## 3 Red Shift of HHG in Resonant Systems
### Four-Step Model of MHOHG in Resonant Systems  
We study numerically the following time-dependent Schrodinger equation (TDSE):  
Figure 1: Illustration of the positive (solid line) and negative (dotted line) chirp of High-order harmonic generation (HHG) by the rapid change of the electric field \(E(t)\) (Equation (5)). The total pulse duration \(\tau\) is 20 optical cycles. \[i\frac{\partial\Psi}{\partial t}=[H_{{}_{0}}+E(t)(z\cos\chi+x\sin\chi)]\Psi \tag{4}\]  
where \(H_{{}_{0}}\) is the field-free Hamiltonian [22], \(\chi\) is the angle between the laser polarization and the molecular axis \(z\) in the \(xz\) plane. The electric field \(\mathbf{E}(t)\) of the laser pulse is given by:  
\[\mathbf{E}(t)=E_{{}_{0}}f(t)\sin(\omega t)\hat{\mathbf{e}}\,,\,\,t\in[-\tau\,/\,2,\tau\,/ \,2] \tag{5}\]  
where the pulse shape \(f(t)=\cos^{2}(\pi t\,/\,\tau)\), \(\tau\) is the total duration of the laser pulses. The molecular ion HeH\({}^{2+}\) has a large ionization potential (\(I_{{}_{p}}=\)2.25 a.u. (61.2 eV)) and a long-life excited state \(2\,p\sigma\) (the life time is around 4 ns [23].) resonant with the ground state. The MHOHG spectra of HeH\({}^{2+}\) are obtained by numerically solving the TDSE. For the computational details of this one-electron system, we refer to reference [21]. The internuclear distance \(R\) is fixed at 4 a.u. (near the excited state minimum \(R=\)3.89 a.u.). The energy of the first excited state \(2\,p\sigma\) is \(-\)1.03 a.u. (28 eV). The initial state for time evolution is the ground state \(1s\sigma\). The laser polarization is along the \(z\) molecular axis. The power spectra of MHOHG is calculated by a Fourier transformation of the dipole momentum in the acceleration form \(d_{{}_{A}}(t)\), as it is the most reliable numerical method for strong field interactions avoiding after pulse transient effects such as in the dipole form [24]. To further probe the temporal structures of MHOHG, we perform a time profile analysis of the harmonic spectra. The time profile of harmonic \(\omega_{{}_{q}}\) is obtained by a wavelet analysis as in signal processing [25; 26; 27]:  
\[d(\omega_{{}_{q}},t)=\int d_{{}_{A}}(t)w_{{}_{t,\omega_{{}_{q}}}}(t^{{}^{ \prime}})dt^{{}^{\prime}}\,,\,\,d_{{}_{A}}(t)=\left\langle\Psi(t)\right|\hat{ \varepsilon}[VV(r)+E(t)]\left|\,\Psi(t)\right\rangle \tag{6}\]  
with the wavelet kernel \(w_{{}_{t,\omega_{{}_{q}}}}(\hat{\cdot})=\sqrt{\omega_{{}_{q}}}W(\omega_{{}_{q} }(\hat{\cdot}-t))\). The mother wavelet we use is a Morlet wavelet:  
\[W(x)=(1/\sqrt{\sigma})e^{i\omega}e^{-x^{2}/2\sigma^{2}} \tag{7}\]  
The MHOHG spectrum of HeH\({}^{2+}\) in a laser field at wavelength 400 nm and intensity \(I=\) 3.5\(\times\)10\({}^{15}\) W/cm\({}^{2}\) with a duration of 15 optical cycles is shown in Figure 2a. The corresponding time profile of the spectrum is presented in Figure 2b. Due to the break of the symmetry of the system, both odd and even order harmonics are generated as illustrated in Figure 2a. A typical difference from the usual HHG spectra of neutral atoms and molecules is a strong resonance around harmonic order 15 in Figure 2a which agrees well with the dressed energy difference between the first excited state and the ground state [20]. One can also observe a strong resonance range around harmonic order 15 in the time profile analysis in Figure 2b. The conversion efficiency of the resonant harmonics is about 3 orders higher than the harmonics in the long plateau, which can be used to generate bright attosecond pulses (trains). The strong resonance in MHOHG (Figure 2a) cannot be explained consistently by the three-step model [5]. We take the resonant EE of long lifetime excited states into account explicitly and generalize the three-step model to a four-step model as presented in Figure 3. The electron is preexcited to a localized long lifetime resonant bound state, tunnel ionized from the resonant excited state, driven by the laser field, followed by a recombination to the ground state to emit MHOHG.