1 Introduction

Recent experimental measurements [1, 2] of momentum distributions of recoil ions produced by double ionization of atoms in intense infrared laser pulses provide complementary information to the data on the total ionization yields in these systems, measured in the past (e.g. [3, 4]). Theoretical analyses of these experiments face considerable difficulties that arise from (i) the highly non-perturbative nature of the laser coupling, (ii) the quantum many-body nature of the atomic system, plus (iii) the non-separable Coulomb correlation between the electrons, ∑_i≠j1/r_ij .

One ab-initio approach to these problems for few-electron systems is to attempt to obtain numerical solutions of the corresponding Schrödinger equation directly. It involves solving 3×N, N=2, 3…, dimensional partial differential equations over realistically large space-time grids. Much progress (especially at higher frequencies) towards numerical solutions of the six-dimensional Schrödinger problem for the double ionization of He have recently been made using state-of-the-art computations (e.g. [5–7] and contributions in this volume).

An alternative ab-initio approach is a systematic approximation method. The recently introduced ‘intense-field many-body S-matrix theory’ (IMST) is designed to this end [8–10]. First, we outline the IMST and clarify its special structural advantages for applications to problems of present interest. Second, we discuss the results of applications (e.g. [8–16]) with special emphasis on the two-electron momentum distributions, analyzed [16] for double ionization of He in a Ti:sapphire laser pulse.

2 TheIMST

In the so-called intense-field many-body S-matrix theory (IMST), the S-matrix (or the probability amplitude for any process) is rearranged in such a way that the dominant features of the process can appear in the leading terms of the S-matrix series. It is interesting to note that, besides providing an ab-initio systematic approximation method for calculating the amplitude of interest, IMST can help in identifying possible mechanisms involved in the process of interest. This can be done by analyzing the Feynman diagrams that are generated by the leading terms of the theory. It is also interesting to note that, in addition to the above use of the theory, IMST can also help in constructing simple models by suggesting appropriate places for introduction of physical hypotheses. The latter possibility is particularly interesting for gaining insights into complex situations that may be otherwise unaccessible to (or extremly impractical for) ab-initio analyses. In this section we briefly present the IMST and point out the motivations behind its construction. We also point out the structural flexibilities of the theory, that lead to the methodical advantages mentioned above.

The Schrödinger equation of an interacting system (e.g. laser + many-body atomic system) can be rewritten as a time-dependent generalization of the Lippmann-Schwinger equation satisfying the initial condition (e.g. the atom in the ground state at an initial time t_i ). To this end, at first, we make the usual partition of the total Hamiltonian of the system H(t) (Hartree atomic units, a.u., are used below),

i.e., as a sum of the initial reference Hamiltonian $H_i^0$ of the unperturbed atom and the initial laser interaction V_i (t) with the atomic electrons, and rewrite the Schrödinger equation

as an integral equation

where we have introduced the total Green’s function (or propagator) G(t, t ^′) by the definition

And, ϕ_i (t) is the solution of the initial Schrödinger problem with the unperturbed Hamiltonian $H_i^0$ (e.g. He atom),

For the problem of double ionization of He, V _i (t) is given by the interaction Hamiltonian of the two electrons coupled to the vector potential of the laser field,

Next, we introduce the final state partitioning of the total Hamiltonian

where, $H_f^0$ (t) is the final reference Hamiltonian (e.g., the sum Hamiltonian of two electrons in the field [17, 18]), and V_f (t) is the final state interaction (e.g., the electron-electron correlation, and also any residual interaction with the nucleus). The final Green’s function satisfies

The total Green’s function is now expanded in terms of the final Green’s function in the form

and is inserted in the expression for the total wavefunction |Ψ(t)> above, Eq. (3), to get

In this way we have arrived at a useful form for the wavefunction that not only satisfies the initial state condition but also is well arranged for computing the transition amplitude, by projection onto any state of the final reference Hamiltonian, even when the latter is not identical to the initial reference Hamiltonian. This is because the projection is orthogonal and hence extracts only one term from the final reference Green’s function (given in the proper state representation). Such a formulation is, therefore, useful not only in the present context but also for many other problems, e.g., the charge exchange reactions, chemical reactions and rearrangement processes in general (which invariably involve unequal reference Hamiltonians in the input and output channels). Thus, projecting onto the final state |ϕ_f (t)>, belonging to $H_f^0$ (t), we write the transition amplitude for the i→f transition as:

This is an exact master equation for the S-matrix, whose two most important features are that (a) the total Green’s function appears between the initial and the final interactions. This is unlike the appearence of the Green’s function at an end, as in the usual ‘prior (or direct time)’ and the ‘post (or time reversed)’ form (see e.g. [20, 21], or the Dyson series of the evolution operator). And (b) the orthogonal projections for the initial and the final states are carried out independently of equal or unequal reference Hamiltonians. This structural flexibility of Eq. (11) permits the introduction of any virtual fragments-propagator of interest, already in the leading terms of the IMST, which is responsible for its potential usefulness for the general many-body reaction problems. Note that, inclusion of the effects of such virtual fragments-channels in the usual ‘prior’ and ‘post’ expansions may only be attemped indirectly, if at all, by summing them to very high orders, if not to infinite orders. We may now introduce the third intermediate partitioning of the total Hamiltonian:

which corresponds to the virtual fragments-Hamiltonian H ₀(t), together with the corresponding propagator G ₀(t, t ^′). For the double ionization problem of present interest, the virtual intermediate fragments can consist of one electron in the Volkov states of all virtual momenta {k}, and a singly charged residual ion in its virtual eigenstates; the corresponding fragments-Hamiltonian is, therefore,

The associated two-particle fragments-Green’s function is [8, 9, 16]

where, θ(t-t ^′) is the Heaviside theta function, and {|${\phi }_j^{+}$ (2)>} is the complete set of residual ionic states. Once G ₀(t, t ^′) is available, we can expand G(t, t ^′) as

substitute it in Eq. (11), and write the resulting amplitude as the desired S-matrix series,

with

……

It is not difficult to show that the ‘prior’ and ‘post’ series could be obtained as special cases of IMST [10]. Moreover, by partial integrations one can show that the first term of the ‘prior’, ‘post’ and IMST series, S ⁽¹⁾ f_i (t), are generally equivalent for long interaction times ([10], footnote 11), but, as discussed above, IMST differs term by term qualitatively from the next term onward. We further note that the first term of the IMST series is the same as that of the so-called KFR theory [22, 23, 21].

3 Two-Electron Sum-Momentum Distributions

Using IMST, we investigate and discuss below the recently obtained momentum distributions of the doubly charged ions of He, both parallel as well as perpendircular to the polarization axis of the laser. The momentum of the doubly charged ions is related to that of the two outgoing electrons via the momentum conservation relation. Thus, neglecting the very small contribution from the momentum of the (long wavelength) laser photons, the overall momentum conservation requires that the sum of the electron momenta parallel to the laser polarization axis, (z-axis), satisfies,

(i): P_par.=−[(k_a)_par.+(k_b)_par.]

and that perpendicular to it satisfies,

(ii): P_perp.≈−[(k_a)_perp.+(k_b)_perp.]

where, P is the momentum of the doubly charged ion, and k _a and k _b are the momenta of the two outgoing electrons.

The most prominent characteristics of the experimental distributions reported [1, 2] are:

(a) the component of the recoil momenta parallel to the laser polarization direction shows a prominent double-hump distribution with a central minimum at peak intensities near the saturation point,

(b) the perpendicular component of the recoil momentum shows a single-hump distribution,

(c) the parallel component distribution is very broad, while the perpendicular distribution is much narrower,

(d) the maximum momentum transfer of the parallel component shows a rather sharp cut-off at a value ${\mathbf{P}}_{\mathit{par}.}\approx \mathit{Re}\left(4\sqrt{U_p}\right)$ for He [1] and a still larger value for Ne [2],

(e) the distributions suggest that both electrons move essentially in the same direction, in contrast to the corresponding distributions for the (γ, 2e) reaction by (weak field) synchrotron photons (that favors the Warnier back-to-back mechanism near zero momentum).

