High-order harmonic generation in the presence of a resonance

We investigate high-order harmonic generation from laser-irradiated systems that support a shape resonance. From the numerical solution of the time-dependent Schrödinger equation, we calculate the harmonic spectra and the time-frequency analysis of the harmonic intensity and phase. The analysis reveals the separate contributions of the short and long trajectories as well as the resonance. A range of harmonics is strongly enhanced by the presence of the resonance irrespective of the pulse length. The signature of the resonance remains significant after coherent summation over intensities as a simple method to simulate macroscopic effects. The time-frequency analysis supports the recently proposed four-step mechanism of the enhanced harmonic generation process.

Figure 1

(a) The two potentials used in this work. The ground-state energies of potentials 1 and 2 are at $-$0.54 and $-$0.58 a.u., as indicated (solid horizontal lines). Shape resonances are found to be at 0.95 and 1.20 a.u., respectively (dot-dashed horizontal lines). (b) Modulus squared of the Fourier-transformed autocorrelation functions showing the resonances of potentials 1 and 2.

Figure 2

(a) Short pulse with a (1-1-1) envelope. (b) Long pulse with a (3.5-2-3.5) envelope.

Figure 3

HHG spectra for (a) potential 1 and (b) potential 2, calculated with different pulse lengths for the potentials shown in Fig. 1a. $n$ is the number of optical cycles in a pulse with intensity $I=4\times {10}^{14}$ W/cm${}^2$.

Figure 4

Time-frequency distribution of the harmonic radiation for (a) potential 1 and (b) potential 2. The laser intensity is $4\times {10}^{14}$ W/cm${}^2$. The thick dashed purple curves show classical trajectories for a monochromatic field at $I=4\times {10}^{14}$ W/cm${}^2$. The black dashed lines show the positions of the resonances.

Figure 5

HHG spectra for a single atom (dashed curve) and after coherent summation over intensities (solid curve). (a) Potential 1. (b) Potential 2.

Figure 6

Time-frequency analysis of the harmonic phase difference between the intensities $I_1=4.14\times {10}^{14}$ and $I_2=4.15\times {10}^{14}$ W/cm${}^2$. (a) Potential 1. (b) Potential 2. The thick dashed purple curves show classical trajectories at $I=4.15\times {10}^{14}$ W/cm${}^2$. The black dashed lines show the positions of the resonances.

Figure 7

Time-frequency distribution of the harmonic radiation after intensity averaging. (a) Potential 1. (b) Potential 2. The thick dashed purple curves show classical trajectories for a monochromatic field at $I=4\times {10}^{14}$ W/cm${}^2$. The black dashed lines show the positions of the resonances.