4 peaks arising from the same ground, excited, and inter- mediate states are not the same in the different spectral groups. We can understand and calculate the spectra using the standard two-photon transition probabilities deriv- able from second-order time-dependent perturbation the- ory. We start by considering a transition of an atom in the ground state with total angular momentum F , ∣ 6S$_{1}$$_{/}$$_{2}$ F 〉 , to an excited state with total angular momen- tum F $^{′′}$, | n $^{′′}$L′′ J ′′ F $^{′′}$〉 via excitation by two cw lasers with angular frequencies ω$_{1}$ and ω$_{2}$ , intensities I$_{1}$ and I$_{2}$ , and wave vectors ⃗ k$_{1}$ and ⃗ k$_{2}$ . Here n ′′ refers to the principal quantum number of the excited state. We further as- sume that ω$_{1}$ is close to the resonant frequency for the 6S$_{1}$$_{/}$$_{2}$ → 6P$_{J}$ ′ transition, with J ′ = 1 / 2 or 3 / 2. The probability of the atom to make a transition is [26] hyperfine level of the intermediate state. For example, the d , e , and f peaks in Fig. 5 all correspond to ex- citation from the the 6S$_{1}$$_{/}$$_{2}$ ( F = 4) ground state to the 8S$_{1}$$_{/}$$_{2}$ ( F ′′ = 4) excited state. The three peaks correspond to excitation through the 6P$_{3}$$_{/}$$_{2}$ ( F ′ = 3 , 4 , 5) intermedi- ate states. We see the groups recur as a different pair of optical modes is resonant with a given transition. This repeating pattern occurs at intervals of δf$_{rep}$ ≈ 1 . 2 kHz, corresponding to a change in the two-photon optical fre- quency of ≈ 1 GHz (i.e., the laser mode spacing). For a given group of three peaks there are three different veloc- ity classes that satisfy the resonance condition for each of the corresponding hyperfine components of the inter- mediate level. For another group of three peaks, the res- onant condition for the components of the intermediate level is satisfied by a different pair of comb modes, and by different velocity classes. Thus, the amplitudes of the where ⃗v is the velocity of the atom, M$_{F}$ , M ′ $_{F}$, and M ′′ F are the projections of the total angular momenta F , F $^{′}$, and F ′′ along the axis of quantization, ⃗ d is the electric dipole operator, ˆ e$_{1(2)}$ is the unit vector along the direction of the polarization for the first(second) laser beam, γ$_{nL}$$_{J}$ is the homogeneous line width of the state | n L$_{J}$ 〉 , and ω$_{nL}$$_{J}$$_{F}$$_{:}$$_{n}$ ′ L ′ J $_{′}$F ′ is the resonant angular frequency of the transition | n L$_{J}$ F 〉 → | n $^{′}$L′ J ′ F $^{′}$〉 . For a given light polar- ization, specified by the vector component in the spheri- cal basis q , the matrix elements for the different magnetic sublevels can be related to reduced matrix elements that do not depend on the magnetic sublevels or the total angular momentum ( F , F $^{′}$, or F $^{′′}$) by use of the Wigner- Eckart theorem and standard Clebsch-Gordan relations as [27] Here I = 7 2 is the nuclear spin and the ( ) and { } terms are the usual 3- J and 6- J symbols. Thus, the transition probabilities for different ground-state an- gular momenta F and excited-state angular momenta F ′′ can be related to a common factor by use of the appro- priate Clebsch-Gordan coefficients. If we assume the two laser beams are counter-propagating then examination of Eq. 2 reveals that for a given transition from the ground state with angular momentum F , through an interme- diate state with angular momentum F $^{′}$, to a final state with F ′′ the transition probability will be a maximum 〈 n $^{′}$L′ J ′ F $^{′}$M ′ $_{F}$| d$_{q}$ | nL$_{J}$FM$_{F}$ 〉 = ( − 1) F ′ − M ′ F ( F ′ 1 F − M ′ F q M$_{F}$ ) ( n $^{′}$L′ J ′ F $^{′}$| | d || nL$_{J}$F ) = ( − 1) F ′ − M ′ $_{F}$+ I +1+ J + F √ (2 F + 1)(2 F ′ + 1) ( F ′ 1 F − M ′ F q M$_{F}$ ) × { J ′ F ′ I F J 1 } ( n $^{′}$L′ J ′ || d || nL$_{J}$ ) . (3) P ( 6S 1 / 2 F, n $^{′′}$L′′ J ′′ F $^{′′}$) ∝ 1 2 F + 1 I 1 I 2 [ ω $^{6S$_{1}$}$/ $_{2}$F : n ′′ L ′′ J $_{′′}$F ′′ − ( ω 1 + ⃗ k 1 · ⃗ v ) − ( ω 2 + ⃗ k 2 · ⃗ v )] 2 + ( γ$_{n}$$_{′′}$$_{L}$$_{′′}$ J ′′ 2 ) 2 × ∑ M$_{F}$ ,M ′′ F ∣ ∣ ∑ F ′ ,M ′ F 〈 n $^{′′}$L′′ J ′′ F $^{′′}$M ′′ $_{F}$| ˆ e 2 · ⃗ d | 6P$_{J}$ ′ F $^{′}$M ′ $_{F}$〉〈 6P$_{J}$ ′ F $^{′}$M ′ $_{F}$| ˆ e 1 · ⃗ d ∣ 6S 1 / 2 F M$_{F}$ 〉 ω $^{6S$_{1}$}$/ $_{2}$F $^{:6 P}$J $_{′}$F ′ − ( ω 1 + ⃗ k 1 · ⃗ v ) − i γ $^{6P}$J ′ 2 ∣ ∣$^{2}$$^{,}$ (2)