3.3 REDUCTION OF THE DIPOLE OPERATOR where wo is the resonant frequency of the lowest-energy transition (i. e, the DI resonance); this approximat expression is valid for light tuned far to the red of the Di line. The STRb polarizabilities are tabulated in Table 6. Notice that the differences in the excited state and ground state scalar polarizabilities are given, rather than the excited state polarizabilities, since these are the quantities that were actually measured experimentally. The polarizabilities given here are in SI units, although they are often given in cgs units(units of cm )or atomic units(units of a3, where the Bohr radius ao is given in Table 1) The SI values can be converted to cgs units via a[cm]=(100h/4TEo (a/)[Hz/(V/cm)]=5.955 213 79(30)x 10-22(a/h)[Hz/(V/cm)2](see [37] for discussion of units), and subsequently the conversion to atomic units is straightforward The level structure of 7Rb in the presence of an external dc electric field is shown in Fig. 7 in the weak-field egime through the electric hyperfine Paschen-Back regime 3.3 Reduction of the Dipole Operator The strength of the interaction between STRb and nearly-resonant optical radiation is characterized by the dipole matrix elements. Specifically, (F mpler)denotes the matrix element that couples the two hyperfine sublevels F mp)and Fmp(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 nd write the matrix element as a product of a Clebsch-Gordan coefficient and a reduced matrix element, using the Wigner-Eckart theorem 38 (F mplerglf'mp)=(FllerllF)(F mFlF'I mp q) Here, q is an index labeling the component of r in the spherical basis, and the doubled bars indicate that the atrix element is reduced. We can also write(34) in terms of a Wigner 3-3 symbol FmlFm)=(Fp-)-1+mVF+(F1F e q-p Notice that the 3-j 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 to a Wigner 6-j symbol, leaving a further reduced matrix element that depends only on the L, S, and quantum numbers 38 (F|erF)≡{ J I FerN T'F (Jer)(-1)F++1+V(2F+1)(2J+1) I. 1 FF I Again, this new matrix element can be further factored into another 6-3 symbol and a reduced matrix element nvolving only the L quantum number (‖er‖J)≡{ L SerL' SJ) L|er2(-1)y+L+1+S②2+1(2+1{L The numerical value of the (J=1/2erlJ'=3/2)(D2)and the(J=1/2 erlJ=1/2)(D1)matrix elements are given in Table 7. These values were calculated from the lifetime via the expression 39 3丌60bc32)+/(Jerl)2 Note that all the equations we have presented here assume the normalization convention ∑ J Mer J"M)H2=∑ J MleralJ'M)2=1(Jlrl) M3.3 Reduction of the Dipole Operator 9 where ω0 is the resonant frequency of the lowest-energy transition (i.e., the D1 resonance); this approximate expression is valid for light tuned far to the red of the D1 line. The 87Rb polarizabilities are tabulated in Table 6. Notice that the differences in the excited state and ground state scalar polarizabilities are given, rather than the excited state polarizabilities, since these are the quantities that were actually measured experimentally. The polarizabilities given here are in SI units, although they are often given in cgs units (units of cm3 ) or atomic units (units of a 3 0 , where the Bohr radius a0 is given in Table 1). The SI values can be converted to cgs units via α[cm3 ] = (100 · h/4πǫ0)(α/h)[Hz/(V/cm)2 ] = 5.955 213 79(30) × 10−22 (α/h)[Hz/(V/cm)2 ] (see [37] for discussion of units), and subsequently the conversion to atomic units is straightforward. The level structure of 87Rb in the presence of an external dc electric field is shown in Fig. 7 in the weak-field regime through the electric hyperfine Paschen-Back regime. 3.3 Reduction of the Dipole Operator The strength of the interaction between 87Rb 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 [38]: 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 [38]: 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 [39] 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)