Development of nonperturbative nonlinear optics models

Development of nonperturbative nonlinear optics models including effects of

high order nonlinearities and of free electron plasma.

Part I: Maxwell-Schrodinger equations coupled with evolution equations for

polarization effects.

Part II: SFA-like nonlinear optics

E. Lorinb,a, M. Lytovab, A. Memarianb, A. D. Bandraukc,a

aCentre de Recherches Mathematiques, Universite de Montreal, Montreal, Canada, H3T 1J4bSchool of Mathematics and Statistics, Carleton University, Ottawa, Canada, K1S 5B6

cLaboratoire de chimie theorique, Faculte des Sciences, Universite de Sherbrooke, Sherbrooke, Canada, J1K 2R1

Abstract

This paper is dedicated to the exploration of non-conventional nonlinear optics models for intense and shortelectromagnetic fields propagating in a gas. When an intense field interacts with a gas, usual nonlinear opticsmodels, such as cubic nonlinear Maxwell, wave and Schrodinger equations, derived by perturbation theorymay become inaccurate or even irrelevant. As a consequence, and to include in particular the effect of freeelectrons generated by laser-molecule interaction, several heuristic models, such as UPPE, HOKE models,etc, coupled with Drude-like models [1], [2], were derived. The goal of this paper is to present alternativeapproaches based on non-heuristic principles. This work is in particular motivated by the on-going debatein the filamentation community, about the effect of high order nonlinearities versus plasma effects due tofree electrons, in pulse defocusing occurring in laser filaments [3], [4], [5], [6], [7], [8], [9]. The motivationof our work goes beyond perturbative models, and is more generally related to models of interaction of anyexternal intense and (short) pulse with a gas. In this paper, two different strategies are developed. Thefirst one is based on the derivation of an evolution equation on the polarization, in order to determine theresponse of the medium (polarization) subject to a short and intense electromagnetic field. Then, we derivea combined semi-heuristic model, based on Lewenstein’s SFA (Strong Field Approximation) model and theusual perturbative modeling in nonlinear optics. The proposed model allows for inclusion of high ordernonlinearities as well as free electron plasma effects. We have developed alternative analytical techniques inorder to properly include free electron and high order nonlinearity effects in polarization models.

Keywords: Nonlinear optics, laser, filamentation, Maxwell-Schrodinger

Contents

1 Introduction 21.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Maxwell-Schrodinger model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 SFA model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Polarization evolution equation for very short pulses 42.1 Evolution equation on the dipole moment: circularly polarized field . . . . . . . . . . . . . . . 42.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Email addresses: [email protected] (E. Lorin), [email protected] (M. Lytova),[email protected] (A. Memarian), [email protected] (A. D. Bandrauk)

Preprint November 25, 2014

3 Polarization evolution equation for longer pulses 143.1 General model under the paraxial approximation . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Geometrical optics approach (two-dimensional case) . . . . . . . . . . . . . . . . . . . . . . . 153.3 Polarization reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 SFA nonlinear optics models 184.1 General non-perturbative approach. How far can we go ? . . . . . . . . . . . . . . . . . . . . 184.2 SFA-like nonlinear optics model: unique continuous state . . . . . . . . . . . . . . . . . . . . 194.3 SFA-like nonlinear optics model: possible extension to multiple continuous states . . . . . . . 29

5 Conclusion 32

1. Introduction

1.1. Introductory remarks

This paper is devoted to the derivation of non-standard nonperturbative nonlinear optics models forintense electromagnetic fields propagating in a gas. This work is strongly motivated by the recurrent debatein the nonlinear optics community, regarding appropriate choice of nonlinear optics models for modeling laserfilaments, in particular appropriate models for high order nonlinearities and plasma of free electrons. Theseissues were already discussed in the celebrated papers [10], [11]. Over the years, several models were proposedand the interested reader can refer to several complete review papers [1], [2], [12]. Although, some ellaboratedmodels (HOKE, UPPE [13], [14]) allow for accurate simulations and analysis of laser filamentation in severalphysical frameworks [13] , [15], [16], [9], [7], [17] at high intensity accurate modeling of the generation andevolution of plasma of free electron and nonperturbative nonlinearities, is still an open problem, which is inparticular studied in [6] [18], [19], [3], [4], [8], [20].The first approach which is proposed here, consists of an extension of a micro-macro model constituted byMaxwell’s equations, ME’s, and Schrodinger equations, TDSE’s, modeling the nonlinear response of a gas toan electromagnetic field, see [21], [22] and [23]. From a practical point of view, this model confronts a majorissue, which is the huge computational cost for computing the polarization using TDSE’s. As a consequencerealistic simulations are only possible on very short propagation distances, which makes this ab-initio modelirrelevant for filamentation for instance. In order to reduce the overall computational complexity, a model isthen proposed which is based on a transport-like equation, modeling the time evolution of polarization. Thisadditional equation allows for a drastic reduction of the number of TDSE’s to be solved, and then of theoverall computational complexity of the numerical model. Although we still do not expect the model to beefficient enough for simulating filamentation, it may be a good candidate for a fundamental understanding ofthis phenomenon [1], [2]. This strategy is first developed in details for circularly polarized pulses and is validfor ultrashort pulses. Other strategies based on geometrical optics or Kirchoff’s formula are also proposedfor more general pulses.In the second part of the paper, we propose to derive from the Strong Field Approximation (SFA) model[24], for intense laser-molecule interaction, a macroscopic nonperturbative nonlinear optics model. As firstorder nonlinearities play an important role in field propagation, Lewenstein’s SFA model, which cannot betreated by 1-D models, is coupled with a traditional perturbative nonlinear optics model [25]. We will alsoshow how far we can go in the explicit and rigorous derivation of the dipole moment including bound andcontinuous states, using the density matrix formalism.

1.2. Maxwell-Schrodinger model

We recall that the Maxwell-Schrodinger model developed in [21], [22] and [23] is based on the coupling ofthe 3D macroscopic ME’s with many TDSE’s under the dipole approximation (electric field is supposed to beconstant in space at the molecular scale). This is valid when the smallest internal wavelengths λmin of theelectromagnetic field are much larger than the molecule size ℓ, that is ℓ ≪ λmin (typically λmin ≈ 800nmwhere ℓ ≈ 0.1nm). We then define the ME’s on a bounded space domain with a boundary Γ, and x′ =(x′, y′, z′)T denotes the electromagnetic field space variable. At the molecular scale, and working under the

Born-Oppenheimer approximation, we will denote by x =(x, y, z

)Tthe TDSE space variable (for electrons).

2

The molecular density is supposed to be constant in time, continuous in space, and is denoted by N (x′).The equations we consider are the following ones:

∂tB(x′, t) = −c∇×E(x′, t),

∂tE(x′, t) = c∇×B(x′, t)− 4π(∂tP(x′, t)

),

∇ ·B(x′, t) = 0,

∇ ·(E(x′, t) + 4πP(x′, t)

)= e(NI −Ne).

(1)

Polarization-TDSE is written as:

P(x′, t) = N (x′)∑m

i=1 Pi(x′, t) = N (x′)

∑mi=1 χΩi

(x′)∫R3 ψi(x, t)xψ

∗i (x, t)dx,

i∂tψi(x, t) = −∇2

x

2ψi(x, t) + x · Ex′

i(t)ψi(x, t) + Vc(x)ψi(x, t)∀i ∈ 1, ..,m

(2)

where VC denotes the “Coulomb” potential. Computation of the TDSE provides complete wavefunctions, in-cluding ionization, that is a continuum spectrum of free electrons propagating in a laser pulse. Such electronscan recombine with the parent ion with maximum energy Ip + 3.17Up, where Ip is the ionization potentialand Up = E2/4mω2, the ponderomotive energy acquired by a particle of mass m in a field E and frequencyω or with neighbours with energies exceeding 3Up, [26], [27]. In (1), Ωi denotes the macroscopic spatialdomain containing a molecule of reference associated to a wavefunction ψi, and Pi denotes the macroscopicpolarization in this domain. In other words, Domain Ωi contains N (x′)vol(Ωi) molecules represented byone single eigenfunction ψi (under the assumption of a unique pure state). Naturally we have ∪mi=1Ωi = Ω.We now assume that the spatial support of ψi is included in a domain ωi ⊂ R3, which is supposed to besufficiently large. We allow free electrons to reach the boundary ωi and we impose absorbing boundaryconditions on ∂ωi. We refer to [23], [26] for a complete description of the geometry of this model. FunctionsχΩi

are defined by χ⊗1Ωiwhere χ is a plateau function and 1Ωi

is the characteristic function of Ωi. FinallyEx′

idenotes the electric field (supposed constant in space) in Ωi.

The overall complexity of this model is huge due to the very large number of TDSE which must be solved inorder to get an accurate description of the medium response. Details can again be found in [26], [27]. In thispaper we propose some models to reduce considerably the complexity of the ME-TDSE model. The principleis based on the derivation of a transport equation satisfied by the polarization vector, which will be coupledto ME’s. Although the model still contains TDSE’s, the evolution equation for the polarization, P, allowsto reduce drastically the number of TDSE’s involved in the model. The most simple polarization evolutionequation is an homogeneous transport equation. More accurate models are then proposed, including, inparticular, the electric field variations. Models for circularly polarized electric pulses, Gaussian beams, aswell as general electric fields are presented.

1.3. SFA model

The second strategy is based on the estimation of the medium polarizationP(x′, t), modeling the responseof a medium to an electromagnetic field, from molecular dipole moments, as a combination of contributionsderived from Lewenstein’s SFA model [24], and from usual perturbative modeling [25]. Classical nonlinearoptics models are derived from ME’s coupled with field-molecule TDSE’s. The TDSE’s are mathematically,solved using perturbation theory, allowing for explicit expressions of molecule dipole moments, then of thepolarization. Although, this approach gives accurate nonlinear models for not too intense electromagneticfields, when ionization occurs, these models can fail for precise description of all the complex nonperturba-tive nonlinear phenomena occurring during laser-molecule interactions (ATI, HHG, plasma of free electrongeneration, etc). Heuristic models are then derived [1], [2], in order to include, free electron contributions.The SFA model is derived from the transitions from free to ground states:

ψL(x, t) = eiIpt(φ0(x) +

∫d3vb(v, t)v

)

3

for a ground state φ0 and ionization potential Ip. It allows for an accurate modeling of laser-moleculeinteraction for intense pulses. The second part of this paper is then devoted to the derivation of a combinedmodel, including the features of both the SFA model and the classical nonlinear optics perturbative models.

1.4. Organization of the paper

The paper is organized as follows. Section 2 is devoted to the modeling of the polarization, based on anevolution equation. More specifically, in Subsection 2.1 transport-like equations modeling the polarizationvector evolution, are derived and are coupled to the ME’s, for circularly ultrashort polarized pulses. Thecase of general 3-d electric fields is proposed in Subsection 2.2. In Appendix, other extensions are proposed,including an evolution equation of the polarization for Gaussian beams. Then, the methodology is extendedfor long pulses in Section 3, under the paraxial and slowly varying envelope approximations Subsection 3.1.Subsection 3.2 is dedicated to geometrical optics techniques for deriving a polarization equation. In Section4, a combined SFA and perturbation approach is then presented. First, Subsection 4.1 is dedicated to thederivation of a Liouville-like equation for free and bound states. However, the complexity of the derivedequations does not allow for an explicit computation of the matrix density, and of the dipole momentwithout using perturbation theory. Then Subsection 4.2 is devoted to SFA-like models including high ordernonlinearities. Concluding remarks are proposed in Section 5.

