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Simple analytical expressions for the analysis of the phase-dependent electromagnetically induced transparency in a double-Λ atomic scheme

We study a double-Λ atomic scheme that interacts with four laser light beams so that a closed loop of radiation-induced transitions is formed. When specific relations for field phases, frequencies and amplitudes are satisfied, coherent superpositions (the so-called ‘dark states’) can be formed in a double-Λ, which leads to the well-known effect of electromagnetically induced transparency (EIT). If the interaction scheme in a double-Λ system is such that a closed loop is formed, the relative phase of the laser light fields becomes very important. We analyze here the effect of the lasers' relative phase on the EIT in double-Λ configuration of levels. The theoretical study of interactions of lasers with a double-Λ atomic scheme is commonly conducted by solving the optical Bloch equations (OBEs). We use here a perturbative method for solving OBEs, where the interaction of lasers with double-Λ is considered a perturbation. An advantage of the perturbative method is that it generally produces simpler solutions, and analytical expressions can be obtained. We present analytical expressions for the lower-order corrections of the EIT signal. Our results show that the EIT by the perturbative method can be approximated by the sum of products of complex Lorentzians. Through these expressions, we see in what way the relative phase affects the overall EIT profile.