Previously we had extensively investigated the intensity dependence of the total double ionization yields using the present theory [8, 9, 11] and using simplified models based on it [10,12–15]. These investigations permitted us to identify a so-called ‘correlation mediated energy-sharing diagram’, that automatically incorporates in the quantum domain the well-known classical ‘rescattering model’ proposed by Corkum [24] (see discussion below and in [13, 16]).

In fact, for the double ionization problem, the leading two terms of the IMST series, Eqs. (17) and (18), provide, respectively, two first rank, and six second rank Feynman diagrams (these are drawn in [8, 9]). For the near infrared Ti:sapphire laser wavelength, λ=800 nm, and the intensity, I=6.6×10¹⁴ W/cm² used for the He experiment [1], the contributions from the first rank diagrams (corresponding to $S_{\mathit{\text{fi}}}^{\left(1\right)}$ ) as well as five other second rank diagrams (belonging to $S_{\mathit{\text{fi}}}^{\left(2\right)}$ ) are found to be much smaller than the remaining correlation mediated energy-sharing diagram. It should be noted that the relative importance of the eight leading diagrams are in general quite different for different domains of the laser parameters, in particular, in different wavelength domains. Thus, for example, one of the two first rank diagrams in fact was found to be dominant at λ=248 nm, in the UV wavelength domain, over the entire range of the intensity in an experiment of interest (see, experiment [25], and its analysis [13, 10]). This is in contrast to the dominant contribution of the correlation mediated energy-sharing diagram in the present case of the infrared Ti:sapphire laser wavelength. This clearly illustrates that IMST can be helpful in determining the relevant mechanism in different parameter domains of the interaction.

The ‘correlation mediated energy-sharing diagram’ is shown in Fig. 1. The quantum-physical picture behind the double ionization process for near infrared wavelengths and for the so-called ‘non-sequential’ domain of intensity, i.e. near and below the so-called saturation intensity I_s (e.g. I_s ≈8×10¹⁴ W/cm² at λ=780 nm, in the double ionization experiment for He, by Walker et al. [3]) can in fact be visualized from this diagram directly by reading it from the bottom to the top, in the indicated direction of the flow of time. Thus, at an initial time t_i the two electrons (say, 1 and 2) are in the ground state of the unperturbed He atom. At a time t ₁ (when the field phase is ϕ ₁=ωt ₁) one of the bound electrons (say, 1) absorbs, via the generalized ATI-like interaction (VAT_I ), a large amount of field energy by a virtual above-threshold ionization (ATI) process, while the other electron propagates in the virtual states of the residual ion. This intermediate ensemble of virtual states corresponds to the two-particle propagator (Green’s function) G ₀(t, t ^′), ({Volkov} ⊗ {ion states}, Eq. (14), [8, 9, 16]). Then at a time t ₂ (field phase ϕ ₂=ωt ₂) the two electrons interact (‘internally collide’) via the electron-electron correlation (V_corr ; that may, in general, include the interaction with the nucleus as well) and share the energy until both of them may have enough energy to escape together from the binding force of the atom, into the final two electron product Volkov states (c.f., [17, 18]), and proceed with the final momenta k_a and k _b to the detectors, at t_f .

 

Fig. 1. ‘Correlation mediated energy-sharing diagram’ for laser induced non-sequential double ionization of He. For interpretation see text.

