  Field-inhomogeneity-induced relaxation of atomic spins confined in vapor
cells with depolarizing walls is studied. In contrast to nuclear spins, such as
noble-gas spins, which experience minimal polarization loss at cell walls,
atomic spins in uncoated cells undergo randomization at the boundaries. This
distinct boundary condition results in a varied dependence of the relaxation
rate on the field gradient. By solving the Bloch-Torrey equation under fully
depolarizing boundary conditions, we illustrate that the relaxation rate
induced by field inhomogeneity is more pronounced for spins with a smaller
original relaxation rate (in the absence of the inhomogeneous field). We
establish an upper limit for the relaxation rate through calculations in the
perturbation regime. Moreover, we connect it to the
spin-exchange-relaxation-free magnetometers, demonstrating that its linewidth
is most sensitive to inhomogeneous fields along the magnetometer's sensitive
axis. Our theoretical result agrees with the experimental data for cells
subjected to small pump power. However, deviations in larger input-power
scenarios underscore the importance of considering pump field attenuation,
which leads to uniformly distributed light shift that behaves as an
inhomogeneous magnetic field.
