A numerical Maxwell-Schr¨odinger model for intense laser ...

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A numericalMaxwell-Schr¨odingermodel for intense laser-molecule interaction and propagation E. Lorin S. Chelkowski A. Bandrauk October 12, 2006 Abstract We present in this paper a numerical Maxwell-Schr¨odinger model and a methodology to simulate intense ultrashort lasers interacting with a 3D H + 2 -gas under and beyond Born-Oppenheimer approxi- mation. After a presentation of the model and a short mathematical study, we will examine some issues on its numerical computation. In particular we will focus on the polarization computation allowing the coupling between the Maxwell and time dependent Schr¨odinger equations (TDSE), on the parallelization, and on the boundary conditions problem for the TDSE. Examples of numerical computations of high order harmonics generation and of electric fields propagations are presented. Keywords: Maxwell, Schr¨ odinger, laser, multiscales problem, transparent boundary conditions. 1 Introduction From the theoretical point of view to the very practical point of view (controlled fusion [35] by inertial confinement, quantum dynamic imaging [10], attosecond pulse generation [4]), there exist numerous applica- tions of ultrashort and intense lasers. Indeed current laser technology allows to create ultrashort pulses with intensities exceeding some molecules and atoms internal electric fields. Typically, for the hydrogen atom the period of circulation is 24.6 attoseconds (10 18 s) and the intensity of its Coulomb field is 3 ×10 16 W/cm 2 with a corresponding electric field E =5 × 10 11 V/m 1 . Current laser intensities can reach around 10 20 W/cm 2 with pulse durations of 50 femtoseconds (10 15 s). The main goal of this work is to study high order non- linear phenomenons (Above Threshold Ionization (ATI), High Harmonics Generation (HHG), filamentation [24], etc) obtained with very intense lasers interacting with molecules and to study their dynamics. HHG is one of these nonlinear phenomenons that appears when the electric field is of same order as the Coulomb potential (ionization) and is created by electron rescattering [1], [20]. It is, morevover, the current source of coherent attosecond pulses [5], [4]: in a first step submitted to an intense electric field the electron leaves the ion vicinity and enters in the ionized continuum with an initial velocity equal to zero [1], [20]. Then the free electron is accelerated by the strong electric field and gains energy. In a last step the electron is driven back into the vicinity of the parent ion and recombined with it, leading to a multiphoton ionization (see [29] for description of HHG or [21] for their numerical aspects) then to the creation of high order harmonics. In this paper we will introduce a precise model to describe this dynamics, including ionization, dissociation of the molecule and including propagation effects in order to take into account the phase matching [14]. Whereas previous works dealt with laser-atom interactions [14], we propose here to study ultrashort pulses interacting with molecules, in particular we will focus on the one-electron H + 2 -molecule [14]. To model phase matching in molecular media such as propagation in the atmosphere [24], the most natural way is to consider a coupling * Centre de Recherche en Math´ ematiques de Montr´ eal (Canada) Universit´ e de Sherbrooke, laboratoire de chimie th´ eorique (Canada) Universit´ e de Sherbrooke, laboratoire de chimie th´ eorique, Centre de Recherche en Math´ ematiques de Montr´ eal, Canada Research Chair (Canada) 1

Transcript of A numerical Maxwell-Schr¨odinger model for intense laser ...

Page 1: A numerical Maxwell-Schr¨odinger model for intense laser ...

A numerical Maxwell-Schrodinger model for intense laser-molecule

interaction and propagation

E. Lorin ∗ S. Chelkowski † A. Bandrauk ‡

October 12, 2006

Abstract

We present in this paper a numerical Maxwell-Schrodinger model and a methodology to simulateintense ultrashort lasers interacting with a 3D H+

2 -gas under and beyond Born-Oppenheimer approxi-mation. After a presentation of the model and a short mathematical study, we will examine some issueson its numerical computation. In particular we will focus on the polarization computation allowing thecoupling between the Maxwell and time dependent Schrodinger equations (TDSE), on the parallelization,and on the boundary conditions problem for the TDSE. Examples of numerical computations of highorder harmonics generation and of electric fields propagations are presented.

Keywords: Maxwell, Schrodinger, laser, multiscales problem, transparent boundary conditions.

