Adiabatic theory of strong-field ionization of molecules including nuclear motion

The adiabatic theory of strong-field ionization of molecules with internuclear motion included into consideration is developed. Two adiabatic regimes in terms of the electronic, nuclear, and laser field timescales are considered. In the first regime, field is the slowest; that is, its timescale is much larger than the electronic and nuclear timescales. The corresponding theory generalizes the adiabatic theory of strong-field ionization of atoms and molecules with frozen nuclei [Phys. Rev. A 86, 043417 (2012)] by treating the internuclear motion on equal footing with the electronic motion. In the second regime, the active electron is the fastest; that is, its timescale is much smaller than that of the nuclei and laser field. The corresponding theory naturally involves the Born-Oppenheimer approximation. The two versions of the adiabatic theory are validated by comparing their predictions with accurate numerical results obtained by solving the time-dependent Schrödinger equation (TDSE) for a model diatomic molecule with one electronic and one internuclear degree of freedom. The adiabatic results are shown to converge to the TDSE results uniformly with respect to the laser field amplitude both in tunneling and over-the-barrier ionization regimes. Two applications of the theory to the analysis of strong-field effects associated with the internuclear motion are discussed.

Figure 1

Illustration of the regions of applicability of AAf [Eq. (36), the wedge-shaped (red) shaded region] and AAnf [Eq. (58), the rectangular (blue) shaded region] in the plane of timescale ratios $T_f/T_e$ and $T_n/T_e$. In this figure, the electronic timescale is chosen as $T_e=2\pi /I_p=12.6$, where $I_p=0.5$. The nuclear timescale is defined by $T_n=2\pi /\mathrm{\Delta }{E}_n$, where $\mathrm{\Delta }{E}_n$ is the energy difference between the ground and first exited vibrational states of the molecule. We have $T_n=331$ and 461 for ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$, respectively [58]; the corresponding ratios $T_n/T_e$ are shown by dashed lines. The field timescale is $T_f=2\pi /\omega$, where $\omega$ is the laser frequency; the ratios $T_f/T_e$ for several characteristic wavelengths from near-infrared (NIR) to terahertz range are indicated.

Figure 2

BO potentials in the present model for light nuclei with $M=3$. The top (red) line shows the ionic potential $U_{\text{ion}}\left(R\right)$ with several of the lowest vibrational energies ${\varepsilon }_v$. The bottom (black) and intermediate (blue) lines show the molecular BO potential $U_{\text{mol}}\left(R\right)$ for the ground (even in $x$) and first excited (odd in $x$) electronic states with all corresponding bound-state molecular energies $E_n$. The horizontal dotted line indicates the boundary of the electronic continuum ${\varepsilon }_0$.

Figure 3

Similar to Fig. 2, but for heavy nuclei with $M=1836$. The top (red) and bottom (black) dashed lines show BO potentials for real ${\mathrm{H}}_2{}^{+}$ [62] and ${\mathrm{H}}_2$ [63].

Figure 4

Solid lines in the top and bottom panels show energies ${\mathcal{E}}_0\left(F\right)$ and ionization rates ${\mathrm{\Gamma }}_0\left(F\right)$, respectively, of molecular (2D) SSs in the present model with $M=3$ (upper red lines) and $M=1836$ (lower black lines). Dashed (blue) lines in the top and bottom panels show the function $U_{\text{ion}}\left(R_{\text{mol}}\right)+{\mathcal{E}}_e(R_{\text{mol}},F)$ defined by the energy of the electronic (1D) SS and the corresponding ionization rate ${\mathrm{\Gamma }}_e(R_{\text{mol}},F)$, respectively, calculated in the present model with $M=1836$ at the equilibrium internuclear distance $R_{\text{mol}}=1.4$.

Figure 5

Partial PEMDs $P_v\left(k\right)$ for the light nuclei model ($M=3$) generated by half-cycle pulses with amplitudes $F_0$ and durations $T$ indicated in the figure. The TDSE results (darker lines) are obtained from Eq. (12) and the AAf results (lighter lines) are calculated using Eq. (57). The top TDSE and AAf lines in each panel correspond to $v=0$; the others follow downward in increasing order of $v$.

Figure 6

Partial PEMDs $P_v\left(k\right)$ for the light nuclei model ($M=3$) generated by two-cycle ($n_{\text{oc}}=2$) pulses with the amplitude $F_0=0.1$ and durations $T$ indicated in the figure. The TDSE results (darker lines) are obtained from Eq. (12) and the AAf results (lighter lines) are calculated using Eq. (57).

Figure 7

Same as in Fig. 6, but for the pulse amplitude $F_0=0.2$.

Figure 8

Total PEMD $P\left(k\right)$ for the heavy nuclei model ($M=1836$) generated by half-cycle pulses with amplitudes $F_0$ and durations $T$ indicated in the figure. The TDSE results (darker lines) are obtained from Eq. (14) and the AAf results (lighter lines) are calculated using Eq. (57).

