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nates for 2 τ < 23 . 4 fs, and the H$_{g}$ (1) mode for 2 τ > 23 . 4 fs, in agreement with the analytical model of Eqs. (2)-(3), and in qualitative agreement with other simulations [12] and experiment [24–28]. 

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In Fig. 3, the agreement between the predictions of 

the analytical model (curves) and the completely inde- pendent DFT-based simulations (points) is truly remark- able. The only adjustable parameter for each curve is the effective polarizability parameter α ′ $_{k}$. 

There is similarly remarkable agreement for the carbon nanotube, as can be seen in Fig. 4. In this case, two pho- ton energies ℏ ω were used: 0 . 8 eV and 2 . 0 eV, which are respectively below and above the excitation gap of 1 . 56 eV for this model nanotube, as discussed above. Figure 4 shows the results for the maximum kinetic energy K max k of the one prominent mode observed in this case. (Re- call that the polarization vector for the pulse was chosen to point across the nanotube, so that only radial modes should be directly excited, and this is a severe constraint for the small (3,3) nanotube, as is also observed in ex- periment [31].) The panels in Fig. 4 correspond to: (a) ℏ ω = 0 . 8 eV, τ = 76 fs; (b) ℏ ω = 0 . 8 eV, E 0 = 9 . 66 V/nm; (c) ℏ ω = 2 . 0 eV, τ = 76 fs; (d) ℏ ω = 2 . 0 eV, E 0 = 9 . 66 V/nm. It is interesting that the analytical model still works in this case even for above-the-gap excitation, but we find less satisfactory agreement for C$_{60}$, and certainly do not expect the model of Eq. (3) to be generally valid when there is a substantial population of electrons in ex- cited states at the end of the pulse. In this case other effects will ordinarily be of dominant importance [4, 11]. 

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As mentioned above, the electronic response in the DFT-based simulations can be characterized by an effec- tive polarizability parameter α ′ k if one fits Eq. (7) to the results of the simulations. If the model leading to Eq. (7) is in fact consistent with the detailed simulations, then the values obtained when (a) varying the electric field amplitude E 0 at constant (FWHM) pulse duration τ and 

