## Abstract

The physics of laser-mater interactions beyond the perturbative limit configures the field of extreme non-linear optics. Although most experiments have been done in the near infrared (*λ*≤1*µ*m), the situation is changing nowadays with the development of sources at longer wavelengths (<5*µ*m), opening new perspectives in the synthesis of shorter XUV attosecond pulses and higher frequencies. The theory of intense-field interactions is based either on the exact numerical integration of the time-dependent Schrödinger equation or in the development of models, mostly based on the strong-field approximation. Recent studies in the short-wave infrared show a divergence between the predictions of these models and the exact results. In this paper we will show that this discrepancy reveals the incompleteness of our present understanding of high-order harmonic generation. We discuss the physical grounds, provide a theoretical framework beyond the standard approximations and develop a compact approach that accounts for the correct scaling of the harmonic yield.

© 2009 Optical Society of America

## 1. Introduction

From a fundamental viewpoint, the interaction of ultraintense lasers with atoms constitutes a paradigmatic example among non-perturbative phenomena. Today’s state-of-the-art laser technology allows, to a given extend, for the control of the laser pulse in terms of intensity, duration and phase characteristics. On the other side, the exact dynamics of the atomic electron can be numerically computed from the Schrödinger equation in the hydrogenic case. The convergence of theory and experiments models this field as a privileged test-ground for general non-perturbative theoretical developments. Among these, S-matrix approaches combined with the strong-field approximation (SFA) [1, 2, 3] are shown to provide an excellent description of the single and multielectron ionization rates, as well as constituting an adequate formalism to study the process of high-order harmonic generation (HOHG). In this later case, a *standard* approach [4, 5, 6] combines SFA with a saddle point approximation to compute the harmonic spectra. The resulting theory offers a semiclassical description in terms of electronic trajectories, which constitute an extraordinary tool for the physical understanding of HOHG. The fundamental concept arising from this description [7] is that harmonics are generated by electrons that have been tunnel ionized, then accelerated by the field and, finally, driven back to the ion (where the radiation is released from the transition to the ground state). As a result, the non-perturbative harmonic spectra is characterised by a plateau of similar harmonic intensities followed by an abrupt cut-off. Calculations of the time-dependent Schrödinger equation (TDSE) confirmed by experiments and S-Matrix approaches, have provided a simple law for the harmonic cut-off frequency (*I _{p}*+3.17

*U*,

_{p}*I*being the ionization energy, and

_{p}*U*=

_{p}*q*

^{2}

*E*

^{2}/4

*mω*

^{2}∝

*λ*

^{2}, the ponderomotive energy). The standard model also succeeds in predicting the mode locking of the highest order harmonics, chirps and modulation of the yields with the intensity [8]. Despite of these achievements, recent studies on the scaling of the harmonic yield with wavelength show a divergence between the predictions of this model and the exact 3-dimensional (3D) TDSE [9, 10]. From the theoretical viewpoint, this issue is of fundamental interest as it constitutes an interesting divergence from the predictions of the standard model. From a practical side, the correct scaling of the harmonic yield is fundamental for the quantitative description of the HOHG with short- and mid-wave infrared sources, which may be used to generate shorter single attosecond pulses [12]. In this paper we shall, first, drive a criticism to these recent works, as they compute the harmonic yield in a spectral region for which the standard theory is not applicable. Second, we will demonstrate that, in its region of applicability, there is still a discrepancy between the standard model and the exact results from the 3D TDSE. In third place, we will demonstrate that the physics behind the scaling law falls beyond the strong-field approximation. Finally, we will develop an S-Matrix formalism using Feshbach operators that goes beyond the SFA. We will show that our model gives good quantitative agreement compared with the exact integration of the 3D TDSE, reproduces the correct scaling laws, and yet offers a compact formulation with reduced computing needs. The recurrence to the concept of field dressing of states to overcome the limitations of the SFA is not new [13, 14, 15, 16]. Field dressing is usually introduced for the description of the electron ionization while in our case affects the lower state of the transition leading to harmonic radiation.

