Following coherent multichannel nuclear wave packets in pump-probe studies of ${\mathrm{O}}_2$ with ultrashort laser pulses

We reexamine the recent pump-probe experiment with ${\mathrm{O}}_2$ using short intense infrared laser pulses theoretically. Using parameters that closely mimic the experimental conditions and taking into account the angle-dependent population redistribution due to resonant coupling between the relevant states, we show that the observed kinetic energy release spectra, including the energy-dependent structure and the quantum beat frequencies, can be accurately reproduced. Our results reveal additional important channels that were missed earlier. In particular, the strong contributions from ${\mathrm{O}}_2{}^{+}a{}^4{\mathrm{\Pi }}_u$ and $b{}^4{\mathrm{\Sigma }}_g^{-}$ states lead to the possibility of observing the interchannel beating. We further demonstrate that, by varying the laser parameters, the coherent nuclear wave-packet motions on different potential energy surfaces (PESs) can be probed and the properties of the PES can be examined. Future experiments with different wavelength lasers are proposed for better probing and controlling nuclear dynamics on different PESs.

Figure 1

(a) Calculated ${\mathrm{O}}_2{}^{+}$ potential energy curves and (b) transition dipole couplings for main channels included in the simulations. The $f{}^4{\mathrm{\Pi }}_g$ curve from Marian et al. [20] and the Morse potential for the $b{}^4{\mathrm{\Sigma }}_g^{-}$ state are also shown as short and long dashed lines, respectively. (c) Angle-dependent rates for ionization from the ground-state ${\mathrm{O}}_2$ to ${\mathrm{O}}_2{}^{+}a{}^4{\mathrm{\Pi }}_u$ and $b{}^4{\mathrm{\Sigma }}_g^{-}$ states, obtained from the MO-ADK at a laser intensity of $1.5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. (d) Same as (c) but accounting for the “redistribution” due to the resonant coupling between $a{}^4{\mathrm{\Pi }}_u$ and $b{}^4{\mathrm{\Sigma }}_g^{-}$ states for a 15-fs laser pulse.

Figure 2

The Franck-Condon factors for $a{}^4{\mathrm{\Pi }}_u$ (a) and $b{}^4{\mathrm{\Sigma }}_g^{-}$ (b) states. (c, d) Vibrational populations in $a{}^4{\mathrm{\Pi }}_u$ and $b{}^4{\mathrm{\Sigma }}_g^{-}$, respectively, after the population “redistribution” for a molecular axis at $\theta ={80}^{\circ }$ with respect to the laser polarization. The laser parameters are the same as for Fig. 1.

Figure 3

Two-channel calculation for the KER spectrum (a), its pump-probe time-delay dependence (b), and the QB spectrum (c) with a 10-fs probe laser pulse with an intensity of $3\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The molecule is assumed to be perfectly aligned along the laser polarization. (d, e, f) Same as (a)–(c), but from the experimental measurements by De et al. [16]. (d), (e), and (f) are adopted from Magrakvelidze et al. [17], copyrighted by the American Physical Society.

Figure 4

Three-channel calculation without population “redistribution” for the KER spectrum (a), its pump-probe time-delay dependence (b), and the QB spectrum (c) with a 15-fs probe laser pulse with an intensity of $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The results have been averaged over ${\mathrm{O}}^{+}$ detection angles near ${90}^{\circ }$ with respect to the laser polarization. The vertical scale in (a) marks the vibrational energies in the $a{}^4{\mathrm{\Pi }}_u$ state after absorbing one photon. (d, e, f) Same as (a), (b), and (c) but for a laser intensity of $3\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The inset in (c) shows a zoom-in of the QB spectrum obtained with a maximum time delay of 700 fs.

Figure 5

(a) KER spectra for fixed angles of $\theta ={75}^{\circ }$, ${80}^{\circ }$, and ${85}^{\circ }$ at a laser intensity of $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The three-channel calculations were carried out without the population redistribution. (b) Same as (a) but for a laser intensity of $3\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 6

Three-channel calculation with population “redistribution” for KER spectrum (a), its pump-probe time-delay dependence (b), and the QB spectrum (c) with a 10-fs probe laser pulse with an intensity of $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The results have been averaged over the detection angles near ${90}^{\circ }$ with respect to the laser polarization. (d, e, f) Same as (a), (b), and (c), but for a 15-fs laser pulse. The black empty circles in (c) and (f) mark the experimental KER peaks from De et al. [16]. In (d) both KER spectra using Morse and current CASSCF potential curves for the $b{}^4{\mathrm{\Sigma }}_g^{-}$ state are shown. The inset in (f) shows the zoom-in of the QB spectrum obtained with the maximum time delay of 700 fs at $\theta ={80}^{\circ }$.

Figure 7

Zoom-in of the KER spectrum for a pump-probe time delay of $[85-150]$ fs for a fixed molecular axis at (a) $\theta ={40}^{\circ }$ and (b) ${80}^{\circ }$. (c) The ion yields versus time delay at KER near 0.2 eV for different molecular alignment angles and laser intensities (as indicated in the labels). Here $I_0=1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. (d) Same as (c) but for the total dissociation yields.

Figure 8

(a, b) Dissociation signals vs QB frequency and molecular axis angle (i.e., ion detection direction with respect to the laser polarization) at a probe laser intensity of $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, for high and low QB frequencies, respectively. (c, d) Same as (a) and (b) but for a probe laser intensity of $3\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. (e) Normalized total dissociation yields vs molecular axis angle. The simple analytical model is shown as the dashed line. See text for more details.

Figure 9

Integrated KER spectrum (a) and its pump-probe time-delay dependence (b) up to 2 eV for a probe laser intensity of $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ at the molecular alignment angle $\theta ={85}^{\circ }$. An additional state ${}^4\mathrm{\Sigma }_u^{+}$ was included in the four-channel calculation. Note that the signals above 1.3 eV (indicated by the horizontal dashed line) have been rescaled up by a factor of 5 for clarity. (c) Schematic of the potential curves most relevant to the KER peak near 1.5 eV, together with their dissociation limits.

Figure 10

Integrated KER spectrum (a), its pump-probe time-delay dependence (b), and QB spectrum (c) with 400-nm laser pulses. A 25-fs probe laser pulse with an intensity of $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ at the molecular alignment angle $\theta ={60}^{\circ }$ was used. Black stars in (a) denote the FC factors (in arbitrary units) for vibrational populations in the ${\mathrm{O}}_2{}^{+}a{}^4{\mathrm{\Pi }}_u$ state after the pump pulse. In (c) the QB signals above 1.7 eV (indicated by the horizontal dashed line) have been multiplied by a factor of 20 for visual clarity. Black circles in (c) are the theoretical estimates for the QB peaks using the CASSCF potential curve for the $a{}^4{\mathrm{\Pi }}_u$ state, while the red triangles indicate the curve from Zhang et al. [22]. (d) Dissociation yields vs QB frequency and molecular axis angle $\theta$.

Figure 11

QB spectrum for the scenario when both ionization from neutral ${\mathrm{O}}_2$ and subsequent dissociation occur during the probe pulse. The results from two-channel and three-channel calculations are shown in (a) and (b), respectively.