{'fraction_non_alphanumeric': 0.04965911960272704, 'fraction_numerical': 0.0126251998989984, 'mean_word_length': 4.452960073428178, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Keldysh theory re-examined: Application of the generalized Bessel functions

A derivation of the ionization rate for the hydrogen-like ion in the strong linearly polarized laser field is presented. This derivation utilizes the famous Keldysh probability amplitude in the length gauge (in the dipole approximation) and without Coulomb effects in the final state of the ionized electron. No further approximations are being made, because the amplitude has been expanded in the double Fourier series in a time domain (with the help of the generalized Bessel functions). Thus, our theory has no other limitations characteristic of the original Keldysh theory. We compare our"exact"theory with the original Keldysh one, studying photoionization energy spectra and total ionization rates. We show breakdown of the original Keldysh theory for higher frequencies. In the barrier-suppresion regime the"exact"Keldysh theory gives results closer to well-known numerical or other analytical results.


I. INTRODUCTION
One of the most important and the most frequently used methods employed to study an ionization in a strong laser field are S -matrix theories [1][2][3]. They are usually called Keldysh, or Keldysh-Faisal-Reiss (KFR) theories [4][5][6][7]. Also the name "strongfield approximation" (SFA) is sometimes preferred [7]. One usually makes the assumption that an ionized electron, due to an action of the laser field, makes a transition from a given initial bound state to a final free state, which can be treated as dominated only by the laser field. In light of the presented considerations the simplest version of the theory neglects completely an effect of a binding potential (short-range or long-range) and uses the Gordon-Volkov wave function [8] as a final state of the ionized electron. In his pioneering work [4] Keldysh identified the so-called adiabaticity parameter  (termed after him) and obtained relatively simple analytical expressions for ionization rates and arbitrary  . The Keldysh parameter  is often used to distinguish between multiphoton ionization (when 1   ) and tunneling ionization (when 1 where  is the laser frequency, F -the amplitude of the laser field, and -the binding energy of the atom (of the nuclear charge Z ) initially in the state described by the well-known   m l n , , quantum numbers (without spin), in the nonrelativistic approach. In Ref. [4] and in the present paper we consider only the hydrogen-like ion (i.e. the initial state is the ground state:   0 , 0 , 1  n ) in the linearly polarized monochromatic laser field. Such a simple theory can be applied only when the laser field is strong enough. An extensive and a very good discussion of applicability conditions of the SFA can be found in Refs. [1,7] (see also Sect. III in Ref. [9]). Based on this, we restrict our interest here only to the domain of the field parameters    [4] which describes ionization rate for 1   has only a character of an estimation and a more accurate theory should take into account Coulomb effects in the final state of the ionized electron. As it was noted by Keldysh and others [10,11], there are two conditions that are necessary to derive final expressions for the ionization rate. These conditions limit the applicability range of the theory: (i) the saddle point approximation to perform the contour integral, and (ii) the small final kinetic momentum approximation. According to Ref. [10], the adiabatic assumption B E   is necessary, to fulfill the condition (i). There have been attempts to derive ionization rate formulas (starting from Eq.(2)) without building on (i) or (ii) [10][11][12]. However, we are aware of no expressions involving the generalized Bessel functions in the length gauge (for linear polarization). In this way our present work fills some gap in the literature. On the other hand, the saddle-point method or semi-analytical methods enabled much progress in the last few decades (see, for example, Refs. [2,3,[13][14][15][16]).
In contrast, the well-known expressions derived in the velocity gauge for  H ion and the   s H 1 atom long ago by Reiss [7] include Bessel functions (the ordinary ones for circular polarization and the generalized ones for linear polarization). These formulas do not rely on the assumptions (i) and (ii). The main aim of the present work is to derive expressions analogous to those in Ref. [7] in the length gauge and investigate their main properties. In the case of circularly polarized laser field such theory has been developed for arbitrary initial states   m l n , , with 1  n and 2  n of the hydrogen-like ion [9,17,18]. S -matrix theories in the above-mentioned two gauges give qualitatively similar results (particularly this concerns the shape of photoelectron energy spectra) for initial states of even parity [19], like the ground states of    (16). In Sec. IV we discuss the results of our numerical calculations (including photoelectron energy spectra and total ionization rates) and we make a comparison of these results with other results, mainly with the original Keldysh theory. In the Appendix we show in details how the Fourier coefficients and the generalized Bessel functions have been calculated in this paper. In the present work we consistently use atomic units (a.u.): , substituting explicitly -1 for the electronic charge. We keep any nuclear charge Z in all the equations given below, but finally, in our numerical calculations, we put Z  1 for the hydrogen atom.

