CoulombCorrected Strong Field Approximation without Singularities and Branch Points^{1}
Abstract
The domain of validity of the CoulombCorrected Strong Field Approximation (CCSFA) is going to be analyzed in relation to the semiclassical dynamics of electrons during ionization of hydrogenlike targets. Our analysis is limited to ionization driven by Tisapphire laser pulses with intensities up to roughly Wcm. For such parameters, the effects related to radiation pressure are small and the laser field can be described in the dipole approximation. By applying the Magnus expansion for the exact retarded electron propagator we obtain an effective action which is free of Coulomb singularities and branch points when the complextime trajectories are used. Furthermore, we show that the classical action is exactly recovered as the asymptotic limit of its effective counterpart. The applicability of such limit is also discussed.
I Introduction
The great English mathematician and philosopher Bertrand Russell (18721970) has once written that ’all exact science is dominated by the idea of approximation.’ One could continue his line of thought by saying that the most beautiful aspects of all exact science are the approximations. It has to be remembered, however, that any approximation has its domain of validity, which is inherently related to it and equally important in its applications. The strongfield physics (SFP) is, in fact, one of the best illustrations of the above statements. Recent developments in this field have led to the creation of new branch of modern science, the attophysics BK (); IK (). One of the most prominent tools in the SFP, the StrongField Approximation (SFA) in photoionization, was originally introduced by Keldysh Keldysh () in the length gauge and further developed by Faisal Faisal () and Reiss Reiss () for other forms of the Schrödinger equation (i.e., in the KramersHenneberger frame kh1 (); kh2 () or in the velocity gauge, respectively). The common feature of those approaches is the approximation of the exact scattering state of the photoelectron by the Volkov solution Volkov (); EJK1998 (). Subsequently, such approximation has been extended to treat the nonrelativistic laserassisted scattering processes Fedorov1 () and its relativistic counterpart Fedorov2 (), where the relativistic form of the Volkov solution EKK2009 (); DiPiazza2012 (); DiPiazza2018 () is fully exploited.
Since the SFA does not offer a proper explanation of various experimental results, it requires further developments. For instance, an attempt to incorporate the interaction of photoelectrons with the parent ion by means of the CoulombVolkov state was undertaken in Tsoar (); Kornev (); Duchateau (). This procedure was successfully applied to the analysis of ionization driven by elliptically polarized laser fields Ferrante (). (Note that similar investigations were recently presented for bicircular laser fields Milo2018 ().) Further generalizations, which fully account for the lowfrequency approximation, were considered in Refs. K1 () and K2 () for the scattering and ionization processes, respectively. This approach, called the CoulombVolkov StrongField Approximation (CVSFA), was recently generalized in PU2018 () by incorporating the density functional theory for the initial bound state of manyelectron atoms, or in ZZ2018 () by applying the parabolic quasiSturmianFloquet approach. Also, the scattering states in the highfrequency approximation were explored Gav1 (); Gav2 (). An alternative approach to the CVSFA, which we call the BornSeries StrongField Approximation (BSSFA), is to apply the Born expansion to the final electron scattering state in both the binding potential and the laser field. Here, the first two terms of the Born series were successfully used in studies of the rescattering process in ionization Milo2018 (); Resc1 ().
Another path of theoretical explorations is related to the eikonal approximation eik1 (); eik2 (); eik3 () and its generalizations eik4 (); KK2015 (); VKK2016 (). In our further discussion, we shall refer to those approaches as the CoulombCorrected StrongField Approximation (CCSFA) ccsfa1 (); ccsfa2 (); ccsfa3 (); ccsfa4 (). In connection to these theoretical investigations is the purely classical analysis of laserassisted atomic or molecular processes classdyn () such as ionization classion1 (); classion2 (), plasma dynamics in atomic clusters classion2a (), and recollisions classresc (). In fact, those works were stimulated by the successful application of the classical dynamics to ionization of Rydberg states by microwave fields classrf1 (); classrf2 ().
The two groups of approaches mentioned above (the CVSFA and BSSFA vs. the CCSFA and the classical analysis) are contradictory when the behavior of the electron wavepacket close to the Coulomb singularity is analyzed. For the first group of approximations (and actually for the full quantum treatment; see, e.g., Nurhuda (); num1 ()) this singularity does not present any conceptual or numerical problem. In contrast, the classical analysis shows severe numerical problems when the electron trajectories approach the singularity (as discussed in classion2 ()). Furthermore, in the CCSFA, when the method of complextime electron trajectories is applied, the Coulomb singularity leads to singular branch points. Such branch points do not exist in the BSSFA or in the exact numerical solution of the Schrödinger equation. The aim of our investigations is to determine the origin of these problems and to derive the effective CCSFA (related to the generalized eikonal approximation and eikonal perturbation theory eik4 (); KK2015 (); VKK2016 ()) which is free of Coulomb singularities and branch points.
