## 3 Results and Discussions
### Peaks with Fractional Harmonic Orders  
Besides the integer order harmonics, the fractional peaks are intricate. Two types of optical responses with fractional order are observed: the companion peaks of the major HHG peaks and the intrinsic optical emission peaks (denoted in Figure 3(a)). The first one originates from the degeneracy of the quasi-energy Floquet states. Since the lifetime of the non-degenerate states is usually different, the interference between these states tends to shift the integer harmonic to a fractional order. It has been noticed before in benzene by TDDFT calculation [25]. We find that these peaks exist when the applied laser field has a large amplitude and short duration time, and will be suppressed when the laser duration is increased. The other type of fractional peak is from the intrinsic emission of the graphene molecule. The peak at the 7.8 harmonic order in Figure 3(a) is a good example. The intensity of this peak is high and comparable to the integer-order HHG peaks, since it represents the first dipole-allowed excitation. However other intrinsic emission peaks can be usually observed when the laser amplitude is small or the frequency is high, in other words, only when the nonlinear effect is small. Otherwise when the amplitude of the laser is large, the signals of the intrinsic emissions are concealed by the HHG signals. We plot the spectra with a small laser amplitude of only 0.002 V/A, see Figure 4. Since these peaks are intrinsic to the graphene molecule itself, their frequencies do not shift with respect to the applied field. By changing the unit of the x-axis in Figure 4 from harmonic order into eV, one can easily find the one-to-one correspondence between these intrinsic peaks and the energy difference of eigenstates. For example, the peaks \(a\)-\(e\) carry photon energies for the optical transitions \(A\)-\(E\) denoted in Figure 2.  
Figure 4: HHG spectra of the non-passivated (**a**) and passivated (**b**) graphene molecules with a weaker laser pulse. The laser field parameters are: \(\hbar\omega=0.9\) eV, \(A=0.002\) V/A for the non-passivated one and \(\hbar\omega=0.28\) eV, \(A=0.002\) V/A for the passivated one. The photon energies of the peaks a-e correspond well to the optical transitions A–E denoted in Figure 2.  
Figure 5: (**a**) Intensity of peaks with harmonic order of 3 and 5 in Figure 2(a) as functions of laser field amplitude. A cubic- and a fifth-order dependence are noticed, respectively. These peaks directly result from the multiphoton process; (**b**) Intensity of the peak with a harmonic order of 2.4 in Figure 3(a). No clear dependence is found; (**c**) and (**d**) Intensities for peaks \(b\) in Figure 3(a) and \(e\) in Figure 3(b). Here the dependence is linear and reveals that the emissions are from the intrinsic excitation. To further distinguish the integer order and the two types of fractional order signals, we examine the intensity of the HHG signal as a function of the field amplitude \(A\) (Figure 5). The intensities of the \(n\)-th order HHG peaks are of \(n\)-th order polynomials of the amplitude, shown in Figure 5(a). It clearly shows the nonlinear properties of the integer order harmonics. This is governed by the relationship [32]: \(\mathsf{D}(\omega)=\sum_{n}\varepsilon^{(n)}E^{n}\), where \(\mathsf{D}\) is the electric displacement, \(\varepsilon^{(n)}\) is the \(n\)-th order dielectric constant, and \(E\) is the electric field. The companion peaks near the integer order harmonics do not show clear relationship with the laser field amplitude (Figure 5(b)). Their intensities generally increase but with high fluctuations. On the other hand, the signal from the intrinsic excitation is easily distinguished from the others, since it reveals a linear dependence (Figure 5(c,d)). The intensity is proportional to the number of photons.