# Time-Dependent, Two-Photon Rabi Frequency Shift Observed in Multi-Frequency Raman Spectra
## 3 Numerical Modeling of Double-Pulse Anti-Stokes Order  
The theory describing MRG by [PERSON] et al. is based on a multi-wave approach where all the Raman orders are mutually coupled together in the process [16]. The Raman orders including the two pumps form a set of frequency components \(\omega_{\rm j}=\omega_{0}+\rm j\omega_{R}\), where \(\omega_{0}\) is the pump frequency and \(\omega_{\rm R}\) is the Raman frequency. The resonant Raman process is described by the interaction of the fields using the two-photon Bloch equations [16; 17], and the authors obtained the following equations describing a generalized Rabi frequency \(\Omega^{\prime}\):  
\[\Omega^{\prime 2}=\Omega^{2}+\Delta^{2}, \tag{1}\]  
with on-resonance Rabi frequency \(\Omega\) and total detuning \(\Delta\) given by:  
\[\Omega\rm e^{i\theta}=\frac{\alpha_{12}}{2\hbar}\rm{\sum_{\rm{j}}V_{\rm{j}}V_{ \rm{j-1}}^{*}}, \tag{2}\]  
\[\Delta=\frac{\partial\theta}{\partial t}+\frac{2\pi(\alpha_{22}-\alpha_{11}) \rm{I}}{\hbar c}+8\omega, \tag{3}\]  
\[\rm{I}=\frac{c}{8\pi}\rm{\sum_{\rm{j}}V_{\rm{j}}V_{\rm{j}}^{*}}, \tag{4}\]  
where the \(\rm{V_{j}}\) are the amplitudes of the j-th Raman order, \(\rm{V_{j}^{*}}\) is the complex conjugate of \(\rm{V_{j}}\), \(\alpha_{\rm{ij}}\) are the transition moments, \(\rm{I}\) is the total intensity of all orders, and \(8\omega\) is the detuning between the frequency separation of the pumps and the Raman frequency. As can be seen from Equation (3), the detuning \(\Delta\) is determined by three terms, the Raman frequency detuning \(\delta\omega\), the time derivative of the Rabi phase \(\theta\), and the two-photon Stark shift due to the coupling of the polarizabilities, \(\alpha_{22}\), \(\alpha_{11}\), and total pulse intensity. The Hickman theory assumed steady state pumps. [PERSON] et al. showed adiabatic population transfer by a time-varying, two-photon laser-induced Rabi frequency shift [17]. The time-dependent  
Figure 1: Schematic diagram of the setup: M—mirror; G—grating; L—lens. two-photon Bloch equations yield manifolds of pairs of coupled states \(\mathrm{E}^{+}\) and \(\mathrm{E}^{-}\) with energies given by  
\[\mathrm{E}^{\pm}(\mathrm{t})=\frac{\hbar}{2}\bigg{[}\Delta(\mathrm{t})\pm\sqrt{ \Omega^{2}(\mathrm{t})+\Delta^{2}(\mathrm{t})}\bigg{]}, \tag{5}\]  
where the detuning \(\Delta(\mathrm{t})\) is given by Equation (3), for time-dependent intensities. As the peak frequency separation is not detuned and kept at \(\delta\omega=0\), we assume the detuning of instantaneous frequency separation with time delay gives the time-dependent phase term in Equation (3). The difference in energy of \(\mathrm{E}^{+}\) and \(\mathrm{E}^{-}\) is then the time-dependent generalized Rabi frequency \(\Omega^{\prime}\) given in Equation (1). In Figure 2, we depict the time-varying shifted energy levels of \(\mathrm{E}^{+}\) and \(\mathrm{E}^{-}\) for three different time delays between the pump pulses. The durations of these demonstration pulses are slightly different so they both can be depicted for all three cases. We use arbitrary values of the polarizabilities for this depiction to demonstrate the different effects of the pump frequency detuning, the Stark shift, and the Rabi frequency. In Figure 2a-c, the pump pulse lags the Stokes pulse by 0 fs, 333 fs, and 666 fs, respectively. The energy levels are shifted by the total detuning and the Rabi frequency following Equation (5). The level shifts corresponding to the three time delays are depicted in Figure 2d-f. In Figure 2d, corresponding to the pump pulses timed together, the level splitting at early and late times is zero because the instantaneous frequency detuning is zero. The maximum shift for both levels occurs at the peak of the pulses due to the coincidence of the peak Stark shift and peak Rabi frequency shift. In Figure 2e,f, the Stokes pulse leads the pump pulse yielding a red-detuning of the instantaneous frequency separation.