Optical pumping of alkali-metal vapor with hyperfine-resolved buffer gas pressure

Abstract

Optical pumping is fundamental to high-precision measurement using thermal alkali-metal atoms in vapor cells. In applications such as spin-exchange-relaxation-free magnetometers, buffer gases (e.g., $\text{N}_2$ or $\text{He}$) are commonly employed to quench fluorescence and mitigate wall relaxation. In the high-pressure limit (e.g., the $\text{N}_2$ pressure $p_{\rm{N}_2}> 1$~atm), where collisional broadening exceeds the hyperfine splitting of the alkali-metal atoms, optical pumping theory provides a clear description of the angular momentum exchange between photons and atomic spins. However, in many magnetic sensing scenarios, this high-pressure condition is not strictly satisfied, rendering the high-pressure approximation inaccurate. Consequently, a precise quantitative understanding of optical pumping under realistic pressures is critical for determining optimal buffer gas parameters, selecting operating points (e.g., pump frequency and intensity), and enhancing system reliability and stability. To address this, we develop a theory of optical pumping in the quasi-high-pressure regime, where collisional broadening is comparable to the ground-state hyperfine splitting. We demonstrate that optical absorption, spin polarization, and magnetic resonance linewidth in this regime differ significantly from those predicted by the high-pressure limit. Our study extends conventional modeling and offers critical guidance for atomic magnetometry operating under realistic buffer gas pressures.

Figures (8)

• Figure 1: Schematic illustration of optical pumping on the alkali-metal D$_1$ line in the HP limit and QHP regime. (a) and (b) show the GS and ES hyperfine multiplets, with different collisional broadening $\Gamma_{\mathrm{brd}}$ illustrated by blue level edges. Each of $F$ and $\bar{F}$ corresponds to either $a=I+1/2$ or $b=I-1/2$ for nuclear spin $I$. Laser detuning from the D$_1$ line reference frequency is denoted by $\Delta_0$, while $\Delta_{F}$ for $F=a$ and $b$ are the detunings corrected from $\Delta_0$ by the GS hyperfine splitting. (c) and (d) present the corresponding absorption cross section $\sigma_0$ for unpolarized $^{87}$Rb, with $\Gamma_{\mathrm{brd}}=3,\,20~\mathrm{GHz}$, respectively. The hyperfine splittings are $\omega^{\mathrm{\{g\}}}_{\mathrm{hf}}=6.8~\mathrm{GHz}$ and $\omega^{\mathrm{\{g\}}}_{\mathrm{hf}}=0.81~\mathrm{GHz}$. Collisional shifts are not included for simplicity. Absorption peaks corresponding to different $F \rightarrow \bar{F}$ transitions are indicated. Doppler broadening $\Gamma_{\mathrm{dop}} = 0.57~\mathrm{GHz}$ is included using the Voigt profile, which is negligible in the two regimes.
• Figure 2: Detuning-dependent factor $Q_{F}$ for $F=a$ and $F=b$ [see Eq. \ref{['eq:Q_F']}]. Solid, dashed, and dotted curves represent $\Gamma_{\mathrm{brd}}=3,~ 10~ \mathrm{and}~ 20\mathrm{~GHz}$. They converge to a common intersection at $\Delta_a=\Delta_b$, yielding $Q_a=Q_b=1$. The thick black dashed curve shows the maximum attainable $Q_{b}$ for varying $\Gamma_{\mathrm{brd}}$.
• Figure 3: Heat-map plot of the superoperator $\mathrm{Re}\left[A_{\mathrm{op}}\right]$ with nuclear spin $I=3/2$ and $\mathbf{s}= \hat{z}$. The underlying matrix elements are arranged with basis labeled by $|F,\,m_F;F,\,m'_{F})$ and presented in the HP limit. They are separated by dashed lines into $m$th- to $n$th- order Zeeman coherence blocks (see Appendix. \ref{['app:blocks']}). The block marked by the solid lines is $\left[A_{\mathrm{op}}\right]_{0,0}$ which governs population pumping, while $\mathrm{Re}\left[A_{\mathrm{op}}\right]_{1,1}$ marked by the dotted lines contributes to evolution of the 1st-order Zeeman coherence. The vertical gray and green bands encode the detuning-dependent $Q_{F}$ for $F=a,\,b$, acting in the QHP regime. Other blocks follow the same way.
• Figure 4: (a) The 1-norm distance between the steady-state population and the STD with the same polarization. The curves correspond to different detuning-dependent factors $Q_F$ for $F=a,\,b$. (b) Electron-spin polarization $P$ versus $\Gamma_{\mathrm{ex}}/\Gamma_{\mathrm{SD}}$, with color coding identical to (a) setting. (c) Steady-state population obtained with $\Gamma_{\mathrm{ex}}/\Gamma_{\mathrm{SD}}=10$, corresponding to the point marked on the red curve in (b). STD with the same polarization is shown and compared. (d) Same as (c) but corresponding to the blue marked point in (b). All results are computed for $I=3/2$ and $s=1$, fixing $R_{\mathrm{op}}/\Gamma_{\mathrm{SD}}=1$.
• Figure 5: (a) Absorption cross section $\sigma_{\mathrm{lin}}$ ($s=0$) versus laser detuning $\Delta_{0}$ with $\Gamma_{\mathrm{brd}}=3~\mathrm{GHz}$. The blue, green, and red curves correspond to $\Phi = 10^{12}$, $10^{16}$, and $10^{20}~\mathrm{cm^{-2}\,s^{-1}}$, respectively. (b) Same as (a), but for a typical HP case, $2\Gamma_{\mathrm{brd}}=20~\mathrm{GHz}$. (c) Integrated spectral area obtained by $\int \sigma_{\mathrm{lin}}\, d\Delta_{0}$, plotted for different $\Gamma_{\mathrm{brd}}$. Relaxation rates, $\Gamma_{\mathrm{ex}}$ and $\Gamma_{\mathrm{SD}}$, correspond to $^{87}$Rb with $\mathrm{N}_2$ buffer gas at $100\,^{\circ}\mathrm{C}$.