## 2. Theoretical approach

The study of the HOHG in the short-wave infrared (SWIR) has been boosted recently by the development of high-power lasers sources in this spectral region, based on parametric amplification. However, the experimental determination of the scaling laws of the harmonic yields is limited by the phase matching sensibility of the propagation process to wavelength [17]. Therefore, the study of the single-atom response still appears to be the most reliable source for the understanding of wavelength-dependent characteristics HOHG.

To our opinion there is an important issue regarding the adequacy of the previous works [9, 10, 11]. As the SFA neglects atomic bound-state excitations, we should expect S-Matrix SFA theories to reproduce faithfully only the higher frequency part of the harmonic spectra and not the whole (this fact is also recognized in [11]). In contrast, references [9, 10, 11] consider a fixed energy interval from 20 eV to 50 eV that falls deeply inside the harmonic plateau as the wavelength increases, where the SFA is in principle invalid. Instead, our choice is to consider a spectral window of 10.2 eV right below the end of the plateau (i.e. attached to the cut-off frequency), which is translated jointly with the spectrum cut-off as the wavelength increases. This particular value corresponds to the energy difference between the 1s and 2p states in H, and therefore the chosen window excludes the possible interference with the harmonics generated through the 2p state. Figure 1(a) shows, for a particular case, the location of the window used in this paper (dashed black lines) and the one used in these previous works (dashed red lines). Figure 1(b) shows the detail of the spectrum in the later case jointly with the results computed for the S-Matrix SFA models in **E**·**r** and **p**·**A** gauges (red and orange graphs), for the model beyond the SFA presented here (green) and the exact 3D TDSE result (blue). The failure of the three models to reproduce accurately the 3D TDSE spectrum in this window becomes apparent and suggest, as discussed above, that the process of harmonic radiation in the inner part of the plateau has a complex nature. On the other hand, the spectra computed at the window near the cut-off show a more accurate resemblance with the 3D TDSE (see figure 2), as will be discussed below.

In table 1 we show the results of the scaling exponents derived from the exact computations of the 3D TDSE for three different laser pulse envelopes (see Appendix A). The yields are computed integrating the harmonic spectra over the two spectral windows discussed above, and multiplying by the pulse length. The scaling of the harmonic power (last column) corresponds to the closest harmonic near the cut-off frequency. The tabled results show stronger dependences of the scaling exponent with the pulse shape for the case of a fixed energy interval (*n*⋍3.9±0.4) than for the translating window near the cut-off (*n*⋍5.6±0.1). This suggests that this latter case is less sensitive to the details of the electromagnetic pulse. Moreover, the difference between the scaling powers of the yields computed at each window suggests a different mechanism for HOHG. In any case, as a main conclusion, our results confirm the discrepancy between the scaling powers of the exact 3D TDSE results (*n*⋍5.6) and the expectations of the S-Matrix SFA models (*n* between 3 and 4, see Appendix B) also for the higher frequency part of the harmonic spectrum.

Next we shall develop an S-Matrix approach beyond the approximations of the standard model. On one hand we will not recur to the saddle point approximation, preserving the full quantum nature of the process and, on the other hand, we will be able to relax the conditions imposed by the SFA. Our model’s results will be compared with the exact 3D TDSE and, also, with the predictions of a general form of the standard model: a SFA S-matrix approach without recurring to the saddle point approximation (referred here as S-Matrix SFA approach). This will allow us to center our discussion around the applicability of the SFA keeping aside the possible issues derived from the application of the saddle point approximation.

Part of our approach is based on a recent work [18, 19], which demonstrated that a proper quantitative agreement with the exact results from the 3D TDSE needs the inclusion of electronic transitions other than the continuum-ground. However, our further work has lead us to a new derivation and interpretation of the theory in which this transition is affected by the field in a way not described by the SFA.

