Collective reflectivity from a one-dimensional dense chain of atoms

We theoretically develop a quantum-mechanical scheme to study propagation of a laser beam through a one-dimensional (1D) dense chain of N effective two-level cold atoms tightly fixed in a free space. Atoms with half a wavelength separation are modeled as a collection of spins with angular momenta ±12, being at equivalent field positions. At an arbitrary driving field, we derive a reflectivity in terms of coherences between N + 1 collective eigenstates of the atoms from the steady-state solutions of the quantum master equation. We unravel that the reflectivity of incident light arises from scattering of electric fields in backward direction from the participating dipoles in the excitation. We identified splitting of reflectivity plotted as a function of detuning of incident light from atomic resonance as the Autler-Townes (AT) effect. Our scheme finds that the width of this reflectivity dip is scaled as a function of a ratio of the driving field frequency to the collective decay rate.