Adiabaticity and diabaticity in strong-field ionization

If the photon energy is much less than the electron binding energy, ionization of an atom by a strong optical field is often described in terms of electron tunneling through the potential barrier resulting from the superposition of the atomic potential and the potential associated with the instantaneous electric component of the optical field. In the strict tunneling regime, the electron response to the optical field is said to be adiabatic, and nonadiabatic effects are assumed to be negligible. Here, we investigate to what degree this terminology is consistent with a language based on the so-called adiabatic representation. This representation is commonly used in various fields of physics. For electronically bound states, the adiabatic representation yields discrete potential-energy curves that are connected by nonadiabatic transitions. When applying the adiabatic representation to optical strong-field ionization, a conceptual challenge is that the eigenstates of the instantaneous Hamiltonian form a continuum; i.e., there are no discrete adiabatic states. This difficulty can be overcome by applying an analytic-continuation technique. In this way, we obtain a rigorous classification of adiabatic states and a clear characterization of (non)adiabatic and (non)diabatic ionization dynamics. Moreover, we distinguish two different regimes within tunneling ionization and explain the dependence of the ionization probability on the pulse envelope.

Figure 1

The energy curves of the two adiabatic states $|{\Psi }_1\rangle$ and $|{\Psi }_2\rangle$ are shown as functions of the parameter $\epsilon$. Through the off-diagonal matrix element $\Delta /2$ a nonadiabatic transition is possible, whereupon the system follows the diabatic states $|1\rangle$ and $|2\rangle$, respectively.

Figure 2

The pure Coulomb potential of the helium atom (solid red line) tilted in the presence of the electric field (dashed green line). The dotted black line denotes the field-free ground-state energy.

Figure 3

The real part of the energy of the first adiabatic eigenstates as a function of a static electric field. The inset magnifies avoided crossings for small electric fields.

Figure 4

(a) The real part of the energy of the diabatic state $|{\Psi }_0^{\left(d\right)}\rangle$ and (b) its tunneling rate, $\Gamma =-2\mathrm{Im}\left(E\right)$, shown as a function of the electric field. (c) The overlap of $|{\Psi }_0^{\left(d\right)}\rangle$ with the field-free ground state.

Figure 5

Comparison of the ground-state populations calculated via numerical solution of the Schrödinger equation and via the rate equation (18) for the distinguished diabatic state for four different photon energies. The pulse intensities are highlighted in the background: the pulse amplitude is $F_0=0.25$ a.u., and the pulse duration is 400 a.u. ($\approx$10 fs).

Figure 6

(a) Ground-state population after the end of the pulse calculated via numerical solution of the Schrödinger equation and from the single diabatic ground state as a function of the photon energy and (b) relative difference between the two results, corresponding to the degree of nondiabaticity of the ionization. The peak field strength is $F_0=0.2$ a.u., and the pulse duration is 400 a.u. The corresponding Keldysh parameter $\gamma$ is shown for different regions.