# Time-Dependent, Two-Photon Rabi Frequency Shift Observed in Multi-Frequency Raman Spectra
## 4 Comparison of Experimental and Numerically Modeled Spectrograms  
To compare with the experimentally measured FROG spectrograms of the first anti-Stokes order, we derived numerical spectrograms constructed from the model, where the first anti-Stokes order comprised two pulses generated through Raman scattering between the different dressed states as depicted in Figure 3. We refer to the first pulse as the Raman pulse and the second pulse as the red-shifted pulse. We assumed that the fields, \(\mathrm{E_{1}}\) and \(\mathrm{E_{2}}\), of both pulses were Gaussian. Adding the two pulses together with an amplitude ratio \(\alpha\), and phases \(\phi_{1}\) and \(\phi_{2}\), we can obtain the total electric field,  
\[\mathrm{E(t)}=\mathrm{E_{1}}(\mathrm{T_{1}},\omega_{1},\phi_{1},\mathrm{t})+ \alpha\mathrm{E_{2}}(\mathrm{T_{2}},\omega_{1},\phi_{2},\mathrm{t}), \tag{6}\]  
where \(\mathrm{T_{1}}\) and \(\mathrm{T_{2}}\) are the pulse durations, \(\omega_{1}\) is the central frequency of the Raman field, \(\mathrm{E_{1}}\). We are assuming that the Raman pulse is given by the pump pulse with the frequency shifted by the Raman frequency. Therefore, the phase \(\phi_{1}\) of the Raman pulse is taken to be the phase of the pump pulse. We also ensure that the phase at the peak of the Raman pulse is 0 so that we can impose a known phase difference between the two pulses. The phase \(\phi_{2}\) is given by the summation of the phase \(\phi_{1}\), the phase given by the time integral of the time-dependent generalized Rabi frequency shift, described in Equation (1), and a constant phase difference. We allowed there to be a constant phase shift of approximately \(\pi\), added to the second pulse. In our initial attempts to model the spectrogram, we used a simpler two-pulse model, where the second pulse could have a different central frequency and we added a third-order phase to allow the central frequency separation to be larger from the peak intensity and we studied the case with maximum red-shifting. We allowed the two pulses to have different peak times and we noted that the optimized double pulse had a red-shifted pulse followed by the Raman pulse. The delay between the two pulses was about 1 ps and dispersion could not account for the red pulse coming ahead of the Raman pulse. Also, assuming that the Raman pulse would be timed to the pump pulses, we realized that this solution having a pulse appear before the pump pulse was unphysical. The appearance of a red pulse ahead of a blue pulse was most likely the result of two overlapping positively chirped pulses with  
Figure 4: The blue line is the instantaneous frequency of the Raman pulse. The orange line is the instantaneous frequency of the red-shifted pulse. a near \(\pi\) phase shift between them. With a \(\pi\) phase shift, a node would appear at the center of the combination of two pulses, with the red front half of the pulses appearing ahead of the trailing combined blue parts of the chirped pulses. Upon closer examination, we concluded that the appearance of a red pulse preceding a blue pulse was likely a consequence of two overlapping positively chirped pulses with an almost \(\pi\) phase shift between them. With a \(\pi\) phase shift, a node formed at the center of the combined pulses, causing the red front half of the pulses to appear ahead of the trailing combined blue portions of the chirped pulses. This intriguing behavior shed light on the complex interplay of phase shifts and chirp effects in our experimental setup. The amplitude ratio between Stark shift and Rabi frequency depends on the polarizabilities, \(\alpha_{12}\), \(\alpha_{11}\), and \(\alpha_{22}\) as shown in Equations (2)-(4). In our experiment, the Raman medium is SF\({}_{6}\), which has calculated values for \(\alpha_{12}\), \(\alpha_{11}\), and \(\alpha_{22}\) of 1.65456 \(\times\) 10\({}^{-31}\) m\({}^{3}\), 1.64378 \(\times\) 10\({}^{-31}\) m\({}^{3}\), and 1.66841 \(\times\) 10\({}^{-31}\) m\({}^{3}\), respectively.