Nonlinear magneto-optical rotation in optically thick media

spontaneous decay rate from the upper state, and ξ representsthe quantum number distinguishing the groundstate (ξ g ) from the excited state (ξ e ). Λ = Λ 0 + Λ repop isthe pumping term, where the diagonal matrixdescribes incoherent ground state pumping (ρ 0atomic density), and2is the〈ξ g J g m|Λ 0 |ξ g J g m〉 =γρ 0(2J g + 1)(3)〈ξ g J g m|Λ repop |ξ g J g m ′ 〉 = γ 0∑m e,m ′ e ,q 〈J g , m, 1, q|J e , m e 〉〈J g , m ′ , 1, q|J e , m ′ e 〉ρ ξ eJ em eξ eJ em ′ e , (4)describes repopulation due to spontaneous relaxationfrom the upper level (see, e.g., Ref. [15]). Here 〈. . .| . . .〉are the Clebsch-Gordan coefficients.The electric field vector is written (see, e.g., Ref. [16])⃗E = 1 []E 0 e iφ (cos ϕ cos ɛ − i sin ϕ sin ɛ)e i(ωt−kz) + c.c. ˆx2+ 1 []E 0 e iφ (sin ϕ cos ɛ + i cos ϕ sin ɛ)e i(ωt−kz) + c.c. ŷ,2(5)where ω is the light frequency, k = ω/c is the vacuumwave number, E 0 is the electric field amplitude, ϕ is thepolarization angle, ɛ is the ellipticity (arctangent of theratio of the major and minor axes of the polarizationellipse), and φ is the overall phase. By substituting (5)into the wave equation( ) ω2c 2 + d2 ⃗Edz 2 = − 4π d 2c 2 dt ⃗ 2 P, (6)where ⃗ P = T r(ρ ⃗ d) is the polarization of the medium, theabsorption, rotation, phase shift, and change of ellipticityper unit distance for an optically thin medium canbe found in terms of the density matrix elements (theseexpressions are given in Ref. [17]). Once the solutionsfor the density matrix are obtained, we will perform anintegration to generalize the result to media of arbitrarythickness.III.THE DOPPLER-FREE CASEWe consider the case of a J = 1 → J = 0 transition,and linearly-polarized incident light, with a magneticfield directed along the light propagation direction(Faraday geometry). Using the rotating wave approximation,the solution of Eq. (1) is obtained, and from this,analytic expressions for thin medium absorption and rotationare found. These expressions can be simplified byassuming that γ ≪ γ 0 . We first consider the case wherethe power-broadened line width is much greater than theDoppler width. In this case, the absorption coefficientper unit length α is found to beα ≈α 0(2∆/γ 0 ) 2 + 2κ/3 + 1 , (7)where ∆ is the light-frequency detuning from resonance,κ = d2 E 2 0¯h 2 γγ 0(8)is the optical pumping saturation parameter, andα 0 ≈ 16π λ2 n (9)is the unsaturated absorption coefficient on resonance,where λ is the transition wavelength and n is the atomicdensity. For Ω ≪ γ, where Ω = gµB is the Larmorfrequency (g is the Landé factor, and µ is the Bohrmagneton), the slope of optical rotation per unit length,dϕ/(dΩdx), (proportional to rotation for small magneticfields) is found to bedϕdΩdx ≈ 1 α 0γ (2∆/γ 0 ) 2 + 2κ/3 , (10)where we have neglected linear optical rotation. Ingeneral, optical rotation can be induced by either lineardichroism or circular birefringence. Analysis of thesteady-state polarization of the ground state shows thatboth processes contribute here: the contribution due tolinear dichroism induced in the medium is given bydϕdΩdx∣ ≈ 1 1α 0dichr.γ 1 + (2∆/γ 0 ) 2 (2∆/γ 0 ) 2 + 2κ/3 , (11)and the contribution due to circular birefringence (arisingdue to alignment-to-orientation conversion in the presenceof both the magnetic field and the strong electricfield of the light [13]) is given bydϕdΩdx∣ ≈ 1 (2∆/γ 0 ) 2 α 0biref.γ 1 + (2∆/γ 0 ) 2 (2∆/γ 0 ) 2 + 2κ/3 . (12)