# Time-Dependent, Two-Photon Rabi Frequency Shift Observed in Multi-Frequency Raman Spectra
## 4 Comparison of Experimental and Numerically Modeled Spectrograms
This characteristic introduces an additional layer of complexity to our analysis, as we must consider this distributed energy when interpreting the results. Shown in Figure 5 are the optimized, simulated chirps of the two pulses comprising the first-order anti-Stokes order, along with the simulated pulse profiles. Also shown are the experimental and simulated spectrograms. In Figure 5a-c, we show the results of one of the shots for experimental pump energy of 1.2, 1.7, and 2.2 mJ, respectively. The optimized simulated results in Figure 5a correspond to a pump energy of 0.52 mJ and a pulse delay of 414 fs. The Raman pulse had a duration of 884 fs and the red-shifted pulse had a duration of 1429 fs. The amplitude of the red-shifted shoulder was 100 times larger for this case. The phase difference was 3.12. The average results for the three sets of profiles gave a Raman pulse duration of 872 fs, a red-shifted pulse duration of 1356 fs, an amplitude ratio of 14 and phase difference of 3.14. The optimized simulated results in Figure 5b correspond to a pump energy of 0.76 mJ and a pulse delay of 359 fs. The Raman pulse had a duration of 848 fs and the red-shifted pulse had a duration of 1010 fs. The amplitude of the red-shifted shoulder was 4.2 times larger for this case. The phase difference was 2.76. The average profiles for the three spectrograms gave a Raman pulse duration of 831 fs, a red-shifted pulse duration of 1084 fs, an amplitude ratio of 5.3, and phase difference of 2.84. The optimized simulated results in Figure 5c correspond to a pump energy of 1.0 mJ and a pulse delay of 365 fs. The Raman pulse had a duration of 865 fs and the red-shifted pulse had a duration of 1485 fs. The amplitude of the red-shifted shoulder was 2.3 times larger for this case. The phase difference was 2.99. As can be seen in Figure 5, the simulated spectrograms match well with the experimental measurements. The chirp of the Raman pulse is given as the chirp of the pump pulse that does not have a perfect linear chirp. It is worth noting that we first modeled the Raman as having a linear chirp that closely matched the pump chirp and that the red-shifted pulse chirp included the same linear chirp but the error between the model and the experiment was not as low as the FROG traces in many cases and so we decided to use the actual pump pulse chirp as measured by the FROG and we were able to reduce the error between the modeled and measured spectrograms. Figure 5: _Cont._  
In our second series of experiments, we aimed to investigate how the instantaneous frequency separation of the pump and Stokes pulses affected the spectrum of the first anti-Stokes order. Figure 6 illustrates the results of these investigations, featuring both simulation outcomes and experimental measurements. Specifically, we focused on three distinct time delays: 0 fs, 333 fs, and 666 fs, all conducted at the average pump energy of 2.2 mJ. For each time delay, we conducted 11 spectrogram measurements. The average errors from FROG reconstructions were as follows for the increasing time delays: 0.0094, 0.0082, and 0.0048, respectively. Meanwhile, the double-pulse model yielded errors of 0.0087, 0.0085, and 0.0054 for the same time delays. Interestingly, the average simulated pump pulse energy across all 33 spectrograms taken during this entire run was 0.76 mJ, which corresponds to just 35% of the expected 2.2 mJ. This discrepancy suggests that coupling losses were likely higher during this experimental run. The simulated chirps, as displayed in Figure 6, indicated slight differences in pump pulse chirp compared to the energy scan experiment. Across the three cases, the average time delays were approximately 58 \(\pm\) 18 fs, 345 \(\pm\) 8 fs, and 709 \(\pm\) 43 fs. The average durations of the Raman pulses closely matched the experimental pump duration, measuring 780 fs for each case. Regarding the red-shifted pulses, their average durations were found to be 1340 fs, 1018 fs, and 1375 fs, respectively, all exceeding the duration of the Raman pulses.