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Note that quantum mechanically speaking, since the electrons are not observed in the above intermediate states in the experiment, contributions to the double ionization amplitude, in principle, must be integrated for all values of Δt=t ₂-t ₁=τ (or the phase difference, Δϕ=ϕ ₂-ϕ ₁), as it naturally occurs in the time integrations of the amplitude of the process (see below). Under appropriate limiting conditions, however, the time integrations over the interval τ may be estimated semiclassically. Under the latter conditions (e.g., U_p ≫ω, ω≪E_B ) a significant contribution can arise from time intervals of the order of the classical return time(s) of the first emitted electron to the core region, with a corresponding return energy of the order of 3U _p. This provides the theoretical connection between the present quantum theory and the classical ‘rescattering model’ of Corkum [24], mentioned above. In the actual calculation below we shall not make the semiclassical approximation of the time integrations but evaluate them exactly after expanding all the Volkov waves in the amplitude in terms of the generalized Bessel functions (see Eqs. (22),(23) below).

For the purpose of actual calculations, we write down the NS double ionization amplitude, following the diagram in Fig. 1, as,

where, the angular brackets stand for the spatial integrals; ${\varphi }_f^V$ (k _a , k _b ; t ₂) is the two-electron product Volkov final state in the field (c.f. [17]), V_corr (t ₂) stands for correlation operator and Ψ _i (t ₂) is the total wavefunction of the system, at time t ₂. In the present approximation, Ψ _i (t ₂) is given by (c.f. Fig. 1),

where, V_ATI (t ₁) is the interaction operator for the virtual ATI-like process at the time t ₁, ϕ^V (k; r, t) is the one-electron Volkov wave function, and ϕ _1S(r ₁, r ₂; t ₁) is the ground state wavefunction of the He atom with binding energy E_B =2.904 (a.u.). An exact evaluation of this amplitude, including all orders of correlation and ATI interaction, is practically an impossible task. We have, therefore, restricted ourselves to the lowest significant terms of the theory by replacing V_corr (t ₂) by 1/r ₁₂ and VAT_I (t ₁) by $\left(-\frac{1}{c}\mathbf{A}\left(t_1\right)\right){\mathbf{p}}_1+\frac{1}{2c^2}{A}^2(t_1))$ This simplification still requires calculation of a formidable multiple integral which we have evaluated approximately as follows. The Jacobi-Anger formula ([26], p. 7) is used to expand the Volkov wavefunctions, ϕ^V (k, r; t), in terms of the corresponding Fourier components defined by the generalized Bessel functions of two arguments (e.g., [19]). This allows us to evaluate the two-fold time integrations, over the instants t ₁ and t ₂ of the two interactions, exactly; for pulses much longer than a laser period (as in the experiments of present interest) this gives,

where,

where ϕ ⁰(k, r) is a plane wave state.

Next, the six-fold space integrations are carried out analytically, the radial integration in k is performed by pole approximation and the integrals over the angles of k and the sum over n are performed numerically. Contribution of the lowest term of the sum over j (corresponding to the ground state of the ionic state) is retained only since it is found to dominate over the contribution from any excited state (c.f. [11]) in the present case. Finally, Monte-Carlo sampling method is used to evaluate the differential rate of double ionization, ${\displaystyle \sum _N}\frac{d{\Gamma }^{\left(N\right)}}{d{\mathbf{k}}_{a}d{\mathbf{k}}_b}$, as a function of the two outgoing momenta k _a and k _b , from the formula,

Computations are carried out for He atom, and the distributions of the parallel and the perpendicular components of the sum of the two momenta, (k _a +k _b ), are determined, in each case by integrating over the remaining variables and summing the contributions from all significant Ns. We recall that under the condition of the experiment the He²⁺ recoil momentum P≈-(k _a +k _b ) [1, 2].

 

Fig. 2. Recoil ion momentum distribution of He²⁺ parallel to the polarization direction, P_par . (experimental data, panel a, [1]) and the sum-momentum of the two outgoing electrons in opposite direction (present theory, panel b, [16]).

