This repository contains the manuscript, the synchronized dataset, and the analysis code for the research paper "Quantum-referenced refractive-index calibration of air at 1762 nm for environmental phase compensation in quantum optical interfaces". The refractive index of air at 1762 nm — the clock transition wavelength of ¹³⁸Ba⁺ — is measured with a dual-wavelength Michelson interferometer whose 1762 nm probe laser is referenced to a single trapped barium ion.
We report an atom-frequency-referenced measurement of the refractive index of air at 1762 nm over a campaign spanning eight months. A dual-wavelength Michelson interferometer referenced to a single trapped ¹³⁸Ba⁺ ion recorded the fringe-count ratio of the two channels together with co-located environmental sensor readings, yielding 145,784 synchronized measurements. The data provide the environmental coefficients ∂n/∂X for temperature, relative humidity, and pressure. Evaluated against the Mathar model on the identical observed conditions, the measured temperature and pressure coefficients agree to within 0.35 % and 0.93 %, respectively; the measured humidity coefficient is smaller in magnitude than the same-condition Mathar surrogate by 24.8 %. The humidity difference is statistically significant in the full dataset but not within the model validity range, and its magnitude is nearly spanned by the gain uncertainty of the humidity sensor; it is reported as an unresolved systematic. The raw Mathar residual is dominated by a static offset (1.166 × 10⁻⁶ in refractive index, 17.46 rad over the 4.2 m round-trip path); after this offset is subtracted, the residuals of the empirical model and the Mathar model agree within about 5 %. The dataset is made openly available.
- Atom-referenced refractometry: refractive-index coefficients of air at 1762 nm measured with a probe laser referenced to a single trapped ¹³⁸Ba⁺ ion
- Multi-month campaign dataset: 145,784 synchronized measurements over January – August 2025 (104 observation segments, duty cycle ~15 %)
- Same-condition model comparison: reproducible protocol comparing measured coefficients with the Mathar model on identical (T, H, P) rows
- Open Data: full synchronized dataset of environmental parameters and interferometric measurements
The synchronized dataset contains 145,784 measurements collected from January to August 2025 at a site in Freiburg, Germany (observed ranges: temperature 17.5–34.6 °C, relative humidity 18.9–45.3 %, pressure 964.6–1006.8 hPa). The sample is not continuous: it consists of 104 observation segments, with gaps arising from ion reloading and interferometer realignment, and the retained observations are weighted toward the warmer months (January: 8.3 %, June–August: 91.7 % of the data). A monthly coverage breakdown is provided in the Supplemental Material and in notebook 05.
Each processed CSV file contains:
time: UTC timestamptemperature: Temperature in °Chumidity: Relative humidity in %pressure: Atmospheric pressure in hPacounts_ratio: Dual-wavelength fringe-count ratio (directly measured)n_1762: Refractive index at 1762 nm derived via n₁₇₆₂ = (f₇₈₀/f₁₇₆₂) · n₇₈₀(T,H,P) / counts_ratio
data/processed/full_data.csv: complete synchronized dataset (145,784 rows, Jan 1 – Aug 20, 2025)data/processed/training_data.csv: Jan 1 – Aug 5, 2025 (108,346 rows)data/processed/validation_data.csv: Aug 5 – Aug 20, 2025 (40,897 rows)
| Parameter | Coefficient | Bootstrap SE (L=156, B=5000) |
|---|---|---|
| Temperature (α_T) | −8.8474 × 10⁻⁷ K⁻¹ | 1.41 × 10⁻⁹ |
| Humidity (α_H) | −1.3152 × 10⁻⁸ %⁻¹ | 1.15 × 10⁻⁹ |
| Pressure (α_P) | +2.5949 × 10⁻⁷ hPa⁻¹ | 9.85 × 10⁻¹⁰ |
The bootstrap standard errors are cross-checked against HAC standard errors (lag 156); the two agree within 3 %. Systematic contributions from the BME280 sensor gain bounds are documented in the Supplemental Material.
