4.2 RESONANCE FLUORESCENCE IN A TWO-LEVEL ATOM 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(=2+1912+22)=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 nd 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 38 Peg)+r (Pge -Peg)-Tpee 2=-(7+1△)ye-2(pc-pg where the Pij are the matrix elements of the density operator p: lb)vl, Q: =-d. Eo/h is the resonant Rabi frequency, d is the dipole operator, Eo is the electric field amplitude(e= Eo cos wit),A: =W -wo is the detuning of the laser field from the atomic resonance, T=1/T is the natural decay rate of the excited state 7: = r/2+c is the"transverse "decay rate(where ?e is a phenomenological decay rate that models collisions) Pge:= Pge exp(-iAt)is a"slowly varying coherence, "and Pge=Peg. 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 (y=r/2), the excited state population settles to the steady state solution pe(t→∞) 1+4(△/)2+2(/ The(steady state) total photon scattering rate (integrated over all directions and frequencies) is then given by rpe(t→∞) I/Isat) /T In writing down this expression, we have defined the saturation intensity Isat such that which gives(with I=(1/2)cEo E2) where e is the unit polarization vector of the light field, and d is the atomic dipole moment. With Isat defined in this way, the on-resonance scattering cross section o, which is proportional to Rsc( A=0/I, drops to 1 /2 of its weakly pumped value go when I= Isat. More precisely, we can define the scattering cross section o as the power radiated by the atom divided by the incident energy flux (i.e, so that the scattered power is al), which from Eq(48)becomes 1+4(△/r)2+(I/at)4.2 Resonance Fluorescence in a Two-Level Atom 11 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, |dˆ| 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 [38], 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) which gives (with I = (1/2)cǫ0E 2 0 ) Isat = cǫ0Γ 2~ 2 4|ˆǫ · d| 2 , (50) where ˆǫ is the unit polarization vector of the light field, and d is the atomic dipole moment. With Isat defined in this way, the on-resonance scattering cross section σ, which is proportional to Rsc(∆ = 0)/I, drops to 1/2 of its weakly pumped value σ0 when I = Isat. More precisely, we can define the scattering cross section σ as the power radiated by the atom divided by the incident energy flux (i.e., so that the scattered power is σI), which from Eq. (48) becomes σ = σ0 1 + 4 (∆/Γ)2 + (I/Isat) , (51)