The general starting point of S-matrix approaches to HOHG is the (exact) definition of the time-dependent wavefunction in terms of Green’s propagators

associated with the splitting of the Hamiltonian in two parts *H*(*t*)=*H _{a}*+

*V*(

_{i}*t*),

*H*being the atomic Hamiltonian, and

_{a}*V*(

_{i}*t*) describing the interaction with a linearly polarized electromagnetic wave,

*V*(

_{i}*t*)=−(

*q*/

*mc*)

*A*(

*t*)

*p*+

_{z}*q*

^{2}/(2

*mc*

^{2})

*A*

^{2}(

*t*) (velocity gauge) or

*V*(

_{i}*t*)=−

*qE*(

*t*)

*z*(length gauge). Following the philosophy of Feshbach formalism, let us consider two general orthogonal projectors

*Q*̂ and

*P*̂. These operators are constructed accordingly to the different boundary conditions at infinity of bound states (

*Q*̂) and continuum states (

*P*̂). By definition, we find

*Q*̂+

*P*̂=1,

*Q*̂

*P*̂=

*P*̂

*Q*̂=0,

*Q*̂

^{2}=

*Q*̂,

*P*̂

^{2}=

*P*̂, and |

*ψ*(

*t*

_{0})〉=

*Q*̂|ψ(

*t*

_{0})〉 as the state is assumed initially to be the ground (note that this does not imply that the projector contains only one state). We further assume [

*H*,

_{a}*Q*̂]≃0 and [

*H*,

*P*̂]≃0. Imposing these definitions, Eq. (1) leads to two coupled equations (one for the bounded part of the wavefunction and other for the free part)

$$+\genfrac{}{}{0.1ex}{}{1}{\overline{h}}{\int}_{{t}_{0}}^{t}\mathrm{dt}\prime \hat{Q}{G}^{+}(t,t\prime )\hat{Q}{V}_{i}\left(t\prime \right)\hat{Q}{G}_{a}^{+}(t\prime ,{t}_{0})\mid \psi \left({t}_{0}\right)\u3009$$

the second term in the *rhs* of Eq. (2) describes the possibility of atomic excitation (ground-state dressing), while the *rhs* of Eq. (3) describes ionization. The SFA consists in setting *Q*̂*V*
* _{i}*(

*t*′)

*Q*̂=0 in Eq. (2), considering that the field interaction leads invariably to ionization, together with the former condition [

*H*,

*P*̂]=0, which prevents from the recombination of an ionized state. Note that these expressions resemble partially to the truncated propagator in [15], however in the present case the ground-state dressing does not involve the continuum states in the time interval (

*t*

_{0},

*t*′).

According to Larmor’s formula (see Appendix B) the harmonic generation is computed evaluating the mean value of the z component of the acceleration operator *a*̂=−(*q*/*m*)*∂V _{c}*/∂

*z*(

*V*being the atomic Coulomb potential). Since the higher frequencies of the harmonic spectrum involve the most energetic transitions (i.e. continuum to bound states), the relevant part of the acceleration corresponds to

_{c}where *a _{b}* and

*a*are two interfering contributions to the total acceleration, associated with transitions between the continuum and the bare atomic ground state and to its field dressing (not considered in the SFA). Respectively,

_{d}$${\int}_{{t}_{0}}^{{t}_{2}}{\mathrm{dt}}_{1}\hat{P}{G}^{+}(t,{t}_{1})\hat{P}{V}_{i}\left({t}_{1}\right)\hat{Q}{G}_{a}^{+}({t}_{1},{t}_{0})\mid \psi \left({t}_{0}\right)\u3009$$

In the following we will demonstrate that the dressing term is essential to for describe accurately the high harmonic spectra as well as to reproduce the correct scaling laws in the SWIR.

To continue our approach, we must give a form to the operators *P*̂*V _{i}*(

*t*

_{1})

*Q*̂,

*P*̂

*G*

^{+}(

*t*,

*t*

_{1})

*P*̂,

*Q*̂

*V*(

_{i}*t*

_{2})

*Q*̂ and

*Q*̂

*G*

^{−}(

*t*,

*t*

_{2})

*Q*̂. In general this is not straightforward, but may be greatly simplified for the study of harmonic generation. High-order harmonics are generated through transitions from continuum to bound states that take place during rescattering. Therefore, to compute the higher frequency part of the spectrum is enough to describe the process during these recollision events. According to the semiclassical theory, the most energetic collisions take place near the zero of the electric field. The electric field being small, we shall assume that the transition lower state at the beginning of the recollision is the atomic ground state. During the collision, as a result of the field interaction, this state evolves perciptibly. Therefore, the bound state at time t results from the evolution of the atomic ground state during the time lapse

*δ t*during which the recollision takes place. Mathematically this is given by Eq. (2) but with the lower limit of the time integral set to

_{s}*t*−

*δ t*instead of

_{s}*t*

_{0}. Consequently, the same substitution is to be done in the lower limit of the outer integral in Eq. (6). For an estimation of the value of

*δ t*, see Appendix C.