The Mathar formulation is evaluated at the identical observed (T, H, P) rows and fitted with the same linear surrogate model, so that both fits share the same design matrix:
| Parameter | This work | Mathar surrogate | Difference |
|---|---|---|---|
| α_T | −8.8474 × 10⁻⁷ K⁻¹ | −8.8164 × 10⁻⁷ K⁻¹ | −0.35 % |
| α_H | −1.3152 × 10⁻⁸ %⁻¹ | −1.7486 × 10⁻⁸ %⁻¹ | +24.8 % (measured smaller) |
| α_P | +2.5949 × 10⁻⁷ hPa⁻¹ | +2.5710 × 10⁻⁷ hPa⁻¹ | +0.93 % |
The humidity difference is statistically significant in the full campaign but not within the Mathar validity domain (10–25 °C). Its direction is opposite to the sign expected from water-vapor dispersion, and its magnitude is nearly spanned by the BME280 humidity-sensor gain uncertainty (a gain at the upper bound accounts for 3.99 × 10⁻⁹ of the 4.33 × 10⁻⁹ difference); it is reported as an unresolved systematic pending an independently calibrated humidity measurement. A Kramers-Kronig evaluation of the HITRAN2020 water-vapor spectrum at the mean campaign conditions yields a coefficient of order 10⁻⁸ %⁻¹ without enhancement over the broadband model.
- Empirical model residual standard deviation: σ_n = 1.84 × 10⁻⁷ (R² = 0.996)
- Blocked out-of-sample validation (five folds, contiguous blocks): held-out RMSE (1.84 ± 0.02) × 10⁻⁷, coinciding with the in-sample residual
- Mathar root-mean-square residual: 1.18 × 10⁻⁶, dominated by a static offset of 1.166 × 10⁻⁶ (17.46 rad over the 4.2 m round-trip path)
- After subtracting the offset: residual standard deviation 1.93 × 10⁻⁷; the empirical model improves on the offset-subtracted model by 4.9 % in amplitude and 9.5 % in variance
Ba-Network/
├── README.md
├── LICENSE-CC-BY-4.0.txt # CC BY 4.0 (documentation, data)
├── main.tex # manuscript source
├── supplemental_material.tex # Supplemental Material source
├── reference.bib
├── submission/
│ ├── cover_letter.tex
│ └── suggested_reviewers.txt
├── prapplied_review_round1/ # referee report and author notes (round 1)
├── figures/ # generated figures
└── zenodo/
├── refractive_index_1762nm_v1.0/ # archived v1 (superseded)
└── refractive_index_1762nm_v2.0/ # current release
├── data/
│ ├── processed/ # full_data.csv, training/validation splits
│ └── derived/models/hitran/
├── code/
│ ├── data_processing/
│ ├── analysis/ # notebooks 00-06
│ └── models/ # Mathar2007, NIST Ciddor, HITRAN HAPI
└── manuscript/
The analysis notebooks under zenodo/refractive_index_1762nm_v2.0/code/analysis/ reproduce every number reported in the manuscript and its Supplemental Material:
| Notebook | Purpose |
|---|---|
| 00 | Campaign statistics (mean/std/range, time-weighted means) |
| 01 | Same-condition Mathar surrogate comparison, blocked CV |
| 02 | Offset-fair improvement (I_amp, I_var) |
| 03 | Uncertainty budget (SM Tables S1a/S1b) |
| 04 | Kramers-Kronig evaluation at campaign mean conditions |
| 05 | Temporal coverage (segments, duty cycle) |
| 06 | Statistical robustness (physical-time ACF, block-length sensitivity) |
Common settings: block length L = 156 samples, B = 5000 replications, random seed 42.
- All analysis notebooks read
data/processed/full_data.csvand import the model modules undercode/models/. - A revised analysis of the submitted results (same-condition Mathar comparison, offset-fair improvement, uncertainty-budget audit, Kramers-Kronig evaluation at the true campaign mean conditions, temporal coverage, and statistical robustness) is provided in the current release; see the manuscript revision history.
- The earlier public description of this dataset (v1) contained claims — a 17.2 % humidity enhancement, agreement with a Kramers-Kronig prediction, and SI traceability — that are retracted in the current version. The raw data are unchanged.
- Data: Creative Commons Attribution 4.0 International (CC BY 4.0) — see
LICENSE.txt - Code: MIT License — see the code header for details
- Wei Wu: wei.wu@physik.uni-freiburg.de
- Ulrich Warring: ulrich.warring@physik.uni-freiburg.de
This research was supported by the European Research Council, Deutsche Forschungsgemeinschaft, QUSTEC Programme, and the Georg H. Endress Foundation.