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 Hamiltonian, 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)
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△Ehfs
3.2 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 groundstate 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)
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 regime
8 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 interaction 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
3.3 REDUCTION OF THE DIPOLE OPERATOR 3.3 Reduction of the Dipole Operator The strength of the interaction between SSRb and nearly-resonant optical radiation is characterized by the dipole matrix elements. Specifically, (F mFer mp) denotes the matrix element that couples the two hyper sublevels F mo) and Fmp(where the primed variables refer to the excited states and the unprimed variables efer to the ground states). To calculate these matrix elements, it is useful to factor out the angular dependence and write the matrix element as a product of a Clebsch-Gordan coefficient and a reduced matrix element, using Eckart theorem 37 (F mpleralf'me)=(Fler FF mFIF'I mp q Here, q is an index labeling the component of r in the spherical basis, and the doubled bars indicate that the matrix element is reduced. We can also write (34) in terms of a Wigner 3-j symbol as Fm|=Fm)=(F|rr(-1-+m2F+1(E1F (35) mF 4-F Notice that the 3-3 symbol (or, equivalently, the Clebsch-Gordan coefficient)vanishes unless the sublevels satisfy mp=mp+g. This reduced matrix element can be further simplified by factoring out the F and F dependence into a Wigner 6-3 symbol, leaving a further reduced matrix element that depends only on the L, S, and J quantum numbers 37 lerF")≡( J I FerJ 'T'F (Jer)(-1)++1+1√②2P+1(2+7JJ1 Again, this new matrix element can be further factored into another 6-3 symbol and a reduced matrix element involving only the L quantum number J‖er‖J)≡{ L SJerL' SJ) (L|er1(-1)y+++s②2+1)(2+1){JJs The numerical value of the (J=1/2erlJ'=3/2)(D2)and the J=1/2erlJ'=1/2)(D1) matrix elements are given in Table 7. These values were calculated from the lifetime via the expression 38 =32+1(pe Note that all the equations we have presented here assume the normalization convention ∑JMrM)2=∑MnlM)2=r)2 here is, however, another common convention(used in Ref. 39) that is related to the convention used here by(Jer l)=v2+1erll'). Also, we have used the standard phase convention for the Clebsch-Gordar coefficients as given in Ref. 137, where formulae for the computation of the Wigner 3-j(equivalently, Clebsch- orda n) and 6-j(equivalently, Racah) coefficients may also be found The dipole matrix elements for specific F mpy-F'mp) transitions are listed in Tables 9-20 as multiples of (JlerIIJ). The tables are separated by the ground-state F number(2 or 3)and the polarization of the transition (where o+-polarized light couples mp - mp=mF+1, T-polarized light couples mp - m=mp,and -polarized light couples mp -mp=mp-1)
3.3 Reduction of the Dipole Operator 9 3.3 Reduction of the Dipole Operator The strength of the interaction between 85Rb and nearly-resonant optical radiation is characterized by the dipole matrix elements. Specifically, hF mF |er|F ′ m′ F i denotes the matrix element that couples the two hyperfine sublevels |F mF i and |F ′ m′ F i (where the primed variables refer to the excited states and the unprimed variables refer to the ground states). To calculate these matrix elements, it is useful to factor out the angular dependence and write the matrix element as a product of a Clebsch-Gordan coefficient and a reduced matrix element, using the Wigner-Eckart theorem [37]: hF mF |erq|F ′ m′ F i = hFkerkF ′ ihF mF |F ′ 1 m′ F qi. (34) Here, q is an index labeling the component of r in the spherical basis, and the doubled bars indicate that the matrix element is reduced. We can also write (34) in terms of a Wigner 3-j symbol as hF mF |erq|F ′ m′ F i = hFkerkF ′ i(−1)F ′−1+mF √ 2F + 1 F ′ 1 F m′ F q −mF . (35) Notice that the 3-j symbol (or, equivalently, the Clebsch-Gordan coefficient) vanishes unless the sublevels satisfy mF = m′ F + q. This reduced matrix element can be further simplified by factoring out the F and F ′ dependence into a Wigner 6-j symbol, leaving a further reduced matrix element that depends only on the L, S, and J quantum numbers [37]: hFkerkF ′ i ≡ hJ I FkerkJ ′ I ′ F ′ i = hJkerkJ ′ i(−1)F ′+J+1+Ip (2F′ + 1)(2J + 1) J J′ 1 F ′ F I . (36) Again, this new matrix element can be further factored into another 6-j symbol and a reduced matrix element involving only the L quantum number: hJkerkJ ′ i ≡ hL S JkerkL ′ S ′ J ′ i = hLkerkL ′ i(−1)J ′+L+1+Sp (2J ′ + 1)(2L + 1) L L′ 1 J ′ J S . (37) The numerical value of the hJ = 1/2kerkJ ′ = 3/2i (D2) and the hJ = 1/2kerkJ ′ = 1/2i (D1) matrix elements are given in Table 7. These values were calculated from the lifetime via the expression [38] 1 τ = ω 3 0 3πǫ0~c 3 2J + 1 2J ′ + 1 |hJkerkJ ′ i|2 . (38) Note that all the equations we have presented here assume the normalization convention X M′ |hJ M|er|J ′ M′ i|2 = X M′q |hJ M|erq|J ′ M′ i|2 = |hJkerkJ ′ i|2 . (39) There is, however, another common convention (used in Ref. [39]) that is related to the convention used here by (JkerkJ ′ ) = √ 2J + 1 hJkerkJ ′ i. Also, we have used the standard phase convention for the Clebsch-Gordan coefficients as given in Ref. [37], where formulae for the computation of the Wigner 3-j (equivalently, ClebschGordan) and 6-j (equivalently, Racah) coefficients may also be found. The dipole matrix elements for specific |F mF i −→ |F ′ m′ F i transitions are listed in Tables 9-20 as multiples of hJkerkJ ′ i. The tables are separated by the ground-state F number (2 or 3) and the polarization of the transition (where σ +-polarized light couples mF −→ m′ F = mF + 1, π-polarized light couples mF −→ m′ F = mF , and σ −-polarized light couples mF −→ m′ F = mF − 1)
4 RESONANCe FLUORESCENCE 4 Resonance fluorescence 4.1 Symmetries of the Dipole Operator Although the hyperfine structure of Rb is quite complicated, it is possible to take advantage of some symmetries of the dipole operator in order to obtain relatively simple expressions for the photon scattering rates due to resonance fuorescence. In the spirit of treating the Di and D2 lines separately, we will discuss the symmetries in this section implicitly assuming that the light is interacting with only one of the fine-structure components at a time. First, notice that the matrix elements that couple to any single excited state sublevel F'mky add up to a factor that is independent of the particular sublevel chosen ∑(F(m+Fm1)2=2+(l as can be verified from the dipole matrix element tables. The degeneracy-ratio factor of (2 +1)/(2 +1)(which is 1 for the Di line or 1 /2 for the D2 line) is the same factor that appears in Eq. ( 38), and is a consequence of the normalization convention(39). The interpretation of this symmetry is simply that all the excited state sublevels decay at the same rate T, and the decaying population"branches" into various ground state sublevels Another symmetry arises from summing the matrix elements from a single ground-state sublevel to the levels in a particular F energy level (41) (2F+1)(2+1)1FF1 This sum SFF is independent of the particular ground state sublevel chosen, and also obeys the sum rule (42) The interpretation of this symmetry is that for an isotropic pump field (i.e, a pumping field with equal components in all three possible polarizations), the coupling to the atom is independent of how the population is distributed among the sublevels. These factors SFp(which are listed in Table 8)provide a measure of the relative strength of each of the F F transitions. In the case where the incident light is isotropic and couples two of the F levels, the atom can be treated as a two-level atom, with an effective dipole moment given by d1aer(F→→F)2=3SF|、列er|J)2 (43) The factor of 1/3 in this expression comes from the fact that any given polarization of the field only interacts with one(of three) components of the dipole moment, so that it is appropriate to average over the couplings rather than sum over the couplings as in(41) levels. If the detuning is large compared to the excited-state frequency splittings, then th th several hyperfine When the light is detuned far from the atomic resonance(A>>r), the light interacts e appropriate dipole strength comes from choosing any ground state sublevel F mF) and summing over its couplings to the excited states. In the case of T-polarized light, the sum is independent of the particular sublevel chosen 2(F+1(2J+F FISIFmelF'1 me O)P This sum leads to an effective dipole moment for far detuned radiation given by ddt, eff-=,)
10 4 Resonance Fluorescence 4 Resonance Fluorescence 4.1 Symmetries of the Dipole Operator Although the hyperfine structure of 85Rb is quite complicated, it is possible to take advantage of some symmetries of the dipole operator in order to obtain relatively simple expressions for the photon scattering rates due to resonance fluorescence. In the spirit of treating the D1 and D2 lines separately, we will discuss the symmetries in this section implicitly assuming that the light is interacting with only one of the fine-structure components at a time. First, notice that the matrix elements that couple to any single excited state sublevel |F ′ m′ F i add up to a factor that is independent of the particular sublevel chosen, X q F |hF (m′ F + q)|erq|F ′ m′ F i|2 = 2J + 1 2J ′ + 1 |hJkerkJ ′ i|2 , (40) as can be verified from the dipole matrix element tables. The degeneracy-ratio factor of (2J + 1)/(2J ′ + 1) (which is 1 for the D1 line or 1/2 for the D2 line) is the same factor that appears in Eq. (38), and is a consequence of the normalization convention (39). The interpretation of this symmetry is simply that all the excited state sublevels decay at the same rate Γ, and the decaying population “branches” into various ground state sublevels. Another symmetry arises from summing the matrix elements from a single ground-state sublevel to the levels in a particular F ′ energy level: SF F ′ := X q (2F ′ + 1)(2J + 1) J J′ 1 F ′ F I 2 |hF mF |F ′ 1 (mF − q) qi|2 = (2F ′ + 1)(2J + 1) J J′ 1 F ′ F I 2 . (41) This sum SF F ′ is independent of the particular ground state sublevel chosen, and also obeys the sum rule X F ′ SF F ′ = 1. (42) The interpretation of this symmetry is that for an isotropic pump field (i.e., a pumping field with equal components in all three possible polarizations), the coupling to the atom is independent of how the population is distributed among the sublevels. These factors SF F ′ (which are listed in Table 8) provide a measure of the relative strength of each of the F −→ F ′ transitions. In the case where the incident light is isotropic and couples two of the F levels, the atom can be treated as a two-level atom, with an effective dipole moment given by |diso,eff(F −→ F ′ )| 2 = 1 3 SF F ′ |hJ||er||J ′ i|2 . (43) The factor of 1/3 in this expression comes from the fact that any given polarization of the field only interacts with one (of three) components of the dipole moment, so that it is appropriate to average over the couplings rather than sum over the couplings as in (41). When the light is detuned far from the atomic resonance (∆ ≫ Γ), the light interacts with several hyperfine levels. If the detuning is large compared to the excited-state frequency splittings, then the appropriate dipole strength comes from choosing any ground state sublevel |F mF i and summing over its couplings to the excited states. In the case of π-polarized light, the sum is independent of the particular sublevel chosen: X F ′ (2F ′ + 1)(2J + 1) J J′ 1 F ′ F I 2 |hF mF |F ′ 1 mF 0i|2 = 1 3 . (44) This sum leads to an effective dipole moment for far detuned radiation given by |ddet,eff| 2 = 1 3 |hJ||er||J ′ i|2 . (45)