The refractive index of a single three-level atom experienced by a single photon

We study the propagation of a quantum field composed of a few photons interacting with a three-level <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:math> -atom driven by a coherent classical field. The quantum field acquires a phase shift, which can be interpreted as a dispersion effect on the photon wave packet and described by the refractive index for quantum fields down to the single-photon level. In this paper, we demonstrate that the phases acquired by quantum fields depend on the number of photons in the quantum states. Notably, the phases differ between single- and two-photon states, enabling the separation of multiphoton states. This finding highlights new applications related to the dispersion of three-level atoms, which are important in advancing quantum information processing and enhancing quantum communication technologies. The results are crucial for long-distance quantum communication and hold potential for developing quantum field-based linear devices such as beam splitters, lenses, and quantum prisms capable of separating different components of quantum fields. The findings can have interesting applications for manipulating and assembling of multiphoton entanglement states.