Doppler-free absorptive signal lineshape of a four-level double λ-type system: Rabi splitting and two-photon effects

Two electromagnetic fields (the stronger called the pump and the weaker called the signal) are allowed to interact with a four-level double.-type system with two pairs of closely spaced levels. The present four-level system could also be viewed as an example of a double resonant system. Under the Doppler-free condition and using a semiclassical formulation of atom-field interaction, we derive the optical Bloch equations involving the said four-level system coupled to two external fields. These coupled optical Bloch equations are unsolvable in closed analytical form. In order to obtain the absorptive lineshapes, we use the usual perturbation method for getting the approximate analytical solution to the coupled optical Bloch equations for the density matrix elements. Through the off-diagonal complex density matrix elements, the introduction of phase angles between the levels participating in dipole-allowed transitions is automatic. It is shown that these phases are appreciable if the applied electromagnetic field is strong. For on-resonance pump conditions, we obtain the usual Rabi splitting. The effects of two-photon resonances are clearly shown for off-resonance pump conditions.