Calculation of the two-photon cross-section for excitation of krypton to the $6p$ shell I am an engineer working on a measurement technique called tagging velocimetry to measure velocities in gas flows. The basic idea is that we excite krypton atoms (that are seeded into the flow) to various energy levels and then we take a picture of the resulting fluoresce that occurs as a results of spontaneous emission. What I am trying to do is to make a population model that would show how many atoms are in each energy level. This way we can compare various excitation schemes and determine which energy level under which conditions will give the most fluorescence.
To do this I need to calculate the two photon cross sections of krypton for two levels $5p[3/2]_2$ and $5p[1/2]_0$ (Racah notation). I found two papers that deal with the two photon excitation rates for krypton, but those are for the $6p$ levels. The first paper is by Bokor and the second one is by Khambatta. 
Now I can follow along the work by Khambatta since he gave a simple formula to evaluate the matrix element (Eq 15 in the paper), and I can evaluate all the other parameters. Therefore I was successfully able to recreate the results in Khambatta and I can apply the same formulation to the $5p$ levels, by using the appropriate values for the various parameters.
The problem is the Bokor paper. In it I do not understand how to evaluate the expression for the matrix element (Eq 3 in the paper). In Eq 3 I understand the 3-j symbol and can evaluate it in Mathemtica etc but I do not understand what the stuff in the <> is or how to calculate it. Specifically what is $\gamma$ and ||$\mu$||. I know the stuff in the <> cannot simply be multiplied as $J_g=0$ and that would make the whole term 0. Could someone tell me how to evaluate that with the given parameters?
Secondly I used the method in Khambatta and applied it to $6p$ states in Bokor (using the parameters in the Bokor paper) and the results are off by factors of 4 and 10. This is concerning as I thought they should be somewhat similar in magnitude and begs the question whose methodology is correct?
Bear in mind that I am a mechanical engineer with no experience in quantum mechanics so please pardon my limited understanding of the subject. 
 A: The Bokor et al. paper isn't really calculating the two-photon cross-section as such; instead, it is basically modelling the two-photon $4p^6 \to 5p^56p^1$ transition as a concatenation of the transitions $4p^6 \to 4p^55s^1$ and $4p^55s^1 \to 4p^5 6p^1$ by assuming that only one term contributes to the second-order perturbation-theory expression, and then obtaining the dipole transition matrix elements of those two transitions from single-photon experiments.
The equation you're worried about,
$$
\langle k|\hat \epsilon \cdot \vec \mu| g \rangle 
=
(-1)^{J_k-M_k} 
\left< \gamma_k J_k \,\middle\|\, \mu \,\middle\|\, \gamma_g J_g \right> 
\begin{pmatrix} J_k & 1 & J_g \\ -M_k & 0 & M_g\end{pmatrix},
\tag 3
$$
doesn't really say all that much, either: it is simply an application of the Wigner-Eckart theorem, which is a standard atomic-physics tool for abstracting away all the trivial directional dependence on the quantization axes and magnetic quantum numbers of the initial final state, as well as the direction of the laser's polarization, so that you're left with a $3j$ symbol that has all the trivial stuff and  a reduced matrix element
$$
\left< \gamma_k J_k \,\middle\|\, \mu \,\middle\|\, \gamma_g J_g \right> 
$$
which contains all the stuff worth calculating. This particular reduced matrix element represents a dipole transition (hence the $\mu$) from the $g$ to the $k$ states; the double mid-bars are used to encode the fact that it's a reduced matrix element instead of a regular one; the $J_k$ and $J_g$ are the angular momenta of the two states, and the $\gamma_k$ and $\gamma_g$ denote the quantum numbers pertaining to the radial dependence of the state and all other relevant non-angular variables.
Thus, what you have there is simply a reduction of the abstract matrix element as used in the calculation to a compact form that can be compared to other configurations:

The dipole matrix element for the $5s[\frac32]_1$-$4p^6({}^1S_0)$ resonance transition may be derived from the measured oscillator strength for this transition19 of $f= 0.219$.

and

The matrix elements for the $6p[\frac12]_0$-$5s[\frac32]_1$ and $6p[\frac32]_1$-$5s[\frac32]_1$ transitions are obtained by using our measured values for the $6p[\frac12]_0$-$6p[\frac32]_1$ radiative lifetimes (see Sec. IV) and previously measured radiative branching rations.20

both of which contain explicit references that presumably contain in-depth calculations of the necessary values.
Hopefully this will be enough to point you in the right direction.
