1. Introduction

The study of the interaction of intense laser pulses with atomic systems has been an area of active research for well over a decade. A number of new and interesting phenomena have been discovered such as high harmonic generation, above threshold ionization, tunnelling ionization and stabilization. For recent reviews of intense-field multiphoton physics see, for example, refs. [1,2]. Stabilization, first suggested by Gersten and Mittleman [3], refers to the decreasing ionization probability, with increasing laser intensity, of an atom interacting with a high frequency laser field [4,5]. In this regime, the atomic structure of the atom becomes strongly distorted and, in addition to stabilization, other phenomena such as light-induced states [6–10] and multiply charged negative ions [11] have been investigated.

Theoretical studies of stabilization have been based on Floquet methods, in particular the high-frequency Floquet theory (HFFT) in the Kramers-Henneberger (K-H) frame (the rest frame of a free electron in the laser field) [4], the direct numerical integration of the time-dependent Schrödinger equation [12–16] and classical simulations [17]. Using Floquet theory, the form of the stabilized electron probability distributions as well as ionization rates for linear and circular laser polarizations have been predicted for the quasi-stationary states of the atom in the laser field [4,5,18,19]. For short laser pulses, explicitly time-dependent methods have been utilized to demonstrate stabilization. These studies have considered one dimensional models [12–14] or atomic hydrogen in a linearly polarized laser pulse [15,16]. For the latter case, obtaining solutions of the time-dependent Schrödinger equation reduces to solving an effective two-dimensional problem. However, if an arbitrarily polarized laser is to be considered, the full three-dimensional problem must be solved. This is computationally demanding [20] and for this reason a limited number of numerical studies have been carried out for atoms interacting with intense laser pulses of arbitrary polarization.

In order to gain insight into laser polarization effects in the stabilization regime, we consider the interaction of a two-dimensional model atom [21, 22] with an intense, high frequency laser pulse of arbitrary polarization. In atomic units (a.u.), which we use throughout, the two-dimensional Schrödinger equation which we solve is

The laser field is described classically with the vector potential A(t) taken to be

where f(t) is the laser pulse envelope, ∈ is the ellipticity parameter of the laser pulse, E ₀ is the peak electric field strength and ω is the angular frequency of the laser. The pulse shape, f(t), is taken to be trapezoidal and is chosen such that displacement

which corresponds to the quiver amplitude of a free, classical electron in the laser field, oscillates about the origin. The maximum amplitude of this displacement is

We use a smoothed Coulomb potential

to avoid problems associated with the singularity at r = 0. The smoothing parameter is a = 0.8, so that the ground state has the same energy as hydrogen, i.e. -0.5. Equation (1) is numerically integrated on an uniform grid using a split step operator method which is described in Protopapas et al [21] and references therein.

Insight into the phenomena of stabilization can be most readily obtained by transforming to the K-H frame which corresponds to the rest frame of an oscillating electron in the laser field. In this frame the influence of the laser-atom interaction is absorbed into the atomic potential,

For laser fields of constant (or slowly varying) amplitude, this potential is periodic in time and can be Fourier expanded. In this case the wavefunction can be written in Floquet-Fourier form and the energies and decay rates of the resonance states of the atom in the laser field are determined by solving the time-independent Floquet eigenvalue problem. When the laser frequency is sufficiently high, a lowest order approximation is obtained by keeping only the static term of the Fourier expansion of this potential. This is the cycle averaged K-H potential given by

where T = 2π/ω. Stationary eigenstates of this potential, which are termed K-H eigenstates,

give the zeroth order approximation to the structure of the atom in the laser field in the HFFT. This approximation is a good one if the frequency of the laser is such that ω ≫ |$E_0^{\text{KH}}$|, where $E_0^{\text{KH}}$ is the energy of the K-H ground state. In the zeroth order approximation, the higher order terms of the Fourier expansion of the potential are neglected. In this case there are no transitions between K-H eigenstates and the bound K-H states are stable against ionization. It is clear from equation (7) that the laser ellipticity will play a major factor in determining the properties of the K-H ground state and hence the stabilization of the atom.

