1. Introduction

The term “stabilization” has been employed with a variety of meanings by different authors. Stabilization is here defined to be that property of atoms in strong laser fields in which continued increases in field intensity will eventually lead to a decrease in the photoionization transition rate. This is consistent with the definition employed by Gavrila [1], who identifies stabilization as a minimum in lifetime rather than a maximum in rate, the inverse of lifetime.

The concept of decreased efficacy of a field in causing transitions as field strength increases is far from new. For example, it was pointed out in the literature many years ago [2,3] that some quantum systems will show a decline in transition rate as field strength increases. There are several qualitative reasons to expect this behavior. One of these reasons is explored in the final section below.

Analytical approaches to the stabilization problem are employed here, since analytical methods provide a unified way to explore the problem, as opposed to general patterns that may emerge from numerical approaches. We start from first principles in stating two related, exact statements of transition amplitudes that apply with complete generality to all perturbative or nonperturbative approaches. This allows us to examine, for the first time, the frequency dependence of approximations that follow from the exact transition amplitudes. The opinion has become widespread in the strong-field community that stabilization in the sense we have defined it is exclusively a high-frequency phenomenon. Our contention is that this notion has arisen only because the strong fields necessary to achieve stabilization exceed computational capacities for exploring low frequency interactions with atoms.

To establish the role of field frequency in nonperturbative analytical approximations, we examine the so-called KFR (Keldysh [4], Faisal [5], Reiss [6]) approximation. It is shown that the “K” method is based on a low-frequency approximation, the “F” method is founded on a high-frequency approximation, and the “R” approach requires only strong fields, with no limitation on the frequency. Hereafter, the “R” method will be referred to as the SFA, or Strong-Field Approximation. The method of Gavrila [1] depends upon high frequencies, and is sometimes called the HFA (High-Frequency Approximation).

As employed here, high frequency means ω>>E_B , where ω is the frequency of the laser field and E_B is the binding energy of the atom. Low frequency refers to ω<<E_B . Atomic units are used.

2. Fundamental transition amplitudes

The notation is adopted that quantum states with no interaction with the laser field have wave functions designated by Φ, and interacting states are given by Ψ. That is, these states satisfy the Schrödinger equations

Boundary conditions in laser experiments are such that the atom is initially and finally in a field-free environment. It is thus useful to define “in-states” [superscript (+)] and “out-states” [superscript (-)] as

Subscripts i and f will identify initial and final states. The overlap of the fully evolving state Ψ_i onto some particular final non-interacting state Φ_f gives the quantum probability amplitude that that final state will occur. One can write, using Eq. (3) for the second element below,

from which it follows that

upon using the equations of motion, Eqs. (1) and (2). In the last element of Eq. (5), the superscript (+) has been dropped as superfluous.

A transition amplitude fully equivalent to the above can be found from the time-reversed point of view that employs the overlap expression

and yields the final transition amplitude

where the superscript (-) has been dropped.

3. Strong-field analytical approximations

3.1 Keldysh approximation

The Keldysh approximation [4] is directly derivable from Eq. (7). Three steps are necessary. First, Keldysh uses the length gauge, so that

Second, on the grounds that the final continuum state of the photoelectron will have its motion dominated by the laser field, Coulomb effects are neglected and

where the Volkov solution for a free particle in the presence of the field is well known. The Volkov solution employed must be in the length gauge, which is a more complicated object than the velocity-gauge Volkov solution. This fact presents enough difficulty that Keldysh introduced his third step, which is to assume large photon orders from the outset. His approximation is therefore restricted to the tunneling case.

The constraint to tunneling for the Keldysh theory means both that it is confined to low frequencies and also that the stabilization domain cannot be explored with the Keldysh theory.

3.2 Faisal approximation

The Faisal approximation [5] has been an enigma, since efforts to associate the fundamental assumptions with a strong-field low-frequency hypothesis have not borne fruit. Faisal’s work begins with the direct-time transition amplitude of Eq. (5). He assumes, as is true of all the strong-field analytical approximations, that the influence of the atomic binding potential can be neglected for the ionized electron, or

where the free-particle final state no longer has the Coulomb influences of the Φ of Eq. (1).

