## 4 Discussion  
To clarify the above point in a simple way, we observe that the slow (Raman) nonlinearity created by an ultrafast pulse can be written as [8; 36; 37]:  
\[\delta n_{R}(t)=\int_{0}^{\infty}R(\vartheta)I(t-\vartheta)d\vartheta \tag{1}\]  
where \(R(\vartheta)\) is the Raman response function of the sample and \(I(t)\) is the pump pulse intensity. \(R(\vartheta)\) was repeatedly modeled in the literature [8; 9; 37] as a sum of contributions of molecular rotational transitions:  
\[R(\vartheta)\propto\sum_{J}F_{J}\sin\bigl{(}2\pi c\omega_{J}\vartheta\bigr{)} \tag{2}\]  
where \(c\) is the speed of light; \(J\) is the rotational quantum number; \(\omega_{J}=2B(2J+3)\) are the rotational wavenumbers of a rigid rotator with \(B\) the rotational constant of the molecule; and \(F_{J}\) are the line intensities, given by:  
\[F_{J}=\bigl{(}\rho_{J+2}-\rho_{J}\bigr{)}Z_{J}(J+2)(J+1)/(2J+3) \tag{3}\]  
in which:  
\[\rho_{J}=\frac{\exp[-chBJ(J+1)/kT]}{\sum_{J}Z_{J}(2J+1)\exp[-chBJ(J+1)/kT]}, \tag{4}\]  
\(h\) is the Planck constant; \(k\) is the Boltzmann constant; \(T\) is the temperature; and \(Z_{J}\) is a degeneracy factor due to nuclear spin. Evaluation of (3) for N\({}_{2}\) at room temperature shows that above \(J=23\) (\(\omega_{J}\cong 187\) cm\({}^{-1}\)), the relative population of rotational modes is less than 0.05. The corresponding rotational period (\(T_{JMAX}\cong 170\) fs) is longer than the laser pulse width (\(\tau_{G}\cong 44\) fs) in our experiment, allowing us to conclude that, as a first approximation, the pump pulse can be considered instantaneous with respect to molecular rotations, and thus its temporal distribution can be approximated as a delta function in the convolution (1), giving \(\delta n_{R}(t)\approx R(t)\). The same approximation applies to the probe pulse, which substantially senses the instantaneous Raman response of the sample. Since the rotational intensities (3) depend on the temperature, the above equations can be used to analyze the experimental data reported in Figure 3a to derive the sample temperature. For a meaningful comparison, however, the data must be interpolated on the rotational frequencies \(\omega_{J}\) and corrected for the proportional dependence of the ISRS spectral signal on the wavenumber [11; 19]. The resultant Raman spectra are shown in Figure 3b, together with the fits obtained using the model expression (3) for \(F_{J}\) with the temperature and an overall normalization constant as fitting parameters. The best-fitting temperatures are 238 \(\pm\) 20, 289 \(\pm\) 19, and 265 \(\pm\) 13 K for 1, 7, and 10 atm N\({}_{2}\) pressure, respectively, which are to be compared with the room temperature T = 295 K. The larger deviation of the fitted temperature at 1 atm from the actual value is clearly produced by the poor SNR, a figure that can be improved as discussed above. Noticeably, the fitting temperatures tend to be systematically lower than the actual value, an effect which can be due to the reduced excitation of the higher-energy rotational levels resulting from a non-sufficiently short pump pulse. This effect could be accounted for by relaxing the approximation of purely impulsive excitation, e.g., calculating the integral (1) using a functional form for the laser pulse intensity different from the delta function. On the other hand, a simplified approach to compensate for this effect is to correct the experimental data by dividing by an excitation efficiency factor \(\exp\left(-\omega^{2}\tau_{G}^{2}/4\right)\)[11]. Applying this correction, the new fitting temperatures are 264 \(\pm\) 23, 321 \(\pm\) 20, and 291 \(\pm\) 14 K, for 1, 7, and 10 atm N\({}_{2}\) pressure, respectively, in satisfactory agreement with the actual value notwithstanding the simplifications and approximations involved in the data analysis. It is worth observing that the assumption of purely impulsive excitation depends on the ratio of the laser pulse width \(\tau_{G}\) to the rotation period of the highest populated rotational level \(T_{MAX}\).