_{s}On the other hand, since the electric field during the most energetic rescatterings is almost zero, we shall consider the polarization of neutral-atom the ground state as a second order effect, and we will reduce the field dressing to a level shift Δ* _{s}* (given in Appendix D) resulting from the classical estimation of the field interaction energy

*V*(

_{i}*τ*), averaged over the scattering time interval

*δt*. Approximating the ground-state by a free wave with negative kinetic energy, we have

_{s}*Q*̂

*H*(

*t*

_{2})

*Q*̂⋍

*p*̂

^{2}/2

*m*+Δ

_{s}*I*̂, and

(*I*̂ being the identity operator), as well as

ε_{0} being the energy of the bare atomic ground state, *H _{a}*|

*ψ*(

*t*

_{0})〉=ε

_{0}|

*ψ*(

*t*

_{0})〉.

The operators *P*̂*V _{i}*(

*t*

_{1})

*Q*̂ and

*P*̂

*G*+(

*t*,

*t*

_{1})

*P*̂ in Eqs. (3)–(6) can be evaluated accordingly to the standard procedure in SFA methods as follows. First we consider a planewave basis, {

**k**}, for the subspace defined by

*P*̂, therefore

Each planewave evolves as Volkov wave of momentum **p**=*h*̄**k** under the influence of the electromagnetic field. Therefore,

with *V _{i}*(

**k**,

*t*

_{1})=−(

*q*/

*mc*)

*A*(

*t*

_{1})

*k*+

_{z}*q*

^{2}/(2

*mc*

^{2})

*A*

^{2}(

*t*

_{1}). In addition we have

were we have introduced the Coulomb factor [20] *C _{F}*/

*rn*=(2

*Z*

^{2}/

*n*

^{2}

*E*

_{0}

*r*) n(here n=1, Z=1), and with

where ε(**k**,τ)=*h*̄^{2}
*k*
^{2}/2*m*−(*q*/*mc*)*A*(*τ*)*k _{z}*+

*q*

^{2}/(2

*mc*

^{2})

*A*

^{2}(

*τ*). Introducing these definitions in Eqs. (5) and (6) (see Appendix E) we can compute the final acceleration as the sum of the transitions of each Volkov state to the bare atom and to the dressed part of the ground state

with |*ψ*(*t*
_{0})〉=|*ϕ*
_{0})〉, the ground state of the atom, and

Note that, in computing the total acceleration (13), the bare and dressed contributions (*a _{b}* and

*a*) interfere destructively, leading to the total acceleration

_{d}The time integral leading to *a _{b}*(

**k**) can be computed numerically very effectively without recurring to the saddle-point approximation and, thus, retaining the full quantum description of the process. This is done by integrating the set of (uncoupled) one dimensional differential equations, each associated with a particular Volkov wave k, that result from differentiating Eq. (14)

Equations (16–17) present a closed formulation for the problem of harmonic generation that (i) falls beyond the standard SFA formulation, since it includes the field dressing of the ground state, and (ii) it is computed without recurring to the saddle point approximation and, therefore, goes beyond the semiclassical description. We will next demonstrate that the quantitative description of the high-order harmonic spectra is only possible if point (i) is fulfilled, and consequently, that the disparity of the predictions for the scaling law of the harmonic yield with the driving laser wavelength reflects the limitations of the standard SFA models. The exploration of the consequences of the saddle point approximation (used also in the standard model) falls out of the scope of this paper.

