Laser-Driven Anharmonic Oscillator: Ground-State Dissociation of the Helium Hydride Molecular Ion by Midinfrared Pulses

The vibrational motion of molecules represents a fundamental example of an anharmonic oscillator. Using a prototype molecular system, ${\mathrm{HeH}}^{+}$, we demonstrate that appropriate laser pulses make it possible to drive the nuclear motion in the anharmonic potential of the electronic ground state, increasing its energy above the potential barrier and facilitating dissociation by purely vibrational excitation. We find excellent agreement between the frequency-dependent response of the helium hydride molecular cation to both classical and quantum mechanical simulations, thus removing any ambiguities through electronic excitation. Our results provide access to the rich dynamics of anharmonic quantum oscillator systems and pave the way to state-selective control schemes in ground-state chemistry by the adequate choice of the laser parameters.

Figure 1

Laser-driven anharmonic oscillator. (a) The ground state ($X{{}^1\mathrm{\Sigma }}^{+}$) of ${\mathrm{HeH}}^{+}$ (solid green) and the vibrational levels of the anharmonic potential of ${\mathrm{HeH}}^{+}$ are compared to a harmonic potential (dashed purple). Midinfrared (MIR) photons can couple these states as consequence of the permanent dipole moment of ${\mathrm{HeH}}^{+}$. The inset enlarges and displays the potential energy curves for the two electronic ground states ($X{{}^1\mathrm{\Sigma }}^{+}$) and the first excited state ($A{{}^1\mathrm{\Sigma }}^{+}$). The large separation makes electronic excitation very unlikely. Panel (b) shows the schematic of the experimental setup. Intense MIR pulses are focused onto a 10 keV ion beam of ${\mathrm{HeH}}^{+}$ and excite the molecule up to dissociation in the electronic ground state. The ionic and neutral fragments are detected in coincidence by a delay line detector. Weak electrostatic fields allow for separation of different fragments. (c) Illustration of a classical trajectory model for laser-induced dissociation of ${\mathrm{HeH}}^{+}$. Shown are the effective potential and the resulting motion on the $X{{}^1\mathrm{\Sigma }}^{+}$ surface for a laser field with $3.2\mu \mathrm{m}$ and 50 fs at a peak intensity of $30\mathrm{TW}/{\mathrm{cm}}^2$.

Figure 2

Frequency-dependent response of ${\mathrm{HeH}}^{+}$. (a) Yield for ${\text{HeH}}^{+}+n\hslash \omega \to \mathrm{He}+{\mathrm{H}}^{+}$ as function of the periodicity of the driving field and laser wavelength (upper abscissa). Measured results are displayed together with one-dimensional time-dependent Schrödinger equation (TDSE) and classical simulations. The computational results have been averaged over a distribution of initial vibrational states ($v=0–4$) according to the temperature of the ${\mathrm{HeH}}^{+}$ source. To account for the lower laser intensity at long wavelengths, simulations have been performed for two intensity values ($1\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ and $7\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$). For better clarity, in the simulation a constant pulse duration of 50 fs was used, although the pulse duration in the experiment varied by a factor of 2 due to technical reasons (see the Supplemental Material [17] for details). Note that longer pulse durations can result in higher dissociation yields, but the assumption of a wavelength-independent pulse duration does not influence the interpretation of this work. In (b), for a larger frequency range, the theoretical dissociation probability of the classical and the quantum mechanical simulations is compared at an intensity of $7\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ for the vibrational ground state ($v=0$) and vibrationally averaged over the states $v=0–4$. The second maximum around $T_L\approx 25\mathrm{fs}$ is attributed to two-photon excitation of the $v=0\to 1$ transition.

Figure 3

Above-threshold dissociation of ${\mathrm{HeH}}^{+}$. (a) Measured kinetic energy release (KER) dependent dissociation yield (blue) for ${\mathrm{T}}_L=6\mathrm{fs}$ ($\lambda =1800\mathrm{nm}$) together with the vibrational and focal-volume-averaged 1D TDSE simulation (red) ($\mathrm{I}\approx 10\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ in the measurement and ${\mathrm{I}}_0=7\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ in the simulation). Along with the $v$-averaged distribution, the contributions of the initial states $v=0$, $v=2$, and $v=4$ are shown (black). Good agreement between measurement and simulation is obtained when averaging over vibrational and rotational states is included (yellow). The inset displays the measured momentum distribution in the polarization plane. (b) same for $T_L=10.7\mathrm{fs}$ (${\lambda }_L=3200\mathrm{nm}$, $\mathrm{I}\approx 1\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ in the measurement and ${\mathrm{I}}_0=1.5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ in the simulation).

Figure 4

Time dependence of the excitation dynamics. (a) Simulated evolution of a wave packet with $v=0$ and $J=0$ on the $X{{}^1\mathrm{\Sigma }}^{+}$ in a $1.8\mu \mathrm{m}$ laser pulse with 50 fs and a peak intensity of $\mathrm{I}=7\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. Displayed is the time-dependent probability density (log scale). (b) shows the initial distribution of different $v$ used in the simulations. In addition, the population of rotational states, which is displayed in (c) for a temperature of 3100 K, is influencing the KER spectrum. In (d), the detailed excitation pathways of vibrational states during the interaction of ${\mathrm{HeH}}^{+}$ for different initial $v$ are shown. $P_{\mathrm{dis}}$ denotes the final dissociation probability for each vibrational state. The major contribution to dissociation comes from $v=0$, $v=2$, and $v=4$. During the interaction, the excitation of the states $v>6$ remains low and is therefore not displayed.