Same Bloch-equation model as the 3D version, viewed top-down instead of in perspective. An ensemble of alkali atoms is optically pumped along x (orange beam), building spin polarization P₀. A field Bz to be measured (perpendicular to this view, shown by the corner indicator) makes the polarization precess, and a probe beam along y (cyan) reads out the transverse component Py via optical rotation — that reading is the magnetometer's output.
ω_L = γ·B_z (γ ≈ 2π·28 Hz/nT, electron spin)
1/T2 = R2,0 + ω_L²τ_se /(1+(ω_L τ_se)²) (spin-exchange broadening, suppressed when ω_L τ_se ≪ 1)
P_y = R_p P0 ω_L / [ R2(R_p+R2) + ω_L² ] (steady-state Bloch solution)
Why "spin-exchange relaxation free"? Random collisions between alkali atoms flip their combined spin state and would normally dephase the ensemble fast. But if collisions happen much faster than the atoms can precess (ω_L τ_se ≪ 1, τ_se = mean time between exchange collisions), each atom's spin gets "motionally narrowed" — it samples many partners before precessing appreciably, and the net dephasing nearly cancels. Raising the vapor density shortens τ_se and pushes more of the ±300 nT range into that regime, which is why the response curve narrows and steepens near B=0 as you raise it. This steady-state formula was checked against a forward-Euler integration of the same coupled Bloch equations at a fine time step (dt=1e-8s) across several parameter sets — the two agree to within 1e-9, confirming the model is self-consistent (its own live 3D twin uses a much coarser per-frame step purely for the visible settling animation, not for the plotted curve).
- Signal field Bz — the tiny field you are trying to detect (this is what a real SERF sensor reports, down to the femtotesla level).
- Vapor density — sets the spin-exchange collision rate 1/τ_se; higher density suppresses relaxation near zero field (SERF regime) but the effect fails once |ω_L τ_se| approaches 1.
- Pump rate Rp — how fast light pumps atoms back into the polarized state; too low and thermal/wall relaxation wins, too high saturates the signal.
- Cell wall relaxation R2,0 — the relaxation floor from wall collisions even with spin-exchange fully suppressed; anti-relaxation coatings push this down, sharpening the zero-crossing slope (the real sensitivity).
Real SERF magnetometers (used in fundamental-physics tests and in magnetoencephalography, MEG) reach ~1 fT/√Hz by running exactly in this suppressed regime — this is also the working principle behind optically-pumped MEG helmets that no longer need liquid-helium-cooled SQUIDs. Drag the cell view to pan and scroll to zoom in on individual atom spins.