Dark-state-induced transparency and the ultranarrow spectrum

Ultranarrow scattering spectra have important applications in precision measurements. Here, we show that the dark state of two atoms coupled to a one-dimensional (1D) waveguide leads to an ultranarrow transparency window with a width significantly smaller than the spontaneous decay rate of the atoms. Specifically, the position of this transparency dip is linearly proportional to the atom-atom distance. This ultranarrow dip can measure the atom-atom distance with an accuracy of one percent of the resonant wavelength. Additionally, the dip depth is highly sensitive to the atom-atom detuning, which could be used to measure small magnetic-field gradients at the nanoscale. The sign of the detuning can be further determined by measuring the second-order correlation function of the reflected photons.