1 Introduction

From the theoretical point of view to the very practical point of view (controlled fusion [35] by inertialconfinement, quantum dynamic imaging [10], attosecond pulse generation [4]), there exist numerous applica-tions of ultrashort and intense lasers. Indeed current laser technology allows to create ultrashort pulses withintensities exceeding some molecules and atoms internal electric fields. Typically, for the hydrogen atom theperiod of circulation is 24.6 attoseconds (10−18s) and the intensity of its Coulomb field is 3×1016W/cm2 witha corresponding electric field E = 5 × 1011V/m−1. Current laser intensities can reach around 1020W/cm2

with pulse durations of ∼ 50 femtoseconds (10−15s). The main goal of this work is to study high order non-linear phenomenons (Above Threshold Ionization (ATI), High Harmonics Generation (HHG), filamentation[24], etc) obtained with very intense lasers interacting with molecules and to study their dynamics. HHG isone of these nonlinear phenomenons that appears when the electric field is of same order as the Coulombpotential (ionization) and is created by electron rescattering [1], [20]. It is, morevover, the current source ofcoherent attosecond pulses [5], [4]: in a first step submitted to an intense electric field the electron leaves theion vicinity and enters in the ionized continuum with an initial velocity equal to zero [1], [20]. Then the freeelectron is accelerated by the strong electric field and gains energy. In a last step the electron is driven backinto the vicinity of the parent ion and recombined with it, leading to a multiphoton ionization (see [29] fordescription of HHG or [21] for their numerical aspects) then to the creation of high order harmonics. In thispaper we will introduce a precise model to describe this dynamics, including ionization, dissociation of themolecule and including propagation effects in order to take into account the phase matching [14]. Whereasprevious works dealt with laser-atom interactions [14], we propose here to study ultrashort pulses interactingwith molecules, in particular we will focus on the one-electron H+

2 -molecule [14]. To model phase matchingin molecular media such as propagation in the atmosphere [24], the most natural way is to consider a coupling

∗Centre de Recherche en Mathematiques de Montreal (Canada)†Universite de Sherbrooke, laboratoire de chimie theorique (Canada)‡Universite de Sherbrooke, laboratoire de chimie theorique, Centre de Recherche en Mathematiques de Montreal, Canada

Research Chair (Canada)

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VACUUMVACUUM

(incident field) (transmitted field)

GAS (H2+)

E (r’,t)

Figure 1: Laser-matter interactions in a H2+ gas

between the Maxwell equations describing the behavior of the electric and magnetic fields and the TDSEdescribing the particles motions. In order to increase the precision of the model, we propose to go beyondthe Born-Oppenheimer approximation taking into account the nuclear motions [17]. The Maxwell equationsand the TDSE are then coupled with the polarization of molecules that describes the relative position ofthe particles constituting the molecule. The model is written first in its whole generality and then someapproximations are proposed in order to solve it numerically efficiently. Note that the model we propose isclose to the “one-dimensional” model presented in [33], [34], [19], [18].In section 2. we will present our approach and we will briefly discuss some existing models. Then we willfocus on the existence of solutions for the Schrodinger equation we consider. Section 3. will be devoted to thepresentation of the numerical and parallel method we have used to solve the coupled system. In particular wewill detail the polarization computation (and how to reduce it in CPU time) and on the numerical boundaryconditions for the TDSE, crucial in this framework. We will demonstrate sclability as a function of numberof particles for the H+

2 gas system. In the last section some numerical results will be presented.

2 Mathematical Model

2.1 Presentation of the Maxwell-Schrodinger model

The TDSE describing the H+2 molecule behavior submitted to a laser field can be written [21] where we use

atomic units (e = h/2π = me = 1):

i∂tψ(r, R, r′, t) =[− 1

2r + Vc(r, R)− i

cA(r′, t) · ∇r +

1

R− 1

4mpR +

‖A(r′, t)‖22c2

]ψ(r, R, r′, t), (1)

where ψ represents the molecular wavefunction. Electron-nuclea, Vc and nuclear ion Vi potentials are givenby

