4 RESONANCe FLUORESCENCE There is, however, another common convention(used in Ref. [ 40) that is related to the convention used here by(JerllJ"=v2J+1(JlerIJ'"). Also, we have used the standard phase convention for the Clebsch-Gordan coefficients as given in Ref. 38, where formulae for the computation of the Wigner 3-3(equivalently, Clebsch- Gordan) and 6-3(equivalently, Racah) coefficients may also be found The dipole matrix elements for specific F me)-F'mp) transitions are listed in Tables 9-20 as multiples of Jer ). The tables are separated by the ground-state F number(1 or 2)and the polarization of the transition (where a+-polarized light couples mF mp= mp + 1, T-polarized light couples mp mp=mp, and -polarized light couples mp -mp=mp-1) 4 Resonance fluorescence 4.1 Symmetries of the Dipole Operator Although the hyperfine structure of 87Rb is quite complicated, it is possible to take advantage of some symmetries of the dipole operator in order to obtain relatively simple expressions for the photon scattering rates due to resonance fluorescence. In the spirit of treating the D1 and D2 lines separately, we will discuss the symmetries in this section implicitly assuming that the light is interacting with only one of the fine-structure components at a time. First, notice that the matrix elements that couple to any single excited state sublevel Fmp) add up to a factor that is independent of the particular sublevel chosen ∑F(m+q) lerglf'm)2 2J+1 F as can be verified from the dipole matrix element tables. The degeneracy-ratio factor of (2J +1)/(2 +1)(which is 1 for the Di line or 1/2 for the D2 line) is the same factor that appears in Eq. ( 38), and is a consequence of the ormalization convention (39). The interpretation of this symmetry is simply that all the excited state sublevels decay at the same rate T, and the decaying population"branches"into various ground state sublevels Another symmetry arises from summing the matrix elements from a single ground-state sublevel to the leve in a particular F energy leve Sr>F+12+{F1 IF mpIF1(mp-9)q)2 (2F+1)(2J+1) This sum SFF, is independent of the particular ground state sublevel chosen, and also obeys the sum rule The interpretation of this symmetry is that for an isotropic pump field (i.e, a pumping field with equal components in all three possible polarizations), the coupling to the atom is independent of how the population is distributed mong the sublevels. These factors SFp(which are listed in Table 8)provide a measure of the relative strength of each of the F- F transitions. In the case where the incident light is isotropic and couples two of the F levels, the atom can be treated as a two-level atom, with an effective dipole moment given by diso.eff(F→→F)P2=SF、er|J/^)P2 The factor of 1/3 in this expression comes from the fact that any given polarization of the field only interacts with one(of three) components of the dipole moment, so that it is appropriate to average over the couplings rather than sum over the couplings as in(41)10 4 Resonance Fluorescence There is, however, another common convention (used in Ref. [40]) that is related to the convention used here by (JkerkJ ′ ) = √ 2J + 1 hJkerkJ ′ i. Also, we have used the standard phase convention for the Clebsch-Gordan coefficients as given in Ref. [38], where formulae for the computation of the Wigner 3-j (equivalently, Clebsch￾Gordan) and 6-j (equivalently, Racah) coefficients may also be found. The dipole matrix elements for specific |F mF i −→ |F ′ m′ F i transitions are listed in Tables 9-20 as multiples of hJkerkJ ′ i. The tables are separated by the ground-state F number (1 or 2) and the polarization of the transition (where σ +-polarized light couples mF −→ m′ F = mF + 1, π-polarized light couples mF −→ m′ F = mF , and σ −-polarized light couples mF −→ m′ F = mF − 1). 4 Resonance Fluorescence 4.1 Symmetries of the Dipole Operator Although the hyperfine structure of 87Rb is quite complicated, it is possible to take advantage of some symmetries of the dipole operator in order to obtain relatively simple expressions for the photon scattering rates due to resonance fluorescence. In the spirit of treating the D1 and D2 lines separately, we will discuss the symmetries in this section implicitly assuming that the light is interacting with only one of the fine-structure components at a time. First, notice that the matrix elements that couple to any single excited state sublevel |F ′ m′ F i add up to a factor that is independent of the particular sublevel chosen, X q F |hF (m′ F + q)|erq|F ′ m′ F i|2 = 2J + 1 2J ′ + 1 |hJkerkJ ′ i|2 , (40) as can be verified from the dipole matrix element tables. The degeneracy-ratio factor of (2J + 1)/(2J ′ + 1) (which is 1 for the D1 line or 1/2 for the D2 line) is the same factor that appears in Eq. (38), and is a consequence of the normalization convention (39). The interpretation of this symmetry is simply that all the excited state sublevels decay at the same rate Γ, and the decaying population “branches” into various ground state sublevels. Another symmetry arises from summing the matrix elements from a single ground-state sublevel to the levels in a particular F ′ energy level: SF F ′ := X q (2F ′ + 1)(2J + 1)  J J′ 1 F ′ F I 2 |hF mF |F ′ 1 (mF − q) qi|2 = (2F ′ + 1)(2J + 1)  J J′ 1 F ′ F I 2 . (41) This sum SF F ′ is independent of the particular ground state sublevel chosen, and also obeys the sum rule X F ′ SF F ′ = 1. (42) The interpretation of this symmetry is that for an isotropic pump field (i.e., a pumping field with equal components in all three possible polarizations), the coupling to the atom is independent of how the population is distributed among the sublevels. These factors SF F ′ (which are listed in Table 8) provide a measure of the relative strength of each of the F −→ F ′ transitions. In the case where the incident light is isotropic and couples two of the F levels, the atom can be treated as a two-level atom, with an effective dipole moment given by |diso,eff(F −→ F ′ )| 2 = 1 3 SF F ′ |hJ||er||J ′ i|2 . (43) The factor of 1/3 in this expression comes from the fact that any given polarization of the field only interacts with one (of three) components of the dipole moment, so that it is appropriate to average over the couplings rather than sum over the couplings as in (41)