3. 2 INTERACTION WITH STATIC EXTERNAL FIELDS 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 and the appropriate form of the interaction energy is Eq.(18). Yet stronger fields induce other behaviors, such as the quadratic Zeeman effect 31], which are beyond the scope of the present discussion The level structure of 85Rb 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 amiltonian in this case is 27, 34, 35 E2--a2E2 3n2-J(J+1) J(2J-1) where we have taken the electric field to be along the z-direction, ao and ag are respectively termed the scalar and tensor polarizabilities, and the second (a2) 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 mpI 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 3m2-F(F+1)B3X(X-1)-4F(F+1)J(J+1 JJ) (2F+3)(2F+2)F(2F-1)J(2.J-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 The static polarizability is also useful in the context of optical traps that are very far off resonance (i.e, several o 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=-12ao E2, 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 36 where wo 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 D, line "x The 85 Rb polarizabilities are tabulated in Table 6. Notice that the differences in the excited state and ground tate 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 a, where the Bohr radius ao is given in Table 1) The SI values can be converted to cgs units via a[cm=(100 h/4eo (a/h)Hz/(v/cm)=5.955 213 79(30) 10-22(a/h)[Hz/(V/cm)21(see [36] for discussion of units), and subsequently the conversion to atomic units straightforward The level structure of s Rb in the presence of an external dc electric field is shown in Fig. 7 in the weak-field gime through the electric hyperfine Paschen-Back regime8 3.2 Interaction with Static External Fields 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 [31], which are beyond the scope of the present discussion. The level structure of 85Rb 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, 34, 35] 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 = −1/2α0E 2 , 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 [36] α0(ω) = ω 2 0 α0 ω 2 0 − ω2 , (33) 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 85Rb 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 [36] for discussion of units), and subsequently the conversion to atomic units is straightforward. The level structure of 85Rb 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