Note that the CCSFA and the classical dynamics follow from the quantum theory in the limit when the Planck constant goes to 0 or the time evolution is very short. However, it must be taken into account that and time have physical dimensions. Thus, in order to provide a sensible physical meaning of this limit, one has to construct a dimensionless parameter out of the Plank constant, the time of evolution, and other relevant physical quantities, such that the CCSFA or the classical dynamics are recovered in such limit. This is going to be done below for the ionization of a hydrogenlike ion driven by laser fields of moderate intensity. Such condition guarantees that the dipole approximation is valid throughout our analysis. For instance, if the Tisapphire laser field (wavelength 800 nm and frequency such that ) is considered, the laser pulse should not exceed intensities of the order of . For larger intensities, relativistic effects related to the radiation pressure press0 (); KKpress () (see, also press1 (); press2 (); press3 (); press4 (); press5 (); press6 ()) and relativistic mass corrections CKKvortex () become already visible.
Our analysis below is based on the Magnus expansion, as opposed to the more general eikonal perturbation theory developed in eik4 (); KK2015 (); VKK2016 (). The Magnus expansion magnus1 (); magnus2 (); magnus2a (); magnus3 () allows one to construct an approximate exponential representation of the propagator of the system. Every order of such expansion corresponds to a partial resummation of infinite terms of an ordinary Born series. Next, we apply this approximation up to the linear term with respect to the binding potential and show under which conditions the classical dynamics is restored. This determines the domain of validity of the ordinary CCSFA ccsfa1 (); ccsfa2 (); ccsfa3 (); ccsfa4 () and it allows us to remove the artificial Coulomb singularities and branch points. We show that the latter arise from the application of the relevant asymptotic expansion beyond its domain of validity. We demonstrate that the effective CCSFA, in the appropriate limits, leads to the exact Born expansion or to the classical (or complextime) dynamics. Moreover, it appears that the effective action is complex and accounts for the electron wave packet spreading in the course of its time evolution. Due to this fact the electron trajectory is also complex even for real times and real initial/boundary conditions, but such that in the classical limit its imaginary part vanishes. Note that the method presented here and in eik4 (); KK2015 (); VKK2016 () allows to incorporate further terms, which are nonlinear with respect to the static binding potential, into the effective action.
Ii General theory and Magnus expansion
We start with general statements about the nonrelativistic quantummechanical evolution of an electron in arbitrary electromagnetic fields. Such evolution, from an initial time up to a final time , is determined by the retarded propagator . If the Hamiltonian of the system, in terms of the ‘primed’ coordinates, is , the propagator satisfies the differential equation (see, e.g., Ref. KK2015 ())
(1) 
As many studies of ionization relate to the momentum or energy distribution of photoelectrons, it is most convenient to analyze the propagator in momentum space. For this reason, we introduce its Fourier transform, , calculated over the nonprimed coordinate,
(2) 
The differential equation for the transformed propagator, , can be easily determined by inserting Eq. (2) into (1). In doing so, we find out that
(3) 
and, as it can be checked, fulfills the retardation and initial conditions,
(4) 
respectively.
As mentioned in magnus2 (), the Magnus expansion finds applications in many subfields of physics including atomic, molecular, and particle physics, quantum electrodynamics, etc. Here, we shall use it to solve the homogeneous equation,
(5) 
while imposing the conditions (4). This is equivalent to solving the inhomogeneous equation (3). Thus, we look for the solution of (5) in the exponential form magnus1 ()
(6) 
where is an a priori unknown function. Such function can be expanded in the power series
(7) 
where is a small and real parameter (see below). The challenge, however, is to determine each term in the sum (7). Below, we illustrate this for the case when can also be represented as a Born series.
First, we separate the total Hamiltonian of the system into two parts, , where describes a relatively weak interaction compared to the main contributing Hamiltonian . This is stressed by a small parameter . Next, we also assume that the transformed propagator can be represented as a Born series with respect to . Specifically, up to the first order in the perturbation, we write it down as
(8) 
where in general , is the th term of the Born expansion. On the other hand, if the Magnus expansion is to be applied, the Fourier transform of the propagator takes the form [see, Eqs. (6) and (7)]
(9) 
Therefore, up to the first order in , we obtain
(10) 
By comparing Eqs. (8) and (10), we conclude that and . Hence, by performing the Born and Magnus expansions of the transformed propagator simultaneously, the terms can be determined. However, it has to be noted that the complexity of the procedure increases with , as for higher orders in a larger number of factors, arising from the series expansion in (9), contribute to the particular term.