Figure 2 shows the higher frequency part of the harmonic spectrum for the six-cycle electromagnetic driving field at wavelengths of 800 nm and 1600 nm (part a and b of the figure). The results of the exact solution of the 3D TDSE, integrated with our code (see appendix A), are plotted in blue colour, the results of the SFA S-matrix models are shown in red (length gauge) and orange (velocity gauge), finally the results of our model are plotted in green. Note that the SFA is known to break the gauge invariance of the theory. This, therefore, imposes a stricter test of our approach, since it must be tested against the SFA S-matrix results in both gauges. On the other hand, the comparison between SFA models and ours is greatly simplified by noting that the formers constitute a simplified version of our model, i.e. setting the dressed contribution *a _{d}*(

*t*) to zero in Eq. (4) or (13).

The most relevant conclusion of this figure is that the model presented in this paper has the best quantitative agreement with the 3D TDSE for both wavelengths. Incidentally, note also in Fig. 2 that the S-matrix SFA model in the length gauge is more accurate for the smaller wavelengths, while the velocity gauge is more accurate for the longer wavelengths. The better accuracy of our approach is a general behaviour at least for wavelengths between 800 and 1600, as it is shown in figure 3. Part (a) of this figure shows the harmonic yields computed from the exact integration of the 3D TDSE (blue) and the present model (green), while part (b) shows the comparison between 3D TDSE (blue) and the S-Matrix SFA models in the length (red) and velocity (orange) gauges. We have also plotted the best fit to *λ*
^{−n} in solid lines. Note that, linked to its better accuracy, our model reproduces also quite accurately the scaling of the yields with the wavelengths. Therefore, as a main result, we conclude that the physical reason for the divergence between the scaling laws predicted by S-Matrix SFA models and the exact 3D TDSE can be attributed to the influence of the electromagnetic field in the ground state.

Table 2 shows the scaling exponents extracted from the three models and the 3D TDSE for the different pulse shapes used in this paper. The yields are computed using the window near the cut-off frequency and, therefore, are quite stable against the change of shape of the pulse. Note that the exponents derived from the S-Matrix SFA (*n*=3.61±0.16 and *n*=3.06±0.06) are also consistent with the expectations of the standard model (*n* between 3 and 4), as discussed above. On the other hand, it is also apparent the discrepancy of the S-matrix SFA models with the exact 3D TDSE results (*n*=5.63±0.22), and with the prediction of our model (*n*=5.41±0.21).

## 3. Conclusion

To great extent, the theoretical understanding of the non-perturbative interaction of atoms with electromagnetic fields has been based on the Strong Field Approximation. Despite of its success in qualitative aspects, we have demonstrated that the correct quantitative description of the high-order harmonic generation falls beyond this approximation. In particular, we have shown that the next order correction to SFA consists of accounting for the interaction of the electromagnetic field with the atomic ground state. Our formalism, based on the introduction of Feshbach operators, leads to a comprehensive account of the later process in the standard framework of the S-matrix formalism. Besides, our model is compact, fully quantum (not semiclassical) as it does no recur to the saddle point approximation, and allows for efficient computation. Our results are tested against the 3D TDSE in a wide range of parameters (pulse shapes and wavelengths) with positive results. Also, we have demonstrated our theory to account for the proper scaling of the harmonic yields at the higher frequencies of the spectrum. Therefore our interpretation of the discrepancy of the scalings of the harmonic yield between the standard model (based on SFA and saddle point approximation) and the exact results of the 3D-TDSE, reported in [9], is the influence of the electromagnetic field in the bound-state dynamics of the atom.

A. Electromagnetic pulses and time-dependent Schrödinger equation.

Along the paper we have used three different linearly polarized electromagnetic field pulses. For simplicity, we refer to them in the text in terms of their duration (4, 6 and 9 cycles), despite each pulse has a different shape for the envelope. The general form of the pulse is (dipole approximation is assumed)

*E*
_{0}(*t*) being pulse envelope: a sin^{2} shape for the 4 cycle pulse, a trapezoidal shape of 6 cycles (with 2 cycles of linear turn-on, 2 cycles of constant amplitude and 2 cycles of linear turn-off), and a trapezoidal shape of 9 cycles (with 1/2 cycle of linear turn-on, 8 cycles of constant amplitude and 1/2 cycle of linear turn-off). The later corresponds to the field used in the previous works [9, 10].

