Light propagation in atomic stratified media: breakdown of the transfer-matrix method at high density

TL;DR

This work assesses the validity of the transfer-matrix approach for light propagation in dense 1D lattices of cold atoms by comparing TM predictions to the exact coupled-dipole model. The authors show that TM accurately reproduces CD results at low density, even for thin, subwavelength layers, but fails at higher densities due to dipole–dipole interactions that induce a collective interlayer response, and crucially, using a high-density refractive index in TM does not restore agreement. They quantify boundary densities around $\rho \sim 0.04$–$0.05\,k^3$ where TM breaks down (with typical rubidium densities in this range), and demonstrate this breakdown in single dense layers as well as in multi-layer Bragg-mirror configurations. The findings provide practical density thresholds for applying TM in 1D atomic photonic systems and guide experimental designs involving ultra-cold atoms and Bragg-like structures.

Abstract

The transfer-matrix method is a standard approach to wave propagation in stratified media. With the advent of cold-atom-based quantum and photonic technologies, several experiments and many proposals consider light propagation in one-dimensional optical lattices, using the transfer matrices as the main tool for the simulation. Here, we study the validity of this method by comparing its results to the microscopic coupled-dipole model, which is exact in the linear-optics regime. We show that the transfer-matrix method works very well at low density, even for thin disordered slices, and breaks down at high density because the dipole-dipole interaction induces a collective response from the atoms such that the properties of one layer are influenced by the others. We determine the boundary values of atomic densities for which this method is still applicable for describing experiments. Our findings are relevant for experimental realizations using ultra-cold atoms.

Figures (9)

• Figure 1: Illustration of the transfer-matrix method. $A,B,C,D$ are the amplitudes of the fields.
• Figure 2: Imaginary part of the refractive index in the medium computed from the reflection and transmission by a thin layer of atoms ($kL=0.25$). The reflection and transmission are computed from the CD model.
• Figure 3: Reflection coefficient by a thick layer of atoms, computed from the CD model (solid lines) and from the TM model (dashed lines). The layer thickness is $k L=2$ (green and black) and $k L=4$ (pink and blue).
• Figure 4: Same as in fig. \ref{['fig.low_thin']} but at higher density, $\rho = 0.1 k^3$ instead of $\rho = 0.001 k^3$.
• Figure 6: Relative error between the CD and the TM models, for the transmission (top) and the reflection (bottom), as a function of the layer thickness. The density is $\rho=0.1k^3$.