Adiabatic theory of ionization by intense laser pulses: Finite-range potentials

The adiabatic theory of ionization of an electron, initially bound in a three-dimensional potential well, by an intense low-frequency laser pulse is developed. The general case of arbitrary time dependence and polarization of the laser field and arbitrary potential which can model an atom or a molecule in the single-active-electron approximation is treated, with the only restriction that the potential should not have a Coulomb tail. The asymptotics of the solution to the time-dependent Schrödinger equation and photoelectron momentum distribution in the adiabatic regime are obtained. Both the adiabatic and the rescattering parts of the wave function and ionization amplitude are considered. These asymptotics are expressed in terms of the Siegert state originating from the initial bound state in the presence of a static electric field and scattering states in the unperturbed potential. They are uniform in terms of the amplitude of the laser field and apply to weak underbarrier as well as strong overbarrier fields, provided the condition defining the region of validity of the adiabatic approximation is fulfilled. The theory is illustrated by calculations which confirm that the adiabatic results converge to the exact ones as the adiabatic parameter tends to zero.

Figure 1

The hatched area shows the region of applicability of the nonrelativistic version of the adiabatic theory. For comparison, the condition of applicability of the Keldysh theory in the tunneling regime is $\epsilon \ll \xi \ll 1$.

Figure 2

Energy (a), ionization rate (b), and projection (46) (c) for the SS that coincides with the ground $1s$ state for $F=0$. Solid lines, the present model potential (47); dashed lines, the Coulomb potential, $V\left(r\right)=-1/r$. The perturbation theory (PT) and weak-field asymptotic theory (AS) results are obtained from Eqs. (48a) and (48b), respectively.

Figure 3

Transverse momentum distribution $|A_{1s}(k_{\perp };F){|}^2$ for the same SS as in Fig. 2 for the present model potential (47). The results for the Coulomb potential look very similar.

Figure 4

The absolute value of the scattering amplitude, $\left|f\right(k,\theta \left)\right|$, for the present model potential (47). The narrow $d$ and broad $f$ resonances are located at $k\approx 0.0529$ and $0.241$, respectively.

Figure 5

Illustration of the function ${\mathbf{k}}_a\left(t\right)=k_a\left(t\right){\mathbf{e}}_z$ and the corresponding classical ${\mathcal{K}}_a$ and quantum $K_a$ supports of the adiabatic part of the PEMD for a few-cycle linearly polarized laser pulse. Solid circles show the classical boundaries of the PEMD.

Figure 6

Exact results and adiabatic approximation (AA) for (a) time-dependent probability to survive in the initial $1s$ state, (b) ionization rate, and (c) projection factor for three half-cycle pulses [Eqs. (143) and (144)] with $F_0=0.1$. The adiabatic results in (a) are obtained from Eq. (110). The “exact” results in (b) and (c) are reconstructed from the exact $P_{1s}\left(t\right)$ using Eqs. (145).

Figure 7

Same as in Fig. 6, but for $F_0=0.4$.

Figure 8

Exact results and adiabatic approximation (AA) for PEMDs $P(k_{\perp },k_z)$ produced by three half-cycle pulses with $F_0=0.1$ (see Fig. 6). The upper classical boundaries of the PEMD $k_z^{\max }$ for these pulses are $2.22$, $4.43$, and $6.65$, respectively.

Figure 9

The cuts of $P(k_{\perp },k_z)$ shown in Fig. 8 at $k_{\perp }=0$ (left) and $k_z=0.5k_z^{\max }$ (right). All panels have the same vertical scale.

Figure 10

Same as in Fig. 8, but for half-cycle pulses with $F_0=0.4$ (see Fig. 7). The values of $k_z^{\max }$ for these pulses are $3.54$, $7.09$, and $10.63$, respectively.

Figure 11

The cuts of $P(k_{\perp },k_z)$ shown in Fig. 10 at $k_{\perp }=0$ (left) and $k_z=0.9k_z^{\max }$ (right). All panels have the same vertical scale.

Figure 12

Exact results and adiabatic approximation (AA) for PEMDs $P(k_{\perp },k_z)$ produced by three one-cycle pulses [Eqs. (143) and (146)] with $F_0=0.1$. The values of $k_z^{\max }$ for these pulses are $2.91$, $4.37$, and $5.83$, respectively.

Figure 13

The cuts of $P(k_{\perp },k_z)$ shown in Fig. 12 at $k_{\perp }=0$ (left) and $k_z=0.6k_z^{\max }$ (right).

Figure 14

Same as in Fig. 12, but for $F_0=0.4$. The values of $k_z^{\max }$ for these pulses are $4.66$, $6.99$, and $9.33$, respectively.

Figure 15

The cuts of $P(k_{\perp },k_z)$ shown in Fig. 14 at $k_{\perp }=0$ (left) and $k_z=0.2k_z^{\max }$ (right).