Figure 9

Same as in Fig. 8, but with the AAf results replaced by the AAnf results calculated using Eq. (73).

Figure 10

Channel ionization probabilities $P_v^{\text{ion}}$ for the same model and pulses as in Figs. 8 and 9. The TDSE results (circles) are obtained from Eq. (13), the AAf results (triangles) are calculated using Eq. (57), and the AAnf (crosses) results are calculated using Eq. (73).

Figure 11

Partial PEMDs $P_v\left(k\right)$ for the same model and pulse as in the bottom right panel in Figs. 8, 9, and 10. The TDSE results (darker lines in both left and right panels) are obtained from Eq. (12), the AAf results (lighter lines in the left panels) are calculated using Eq. (57), and the AAnf results (lighter lines in the right panels, almost indistinguishable from the TDSE results) are calculated using Eq. (73).

Figure 12

Probabilities $P_n$ that the molecule survives in a bound state as functions of the energy $E_n$ of the state for the same model and pulses as in Figs. 8, 9, and 10. The solid (open) circles show the TDSE results obtained from Eq. (10) for bound states with even (odd) symmetry in $x$, corresponding to the ground (first excited) electronic state within the BOA. In the right panels, the TDSE results for the odd states are multiplied by a factor indicated in the figure. The crosses show the AAnf results for the even states obtained from Eq. (71).

Figure 13

Total PEMD $P\left(k\right)$ for the heavy nuclei model ($M=1836$) generated by half-cycle pulses with amplitudes $F_0$ and durations $T$ indicated in the figure. The AAf (lower lines in the left panels and upper lines in the right panels) and AAnf results are calculated using Eqs. (57) and (73), respectively.

Figure 14

Total PEMD $P\left(k\right)$ for the heavy nuclei model ($M=1836$) generated by two-cycle ($n_{\text{oc}}=2$) pulses with the amplitudes and durations indicated in the figure. The TDSE results (darker lines) are obtained from Eq. (14) and the AAnf results (lighter lines) are calculated using Eq. (73).

Figure 15

Channel ionization probabilities $P_v^{\text{ion}}$ for the same model and pulses as in Fig. 14. The TDSE results (circles) are obtained from Eq. (13) and the AAnf results (crosses) are calculated using Eq. (73).

Figure 16

Partial PEMDs $P_v\left(k\right)$ for the same model and pulse as in the bottom right panel in Figs. 14 and 15. The TDSE results (darker lines) are obtained from Eq. (12) and the AAnf results (lighter lines) are calculated using Eq. (73).

Figure 17

Probabilities $P_n$ that the molecule survives in a bound state as functions of the energy $E_n$ of the state for the same model and pulses as in Figs. 14 and 15. The solid (open) circles show the TDSE results obtained from Eq. (10) for states with even (odd) symmetry in $x$. The crosses show the AAnf results for the even states obtained from Eq. (71).

Figure 18

Partial AAnf PEMDs for the heavy nuclei model with $M=1836$ generated by a pulse with $\omega =0.057$, $n_{\text{oc}}=19$, and $F_0=0.05$ calculated using Eq. (73). In the top panel, the solid (blue) lines show the PEMDs $P_v^{\text{nf}}\left(k\right)$ as functions of $k^2/2+{\varepsilon }_v$. The dashed lines show the corresponding averaged PEMDs defined by Eq. (92). The vertical (gray) lines indicate the energies where the condition (91) is fulfilled. The bottom panel shows the ratio (93) in the plane of the electronic $k^2/2$ and molecular ion ${\varepsilon }_v$ energies. The conditions (91) and (94) are fulfilled along the solid (gray) lines and dashed (red) lines in this panel, respectively. The results include only electrons ejected in the positive direction $k>0$. For multicycle pulses, the $k<0$ results are practically identical.

Figure 19

Solid circles show the TDSE results for the channel ionization probabilities $P_v^{\text{ion}}$ for the heavy nuclei model with $M=m_p$, $2m_p$, and $4m_p$ (from top to bottom), where $m_p=1836$, obtained from Eq. (13) for a few-cycle pulse with $\omega =0.057$, $n_{\text{oc}}=2$, and $F_0=0.15$. The results are shown as functions of the ionic energy ${\varepsilon }_v$. Crosses show the corresponding AAnf results for the largest nuclear mass $M=4m_p$ calculated using Eq. (73). The solid line shows the unperturbed ground-state nuclear probability density $|{\mathrm{\Psi }}_0\left(R_v\right){|}^2$ for $M=4m_p$, where the ionic turning point $R_v$ is a function of ${\varepsilon }_v$ defined by $U_{\text{ion}}\left(R_v\right)={\varepsilon }_v$. The solid line is normalized to the AAnf results at the maximum.