Quantum interference of high-order harmonics from mixed gases

We present a theoretical study about the interference of the harmonics generated by a mixture of two gases, He-Ne. Our model is based on the electron quantum paths, a discrete number of electron trajectories, and continuum-bound transitions. A laser with intensity around ${10}^{14}\text{W}/{\text{cm}}^2$ that interacts with a mixture of gases, He-Ne, produces an interference that is destructive at the low-order harmonics and oscillates between constructive and destructive near to cutoff. This destructive interference at high-order harmonics may be used to explore other transitions, which are currently hidden. At low-order harmonic frequencies, our numerical results are in very good agreement with experimental data. At higher-order harmonics, where there are no experimental data, comparison is with a Schrödinger solver.

Figure 1

Phase of Ne harmonics calculated from the Fourier transform of the Eq. (5) with an intensity of $I=4\times {10}^{14}\text{W}/{\text{cm}}^2$. Comparison using different dipole transition matrices $\langle k\left|r\right|{\mathrm{\Psi }}_0\rangle$: STO is based on Slater-type orbitals for Ne (11), GAUS uses a dipole matrix based on a Gaussian wave function, and Qprop is a Schrödinger solver.

Figure 2

(a) The solid black line is the spectrum obtained with a mix of gases in which the number of He atoms is 33 times higher than the number of Ne atoms for the intensity $4.0\times {10}^{14}\text{W}/{\text{cm}}^2$. The 61st harmonic appears practically annihilated, as is calculated in Fig. 7. (b) Constructive interference for the 59th harmonic.

Figure 3

Relative phase $\mathrm{\Delta }\varphi ={\varphi }_{\mathrm{He}}-{\varphi }_{\mathrm{Ne}}$ with a laser intensity of $I=4.0\times {10}^{14}\text{W}/{\text{cm}}^2$ and $\lambda =800$ nm. QP: Relative phase calculated by means of quantum paths. Expt.: Experimental data from Kanai et al. [24].

Figure 4

For the higher-order harmonics where there are no experimental data, we use a reference to a Schrödinger solver (Qprop). $d\left(\omega \right)$ from Eq. (16) is replaced by $-d^{*}\left(w\right)$ to compare with the fast Fourier transform of Qprop because it uses a negative exponential complex.

Figure 5

(a) Difference of phase from He and Ne. QP1: Selections of trajectories that recombine in the first visit to the parent ion [Eq. (29)]. (b) Dipole modulus, in atomic units, when the electron recombines in same cycle, $0<\mathrm{Re}[{\tau }_r-{\tau }_i]<T$, i.e., the first parent visit. $T<\mathrm{Re}[{\tau }_r-{\tau }_i]<2T$ represents when the electron recombines in the second parent visit, and $2T<\mathrm{Re}[{\tau }_r-{\tau }_i]<3T$ represents when the electron recombines in the third parent visit. The short trajectories are plotted with. Both trajectories converge at the cutoff.

Figure 6

This figure shows the value of the harmonic produced when the electron is set free at $t_i$ and when it is recombined at $t_r$. The $t_i$ and $t_r$, from long and short paths, produce the same frequency. The figure contains the times from two gases He and Ne for two intensities: $3.0\times {10}^{14}$ and $4.0\times {10}^{14}\text{W}/{\text{cm}}^2$. The times correspond to the electron's first visit to the parent ion. Data are extracted from cycle 6 of the electric field.

Figure 7

The dashed line represents $\mathrm{\Delta }\varphi =d_{l\left(\mathrm{He}\right)}^1\left(\omega \right)-d_{l\left(\mathrm{Ne}\right)}^1\left(\omega \right)$ for long trajectories in the first visit to the parent ion. The dot-dashed line represents $\mathrm{\Delta }\varphi =d_{s\left(\mathrm{He}\right)}^1\left(\omega \right)-d_{s\left(\mathrm{Ne}\right)}^1\left(\omega \right)$ for short trajectories. The solid line oscillates around the dash line, and it is enough to predict the relative phase in the extremes.

Figure 8

The simplification $\mathrm{\Delta }{I}_p\mathrm{\Delta }t$ is based on the long paths. Both models have similar values at low-order harmonics, where the modulus of the dipole moment, from the long paths, is predominant. At the harmonics near the cutoff, the 53rd to the 65th, where there is a predominant contribution to the first visit to the parent ion, the approximation $\mathrm{\Delta }{I}_p\mathrm{\Delta }t$ fails for those harmonics where the interference is destructive, although it produces a result similar to that of QP1 for most of the harmonics: 53rd, 57th, 59th, and 65th.

Figure 9

Interference due to the mixture He-Ne for several density ratios (He/Ne): 0, 10, 20, and 30. Some harmonics have constructive interference compared with the case of only neon, and others have destructive interference.