Vc(r, R) = − 1√x2 + (y −R/2)2 + z2

− 1√x2 + (y +R/2)2 + z2

, Vi(R) =1

R. (2)

The electronic position in the nuclear center of mass coordinates is denoted by r = (x, y, z)T . R representsthe nuclear relative position and mp denotes the H+ mass. We suppose in this paper that the nuclear motionis one-dimensional. The variable r′ denotes the spatial dependency of the electric and magnetic fields, andA represents the electromagnetic potential. Note that in our orthogonal coordinates, ex = ex′ , ey = ey′ ,ez = ez′ . Next, if the laser wavelength is large enough we can assume that the electric and magnetic fieldsare constant in space at the molecular scale λ/ℓ >> 1 (this corresponds to the dipole approximation [31], [9])

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H+H+

R

y

z’z

x

e−

Figure 2: Coordinates.

allowing us to reduce the numerical and mathematical complexity of the problem. Indeed, in this situationthe Schrodinger equation can be written:

i∂tψr′(r, R, t) =

[− 1

2r + Vc(r, R)− i

cAr

′(t) · ∇r +1

R− 1

4mpR +

‖Ar′(t)‖2

2c2

]ψr

′(r, R, t), (3)

r′ is now a parameter denoting the molecule position in the “Maxwell domain”. For each r′ in the gas, onehas to solve a 4D-TDSE. Let us remark that this approximation leads to [r,Ar

′(t) · ∇r] = 0; a very usefulfeature for applying an error numerical splitting.

The previous TDSE is given is the so-called velocity gauge. Another possible formulation is in the so-calledlength gauge. Applying the unitary transformation

ψ 7→ ψ exp(− i

2c2

∫ t

0

‖Ar′‖2(s)ds+

i

cr ·Ar

′(t)), (4)

we get a new formulation of the TDSE [9], [15]:

i∂tψr′(r, R, t) =

[− 1

2r + Vc(r, R) + r ·Er

′(t) + Vi(R)− 1

mpR

]ψr

′(r, R, t). (5)

with Er′(t) = −c∂tAr

′(t)dt denoting the electric field.

Ideally the electric field dynamics modeling is given by the microscopical Maxwell equations coupled withthe Schrodinger equations and will be done in a future work. A full description of the microscopical Maxwellequations can be found in [26]. A close approach is proposed in [16], where the authors study the micro-scopical Maxwell equations but coupled with classical dynamics equations to describe the particles motions.We will here consider the macroscopical Maxwell equations that correspond to a spatial average of the mi-croscopical ones. These equations are typically valid in a domain equal or exceeding a size of 10−18cm3 witha sufficiently high molecular density (see again [26]). The macroscopical 3D-Maxwell equations written inatomic units are:

∂tB(r′, t) = −c∇r′ ×E(r′, t),

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

∇r′ ·

(E(r′, t) + 4πP(r′, t)

)= 0,

∇r′ ·B(r′, t) = 0.

(6)

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Under the dipolar approximation the polarization P for a molecule located at r′ is given by:

P(r′, t) = −n(r′)

∫ψr

′(r, R, t) · r · ψ∗r′(r, R, t)dRdr, (7)

where n(r′) denotes the molecular density defined below. The sign comes from the electron charge. The po-larization P is then the coupling term between the Schrodinger and Maxwell equation. The model equationsare then given by (5), (7), (6). A full description of the numerical polarization modeling will presented insection 3.4

The Maxwell domain is denoted by ΩM . The molecules are located in a sub-domain Ωm and the vac-uum regions are denoted by Ω0: ΩM = Ωm ∪ Ω0. Initially the laser-pulse of frequency ω and intensity I isdefined as:

E(r′, 0) = E0(r

′)fω(r′), r′ ∈ Ω0,E(r′, 0) = 0, elsewhere.

The function f is usually a sinusoidal, and E0 is a gaussian function E0(r′) = Ie−cx(x′−xc)

2−cy(y′−yc)2−cz(z′−zc)

2

ey′

MOLECULES

LASERAT TIME T=0

VACUUM

VACUUM

−L Ll−l −a az

n(z) (density)

z’

y

H+

H+

Figure 3: Physical configuration: 1D-cut

even if for usual cases the beam is sufficiently large so that along the propagation axis, we can again neglectthe x′ and y′ dependencies. The quantities cx, cy, cz, depend on the pulse shape.The harmonics spectra of the transmitted field, denoted by ET , possesses in theory [29], a frequency cut-offdenoted by ωc (see fig. 4) where ωc = Ip + 3.17Up = Nω for recollision of the electron with the parention [35], [20]. Ip is the ionization potential, Up = I/4ω2 (a.u.) for intensity I and frequency ω. N is theharmonic order at the cut-off frequency. Collisions with neighboring ions such as in molecules produce largerorders N due to large collision energies exceeding 3.17Up [3]. It is then possible to filter this field around ωc

and to pick-up a shorter “intense” pulse denoted by EF :

EF (t) =1

∫ ωc+∆ω

ωc−∆ω

ET (ω)eiωtdt.

