# Time-Dependent, Two-Photon Rabi Frequency Shift Observed in Multi-Frequency Raman Spectra
## 3 Numerical Modeling of Double-Pulse Anti-Stokes Order
The total detuning is reduced by the Stark shift as the pulse intensity increases, causing a reduction in frequency separation of the levels as shown in Figure 2e,f. With the larger time delay as shown in Figure 2c,f, the constant detuning caused by the pump delay increases, but the maximum Stark shift is lower and the peak Rabi frequency is significantly reduced. The work reported here demonstrates that the red-shifted spectral peak observed in transient MRG spectra has a spectral chirp that matches what would be expected from linear Raman scattering between manifolds of time-varying two-photon Stark-shifted states. The Hickman theory only allows for a closed system, with all the interactions coming from the  
Figure 2: (**a–c**) The relative timing and pulse durations of the two pump pulses. (**d–f**) The energy levels resulting from the two-photon Bloch equations for the corresponding time delays shown above each figure. third order, nonlinear process at the Raman frequency. This theory then does not account for the frequencies that could come from linear Raman scattering occurring between the lower energy state of one manifold and the higher energy state of another as depicted in Figure 3. In Figure 3a, we show the linear Raman process for the unshifted states having the same Raman frequency as the nonlinear Raman signal. Also shown in Figure 3a are the Rabi frequency shifts of the molecular levels for the case of constant pump fields. In Figure 3b,c, we show that if we assume that initially the population is in one state, which would be the E\({}^{-}\) state for red-detuned pumps, then the linear Raman process could lead to two Stokes frequencies being generated between the dressed states. For the time-varying coupled states, the frequency difference between the two Stokes pulses also then varies in time. In Figure 3b, we show the Raman scattering between the shifted levels when the pump pulses overlap temporally. The pump pulse is depicted by a black arrow and population is pumped from the lower E\({}^{-}\) level to a virtual level. The usual Stokes signal given by a shift of the Raman frequency is depicted by the green arrow and moves the population to the higher E\({}^{-}\) level. The red-shifted Stokes pulse is depicted by a red arrow and takes the population to the higher E\({}^{+}\) level. For the case of timed pump pulses, there are no shifts at early and late times. The arrows depict the maximum shift in frequency for the two Stokes pulses. Three different times during the pumping time are depicted in Figure 3c, for the case of time-delayed pump pulses. The frequency of the usual Stokes pulse, given by the scattering between the E\({}^{-}\) states, is constant and shifted from the pump frequency by the Raman frequency. We will therefore refer to this pulse as the Raman pulse. However, the second Stokes pulse will have a time-varying frequency with the maximum frequency difference between the two Stokes pulses occurring at the center time between the two pulses. We will refer to this pulse as the red-shifted pulse. In this case, due to the Stark shift, the frequency difference between the Raman pulse and the red-shifted pulse can be reduced from the frequency shift that caused by pump detuning. The frequency separation will again increase at the center because of the Rabi frequency shift. We only show the Stokes order being produced. This would imply that the red-shifted frequency observed on the first anti-Stokes comes from the red-shifted Stokes pulse pumped by the second anti-Stokes. We could have also shown linear anti-Stokes being generated if the pump pulse started on the upper manifold of states, causing two anti-Stokes pulses from transitions to either E\({}^{+}\) or E\({}^{-}\) on the lower  
Figure 3: (**a**) Linear Raman scattering diagram between regular Raman states. Also shown is the Rabi splitting of the levels to states E\({}^{+}\) and E\({}^{-}\) for constant electric fields.