Nonlinear magneto-optical rotation in optically thick media

4the optimum initial saturation parameter κ 0 ≈ 1.8α 0 l,we have from Eq. (14) that κ(x) ≫ 1 for all x. For theoptimum light intensity, then,√(δB) −1 ≈ 0.8 gµω Atα 0 l¯hc≈ 1.1 gµ¯hγ√Alntγ . (18)This result is consistent with the general observation [3]that optimized NMOR provides a method for measuringa spin-system at the standard quantum limit (SQL) givenby spin-projection noise [23]. The sensitivity is then expectedto scale as the square root of the product of thenumber of available atoms and the spin-relaxation time(see, e.g., Ref. [18], Sec. 3.1.1), which is indeed the result(18).IV.THE DOPPLER-BROADENED CASENow we consider the case where the Doppler width Γis much greater than the power-broadened line width. Inthis case, we need to average over the over the atomic velocitydistribution, which is equivalent to averaging overthe Doppler-free spectral profiles. On resonance with theDoppler broadened transition, the absorption coefficientis given byα DB ≈ 1 √ πΓ∫ ∞−∞≈ α ∫ ∞0√ πΓ=√3π8γ 0Γ−∞α(∆)d∆d∆(2∆/γ 0 ) 2 + 2κ/3α 0√ κ. (19)In this case the Doppler-broadened unsaturated absorbtioncoefficient is given in terms of the Doppler-free unsaturatedabsorbtion coefficient by√ πα 0 | DB=2γ 0Γ α 0. (20)Comparing Eqs. (19) and (7), we see that we have reproduceda well-known result (see, e.g., Ref. [19], Sec.7.2.1) that resonant absorption falls as 1/κ for Dopplerfreemedia and as 1/ √ κ for Doppler-broadened mediawhen κ ≫ 1. The change in κ per unit length isdκ(x)dx= −α DBκ(x)√6π γ 0≈ −4 Γ α √0 κ(x), (21)and solving for κ as a function of position,( √ ) 2√ 6π γ 0κ(x) ≈ κ0 −8 Γ α 0x . (22)Note that the behavior of the saturation parameter as afunction of distance is different in the Doppler-broadenedcase compared to the Doppler-free case [cf. Eq. (14)]where the saturation parameter falls approximately linearlywith distance.Taking the average of small-field rotation over theDoppler distribution givesdϕdΩdx∣ ≈ 1 ∫ ∞√DBπΓ −∞≈ α ∫ ∞0√ πγΓ≈√6π4γ 0Γ−∞dϕdΩdx (∆)d∆1(2∆/γ 0 ) 2 + 2κ/3 d∆α 0γ √ κ . (23)We see that the rotation per unit length scales as 1/ √ κ,in contrast to the 1/κ scaling for the Doppler-free case[similar to the situation with absorption, Eqs. (7,19)].This is because the number of atoms producing the effectis, in a sense, not fixed; with increasing light power, alarger fraction of the Doppler distribution is involved.Substituting (22) into Eq. (23) to find the contributionto the slope as a function of position givesdϕdΩdx (x) ∣ ∣∣∣DB≈ 2 γα 08 √6πΓγ 0√κ0 − α 0 x . (24)It is interesting to note that while the light power andthe rotation slope per unit length behave differently inthe Doppler-broadened case compared to the Dopplerfreecase, Eq. (24) has the same functional form as forthe Doppler-free case [Eq. (15)].Integrating over the length of the medium, we obtain∫dϕl∣ dϕ ∣∣∣DBdΩ∣ ≈DBdΩdx (x) dx≈ 2 γ0∫ l0√8Γ6π γα 0√κ0 − α 0 x dx(= 2 √γ ln κ0√κ0 − √ 6π γ 0Γ8α 0 l). (25)The behavior of (25) as a function of α 0 l is qualitativelysimilar to that of Eq. (16) shown in Fig. 2. However, thedependence on κ 0 is different.The magnetometric sensitivity is given by(δB) −1DB= gµω¯hc≈ gµω¯hc= gµω¯hc√dϕ2AtdΩ∣ DBπ γκ(l)√8Atπγ√ √32( √ ) ( √ )√ 6π γ 0κ0 −8 Γ α κ00l ln √κ0 − √ 6π γ 0Γ8α 0 l( )At γ 0pγ Γ α 0l(p − 1) ln , (26)p − 1