Fundamental trade-off between the speed of light and the Fano factor of photon current in three-level lambda systems

Electromagnetically induced slow-light medium is a promising system for quantum memory devices, but controlling its noise level remains a major challenge to overcome. This work considers the simplest model for such medium, comprised of three-level $\Lambda$-systems interacting with bosonic bath, and provides a new fundamental trade-off relation in light-matter interaction between the group velocity of light and the Fano factor of photon current due to radiative transitions. Considering the steady state limits of a newly derived Lindblad-type equation, we find that the Fano factor of the photon current maximizes to 3 at the minimal group velocity of light, which holds true universally regardless of detailed values of parameters characterizing the medium.