4 RESONANCe FLUORESCENCE 4 Resonance fluorescence 4.1 Symmetries of the Dipole Operator Although the hyperfine structure of Rb 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 fuorescence. In the spirit of treating the Di 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'mky add up to a factor that is independent of the particular sublevel chosen ∑(F(m+Fm1)2=2+(l as can be verified from the dipole matrix element tables. The degeneracy-ratio factor of (2 +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 normalization 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 levels in a particular F energy level (41) (2F+1)(2+1)1FF1 This sum SFF is independent of the particular ground state sublevel chosen, and also obeys the sum rule (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 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 d1aer(F→→F)2=3SF|、列er|J)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) levels. If the detuning is large compared to the excited-state frequency splittings, then th th several hyperfine When the light is detuned far from the atomic resonance(A>>r), the light interacts e 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(F+1(2J+F FISIFmelF'1 me O)P This sum leads to an effective dipole moment for far detuned radiation given by ddt, eff-=,)10 4 Resonance Fluorescence 4 Resonance Fluorescence 4.1 Symmetries of the Dipole Operator Although the hyperfine structure of 85Rb 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). 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)