Direct measurement of atomic number density using Single Pass Absorption Spectroscopy (SPAS)

TL;DR

Directly measures atomic number density in warm rubidium vapors using SPAS and a Lindblad-density-matrix framework, extracting $N$ as the sole free parameter by fitting absolute absorption spectra with experimental inputs such as $T$, beam diameter, and power. The model accounts for optical pumping, transit-time broadening, and Doppler broadening, by solving the steady-state density matrix and computing $\alpha_{ij}(\Delta,T)=k\,\mathrm{Im}\{\chi_{ij}(\Delta,T)\}$ and the Beer–Lambert transmission. Validation across 100 mm and MEMS cells, isotopes $^{85}$Rb and $^{87}$Rb, temperatures 293–343 K, and powers up to $2I_{sat}$ shows $R^2>0.99$ fits and good agreement with Alcock et al.'s vapor-pressure model. The dark-current baseline correction enables accurate absolute transmission in dilute, weak-absorption regimes, making this approach practical for miniature quantum sensors and metrology applications, with potential extensions to other species and field sensing.

Abstract

We demonstrate a direct measurement of the atomic number density of alkali atoms using single-pass absorption spectroscopy (SPAS). We developed our methodology based on modeling the absolute absorption spectra of warm rubidium (Rb) vapor and infer the atomic number density from SPAS measurements. The model combines the Lindblad formalism with a density matrix approach and incorporates experimentally measurable parameters such as laser beam power, laser beam diameter, and cell temperature. The framework explicitly incorporates optical pumping and transit-time broadening effects and shows excellent agreement ($> 99\%$) with experimental data using rubidium vapor cells across a wide range of temperature ($293$-$343$ K), laser powers ($\sim 0.2~I_{sat}$ -$~ 2~I_{sat}$), and cell lengths ($2$-$100$ mm). To ensure accurate quantification of absolute absorption measurements through the dilute atomic vapor, we measure and subtract the dark current of the photodetectors. This dark current is recorded in the absence of any light incident on the photodetector, to obtain accurate baseline corrections. This approach ensures an accurate determination of the atomic number density, even in the weak absorption regime. It provides a suitable alternative to the high-temperature, saturation-based method for baseline correction and enables the precise determination of the atomic number density in dilute atomic vapor for quantum technology applications in communication, sensing, and metrology using miniature atomic vapor cells. Furthermore, the methodology can be extended to determine the concentration of harmful gases and gaseous pollutants in urban, rural, as well as industrial environments.

Figures (9)

• Figure 1: (a) [Left] Generalized four-level atomic system comprising a ground state $|0\rangle$ and three excited states $|1\rangle$, $|2\rangle$, and $|3\rangle$. Transitions are driven by classical fields with Rabi frequencies $\Omega_{01}$ , $\Omega_{02}$ , and $\Omega_{03}$ , and corresponding detunings $\Delta_1$ , $\Delta_2$ , and $\Delta_3$ . Spontaneous emission processes occur with decay rates $\Gamma_{10}$ , $\Gamma_{20}$, and $\Gamma_{30}$ . (b) [Right] Schematic energy level diagram representing the hyperfine structure of rubidium atoms. Transitions originate from two ground hyperfine states, $|0_a\rangle$ and $|0_b\rangle$, coupling to excited states $|1\rangle$ to $|4\rangle$ as allowed by dipole selection rules. States $|2\rangle$ and $|3\rangle$ are common to both pathways. Decay channels follow branching ratios defined by $\Gamma_b$ and $\Gamma_d$ (from $|3\rangle$) and $\Gamma_c$ and $\Gamma_e$ (from $|2\rangle$).
• Figure 2: Schematic of the experimental setup for absolute absorption spectroscopy and frequency calibration. The absorption signal transmitted through the rubidium vapor cell is detected using photodetector (PD-1), while the reference transmission signal from the Fabry–Perot interferometer (FPI) is monitored by photodetector (PD-2). Both signals are simultaneously recorded with a digital storage oscilloscope (DSO). The piezoelectric transducer (PZT) scan signal from the external cavity diode laser (ECDL) controller is used as the trigger input to synchronize the oscilloscope trace with the frequency scan. Optical components include mirrors (M1–M5), neutral density filters (NDF), lenses (L1–L3), polarizing beam splitters (PBS), an electro-optic modulator (EOM), and a beam dump (BD). The vapor cell temperature is stabilized using a temperature controller (TEC) and resistive foil heaters monitored by four NTC (Negative Temperature Coefficient) temperature sensors.
• Figure 3: Experimental and fitted absorption spectra as a function of detuning for three different temperatures: $20.3^\circ$C (red solid line for fit, blue dots for experiment), $39.5^\circ$C (black dashed line for fit, orange dots for experiment), and $66.5^\circ$C (blue dotted line for fit, green dots for experiment). The lower panels show the corresponding residuals (experimental minus fitted). The fitted spectra exhibit excellent agreement with experimental data across all temperatures, with residuals remaining within $\pm 0.25$ in absolute transmission.
• Figure 4: Comparison of simulated and experimental transmission spectra of the rubidium MEMS vapor cell at $34.2^\circ$C, $51.1^\circ$C, and $59.1^\circ$C. Lower panels show the residuals, demonstrating excellent agreement across the temperature range.
• Figure 5: Rubidium vapor number density as a function of temperature obtained from the 100 mm cell (purple circles) and MEMS cell (red diamonds), compared with the empirical model of Alcock et al. Alcock01071984 (blue dashed line). The vertical red dashed line marks the solid–liquid phase transition at 312.45 K. Insets (a)--(c) highlight the temperature ranges 291--295 K, 311--315 K, and 339--342 K, respectively, showing excellent agreement between the experimentally measured and theoretically predicted number densities within experimental uncertainties.