# Time-Dependent, Two-Photon Rabi Frequency Shift Observed in Multi-Frequency Raman Spectra
## 1 Introduction  
Multi-frequency Raman generation (MRG), a promising nonlinear optical technique for generating high-energy ultra-short pulses, produces a broad spectrum with discrete Raman spectral orders [1]. The coherence between different Raman orders makes it possible to generate ultra-short pulses [2]. Many MRG research efforts are dedicated to the generation of more Raman orders and the attainment of shorter laser pulses through the coherent combination of these orders. [PERSON] et al. observed that the efficiency of MRG is enhanced with red-detuned pumps and attributed this effect to the Stark shifting of energy levels [3]. By phasing seven Raman orders with a prism dispersion line and liquid crystal phase modulator, [PERSON] and colleagues obtained pulses with a duration of 1.6 fs [4]. [PERSON] et al. generated a nearly Fourier transform limited pulse with duration of 1.78 fs by simply inserting plates with different thicknesses on the optical axis [5], and they reported that pulses as short as 728 attoseconds could be achieved by combining the Raman orders [6]. The high-energy ultra-short pulses would find applications in nonlinear optical experiments in molecular dynamics. Our group has been investigating MRG in the transient regime, where the pumpp pulse duration is similar to the dephasing time of the Raman material. In transient MRG experiments, the pumps are linearly chirped to either reduce nonlinear effects such as self-phase modulation [7, 8] or to have the ability to easily change the instantaneous frequency separation of the pumps by varying the time delay [9, 10]. During our previous MRG experiments, we noticed that the individual Raman orders become double-peaked when the instantaneous frequency separation of the pump and Stokes is red-detuned from resonance. We noted that the shoulders increased and broadened nonlinearly with increased laser intensity. The secondary peaks appeared only on the red side of the Raman peaks. This phenomenon can also be seen in other groups' experiments with hydrogen [8], diamond [11], and PbWO\({}_{4}\) crystal [12], but no further discussions were made by the authors. [PERSON] et al. successfully synthesized nearly single-cycle sub-5 femtosecond pulses by using Raman sidebands [13]. To date, single-femtosecond pulses have not been generated in the transient regime, where the emergence of a red-shifted spectrum has significantly complicated the spectral characteristics of individual orders,posing challenges for pulse synthesis. This is particularly evident when the detuning is set to have the maximum number of orders that if perfectly compressed would give the shortest pulse duration. Our primary interest lies in understanding the red-shifted spectrum, with a particular focus on unraveling the associated phases. This knowledge may help us in improving pulse compression. Additionally, the broadened bandwidth of individual orders offers the potential to enhance the energy of the central ultra-short pulse. We have investigated these red-shifted spectra under different instantaneous frequency separations and input energies. The results showed that the red-shifted spectrum is not a result of four-wave mixing but could be the result of a two-photon dressed-state Rabi frequency shift [10]. To further determine if the anti-Stokes orders comprise a Raman pulse and a second pulse resulting from Raman scattering between two-photon Rabi frequency-shifted states, we have now applied standard second harmonic frequency-resolved optical gating (FROG) [14] to determine the electric field of the broadened first anti-Stokes Raman order. We found out that the reconstructed electric field from the second harmonic FROG was too complicated to provide a good understanding of our red-shifted phenomenon. We expect that this complicated structure comes from the broadened order and it is a result of two pulses overlapping both spectrally and temporally. We have therefore analyzed the measured spectrograms using a double-pulse model.