See for instance, [32], [2] or [4] for the control aspects. Note that, this problem constitutes typically amultiscale physical problem as the characteristic length and time of the TDSE are much smaller than theMaxwell equations characteristic ones.

There exist many models describing the physical phenomenon presented above. The most simplest onesare the nonlinear Maxwell models, and consist in calculating the polarization as an expansion of the suscep-tibility (linear, quadratic, cubic, and so on.):

P(r′, t) = Ξ(1) · E + Ξ(2) · E2 + Ξ(3) · E3 + · · · ,

instead of deducing it from the molecular wave functions. These models allow for example to simulate thebehavior of low intensity lasers with low harmonic generation. As our transmitted fields possess high order

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1e-35

1e-30

1e-25

1e-20

1e-15

1e-10

1e-05

10 20 30 40 50 60

"dipolar acceleration"

Figure 4: Typical harmonics spectra (dipolar acceleration for a single H+2 molecule with λ = 800nm and

I = 1014). In abscissa: harmonics orders (1 corresponds to the laser frequency). In ordinate: harmonicsintensities.

harmonics these models are not applicable here due to the high nonlinear regime.

The Maxwell-Bloch equations constitute a more precise model where the polarization P is obtained fromthe TDSE via the Bloch equation coupled with the Maxwell equations. More precisely, supposing the nucleifixed, the TDSE can be written in a compact form:

i∂tψ =[H0 + L(E)

]ψ,

where H0 is the free Hamiltonian (without laser) and L(E) denotes the laser operator and E the electricfield. We then compute the N first eigenvalues and associated eigenvectors of the Hamiltonian operator(energy levels (ωj)j and associated eigenfunctions, (φj)j), that is, supposing no degeneracy :

H0φj = φjωj , ∀j ∈ 1, · · · , N.

We can then deduce the so-called dipolar matrix µ ∈MN (C3) obtained using the position operator and theeigenfunctions of the free Hamiltonian. Then we have to solve a set of differential equations on the electronicdensity matrix ρ = (ρjk)jk ∈MN (C):

∂tρjk = −i(ωj − ωk)ρjk −2iπ

h[V,ρ], (j, k) ∈ 1, · · · , N2,

ρ(·, t = 0) = (∫φjφ

∗k)jk.

with V = −µ ·E under the dipolar approximation, and where [·, ·] is the Lie bracket. Finally denotingby N the molecular density, the polarization is given by: P = N tr(µ · ρ). A detailed numerical study ofthis model can be found in [13] and [37]. To understand the validity of this model, let us observe that thepolarization P is obtained from the energy levels of the free Hamiltonian. As is well known the computationof the eigenelements of the Hamiltonian operator is numerically very costly, so that N is in practice limitedto very small numbers (less than 5 usually). With such a limited model it is then a priori not possible toreach the continuous part of the Hamiltonian to compute high order harmonics with very intense laser pulses(typically of the order of the Coulomb potential intensity).

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The Maxwell-TDSE system under the slowly varying envelope approximation consists in computing thefollowing. For instance, in the case of a one-dimensional propagation in z′, one first solves the TDSE (see[28]) for a given molecule and we deduce its dipolar acceleration a(t) defined here as:

a(t) =

∫ψ(r, R, t)

(− ∂V (r, R)

∂y− E(t)

)ψ∗(r, R, t)drdR. (8)

Then we approximate the polarization for all z′ by:

∂2ttP (z′, t) = Na

(t− z′/c+ φ(z′, t)

), (9)

where φ denotes the electric field envelope phase. The polarization P is then coupled with the Maxwellequations. The main advantage of this approximate technique comes from the fact that it is only necessaryto solve one single TDSE to deduce the polarization P in the whole gas. However such a strong approximationinduces a “loss of informations” difficult to evaluate, [28].