Up to now our analysis has been very general. In the following, the Hamiltonian will describe an electron interacting with an oscillating laser field, which in the dipole approximation is defined by the vector potential , and with a general timedependent binding potential, , in the KramersHenneberger frame.
Iii Magnus and Born expansions in the KramersHenneberger frame
Our aim is to derive an approximate expression for the propagator in the KramersHenneberger (KH) frame. Such frame is particularly suitable to perform our calculations, as the interaction with the scalar and vector potentials is reduced to a single term in the Hamiltonian.
The retarded Schrödinger propagator in the KramersHenneberger (or accelerating) frame kh1 (); kh2 (), here denoted as , satisfies the differential equation [see, Eq. (1)]
(11) 
where and is the free particle Hamiltonian. is the socalled displacement vector, and it relates to the oscillating electric field such that . This, in turn, suggests that determines the classical trajectory of a free electron in the light field, and corresponds to its acceleration kh2 (). In this paper, it is assumed that the laser pulse acts over a finite period of time (from up to ), which means that for and . We also choose the vector potential such that , and assume that the displacement vanishes at times .
We proceed with expanding the propagator in a Born series. This is done by using repeatedly the LippmannSchwinger equation,
(12) 
where is the freeparticle propagator. In the firstorder Born approximation, we obtain that
(13) 
Note that such general iterative procedure leads to a series expansion of the total propagator in powers of the binding potential. This is the essence of the BSSFA.
Since we are interested in deriving an expression for the propagator in momentum space, we perform the Fourier transform of in (13) with respect to . In addition, since our main focus is the photoionization of atoms/ions, the binding potential is static. This means that and we can use
(14) 
As a result,
(15) 
where one can derive that . Moreover, in Eq. (15), we have introduced a classical trajectory of an electron in the laser field, . Note that the second term in the square bracket in Eq. (15) can be related to an effective potential experienced by the electron,
(16) 
Having this in mind, we rewrite (15) in a more compact form,
(17) 
which represents the Magnus expansion of the propagator in momentum space (10) with
(18) 
It allows us to formally write down that, up to the linear term in the potential,
(19) 
This expression is fundamental for our further analysis. As we will demonstrate shortly, the fact that the propagator depends on an effective potential rather than on the classical one, has very important consequences in the photoionization dynamics.
iii.1 Effective CCSFA
In order to define our effective CCSFA, let us modify the classical trajectory of the electron in the laser field by a correction that is linear in . Namely, we introduce an effective trajectory which satisfies the same boundary conditions as the classical trajecotry; namely, and . By making a transformation from the KH to the laboratory frame, we obtain that, up to linear terms in , the propagator (19) is given by
(20)  
(21) 
where the subscript stands for the velocity gauge and where , for fixed and , is a function of and . The functional in (21) is given by
(22) 
which can be recognized as the classical action of an electron interacting with the laser field and the scalar effective potential . This, in turn, implies that the dynamics of the system is governed by the effective (and complex) Newton equation
(23) 
, being a complex function, differs from the classical potential as it contains important quantum corrections (see, Sec. IV). Furthermore, the effective potential (16) depends explicitly on the initial time , meaning that the Newton equation (23) contains information about past events. In other words, contrary to its classical counterpart, the effective quantummechanical evolution is nonMarkovian.
Now, we present for the sphericallysymmetric Coulomb potential . Here, is the finestructure constant, is the atomic number, and is the speed of light. From the definition (16), it follows that (see, Refs. KK2015 (); VKK2016 ())
(24) 
where . The mathematical and physical consequences of this expression are analyzed below.
Iv Classical limit of the effective CCSFA and its range of validity
The propagator in the effective CCSFA [Eq. (19)] depends on the effective potential (24), which differs from the pure Coulomb one. As a result, the effective interaction already accounts for the spreading of the electron wave packet and quantum diffusion effects KK2015 (); VKK2016 (). Such effects, while negligible at infinitesimallyshort time intervals, become relevant at the characteristic timescale at which photoionization takes place (see below). In addition, is an entire function of , i.e., it does not have singularities or branch points when the electron trajectories come back to the parent ion. This is in contrast to the pure Coulomb potential, as it is singular at the origin of coordinates. To see this better, consider the powerseries expansion of the error function (see, Ref. olver2010 ()) in Eq. (24),
(25) 
Thus, the effective potential can be expressed as an absolutely convergent power series in the whole space,
(26) 
and, in particular, , provided that . On the other hand, for large arguments, the effective potential can be analyzed by making use of the asymptotic expansion of the complementary error function, (see, Ref. olver2010 ()). In doing so, we find that for large (or, equivalently, for large ), , i.e., the Coulomb interaction is recovered when the argument of the effective potential is large enough.