2. Polarization evolution equation for very short pulses

2.1. Evolution equation on the dipole moment: circularly polarized field

We assume first that the external electromagnetic field is circularly polarized, which satisfies in vacuum,the following equations:

∂tEx′(z′, t) = −c2∂z′By′(z′, t)∂tEy′(z′, t) = +c2∂z′Bx′(z′, t)∂tBx′(z′, t) = +∂z′Ey′(z′, t)∂tBy′(z′, t) = −∂z′Ex′(z′, t)

(3)

A solution to these (Maxwell) equations is

Ex′(z′, t) = E0f(kz′ − ω0t) sin(kz

′ − ω0t)Ey′(z′, t) = −E0f(kz

′ − ω0t) cos(kz′ − ω0t)

Bx′(z′, t) = −B0f(kz′ − ω0t) cos(kz

′ − ω0t)By′(z′, t) = B0f(kz

′ − ω0t) sin(kz′ − ω0t)

where f is the envelope of the initial electromagnetic field, k = ω0/c. In Fig. 1, we illustrate an exampleof a pulse we will consider, where we use atomic units: E0(a.u.) = 5 × 109V · cm−1 corresponding toI = cE2

0/8π = 3.5 × 1016W · cm−2, T (a.u.) = 24 × 10−18s = 24asec, a0 = 0.0529nm, e = ~ = me = 1.Within the ME-TDSE model, interaction with a molecule requires the solution to TDSE:

i∂tψ = −1

2ψ + Vc(x)ψ + x ·Ex′(t)ψ.

with x = (x, y), Ex′ = (Ex′ , Ey′). In velocity gauge [28], this is written equivalently as:

i∂tψ = −1

2ψ + Vc(x)ψ + iAx′(t) · ∇ψ +

‖Ax′(t)‖2

4ψ.

with Ax′ = (Ax′ , Ay′), Ex′ = −∂tAx′ . The computation of the wavefunction ψ of a molecule “located inx′”, allows to deduce the dipole moment d as follows

d(x′, t) =

∫

R2

|ψ(x, t)|2xdxdy

4

0100

200300

400500

600700

−4

−3

−2

−1

0

1

2

3

4

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time in a.u.

Ex in a.u.

Ey in

a.u

.

Figure 1: A circularly polarized pulse, E(x′, y′, t)

Following [29] and [30], we now derive an evolution equation for d. Assuming that the wavefunction ofmolecule m1 “located” in x′

1 = (x′1, y′1, z

′1) is known, a sequence of evolution equations to estimate the dipole

moment of molecule m2 “located” in x′2 = (x′1, y

′1, z

′2) with z

′2 > z′1, can be derived as follows.

Model 1. If |∆z′| := |z′2 − z′1| is small enough, we may assume that a circularly polarized electromag-

netic field interacting with molecule m2 is almost identical (up to a time delay) to the one which moleculem1 is subject to. Note that overall, we obviously do not assume that the electromagnetic field propagatesas in a linear medium (or vacuum) in Maxwell’s equations. In that case, let us define ψi the wavefunctionof molecule mi

i∂tψi = −1

2ψi + Vc(x)ψi + x ·Ei(t)ψi.

and d(x′i, t) the corresponding dipole moment, with i = 1, 2. In addition Ei(t) = E(x′

i, t) denotes the electricfield that molecule mi is subject to. The above assumption mathematically implies: E2(t) = E1

(t−∆z′/vg

)

and as a consequence

d(x′2, t) = d

(x′1, t−

∆z′

vg

)(4)

where vg is the group velocity (c in vacuum). Then the polarization P satisfies for z′ ∈ [z′1, z′2] the following

transport equation

∂tP(x′, t) + vg∂z′P(x′, t) = 0 (5)

with initial data P(x′1, ) = N0d(x

′1, ·). The model is then purely macroscopic (except for the computation of

the initial data for P), see Fig 2. This model is applicable as long as |∆z′| is small enough, or if the molecule

5

density is small enough, that is as long as the effect of the medium on E during the pulse propagation fromx′1 to x′

2 is sufficiently negligible to not be included in the dipole moment calculation of molecule m2. In orderto include medium effects on E in the propagation from x′

1 to x′2 an improvement of the model is necessary.

In particular, this approach will allow to consider larger propagation lengths.

TDSE calculationfor initial data

Macroscopic modeling Macroscopic modeling Macroscopic modeling

TDSE calculation TDSE calculationfor initial data for initial data

z

Figure 2: Macroscopic model with initial data from TDSE

Model 2. We now assume that the effect of the medium is sufficiently strong to make the assumptionE2(t) = E2

(t−∆z′/vg

)inaccurate. In that case, we have to include E2(t)−E1

(t−∆z′/vg

)in the interac-

tion of the field with molecule m2. We define (see also the remark below about nonlinear modeling):

∆E(t) =(∆E

(x)(t),∆E(y)(t)

):= E2(t)−E1

(t−

∆z′

vg

)

Now assume that ψ1 the solution to:

i∂tψ1 = −1

2ψ1 + Vc(x)ψ1 + x · E1(t)ψ1

from which dipole moment d(x′1, t), is obtained and that ii)

c(x)(x′1, t) =

∫R2 x

2x|ψ1(x, t)∣∣2dxdy,

c(y)(x′1, t) =

∫R2 y

2x|ψ1(x, t)∣∣2dxdy,

c(xy)(x′1, t) =

∫R2 xyx|ψ1(x, t)

∣∣2dxdy

(6)

Then, in order to solve:

i∂tψ2 = −1

2ψ2 + Vc(x)ψ2 + x · E2(t)ψ2 (7)

for t ∈ [ta, tb], we solve, using an operator splitting method (from Trotter-Kato’s formula)

i∂tψ2 = −1

2ψ2 + Vc(x)ψ2 + x · E1

(t−

∆z′

vg

)ψ2, t ∈ [ta, t

∗b ],

ψ2(·, ta) = φ0(·)

then

i∂tψ2 = x ·∆E(t)ψ2, t ∈ [ta, tb],

ψ2(·, ta) = ψ2(·, t∗b)

6

where t∗b = tb. Using that

ψ1

(·, ta −

∆z′

vg

)= φ0(·)

The solution to this equation is approximated by:

ψ2(x, tb) ≈ ψ1

(x, tb −

∆z′

vg

)(1− i∆Tx ·∆E(ta)

)

where ∆T = tb − ta. Note that the choice of approximating ∆E at ta, is arbitrary (anytime time t in [ta, tb]would be acceptable, in principle). That is

|ψ2(x, tb)|2 ≈

∣∣∣ψ1

(x, tb −

∆z′

vg

)∣∣∣2(

1 + ∆T 2(x ·∆E(ta)

)2)(8)

So that

d(x′2, tb) ≈ d

(x′1, tb −

∆z′

vg

)+∆T 2

((∆E

(x)(ta))2c(x)

(x′1, tb −

∆z′

vg

)

+(∆E

(y)(ta))2c(y)

(x′1, tb −

∆z′

vg

)+ 2∆E

(x)(ta)∆E(y)(ta)c

(xy)(x′1, tb −

∆z′

vg

))(9)

and from which we can evaluate P(x′2, tb) = N0(x

′2)d(x

′2, tb).

The operator splitting used above induces an error between the approximate ψ(a)2 (·, tb) (computed in (8)),

and exact wavefunction ψ(e)2 (·, tb) solution to

i∂tψ2 = −1

2ψ2 + Vc(x)ψ2 + x · E2(t)ψ2

is of the form O((tb − ta)2x ·∆E(ta)ψ(·, ta)

). As a consequence, we can evaluate the error between the

exact polarization P(e)(x′2, tb)

P(e)(x′2, tb) = N0(x

′2)d

(e)2 (x′, t) = N0(x

′2)

∫

R2

|ψ(e)2 (x, t)|2xdxdy

and approximate polarization P(a)(x′2, tb) computed from (9).

Note that from a practical point of view this approximation is accurate in principle, for (tb− ta)small enough as the splitting error leads to

∣∣∣P(e)(x′2, tb)−P(a)(x′

2, tb)∣∣∣ = O

(N0(x

′2)(tb − ta)

4∣∣∣∆E

∣∣∣2

∞

)

In practice however, tb− ta is large, corresponding to the overall computational time for TDSE.Typically for a Nc-cycle pulse of wavelength λ, as (tb − ta) ≈ Ncλ/vg

∣∣∣P(e)(x′2, tb)−P(a)(x′

2, tb)∣∣∣ = O

(N0(x

′2)[Ncλ

vg

]4∣∣∣∆E

∣∣∣2

∞

)(10)

This makes, in practice, this approximation only relevant for very short pulses, or equivalently

for∣∣∣∆E

∣∣∣∞

small enough.

7

We deduce from this estimate that the polarization computed above is accurate as long asN0(x′2)(∆z/vg

)4∣∣∆E∣∣2∞

is small enough. This means that: for low density medium and/or short enough propagation length, and/orsmall electric field variation approximation P(a)(x′

2, tb) is accurate.

Under the above assumptions, a macroscopic model can then be derived from the above calculus. Weset

Q(x,y,xy)(·, t) := N0(·)c(x,y,xy)(·, t)

and

∂tP(x′, t) + vg∂zP(x′, t) =∫ t

0

[∆E(x′, s′),∆E(x′, s′)

]Qds′

∂tQ(x′, t) + vg∂zQ(x′, t) = 0

(11)

where for Ω ∈M42(R) such that

Ω =

Ω(x) Ω(xy)

Ω(xy) Ω(y)

with Ω(x,y,xy) in M21(R), [·, ·]Ω :M21(R)×M21(R)→M21(R) is defined as follows. For Γ,∆ ∈M21(R):

[Γ,∆

]Ω= ∆xΓxΩ

(x) +∆yΓyΩ(y) +

(∆xΓy +∆yΓx

)Ω(xy)

The initial data (at t = ta) satisfies

P(x′, ta) = P(x′1, ta −

z′ − z′1vg

)

Q(x′, ta) = Q(x′1, ta −

z′ − z′1vg

) (12)

with Q matrix function with values in M42(R)

Q(x′, t) =

Q(x)(x′, t) Q(xy)(x′, t)

Q(xy)(x′, t) Q(y)(x′, t)

and

∆E(x′, t) = E(x′, t)−E(x′, t−

∆z′

vg

)

This set of equations is then coupled to ME’s (only): (11), (12), (1). From x′1 to x′

2, the model is thenpurely macroscopic. Schrodinger’s equation is only solved to determine the initial data of P and Q at x′

1.This approach allows to take into account variations (including linear, nonlinear medium effects as well asdiffraction) of electromagnetic field which may occur during the propagation from x′

1 to x′2. Note that the

RHS of (11), can in particular be interpreted as the polarization change due to diffraction, and mediumnonlinear effects.

The evolution equation for P is then coupled to ME’s (1) for modeling the propagation of the laser pulseover (z′1, z

′2). Polarization at (x′, y′, z′) for z′ > z′2, is then computed again using the TDSE from (2). More

specifically, the domain is decomposed in subdomains in the z′ direction, and only at certain locations ofeach sudomain, the TDSE’s are computed to evaluate the dipole moment. At other points, the macro-scopic evolution equations on P are used. This methodology is summarized in Fig. 3, where the dipole

8

TDSE

TDSE

TDSE

TDSE

TDSE

TDSE

TDSE

TDSE

Z Z

Local propagation of the dipole moment/polarization

a b

Figure 3: Spatial evolution of polarization

moment/polarization is computed from TDSE only for certain z′ (Za, Zb on the figure). Elsewhere, theevolution equation for P, allows a “cheap” (as fully macroscopic) computational evaluation. Computationaldetails can be found in [29], where this strategy is presented in 1-d dimension, and where it is shown thata reduction of almost two orders of magnitude in the computational complexity can be reached. From apractical point of view, the range of application of the presented approach is then limited only to very shortpulses. We present in Appendix A, an extension of the method for Gaussian beams.

Remark 2.1 (Estimation of the group velocity). In our models, to estimate the velocity group vg in agiven medium and for which n0, n2 are approximately known (from χ(1), χ(3)), we use a standard approach.Start from n ≈ n0 + n2|E|2, where n0 (resp. n2) is the linear (resp. second nonlinear) refractive index andng, the group velocity

vg =c

ng(ω)=

c

n(ω) + ω∂n(ω)

∂ω

In first approximation, we can take vg ≈ c/√1 + χ(1), where χ(1) is the instantaneous linear susceptibility.

