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We make a quantitative investigation on the tunneling and multi-photon channels in the transition regime from Keldysh parameter γ < 1 to γ > 1 by numerically solving the time-dependent Schrödinger equation (TDSE). A method is proposed to separate the contributions of those ionization channels based on the characteristics of the current. By analysing the dependence of the ionization rate on the Keldysh parameter γ, we identify a field independent transition point at γ ≈ 2, which is different from the well-accepted consensus of γ ≈ 1, from adiabatic to nonadiabatic regime.
© 2016 Optical Society of America
1. Introduction
After more than thirty years intensive investigations, atomic photoionization under intense laser irradiation could be considered as the following well-understood process (see, for example, the review articles [1–3]): In the low-intensity and short-wavelength limit, multi-photon absorption dominates, while in the high-intensity and long-wavelength limit, electron’s ionization dynamics is governed predominantly by adiabatic tunneling process. Based on the pioneering work by Keldysh, the above two limits can be identified by a single dimensionless parameter (called as Keldysh parameter), which is defined as the ratio between the frequency of laser light ω and the frequency ωt of electron tunneling through the potential barrier formed by Coulomb potential and electric field [4],
where Ip is the ionization potential, F is the amplitude of the electric field. The Keldysh parameter is also a easy-to-use tool for experimentalists to identify which regime their experiments fall in. When γ ≫ 1, the ionization process is in the multi-photon regime, while for γ ≪ 1 where the field almost does not change during tunneling, it is in the adiabatic tunneling regime. However, if the adiabatic condition is not well satisfied the nonadiabatic effects may arise, especially for γ ~ 1.It is interesting to note that even nonadiabatic effects [5–10] have been theoretically predicted in a transition regime, where γ is around one, adiabatic tunneling theory is still used to interpret the current experimental phenomena (see, for example, [12, 13] among many others) which are performed in the transition regime. Actually, there are different criteria to define the nonadiabaticity. It can be defined as the deviations to the ionization rate and the momentum distribution of the adiabatic tunneling model [7, 9], or it can be related to the electron reaching the continuum with non-vanishing velocity [8]. Until very recently, there are a few experiments tried to test the validity of adiabatic approximations in different criteria, but with contradictory conclusions [14–16]. The authors in Ref. [14] found substantial, but not total, agreement between their results and the predictions of the adiabatic tunneling model by measuring the momentum distribution. The result in Ref. [15] supported the existence of the non-vanishing initial velocity at the tunneling exit, while Refs. [16] confirmed the validity of vanishing velocity approximation for a relative large Keldysh parameter. So the situation becomes even more confused: Is it reasonable to ignore the nonadiabatic effects? If not, what is the condition, wherein the nonadiabatic effects become important?
To make a clear answer to the above question, a quantitative investigation on the multi-photon ionization (MPI) and tunneling ionization (TI) channels is needed. Quantitative estimation on the contributions from these two channels are hard for current experimental techniques, and out of the ranges of normal theory which is usually based on some approximations such as the strong field approximation (SFA) [7–11]. Therefore, the best candidate is the numerical simulation of time-dependent Schrödinger equation (TDSE), which is just like a numerical experiment and can be controlled easily to exclude the undesirable effects.
Since the MPI and TI channels co-exist with comparable contributions, a appropriate criterion is needed to identify the two channels in the transition regime of γ ∼ 1. It is well-known that the emission directions of the electron from these two channels are quiet different. Electrons from TI prefer to emit along the direction of the instantaneous external laser field, while MPI is believed to be independent of the phase of the external laser field [8]. Therefore, it is reasonable to define the two channel as follows: the isotropic contribution in the spatial current distribution is attributed to the MPI channel while the field-direction-dependent remainder makes up the TI channel.
In this paper, we make a quantitative investigation on the MPI and TI channels in the transition regime of γ ∼ 1. A method is proposed to separate the contributions from these two ionization channels based on different characteristics of the instantaneous current. A field independent transition point from adiabatic to non-adiabatic regime at γ ≈ 2 is found, which is different from the consensus of γ ≈ 1.
