Spin dynamics and noise spectrum for alkali-metal atoms of arbitrary polarization in the spin-exchange-relaxation-free regime

Theoretical studies on the spin dynamics of hot alkali-metal vapors typically require numerical simulations of the density-matrix equation, which is nonlinear because of spin-exchange collisions. In this study we derive a set of effective hyperfine Bloch equations (EHBEs) for systems with arbitrary polarization in the spin-exchange-relaxation-free (SERF) regime. These EHBEs provide an intuitive framework for understanding the spin dynamics in the SERF regime and yield simple solutions that are in excellent agreement with the results of density-matrix simulations. A stochastic EHBE model is created by adding spin noise caused by fundamental quantum fluctuations. This yields an optical-spin-noise spectrum that includes the intrinsic spin-projection noise and the backaction light-shift noise from Faraday rotation measurements. The spectrum is a summation of several absorptive and dispersive components, each of which has a clear physical origin and a simple algebraic expression. This provides a benchmark for calibrating future spin-squeezing experiments in the SERF regime.