## 1 Introduction
They derived the two-photon optical Bloch equations and showed that the states are split into two coherent states separated by the two-photon Rabi frequency \(\Omega^{\mathrm{z}}\), which is given by:  
\[\Omega^{\mathrm{z2}}=\Omega^{\mathrm{z}}+\Delta^{\mathrm{z}} \tag{1}\]  
where:  
\[\Omega e^{i\theta}=\frac{\alpha_{\mathrm{12}}}{2\hbar}\sum_{j}V_{j}V_{j-1}^{*} \tag{2}\]  
\[I=\frac{c}{8\pi}\sum_{j}V_{j}V_{j-1} \tag{3}\]  
\[\Delta=\frac{\partial\theta}{\partial t}+\frac{2\pi(\alpha_{\mathrm{22}}- \alpha_{\mathrm{11}})}{\hbar c}I+\delta\omega \tag{4}\]  
where \(\mathrm{V_{j}}\) are the electric field amplitudes of the Raman orders, \(\alpha_{\mathrm{ij}}\) are the transition moments and \(\delta\omega\) is the detuning between the peak frequency separation of the two pump beams and the Raman transition. The term I, is related to the intensity of each MRG order and leads to a Stark shift of each molecular level through a single-color, two-photon interaction. The small difference between the transition moments \(\alpha_{\mathrm{22}}\) and \(\alpha_{\mathrm{11}}\), leads to a change in the Raman transition frequency. It is this Stark shifting that leads the Raman transition to being slightly red-shifted when pumped at high intensity and explained the [PERSON] observation [10] of more Raman orders generated with red-detuned pump beams. This Stark shift term needs to be included in the detuning term, \(\Delta\). The interaction of the Raman transition with adjacent Raman orders leads to the complex two-photon Rabi frequency \(\Omega\mathrm{e}^{\mathrm{i0}}\). A time varying \(\theta\) gives a frequency detuning term d\(\theta\)/dt. When using linearly chirped pump pulses, the d\(\theta\)/dt term is linearly dependent on both the chirp rate and time delay between the pulses. In [PERSON]'s work [9], a single linearly chirped pump pulse was split into two identical pulses that were then combined with a time delay between the two pulses. In this case, the pump pulses have the same peak frequency making the \(\delta\omega\) term equal to the Raman transition frequency. The d\(\theta\)/dt from the time delay was used to cancel this detuning. In our experiments, we set the two pump frequencies such that \(\delta\omega\) equals zero and the time delay between the linearly chirped pulses detunes the frequencies away from the low intensity resonance. We assume the two pump pulses have the same linear chirp. Using the two-photon optical Bloch equations with multiple frequency inputs has proven successful in determining the number of Raman orders in both the adiabatic regime [10] and in the transient regime [12]. However, this multi-frequency theory does not allow for any extra peaks to be derived because all possible frequency components, \(\omega_{\mathrm{j}}\), are input and only their amplitudes, \(\mathrm{V_{j}}\) grow with the Raman gain as a function of propagation length. In order to have extra peaks grow in a theoretical model, the input would have to be a continuum spectrum. Theoretically modeling the observed behavior is beyond the scope of this paper. We do consider what would be expected from a two-photon dressed state picture for the Raman transition. To do this, we will first consider single photon Stark shifting and look at the effect of a weak probe beam tuned through the Stark shifted transition [13]. In his book, [PERSON] has derived the case for a two-level atom, which is driven by a near resonant, single monochromatic beam. The pump frequency is slightly red-detuned from the transition, that is,the laser frequency is slightly less than the transition. For red-detuning, the population would initially be in the lower coherent state. The calculation of the absorption of a probe beam as its frequency is varied across the transition frequency shows gain on the Stokes side of the transition at a frequency detuning given by the one-photon Rabi frequency and loss on the anti-Stokes side also at the frequency shift given by the Rabi-frequency.