Light Propagation Breakdown at High Density Limits Transfer-Matrix Method Accuracy

Scientists are increasingly investigating light propagation through layered atomic systems, driven by advances in cold-atom and photonic technologies. Igor M. Sokolov (Ioffe Institute), William Guerin (Université Côte d’Azur, CNRS, Institut de Physique de Nice), and colleagues have now rigorously tested the widely used transfer-matrix method against a precise coupled-dipole model to determine its limits. Their research reveals the transfer-matrix method accurately predicts light behaviour at low atomic densities, but fails as density increases , a breakdown caused by significant atom-atom interactions creating a collective response. This discovery is crucial for interpreting experimental results involving ultra-cold atoms and will guide the development of more accurate modelling techniques for one-dimensional optical lattices.

However, the study reveals a critical breakdown at higher densities, pinpointing that dipole-dipole interactions induce a collective atomic response, effectively meaning the properties of one layer become influenced by its neighbours. The team rigorously determined the precise boundary values of atomic densities for which the transfer-matrix method remains applicable in experimental settings.

This work is particularly relevant for ongoing efforts to realise quantum technologies using ultra-cold atoms, offering a crucial assessment of modelling techniques. Experiments involving microscopic samples of atoms often rely on numerical solutions or, in the linear-optics regime, coupled-dipole equations, but these become computationally impractical for larger samples containing millions of atoms. Consequently, macroscopic theories are essential for simulating light propagation through disordered samples, often employing the Beer-Lambert exponential law to compute transmission. While simple, this law effectively explains observed collective effects on scattered light, radiation pressure, and localization phenomena.

Transfer-matrix and coupled-dipole modelling of optical lattices

Researchers employed the coupled-dipole model as a benchmark, solving the linear-optics regime exactly for microscopic samples to provide a ground truth for comparison with the transfer-matrix approach. This involved modelling each atom as an oscillating dipole and calculating the collective response to incident light, a computationally intensive process for larger systems. Specifically, the team calculated the scattering cross-section of individual atoms, defined as σsc = (k4/6π)|α|2, where λ = 2π/k represents the wavelength of the transition and α denotes the atomic polarizability, a crucial parameter for modelling light-atom interactions. Furthermore, scientists determined the complex refractive index, n = p 1 + ρα, using the calculated atomic polarizability α = 6π k3 × 2∆/Γ + i 1 + 4∆2/Γ2, and incorporated this into the transfer-matrix model to predict light propagation, a standard procedure at low densities. This innovative methodology enabled the determination of boundary values for atomic densities, defining the range where the transfer-matrix method provides a reliable approximation for experimental realizations involving ultra-cold atoms.

Transfer-matrix accuracy limits in atomic lattices

Experiments utilising ultra-cold atoms will directly benefit from these newly defined boundaries for applicability. However, measurements confirm a breakdown in accuracy as atomic density increases, attributable to dipole-dipole interactions inducing a collective atomic response, effectively meaning the properties of one layer influence those adjacent to it. This unexpected finding suggests that dipole-dipole interactions extend beyond individual layers, influencing the overall propagation characteristics. Calculations relied on the complex refractive index, defined as the square root of 1 plus the atomic polarizability multiplied by atomic density.

The team modelled the transmission through atomic layers of varying thickness, utilising a transfer matrix to represent each interface and the layer itself. This breakthrough delivers a crucial understanding of the limitations inherent in simplified modelling techniques for dense atomic systems. Measurements confirm that at high densities, the collective behaviour of atoms, driven by dipole-dipole interactions, cannot be accurately captured by methods relying on individual atomic properties. Future research will likely focus on developing more sophisticated models that account for these collective effects, paving the way for improved control and manipulation of light in quantum technologies, including atomic clocks, quantum memories, and novel lasing schemes.

Transfer-matrix limits at atomic density thresholds reveal critical

This research compared the transfer-matrix method with the more accurate, albeit computationally intensive, coupled-dipole model to determine limitations of the simpler approach. Researchers found that at densities exceeding approximately 0.05k3 (around 2.6x 10 12cm -3 for rubidium atoms), the transfer-matrix method introduces a relative error of roughly 20% in calculating reflection and transmission coefficients. This breakdown occurs because increased atomic density induces strong dipole-dipole interactions, creating collective atomic responses that invalidate the assumptions underlying the transfer-matrix approach, specifically, the use of textbook formulas for refractive index and the independent treatment of each layer. These findings are crucial for experiments employing ultra-cold atoms, potentially enabling investigations of photonic properties in denser atomic systems than previously explored. The authors acknowledge a limitation in data availability, stating that the underlying data are not currently public but can be requested. Future research could focus on developing modified transfer-matrix techniques or alternative computational methods capable of accurately modelling light propagation in highly dense atomic media, potentially unlocking new avenues for exploring quantum phenomena and photonic applications.