3.3 REDUCTION OF THE DIPOLE OPERATOR 3.3 Reduction of the Dipole Operator The strength of the interaction between SSRb and nearly-resonant optical radiation is characterized by the dipole matrix elements. Specifically, (F mFer mp) denotes the matrix element that couples the two hyper sublevels F mo) and Fmp(where the primed variables refer to the excited states and the unprimed variables efer to the ground states). To calculate these matrix elements, it is useful to factor out the angular dependence and write the matrix element as a product of a Clebsch-Gordan coefficient and a reduced matrix element, using Eckart theorem 37 (F mpleralf'me)=(Fler FF mFIF'I mp q Here, q is an index labeling the component of r in the spherical basis, and the doubled bars indicate that the matrix element is reduced. We can also write (34) in terms of a Wigner 3-j symbol as Fm|=Fm)=(F|rr(-1-+m2F+1(E1F (35) mF 4-F Notice that the 3-3 symbol (or, equivalently, the Clebsch-Gordan coefficient)vanishes unless the sublevels satisfy mp=mp+g. This reduced matrix element can be further simplified by factoring out the F and F dependence into a Wigner 6-3 symbol, leaving a further reduced matrix element that depends only on the L, S, and J quantum numbers 37 lerF")≡( J I FerJ 'T'F (Jer)(-1)++1+1√②2P+1(2+7JJ1 Again, this new matrix element can be further factored into another 6-3 symbol and a reduced matrix element involving only the L quantum number J‖er‖J)≡{ L SJerL' SJ) (L|er1(-1)y+++s②2+1)(2+1){JJs The numerical value of the (J=1/2erlJ'=3/2)(D2)and the J=1/2erlJ'=1/2)(D1) matrix elements are given in Table 7. These values were calculated from the lifetime via the expression 38 =32+1(pe Note that all the equations we have presented here assume the normalization convention ∑JMrM)2=∑MnlM)2=r)2 here is, however, another common convention(used in Ref. 39) that is related to the convention used here by(Jer l)=v2+1erll'). Also, we have used the standard phase convention for the Clebsch-Gordar coefficients as given in Ref. 137, where formulae for the computation of the Wigner 3-j(equivalently, Clebsch- orda n) and 6-j(equivalently, Racah) coefficients may also be found The dipole matrix elements for specific F mpy-F'mp) transitions are listed in Tables 9-20 as multiples of (JlerIIJ). The tables are separated by the ground-state F number(2 or 3)and the polarization of the transition (where o+-polarized light couples mp - mp=mF+1, T-polarized light couples mp - m=mp,and -polarized light couples mp -mp=mp-1)3.3 Reduction of the Dipole Operator 9 3.3 Reduction of the Dipole Operator The strength of the interaction between 85Rb and nearly-resonant optical radiation is characterized by the dipole matrix elements. Specifically, hF mF |er|F ′ m′ F i denotes the matrix element that couples the two hyperfine sublevels |F mF i and |F ′ m′ F i (where the primed variables refer to the excited states and the unprimed variables refer to the ground states). To calculate these matrix elements, it is useful to factor out the angular dependence and write the matrix element as a product of a Clebsch-Gordan coefficient and a reduced matrix element, using the Wigner-Eckart theorem [37]: hF mF |erq|F ′ m′ F i = hFkerkF ′ ihF mF |F ′ 1 m′ F qi. (34) Here, q is an index labeling the component of r in the spherical basis, and the doubled bars indicate that the matrix element is reduced. We can also write (34) in terms of a Wigner 3-j symbol as hF mF |erq|F ′ m′ F i = hFkerkF ′ i(−1)F ′−1+mF √ 2F + 1  F ′ 1 F m′ F q −mF  . (35) Notice that the 3-j symbol (or, equivalently, the Clebsch-Gordan coefficient) vanishes unless the sublevels satisfy mF = m′ F + q. This reduced matrix element can be further simplified by factoring out the F and F ′ dependence into a Wigner 6-j symbol, leaving a further reduced matrix element that depends only on the L, S, and J quantum numbers [37]: hFkerkF ′ i ≡ hJ I FkerkJ ′ I ′ F ′ i = hJkerkJ ′ i(−1)F ′+J+1+Ip (2F′ + 1)(2J + 1)  J J′ 1 F ′ F I  . (36) Again, this new matrix element can be further factored into another 6-j symbol and a reduced matrix element involving only the L quantum number: hJkerkJ ′ i ≡ hL S JkerkL ′ S ′ J ′ i = hLkerkL ′ i(−1)J ′+L+1+Sp (2J ′ + 1)(2L + 1)  L L′ 1 J ′ J S  . (37) The numerical value of the hJ = 1/2kerkJ ′ = 3/2i (D2) and the hJ = 1/2kerkJ ′ = 1/2i (D1) matrix elements are given in Table 7. These values were calculated from the lifetime via the expression [38] 1 τ = ω 3 0 3πǫ0~c 3 2J + 1 2J ′ + 1 |hJkerkJ ′ i|2 . (38) Note that all the equations we have presented here assume the normalization convention X M′ |hJ M|er|J ′ M′ i|2 = X M′q |hJ M|erq|J ′ M′ i|2 = |hJkerkJ ′ i|2 . (39) There is, however, another common convention (used in Ref. [39]) 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. [37], 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 (2 or 3) 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)