3. 2 INTERACTION WITH STATIC EXTERNAL FIELDS which leads to a hyperfine energy shift of △Ehfs= -Husk+Bhfs 2K(K+1)-2(I+1)J(J+1) 4I(2I-1)J(2J-1) +Ch 5K2(K/4+1)+K[I(I+1)+J(J+1)+3-3I(+1)J(J+1)-5(I+1)J(J+1) I(-1)(2-1)J(J-1)(2J-1) where K=F(F+1)-I(I+1)-J(J+1) Ahfs is the magnetic dipole constant, Bhfs is the electric quadrupole constant, and Chfs is the magnetic octupole constant(although the terms with Bhfs and Chfs apply only to the excited manifold of the D transition and not to the levels with =1/2). These constants for the Rb D line are listed in Table 5. The value for the ground state Anfs constant is from [26], while the constants listed for the 52P3/2 manifold are averages of the values from 26 and [10]. The Ahfs constant for the 5-P1/2 manifold is the average from the recent measurements of [10] and 11. These measurements are not yet sufficiently precise to have provided a nonzero value for Chfs, and thus it is not listed. The energy shift given by(16)is relative to the unshifted value(the"center of gravity")listed in Table 3. The hyperfine structure of sRb, along with the energy splitting values, is diagrammed in Figs. 2 and 3 3.2 Interaction with static External fields 3.2.1 Magnetic Field Each of the hyperfine(F)energy levels contains 2F +I magnetic sublevels that determine the angular distribution of the electron wave function. In the absence of external magnetic fields, these sublevels are degenerate. However when an external magnetic field is applied, their degeneracy is broken. The Hamiltonian describing the atomic interaction with the magnetic field is B (gsS+gL+gD·B (18) =2(gsS2+9L2+9L2)B2, if we take the magnetic field to be along the z-direction (i.e, along the atomic quantization axis In this Hamilton- nian, the quantities s, gL, and g, are respectively the electron spin, electron orbital, and nuclear "g-factors "that account for various modifications to the corresponding magnetic dipole moments. The values for these factors are listed in Table 6, with the sign convention of (26. The value for gs has been measured very precisely, and the value given is the CODATA recommended value. The value for gr is approximately 1, but to account for the finite nuclear mass, the quoted value is given by 9L (19) mnt which is correct to lowest order in me/muc, where me is the electron mass and muc is the nuclear mass 29 The nuclear factor g, accounts for the entire complex structure of the nucleus, and so the quoted value is an experimental measurement 26 If the energy shift due to the magnetic field is small compared to the fine-structure splitting, then is a good uantum number and the interaction hamiltonian can be written H (gJJ2+91I2)B2 Here, the Lande factor g, is given by (291 (J+1)-S(S+1)+L(L+1),J(+1)+S(S+1)-L(L+1) +9s 2J(J+1 J(J+1)+S(S+1)-L(L+1 J(J+1)6 3.2 Interaction with Static External Fields which leads to a hyperfine energy shift of ∆Ehfs = 1 2 AhfsK + Bhfs 3 2K(K + 1) − 2I(I + 1)J(J + 1) 4I(2I − 1)J(2J − 1) + Chfs 5K2 (K/4 + 1) + K[I(I + 1) + J(J + 1) + 3 − 3I(I + 1)J(J + 1)] − 5I(I + 1)J(J + 1) I(I − 1)(2I − 1)J(J − 1)(2J − 1) , (16) where K = F(F + 1) − I(I + 1) − J(J + 1), (17) Ahfs is the magnetic dipole constant, Bhfs is the electric quadrupole constant, and Chfs is the magnetic octupole constant (although the terms with Bhfs and Chfs apply only to the excited manifold of the D2 transition and not to the levels with J = 1/2). These constants for the 85Rb D line are listed in Table 5. The value for the ground state Ahfs constant is from [26], while the constants listed for the 52P3/2 manifold are averages of the values from [26] and [10]. The Ahfs constant for the 52P1/2 manifold is the average from the recent measurements of [10] and [11]. These measurements are not yet sufficiently precise to have provided a nonzero value for Chfs, and thus it is not listed. The energy shift given by (16) is relative to the unshifted value (the “center of gravity”) listed in Table 3. The hyperfine structure of 85Rb, along with the energy splitting values, is diagrammed in Figs. 2 and 3. 3.2 Interaction with Static External Fields 3.2.1 Magnetic Fields Each of the hyperfine (F) energy levels contains 2F + 1 magnetic sublevels that determine the angular distribution of the electron wave function. In the absence of external magnetic fields, these sublevels are degenerate. However, when an external magnetic field is applied, their degeneracy is broken. The Hamiltonian describing the atomic interaction with the magnetic field is HB = µB ~ (gSS + gLL + gI I) · B = µB ~ (gSSz + gLLz + gI Iz)Bz, (18) if we take the magnetic field to be along the z-direction (i.e., along the atomic quantization axis). In this Hamilto￾nian, the quantities gS, gL, and gI are respectively the electron spin, electron orbital, and nuclear “g-factors” that account for various modifications to the corresponding magnetic dipole moments. The values for these factors are listed in Table 6, with the sign convention of [26]. The value for gS has been measured very precisely, and the value given is the CODATA recommended value. The value for gL is approximately 1, but to account for the finite nuclear mass, the quoted value is given by gL = 1 − me mnuc , (19) which is correct to lowest order in me/mnuc, where me is the electron mass and mnuc is the nuclear mass [29]. The nuclear factor gI accounts for the entire complex structure of the nucleus, and so the quoted value is an experimental measurement [26]. If the energy shift due to the magnetic field is small compared to the fine-structure splitting, then J is a good quantum number and the interaction Hamiltonian can be written as HB = µB ~ (gJ Jz + gI Iz)Bz. (20) Here, the Land´e factor gJ is given by [29] 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)