Carrier-Envelope Phase Control over Pathway Interference in Strong-Field Dissociation of ${\mathbf{H}}_2^{\mathbf{+}}$

The dissociation of an ${\mathrm{H}}_2^{+}$ molecular-ion beam by linearly polarized, carrier-envelope-phase-tagged 5 fs pulses at $4\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ with a central wavelength of 730 nm was studied using a coincidence 3D momentum imaging technique. Carrier-envelope-phase-dependent asymmetries in the emission direction of ${\mathrm{H}}^{+}$ fragments relative to the laser polarization were observed. These asymmetries are caused by interference of odd and even photon number pathways, where net zero-photon and one-photon interference predominantly contributes at ${\mathrm{H}}^{+}+\mathrm{H}$ kinetic energy releases of 0.2–0.45 eV, and net two-photon and one-photon interference contributes at 1.65–1.9 eV. These measurements of the benchmark ${\mathrm{H}}_2^{+}$ molecule offer the distinct advantage that they can be quantitatively compared with ab initio theory to confirm our understanding of strong-field coherent control via the carrier-envelope phase.

Figure 1

(a) Schematic view of the experimental setup. The laser beam is split into two arms, and in each arm, the dispersion is compensated by a pair of silica wedges. The polarization (indicated by arrows in the path) of the strong arm (80%) is rotated by a broadband $\lambda /2$ wave plate and is focused into the imaging spectrometer, where it intersects an ${\mathrm{H}}_2^{+}$ beam. The weak arm is focused into a single-shot stereographic ATI CEP meter [40, 41], which is used to monitor the CEP at the full repetition rate and CEP tag the molecular data. (b) CEP-integrated dissociation yield as a function of KER and $\cos \theta$. (c) Measured parametric asymmetry plot, from which the CEP is determined (linear color scale).

Figure 2

(a) KER spectrum of ${\mathrm{H}}_2^{+}$ dissociation by 5 fs, $4\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ laser pulses, averaged over $\varphi$. The upper (purple) and middle (pink) shaded regions indicate the regions where the highest asymmetries are observed, and the lower (blue) shaded region indicates the losses into the Faraday cup. (b) The corresponding asymmetry map showing the dependence of $A(\mathrm{KER},\varphi )$ on KER and CEP. The data are shown for fragments within a cone $\Delta \cos \theta =0.2$ around the polarization axis. For each KER bin, the asymmetry is shifted to oscillate around zero. (c) The asymmetry parameter integrated over the indicated energy regions, fit to sinusoidal curves (see the text). (d) The dependence of the asymmetry amplitude $\alpha$ and fraction of the total counts $F$ within the high energy range on the angular range $\Delta \cos \theta$ (lines to guide the eye).

Figure 3

Calculated (a) $\varphi$-averaged dissociation probability $dP/dE$ as a function of KER with the same shaded regions as in Fig. 2a. (b) $A(\mathrm{KER},\varphi )$ as a function of KER and $\varphi$ for the dissociation of ${\mathrm{H}}_2^{+}$. The asymmetry, averaged over the (c) low and (d) high KER regions replotted from Fig. 2c. The solid light blue lines are theoretical predictions for the same KER regions, with the estimated theoretical error in dark blue [32]. The calculations include Franck-Condon factors and intensity averaging.

Figure 4

Intensity-averaged dissociation probability $dP/dE$ as a function of KER for ${\mathrm{H}}_2^{+}$ in select initial vibrational levels (as indicated), weighted by their Franck-Condon factors. The $2p{\sigma }_u$ (solid lines) and $1s{\sigma }_g$ (dashed lines) dissociation probabilities have comparable magnitudes for the (a) low (shaded pink) and (b) high (shaded purple) KER regions exhibiting high asymmetry.