Note that a precise estimate of the group velocity is in fact not essential (except for Model 1), but allows for areduction of

∣∣∆E∣∣∞

and we can then expect a reduction of the splitting error, in particular when a low-orderoperator splitting is used (Model 2), and then a more accurate modeling of the polarization evolution.

We conclude this subsection by a remark regarding the backward propagation.

Remark 2.2. As we are interested in multidimensional electromagnetic field propagation, it is important tomention that backward propagation should be included in the derivation of the polarization evolution equation.We should then derive a second PDE of the form

∂ttP− v2gP = S(E)

with initial data to determine from quantum TDSE. This is possible extending the arguments developedabove. This time, we will use: P ≈ χ(1)E+ χ(3)E3, where χ(1) and χ(3) are estimates of the first and third

9

instantaneous susceptibilities. That is

E ≈P

χ(1)

1

1 +χ(3)

χ(1)|E|2

=P

χ(1)

(1−

χ(3)

χ(1)|E|2

). (13)

Then substituting E from (13), in the wave equation, and including the current density J:

∂2tE− c2E−∇(∇ · E) = −4π(∂2tP+ ∂tJ)

Note that J satisfies the following evolution equation [2], [21], where νe is the collision frequency:

∂tJ+ νeJ =e2ρ

me

E

Thus, we get a general equation for P, which writes:

∂2tP− v2gP = −

c2χ(3)

χ(1)(1 + 4πχ(1))

[(P|E|2)−

1

c2∂2t (P|E|

2)

]−

4πχ(1)

1 + 4πχ(1)∂tJ (14)

This equation is then coupled to the usual wave equation, and the initial data P(·, 0), ∂tP(·, 0) are computedfrom TDSE’s, following a similar approach as above. Note that (14) is a general wave equation for P fromwhich we can derive for instance (5) or (11).

To illustrate this model, its strengths and its limits, we propose a numerical example which consists of thecomparison of harmonic spectra of an electric field solution to a non-homogeneous 1-d ME’s, coupled withSchrodinger equations. Molecules are supposed to be aligned, and we compare the electric field spectra,when the response (polarization) of the medium is computed from, respectively, 1024, 256, 64, 16, 4 and 2TDSE’s. The physical data are as follows:

• N0 = 1.2× 10−5 mol·(volume unit)−1 in atomic unit.

• number of cycles ≈ 7, and wavelength 800nm.

• the total propagation length is ≈ 30µm, including ≈ 10µm in the gas.

As expected the spectra are quite close, even when the polarization is computed from 4 TDSE’s versus 1024,see Fig. 4. In Fig. 5, we represent in logscale, the L2-norm of the error with respect to the spectrum ofreference (computed from 1024 TDSE’s), as well as the estimation of the CPU/storage gain. Roughly, wecan estimate that dividing by a factor N the number of the TDSE to solve (compared to a full Maxwell-Schrodinger model), allows to reduce by a factor N the overall computational complexity and data storage.

We now propose the same example (same data) except that the medium is 5 times denser that in theprevious example. In that case, we expect that nonlinearities will deteriorate the numerical solution, seeFig. 6. In that case, a good representation of the medium requires more TDSE’s, which increases the overallcomplexity of the simulation. More advanced models should then derived to include nonlinearities.

2.2. Generalization

This section is devoted to a generalization of the methodology which was developed above for circularlypolarized and Gaussian fields (Appendix). We consider the full 3-D Maxwell equations coupled with TDSE’s.We are first interested in computing the dipole moment of molecule mα, located in x′

α at time t. In this goalwe have to evaluate the wavefunction ψα, solution to the following TDSE

i∂tψα(x, t) =(−

1

2+ Vc(x) + x · Eα(t)

)ψα(x, t)

10

2 4 6 8 10 12 14 16

10−4

10−2

100

102

104

Harmonic order (ω/ω0

inte

nsity

in lo

gsca

le

Electric field Harmonics from N TDSE computation

2 TDSE: speed−up by factor 5124 TDSE: speed−up by factor 25616 TDSE: speed−up by factor 6464 TDSE: speed−up by factor 16256 TDSE: speed−up by factor 81024 TDSE: no gain

Figure 4: Electric field harmonic spectrum comparison for 4, 16, 64, 256 and 1024 TDSE’s

0 50 100 150 200 250 30010

−2

10−1

100

101

102

103

Number of TDSE

L2−norm and CPU/Storage gain

L2−norm errorGain factor

Figure 5: CPU/storage gain and L2−norm error, as function of TDSE’s

and deduce the corresponding dipole moment d(x′α, t) = dα(t). We assume that Eα(t) ≈

∑Li=1 αiEi(t− T )

where T is some positive real number such that t − T > 0. This approximation comes typically fromKirchhoff’s formula:

E(x′α, t) =

1

4π

∂

∂t

(t

∫

|ξ|=1

E(x′α + c(t− T )ξ

)dSξ

)+

t

4π

∫

|ξ|=1

∂tE(x′α + c(t− T )ξ

)dSξ

11

2 4 6 8 10 12 14 16 18 2010

−4

10−3

10−2

10−1

100

101

102

103

104

105

Harmonic order (ω/ω0

inte

nsity

in lo

gsca

le

Electric field Harmonics from N TDSE computation

16 TDSE: speed−up by factor 6464 TDSE: speed−up by factor 16256 TDSE: speed−up by factor 81024 TDSE: no gain

Figure 6: Electric field harmonic spectrum comparison, for 4, 16, 64, 256 and 1024 TDSE’s.

We set χ(·, t) :=∑L

j αjφj(·, t− T ) for t > T , where φj is solution to

i∂tφj(x, t) =(−

1

2+ Vc(x) + x · Ej(t)

)φj(x, t)

which are assumed known at time t−T , as well as dj(t−T ). These are wavefunctions of molecules “located”on the sphere of radius c(t− T ) and center x′ according to Kirchhoff’s formula. Thus

i∂tχ(x, t) =(−

1

2+ Vc(x)

)χ(x, t) +

L∑

j=1

αjx ·Ej(t− T )φj(x, t− T )

that we rewrite

i∂tχ(x, t) =(−

1

2+ Vc(x) + x ·Eα(t)

)χ(x, t) + F (x, t)

where

F (x, t) = x ·L∑

j=1

αj

(Ej(t− T )−Eα(t)

)φj(x, t− T )

We denote

i∂tψα(x, t) =(−

1

2+ Vc(x) + x · Eα(t)

)ψα(x, t)

We denote dβ(t) the dipole moment associated to a virtual molecule of Wavefunction χ, that is:

dβ(t) =∫R3 |χ(x, t)|

2xdxdydz =∑L

i,j=1 αiαj

∫R3 φi(x, t− T )φj(x, t− T )xdxdydz

12

Now we introduce d(c)β (t), d

(u)β (t) such that

dβ(t) = d(c)β (t) + d

(u)β (t)

with

d(c)β (t) =

∑Lj=1 α

2jdj(t− T )

d(u)β (t) =

∑Li6=j;i,j=1 αiαj

∫R3 φi(x, t− T )φj(x, t− T )xdxdydz

We also have

ψα(x, t) = χ(x, t) + εα(x, t)

where εα satisfies the equation

i∂tεα(x, t) =(−

1

2+ Vc(x) + x ·Eα(t)

)εα(x, t)− F (x, t)

with null initial condition.

εα(x, t) = −i

∫ t

T

[(−

1

2+ Vc(x) + x ·Eα(s)

)εα(x, s)

]ds+ i

∫ t

T

F (x, s)ds

In the following, we denote

ηj(x, t) := i

∫ t

T

(Ej(s− T )−Eα(s)

)φj(x, s− T )ds

so that∫ t

T

F (x, s)ds =

L∑

j=1

αjx · ηj(x, t)

We now use the approximation

ψα(x, t) ≈ χ(x, t) +L∑

j=1

αjx · ηj(x, t)

and

x(t) =∑L

i,j=1 αiαj

∫R3

(x · ηj(x, t)

)(x · ηi(x, t)

)xdxdydz

−∑L

j=1 αj

∫R3 x ·

(ηj(t)χ(t) + ηj(t)χ(t)

)xdxdydz

which is again justified by the assumption of non-interaction of electrons attached to distinct molecules.

dα(t) = d(c)β (t) + d

(u)β (t) + x(t)

Now we argue that in first approximation x and d(u)β can be neglected. Indeed, the Maxwell-Schrodinger

model is derived assuming that molecules mi and mj for i 6= j, do not interact, that is 〈φi, φj〉 = 〈ηi, ηj〉 = 0.That is

dα(t) ≈ d(c)β (t) =

L∑

j=1

α2jdi(t− T )

This equation leads to an expression of the polarization of the form

P(x′α, t) ≈

L∑

j=1

α2jd(x

′i, t− T )

where the xi lie on a sphere of radius cT and center x′α. From a practical point of view, this relation is of

little interest, as the dipole moment computation in one single location requires computations in several loci.However, from a more theoretical view point, it gives interesting information about the global picture.

13

3. Polarization evolution equation for longer pulses

The next two subsections are dedicated to the derivation of an evolution equation for the polarizationunder the paraxial and slowly varying envelope approximations. We then no more assume that the pulseduration is ultrashort.

3.1. General model under the paraxial approximation

The interaction of the laser field with the medium breaks the possible symmetry of the incoming pulse.We here propose an extension of Subsection 2.1, when paraxial approximation is assumed [31]. We write theelectric field propagation in direction ez′ as

E(x′⊥, z

′, t) = A(x′⊥, z

′, t)ei(kz′z′−ωt)ez′

Say at time ta and in (x′⊥,1, z

′1), the initial polarization P(x′

⊥,1, ta) and its derivative ∂tP(x′⊥,1, ta), will be

computed from a TDSE

i∂tψ1 = −1

2ψ1 + Vc(x)ψ1 + x ·E(t)ψ1

with initial data ψ1(t = ta) = φ. Now we seach for an evolution equation for P starting from (14) but underthe paraxial approximation. In this goal we search for P and J in the form

P(x′⊥, z

′, t) = Π(x′⊥, z

′, t)ei(kz′z′−ωt)ez′ (15)

J(x′⊥, z

′, t) = Λ(x′⊥, z

′, t)ei(kz′z′−ωt)ez′ (16)

where A,Π,Λ are the slowly varying complex amplitudes. We intend to rewrite the wave equation coupledwith (14) under the SVAE. We first get

E =

(⊥A+ ∂2z′A+ 2ikz′∂z′A− k2z′A

)ei(kz′z

′−ωt)ez′

∂2tE =(∂2tA− 2iω∂tA− ω

2A)ei(kz′z

′−ωt)ez′

Now as

∂tJ =(∂tΛ− iωΛ

)ei(kz′z

′−ωt)ez′

the continuity equation can be rewritten

∂tΛ = (iω − νe)Λ +e2ρ

me

A

Thus we also have

∂tJ =(− νeΛ +

e2ρ

me

A)ei(kz′z

′−ωt)ez′

We now rewrite the nonlinearity in (14) (RHS), as follows

(∂2t − c

2)(P|E|2

)=

(∂2t(ei(kz′z

′−ωt)Π|A|2)− c2⊥(Π|A|2)

−c2∂2z′(ei(kz′z′−ωt)Π|A|2)

)ei(kz′z

′−ωt)ez′

14

Now we rewrite

⊥

(Π|A|2

)= |A|2⊥Π+Π⊥|A|2 + 2∇⊥|A|2 · ∇⊥Π

∂2z′

(ei(kz′z

′−ωt)Π|A|2)

= ei(kz′z′−ωt)

(− k2Π|A|2 + 2ik|A|2∂z′Π+ 2ikΠ∂z′ |A|2

+2(∂z′ |A|2

)(∂z′Π

)+Π∂2z′ |A|2 + |A|2∂2z′Π

)

∂2t(ei(kz′z

′−ωt)Π|A|2)

= ei(kz′z′−ωt)

(− ω2Π|A|2 − 2iω|A|2∂tΠ− 2iωΠ∂t|A|2

+2(∂t|A|2

)(∂tΠ

)+Π∂2t |A|

2 + |A|2∂2tΠ)

We now apply the SVEA that is

∣∣∂2z′A∣∣ ≪ kz′ |∂z′A| ≪ k2z′ |A|∣∣∂2z′Π∣∣ ≪ kz′ |∂z′Π| ≪ k2z′ |Π|∣∣∂2tA∣∣ ≪ ω|∂tA| ≪ ω2|A|∣∣∂2tΠ∣∣ ≪ ω|∂tΠ| ≪ ω2|Π|

Then, we get a full model.