In addition, the pulse turn-on time is influential in governing the extent of stabilization of the atom [15,23] and the electron wavepacket dynamics in the intense, high frequency regime. When the laser pulse turn-on is long, the ground state of the atom evolves adiabatically into the ground state of the K-H potential. However, such a turn-on results in a high degree of ionization before the atom experiences the intensities required for the zeroth order HFFT to be valid, and hence stabilization to occur. On the other hand, if the laser pulse turn-on is too rapid, adiabatic evolution is excluded and there will be substantial “shake-off” ionization as well as the population of excited bound K-H states due to the fact that the wavepacket cannot respond quickly enough to the laser turn-on. Before considering these turn-on effects in more detail, we investigate in the following section the ellipticity dependence of the stabilized wavepackets.

2. Ellipticity dependence of wavepacket structure and corresponding K-H ground states

Insight into the dynamics of laser-atom interactions in the stabilization regime can be obtained by studying the structure of the localized wavepackets. We start by comparing the wavepackets formed by lasers having different ellipticities. In Fig. 1 we have plotted the magnitude squared of the stabilized wavepackets as a function of ellipticity with the ellipticity varying from 0.1 to 0.9 in steps of 0.1. Note the same color-scale is used for each of the plots in the figure. The angular frequency of the laser is ω = 1 and the electric field strength is E ₀ = 15 so that α ₀ = 15. The laser pulse shape consists of a 2 cycle turn-on followed by 10 cycles of constant amplitude. The snapshots in Fig. 1 are taken at the end of the 12th cycle.

From Fig. 1 we observe that as the ellipticity is increased from ∈ = 0.1 to ∈ = 0.9 there is a dramatic change in the structure of the wavepackets, both in the localized part and the extended part which corresponds to ionization. For small ellipticities we find that there is a tridental peak structure. As the ellipticity is increased the localization of the central peak reduces and becomes less intense than the two outer peaks. There is a clear change in the structure between ∈ = 0.3 to ∈ = 0.4, where the central peak divides and forms the beginnings of the “kidney” shaped localizations which becomes prominent for ∈ = 0.5 [22]. Another transition occurs from ∈ = 0.4 to ∈ = 0.6 in which the two outer peaks become less intense and the kidney-like localizations elongate leading to the onset of a toroidal structure. For these higher ellipticities a central node also becomes apparent.

Focusing now on the ionizing part of the wavepacket, and first considering the case of ∈ = 0.1, we can see waves of ionization extending out from each of the two outer peaks. In addition, four minima extending diagonally out from centre of the wavepacket can be distinguished. For ∈ = 0.4 and greater, the structure in the ionizing wavepacket now appears to be originating from the entire structure and there is marked reduction in the fringe structure. In the final case, ∈ = 0.9, we observe ionization as concentric circles of probability density originating from the center of the structure, with ionization occurring nearly isotropically. The structure in the ionizing part of the wavepacket is discussed further in the following section. We note that there is a residual anisotropy in the ionizing wavepacket which is primarily due to the laser pulse turn-on. Also, there is an offset in the position of the localized wavepacket with increasing ellipticity due to the instantaneous electric field along the y-axis.

 

Figure 1. Snapshots of the magnitude squared of the wavepackets as a function of ellipticity, taken after 12 cycles of a laser pulse having angular frequency ω = 1 and peak electric field E ₀ = 15. Distances (X,Y) are shown in atomic units (a.u.).

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In Fig. 2 we show the K-H ground states for each of the ellipticities considered in Fig. 1. For ∈ = 0.1 the ground state is essentially a dichotomous structure. As the ellipticity is increased the K-H potential forms saddle points [22] along the y-axis and the dichotomous structure of the wavefunction becomes less localized. As the ellipticity approaches 1.0, the saddle points in the potential become deeper until a single circular minimum is formed, which results in a circular symmetric K-H ground state. In Fig. 2, note the different color scales.

By comparing the K-H ground states and the stabilized wavepackets, we are able to obtain an indication of the extent of adiabaticity in the evolution of the system. For each ellipticity the global structures found in the K-H ground state are present in the wavepacket, implying that a portion of the initial Coulomb ground state has evolved into the K-H ground state. However, due to the non-adiabaticity of our pulse turn-on, population has been excited into higher K-H states. In each case the superposition of excited K-H states manifests itself by the additional structure observed in the wavepackets and the form of the structure will depend on the time of the snapshot. In particular, in the first row of Fig. 1 the excited K-H states manifest themselves as a node at the center of the wavepacket. As the ellipticity is increased, the emergence of kidney shaped lobes can be seen. Finally, in the third row, significant probability is located at the center of the wavepacket and is separated by a trough from the outer ring structure corresponding to the position of K-H ground states.

