Quantum interference effects in a $\mathrm{\Lambda }$-type atom interacting with two short laser pulse trains

We study the quantum interference between the excitation pathways in a three-level $\mathrm{\Lambda }$-type atom interacting with two short laser pulse trains under the conditions of electromagnetically induced transparency. The probability amplitude equations which describe the interaction of a three-level $\mathrm{\Lambda }$-type atom with two laser pulse trains are numerically solved. We derive analytical expressions for the population of the upper excited state for resonant laser pulse trains with a rectangular temporal profile. By varying the parameters of the laser pulse trains such as area of a single pulse, detuning, repetition period, and number of individual pulses, we analyze the quantum interference between the excitation pathways in terms of the upper excited state population.