∂tA+ c∂z′A =ic

2kz′

⊥A+ 4π

(−∂tΠ+

ikz′c

2Π +

iνe

2ωΛ−

iρ

ωA

)

∂tΠ+c

1 + 4πχ(1)∂z′Π =

ic

2kz′

(1 + 4πχ(1)

)⊥A− 4πχ(1)Π+2iπχ(1)νe(

1 + 4πχ(1))ckz′

Λ−2iπχ(1)ρ(

1 + 4πχ(1))ckz′

A

−2ic

kz′

(1 + 4πχ(1)

)χ(3)

χ(1)

[|A|2⊥Π+Π⊥|A|

2 + 2(∇⊥|A|2) · (∇⊥Π)

]

−χ(3)

χ(1)(1 + 4πχ(1)

) [|A|2(∂tΠ+ c∂z′Π) + Π(∂t|A|2 + c∂z′ |A|2)]

∂tΛ = (iω − νe)Λ + ρA

(17)

The interest of this model is that it gives an accurate description of the polarization envelope (then of theelectric field envelope). Again, the evolution equation on Π is used, only in a localized spatial region, andfrom initial data P(·, ta), ∂tP(·, ta), computed from TDSEs following the same technique as the one describedin Section 2, except that we can now consider much larger propagation distances. Naturally from (17) it ispossible to derive more simple models neglecting certains terms of the RHS.

3.2. Geometrical optics approach (two-dimensional case)

We here discuss an approach based on the geometrical optics techniques which allows to derive a moresimple model than (17). We will follow strategy and notation from [32], in order to reduce TDSE computa-tions. Starting first from E(x′, t) = Re

[A(x′, t)ei(k0SE(x′)−ωt

]ez′ then working in the moving frame z′ ← z′

and t′ ← t′ − z′/vg, allows to get ride of the time dependence in the envelope calculation, with x′ = (x′, z′)

E(x′) = A(x′)eik0SE(x′)ez′

which satisfies the following Helmholtz equation

∇2E(x′) + k2(x′)E(x′) = 0

15

with k2(x′) = k20(1+F (|U(x′)|2)

), and F a function, modeling to the medium response to the electric field.

Note that F null corresponds to a pulse propagation in vacuum. The eikonal and transport equations whichfollow, by identifying terms in k0 and k20 , are

∇SE(x

′) · ∇SE(x′) = 1 + F

(A2(x′)

),

2∇SE(x′) · ∇A(x′) +A(x′)∇2SE(x

′) = 0

Then, under the paraxial approximation, the envelope can be rewritten U(x′) = A(x′)eik0SU (x′), assuming

k0 ≫ 1 and SU = SE − z′ where SU (x′) = F

(|U0|2

)z′. Ray trajectories are given by

(x′(σ), σ

), σ ∈ R

,

where

dx′

dσ(σ) = ∇SU

(x′(σ)

)

According again to [32], we can assume z′ ≈ σ (high power beam) so that we can parameterize the rays inz′ (direction of propagation of the pulse).

The main idea is now to use rays in order to reduce the TDSE computations.

dS

S=constant

Ray

dS

S=constant

U at z’ = 0

U at z’ > 0

Figure 7: Ray path

1. The starting point is to model SU (x′) = z′ + F

(|U(x′)|2

), where F is a medium dependent function,

modeling the Kerr effect (self-focusing), such that:

F (|U |2) = χ(3)|U |2

The susceptibility χ(3) is medium, as well as time and space dependent. In first approximation χ(3) canbe taken constant. However, it is possible to more precisely determine its value via TDSE computation[3]. We can rewrite Fα, such that

Fα(|U |2) = χ(3)

α |U |2

where χ(3)α is computed from a TDSE for a molecule “located” at (x′α, 0). In vacuum, F is set to zero.

2. Determine the ray trajectories assuming that z′ ≈ σ:

dx′

dz′(z′) = −

1

2∂x′SU

(x′(z′), z′

)(18)

with x′(0) = x′0 given.

16

3. Determine A along the rays, from the transport equation

dA

dz′(x′(z′)

)= −

1

2A(x′(z′)

)∂x′SU

(x′(z′), z′

)(19)

From there, U is deduced along the rays: U(x′(z′), z′

)= A

(x′(z′), z′

)eik0SU

(x′(z′),z′

). In practice, the

electric field will be computed from ME. However, this information is relevant from a practical pointof view, in order to estimate the polarization.

4. We assume that for a molecule “located” at x′α, there exists a trajectory

x′α(σ), σ > 0

, passing

through that point. Except in vacuum, U(x′) = U(x′α(σ)

)6= U

(x′α, 0)

). In the moving frame, the

electric field the molecule is subject to, is identical to the one applied to a molecule “located” at (0, z′β),

with z′β 6 z′α. More specifically, there exists a level set denoted by Cα(σ), of normal vector ∇SU

(C(σ)

)

and passing though x′α, which intersects the line x′ = 0, at say z′β. Along this curve, the electric field

is constant, and in particular: E(x′α(σ)

)= E

(0, z′β

)as well as P

(x′α(σ)

)= P

(0, z′β

). We may assume

that P(0, z′β

)was evaluated from a direct TDSE computation. From this remark, we can construct a

continuous equation, modeling the time evolution of the polarization. As along Cα(σ), the polarizationis constant, we obviously get:

∂σ

P(C(σ)

)= 0

By assumption z′ ≈ σ, so that

∂z′

P(C(z′)

)=

dC(z′)

dz′· ∇P

(C(z′)

)= 0

or equivalently in the moving frame

∂z′P(C(z′)

)+dCx′(z′)

dz′∂x′P

(C(z′)

)= 0

In the fixed frame we get

∂tP(C(z′), t

)+ vg∂z′P

(C(z′), t

)+ vg

dCx′(z′)

dz′∂x′P

(C(z′), t

)= 0 (20)

The equation models, along the level sets, the evolution of the polarization, taken into account, thepropagation and nonlinear effects. Diffraction effects are here assumed negligible.

From the above analysis we can then determine the rays, x′(σ), as well as U(x′(σ)

)and SU

(x′(σ)

)(U and

SU along the rays), starting from any (x′

⊥, 0). We denote by x′α(σ) the ray:

dx′α

dσ(σ) = ∇SU

(x′(σ)

), x′

α(0) = (x′⊥,α, 0)

At for any x′ such that x′ = x′α(σ), a molecule located at x′, will be subject to the field

E(x′, t) = U(x′α(σ), t

)cos(ωt− k0SU

(x′α(σ)

))ez′

For (0, z′β), such that U(x′α(σ), t

)= U

((0, z′β), t

), then

E(x′, t) = U((z

′

β , 0))cos(ωt− k0SU

((0, z′β)

))ez′

From a practical view point, it is then necessary to evaluate the level sets C(σ). in order to solve (20). Inconclusion, geometrical optics is not used here to directly update the electric field, but (only) to determinethe nonlinear response of the medium in the wave equation, without a direct TDSE computation. Note thatin vacuum, F = 0, dA

(x′(σ)

)/dσ ≈ 0, that is A is constant along the ray (= A

(x′(0)

)) and only the phase

SU

(x′(σ)

)evolves.

17

3.3. Polarization reconstruction

Polarization is often deduced from TDSE in the hypothesis of a unique pure state, that is at (x′, t),polarization is given by

P(x′, t) = N0(x′)d(x′, t)

where d(x′, t) is the dipole moment of a molecule “located” in x′. A natural extension to p pure statesconsists of setting

P(x′, t) = N0(x′)

p∑

l=1

d(l)(x′, t)

where d(l)(x′, t) = 〈ψ(l)|x|ψ(l)〉 and ψ(l) is solution to

i∂tψ(x, t) =(H0 + x · Eα(t)

)ψ(x, t), ψ(x, 0) = φl(x).

with H0φl = εlφl. From a practical point of view, solving this p TDSE’s can easily be implemented inparallel (multi-threading for instance) as each TDSE computation is done independently.

4. SFA nonlinear optics models

The method which is developed in this section consists of coupling bound and free states in the solving ofTDSE, in order to determine molecule dipole moments. More specifically, the overall strategy is to determinethe bound state contribution combining the usual perturbative approach for weak fields, and a Lewenstein’sSFA approach for the free state contribution, as well as the bound-continuous state interactions; howeverwe do not limit the interaction to free states of the continuum with the ground state like in [24]. In fine, wedetermine an explicit approximate solution to the TDSE, and of the dipole moment allowing to accuratelymodel the polarization in Maxwell’s equations. We refer mainly to [25] for the notation and derivation ofperturbative nonlinear optics modeling and [24] for the SFA model. Before presenting this model, we firstderive a general Liouville equation, Subsection 4.1, including bound and continuous states, from which, inprinciple we could derived macroscopic polarization. However, due to the complexity of the derived model,it is expected of poor interest from a practical point of view. 1

4.1. General non-perturbative approach. How far can we go ?

In this subsection, we consider the general situation:

i∂tψ =(H0 + x · E(t)

)ψ, ψ(x, 0) = φ0

We consider the case a unique pure state. We search for a wavefunction in the general form:

ψ(x, t) =∑

n

cn(t)φn(x) +

∫

σc

c(t, η)φc(x, η)ρc(η)dη (21)

where i) σc denotes the continuous spectrum of H0, ii) ρc denotes the spectral density in σc, iii) φc satisfiesH0φc(x, λ) = λφc(x, λ) (where “‖φc(·, λ)‖L2 = +∞”, and finally iv) c(t, λ) := 〈φ(·, λ)|ψ(·, t)〉. Assumingthat the full spectrum (including eigenelements) is known, we have to determine

(cn(t)

)n,(c(t, λ)

)λas well

as ρ(λ). In the following, the discrete spectrum elements of H0 are denoted εn. For any n ∈ N:

i〈∂tψ|φn〉 = 〈(H0 + x · E(t)

)ψ|φn〉

and for any λ ∈ σc

i〈∂tψ|φc(·, λ)〉 = 〈(H0 + x · E(t)

)ψ|φc(·, λ)〉

18

We set

Hnm = E(t) · 〈xφn|φm〉

Hn(λ) = E(t) · 〈xφn|φc(·, λ)〉

Kn(λ) = 〈E(t) · 〈xφc(·, λ)|φn〉

H(λ, µ) = E(t) · 〈xφc(·, λ)|φc(·, µ)〉

D(λ, µ) = 〈φc(·, λ)|φc(·, µ)〉

and

ρmn(t) = c∗m(t)cn(t), ρmλ(t) = c∗m(t)c(t, λ), ρλm(t) = c∗(t, λ)cm, ρλµ(t) = c∗(t, λ)c(t, µ)

We prove in Appendix B, that for all m, n, λ, µ:

ρmn(t) = i∑

ν Hmνρνn(t)− ρmν(t)Hνn + i∫σc

(ρη(t)K

∗m(η)ρ∗c(η)− ρmη(t)Kn(η)ρ(η)

)dη

ρmλ(t) = i∑

n

[ρnλ(t)H

∗nm − ρmn(t)Hn(λ)

]

+i∫σc

[K∗

n(η)ρ∗c(η)ρηλ(t)−H(η, λ)ρc(η)ρmη(t)

]dη

ρλµ(t) = i∑

n

[ρnµ(t)H

∗n(λ) − ρλn(t)Hn(µ)

]

+i∫σc

[H∗(η, λ)ρ∗c(η)ρηµ(t)−H(η, µ)ρc(η)ρλη(t)

]dη

Remark 4.1. This system is a Liouville-like equation which is comparable to the usual one, which does notinclude the continuum of the Hamiltonian spectrum and which writes

ρmn = −i

~

[H, ρ

]mn

from which is deduced Tr(ρx) =∑

nm ρnmxmn in the classical theory.