In the upper panel of Fig. 1 we show the real (blue line) and imaginary (green line) parts of together with the curve (red line). While the first function determines the behavior of the effective potential [Eq. (24)], the second one defines the classical Coulomb interaction. It can be seen that the imaginary part of goes fast to zero for large values of . In contrast, behaves in a very similar way as , when . This is in agreement with our previous analysis which established that, in the limit of large arguments, the effective potential and the classical one coincide. On the other hand, for small values of the situation changes drastically. First, is nonzero and remains finite. Second, the general behavior of and differ considerably: while the former remains bounded and reaches values near to one, the latter increases rapidly and presents a singularity at . This agrees with our earlier conclusions that the effective potential is an entire function of and does not contain singularities or branch points. The dramatic differences in the behavior of the two potentials near the origin of coordinates () can be attributed to the inherent quantummechanical nature of photoionization; diffusion and spreading of the wave packet play an important role in the process and they are neglected in a classical treatment or CCSFA.
As we have shown, only the asymptotic form of (which is valid provided that is large) has singularities and branch points at small values of . A very similar situation is met in the theory of special functions. As an example, consider the Bessel function , which is an entire function of (cf., Fig. 2). Its power series expansion,
(27) 
is absolutely convergent and wellbehaved for all complex arguments . However, its asymptotic behavior, for , might indicate that exhibits the singularity and the branch point at . This is not the case because the asymptotic form is not applicable there. We illustrate this in Fig. 2, where (blue line) and its asymptotic expansion (red line) are presented.
Now, an important question arises: What are the arguments of for which the effective potential can be well approximated by its asymptotic (classical) form? This can happen for real such that . In the lower panel of Fig. 1 we present the real and imaginary parts of this function (blue and green lines, respectively). Hence, one can estimate that the condition is quite well fulfilled for , i.e., the classical form of the effective potential can be applied provided that
(28) 
This condition is satisfied if either:

For fixed and , the particle mass is very large. In fact, such statement is the essence of the BornOppenheimer approximation, i.e., it is assumed that the dynamics of massive particles can be approximated by the classical treatment.

For fixed and , the trajectories do not come back close to the origin of coordinates. This limit is justified, for instance, in the ionization of Rydberg states (RS) by microwave fields. If the distance between the electron and the nucleus is approximately equal to the Bohr radius of the RS (with principal quantum number ), and corresponds to its orbital period, the classical limit is applicable provided that [see, Eq. (28)]. Thus, for the electron dynamics is determined by the classical evolution classrf1 (); classrf2 ().

In scattering processes the condition (28) is equivalent to , where is the electron momentum and is the impact parameter. This is, in fact, the condition of applicability of the traditionallyused eikonal approximation. Hence, such approximation is valid provided that and/or are sufficiently large and it excludes the case of backward scattering.
It follows from here that the asymptotic expansion of the effective potential can be applied, for instance, to ionization driven by elliptically polarized laser fields, as the electron trajectories do not return to the parent ion. The situation is different when a linearly polarized pulse is considered. In such case, the electron can come back to the origin of coordinates at times close to half of the laser cycle, i.e., , where is the carrier frequency of the laser field. This means that the asymptotic limit (i.e., the classical theory or the CCSFA) is only justified provided that , in atomic units. For a typical frequency of eV a.u., we have or, equivalently, a.u. This means that the electron trajectory should be far away from the parent ion. The range of applicability of the classical description of ionization is improved for Xray laser pulses. For instance, for eV the electron can approach the origin of coordinates up to a distance of the order of , where is the Bohr radius.
V Conclusions
Both CCSFA and the classical approach are asymptotic limits of the quantum theory based on the Schrödinger equation. Therefore, they can be used only within their domains of validity. We suggest that, beyond those domains, either the methods based on the Born expansion or the effective CCSFA with the regular complex effective action should be used. Our analysis has been limited to the case when the effective action is a linear functional of the binding potential. However, by considering higher terms in the Born expansion or the eikonal perturbation theory, one can derive further modifications of the effective CCSFA which are consistent with the quantum theory.
Acknowledgments
This work is supported by the National Science Centre (Poland) under Grant No. 2014/15/B/ST2/02203.
References
Footnotes
 This work was presented at the 27th Annual International Laser Physics Workshop (Nottingham, July 1620, 2018).
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