The use of the direct-time amplitude requires a knowledge of the initial bound state in the presence of the strong laser field, thus eschewing the advantage of the time-reversed amplitude, Eq. (7), that requires no such knowledge. The Kramers-Henneberger transformation [7]U^KH (actually the inverse transformation) is applied to the initial state, giving

where α⃗(t) is the classical displacement of the atomic electron from its center of oscillation in the field of the laser, treated here in the dipole approximation. The next step is to neglect this displacement,

The approximation in Eq. (12) is a high-frequency approximation, in that it is justified only if the oscillation α occurs at too high a frequency to permit the atom to follow the motion [1]. The final conclusion is that the Faisal transition amplitude,

is a strong-field, high-frequency approximation.

3.3 Strong-field approximation (SFA)

The SFA is based on the strong-field assumption that, in the time-reversed transition amplitude Eq. (7), one need only make the replacement

This requires only that the laser field be strong, and makes no reference to field frequency.

It is known [6] that the SFA transition amplitude yields the tunneling result in the low frequency limit, and that detailed behavior of strong-field, low-frequency photoionization is accurately predicted [8]. It is also easy to show that the SFA amplitude can be transformed to the Faisal amplitude (13) using the fact that the dipole-approximation Volkov solution can be written as a Kramers-Henneberger transformation of the free-particle wave function. The SFA thus possesses both low- and high-frequency behavior. This is further verified by the close correspondence between rates calculated with the HFA [9] and with the SFA [10].

4. Strong-field stabilization

4.1 Transition rates

Figure 1 shows monochromatic transition rates calculated with the SFA both relativistically and nonrelativistically for ionization of hydrogen by a circularly polarized laser field. Figure 1 gives two high frequency and two low frequency examples. The sharp drop in rate is clearly in evidence for all frequencies beyond a certain intensity that we shall call the stabilization intensity I_stab . Matters are less clear for an actual experiment, since ionization will saturate before I_stab is reached. Laboratory observation of stabilization will be considered elsewhere.

Relativistic effects are seen to enhance the stabilization effect. Earlier calculations [11] that showed some ambiguity in the relativistic enhancement effect have been refined. It is now clear that relativistic rates are less than or equal to nonrelativistic rates at all intensities for circular polarization.

Relativistic SFA results follow from the Dirac version [12] of Eq. (7),

where the bar over Ψ indicates the Dirac adjoint Ψ^†γ⁰ the γ^µ are the Dirac gamma matrices; µ=0,1,2,3; A ^µ is the four-vector potential of the laser field; and a time-favoring real metric is used. Under relativistic conditions, the neglect of Coulomb effects on the ionized electron is a well-justified approximation. One then uses the Dirac Volkov solution for the final-state wave function, uses known solutions (or superpositions thereof) to the hydrogen-atom Dirac equation for the initial-state wave function, and obtains thereby a completely relativistic Dirac transition amplitude. Details of this procedure are given in Ref. [12]. Additional examples as well as an extension to linear polarization are to be found in Refs. [11,13]. An important feature true for both relativistic and nonrelativistic calculations is that I_stab is approximately the same for both circular and linear polarizations.

 

Fig. 1. Atomic hydrogen monochromatic ionization rates in a circularly polarized laser beam.

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4.2 Comparison with alternative predictions of Istab

It has only been very recently that predictions for I_stab have been calculated (by a method other than the SFA) for frequencies encompassing both the low and high frequency domains. Popov et al. [14] have devised a one-dimensional numerical method for this task, and apply it to a hydrogenic state with a binding energy of 0.75 eV, or 0.0276 a.u. Theoretical studies of stabilization show that I_stab ∝ω ³ for high frequencies [15], and I_stab ∝$E_B^3$ for low frequencies [16]. That is, the important feature is the energy. Hence, three-dimensional SFA predictions were calculated for the same binding energy as employed in Ref. [14]. Comparison of the two sets of results is shown in Fig. 2.

