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Photonics

Accuracy and Validity Range of a Simplified Lorentzian Approximation for Pressure-Broadened 87Rb Hyperfine Spectra

Shangtao Jiang, Tengyue Wang, Xuyang Qiu, Heng Yuan

Published Feb 26, 2026• 1 citation
Publisher

Abstract

Reliable modeling of pressure-broadened spectra is essential for maintaining physical consistency in alkali vapor-cell diagnostics. In this work, we investigate the low-power absorption spectra of isotopically enriched 87Rb vapor cells at the D1 and D2 transitions, systematically comparing three fitting algorithms: single Lorentzian, hyperfine-resolved Voigt, and the doublet Lorentzian approximation. Experiments were performed across optical intensities from 40 nW to 10 mW, buffer-gas pressures from 200 Torr to 520 Torr, and temperatures between 370 K and 390 K. It is shown that when the pressure broadening dominates over Doppler broadening and the optical intensity remains below the saturation regime, the reduced doublet Lorentzian model achieves a fitting accuracy of R2 > 0.996. Under the pressure conditions of 350 Torr, the fitting error for both the doublet and Voigt approximations remains below 0.3% for the D2 line and below 0.1% for the D1 line. At the pressures of 520 Torr and under elevated optical intensities, the spectrum evolves toward a single-peaked profile, rendering a single Lorentzian model sufficient. The quantitative applicability boundary of the doublet approximation in 87Rb vapor cells is established, defining the operational regime where hyperfine-resolved modeling can be reduced under collision-dominated conditions in NMRG systems.

DOI
10.3390/photonics13030221

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Reliable modeling of pressure-broadened spectra is essential for maintaining physical consistency in alkali vapor-cell diagnostics. In this work, we investigate the low-power absorption spectra of isotopically enriched 87Rb vapor cells at the D1 and D2 transitions, systematically comparing three fitting algorithms: single Lorentzian, hyperfine-resolved Voigt, and the doublet Lorentzian approximation. Experiments were performed across optical intensities from 40 nW to 10 mW, buffer-gas pressures from 200 Torr to 520 Torr, and temperatures between 370 K and 390 K. It is shown that when the pressure broadening dominates over Doppler broadening and the optical intensity remains below the saturation regime, the reduced doublet Lorentzian model achieves a fitting accuracy of R2 > 0.996. Under the pressure conditions of 350 Torr, the fitting error for both the doublet and Voigt approximations remains below 0.3% for the D2 line and below 0.1% for the D1 line. At the pressures of 520 Torr and under elevated optical intensities, the spectrum evolves toward a single-peaked profile, rendering a single Lorentzian model sufficient. The quantitative applicability boundary of the doublet approximation in 87Rb vapor cells is established, defining the operational regime where hyperfine-resolved modeling can be reduced under collision-dominated conditions in NMRG systems.
Abstract We theoretically and experimentally investigate the Doppler-broadened absorption spectrum of the D2-line (52 S 1/2 → 52 P 3/2) in thermal (room-temperature) Rb vapour over a wide range of probe intensity, probe diameter, and vapour cell temperature to understand the effect of hyperfine pumping and transit-relaxation on the absorption lineshape. We present a relatively simple but comprehensive five-level rate equation model which incorporates optical pumping of the atomic population into the other ‘dark’ ground hyperfine level (leading to absorption saturation), including contributions of the three closely spaced (within one Doppler linewidth) hyperfine levels in the excited state 52 P 3/2 (allowed by the dipole transition selection rules). D2-line transmission spectra predicted by our model show excellent agreement (rms error < 5 % ) with the experimental data for a wide range of probe intensities (from ∼ 0.001 I s a t 0 all the way up to ∼ 10 I s a t 0 , where I sat0 is the usual two-level saturation intensity for the respective atomic transition), vapour cell temperatures (24.2°C–53.5 °C) and beam diameters (1/e 2 width ∼ 1 − 6 mm). Our model also takes into account the finite transit time of atoms across the probe beam cross-section, and correctly predicts the dependence of effective saturation intensity on probe diameter (up to ∼ 10 × lower saturation intensity as compared to I sat0 for few-mm beam sizes). This ab initio five-level model can thus predict accurate Doppler lineshapes for any given experimental parameter set for a linearly polarized probe without any fitting parameters, and can be easily applied to any other atomic system by an appropriate change of the atomic constants in the rate equations.
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The normalized fluorescence spectrum of the Li 6708-\AA{} resonance line perturbed by the noble gases has been measured for 100 \AA{} either side of the resonance line, at a temperature of 400 \ifmmode\pm\else\textpm\fi{} 10\ifmmode^\circ\else\textdegree\fi{}C. The noble-gas pressure range of 100-1800 Torr was investigated. It was observed that the line wings scale linearly with this pressure for $2<|\ensuremath{\Delta}\ensuremath{\lambda}|<100$ \AA{}, and it was was demonstrated that a Lorentzianbroadened line core can be observed for this pressure range. The Li density was \ensuremath{\sim} 1010{10}^{10}1010/cm3{\mathrm{cm}}^{3}cm3 and was varied to establish the fluorescence spectrum in the limit of no radiation entrapment. The central region of the line has been measured with high resolution and compared to calculated profiles due to the various fine structure and isotopic line components, each broadened by a convolution of Lorentz, Doppler, and instrumental profiles. The shift and Lorentz broadening rates, as well as the wavelengths for transition to non-Lorentzian behavior, are reported. In contrast to the heavier alkali-metal-noble-gas cases, no distinctive far-wing features such as satellites or sharp inflections are observed. This is consistent with the much larger collision velocities for the Li case.
Spectroscopy of alkali-metal--buffer-gas mixtures at high pressures from single-digit to several hundred bars in the regime of substantial collisional broadening is relevant in a wide range of fields, ranging from collisional redistribution cooling to laboratory astrophysics. Here we report on spectroscopic measurements of dense rubidium--noble-gas mixtures recorded in a pressure cell equipped with soldered sapphire optical viewports, which allows for the controlled realization of extreme conditions of high temperature and high pressure in a table top laboratory experiment. In the gas cell, we have recorded absorption and emission spectra of rubidium subject to 188 bars of helium buffer-gas pressure at 548-K temperature. The spectra to good accuracy follow a Boltzmann-type Kennard-Stepanov frequency scaling of the ratio of absorption and emission spectral profiles. Further, the long optical path length in the cell allowed us to both record spectra of rubidium-argon mixtures at moderate temperatures and high pressures and observe redistributional laser cooling in this system.
We present ab initio reduced rate equation models to study the interplay of Zeeman and hyperfine pumping, which determines the Doppler-broadened absorption spectra of an arbitrarily polarized probe laser, in thermal atomic vapor. Our simple models (with maximum seven levels) accurately predict the dependence of probe absorption on its polarization for the 87Rb D2 line (residual error <2%) without any fitting parameters, while being ∼100× less computational resource intensive as compared to numerically complex full multi-level models that treat all (up to 23) the magnetic (m F ) sub-levels present in the system individually. We demonstrate that either Zeeman or hyperfine pumping dominates depending on whether the atomic transitions being probed are primarily closed or open.