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Fig. 3. The same as in Fig. 2 but perpendicular to the polarization direction; panel a (experiment, [1]), panel b (theory, [16]).

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We show in Fig. 2 the experimental result (panel a) obtained for the recoil momentum of He⁺⁺, parallel to the (linear) polarization axis by Weber et al. [1], in the field of a Ti:sapphire laser pulse of 200 fs, and a peak intensity of 6.6×10¹⁴ W/cm² and λ=800 nm. They are compared with the corresponding results of the present theory [16] in panel b. We note that the experimental data are available only in arbitrary units. For the sake of comparision of the relative values of the experimental and theoretical distributions, we have therefore scaled the theoretical results to match at one point, namely the maximum of the distribution (at -(k a+k _b )=-2 a.u.), with the experimental data. This one-point fit determines the relative scale for the entire theoretical set in the Figure. As it can be seen from the comparison, all the essential features of the experimental distribution of the parallel component of the recoil momentum are reproduced by the theoretical results. Thus, both the distributions show a double-hump structure with a central minimum. The positions and the heights of the two maxima are also well reproduced. We observe that the maximum size of the recoil ion momentum (cut-off momentum) of the experiment and the calculation of the sum electron-momenta also agree well with each other and in fact in both cases it is as large as ≈5 a.u. Finally, we note that there is a quantitative difference at the minimum of the distribution. This suggests that either the uncertainty in the intensity measurement and momentum resolution [27] and/or the higher order contributions in the theory might be involved here.

 

Fig. 4. Sum-momentum distributions calculated without the final state ‘Volkov dressing’ of the two outgoing electrons: (a) parallel case (theory), c.f. Fig. 2; (b) perpendicular case (theory), c.f. Fig. 3.

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In Fig. 3 we show the distributions for the perpendicular component of the recoil momentum. Again the essential features of the experimental distribution (panel a) are reproduced by the theoretical calculation for the sum momentum of the electrons (panel b). Moreover, comparison of Fig. 2 with Fig. 3 shows that the theoretical calculations are consistent with the observation with respect to relative widths in the parallel and perpendicular directions, the latter being much narrower than the former.

What is the origin of the double-hump structure in the parallel case and its absence in the perpendicular case? To gain a greater insight into their origin we calculated the corresponding distributions by deliberately neglecting the final-state interaction of the two electrons with the laser pulse, i.e. dropping the ‘Volkov dressing’ of the two electron in the final state. This is easily done in the present theory by replacing the final state two-electron Volkov wavefunction, by two free (plane) waves of momenta k _a and k _b , and keeping everything else the same.

The result so obtained for the parallel component is shown in Fig. 4a. A comparison of the theoretical calculation with (Fig. 2b) and without (Fig. 4a) the Volkov dressing in the final state, clearly shows that the double-hump character of the parallel component of the distribution collapses into a single-hump structure in the absence of the final state Volkov dressing. This unequivocally suggests that the final-state field interaction of the two electrons is primarily responsible for the double-hump structure of the observed distribution in this case. In Fig. 4b we show the corresponding result of calculation for the perpendicular component in the absence of the final state laser interaction. Comparison of this distribution with that of Fig. 3b shows that in this (perpendicular) case the presence of the final state laser interaction does not play a significant role. This is as might be expected in the present case, since the force due to the electric field is negligible in the direction perpendicular to the polarization direction. This is also consistent with the observed narrow width of the perpendicular distribution due to the absence of significant laser coupling in the final state. Therefore, in this case we have the interesting situation of observing the double ionization in the laboratory as if the laser field is switched off as soon as the two electrons are freed from the bound atomic system.

We may finally ask, what determines the observed large width of the parallel distributions in Fig. 2? From the explicit expression of the transition matrix element, Eq. (23), and using the properties of the (generalised) Bessel functions involved, one could derive [16] the following expression for the cut-off momentum of the distribution,

where E _B is the binding energy. For the case of the distribution shown in Fig. 2 use of this formula predicts the cut-off momentum ≈5 a.u., in good agreement with both the experimental value, and the numerical result. Satisfactory agreement between the above cut-off formula and the available experimental values at other intensities for He [1] and Ne [2] has also been found [16].