The above equation contains, in principle, all the state transitions. However, from a practical point of view,it is of moderate interest, due to its complexity. The following section is devoted to an analogue approach,except that the integral over the continuum in (21), is approximated by SFA.

4.2. SFA-like nonlinear optics model: unique continuous state

The principle of the following model is to include perturbative and nonperturbative contribution, usingin particular on Lewenstein’s SFA model, which has the ability to accurately capture phenomena such asionization and high order harmonic generation. We search for a wavefunction ψ solution to

i∂tψ =(H0 + V (x, t)

)ψ, ψ(x, 0) = φ0(x)

whereH0 denotes the laser-free Hamiltonian and V the electric potential, as the sum of bound and bound-freestate contributions:

ψ(x, t) = ψB(x, t) + ψL(x, t)

and where the purely bound part is of the form

ψB(x, t) =∑

l∈N

λlψ(l)B (x, t)

19

with

ψ(l)B (x, t) =

∑

k∈N

a(l)k (t)φk(x)e

−iωkt

where we have denoted φk the eigenvectors of the field-free Hamiltonian: H0φk = εkφk and ωk = ~εk. Weassume that

ψ(0)B (x, t) =

(1− α(t)eiIpt

)φ0(x)e

−iω0t

where α(t) is a time-dependent complex. The bound-continuous part is of the form of a SFA:

ψL(x, t) = eiIpt(α(t)φ0(x) +

∫d3pb(p, t)eip·x

)

with Ip the ionization potential and where ψL(x, 0) = α0φ0(x) and b(p, 0) = 0. A natural choice for α isthen the depletion coefficient. According to [24], uncoupling the equation in α and the TDSE, α can bemodeled by the following equation

α(t) = −γ(t)α(t), α(0) = α0 ∈ (0, 1)

where γ is defined by (52) in [24] and is naturally maximal at the electric field peaks. In practice, and asproposed in that paper γ will be approximated by its average γ ∈ C with positive real part, so that

α(t) ≈ α0e−γt

Now, we set

δ := γ − iIp

so that

eiIptα(t) ≈ α0e−δt (22)

The unknowns of the problem are (al)l>1 and b. Now by linearity, we have

i∂tψB,L =(H0 + V (x, t)

)ψB,L

Parameter α is chosen such that initially

ψ(x, 0) = ψB(x, 0) + ψL(x, 0) = (1− α0)φ0(x) + α0φ0(x) = φ0(x)

Naturally, we first have:

ψ(0)B (x, t) ≈

(1− α0e

−δt)φ0(x)e

−iω0t (23)

Coefficients (al)l>1 will be searched following the usual perturbative method as presented in [25]. Denotingωl = ~εl, for all l

a(N)m (t) =

1

i~

∑

l∈N

∫ t

−∞

Vml(t′)a

(N−1)l (t′)eiωmlt

′

dt′

where

Vml(t) = 〈φm|x · E(t)|φl〉 =

∫x ·E(t)φ∗mφld

3x

and the transition frequencies are denoted ωmg = ωm − ωg. We deduce that for E(t) =∑

p∈NE(ωp)e

−iωpt,and using (22)

a(1)m (t) ≈1

~

∑

p∈N

µmg · E(ωp) 1

ωmg − ωp

−α(t)eiIpt

ωmg − ωp + iδ

ei(ωmg−ωp)t

then a(2)m , etc, see [25] for details.

20

Remark 4.2 (Important remark on the contribution of χ(l) in SFA’s Lewenstein.). As it is well-known, SFA’s Lewenstein model contributes to χ(l). Then, it is important to separate the contribution ofthe susceptibility tensors from the bound states and from SFA’s Lewenstein function. In this goal, we willsum (an)n only over the bound states, omitting the contribution from ”bound-continuum” from the usualperturbative approach, as this contribution is already present in SFA. We denote the corresponding indicesby B (sum over the bound states only).

For instance

a(2)n (t) ≈

1

~2

∑(p,q)∈N2

∑m∈B

(µmg · E(ωp)

)(µnm · E(ωq)

)

× 1

(ωng − ωp − ωq)(ωmg − ωp)−

α(t)eiIpt

(ωmg − ωp + iδ)(ωng − ωp − ωq + iδ)

ei(ωng−ωp−ωq)t

and similarly

a(3)ν (t) ≈

1

~3

∑(p,q,r)∈N3

∑(m,n)∈B2

(µνn ·E(ωr)

)(µnm · E(ωq)

)(µmg · E(ωp)

)

× 1

(ωνg − ωp − ωq − ωr)(ωng − ωp − ωq)(ωmg − ωp)

−α(t)eiIpt

(ωνg − ωp − ωq − ωr + iδ)(ωng − ωp − ωq + iδ)(ωmg − ωp + iδ)

ei(ωng−ωp−ωq−ωr)t

In the above formula, if m > 1 (recall that in atomic unit the electronic charge is e = 1):

µmg = −

∫φmxφ0d

3x

We deduce:

ψ(1)B (x, t) =

∑m∈B

a(1)m φm(x)e−iωmt

=∑

m∈B

∑p∈N

(µmg ·E(ωp)

)[ 1

ωmg − ωp

−α(t)eiIpt

ωmg − ωp + iδ

]φm(x)e−i(ω0+ωp)t

ψ(2)B (x, t) =

∑(m,n)∈B2

∑(p,q)∈N2

(µnm ·E(ωq)

)(µmg ·E(ωp)

)[ 1

(ωmg − ωp)(ωng − ωp − ωq)

−α(t)eiIpt


]φn(x)e

−i(ω0+ωp+ωq)t

ψ(3)B (x, t) =

∑(m,n,ν)∈B3

∑(p,q,r)∈N3

(µνn · E(ωr)

)(µnm ·E(ωq)

)(µmg ·E(ωp)

)

×

[1

(ωmg − ωp)(ωng − ωp − ωq)(ωνg − ωp − ωq − ωr)

−α(t)eiIpt

(ωmg − ωp + iδ)(ωng − ωp − ωq + iδ)(ωνg − ωp − ωq − ωr + iδ)

]φν(x)e

−i(ω0+ωp+ωq+ωr)t

Now regarding the continuum, we have

∂tb(p, t) = −i(p2

2+ Ip

)b(p, t)−E(t) · ∇pb(p, t)− iα(t)E(t)dg(p)

21

where dg(p) = 〈φ0|x|p〉. We can then show (see [24], and method of characteristics) that

b(p, t) = −i

∫ t

0

dt′α(t′)E(t) · dg

(p−A(t) −A(t′)

)exp

(− i

∫ t

t′dt′′(p−A(t)−A(t′′)

)2/2 + Ip

)

From there, we can give an approximate expansion of the overall dipole moment defined as

d(t) = 〈ψ|µ|ψ〉

using that

d(t) = 〈ψB|µ|ψB〉+ 〈ψL|µ|ψL〉+ 〈ψB|µ|ψL〉+ 〈ψL|µ|ψB〉

= dBB(t) + dLL(t) + dBL(t)

with

dBB(t) =∑

l∈N

λld(l)BB(t) (24)

In order to simplify the presentation, we will assume that damping phenomena are not incorporated in themodel, which allows to deal with real transition frequencies. Extension to complex transitions is possiblefollowing [25]. We then get

d(0)BB(t) = 0 (25)

then

d(1)BB(t) = 〈ψ

(0)B |µ|ψ

(1)B 〉+ 〈ψ

(1)B |µ|ψ

(0)B 〉 (26)

leading, from E∗(ωp) = −E(ωp) to

d(1)BB(t) =

1

~

∑m∈B

∑p∈N

µgm

(µmg · E(ωp)

)(1− α∗(t)e−iIpt

ωmg − ωp

−α(t)eiIpt

(1− α∗(t)e−iIpt

)

ωmg − ωp + iδ

)e−iωpt

+(µmgE(ωp)

)∗µmg

(1− α(t)eiIptωmg − ωp

−α∗(t)e−iIpt

(1− α(t)eiIpt

)

ωmg − ωp − iδ∗

)eiωpt

(27)

This can be rewritten

d(1)BB(t) =

1

~

∑m∈B

∑p∈N

µgm

(µmg · E(ωp)


ωmg − ωp

−α(t)eiIpt


)

ωmg − ωp + iδ

)

+(µgm ·E(ωp)

)µmg

(1− α(t)eiIptωmg + ωp

−α∗(t)e−iIpt

(1− α(t)eiIpt

)

ωmg + ωp − iδ∗

)e−iωpt

(28)

Similarly

d(2)BB(t) = 〈ψ

(0)B |µ|ψ

(2)B 〉+ 〈ψ

(2)B |µ|ψ

(0)B 〉+ 〈ψ

(1)B |µ|ψ

(1)B 〉 (29)

From above, we deduce that

〈ψ(0)B |µ|ψ

(2)B 〉 =

1

~2

∑(p,q)∈N2

∑(m,n)∈B2 µgn(µnm · E(ωq))(µmg ·E(ωp))

×

1− α∗(t)e−iIpt



)α(t)eiIpt


e−i(ωp+ωq)t

22

and

〈ψ(1)B |µ|ψ

(1)B 〉 =

1

~2

∑(p,q)∈N2

∑(m,n)∈B2

(µng ·E(ωq)

)∗µnm

(µmg · E(ωq)

)

×( 1

ωmg − ωp

−α(t)eiIpt

ωmg − ωp + iδ

)( 1

ωng − ωq

−α∗(t)e−iIpt

ωng − ωq − iδ∗

)e−i(ωp−ωq)t

So that:

d(2)BB(t) =

2

~2

∑(p,q)∈N2

∑(m,n)∈B2 Re

µgn

(µnm · E(ωq)

)(µmg · E(ωp)

)

×( 1− α∗(t)



)α(t)eiIpt


)e−i(ωp+ωq)t

+1

~2

∑(p,q)∈N2

∑(m,n)B2

(µng · E(ωq)

)∗µnm

(µmg ·E(ωq)

)

×( 1

ωmg − ωp

−α(t)eiIpt

ωmg − ωp + iδ

)( 1

ωng − ωq

−α∗(t)e−iIpt

ωng − ωq − iδ∗

)e−i(ωp−ωq)t

(30)

or written similarly

d(2)BB(t) =

1

~2

∑(m,n)∈B2

∑(p,q)∈N2

µgn


)(µnm · E(ωq)

)(µmg · E(ωp)

)

×1

(ωmg − ωp)(ωng − ωp − ωq)−

α(t)eiIpt


+µnm

(µgn · E(ωq)

)(µmg · E(ωp)

)( 1

ωng + ωq

−α∗(t)e−iIpt

ωng + ωq − iδ∗

)(1

ωmg − ωp

−α(t)eiIpt

ωmg − ωp + iδ

)

+µng

(1− α(t)eiIpt

)(µmn ·E(ωq)

)(µgm ·E(ωp)

)

×1

(ωmg + ωp)(ωng + ωp + ωq)−

α∗(t)e−iIpt

(ωmg + ωp − iδ∗)(ωng + ωp + ωq − iδ∗)

e−i(ωp+ωq)t

Similarly, one can deduce d(3)BB , from

d(3)BB(t) = 〈ψ

(0)B |µ|ψ

(3)B 〉+ 〈ψ

(3)B |µ|ψ

(0)B 〉+ 〈ψ

(1)B |µ|ψ

(2)B 〉+ 〈ψ

(2)B |µ|ψ

(1)B 〉 (31)

where

〈ψ(0)|µ|ψ(3)〉 =∑

(m,n,ν)∈B3

∑(p,q,r)∈N3 µgν


)

×(µνn.E(ωr)

)(µnm · E(ωq)

)(µmg · E(ωp)

)

×( 1


−α(t)eiIpt


)e−i(ωq+ωp+ωr)t

23

and

〈ψ(1)|µ|ψ(2)〉 =∑

(m,n,ν)∈B3

∑(p,q,r)∈N3 µmn

(µgν · E(ωr)