 

Figure 2. The K-H ground states for α ₀ = 15 and ellipticities corresponding to those of Fig. 1. Distances (X,Y) are shown in atomic units (a.u.).

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3. Turn-on effects

As mentioned above, the laser pulse shape is important in determining the extent of shake-off ionization, the number of excited K-H states populated and the amount of ionization before the atom is stabilized. We now investigate these effects using ramped laser pulse turn-on times of t_on = 1, 4, 8 and 12 cycles, and for the ellipticities ∈ = 0, 0.5 and 1. We plot the magnitude squared of the wavepacket at the end of 10, 20 and 30 cycles using two color scales; one chosen to emphasize the localized part of the wavepacket and the other to illustrate the ionization dynamics. As previously, we take ω = 1 and E ₀ = 15.

We begin by considering the linear case, ∈ = 0. In Fig. 3, snapshots of the magnitude squared of the wave functions are plotted with a color scale which emphasizes the localized structure. For a one-cycle turn-on, shown in the first row, it is clear from the evolution of the wavepacket that, in addition to the K-H ground state, significant population is also present in the K-H excited states, as can be seen in the time-dependence of the localized wavepacket density. Very similar behavior was observed by Kulander et al. [15,16] in time-dependent calculations on atomic hydrogen in intense, linearly polarized, high frequency laser pulses. For t_on = 8 and 12, the central peaks are less well defined, which would be expected from the more adiabatic pulse turn-ons. There is a greater resemblance to the K-H ground states in these latter cases. This can be quantified by comparing the projections of the wavepackets after 30 cycles onto the corresponding K-H ground state. The magnitude squared of the projections, |⟨Ψ(t)|${\mathrm{\Psi }}_0^{\mathit{\text{KH}}}$⟩|², for the case of a 1, 4, 8 and 12 cycle turn-on are, respectively, 0.106, 0.089, 0.399 and 0.433. This clearly demonstrates that significantly more population evolves into the K-H ground state for longer pulse turn-ons.

In Fig. 4 we plot the magnitude squared of the wavepackets as in Fig. 3, however now with a darker color scale to emphasize the structure of the ionizing part of the wavepacket. We initially consider the first row, corresponding to a 1 cycle turn-on. Comparing the wavepackets after 10 and 30 cycles, large differences can be seen. For such a short turn-on, much ionization occurs via shake-off, with primarily very low energy electrons being emitted. The dark lobes appearing along the polarization axis are due to shake-off ionization. The clear fringe structures emanating from the localized part of the wavepacket are due to photoionization. Photoionization occurs when either of the two outer peaks of the localized wavepacket (see Fig. 3), which are comprised of the K-H ground and excited states, overlap with the core. This overlap occurs a half a cycle out of phase for each of these two peaks. The resulting rings of ionization are therefore out of phase and in the first approximation the fringe structure can be regarded as an incoherent superposition of ionizing wavepackets originating from these peaks. As the laser pulse turn-on is increased from 4 cycles to 12 cycles, ionization resulting from shake-off becomes less dominant. A large degree of photoionization now occurs during the turn-on, as can be clearly seen for the 12 cycle turn-on. After 30 cycles, photoionization fringes are the most prominent feature in the structure of ionizing part of the wavepacket. Comparing with the other wavepackets at 30 cycles, we see that as the turn-on increases the two interference patterns extending from the edges of the localized part of the wavepacket, which are most evident in the 1-cycle case, gradually disappear. These patterns can either be due to remnants of shock excitation or photoionization from excited K-H states. While it is difficult to deduce specific details of the angular distributions of the ionized electron, we can see that for short laser turn-ons the angular distributions will have maxima occurring along the polarization axis, whereas for long turn-ons the distribution will be more isotropic.