 

Fig. 5. Results of model calculations for the yields of He⁺⁺-ions at λ=780 nm and τ=160 fs assuming a collision energy equal to the rescattering energy (dashed line) and to the back-scattering energy (solid line), in comparision with experimental data [3].

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4 Models of Total Ionization Yields

We may note that besides the systematic calculations [8, 9, 11], earlier we have also developed simple model formulae, starting from the amplitude (Eqs. (22,23)) and making additional simplifying physical assumptions, such as incoherent photonic contribution (incoherent n-sum), semi-empirical ‘e - 2e’ rates at collison energies given by the maximum classical energy [12] or the back-scattering energy [13–15], that provided excellent double and multiple ionization yields when compared with the available experimental data, in a wide variety of cases [12–15].

In Fig. 5 we show the corresponding results of model calculations for double ionization of He at λ=780 nm and a pulse duration of τ=160 fs using the model of collision energy for the ‘e - 2e’ collision to be equal to the ‘rescattering energy’ (of the order of 3U_p , cf. [12]), or the back-scattering energy (of the order of 8U_p , cf. [13–15]); the actual intermediate energies, of course, will lie at all values, in between and including them. In Fig. 5 we compare the results of calculation with the data of the intensity dependence of the total ion yields [3]. Quantitatively speaking the results from these two model assumptions (rescattering: dashed line, back-scattering: solid line) agree except at the lowest intensities (in the so-called ‘multiphoton’ regime). In the latter regime an internal collision energy of the order of the rescattering energy (≈3U_p ) may not be quite enough as a higher collision energy of the order of the back-scattering energy (≈8U_p ) apparently fare better.

However, the qualitative yields from these two model assumptions are rather similar. We may ask, why do the two significanly different assumptions of the collision energy do not make much difference in the ion yields? One reason for this seems to lie in the nature of the ‘e - 2e’ rates as a function of the collision energy, which is shown for He⁺ ions in Fig. 6, as calculated from the well-known formula given by Lotz [28]. Note that the ‘e - 2e’ rate increases rapidly at first and remains virtually constant at higher collision energies. This appears to be the reason for the relative insensitivity of the ion yields, specially at not too low intensities, with respect to the assumption of the two collision energies. Furthermore, it should be noted that the actual value of the near constant rate is Γ^(e-2e)≈3×10^-3 a.u., is close to the magnitude of the ratio between the double to single ionization yields of He of about 2×10^-3, that has been observed [3] for intensities not too far from the saturation intensity.

 

Fig. 6. ‘e - 2e’ rates as a function of the collision energy for He⁺ ions, as calculated from the Lotz formula [28].

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5 Conclusion

We have given a brief account of the intense-field many-body S-matrix theory (IMST) and pointed out its methodical advantages over the other forms of S-matrix series. The theory is applied to the recently observed recoil-ion momentum distribution from double ionization of He in an intense femtosecond laser pulse. The observed characteristics of the distributions, both parallel and perpendicular to the polarization axis, are shown to be reproduced by the sum-momentum distributions of the two ejected electrons. It is shown that the correlated energy-sharing between the two electrons in the intermediate state and the ‘Volkov-dressing’ of them in the final state, can account for the observed characteristics. Besides these ab-initio analysis of the momentum distributions, we also give examples of two models of total ionization yields with two different physical hypotheses for the ‘e-2e’ collision-energy and suggest a possible reason for their relative insensivity toward the total yields. To conclude, the basic mechanism of double ionization of He in intense infrared laser pulses may be summarized succinctly as a symbolic equation: Double ionization = ‘ATI’ + ‘e - 2e’-collision + ‘Volkov-dressing’.