Determination of atomic number density in MEMS vapor cells via single-pass absorption spectroscopy (SPAS)

Micro-electro-mechanical systems (MEMS)-based (chip-scale) alkali vapor cells are key components in emerging quantum technologies, where device performance critically depends on the atomic number density. Thus, it is important to have an accurate estimate of the atomic number density in MEMS-based alkali vapor cells to optimize light-matter interactions and design efficient quantum sensing systems. Here, a quantitatively validated method is presented for determining the rubidium (Rb) atomic number density in warm vapor using Single-Pass Absorption Spectroscopy (SPAS). The absolute transmission spectra are measured and modeled using the 780.24~nm and 420.29~nm transitions in Rb-filled MEMS vapor. The theoretical model employs a density-matrix formalism within the Lindblad framework and incorporates directly measurable experimental parameters, such as laser beam power, diameter, and cell temperature. The model explicitly accounts for optical pumping, Doppler broadening, and transit-time broadening effects and exhibits quantitative agreement ($> 99\%$) with experimental spectra over a broad range of temperatures (293-353~K), laser probe powers of approximately 10~$\mu$W-100~$\mu$W at the 780.24~nm transition and 8~$\mu$W-80~$\mu$W at the 420.29~nm transition, and cell lengths (2--100~mm). This method demonstrates a practical and reliable approach for determining the density of alkali vapor cells for quantum sensing, metrology, and quantum communication applications.