3. 2 INTERACTION WITH STATIC EXTERNAL FIELDS In order to avoid a sign ambiguity in evaluating(26), the more direct formula E=1/2m1m)=△B2+1+2(+29)nB 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= mp for small magnetic fields, we obtain c=(-0)1B2 to second order in the field strength. If the magnetic field is sufficiently strong that the hyperfine Hamiltonian is negligible compared to the inter- action Hamiltonian, then the effect is termed the normal Zeeman effect for hyperfine structure. For even stronger fields, there are Paschen-Back and normal Zeeman regimes for the fine structure, where states with different J can mix, and the appropriate form of the interaction energy is Eq.(18). Yet stronger fields induce other behaviors such as the quadratic Zeeman effect 32, which are beyond the scope of the present discussion The level structure of 87Rb in the presence of a magnetic field is shown in Figs. 4-6 in the weak-field(anomalous Zeeman) regime through the hyperfine Paschen-Back regime 3.2.2 Electric Fields An analogous effect, the dc stark effect, occurs in the presence of a static external electric field. The interaction Hamiltonian in this case is 27, 35, 36 1m232-J(J+1) J(2J-1) where we have taken the electric field to be along the z-direction, ao and a are respectively termed the scalar nd tensor polarizabilities, and the second (a) term is nonvanishing only for the J= 3/2 level. The first term shifts all the sublevels with a given J together, so that the Stark shift for the J=1/2 states is trivial. The only mechanism for breaking the degeneracy of the hyperfine sublevels in(30) is the J2 contribution in the tensor erm. This interaction splits the sublevels such that sublevels with the same value of mFI remain degenerate An expression for the hyperfine Stark shift, assuming a weak enough field that the shift is small compared to the △E 1Fmp)=-2 E E F+3(2F+2)F(2F-1((+ F(F+1)3X(X-1)-4F(F+1) 〓a2 1) X=F(F+1)+J(J+1)-I(I+1) For stronger fields, when the Stark interaction Hamiltonian dominates the hyperfine splittings, the levels split according to the value of m, leading to an electric-field analog to the Paschen-Back effect for magnetic fields. to many nm away from resonance, where the rotating-wave approximation is invalid), since the optical potentia.? The static polarizability is also useful in the context of optical traps that are very far off resonance(i.e, seve given in terms of the ground-state polarizability as V=-1/2aoE, where E is the amplitude of the optical field. a slightly more accurate expression for the far-off resonant potential arises by replacing the static polarizability with the frequency-dependent polarizability 378 3.2 Interaction with Static External Fields 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) to second order in the field strength. If the magnetic field is sufficiently strong that the hyperfine Hamiltonian is negligible compared to the inter￾action Hamiltonian, then the effect is termed the normal Zeeman effect for hyperfine structure. For even stronger fields, there are Paschen-Back and normal Zeeman regimes for the fine structure, where states with different J can mix, and the appropriate form of the interaction energy is Eq. (18). Yet stronger fields induce other behaviors, such as the quadratic Zeeman effect [32], which are beyond the scope of the present discussion. The level structure of 87Rb in the presence of a magnetic field is shown in Figs. 4-6 in the weak-field (anomalous Zeeman) regime through the hyperfine Paschen-Back regime. 3.2.2 Electric Fields An analogous effect, the dc Stark effect, occurs in the presence of a static external electric field. The interaction Hamiltonian in this case is [27, 35, 36] HE = − 1 2 α0E 2 z − 1 2 α2E 2 z 3J 2 z − J(J + 1) J(2J − 1) , (30) where we have taken the electric field to be along the z-direction, α0 and α2 are respectively termed the scalar and tensor polarizabilities, and the second (α2) term is nonvanishing only for the J = 3/2 level. The first term shifts all the sublevels with a given J together, so that the Stark shift for the J = 1/2 states is trivial. The only mechanism for breaking the degeneracy of the hyperfine sublevels in (30) is the Jz contribution in the tensor term. This interaction splits the sublevels such that sublevels with the same value of |mF | remain degenerate. An expression for the hyperfine Stark shift, assuming a weak enough field that the shift is small compared to the hyperfine splittings, is [27] ∆E|J I F mF i = − 1 2 α0E 2 z − 1 2 α2E 2 z [3m2 F − F(F + 1)][3X(X − 1) − 4F(F + 1)J(J + 1)] (2F + 3)(2F + 2)F(2F − 1)J(2J − 1) , (31) where X = F(F + 1) + J(J + 1) − I(I + 1). (32) For stronger fields, when the Stark interaction Hamiltonian dominates the hyperfine splittings, the levels split according to the value of |mJ |, leading to an electric-field analog to the Paschen-Back effect for magnetic fields. The static polarizability is also useful in the context of optical traps that are very far off resonance (i.e., several to many nm away from resonance, where the rotating-wave approximation is invalid), since the optical potential is given in terms of the ground-state polarizability as V = −1/2α0E2 , where E is the amplitude of the optical field. A slightly more accurate expression for the far-off resonant potential arises by replacing the static polarizability with the frequency-dependent polarizability [37] α0(ω) = ω 2 0 α0 ω 2 0 − ω2 , (33)