## 2 Experimental Setup  
The experimental setup is shown in Figure 1. We used a Ti:Sapphire amplifier, which outputs 1 mJ per 40 fs pulse (at 1 kHz repetition rate) with central wavelength at 806 nm. Using a beamsplitter (60:40 (R:T)), the pulse was divided into two parts. Forty percent of the beam was used as the pump beam for Raman generation while the other part was used to pump an optical parametric amplifier (OPA). The second harmonic (900 nm) of the idler beam from the OPA was used as the Stokes beam (we follow the coherent anti-Stokes Raman scattering convention and denote the shorter 806 nm wavelength beam as pump and the longer wavelength 900 nm as the Stokes beam). The pulses were vertically polarized. The power of the pump was around 10 mW and the power of the Stokes was around 2 mW. The intensity of the pump at the focus was around 2 \(\times\) 10\({}^{11}\) W/cm\({}^{2}\)[21]. Coherent Raman sidebands were generated when the two beams were crossed on the crystal (1 mm thick synthetic single-crystal diamond) at 3.7 degrees. The angle was theoretically calculated under the standard phase matching conditions considering the higher order Raman generation. It was further optimized experimentally using the generation of the spectrum and spatial beam profile of the sidebands. Further details on the properties of sideband generation in diamond can be found to in Reference [22]. The robustness of the sidebands generation in a Raman active crystal in our experimental setup has also been studied in Reference [23]. In Figure 1, the top picture shows the Raman sidebands generated from our Raman active crystal (the sidebands were generated in PbWO4).  
We designed a 2f-2f reflection scheme to characterize the relative spectral phases of the Raman sidebands. Specifically, we used concave spherical mirrors to reflect the beams back to the same crystal. In our experiment, we chose two spherical mirrors with a focal length of 10 cm and kept them about 20 cm away from the crystal. One mirror was to reflect the pump and Stokes beams and the other was to reflect the higher order Raman sidebands (in this paper, AS3, AS4, AS5, AS6, and AS7 are reflected back to the crystal). In this configuration, the beams were re-focused back to the same spot of the crystal and phase matching was automatically fulfilled. Consequently, the interaction between the beams was maximized. The beams were re-focused back to the crystal with an offset from their incident spots such that two spots were just distinguishable from each other. A translation stage was used to finely adjust the distance between the spherical mirror and the crystal. In the experiment, the mirror used to reflect the sidebands was put on the translation stage. The position of the mirror that reflects the pump and Stokes beams was fixed. The configuration enabled us to adjust the relative spectral phases between the pump and Stokes beams and the sidebands. As a result, an interferogram was produced as a function of the relative spectral phases due to the coherent Raman interaction [21]. It was recorded with a 200 nm scanning step of translation stage, which corresponded to a 1.3 fs time delay between the two spherical mirrors. When presenting the experimental data, we converted the scanning of the translation stage to the relative time delay.  
Figure 1: Experimental setup schematic. The Raman sidebands are generated from a Raman active crystal. We used a spherical mirror to reflect the sidebands and the pump and Stokes beams back to the crystal. The top is a picture of the Raman sidebands taken by a camera.