${\text{H}}_2$ double ionization with few-cycle laser pulses

The temporal dynamics of double ionization of ${\text{H}}_2$ has been investigated both experimentally and theoretically with few-cycle laser pulses. The main observables are the proton spectra associated with the ${\text{H}}^{+}+{\text{H}}^{+}$ fragmentation channel. The model is based on the time-dependent Schrödinger equation and treats the electronic and nuclear coordinates on the same level. Therefore it allows the ultrafast nuclear dynamics to be followed as a function of the laser pulse duration, carrier-envelope phase offset, and peak intensity. We mainly report results in the sequential double-ionization regime above $2\times {10}^{14}\text{W}{\text{cm}}^{-2}$. The proton spectra are shifted to higher energies as the pulse duration is reduced from 40 down to 10 fs. The good agreement between the model predictions and the experimental data at 10 fs permits a theoretical study with pulse durations down to a few femtoseconds. We demonstrate the very fast nuclear dynamics of the ${\text{H}}_2{}_{+}$ ion for a pulse duration as short as 1 fs between the two ionization events, giving $\text{H}_2{}^{+}$ from ${\text{H}}_2$ and ${\text{H}}^{+}+{\text{H}}^{+}$ from $\text{H}_2{}^{+}$. The carrier-envelope phase offset plays a significant role only for pulse durations shorter than 4 fs. At 10 fs, the laser intensity dependence of the proton spectra is fairly well reproduced by the model.

Figure 1

(Color online) Wavelength spectra before (dashed curve) and after (full curve) the hollow fiber filled with argon gas. The argon pressure is 700 mbar. The laser pulse energy and duration are, respectively, $600\mu \text{J}$ and 40 fs.

Figure 2

(Color online) Interferometric autocorrelation (IAC) of 10 fs laser pulses. The experimental and calculated signals are given, respectively, by the full and dotted curves. The IAC curve is calculated using the Fourier transform of the frequency domain electric field, assuming a constant spectral phase.

Figure 3

(Color online) Covariance map from ${\text{H}}_2$ recorded with linearly polarized 40 fs laser pulse at $I=2\times {10}^{14}\text{W}{\text{cm}}^{-2}$ and $p\left({\text{H}}_2\right)=5\times {10}^{-8}\text{mbar}$. The collection electric field is $25\text{V}{\text{cm}}^{-1}$. The bottom and left panels represent the usual proton time-of-flight spectrum and the central panel represents the covariance map. The f1, f2, and f3 peak labels correspond to protons ejected toward the detector. The b1, b2, and b3 labels are associated with protons ejected away from the detector.

Figure 4

(Color online) Proton spectra from ${\text{H}}_2$ recorded with linearly polarized 40 fs laser pulse at $I=2\times {10}^{14}$ and $2\times {10}^{15}\text{W}{\text{cm}}^{-2}$. For each laser intensity, the full curve represents the total proton spectrum and the dashed curve represents the covariance proton spectrum from the ${\text{H}}^{+}+{\text{H}}^{+}$ dissociation channel.

Figure 5

(Color online) Calculated potential curves of the electronic ground states of ${\text{H}}_2$ and $\text{H}_2{}^{+}$. Squares are obtained from the model presented in the paper and full curves are the ab initio energies reported in Refs. [30, 31].

Figure 6

(Color online) Soft Coulomb parameters. The full and the dashed curves represent, respectively, the electron-nuclei and the electron-electron smoothing parameters.

Figure 7

(Color online) Probability density from the vibrational ground state of ${\text{H}}_2$. Full curve: This model. Squares: Calculated from the ab initio ground electronic state $v=0$ of ${\text{H}}_2$ given in Ref. [31].

Figure 8

(Color online) Partition zones of the electron coordinates $z_1$ and $z_2$. The three regions ${\Gamma }_0$, ${\Gamma }_1$, and ${\Gamma }_2$ correspond, respectively, to the ${\text{H}}_2$, $\text{H}_2{}^{+}+e^{-}$, and $\text{H}_2{}^{2+}+e^{-}+e^{-}$ systems.

Figure 9

(Color online) Proton spectra from ${\text{H}}_2$ at $4\times {10}^{14}\text{W}{\text{cm}}^{-2}$. Full curve: Experimental with a pulse duration of 10 fs. Dotted curve: Theoretical with a pulse duration of 10 fs. Dashed curve: Theoretical with a pulse duration of 12 fs.

Figure 10

(Color online) Proton spectra from ${\text{H}}_2$ at $8\times {10}^{14}\text{W}{\text{cm}}^{-2}$ recorded with pulse durations of 40 and 10 fs.

Figure 11

(Color online) Normalized proton spectra from ${\text{H}}_2$ calculated at $8\times {10}^{14}\text{W}{\text{cm}}^{-2}$ for different pulse durations. From top to bottom: Dirac-“0,” 1, 2, 4, and 10 fs. The Dirac-“0”-fs pulse duration spectrum represents what is expected from an instantaneous two-electron ionization of ${\text{H}}_2$.

Figure 12

(Color online) Optimization of the ultrashort pulse duration. Proton spectra from ${\text{H}}_2$ recorded at $2\times {10}^{15}\text{W}{\text{cm}}^{-2}$ with different thicknesses of fused silica from 0 to 5 mm inserted in the beam path before the ion spectrometer.

Figure 13

(Color online) Calculated proton spectra from ${\text{H}}_2$ at $8\times {10}^{14}\text{W}{\text{cm}}^{-2}$ as a function of the carrier-envelope phase $\phi$ and the pulse duration. Full curves: $\phi =0$. Dashed curves: $\phi =\pi /2$. The left (a), middle (b), and right (c) top panels correspond to different pulse durations of, respectively, 1, 2, and 4 fs. The vertical arrows mark the positions of the different peaks. The corresponding electric field is represented in the bottom panel (d) as a function of time for a pulse duration of 1 fs.

Figure 14

(Color online) Proton spectra from ${\text{H}}_2$ at different laser intensities $8\times {10}^{14}$, $4\times {10}^{14}$, $2\times {10}^{14}$, and ${10}^{14}\text{W}{\text{cm}}^{-2}$ from top to bottom. Full curves: Experiment. Dashed curves: Theory.