The conditions for applicability of the SFA are U_p >>E_B , where U_p is the ponderomotive energy. The values of I_stab predicted by the SFA in the neighborhood of ω=E_B do not satisfy that constraint. Only those stabilization intensities are plotted in Fig. 2 that satisfy SFA validity conditions, and hence a gap occurs. Plainly, the SFA and the results of Ref. [14] are in excellent agreement for both low and high frequencies.

4.3 Physical explanation for stabilization

There are numerous ways to understand stabilization, but one of the simplest follows from Eq. (6). This equation gives the probability amplitude for ionization as the quantum overlap of the undisturbed initial state of the atom and a possible final continuum state for the photoelectron as projected back in time. When Coulomb effects on the continuum electron are negligible, as they are for fields strong enough to induce stabilization, the predominant part of the probability amplitude for $\underset{t\to -\infty }{\lim }$ Ψf is approximately just ${\mathrm{\Psi }}_f^{\mathit{\text{Volk}}}$ (t) at any t. Consider the nonrelativistic circular polarization case. The initial atom can be viewed as a spherical Bohr atom, and the final photoelectron will be in a circular orbit around the atom with essentially the classical energy and angular momentum of a free electron in the laser field. The cylindrical symmetry of the final orbit is imposed by the symmetry of the problem, and the number of photons absorbed to provide the initially bound electron with its final energy in the laser field is also appropriate to provide the final angular momentum. This is illustrated by the sketch in Fig. 3(a). As the field intensity increases, the classical radius of the orbit grows much larger than the size of the atom, the wave function overlap decreases, and stabilization sets in.

 

Fig. 2. Value of the stabilization intensity as a function of field frequency for ionization from a hydrogenic atom with binding energy 0.75 eV. The Popov et al. data are from Ref. [14], and proportions in this figure mimic those of Fig. 10 of this reference. See text for explanation of the gap in SFA results.

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4.4 Relativistic effects on stabilization

With circular polarization, the momentum of the absorbed photons displaces the plane of the circulating photoelectron in the forward direction, so that the circular orbit now lies on the surface of a circular cone with the atom at the vertex. The overlap of final and initial states is somewhat reduced. This effect is illustrated in Fig. 3(b), with the resulting diminution of overlap of Φ _i and Ψ _f leading to the reduced relativistic rate seen in Fig. 1.

For linear polarization, the relativistic path of the free electron is just the well-known “figure-8” as seen in the laboratory frame. The effect is to cause the ionized electron to “walk” away from the atom, thus reducing the wave function overlap expressed in Eq. (6).

The above conclusions differ from those obtained by Kylstra et al. [17], who predict that magnetic field effects will suppress stabilization. This contrast poses an extremely interesting contradiction that needs to be resolved. In support of the conclusions given here, we can make several remarks. First, Kylstra et al. use a two-dimensional theory to calculate the explicitly three-dimensional phenomena that are a necessary consequence of coupled electric and magnetic fields. Second, it is not clear that it is consistent to introduce retardation effects into the nonrelativistic Schrödinger equation [18] as done by Kylstra. The procedures of Refs. [11–13] are three-dimensional, fully relativistic, and subject only to the mild (for intensities like I_stab ) constraint of Eq. (14). Finally, we remark that the qualitative reasoning employed above and illustrated in Fig. 3 follows from the exact expression in Eq. (6), and is independent of any specific calculational method. Basically, Eq. (6) tells us that because the wave function overlap between the initial atom and a final relativistic free electron is reduced from the nonrelativistic situation, then as intensity increases the atom is denied an increasing portion of the probability of making a transition to an ionized state, and is thus more strongly stabilized relativistically.

 

Fig. 3. Ionization by circularly polarized light. (a) Nonrelativistically, the electron circulates around the atom in a circular orbit in a plane perpendicular to the propagation direction, with its classical energy and radius of motion. (b) Relativistically, the plane of the orbit is displaced forward by the momentum of the number of photons that must be absorbed to supply the ponderomotive energy of the electron.

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