The exact results shown in this paper correspond to the integration of the 3D time-dependent Schrödinger equation for the Hydrogen atom in the length gauge. We have used our own code based in finite differences and projection into partial waves.

B. Estimation of the scaling law from the standard model.

In the standard approach the harmonic dipole amplitude, Eq. (18) in [4], is expressed as an integral over the rescattering time *τ*. The higher harmonics have associated typical rescattering times of the order of 3*T*/4 (*T* being the laser period). Therefore the first factor in the integral contributes approximately as *T*
^{−3/2}∝*λ*
^{−3/2} to the dipole amplitude. On the other hand, an additional *ω*
^{−1}
_{0}∝*λ* factor comes from the time integration of the remaining oscillating functions. Consequently, in the standard model, the dipole amplitude can be expected to be proportional to *λ*
^{−1/2}.

The power radiated at a given harmonic can be computed using Larmor’s formula as

*a _{q}* being the dipole acceleration (second derivative of the dipole amplitude, therefore scaling as

*λ*

^{−5/2}). Consequently, the estimation of the standard model for the harmonic power is a scaling

*λ*

^{−5}. As pointed out in [9], the experimental accessible quantity is the harmonic yield (energy radiated during the interaction time) instead of the power. The present and, also, the previous studies [9, 10] of the SWIR scaling laws consider laser pulses with constant number of cycles but different wavelengths. In this case, the total interaction time scales as

*λ*, and the yield at a given harmonic frequency (product of the harmonic power times the interaction time) would scale as

*λ*

^{−4}, according to the above discussion. Note that the same scaling (

*λ*

^{−4}) is also expected if we consider the yield over a fixed energy interval. We must note, however, that previous works [9, 10] consider that the expectation of the standard model is

*λ*

^{−3}. In any case, our numerical computations confirm a scaling exponent between −3 and −4, which is in conflict with the exact TDSE (≃−5.6), as is demonstrated in this paper.

C. Evaluation of the time lapse of rescattering.

The time lapse during which the rescattering takes place can be estimated by computing the characteristic time in which the wavepacket of the ionized electron crosses the coordinate origin. Assuming that the electron has tunnelled out, the expansion of the wavepacket in free space after some time τ can be estimated as $\sqrt{2\mid {\epsilon}_{0}\mid \u2044m}\times \tau $. The typical time for the excursion of the electron through the continuum is ⋍(3/4)2*π*/*ω*
_{0}, therefore the size of the wavepacket of the returning electron at the rescattering time is $\simeq (3\u20444)(2\pi \u2044{\omega}_{0})\sqrt{2\mid {\epsilon}_{0}\mid \u2044m}$. The time lapse of rescattering is given by this size divided by the velocity of the electron at rescattering event (aprox. $\sqrt{2\times 3.17{U}_{p}\u2044m}$), therefore

D. Evaluation of the energy level shift Δ* _{s}*.

The energy shift is evaluated as the time average of the interaction *V _{i}*(

*τ*) over the collision time

*δt*,

_{s}where kz is a relevant momentum of the state. Assuming a monochromatic field, $A\left(t\right)=(2c\u2044q)\sqrt{m{U}_{p}}\mathrm{cos}\left({\omega}_{0}t+\varphi \right)$, and for the particular times when the higher energy rescattering takes place (field near 0, i.e. *ω*
_{0}
*t*+*ϕ*≃*nπ*), we find 〈cos(*ω*
_{0}
*t*+*ϕ*)〉≃sin(*ω*
_{0}
*δ t _{s}*)/

*ω*

_{0}

*δ t*and 〈cos

_{s}^{2}(

*ω*

_{0}t+

*ϕ*)〉≃1/2+sin(2

*ω*

_{0}

*δ t*)/4

_{s}*ω*

_{0}

*δ t*, therefore

_{s}To compute *k _{z}*, we shall approximate the dressed state energy to the instantaneous energy of the bound state in the electromagnetic field 〈

