Enhanced harmonic generation in double-well potentials

Recently, a new three-level structure, called the $\Gamma$ system, which shows strong multiphoton coupling and harmonic generation, has been identified and studied. Under certain circumstances, the $\Gamma$ system appears in one-dimensional (1D) double-well potentials, suggesting that strong multiphoton excitation and harmonic generation may be seen in molecules at favorable internuclear separations, as various aspects of molecules in strong fields can be modeled by 1D potentials. However, even 1D potentials will have additional states, besides the $\Gamma$ system, which may interfere with these multiphoton processes. This paper presents numerical solutions to the time-dependent Schrödinger equation (TDSE) of two electrons in a double-well potential approximating a dication ($A_2{}^{2+}$) at various internuclear separations $R$. These calculations are compared to two- and three-level approximations of the full energy level structure. We show (1) the presence of the $\Gamma$ system in the 1D double-well potential; (2) that harmonic generation is enhanced at certain $R$; (3) that the enhancement is due to the $\Gamma$ system by showing that the three-level system shows the enhancement, while the two-level system does not. This goes beyond the work of Berent and Parzyński [Phys. Rev. A 82, 023804 (2010)] by demonstrating that the additional levels in the full TDSE calculation do not disrupt this response. In addition, we investigate a neutral double-well potential ($A_2$) with both TDSE and five-level calculations. While the level structure is not as clear as the dication, enhanced harmonic generation is still present, from which we conclude that the $\Gamma$ system inherent in the double-well potential still allows for a strong multiphoton coupling, even in the presence of other levels. From these calculations, we expect that neutral diatomic molecules at intermediate $R$ should be a strong source of harmonic generation when exposed to intense laser fields. Indeed, in both charge states, the harmonic radiation is highly nonperturbative, as the power in the harmonics grows with harmonic order from 3 to 17, under certain conditions.

Figure 1

Five lowest potential-energy curves for model diatomic molecules (a) $A_2$ and (b) $A_2{}^{2+}$.

Figure 2

Allowed dipole couplings between the various states of (a) $A_2$ and (b) $A_2{}^{2+}$.

Figure 3

Ionization probability as a function of $R$ for (a) $A_2$ and (b) $A_2^{2+}$ for various electric-field strengths $E_o$. Note that ${10}^{-10}$ is the numerical noise level for ionization probability in these calculations.

Figure 4

Harmonic generation from $A_2{}^{2+}$ for (a) $R=1$, $E_o=0.05$ and (b) $R=4$, $E_o=0.05$. The positions of the first few harmonics are indicated in (a). Various resonant transitions can be seen, such as the 33-eV peak in (a).

Figure 5

Harmonic generation from $A_2{}^{2+}$ for (a) $R=1$, $E_o=0.15$ and (b) $R=4$, $E_o=0.15$.

Figure 6

Harmonic generation from $A_2{}^{2+}$ with $E_o=0.15$ for (a) two-level, (b) three-level, and (c) full TDSE calculations.

Figure 7

Harmonic generation from $A_2$ for (a) $R=1$, $E_o=0.02$ and (b) $R=5$, $E_o=0.02$, using the full TDSE and a five-level approximation.

Figure 8

Harmonic generation from $A_2$ for (a) $R=1$, $E_o=0.075$ and (b) $R=5$, $E_o=0.075$, using the full TDSE and a five-level approximation.

Figure 9

Harmonic generation from $A_2$ for (a) $E_o=0.05$ and (b) $E_o=0.075$, using a five-level approximation, and (c) $E_o=0.05$ and (d) $E_o=0.075$, using the full TDSE calculations.

Figure 10

Level diagrams for $A_2$ and $A_2^{2+}$ showing the nonzero dipole couplings at large $R$. The thick lines correspond to the $\Gamma$ system.