Hidden identity in a generic Λ system: applications to coherent population trapping

Abstract

A generic three-level Λ system is the source of a group of interesting quantum optical phenomena from coherent population trapping (CPT) to electromagnetic induced transparency. However, in contrast to its cousin, the two-level system, analytic solutions are difficult to acquire, except certain special cases. Here, we show that there exists a hidden identity in a generic Λ system as far as the rotating wave approximation is valid. This identity links the squares of the probability amplitudes in three states through the laser detuning phase (product of the detuning and time) and the Rabi phase. It is independent of the laser pulse wave forms and time delay. An application to CPT not only recovers all the previous asymptotic results, but also explains analytically why in the stimulated Raman adiabatic passage, the Stokes pulse must precede the pump pulse. More importantly, our results show that while a longer probe pulse is always favoured, a not too short probe pulse is still able to induce CPT because the time delay between the probe and coupling pulses can make up the difference in the formation of dark states. These can be directly tested experimentally.