2. Theory and numerical methods
We employ the numerical solution of the time-dependent Schrödinger equation (TDSE) for simulation (details of the TDSE method can be seen elsewhere [17, 18]). The Hamiltonian of a hydrogen atom driven by a linearly polarized laser field is
where F (t) = F sin (ωt) is the time-dependent electric field linearly polarized along the z direction. To avoid the undesired effects such as rescattering, the wave functions at the first maximum of the electric field are used for analysis, and the electric field is zero outside the first quarter cycle. The solution of the TDSE can be expanded in spherical harmonicsDue to the cylindrical symmetry of the system, the wave function can be reduced to a two-dimensional one:
After substituting Eq. (4) into the current density formula
we obtain . where er and eθ are the unit vectors in the radial direction and angular direction respectively.3. Results and discussions
To avoid disturbing the ionization process we are interested in here, the rescattering effect, the intra- and inter-cycle interference effects as well as other multi-cycle effects are better to be excluded. Therefore, in all the results of this paper, the wave functions at the first maximum of the electric field, i.e., after an evolution of only a quarter optical cycle for a sine field, are used for analysis.
In Figs. 1 and 2, we present the electron density distribution just at the first maximum of the electric filed. In Fig. 1 the electric field is fixed (F = 0.03 a.u.) while the frequency ω is changing to ensure the Keldysh parameter γ ranging from 4 to 0.5, i.e., from the MPI regime to the TI regime. The distribution in Fig. 1(a) shows a mirror symmetry with respect to z = 0, which is a typical pattern of the MPI channel. It is noted that the distribution in Fig. 1(a) exhibits a dip along the x -axis. This is mainly due to the linear polarization of the laser field. The angular distribution will show a minimum in the direction perpendicular to the polarization even for the single-photon ionization process under linearly polarized laser field. With γ decreasing, this symmetry is gradually being broken, and more probability can be found in the left part, i.e. the opposite direction of the external field. This symmetry broken process is due to more ionization from TI channel. Then, a co-exist of MPI and TI style can be found in Figs. 1(b) and 1(c). When γ decreases to 0.5 as depicted in Fig. 1(d), the symmetry is completely broken, which is a typical pattern of the TI channel. In Fig. 2 the fixed electric field increases to F = 0.05 a.u. while the Keldysh parameter γ varies the same as in Fig. 1. The transition behavior of the density distribution in this higher field situation is the same as that in Fig. 1 except for a minor difference when γ < 1. Much more distributions are found beyond the left boundary of the center red part in Figs. 2(c) and 2(d) compared with Figs. 1(c) and 1(d), which indicates a more strong TI process due to the high field.
Fig. 1 Density distributions with different Keldysh parameter γ. The peak value of the sinusoidal field F = 0.03a.u.. The ionization potential is 0.5 a.u.. The frequency (a) ω = 0.12a.u.; (b) ω = 0.06a.u.; (c) ω = 0.03a.u.; (d) ω = 0.015a.u..
Fig. 2 Density distributions with different Keldysh parameter γ. The peak value of the sinusoidal field F = 0.05a.u.. The ionization potential is 0.5 a.u.. The frequency (a) ω = 0.2a.u.; (b) ω = 0.1a.u.; (c) ω = 0.05a.u.; (d) ω = 0.025a.u..
Compared with density distributions, the current distributions can provide more deep insight on the ionization process. Based on Eq. (5), we calculate the current distributions corresponding to Figs. 1 and 2, as shown in Figs. 3 and 4. The arrows indicate the direction and the color represents the magnitude of the current. Close inspection shows that all the distributions can be divided into two parts with different behaviors: the inner part and the outer part. Black dashed lines whose radius is defined by the tunneling exit in the tunneling model, are plotted as a boundary to indicate these two parts in Figs. 3 and 4. According to the tunneling model, once the electron moves out the tunneling exit, it is ionized. The very complex current in the inner part may relate to the polarization process or other under-barrier processes which are not interested in here, and we mainly focus on the ionization process which occurs in the outer part.
Fig. 3 Current distributions with different Keldysh parameter γ. The peak value of the sinusoidal field F = 0.03. The black dashed lines: see text. The parameters are the same as in Fig. 1.
Fig. 4 Current distributions with different Keldysh parameter γ. The peak value of the sinusoidal field F = 0.05. The black dashed lines: see text. The parameters are the same as in Fig. 2.
The current distribution provides a very clear picture of the ionization process. In MPI regime as depicted in Figs. 3(a) and 4(a), the current distribution shows a good symmetry with respect to the z = 0 plane, and the emission is along the radial direction. With decreasing γ, the symmetry is broken. The current gradually turns to the opposite direction of the electric field, meanwhile the magnitude of the current in the region of z < 0 increases dramatically. When γ drops to 0.5, which indicates a deep tunneling regime, almost all the arrows completely point to the polarization direction. Therefore the current can be used to identify the different ionization channels. In the following we will try to separate the contributions from the two ionization channels based on the above analysis.
