3. 2 INTERACTION WITH STATIC EXTERNAL FIELDS Here, the Lande factor g, is given by [30 J(J+1)-S(S+1)+L(L+1),J(J+1)+S(S+1)-L(L+1) 2J(J+1) 2J(J+1) J(J+1)+S(S+1)-L(L+1) where the second, approximate expression comes from taking the approximate values gs a 2 and g a 1. The expression here does not include corrections due to the complicated multielectron structure of Rb [30 and QED effects[31], so the values of g, given in Table 6 are experimental measurements [26](except for the 52P1/2 state value, for which there has apparently been no experimental measurement) If the energy shift due to the magnetic field is small compared to the hyperfine splittings, then similarly F a good quantum number, so the interaction Hamiltonian becomes s HB=HBgF F: B2. where the hyperfine Lande g-factor is given by F(F+1)-I(I+1)+J(J+1 2F(F+1) 2F(F+1) (23) F(F+1)-I(I+1)+J(J+1) 2F(F+1) The second, approximate expression here neglects the nuclear term, which is a correction at the level of 0. 1% since gr is much smaller than g For weak magnetic fields, the interaction Hamiltonian HB perturbs the zero-field eigenstates of Hhfs. To lowest order, the levels split linearly according to 23 △E AB g The approximate gp factors computed from Eq(23)and the corresponding splittings between adjacent magnetic sublevels are given in Figs. 2 and 3. The splitting in this regime is called the anomalous Zeeman effect For strong fields where the appropriate interaction is described by Eq.(20), the interaction term dominates the hyperfine energies, so that the hyperfine Hamiltonian perturbs the strong-field eigenstates m, I mi. The energies are then given to lowest order by 3 EyJmJ I mi)=Ahfsmjm,+ Bhfs 3(m;m)2+m;m-I(I+1)J(J+1) 2J(2J-1)I(21-1) +p(9m+9nm)B2.(25) The energy shift in this regime is called the Paschen-Back effect. For intermediate fields, the energy shift is more difficult to calculate, and in general one must numerically diagonalize Hhfs HB. A notable exception is the Breit- Rabi formula 23, 32, 34, which applies to the ground- state manifold of the d transition E1=1/2 mIMi= 2(27+1)+9AmB±AE In this formula, A Ehfs =Ahfs(I +1/2)is the hyperfine splitting, m=mtmy=mrt1/2(where the t sign is aken to be the same as in(26), and △Ehfs3.2 Interaction with Static External Fields 7 Here, the Land´e factor gJ is given by [30] gJ = gL J(J + 1) − S(S + 1) + L(L + 1) 2J(J + 1) + gS J(J + 1) + S(S + 1) − L(L + 1) 2J(J + 1) ≃ 1 + J(J + 1) + S(S + 1) − L(L + 1) 2J(J + 1) , (21) where the second, approximate expression comes from taking the approximate values gS ≃ 2 and gL ≃ 1. The expression here does not include corrections due to the complicated multielectron structure of 87Rb [30] and QED effects [31], so the values of gJ given in Table 6 are experimental measurements [26] (except for the 52P1/2 state value, for which there has apparently been no experimental measurement). If the energy shift due to the magnetic field is small compared to the hyperfine splittings, then similarly F is a good quantum number, so the interaction Hamiltonian becomes [32] HB = µB gF Fz Bz, (22) where the hyperfine Land´e g-factor is given by gF = gJ F(F + 1) − I(I + 1) + J(J + 1) 2F(F + 1) + gI F(F + 1) + I(I + 1) − J(J + 1) 2F(F + 1) ≃ gJ F(F + 1) − I(I + 1) + J(J + 1) 2F(F + 1) . (23) The second, approximate expression here neglects the nuclear term, which is a correction at the level of 0.1%, since gI is much smaller than gJ . For weak magnetic fields, the interaction Hamiltonian HB perturbs the zero-field eigenstates of Hhfs. To lowest order, the levels split linearly according to [23] ∆E|F mF i = µB gF mF Bz. (24) The approximate gF factors computed from Eq. (23) and the corresponding splittings between adjacent magnetic sublevels are given in Figs. 2 and 3. The splitting in this regime is called the anomalous Zeeman effect. For strong fields where the appropriate interaction is described by Eq. (20), the interaction term dominates the hyperfine energies, so that the hyperfine Hamiltonian perturbs the strong-field eigenstates |J mJ I mI i. The energies are then given to lowest order by [33] E|J mJ I mI i = AhfsmJmI + Bhfs 3(mJmI ) 2 + 3 2mJmI − I(I + 1)J(J + 1) 2J(2J − 1)I(2I − 1) + µB(gJ mJ + gI mI )Bz. (25) The energy shift in this regime is called the Paschen-Back effect. For intermediate fields, the energy shift is more difficult to calculate, and in general one must numerically diagonalize Hhfs + HB. A notable exception is the Breit-Rabi formula [23, 32, 34], which applies to the ground￾state manifold of the D transition: E|J=1/2 mJ I mI i = − ∆Ehfs 2(2I + 1) + gI µB m B ± ∆Ehfs 2  1 + 4mx 2I + 1 + x 2 1/2 . (26) In this formula, ∆Ehfs = Ahfs(I + 1/2) is the hyperfine splitting, m = mI ± mJ = mI ± 1/2 (where the ± sign is taken to be the same as in (26)), and x = (gJ − gI )µB B ∆Ehfs . (27)