*ϕ*

_{0}|

*H*(

*t*)|

*ϕ*

_{0}〉=

*ε*

_{0}+2

*U*cos

_{p}^{2}(

*ω*

_{0}

*t*+

*ϕ*) (|

*ϕ*

_{0}〉 being the ground state of the atom). Note from the virial theorem that 〈

*V*(

*r*)〉=2ε

_{0}. On the other hand, as the electric field during the collision is small, the departure from equilibrium is harmonic and, therefore, half of the interaction energy is invested into kinetic energy. Therefore, assuming energy equipartition, the kinetic energy along the field polarization direction is

therefore

After time-averaging over the rescattering time *δt _{s}*, the solution of

*k*reads as

_{z}E. From Eqs. (5–6) to Eqs. (14–15)

First let us consider the undressed part of the acceleration (5) to derive Eq. (14). Using the idempotence property *P*̂^{2}≡*P*̂ and the definition (9) in Eq. (5), we have

Assuming that the initial state is the ground state of the atom |*ψ*(*t*
_{0})〉=|*ϕ*
_{0}〉, therefore ${G}_{a}^{+}(t,{t}_{0})=-i{e}^{-i{\epsilon}_{0}\left(t-{t}_{0}\right)\u2044\overline{h}}$, and using Eqs. (10) and Eqs. (12) we readily obtain Eq. (14) for every momentum component **k**.

The derivation for the dressed part of the acceleration (6) to derive Eq. (15) follows similarly. The first step leads to

$$\int d\mathbf{k}\mid \mathbf{k}\u3009\u3008\mathbf{k}\mid {\int}_{{t}_{0}}^{{t}_{2}}{\mathrm{dt}}_{1}\hat{P}{G}^{+}(t,{t}_{1})\hat{P}\hat{P}{V}_{i}\left({t}_{1}\right)\hat{Q}{G}_{a}^{+}({t}_{1},{t}_{0})\mid \psi \left({t}_{0}\right)\u3009$$

$$\int d\mathbf{k}\mid \mathbf{k}\u3009{\int}_{{t}_{0}}^{{t}_{2}}{\mathrm{dt}}_{1}{e}^{-i{\int}_{{t}_{1}}^{t}d\tau \epsilon (\mathbf{k},\tau )}{V}_{i}(\mathbf{k},{t}_{1}){e}^{-i{\epsilon}_{0}\left({t}_{1}-{t}_{0}\right)\u2044\overline{h}}\u3008\mathbf{k}\mid {r}^{-n}\mid \psi \left({t}_{0}\right)\u3009$$

Being tunnel ionized near the field maximum, a small momentum characterizes the rescattering electron and, therefore, the transition amplitude is characterized by the long-range behaviour of the bound-state wavefunction. In the long range limit, the operators *p*̂^{2} and *a*̂ approximately commute and we may write

$${\int}_{{t}_{0}}^{{t}_{2}}{\mathrm{dt}}_{1}{e}^{i{\epsilon}_{0}\left({t}_{2}-{t}_{1}\right)\u2044\overline{h}}{e}^{-i{\int}_{{t}_{1}}^{t}d\tau \epsilon (\mathbf{k},\tau )}\u3008{\varphi}_{0}\mid \hat{a}\mid \mathbf{k}\u3009{V}_{i}(\mathbf{k},{t}_{1})\u3008\mathbf{k}\mid {r}^{-n}\mid \psi \left({t}_{0}\right)\u3009$$

using Eq. (14), we have

We expect our model to reproduce the higher frequency part of the acceleration spectrum, that corresponds to the lower frequencies of the product ${e}^{i{\epsilon}_{0}\left(t-{t}_{2}\right)\u2044\overline{h}}{e}^{-i{\int}_{{t}_{2}}^{t}d\tau \epsilon (\mathbf{k},\tau )}{a}_{b}(\mathbf{k},{t}_{2})$. Therefore we may consider this as a slowly varying part of the integral and approximate

The integral of the exponential function can be computed taking into account that the lower limit (*t*−*δ t _{s}*) does not contribute, as the wavefunction overlap before the rescattering is negligible. Therefore

## Acknowledgments

We thank A. Zaïr for fruitful discussions. This work has been supported by the Spanish Ministerio de Ciencia e Innovación (FIS2005-01351, Consolider program SAUUL, CSD2007-00013) and by the Junta de Castilla y León (SA146A08).

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