4.2 RESONANCE FLUORESCENCE IN A TWO-LEVEL ATOM When the light is detuned far from the atomic resonance(4> r), the light interacts with several hyperfine levels. If the detuning is large compared to the excited-state frequency splittings, then the appropriate dipole strength comes from choosing any ground state sublevel F mF) and summing over its couplings to the excited states. In the case of T-polarized light, the sum is independent of the particular sublevel chosen 2(2F+1)(2J+1F,FIKFmelF'lme O) This sum leads to an effective dipole moment for far detuned radiation given by ae=(|)2 45 The interpretation of this factor is also straightforward. Because the radiation is far detuned, it interacts with the full J-J transition; however, because the light is linearly polarized, it interacts with only one component of the dipole operator. Then, because of spherical symmetry, ld 2= ler(2=e(=12+1912+212)=3e21212.Note that this factor of 1/3 also appears for o+ light, but only when the sublevels are uniformly populated(which,of course, is not the equilibrium configuration for these polarizations). The effective dipole moments for this case and the case of isotropic pumping are given in Table 7. 4.2 Resonance fluorescence in a two-Level atom two-level atom interacting with a monochromatic field is described by the optical Bloch equations/g atoms. A In these two cases, where we have an effective dipole moment, the atoms behave like simple two-level (Pge -peg)+TPee (46) pge=-(+i△)jg Pgg) where the Pij are the matrix elements of the density operator p: l v1, 32: =-d. Eo/h is the resonant Rabi frequency, d is the dipole operator, Eo is the electric field amplitude(e= Eo cos wnt),A: =Wr -wo is the detuning of the laser field from the atomic resonance, T= 1/T is the natural decay rate of the excited state =r/2+e is the "transverse"decay rate(where %e is a phenomenological decay rate that models collisions) made the rotating-wave approximation and used a master-equation approach to model spong ations,we have Additionally, we have ignored any effects due to the motion of the atom and decays or couplings to other auxil states. In the case of purely radiative damping(y=r/2), the excited state population settles to the steady solution (t→∞) 1+4(△/)2+2(g/)2 The(steady state) total photon scattering rate(integrated over all directions and frequencies) is then given by (t→∞): (/Isat) 1+4(△/T)2+(I/a) In writing down this expression, we have defined the saturation intensity Isat such that4.2 Resonance Fluorescence in a Two-Level Atom 11 When the light is detuned far from the atomic resonance (∆ ≫ Γ), the light interacts with several hyperfine levels. If the detuning is large compared to the excited-state frequency splittings, then the appropriate dipole strength comes from choosing any ground state sublevel |F mF i and summing over its couplings to the excited states. In the case of π-polarized light, the sum is independent of the particular sublevel chosen: X F ′ (2F ′ + 1)(2J + 1)  J J′ 1 F ′ F I 2 |hF mF |F ′ 1 mF 0i|2 = 1 3 . (44) This sum leads to an effective dipole moment for far detuned radiation given by |ddet,eff| 2 = 1 3 |hJ||er||J ′ i|2 . (45) The interpretation of this factor is also straightforward. Because the radiation is far detuned, it interacts with the full J −→ J ′ transition; however, because the light is linearly polarized, it interacts with only one component of the dipole operator. Then, because of spherical symmetry, | ˆd| 2 ≡ |erˆ| 2 = e 2 (|xˆ| 2 + |yˆ| 2 + |zˆ| 2 ) = 3e 2 |zˆ| 2 . Note that this factor of 1/3 also appears for σ ± light, but only when the sublevels are uniformly populated (which, of course, is not the equilibrium configuration for these polarizations). The effective dipole moments for this case and the case of isotropic pumping are given in Table 7. 4.2 Resonance Fluorescence in a Two-Level Atom In these two cases, where we have an effective dipole moment, the atoms behave like simple two-level atoms. A two-level atom interacting with a monochromatic field is described by the optical Bloch equations [39], ρ˙gg = iΩ 2 (˜ρge − ρ˜eg) + Γρee ρ˙ee = − iΩ 2 (˜ρge − ρ˜eg) − Γρee ρ˜˙ ge = −(γ + i∆)˜ρge − iΩ 2 (ρee − ρgg), (46) where the ρij are the matrix elements of the density operator ρ := |ψihψ|, Ω := −d · E0/~ is the resonant Rabi frequency, d is the dipole operator, E0 is the electric field amplitude (E = E0 cos ωLt), ∆ := ωL − ω0 is the detuning of the laser field from the atomic resonance, Γ = 1/τ is the natural decay rate of the excited state, γ := Γ/2 + γc is the “transverse” decay rate (where γc is a phenomenological decay rate that models collisions), ρ˜ge := ρge exp(−i∆t) is a “slowly varying coherence,” and ˜ρge = ˜ρ ∗ eg. In writing down these equations, we have made the rotating-wave approximation and used a master-equation approach to model spontaneous emission. Additionally, we have ignored any effects due to the motion of the atom and decays or couplings to other auxiliary states. In the case of purely radiative damping (γ = Γ/2), the excited state population settles to the steady state solution ρee(t → ∞) = (Ω/Γ)2 1 + 4 (∆/Γ)2 + 2 (Ω/Γ)2 . (47) The (steady state) total photon scattering rate (integrated over all directions and frequencies) is then given by Γρee(t → ∞): Rsc =  Γ 2  (I/Isat) 1 + 4 (∆/Γ)2 + (I/Isat) . (48) In writing down this expression, we have defined the saturation intensity Isat such that I Isat = 2  Ω Γ 2 , (49)