WGQED1D: a microscopic N-atom waveguide-QED toolkit for single-photon scattering and weak-measurement experiments

Technical Summary of WGQED1D

Problem Statement
The propagation of weak light through an atomic ensemble is a fundamental problem in quantum optics, relevant to phenomena such as electromagnetically induced transparency, slow-light propagation, and negative post-selected dwell times. While macroscopic linear-response theory (e.g., Beer–Lambert absorption and Kramers–Kronig dispersion) is well-established, bridging the gap between the microscopic dynamics of $N$ individual atoms and the integrated optical response of a cloud requires a careful input–output treatment. The authors note that straightforward implementations of standard formulas often yield quantitative disagreements with the Beer–Lambert limit by factors of order unity. These discrepancies are attributed to subtle convention choices regarding the bidirectional structure of waveguide coupling and the calibration between optical depth and per-atom coupling efficiency.

Methodology
The paper introduces WGQED1D, a lightweight, validated Python toolkit designed for the microscopic simulation of single-photon scattering by $N$ two-level atoms in a one-dimensional bidirectional waveguide. The methodology relies on the following core components:

1. Physical Model: The system is modeled in the single-excitation subspace, justified by single-photon inputs and low saturation. The dynamics are governed by a standard non-Hermitian effective Hamiltonian ($H_{\text{eff}}$) acting on $N$ complex amplitudes ($b_j$). This Hamiltonian includes:
   • Collective dipole–dipole interactions mediated by retarded photon exchange ($e^{ik_0|z_j-z_k|}$).
   • Diagonal loss to free space.
   • A driving term representing the input Gaussian wavepacket.
2. Input–Output Relation: The toolkit reconstructs the output field ($\phi_{\text{out}}$) from atomic amplitudes using a specific input–output relation that accounts for the phase accumulation of atoms relative to the cloud boundary.
3. Critical Calibration: A central methodological contribution is the identification and verification of a specific calibration between the per-atom waveguide coupling efficiency ($\beta = \Gamma_{1D}/\Gamma$) and the target optical depth ($OD$). The authors establish that quantitative agreement with the free-space Beer–Lambert limit requires $\beta = OD/(2N)$. This factor of two arises from the bidirectional nature of the waveguide, where atoms emit forward and backward with equal probability, contrasting with unidirectional ("chiral") idealizations.
4. Numerical Implementation: The toolkit uses deterministic evolution via an explicit Runge–Kutta 4(5) method. It operates in $O(N^2)$ memory and operations per timestep, utilizing dense linear algebra. It includes spectral analysis tools to extract the post-selected atomic excitation time ($\tau_T$) using the Wiseman–Steinberg weak-value formula.

Key Contributions

• WGQED1D Toolkit: A reproducible, open-source Python module (~400 lines) that implements the microscopic model, input–output reconstruction, and spectral weak-value analysis. It depends only on standard scientific libraries (NumPy, SciPy, Matplotlib).
• Microscopic Calibration: The explicit derivation and numerical verification of the relationship $\beta = OD/(2N)$, which resolves systematic quantitative disagreements between microscopic simulations and macroscopic Beer–Lambert predictions.
• Validation Infrastructure: The paper documents three distinct validations:
  1. Agreement with the analytical pulse-averaged Beer–Lambert transmission across optical depths $OD \in [0.25, 3]$ with errors $<0.1\%$.
  2. Verification of the Wiseman–Steinberg weak-value formula for post-selected excitation times.
  3. Direct comparison with experimental data from Angulo et al. (2026) regarding negative atomic excitation times.

Results

• Beer–Lambert Agreement: Simulations with $N=500$ atoms reproduce the analytical Beer–Lambert transmission to better than 0.1% across the tested range of optical depths, confirming the validity of the $\beta = OD/(2N)$ calibration in the dilute limit.
• Weak-Value Extraction: The toolkit successfully extracts post-selected excitation times ($\tau_T$) consistent with the analytical weak-value formula. Discrepancies are within 10% and are attributed to finite spectral resolution and collective finite-$N$ corrections at high optical depths.
• Experimental Comparison: When compared to Angulo et al.'s data on negative post-selected dwell times in cold rubidium clouds, the toolkit reproduces three of six data points within their $1\sigma$ error bars. For the remaining points (mostly at $OD=4$), the simulation tracks the qualitative trend and, in some cases, aligns more closely with experimental measurements than the linear-response analytical formula.
• Performance: The toolkit runs efficiently on standard hardware, producing converged results for realistic ensembles ($N \sim 200–1000$) in seconds to minutes.

Significance and Claims
The paper positions WGQED1D as a "reproducible numerical platform" for studying weak measurement, anomalous dispersion, group delay, and post-selected observables in single-photon transmission through optically thick atomic clouds. The authors emphasize that the toolkit prioritizes numerical reliability and verifiability over raw performance, offering a direct line-by-line match to the governing equations.

The significance of the work lies in providing a validated tool that correctly handles the subtleties of bidirectional waveguide coupling, thereby enabling researchers to perform microscopic simulations that quantitatively match macroscopic limits and recent experimental observations. The authors explicitly state that for users requiring larger system sizes ($N \gtrsim 10^4$), the algorithmic structure can be extended to sparse linear algebra or GPU backends, though these are not currently implemented. The toolkit is released under the MIT license to facilitate open scientific inquiry.