II. KELDYSH THEORY
In the Keldysh theory [4] one starts from the following matrix element, which is the approximate probability amplitude of strong-field ionization in the length gauge: where the initial, ground state of a hydrogen-like ion is described by the well-known wave function in position space. In the length gauge the laser-atom interaction Hamiltonian is being the electric field vector, and the Gordon-Volkov state [8] in this gauge is where p  is kinetic (asymptotic) momentum of the ionized outgoing electron and its canonical momentum is given by (5) In the dipole approximation magnetic-field component of the laser is zero and the  is the polarization vector and 0 some arbitrary initial phase of the laser field). Starting from Eq. (1), Keldysh [Eq. (20) in Ref. [4]] obtained the following ionization rate formula (for 1 In the above expression n -photon contributions have been summed up owing to the procedure valid when 0   (going over from summation over n to integration). Without this procedure, the ionization rate (6) has the general form where n W are partial ionization rates corresponding to absorption of exactly n photons above threshold. n W are given by Eqs. (16), (18), and (19) in Ref. [4]. To obtain Eq. (6) one also has to make a Taylor expansion around 0   , leaving only the lowestorder term in the pre-exponential factor and the two lowest-order terms in the exponent of Eq. (16) in Ref. [4]. Now then, one can also write down "an intermediate" expression, which does not involve "a multiphoton" summation, but contains all the complicated dependence on the Keldysh parameter  : results. We have also verified numerically that for const F  and 0   Eq. (8) goes to finite value (like Eq. (6)), as should be. To obtain the most accurate expression (7), one has to include the missing factor  / 2 as well. Let us denote these three ionization rates as "Keldysh 1" (Eq. (7)), the most accurate), "Keldysh 2" (Eq. (8)), and "Keldysh 3" (Eq. (6), the least accurate), respectively.

III. KELDYSH THEORY WITH THE GENERALIZED BESSEL FUNCTIONS
Using Eqs. (3)-(5) one can show [9] that the ionization probability amplitude (2) may be presented as follows where the momentum-space wave function of the initial ground state of the   s H 1 atom is equal to 7 To calculate the ionization probability amplitude (9) we assume that the laser field propagates along the x axis and the polarization vector   is parallel to the z axis. Following Reiss [1,7] (and our earlier works, for example [9]) we introduce the z parameter such that ). The product of the first two factors in the integrand of Eq.
and may be expanded in the above Fourier series. It appears that the coefficients can be calculated analytically with the help of the residue theorem (see the Appendix). The exponential factor in Eq. (9) may be also expanded, in the standard way, using the Fourier-Bessel expansion: for linearly polarized field (we use the same convention with respect to   b a J n , as in Ref. [7]).
When we apply expansions (11) and (12), the amplitude (9) turns out to be proportional to The integral upon time leads to the well-known distribution (a particular model of Dirac  function), which enables applying standard procedure. The differential ionization rate   p w  , which is the transition probability per unit time and unit volume in the canonical momentum (  p ) space, can be found from To obtain the total ionization probability per unit time W , one has to integrate the differential ionization rate over all the possible final momenta of the outgoing electron.

The final result is
where . The minimal number of photons absorbed is , and the kinetic energy of the ionized outgoing electron is The symbol   .. denotes integer part of the (positive) number inside. In deriving equation (16) we have used the following property of the generalized Bessel functions: , and the relation: (see the Appendix for some other details).
Let us note that analogous expression in the velocity gauge is much simpler [7] and contains only one summation. In our notation it is given below ( a and b as in Eq. (16)): Equations (16) and (18) show total ionization rates as a sum over partial N -photon ionization rates, where the kinetic energy of outgoing electron is given by Eq. (17).
The same concerns Eq. (7), but there the index n denotes the number of photons above threshold. Thus . This is obvious, if we look at the argument of the Dirac  function from Eq. (14) in Ref. [4].

IV. NUMERICAL RESULTS AND CONCLUSION
The  Fig. 1 to Fig. 3. Comparing upper two curves each time, one can see that agreement between "Keldysh 1" and "Exact Keldysh" is quite satisfactory, particularly for lower photoelectron energies. This is understandable, because the small final kinetic momentum approximation is utilized in "Keldysh 1". According to this approximation only electrons with sufficiently small final kinetic energy E should mostly contribute to the total ionization rate ( .
). This is hardly satisfied only in Fig. 3 This line is shown in our log-log plots as a slanted dashed line. Figures 8 and 9 resemble Fig. 6 from Ref. [22]. (However, let us notice that numerical ionization rates in the latter figure indicate that the proper ionization rate should be a concave function in log-log plot, but not a ruled line.) It seems that a significant theoretical progress was attained about ten years later [14]. In Figs. 8 and 9 for comparison we show also the ionization rate of Popruzhenko et al. (see Eqs. (5)- (7) 11 from Ref. [14]), which is valid for any Keldysh parameter and takes into account Coulomb effects in the final state of the ionized electron. Curiously enough, our "Exact Keldysh" curve typically lies about one order of magnitude below the curve denoted as "Ref. [14]" and the distance between these two curves decreases with increasing F . In Figs. 8 and 9 we show also for comparison the above-mentioned three " Keldysh

ACKNOWLEDGMENTS
The present paper has been supported by the University of Łódź.

APPENDIX
is always a real number. Let us introduce new parameters: where  C denotes a circle of a unit radius, with the center in the origin, in the complex s plane. The "plus" means that we circulate the circle in a counter-clockwise direction.
To utilize the residue theorem, we have to solve the equation: