature before. In particular, we show that the widespread viewpoint that the CPT resonance shift is determined by usual ac Stark shifts of the lower levels is, in general, fundamentally incorrect, since the contribution to the light shift due to beats at the frequency $\Delta_{\rm hfs}$ between different off-resonant frequency components can be comparable (or even dominate) with the standard ac Stark shift. Therefore, even if we have detailed information on the modulated field spectrum (e.g. using a spectrum analyzer), this is, in general, absolutely insufficient to determine the light shift of CPT resonance.