We now turn our attention to the influence of the laser turn-on for the ellipticities ∈ = 0.5 and ∈ = 1 and contrast these with the linear case. We start by considering the structure of the localised wavepackets. As in the linear case, for a one cycle turn-on it is evident that there is a significant amount of population in the excited K-H states as can be seen in the pronounced time dependence in the first row of plots in Figs. 5 and 7. In addition, for the twelve cycle turn-on, the wavepacket displays far less structure and resembles more closely the corresponding K-H ground states. Note that the wavepacket is displaced along the negative y-axis due to the instantaneous electric field along this axis, and the whole wavepacket follows an elliptical (Fig. 5) or a circular (Fig. 7) trajectory about the origin. The magnitude squared of the projections of the wavepackets after 30 cycles onto the K-H ground state, |⟨Ψ(t)|${\mathrm{\Psi }}_0^{\mathit{\text{KH}}}$⟩|² , for the elliptical polarization and 1, 4, 8 and 12 cycle turn-ons are, respectively, 0.046, 0.057, 0.311 and 0.403. For circular polarization, the magnitude squared of the projections are 0.021, 0.007, 0.186 and 0.400, respectively, for each of the turn-on times considered.

As in the linear case, to illustrate the ionization process we have plotted the wavepackets shown in Figs. 5 and 7, respectively, with a darker color scale in Figs. 6 and 8. Again the general behavior as a function of the laser turn-on is similar to that of the linear case; at t = 10 cycles, for short turn-on times a substantial amount of shake-off ionization is observed while at longer times a fringe structure due to photoionization can be clearly seen for all turn-ons. In particular for ∈ = 0.5, and when t = 30 cycles we still observe remnants of shake-off ionization along the x-axis and an angular interference pattern. In addition, also present are circular fringes corresponding to photoionization, emanating from the localized structure. As in the linear case these photoionization fringes appear when the localized part of the wavepacket (most notably the K-H ground state) collides with the core, which again occurs twice per cycle. The difference in the observes fringe structure is due to the fact that the K-H ground state is more delocalized and follows an elliptical trajectory. The blurring of the fringe pattern is a consequence of the fact that the localized part of the wavepacket overlaps with the core for a longer length of time, and therefore ionization occurs for a longer period of time per half cycle.

 

Figure 3. Snapshots of the magnitude squared of the wavepackets taken after 10, 20 and 30 cycles of a laser pulse with turn-on times of 1, 4, 8 and 12 cycles, angular frequency ω = 1, peak electric field E ₀ = 15 and linear polarization. The color scale is chosen to emphasize the structure of the localized part of the wavepackets. Distances (X,Y) are shown in atomic units (a.u.).

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Figure 4. Same as in Fig. 3 except the color scale is chosen to emphasize the ionizing part of the wavepackets.

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For the circular case, ∈ = 1, and when t = 30 cycles, we observe a fringe structure resembling rings due to photoionization originating from the localized structure. However, on closer inspection we see that these rings are actually a spiral. Indeed, for circular polarization there is always a portion of the localized part of the wavepacket which overlaps with the core. Therefore, there is continual ionization from the localized part of the wavepacket that is instantaneously located at the origin. After 30 cycles, we note that as the turn-on time becomes longer there is a reduction in the intensity of ionizing part of the wavepacket. This is most probably due to the remnants of the low energy electrons produced by shake-off ionization during the turn-on.

We now investigate the ionization of the wavepackets as a function of the laser pulse turn-on and laser ellipticity by monitoring the norm of the wavepacket. The norm decreases in time due to the application of absorbing boundaries which remove the ionizing part of the wavepacket, thereby removing reflections. In Fig. 9, we plot the normalization, as a function of time, on a log scale for ellipticities ∈ = 0.0, 0.5, 1.0 and pulse turn-on times of 1, 4, 8 and 12 cycles.

Overall, similar behavior for the 1 and 4 cycle turn-ons and the 8 and 12 cycle turn-ons is observed. In the later case, the curves display two distinctive features. Namely, a large degree of ionization occurs during approximately the first twenty cycles, after which the slope of the curves decreases markedly. This initial ionization occurs mainly during the first few cycles, before the stabilized wavepacket has been established. These photoelectrons have energies of approximately 0.5 and reach the boundary quickly. Hereafter, photoionization is from the stabilized wavepacket and the rate of change of the norm decreases markedly. From the slope of the curves, we can conclude that for circular polarization the wavepacket is more stable against ionization. An indication of the extent of the stability of the atom in the laser field as a function of the laser polarization can be deduced from the energy of the ground state of the corresponding K-H potential. Roughly speaking, the higher the K-H ground state energy, the more stable the atom is against ionization. We have calculated the energies $E_0^{\text{KH}}$ for ∈ = 0, 0.5 and 1, with the respective results -0.133, -0.100 and -0.077. This supports our above conclusion obtained from the calculated norms of the wavefunction.

