1/N Atom number scaling of photon-shot-noise-limited Faraday detection of atomic spin ensembles

Off-resonant Faraday rotation of atomic spins is frequently utilized as a means of measuring atomic spin polarizations in a dense vapor, For optically thick samples, the achievable photon shot noise sensitivity can be below the fundamental limits set by quantum projection noise. This is because each photon scattered by the atoms produces a photon difference signal (in a balanced Faraday detector) amplified by the square root of the optical depth [1].  Curiously, the scaling of the photon-shot noise with atom number N is 1/N in an optically thick, well-optimized detection system.

[1] M. V. Romalis, "Quantum noise in atomic magnetometers", in Optical Magnetometry, Cambridge University Press (2013), eds. D. F. J. Kimball and D. Budker