Bridging classical and quantum approaches in optical polarimetry: Predicting polarization-entangled photon behavior in scattering environments

TL;DR

The paper develops a generalized Monte Carlo framework that unifies classical optical polarimetry and quantum polarization-based sensing by tying Wolf's coherency matrix to quantum density matrices via the Bethe-Salpeter equation and Jones-vector formalism. It extends Monte Carlo photon-packet tracing to polarization-entangled two-photon states, enabling prediction of quantum-state evolution in turbid media and validating predictions with tissue-mimicking phantoms. Experimental results show high fidelity (roughly 91–98%) between measured and simulated density matrices across scattering conditions, demonstrating robust entanglement preservation and the potential for quantum-enhanced diagnostics in biomedical, environmental, and remote sensing contexts. The approach is scalable to multi-photon scenarios and lays groundwork for optimal measurement protocols in quantum polarimetry and related quantum technologies. Overall, the work bridges classical and quantum polarimetry in scattering environments and provides a practical tool for predicting and diagnosing quantum states of light in turbid media.

Abstract

We explore quantum-based optical polarimetry as a potential diagnostic tool for biological tissues by developing a theoretical and experimental framework to understand polarization-entangled photon behavior in scattering media. We investigate the mathematical relationship between Wolf's coherency matrix in classical optics and the density matrix formalism of quantum mechanics which allows for the extension of classical Monte Carlo method to quantum states. The developed generalized Monte Carlo approach uniquely integrates the Bethe-Salpeter equation for classical scattering, the Jones vector formalism for polarization, and the density matrix approach for quantum state representation. Therefore, this unified framework can model both classical and quantum polarization states, handle multi-photon states, and account for varying degrees of entanglement. Additionally, it facilitates the prediction of quantum state evolution in scattering media based on classical optical principles. The validity of the computational model is experimentally confirmed through high-fidelity agreement between predicted and measured quantum state evolution in tissue-mimicking phantoms. This work bridges the gap between classical and quantum optical polarimetry by developing and validating a comprehensive theoretical framework that unifies these traditionally distinct domains, paving the way for future quantum-enhanced diagnostics of tissues and other turbid environments.

Figures (5)

• Figure 1: Schematic representation of possible trajectories of the photon packets passing through a scattering medium. Notation $\vert H \rangle$ and $\vert V \rangle$ addresses the problem of entangled state modelling, where each trajectory can be followed by a photon with both horizontal $\vert H\rangle$ and vertical $\vert V\rangle$ polarization. In turn, $\alpha_{H}\vert H\rangle + \alpha_{V}\vert V\rangle$ and $\beta_{H}\vert H\rangle + \beta_{V}\vert V\rangle$ correspond to the resultant polarization states for initial either $\vert H \rangle$ or $\vert V \rangle$ input states (for details on such notation refer to Sec. \ref{['sec:polTracing']} and Sec. \ref{['sec:simApproaches']}). $\mu_s$, $\mu_a$, $g$, and $n$ define the properties of the medium: scattering coefficient, absorption coefficient, scattering anisotropy factor, and refractive index, correspondingly Tuchin2015book. One of the trajectories corresponds to a snake photon path and features ladder diagrams for visualization of the iterative solution of the Bethe-Salpeter equation. Here, $G$ denotes the propagator of the Bethe-Salpeter equation and $p$ stands for the scattering phase function (adapted from Ref. RoyalSoc2005).
• Figure 2: Scattering scenario considered in this study, conceptual sketch. A pair of polarization-entangled photons is guided so that one of the partner photons interacts with a turbid medium. Another photon remains unchanged. Quarter-wave plates and linear polarizers enable polarization projective measurements for two-photon state reconstruction upon detection and coincidence events counting.
• Figure 3: Density matrix of the two-photon state after interaction of one of its partner photons with ZnO-based tissue phantom of $d/l^* \approx$ 1 in one of the arms. (a) Measured and (b) computed with Eq. (\ref{['eq:averagedDensityMatrix']}) with account for the initial state impurity (see Supplementary Material for details). Simulation parameters are selected to be identical to the measured sample's properties. Theoretical estimate also includes a fit for phase delay equal to $\delta=-\lambda/14$ induced by the possible birefringence of the sample. The obtained fidelity between the measured and simulated matrices is 91%.
• Figure 4: Measured (barplot) and modelled (diamonds) density matrix from Fig. \ref{['fig:scattering']} reshaped to vectors. Error bars represent the error estimation of the experimentally retrieved density matrix elements Kwiat2001.
• Figure 5: Evolution of the polarization-entangled two-photon state due to interaction with a scattering medium in terms of concurrence, linear entropy, purity, and dephasing of the output state vs effective thickness of the scattering medium $d/l^*$. Simulation results are shown with shaded stripes for different quality levels of the initial probing state ($C_{pr}$ = 1.0, 0.95, 0.90, 0.85, 0.80, 0.75 and 0.70). Experimentally measured points (diamonds, $C_{pr}$ = 0.88$\pm$0.01) with error estimation Kwiat2001 and best fitting simulated outcome (dashed line, $C_{pr}$ = 0.90) for $d/l^*$ = 0.003, 0.135, 0.287, 0.465, 0.733, and 1.002.