3. 2 INTERACTION WITH STATIC EXTERNAL FIELDS here 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 sRb[29 and QED effects 130, so the values of g, given in Table 6 are experimental measurements (26(except for the 5-P1/2 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 31] HB=HB gp F2 B2 where the hyperfine Lande g-factor is given by F(F+1)-I(I+1)+J(+1),F(F+1)+I(I+1)-J(J+1) 2F(F+1) 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 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 Fmp)=AB 9F B 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 my I mi. The energies are then given to lowest order by 32 3(mJm)2+3mymr-I(I+1)J(+ EyJmJ I mi)= Hfsm mr t Bhfs (2J-1)r(21-1) +He(g,mj+gr mI)B 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 numericall diagonalize Hhfs+HB. A notable exception is the Breit- Rabi formula 23, 31, 33, which applies to the ground- tate manifold of the d transition 2(27+1+9,AnmB±≌Es △Ehfs Ama 1/2 2I+1 In this formula, A Ehfs= Anfs(I+1/2)is the hyperfine splitting, m=mr tmj=mr+1/2(where the t sign taken to be the same as in(26), and △Ebs In order to avoid a sign ambiguity in evaluating(26), the more direct formula △Eh1 (g+2I91)BB can be used for the two states m=+(I +1/ 2). The Breit-Rabi formula is useful in finding the small-field shift of the"clock transition"between the mp=0 sublevels of the two hyperfine ground states, which has no first-order Zeeman shift. Using m= mg for small magnetic fields, we obtain clock (0=9)hB 2h△Ehfs3.2 Interaction with Static External Fields 7 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 85Rb [29] and QED effects [30], 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 [31] 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 [32] 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, 31, 33], 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) In order to avoid a sign ambiguity in evaluating (26), the more direct formula E|J=1/2 mJ I mI i = ∆Ehfs I 2I + 1 ± 1 2 (gJ + 2IgI )µB B (28) can be used for the two states m = ±(I + 1/2). The Breit-Rabi formula is useful in finding the small-field shift of the “clock transition” between the mF = 0 sublevels of the two hyperfine ground states, which has no first-order Zeeman shift. Using m = mF for small magnetic fields, we obtain ∆ωclock = (gJ − gI ) 2µ 2 B 2~∆Ehfs B 2 (29)