We rigorously derive the formula of the coherent-population-trapping (CPT) resonance profile by solving the Liouville equation for the density matrix in a four-level double-$\mathrm{\Lambda }$ scheme excited by a bichromatic light field. We demonstrate that the CPT resonance profile can be expressed as the sum of a symmetric and an antisymmetric Lorentz function, where the relative weights depend on the one-photon detuning. Additionally, we explicitly resolve the ac Stark shift and the line-shape-asymmetry-induced shift from the CPT resonance formula. Our theoretical calculations, based on a four-level double-$\mathrm{\Lambda }$ scheme for the $(-1,1)$ CPT resonance between ground states of $|F=3,m=-1\rangle$ and $|F=4,m=1\rangle$, successfully reproduces the observed resonance profile and light shifts under experimental conditions involving $\mathrm{lin}\parallel \mathrm{lin}$ polarization on the $D_1$ line of ${}^{133}\mathrm{Cs}$ atoms. Furthermore, the dependence of the light shift on the one-photon detuning is explained by our theoretical framework.

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• Atomic, optical & lattice clocks
• Coherent population trapping