)(µnm ·E(ωq)

)(µmg ·E(ωp)

)

×

(1

ωνg + ωr

−α∗(t)e−iIpt

ωνg + ωr − iδ∗

)(1


−α(t)eiIpt


)e−i(ωp+ωq+ωr)t

and

〈ψ(2)|µ|ψ(1)〉 =∑

(m,n,ν)∈B3

∑(p,q,r)∈N3 µnm

(µνg · E(ωr)

)(µmn ·E(ωq)

)(µgm ·E(ωp)

)

×

(1

ωνg − ωr

−α(t)eiIpt

ωνg − ωr + iδ

)(1

(ωmg + ωp)(ωng + ωp + ωq)

−α∗(t)e−iIpt


)e−i(ωp+ωq+ωr)t

and

〈ψ(3)|µ|ψ(0)〉 =∑

(m,n,ν)∈B3

∑(p,q,r)∈N3 µνg

(1− α(t)eiIpt

)(µnν .E(ωr)

)(µmn ·E(ωq)

)(µgm ·E(ωp)

)

×

1

(ωmg + ωp)(ωng + ωp + ωq)(ωνg + ωp + ωq + ωr)

−α∗(t)e−iIpt

(ωmg + ωp − iδ∗)(ωng + ωp + ωq − iδ∗)(ωνg + ωp + ωq + ωr − iδ∗)

e−i(ωq+ωp+ωr)t

Now from [24] and denoting dg(t) = 〈p|x|φ0〉, we have

dLL(t) = i

∫ t

0

∫dτd3vE(t− τ) · dg

(v −A(t− τ)

)µ∗g

((v −A(t)

)e−iS(v,t,τ)−δ∗t−δ(t−τ) (32)

where

S(v, t, τ) :=

∫ t

τ

ds[ (v−A(s))2

2+ Ip

]

Naturally (29) and (32) are trivially deduced from the usual theory. We have now to evaluate

dBL(t) = 〈ψB |µ|ψL〉+ 〈ψL|µ|ψB〉

In this goal, we will follow a strategy close the one developed by [24]. That is:

dBL(t) =∑

l∈N

λld(l)BL(t) (33)

with

d(l)BL(t) = 2Re

〈ψ

(l)B (x, t)|µ|ψL(x, t)〉

(34)

where

〈ψ(l)B (x, t)|µ|ψL(x, t)〉 =

∑

m∈B

∫

R3

d3x(a(l)m (t)

)∗φm(x)eiωmtµeiIpt

(α(t)φ0(x) +

∫

R3

d3pb(p, t)eip.x)

24

and

d(l)BL(t) = 2Re

∑

m∈B

((a(l)m (t)

)∗ei(ωm+Ip)tµmgα(t) +

∫

R3

d3p(a(l)m (t)

)∗b(p, t)ei(ωm+Ip)tdm(p)

)(35)

where for all l > 1

dl(p) = 〈φl|µ|p〉

and

b(p, t) = −i

∫ t

0

E cos(τ) · d(p(t) +A(t)−A(τ)

)exp(−

∫ t

τ

i((p(t) +A(t)−A(s))2

2+ Ip

)ds)dτ (36)

with for l > 1, dl(p) which is negligible. Indeed, as mentioned before in SFA model an implicit assumptionis that the matrix elements of the Hamiltonian between bound states (except for the ground state) and freestates are negligible. It is however possible to estimate these contributions within our model. Moreover, asecond model Subsection 4.3, will be developed to go beyond this assumption. Now for l = 0

d(0)BL(t) = −i

∫

R3

d3v

∫ t

0

dτ(1− α0e

−δ∗t)E(τ) · d∗

g

(v−A(t)

)dg

(v−A(τ)

)eiS(v,t,τ) (37)

In addition, for the l > 1 we have

d(l)BL(t) =

∑

m∈B

(a(l)m (t)

)∗ei(ωm+Ip)tdmgα(t) (38)

We can now estimate the susceptibility tensors from bound-free state contribution. More specifically, forl = 1, we have

d(1)BL(t) =

∑m∈B

∑p∈N

[(µgm ·E(ωp)

)( 1

(ωmg + ωp)−

α∗(t)e−iIpt

(ωmg + ωp − iδ∗)

)µmgα(t)

+(µmg · E(ωp)

)( 1

(ωmg − ωp)−

α(t)eiIpt

(ωmg − ωp + iδ)

)µgmα(t)

+∫R3 d

3p (µgm ·E(ωp))( 1

(ωmg + ωp)−

α∗(t)e−iIpt


)b(p, t)dm(p)

+∫R3 d

3p(µmg ·E(ωp)

)( 1

(ωmg − ωp)−

α(t)eiIpt


)b∗(p, t)d∗

m(p)

]e−iωpt

(39)

For the 2nd order dipole moment we have

d(2)BL(t) =

∑(m,n)∈B2

∑(p,q)∈N2

[(µmn · E(ωq)

)(µgm · E(ωp)

)[ 1


−α∗(t)e−iIpt


]µngα(t)

+(µnm · E(ωq)

)(µmg · E(ωp)

)[ 1


−α(t)eiIpt


]µ0nα

∗(t)

+∫R3 d

3p(µmn ·E(ωq)

)(µgm ·E(ωp)

)[ 1


−α∗(t)e−iIpt


]b(p, t)dn(p)

+∫R3 d

3p(µnm ·E(ωq)

)(µmg ·E(ωp)

)[ 1


−α(t)eiIpt


]b∗(p, t)d∗

n(p)

]e−i(ωp+ωq)t

(40)

25

Finally for the 3rd order bound-free dipole moment we have

d(3)BL(t) =

∑(m,n,ν)∈B3

∑(p,q,r)∈N3

[(µnσ.E(ωr)

)(µmn ·E(ωq)

)(µgm ·E(ωp)

)

×

(1


−α∗(t)e−iIpt


)µνgα(t)

+(µσn ·E(ωr)

)(µnm · E(ωq)

)(µmg · E(ωp)

)

×

(1


−α(t)eiIpt


)µgνα(t)

+∫R3 d

3p(µnσ.E(ωr)

)(µmn · E(ωq)

)(µgm · E(ωp)

)

×

(1


−α∗(t)e−iIpt


)b(p, t)dν(p)

+∫R3 d

3p(µσn · E(ωr)

)(µnm · E(ωq)

)(µmg · E(ωp)

)

×

(1


−α(t)eiIpt


)b∗(p, t)d∗

ν(p)]e−i(ωp+ωq+ωr)t

(41)

We then get

P(x′, t) = PBB(x′, t) +PBL(x

′, t) +PLL(x′, t) (42)

with

PBB(x′, t) = NdBB(t), PBL(x

′, t) = NdBL(t), PLL(x′, t) = NdLL(t)

where dBB(t), dBL(t) and dLL(t) are approximated from (24), (27), (44) then (32) and finally (34), (40),(41). To conclude we take λ = 1. In fine, P has computed in (42) is used in (1).

Remark 4.3. The contribution dBL is estimated using [24], where transitions from “continuum-continuum”

and “continuum-excited states” are neglected. As a consequence, the contribution d(l)BL, for l > 1 may be, in

principle, neglected as by construction of SFA model, these transitions were neglected. In the next section,we will show how to improve the modeling, including contribution more “continuum-bound state” transitions.

26

That is

PBB(x′, t) ≈

N

~

∑m∈B

∑p∈N

µmg ·

(µmg · E(ωp)


ωmg − ωp

−α(t)


)

ωmg − ωp + iδ

)

+(µgm · E(ωp)

)µmg

( 1− α(t)

ωmg + ωp

−α∗(t)e−iIpt

(1− α(t)eiIpt

)


)e−iωpt

+2N

~2

∑(p,q)∈N2

∑(m,n)∈B2 Re

µgn

(µnm · E(ωq)

)(µmg ·E(ωp)

)

×( 1− α∗(t)e−iIpt


(1− α∗(t)

)α(t)eiIpt


)e−i(ωp+ωq)t

+N

~

∑(p,m)∈N×B

µmg

(µmg ·E(ωp)

)( 1− α∗(t)

ωmg − ωp

−α(t)eiIpt


)

ωmg − ωp + iδ

)

+(µgm · E(ωp)

)· µmg

( 1− α(t)

ωmg + ωp

−α∗(t)e−iIpt

(1− α(t)eiIpt

)


)e−iωpt

PLL(x′, t) ≈ iN∫ t

0

∫dτd3pE(t− τ) · dg

(p−A(t− τ)

)d∗g

((p−A(t)

)e−iS(p,t,τ)−δ∗t−δ(t−τ)

PBL(x′, t) ≈ −2NRe

i∫R3 d3v

∫ t

0dτ(1− α0e

−δ∗t)E)(τ) · d∗

g

(v−A(t)

)dg

(v−A(τ)

)eiS(v,t,τ)

Now a bit more detailed analysis is necessary to estimate the susceptibility tensors.

First, the linear susceptibility χ(1)BB, is given through P

(1)BB,i =

∑j χ

(1)ij,BBEj(ωp) by

χ(1)BB,ij(t) =

N

~

∑

m∈B

µimgµ

jmg

( 1− α∗(t)

ωmg − ωp

−α(t)

(1− α∗(t)

)

ωmg − ωp + iδ

)+ µj

gmµimg

( 1− α(t)

ωmg + ωp

−α∗(t)

(1− α(t)

)


)

Assuming that α = 0, the linear susceptibility is the same usual (from perturbation theory).Next, using again [25]’s notations, and

P(2)BB,i =

∑

jk

∑

pq

χ(2)BB,ijk

(ωp + ωq, ωq, ωp

)Ej(ωq)Ek(ωp)

the second-order susceptibility is given by

χ(2)BB,ijk(ωp + ωq, ωq, ωp) =

N

~2PI

∑(m,n)∈B2

(µignµ

jnmµ

kmg

×( (

1− α∗(t)e−iIpt)


α(t)eiIpt(1− α∗(t)e−iIpt

)


)

+µjgnµ

inmµ

kmg

(1

ωng + ωq

−α∗(t)e−iIpt

ωng + ωq − iδ∗

)(1

ωmg − ωp

−α(t)eiIpt

ωmg − ωp + iδ

)

+µjgnµ

knmµ

img

×( (

1− α(t)eiIpt)


α∗(t)e−iIpt(1− α(t)eiIpt

)


))

27

where PI the intrindic permutation operator.