For short turn-ons, i.e. 1 and 4 cycles, the norm decreases at approximately the same rate. Here the ionization is predominantly due to shake-off, which populates very low energy continuum states, and therefore even after 30 cycles no clear decrease in the rate of change of the norm is seen due to the fact that the low energy electrons require more time to reach the boundary. Our results indicate that there is a larger degree of ionization for circular polarization, as opposed to linear. This is due to a smaller overlap between the field free ground state and the excited K-H states of the circular K-H potential. Indeed the magnitude squared of the projections of the initial ground state onto the respective K-H ground states for ∈ = 0.0, 0.5 and 1.0 are 0.119, 0.051 and 0.022.

Finally, we point out that from a time-dependent point of view, the amount of photoionization which occurs during the flat part of the pulse can be attributed to two factors: the time scales in which the localized structure overlaps with core and the density of the localized part of the wavefunction.

 

Figure 5. Snapshots of the magnitude squared of the wavepackets taken after 10, 20 and 30 cycles of a laser pulse with turn-on times of 1, 4, 8 and 12 cycles, angular frequency ω = 1, peak electric field E ₀ = 15 and ellipticity ∈ = 0.5. The color scale is chosen to emphasize the structure of the localized part of the wavepackets. Distances (X,Y) are shown in atomic units (a.u.).

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Figure 6. Same as in Fig. 5 except the color scale is chosen to emphasize the ionizing part of the wavepackets.

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Figure 7. Snapshots of the magnitude squared of the wavepackets taken after 10, 20 and 30 cycles of a laser pulse with turn-on times of 1, 4, 8 and 12 cycles, angular frequency ω = 1, peak electric field E ₀ = 15 and circular polarization. The color scale is chosen to emphasize the structure of the localized part of the wavepackets. Distances (X,Y) are shown in atomic units (a.u.).

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Figure 8. Same as in Fig. 7 except the color scale is chosen to emphasize the ionizing part of the wavepackets.

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Figure 9. (a) Norm of the wavefunction on a log scale as a function of time with laser pulse turn-on times of 1, 4, 8 and 12 cycles and linear (a), elliptical, ∈ = 0.5, (b) and circular polarization (c).

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As discussed above, for the linear case we have relatively short overlap times, however the wavepacket is relatively well localized. In contrast, for the circular case, we have continual overlap, but the localized wavefunction density is much lower. These qualitative arguments can be used to further explain the observed differences in stabilization for the linear and circular cases and, from a time-dependent point of view, how increasing the intensity of the laser, and hence increasing the excursion amplitude of the electron, reduces the photoionization probability [24].

4. Conclusions

Using a two-dimensional model, we have investigated the influence of the laser ellipticity and the laser turn-on on the evolution of atomic wavepackets in the high intensity, high frequency stabilization regime. In this regime, we see that the K-H frame transformation, in particular the cycle averaged K-H potential, provides a useful framework for understanding the main features of the laser-atom dynamics. This has been illustrated by comparing the wavepackets calculated using the TDSE and the K-H ground states as a function of the laser ellipticity. The differences between the localized part of the wavepacket with the K-H ground state gives a measure of how adiabatic the evolution of the system is.

The relative contributions of photoionization and shake-off ionization to the total ionization probability is determined by the duration of the laser pulse turn-on and turn-off. By examining the probability distribution of the wavepackets for different pulse turn-ons and at different times during the laser pulse, insight into the ionization dynamics has been gained. For short laser pulse turn-ons, the wavefunction cannot evolve adiabatically. In this case, ionization occurs primarily by shake-off during the turn-on. Significant probability is also transferred to excited K-H states. It would appear that the survival probabilities are larger for linear polarization. This is due to the fact that the overlap of the ground state and the K-H bound states is larger for linear polarization than for circular. In contrast, for long laser pulse turn-ons, ionization occurs essentially by photoionization during the turn-on. Our results indicate that it is easier to achieve adiabatic evolution for linear polarization. However, in the adiabatic regime, the circular case is more stable against ionization.

While the stabilization dynamics are complicated and depend strongly on the laser pulse turn-on duration, insight into the photoionization process is obtained from our time-dependent results. In particular, it allows us to observe when ionization occurs, namely, when the localized wavepacket overlaps, or collides, with the core giving rise to the fringe structure seen in the ionizing part of the wavepacket.