Coherent Population Trapping in An Inhomogeneously Broadened \(\Lambda\)-System with Multiple Excited States

We present a theoretical model using density matrix approach to study the phenomenon  of coherent population trapping (CPT) in the \(\Lambda\)-type Doppler broadened system with multiple excited states. A multi-level \(\Lambda\)-system is formed by considering the closely spaced hyperfine levels in the D2 line of \(^{87}\)Rb. The presence of closely spaced hyperfine levels affects the transparency window and cause asymmetry in absorption profiles. We observe the  sharp CPT dip when the frequency difference of applied fields is equal to the frequency separation between two ground levels. This dip demonstrates that the system is trapped in the dark state. By performing a thermal averaging in the Doppler-broadened \(\Lambda\)-system, we have shown that the transparency window  becomes narrower and its linewidth decreases at higher temperature. This study has potential applications in precision measurements due to the small linewidth of the transparency window.