Analytical model for polarization transfer during gas-phase collision events in spin-exchange optical pumping: Spin-$\frac{1}{2}$ $^{129}$Xe versus spin-$\frac{3}{2}$ $^{131}$Xe
Spin-exchange optical pumping (SEOP) is a method for producing spin-hyperpolarized noble gas nuclei, such as 129Xe and 131Xe, which are used in various magnetic resonance applications from fundamental physics to quantum sensing and medical imaging. In SEOP, optically polarized alkali-metal atoms transfer their spin polarization to the noble gas nuclei in gas-phase collision events via the hyperfine coupling (HFC) between the alkali valence electron and the noble gas nucleus. While the polarization transfer physics of spin $I = 1/2$ nuclei, such as 129Xe, is relatively well understood, that of spin $I > 1/2$ nuclei, such as 131Xe ($I = 3/2$), has been far less studied, and no rigorous theoretical model has been presented to date. To this end, we derive a simple analytical model for the upper limit, neglecting relaxation, of the SEOP polarization transfer, applicable to noble gases with arbitrary nuclear spin. Analytical evaluation of the Baker-Campbell-Hausdorff expansion for the time evolution of the spin density operator $\hat{\rho}(t)$ reveals that only even-order terms in the HFC contribute to the polarization transfer, with the leading-order quadratic term being the most significant. We obtain a result similar to that derived for the spin-exchange cross section by Herman [Phys. Rev. 137, A 1062 (1965)], but in a more general framework for the time evolution of $\hat{\rho}(t)$ that is also more familiar to magnetic resonance researchers. The model is applied to understand the difference in the polarization transfer efficiency between 129Xe and 131Xe, yielding results in agreement with previous experiments. We also validate the model by comparison to detailed numerical multiscale simulations of the SEOP process, where full quantum-chemically computed spin Hamiltonians sampled from molecular dynamics simulations of the gas-phase collision events are used to propagate the spin dynamics.