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 that
4.2 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)
4.3 OPTICAL PUMPING which gives(with I=(1/2)cEoE0) where a 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 a, which is proportional to Rsc(A=0/I, drops to 1/2 of its weakly pumped value o 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 aI), which from Eq(48)becomes 1+4(△/r)2+(I/Ia) where the on-resonance cross section is defined by 21sat (52) Additionally, the saturation intensity(and thus the scattering cross section)depends on the polarization of the pumping light as well as the atomic alignment, although the smallest saturation intensity (Isat(mF=+2-m'p=+3) discussed below) is often quoted as a representative value. Some saturation intensities and scattering cross sections corresponding to the discussions in Section 4.1 are given in Table 7. A more detailed discussion of the resonance Huorescence from a two-level atom, including the spectral distribution of the emitted radiation, can be found in Ref. 39 4.3 Optical Pumping If none of the special situations in Section 4.1 applies to the fluorescence problem of interest, then the effects of optical pumping must be accounted for. A discussion of the effects of optical pumping in an atomic vapor on the saturation intensity using a rate-equation approach can be found in Ref. 41]. Here, however, we will carry out an analysis based on the generalization of the optical Bloch equations(46)to the degenerate level structure of alkali atoms. The appropriate master equation for the density matrix of a Fg- Fe hyperfine transition is [ 1, 42-44 a ∑?(ma,mg) g, P B 6B∑2(m,m)lm, field) +6g∑9'(ma,ma)Am,Bm-6∑?'(m,mx)am,gms doe r p (ma+q),e(ma+q) )IFg I ma g)(Fe(mg+q)lFg 1 mg g)
12 4.3 Optical Pumping 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) where the on-resonance cross section is defined by σ0 = ~ωΓ 2Isat . (52) Additionally, the saturation intensity (and thus the scattering cross section) depends on the polarization of the pumping light as well as the atomic alignment, although the smallest saturation intensity (Isat(mF =±2→ m′ F =±3), discussed below) is often quoted as a representative value. Some saturation intensities and scattering cross sections corresponding to the discussions in Section 4.1 are given in Table 7. A more detailed discussion of the resonance fluorescence from a two-level atom, including the spectral distribution of the emitted radiation, can be found in Ref. [39]. 4.3 Optical Pumping If none of the special situations in Section 4.1 applies to the fluorescence problem of interest, then the effects of optical pumping must be accounted for. A discussion of the effects of optical pumping in an atomic vapor on the saturation intensity using a rate-equation approach can be found in Ref. [41]. Here, however, we will carry out an analysis based on the generalization of the optical Bloch equations (46) to the degenerate level structure of alkali atoms. The appropriate master equation for the density matrix of a Fg → Fe hyperfine transition is [1, 42–44] ∂ ∂tρ˜α mα, β mβ = − i 2 δαe X mg Ω(mα, mg) ˜ρg mg, β mβ − δgβ X me Ω(me, mβ) ˜ρα mα, e me + δαg X me Ω ∗ (me, mα) ˜ρe me, β mβ − δeβ X mg Ω ∗ (mβ, mg) ˜ρα mα, g mg (pump field) − δαeδeβ Γ ˜ρα mα, β mβ − δαeδgβ Γ 2 ρ˜α mα, β mβ − δαgδeβ Γ 2 ρ˜α mα, β mβ + δαgδgβ Γ X 1 q=−1 h ρ˜e (mα+q), e (mβ+q) hFe (mα + q)|Fg 1 mα qihFe (mβ + q)|Fg 1 mβ qi i (dissipation) + i(δαeδgβ − δαgδeβ) ∆ ˜ρα mα, β mβ ) (free evolution) (53)