Finally the expression of the third-order susceptibility χ(3)BB,kjih can be deduced from above calculation

through:

P(3)BB,k(ωp + ωq + ωr) =

∑

hij

∑

pqr

χ(3)BB,kjih

(ωp + ωq + ωr, ωr, ωq, ωp

)Ej(ωr)Ei(ωq)Eh(ωp)

by

χ(3)BB,kjih(ωp + ωq + ωr, ωr, ωq, ωp) =

N

~3PI

∑(m,n,ν)∈B3

(µkgνµ

jνnµ

inmµ

hmg

×( (

1− α∗(t)e−iIpt)


−


)α(t)eiIpt


)

+µjgνµ

kνnµ

inmµ

hmg

(1

ωνg + ωr

−α∗(t)e−iIpt

ωνg + ωr − iδ∗

)

×

(1


α(t)eiIpt


)

+µjgνµ

iνnµ

knmµ

hmg

(1

ωνg + ωr

−α(t)eiIpt

ωνg − ωr + iδ

)

×

(1


α∗(t)e−iIpt


)

+µkgνµ

jνnµ

inmµ

hmg

×

( (1− α(t)eiIpt

)


−α∗(t)e−iIpt

(1− α(t)eiIpt

)


))

Now regarding the bound-free susceptibility tensors we have:

χ(1)BL,ij(ωp) =

N

ε0

∑

m∈B

∑

p∈N

[µjgmµ

img

(1

(ωmg + ωp)−

α∗(t)e−iIpt


)α(t)

+µjmgµ

igm

(1

(ωmg − ωp)−

α(t)eiIpt


)α∗(t)

+

∫

R3

d3p µjgmd

im(p)

(1

(ωmg + ωp)−

α∗(t)e−iIpt


)b(p, t)

+

∫

R3

d3p µjmg(d

im(p))∗

(1

(ωmg − ωp)−

α(t)eiIpt


)b∗(p, t)

]

28

then

χ(2)BL,ijk(ωr, ωq, ωp) =

∑

(m,n)∈B2

[µjmnµ

kgmµ

ing

[1

(ωmg + ωp)

1

(ωng + ωp + ωq)−

1


α∗(t)e−iIpt

(ωng + ωp + ωq − iδ∗)

]α(t)

+µjnmµ

kmgµ

i0n

[1

(ωmg − ωp)

1

(ωng − ωp − ωq)−

α(t)eiIpt


]α∗(t)

+

∫

R3

d3p µjmnµ

kgmd

in(p)

[1


−α∗(t)e−iIpt


]b(p, t)

+

∫

R3

d3p µjnmµ

kmg(d

in)

∗(p)

[1


−α(t)eiIpt


]b∗(p, t)

]

and

χ(3)BL,kjih(ωσ, ωr, ωq, ωp) =

∑

(m,n,ν)∈B3

∑

(p,q,r)∈N3

[µjnσµ

imnµ

hgmµ

kνg

(1


−α∗(t)e−iIpt


)α(t)

+µjσnµ

inmµ

hmgµ

kgν

(1


−α(t)eiIpt


)α(t)

+

∫

R3

d3p µjnσµ

imnµ

hgmd

kν(p)

(1


−1


1

(ωng + ωp + ωq − iδ∗)

α∗(t)e−iIpt

(ωνg + ωp + ωq + ωr − iδ∗)

)b(p, t)

+

∫

R3

d3p µjσnµ

inmµ

hmg(d

kν)

∗(p)×

(1


−α(t)eiIpt


)b∗(p, t)

]

and where ωσ = ωp + ωq + ωr.

Interpretation and simplification of these tensors, such as the third harmonic generation from χ(3)(3ω),will be analyzed is a forthcoming paper.

4.3. SFA-like nonlinear optics model: possible extension to multiple continuous states

The use of the SFA modeling implies (implicitly) that some transitions are (by construction) neglected:“continuum-continuum” and “continuum-excited states” transitions [24]. In the following section, we proposeto formally include more properly and accurately some of these transitions, which will allow us to modelmore accurately dBL and dLL. In this goal, we will search for ψ in the form:

ψ(x, t) = ψB(x, t) + ψL(x, t)

29

where the purely bound part is of the form

ψB(x, t) =∑

l∈N

λlψ(l)B (x, t)

with

ψ(l)B (x, t) =

∑

k∈N

a(l)k (t)φk(x)e

−iωkt

where we have denoted φk the eigenvectors of the field-free Hamiltonian: H0φk = εkφk. The bound-continuous part is of the form of SFA:

ψL(x, t) =∑

l∈N

λlψ(l)L (x, t)

where for l > 0

ψ(l)L (x, t) = eiI

(l)p t(αl(t)φl(x) +

∫d3pb(l)(p, t)eip·x

)

and with αl(t) a time-dependent parameter and I(l)P is defined as a relative ionization potential, from Level

l energy

I(l)p = Ip − (ωl − ω0) = −ωl (43)

In addition, we impose:

• for all l > 0, b(l)(p, 0) = 0,

• for l = 0, α0(0) ∈ (0, 1) and for l > 1, αl(0)≪ 1.

d(t) = 〈ψB|µ|ψB〉+ 〈ψL|µ|ψL〉+ 〈ψB|µ|ψL〉+ 〈ψL|µ|ψB〉

= dBB(t) + dBL(t) + dLL(t)

with

dBB(t) =∑

l∈N

λld(l)BB(t) (44)

which takes the same value as above (28), (30), (31). Then

dBL(t) =∑

(k,l)∈N2

λk+ld(kl)BL (t), dLL(t) =

∑

(k,l)∈N2

λk+ld(kl)LL (t)

where

d(kl)BL (t) =

∑

(k,l)∈N2

(〈ψ

(k)B |µ|ψ

(l)L 〉+ 〈ψ

(k)L |µ|ψ

(l)B 〉)

(45)

and

d(kl)LL (t) =

∑

(k,l)∈N2

(〈ψ

(k)L |µ|ψ

(l)L 〉+ 〈ψ

(k)L |µ|ψ

(l)L 〉)

(46)

As dBL,LL is computed according to mathematical assumptions from [24], the following terms are neglected:

〈ψ(k)B |µ|ψ

(l)L 〉 ≈ 0 and 〈ψ

(k)L |µ|ψ

(l)L 〉 ≈ 0 for k 6= l. Then for all l:

dBL(t) ≈∑

l

λ2ld(ll)BL(t) = 2

∑

l

λ2lRe〈ψ(l)B |µ|ψ

(l)L 〉

30

where

d(ll)BL(t) = 2Re

∑

m∈B

((a(l)m (t)

)∗ei(ωm+I(l)

p )tdmgαl(t) +

∫

R3

d3p(a(l)m (t)

)∗b(l)(p, t)ei(ωm+I(l)

p )tdm(p)

)(47)

where dl(p) = 〈p|µ|φl〉. Regarding the continuum-continuum contribution

dLL(t) ≈∑

l

λ2ld(ll)LL(t) = 2

∑

l

λ2lRe〈ψ(l)L |µ|ψ

(l)L 〉

According to (32), we get

d(ll)LL(t) = i

∫ t

0

∫dτd3pE(t− τ) · dl

(p−A(t− τ)

)d∗Ll

((p−A(t)

)e−iS(p,t,τ)−δ∗l t−δl(t−τ)

We get a full description of macroscopic polarization.

PlBB(x

′, t) ≈N

~

∑(p,m)∈N×B

µmg ·

(µmg · E(ωp)

)( 1− α∗l (t)

ωmg − ωp

−αl(t)

(1− α∗

l (t))

ωmg − ωp + iδ

+(µgm · E(ωp)

)· µmg

( 1− αl(t)

ωmg + ωp

−α∗l (t)

(1− αl(t)

)


)e−iωpt

+2N

~2

∑(p,q)∈N2

∑(m,n)∈B2 Re

µgn(µnm · E(ωq))(µmg ·E(ωp))

×( 1− α∗

l (t)

(ωng − ωp − ωq)(ωmg − ωp))−

(1− α∗

l (t))αl(t)e

iIpt


)e−i(ωp+ωq)t

+N

~

∑(p,m)∈N×B

µmg ·

(µmg ·E(ωp)

)( 1− α∗l (t)

ωmg − ωp

−αl(t)

(1− α∗

l (t))

ωmg − ωp + iδ

)

+(µgm · E(ωp)

)· µmg

( 1− αl(t)

ωmg + ωp

−α∗l (t)

(1− αl(t)

)


)e−iωpt

PlLL(x

′, t) ≈ iN∑

l∈N

∫ t

0

∫dτd3pE(t− τ) · dl

(p−A(t− τ)

)d∗Ll

((p−A(t)

)e−iS(p,t,τ)−δ∗l t−δl(t−τ)

PlBL(x

′, t) ≈ 2N∑

l∈NRe∑

m∈B

((a(l)m (t)

)∗ei(ωm+I(l)

p )tdmgαl(t)

+∫R3 d

3p(a(l)m (t)

)∗b(l)(p, t)ei(ωm+I(l)

p )tdm(p)

)

Remark 4.4. One can also consider wavefunction of the type

ψL(r, t) = αm(t)φm(r) + αn(t)φn(r) +

∫

R3

d3p b(mn)(p, t)eip·r (48)

By plugging this expression into the Schrodinger equation and taking the inner product with |φm(r)〉, |φn(r)〉

31

and |p〉 respectively, we get:

iαm(t) = Emαm(t) + αn(t)E(t) · µmn

+

∫

R3

d3p b(mn)(p, t)〈φm|x|p〉 ·E(t) (49)

iαn(t) = Enαn(t) + αm(t)E(t) · µnm

+

∫

R3

d3p b(mn)(p, t)〈φn|x|p〉 · E(t) (50)

ib(p, t) =(am(t)E(t) · d(m)

x (p) + an(t)d(n)x (p)

)

−p2

2b(mn)(p, t)E(t)∂px

b(mn)(p, t) (51)

assuming am(t) and an(t) are constant in time, can be solved using the method of characteristics as shownabove. Doing so implicitly means that the rate of change of am(t) and an(t) are much less than that ofb(mn)(p, t). One acquires a coupled ODE with time dependent coefficients.

Alternatively, we can propose a close approach, which may be cheaper numerically, as follows: we searchfor ψL in the form of a discrete sum (which is then an approximation of the integral over the continuum).Let us denote by W, the corresponding indices and including as well, the ground state. That is

ψL(x, t) = cg(t)φg(x) +∑

m∈W

cm(t)φm(x)

Plugging in the system in the TDSE gives:

icg(t) = εgcg(t) +∑

m∈WE(t) · 〈φm(x)|x|φg〉cm(t)

icm(t) = εmcm(t) +E(t) · 〈φm(x)|x|φg(x)〉cg(t)

neglecting again the continuum-continuum transitions. Then ψL is combined with ψB are proposed in theabove sections.

5. Conclusion

This paper was devoted to the derivation of two non-classical nonlinear optics models. In both cases,contribution of bound-bound, bound-continuous and to a certain extent continuous-continuous state tran-sitions are evaluated in order to determine the macroscopic polarization in Maxwell’s equations. Althoughthe first model, fully non-pertrubative, is an important improvement of the Maxwell-Schrodinger model interm of computational complexity, it however still contains microscopic components: TDSE computationshave to be performed to determined initial polarizations. In addition, from a practical point of view in thisconfiguration, it is valid only for very short pulses. The second model, although fully macroscopic is howeversemi-empirical and partially perturbative.In order to validate and to analyze these models, we plan, in a close future, to test them on realistic situations.In particular simple filamentation simulations will be performed. The inclusion of nonlinearities coming frombound states as well as free electrons, is essential in order to describe accurately filament dynamics [1], [2].

32

Appendix A. (Gaussian beam). We consider the extension to Gaussian beams of the method developedabove. Using [25]’s notations (see also [33]), we denote by w0 the beam waist, λ its wavelength, and A itsmaximal amplitude, see Fig. 8, and [22] for details about this Gaussian pulse. We assume that

E(x′, t) = A(x′)ei(kz′−ωt)ez′

where x′ = (r′, z′) and

A(r′, z′) =A

1 + iξexp

(−

r′2

w20(1 + iξ)

)

with ξ = 2z′/b and b = kw20. The beam power is given by

P =

∫nc

2π|A|2 =

1

4nω0|A|

2

Assuming that the dipole moment d(x′1, t) at x

′1 := (r′1, z

′1) for any r

′1 is known, from the solution to

Figure 8: Gaussian beam intensity at different propagation times, in vacuum, in the (x, z)-plan, I = 1013W·cm−2

i∂tψ = −1

2ψ + Vcψ + x · Ex′

1ψ

we construct d(x′2, t) with x′

2 := (r′1, z′2) as follows. For |∆z

′|/b = |z′2 − z′1|/b small enough, we next rewrite

A(r′1, z′2) as a function of A(r′1, z

′1)

A(r′1, z′2) =

A

1 + iξ2exp

(−

r′21w2

0(1 + iξ2)

), ξ2 =

2z′2b

which can be rewritten, denoting ∆ξ = 2∆z′/b

A(r′1, z′2) =

A

1 + iξ1 + i∆ξexp

(−

r′21w2

0(1 + iξ1 + i∆ξ)

)

and

A(r′1, z′2) = A(r, z′1)

1 + iξ1


(−r′21ω20

( 1

1 + iξ1 + i∆ξ−

1

1 + iξ1

))

= A(r, z′1)1 + iξ1


(−

r′21ω20(1 + iξ1)

)exp

( 1 + iξ1

1 + iξ1 + i∆ξ− 1)

