# Time-Dependent, Two-Photon Rabi Frequency Shift Observed in Multi-Frequency Raman Spectra
## 5 Discussion  
As discussed in the Introduction, a second harmonic FROG iteration program does not give unique results for complicated electric fields. We were not able to obtain consistent FROG results for the different shots with the same experimental conditions and so we looked for a different method to glean the electric field from the measured spectrograms. We first tried simple double-pulse models that had the Raman pulse with a simple linear chirp and the shifted pulse having a central frequency shifted by some amount and third-order phase to account for a time-varying frequency separation. This gave unphysical results. We then used the theoretical model for the generalized two-photon Rabi frequency-shifted dressed states, included in the theory describing MRG. We looked to see if the complicated measured spectrograms would give more consistent results, if we assumed one of the pulses came from the normal Raman scattering process and a second pulse was generated through Raman scattering between the phased and anti-phased dressed states. This model did indeed give more consistent results for each of the experimental conditions and the changes in the red-shifted pulse corresponded to what would be expected with the model. As can be seen in the plots of pulse chirps shown in Figure 5, with increasing pump energy, there is an increasing dip in the frequency of the red-shifted shoulder at half the pump pulse delay, corresponding to the time of the maximum generalized Rabi frequency. The pump energy derived in the model for the three different energy cases varied by the same ratios as the experimental values. The simulated pump energy was 0.44 of the energy measured before the pressure cell containing the hollow fiber and so this value is very close to the experimental value. The average time delay across all nine shots for this run was \(357\pm 72\) fs which falls within the experimental accuracy and jitter levels. The results shown in Figure 6 show that the frequency difference between the pulses near the peaks of the pulses at time zero does not change much as the pulse delay increases. Rather, the shape of the frequency chirp of the red-shifted pulse changes significantly. For the timed case, shown in Figure 6a, the optimized numerical result shows that the two pulses are overlapped spectrally until the intensity of the pulses becomes significant. This is expected from the model, as there is zero total detuning at the edges of the pulse because the pump pulses are resonant with the Raman transition and the generalized Rabi frequency would be zero. We also see the maximum dip in frequency in the red-shifted pulse at the center for the timed case as expected, because both the Stark shift and the on-axis Rabi shift are maximized in this case where the two pump pulses coincide. It can be seen in Figure 6, with increasing time delay, that the frequency shift between the pulses at early and late times increases with increasing delay between the pump pulses and the frequency dip both decreases and occurs further from the peak of the pulses. At the longest time delay shown in Figure 6c, the two pulses resemble what would be seen if the second pulse came simply from four-wave mixing as the overlap of the two pulses is small, resulting in low Rabi frequency shifts. The simulated results show good agreement between the simulation and the experimental results and so indicate that the secondary pulses are indeed a result of Raman scattering between the dressed states. The simulations also give pulse profiles and phase differences between the pulses that we cannot yet address with the theory of Raman scattering between different dressed states. We have not carried out a theoretical calculation that could elucidate a possible p phase shift. We expect it could be a result of the Raman pulse being caused by scattering between like states and the red-shifted pulse coming from scattering between phased and anti-phased states.