Let’s safely assume that electrons beyond the tunneling exit are ionized. The ionization rate can be calculated by integrating the current over a sphere surface S with radius at the tunneling exit. The outer turning point is obtained by reducing the three-dimensional Schrödinger equation to one-dimensional one in parabolic coordinates [19]. From Figs. 3 and 4 we know that the MPI current shows a good symmetry with respect to the z = 0 plane and is along the radial direction, while the TI current appears only in the z < 0 region and is along the opposite direction of the electric field. If we divide the whole sphere into two hemispheres S1 (z > 0) and S2 (z < 0), then both TI and MPI current go through S2, but only MPI current go through S1. Further considering the symmetry property of MPI, the rate of MPI can be obtained by integrating the current along the radius direction over S1 and then double it
where r0 corresponds to the outer tunneling exit. In hemispheres S2 both the TI and MPI contribute to the current, so the rate of TI is equal to the integration of the current in the −z direction minus the rate of MPI in this hemisphere .With the help of Eq. (6), the ionization rates from different channels are shown in Fig. 5. It is interesting to note that the ionization rates of the two channels show completely different dependence on Keldysh parameter γ. With increasing γ, the MPI rate first increases quickly and then achieves a plateau, in contrast, the TI rate almost does not change first but decreases quickly then. Since the electric field is fixed here, the rate is actually changing with the light frequency. The characteristics in Fig. 5 can be understood as follows. For the MPI channel, the N-photon ionization rate is in direct ratio to the Nth power of the laser intensity according to perturbation theory. In the present case the laser intensity is smaller than one unit, so the rate increases quickly with decreasing photons involved. However, when the frequency increases further, the above effect becomes saturated and the ionization rate reaches a plateau and even decreases with increasing frequency. This can be explained by considering that the MPI channel needs a duration of time to establish (relevant analysis will be presented elsewhere). When the time of the field decreasing approaches this establishing time, the rate of the MPI becomes saturated. When the laser frequency increases further, the rate starts to decrease.
For the TI channel, according to ADK theory [20] which is based on the adiabatic approximation, the TI rate depends only on the electric field. Therefore, in the region of small γ where the adiabatic approximation is valid, the TI rate almost does not change for a fixed electric field as shown in Fig. 5(b). However, when γ keeps increasing, the adiabatic approximation is not valid any more, the rate of non-adiabatic TI decreases quickly. It is interesting to note that there is a clear transition point from adiabatic to non-adiabatic regime at γ ≈ 2 which is indicated by the dashed line in Fig. 5(b). This is different from the consensus of γ ≈ 1. It is not surprising that the rates of the two channels increase with increasing electric field, but the dependence on γ and the transition point at γ 2 seem electric field independent.
In Fig. 6 we show the dependence of the ratio of MPI to TI on the Keldysh parameter γ. Generally, the ratio increases quickly with increasing γ. If we assume that the contribution of MPI can not be ignored once the ratio exceeds 0.1, then it is easy to read a Keldysh parameter above which the MPI channel has to be considered in Fig. 6. For the lower field of F = 0.03 the corresponding Keldysh parameter is γ = 2.4 while for higher field of 0.05 it is γ = 3. All these values are larger than the transition point of γ ≈ 2. Therefore, the adiabatic tunneling model is probably accurate enough to describe the ionization process in the region of γ < 2.
Fig. 6 The ratio of the MPI rate to the TI rate at different electric fields. The ionization potential is 0.5 a.u..
4. Conclusions
In conclusion, we make a quantitative investigation on the tunneling and multi-photon channels in the transition regime γ ∼ 1. We find that current of the two channels shows very different characteristics. The MPI current is in the radial direction and shows a good symmetry with respect to the z = 0 plane (the laser is polarized along z axis), while the TI current is in the opposite direction of the electric field and appears only in the z < 0 region. Based on the above characteristics, we propose a method that can be used to separate the contributions of the two channels. Our results show that the ionization rates of the two channels show distinct dependence on the Keldysh parameter γ. With increasing γ, the MPI rate first increases quickly and then shows a plateau, in contrast, the TI rate almost does not change first but decreases quickly then. And we identify a field independent transition point from adiabatic to non-adiabatic regime at γ ≈ 2 which is different from the well-accepted consensus of γ ≈ 1. Our results can be tested by measuring the angular-dependent momentum spectrum if the carrier-envelope phase-locked few-cycle laser pulse is applied.
Funding
National Key program for S and T Research and Development (No. 2016YFA0401100); National Basic Research Program of China Grant (2013CB922201); National Natural Science Foundation (NSFC) (11274050, 11334009, 11425414, 11374197, 11447114, 11504215).
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