# Time-Dependent, Two-Photon Rabi Frequency Shift Observed in Multi-Frequency Raman Spectra
## 4 Comparison of Experimental and Numerically Modeled Spectrograms
With these values, we can determine that the amplitude ratio between the Stark shift and Rabi frequency is approximately 1:60. Hence, the Rabi frequency dominates the Stark shifting in SF\({}_{6}\). For the calculation of the intensity-dependent generalized Rabi frequency term, we assumed the pump pulses have Gaussian shape, with the experimentally measured pulse duration. The beam area was given by the diameter of the hollow fiber, which was 150 \(\upmu\)m. We measured the energy before the hollow fiber and performed the experiments at three average energies of 1.2 mJ, 1.7 mJ, and 2.2 mJ. We also measured that \(\sim\)30% of the light was transmitted through the fiber, with most of the loss coming from fiber coupling losses and Fresnel losses at both cell windows. In the model, we used the average energy before the fiber and used a variable energy coefficient to account for the losses and energy jitter of the laser. To minimize the error between the experimental and modeled spectrograms, several of the variables were iterated. Within the generalized Rabi frequency term, we allowed the pump energy and time delay between pulses to vary around the average experimental values. This accounts for the experimental jitter of the laser in energy and time delay. In addition to these variables, we iterated the values of pulse durations T\({}_{1}\) and T\({}_{2}\), the amplitude ratio of the two fields, and the constant phase shift between the pulses. We compared the error with that given by the standard second harmonic FROG iteration and obtained comparable errors to the double-pulse model, even though we limited the profiles of both pulses to be Gaussian in the simulation. We initiated our analysis by comparing the model's predictions to experimental results obtained at a pump pulse delay of 333 fs, which corresponds to the point where the maximum number of Raman orders is generated. In this specific experimental run, we acquired three spectrograms for each energy setting. Notably, for the total pump energy of 2.2 mJ, one of the three spectrograms exhibited distinct characteristics compared to the other two. Strikingly, it closely resembled the spectrogram observed when the pump pulses were temporally synchronized. However, to obtain a result with minimal error for this spectrogram, the model required a significantly lower pump energy and a much smaller time delay between the pulses. We attribute this discrepancy to experimental timing and energy fluctuations, which can introduce variability into the measurements. For the other two pulse energies, the three measured spectrograms exhibited consistent patterns. The average FROG reconstruction errors across the three shots for the pump energies of 2.2 mJ, 1.7 mJ, and 1.2 mJ were 0.0059, 0.0058, and 0.0135, respectively. To further gauge the accuracy of our model, we assessed the error between the double-pulse model predictions and the experimental spectrograms for the same three pulse energies. The resulting errors were 0.0064, 0.0056, and 0.0065, respectively. Throughout the nine shots, the average delay time was determined to be 357 fs, with a standard deviation of 72 fs. This standard deviation aligns with our experimental timing accuracy and jitter, reflecting the precision of our timing measurements. Additionally, the average pump energy used in the model to calculate the generalized Rabi frequencies was found to be 0.52, 0.75, and 0.98 for the cases with pump energies of 1.2 mJ, 1.7 mJ, and 2.2 mJ, respectively. These model-predicted energies corresponded to approximately 43%, 44%, and 45% of the measured energies before the fiber. This alignment is reasonably consistent with the 30% measured energy output, with the understanding that some energy loss may occur at the exit window. Moreover, it is worth noting the presence of white light emitted along the length of the fiber, which means our measurements represent an average of the energy distributed along the fiber.