Coulomb and dipole effects in tunneling ionization of molecules including nuclear motion

We study tunneling ionization in a one-dimensional three-body model of a molecule, treating both electronic and nuclear degrees of freedom exactly. In a recent paper [Phys. Rev. Lett. 111, 153003 (2013)] it was demonstrated using a finite-range potential that the Born-Oppenheimer approximation breaks down for fields weaker than a critical field $F_{\mathrm{BO}}$ when describing tunneling ionization of molecules, but works for stronger fields. It was also demonstrated that the weak-field asymptotic theory allows for an accurate description in this weak-field limit. In the present paper, we consider a potential with a Coulomb tail and nonzero dipole moment, modeling polar molecules. Our study shows that the conclusions of the aforementioned paper also hold for this potential.

Figure 1

Three-body center-of-mass (c.m.) frame with the two heavy nuclei of mass $m_1$ and $m_2$ [large (blue) circles] at $x_1$ and $x_2$ respectively and the electron of mass $m_3$ [small (red) circle] at $x_3$. The internuclear coordinate is $R=x_2-x_1$.

Figure 2

Solid lines: BO potentials for ${\mathrm{H}}_2{}^{+}$ [37] and ${\mathrm{H}}_2$ [38, 39] as functions of the internuclear distance $R$. Dashed line: present model internuclear potential $U\left(R\right)$. Dashed-dotted line: BO potential for the three-body system $U\left(R\right)+E_e(R;0)$ using the FRP [Eq. (43)]. Dotted line: $U\left(R\right)+E_e(R;0)$ using the CTP [Eq. (44)]. Shaded areas: lowest vibrational state ${\varphi }_0\left(R\right)$ (upper) and BO vibrational ground state $\chi \left(R\right)$ [lower, Eq. (38b)] using the CTP.

Figure 3

Solid (red) [dashed (blue)] line: the FRP [Eq. (43)] [CTP, Eq. (44)] at $R_0=1.4$. Shaded red (blue) area bounded by solid (red) [dashed (blue)] line: ground-state electronic wave function at this $R_0$ for the FRP (CTP), the horizontal solid (black) line is at the electronic energy $E_e(R_0;0)$ of these ground states (the energies are indistinguishable on the scale of the figure).

Figure 4

Upper panel: Real part of the energy $\mathcal{E}=Re\left(E\right)$, for the SS that is the analytic continuation of the field-free ground state for the FRP [Eq. (43)] and CTP [Eq. (44)]. Lower panel: The exact rates $\Gamma =-2Im\left(E\right)$ for the same states.

Figure 5

Total ionization rates of the ground state divided by the field factor for the zeroth channel $W_0\left(F\right)$ [Eq. (36a)]. Solid (black) lines are exact rates, dashed (blue) lines are WFAT rates, and dash-dotted (red) lines are BO rates. Arrows indicate the boundary of the region of validity of the BO approximation [Eq. (40)]. Panel (a) is for the FRP [Eq. (43)] and panel (b) is for the CTP [Eq. (44)].

Figure 6

Exact partial rates ${\Gamma }_v={\left|f_v\right|}^2$ divided by the WFAT partial rates ${\Gamma }_v^{\mathrm{WFAT}}$ [Eq. (46)]. The solid (red) line is the first channel, the rest follow in order downwards. The dotted horizontal line is a guide for the eye. Panel (a) is for the FRP [Eq. (43)] and panel (b) is for the CTP [Eq. (44)].

Figure 7

Real and imaginary parts as well as norm square of the wave function for the FRP and CTP [Eqs. (43) and (44)] at $F=0.06$ in the under-the-barrier regime. The outgoing wave is smaller than the part of the wave function closer to the nuclei, so for $x<-5$ the wave function has been multiplied by a factor of 200 or 30 to make it visible. The dotted (purple) line is an isocontour showing where $V(x;R)+U\left(R\right)+(QR+qx)F=Re\left[E\right(F\left)\right]$. It marks the boundary between the classically allowed and forbidden regions.

Figure 8

Real and imaginary parts as well as norm square of the wave function for the FRP and CTP [Eqs. (43) and (44)] at $F=0.26$ in the over-the-barrier regime. The FRP wave function has been divided by 5 to make it on the same scale as the CTP wave function. The dotted (purple) line is an isocontour showing where $V(x;R)+U\left(R\right)+(QR+qx)F=Re\left[E\right(F\left)\right]$. It marks the boundary between the classically allowed and forbidden regions.

Figure 9

Exact ionization rates for different dipoles using the CTP [Eq. (44)]. For $q_1<0.5$ the dipole and the field are parallel, for $q_1>0.5$ they are antiparallel.

Figure 10

Ionization rates for different dipoles using the CTP [Eq. (44)]. The horizontal arrows indicate the orientation of the total dipole moment [Eq. (31)] with respect to the field. The vertical arrow indicates the boundary of the region of validity of the BO approximation [Eq. (40)]. Solid lines: exact rates. Dashed lines: WFAT rates. Dashed-dotted lines: BO rates.

Figure 11

Exact partial rates ${\Gamma }_v={\left|f_v\right|}^2$ divided by the WFAT partial rates ${\Gamma }_v^{\mathrm{WFAT}}$ [Eq. (46)], for different dipoles using the CTP [Eq. (44)]. The solid (red) line is the first channel, the rest follow in order. The dotted horizontal line is a guide for the eye.