Analytical expressions for parameters of the dark resonance in a vacuum vapour cell

In spite of great number of papers devoted to theoretical and experimental study of the coherent population trapping phenomenon (CPT), still there is a lack of analytical results for the properties of the CPT-caused dark resonance in some principal cases. We will present the results of our study for single and double A schemes of atomic energy levels, which are of great importance for various applications of CPT (atom clocks, magnetometers, etc.). Analytical expressions for width and amplitude of the dark resonances have been obtained, which are valid for wide range of light wave intensities. The results concern closed as well as open A schemes. We will also discuss the difference between the two A schemes and state the condition when a double scheme can be effectively treated as a single A scheme. The analytical approximate formulas obtained have been compared with exact numerical solutions based on the optical Bloch equations as well as experimental.