Using |∆ξ| ≪ 1, we then have

1 + iξ1

1 + iξ1 + i∆ξ= 1−

i∆ξ

1 + iξ1−

∆ξ2

(1 + iξ1)2+ · · ·

and as a consequence

A(r′1, z′2) ≈ A(r, z′1)

(1−

i∆ξ

1 + iξ1−

∆ξ2

(1 + iξ1)2

)(1 +

r′21ω20(1 + iξ1)

( i∆ξ

1 + iξ1+

∆ξ2

(1 + iξ1)2))

33

This gives a useful relation between A(r′1, z′1) and A(r

′1, z

′2). Denoting, now

ε(ξ1, r′1,∆z

′) = −i∆ξ

1 + iξ1+

ir′21 ∆ξ

(1 + iξ1)2ω20

−∆ξ2

(1 + iξ1)2+

2r′21 ∆ξ2

(1 + iξ1)3ω20

we obtain

A(r′1, z′2) ≈ A(r

′1, z

′1)(1 + ε(ξ1, r

′1,∆z

′))

In general |A(r′1, z′2)| 6≈ |A(r

′1, z

′1)|, so that d(x′

2, t) 6≈ d(x′1, t − ∆z′/vg

). However, we can define ∆ρ′1

from

|A(r′1, z′2)| ≈ |A(r

′1 +∆ρ′1, z

′1)| (52)

so that d(x′2, t) ≈ d

(x′1, t−

∆z′

vg

), where x′

1 = (r′1 +∆ρ′1, z′1). In order to determine ∆ρ′1, we expand

A(r′1 +∆ρ′1, z′1) = A(r′1, z

′1) exp

(−

2r′1∆ρ′1 +∆ρ′21

w20(1 + iξ1)

)

≈ A(r′1, z′1)(1−

2r′1∆ρ′1 +∆ρ′21

w20(1 + iξ1)

+(2r′1∆ρ

′1 +∆ρ′21 )

2

2w40(1 + iξ1)2

)

We set

δ(ξ1,∆ρ′1) := −

2r′1∆ρ′1 +∆ρ′21

w20(1 + iξ1)

+(2r′1∆ρ

′1 +∆ρ′21 )

2

2w40(1 + iξ1)2

so that

A(r′1 +∆ρ′1, z′1) ≈ A(r

′1, z

′1)(1 + δ(ξ1, r

′1,∆ρ

′1))

We now determine ∆ρ′1 such that (52) holds, that is ∆ρ′1 is such that

δ(ξ1, r′1,∆ρ

′1) = ε(ξ1, r

′1,∆z

′)

As a first approximation

∆ρ′1 ≈ ∆z′Γ(r′1, z′1, ω0, b)

where we have set, recalling that ξ1 = 2z′1/b

Γ(r′1, z′1, ω0, b) := ±

ω20

r′1b

∣∣∣1−r′21

(1 + 2iz′1/b)ω20

∣∣∣

In Fig. 9, we represent the dipole moment propagating from x′1 to x′

2. We then deduce that

P(x′2, t) ≈ P

(x′1, t−

∆z′

vg

)

A corresponding quasilinear model is, for (r′, z′) ∈ R∗+ × (z′1,∞)

∂tP(r′, z′, t) +V(E, ω0, b) · ∇P(r′, z′, t) = 0

with initial data P(x′1, 0) assumed to be known and where V =

(Γ(r′, z′, ω0, b), vg

).

We conclude this section by a remark on a more accurate approximation for P(x′2, t).

34

z’

∆ρ∆z’

r’

Figure 9: Dipole moment evolution

Remark 5.1. In order to include electromagnetic field variation between x′1 and x′

2, it is possible to use anapproach similar as the one proposed in Model 2 of Section 2.1. For the same reason as above, we get fort ∈ (ta, tb].

P(x′2, t) ≈ P

(x′1, t−

∆z′

vg

)+∆T 2

((∆E

(x)(ta))2c(x)

(x′1, t−

∆z′

vg

)

+(∆E

(x)(ta))2Q(y)

(x′1, tb −

∆z′

vg

)+ 2∆E

(x)(ta)∆E(y)(ta)Q

(xy)(x′1, tb −

∆z′

vg

))

where

∆E(ta) =(∆E

(x)(ta),∆E(y)(ta)

):= Ex′

2(ta)−E

x′

1

(t−∆z′/vg

)

and Q(x,y,xy) are defined in (6). Again, for z′ > z′1, we can model the evolution by

∂tP(x′, t) + vg∂zP(x′, t) + Γ(r′, z′, ω0, b)∂r′P(r′, z′, t) =∫ t

0〈Q(x′, s′) ·∆E(x′, s′),∆E(x′, s′)〉ds′

∂tQ(x)(x′, t) + vg∂zQ

(x)(x′, t) + Γ(r′, z′, ω0, b)∂r′Q(x)(r′, z′, t) = 0

∂tQ(y)(x′, t) + vg∂zQ

(y)(x′, t) + Γ(r′, z′, ω0, b)∂r′Q(y)(r′, z′, t) = 0

∂tQ(xy)(x′, t) + vg∂zQ

(xy)(x′, t) + Γ(r′, z′, ω0, b)∂r′Q(xy)(r′, z′, t) = 0

with initial data P(x1, 0).

An alternative approach can be derived from simple expansions as follows.

Remark 5.2. Starting again from A(r′ + ∆ρ′, z′1) but without using the Gaussian shape of the pulse, wefirst get A(r′ +∆ρ′, z′1) ≈ A(r

′, z′1)+∆ρ′∂rA(r′, z′1). Similarly in A(r′, z′2) ≈ A(r

′, z′1)+∆z′∂z′A(r′, z′1). We

deduce in first approximation that for a given ∆z′, |∆ρ′| ≈∣∣∣∆z′∂z′A(r′, z′1)/∂r′A(r

′, z′1)∣∣∣.

Appendix B. (Generalized Liouville equation). For any n ∈ N:

i〈∂tψ|φn〉 = 〈(H0 + V (x, t)

)ψ|φn〉

and for any λ ∈ σc

i〈∂tψ|φc(·, λ)〉 = 〈(H0 + V (x, t)

)ψ|φc(·, λ)〉

35

First, we get

i∂tψ(x, t) =∑

n

cn(t)φn(x) +

∫

σc

∂tc(t, η)φc(x, η)ρc(η)dη

and

(H0 + V (x, t)

)ψ(x, t) =

∑

n

(εn + V (x, t)

)cn(t)φn(x) +

∫

σc

(η + V (x, t)

)c(t, η)φc(x, η)ρc(η)dη

As 〈φn|φm〉 = δnm,

i〈∂tψ|φm〉 = icm(t) + i

∫

σc

〈φc(·, η)|φm〉∂tc(t, η)ρc(η)dη

and

i〈∂tψ|φc(·, µ)〉 = i∑

n

cn(t)〈φn|φc(·, µ)〉+ i

∫

σc

〈φc(·, η)|φc(·, µ)〉∂tc(t, η)ρc(η)dη

Similarly

〈(H0 + V (x, t)

)ψ|φm〉 =

∑

n

cn(t)〈(εn + V (x, t)

)φn|φm〉+

∫

σc

c(t, η)〈(η + V (x, t)

)φc(·, λ)|φm〉ρc(λ)dλ

and

〈(H0 + V (x, t)

)ψ|φc(·, µ)〉 =

∑

n

cn(t)〈(εn + V (x, t)

)φn|φc(·, µ)〉+

∫

σc

(η + V (x, t)

)c(t, η)〈φc(·, η)|φc(·, µ)〉ρc(η)dη

We set

Hnm := 〈(εn + V (x, t)

)φn|φm〉 = εnδnm + 〈V (x, t)φn|φm〉

Hn(λ) := 〈(εn + V (x, t)

)φn|φc(·, λ)〉 = 〈V (x, t)φn|φc(·, λ)〉

Kn(λ) := 〈(λ+ V (x, t)

)φc(·, λ)|φn〉 = 〈V (x, t)φc(·, λ)|φn〉

H(λ, µ) := 〈(λ+ V (x, t)

)φc(·, λ)|φc(·, µ)〉 = 〈V (x, t)φc(·, λ)|φc(·, µ)〉

D(λ, µ) := 〈φc(·, λ)|φc(·, µ)〉

Dn(µ) := 〈φn|φc(·, µ)〉

So that

icm(t) + i∫σcD∗

m(η)ρc(η)∂tc(t, η)dη =∑

n cn(t)Hnm +∫σcc(t, η)Kn(η)ρc(η)dη

i∑

n cn(t)Dn(µ) + i∫σcD(η, µ)ρc(η)∂tc(t, η)dη =

∑n cn(t)Hn(µ) +

∫σcH(η, µ)ρc(η)c(t, η)dη

As the operator H0 + V (x, t) is self-adjoint, we deduce that: D(λ, µ) = 0 if λ 6= µ and Dn(λ) = 0 then

i∑

n

cn(t)Dn(µ) + i

∫

σc

D(η, µ)ρc(η)∂tc(t, η)dη = i∂tc(t, µ)

This gives us a new set of equations:

icm(t) =∑

n cn(t)Hnm +∫σcc(t, η)Kn(η)ρc(η)dη

i∂tc(t, µ) =∑

n cn(t)Hn(µ) +∫σcH(η, µ)ρc(η)c(t, η)dη

36

As V (x, t) = x ·E(t)

Hnm = E(t) · 〈xφn|φm〉

Hn(λ) = E(t) · 〈xφn|φc(·, λ)〉

Kn(λ) = 〈E(t) · 〈xφc(·, λ)|φn〉

H(λ, µ) = E(t) · 〈xφc(·, λ)|φc(·, µ)〉

D(λ, µ) = 〈φc(·, λ)|φc(·, µ)〉

We are interested in d(t) = 〈x〉 =∫|ψ|2xd3x. In this goal, we set

dmn(t) = 〈φm|x|φn〉, ρmn(t) = c∗m(t)cn(t), ρmλ(t) = c∗m(t)c(t, λ), ρλm(t) = c∗(t, λ)cm, ρλµ(t) = c∗(t, λ)c(t, µ)

In order to compute d or equivalently Tr(ρx), we derive a system of differential equiations. First, as

ρmn(t) = c∗m(t)cn(t) + c∗m(t)cn(t)

then

c∗n(t)cm(t) = −i∑

ν

c∗n(t)cν(t)Hνm − i

∫

σc

c∗n(t)c(t, η)Km(η)ρc(η)dη

and

c∗n(t)cm(t) = i∑

ν

cm(t)c∗ν(t)Hνn + i

∫

σc

cm(t)c∗(t, η)K∗n(η)ρ

∗(η)dη

so that

ρmn(t) = i∑

ν

Hmνρνn(t)− ρmν(t)Hνn + i

∫

σc

(ρnηn(t)K

∗m(η)ρ∗c(η) − ρmη(t)Kn(η)ρc(η)

)dη

Similarly, we have to evaluate ∂tρλ,µ(t). From

ρλµ(t) = ∂tc∗(t, λ)c(t, µ) + c∗(t, λ)∂tc(t, µ)

We then get

ρλµ(t) = i∑

n

[ρnµ(t)H

∗n(λ)− ρλn(t)Hn(µ)

]

+i∫σc


]dη

Similarly

ρmλ(t) = c∗m(t)c(t, λ) + c∗m(t)∂tc(t, λ)

so that

ρmλ(t) = i∑

n

[ρnλ(t)H


]

+i∫σc

[K∗


]dη

37

That is, we get the following system, for all m, n, λ, µ:

ρmn(t) = i∑

ν Hmνρνn(t)− ρmν(t)Hνn + i∫σc

(ρη(t)K

∗m(η)ρ∗c(η)− ρmη(t)Kn(η)ρ(η)

)dη

ρmλ(t) = i∑

n

[ρnλ(t)H


]

+i∫σc

[K∗


]dη

ρλµ(t) = i∑

n

[ρnµ(t)H

∗n(λ) − ρλn(t)Hn(µ)

]

+i∫σc


]dη

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Development of nonperturbative nonlinear optics models