2.2 Existence and regularity for the TDSE model

In this section we are interested in the existence and regularity of solutions for the TDSE (5). To do this,we will use the Fourier transform set of H1:

H1 =u ∈ L2(R4),

R4

(1 + ‖(r, R)‖2

)|u(r, R)|2drdR <∞

From Baudouin and Puel [12], [11], we can easily deduce that for L ∈ L∞(0, T ;C2

b (R4))

and if u0 ∈ H1∩H1,

then there exists a unique solution u in C0(0, T ;H1 ∩H1

)such that

(i∂t +

y

2+R

mp+ L(r, R, t)

)u(r, R, t) = 0, u(r, R, 0) = u0(r, R).

Furthermore, for K > 0 such that

‖V ‖L∞

(0,T ;C2

b(R4)

) 6 K

then there exists CT,K such that

‖u‖C0

(0,T ;H1∩H1

) 6 CT,K‖u0‖H1∩H1.

Considering now (5), let us suppose that for all T > 0, E ∈ L∞(0, T ) and ∂tE ∈ L1(0, T ). The laserfield r · E(t) we consider is a priori non-zero in the whole space it is defined for all r ∈ R3. Physically it isobviously not true, so that we will consider a function χ defined as follows: χ : (r, R) 7→ χ(r, R) in C2

b (R4)and χ(r, R) = r on a compact set Ω1 of R4 and χ(r, R) is zero outside a set Ω2 containing strictly Ω1. Sucha function can easily be constructed by convolution of y and a plateau function. Then we can prove:

Theorem 2.1 Let us consider the following TDSE:

i∂tψ(r, R, t) =[− 1

2y + Vc(r, R)− 1

mpR + Vi(R)− χ(r, R) ·E(t)

]ψ(r, R, t).

∀T > 0, suppose that E ∈ L∞(0, T ) and ∂tE ∈ L1(0, T ). Then there exists CT > 0 such that, for all

ψ0 ∈ H1 ∩H1, there exists a solution unique ψ ∈ L∞(0, T ;H1 ∩H1) and

‖ψ(t)‖L∞(0,T ;H1∩H1) 6 CT ‖ψ0‖H1∩H1.

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The proof is based an energy estimate in H1 ∩H1 via a Gronwall inequality and a compacity argument toextract a convergent subsequence. As it can be derived with some elementary modifications from the proofof theorem 2.1 in [11] and is very close of existing results, for instance [25] we present it in appendix.

We can in fact improve the previous results. Under regularity assumptions on the electric field we couldprove (using again [12]) that for u0 ∈ H2

mp∩H2 the existence of a unique solution in L∞(0, T ;H2

mp∩H2).

The existence of solutions for the coupled systems Maxwell-Schrodinger has yet to be proved; in partic-ular the regularity of the polarization (linking the two systems) is useful in our numerical approach. We canexpect that the polarization regularity on r′ comes from the incoming electric field spatial regularity. Someinteresting results about this coupling can be found for instance in [22], where the existence of global smoothand weak Maxwell-Bloch solutions is presented.

3 Numerical approach

We describe in this section our numerical method to approximate the Maxwell-Schrodinger system presentedabove. We will focus on the polarization computation that allows the coupling between the Maxwell andTDSE and on the boundary conditions for the TDSE and finally on the parallelization of the coupled system.Suppose to simplify the notations that the Maxwell computational domain is given by ΩM = [−Lx, Lx] ×[−Ly, Ly]× [−Lz, Lz], Lx, Ly, Lz > 0 and the molecules are located in Ωm = [−ℓx, ℓx]× [−ℓy, ℓy]× [−ℓz, ℓz],and the vacuum region is Ω0 = [−Lx, Lx]× [−Ly, Ly]× [−Lz, Lz]× /[−ℓx, ℓx]× [−ℓy, ℓy]× [−ℓz, ℓz]. In thisframework, introducing some positive constants 0 < a < ℓ < L, the molecular density can be given by:

n(r′) =

0, if ℓ < ‖r′‖ < L,n0 exp

(− (‖r′‖ − a)2

)if a < ‖r′‖ < ℓ,

n0 if ‖r′‖ 6 a,

where n0 ∈ R∗+ (mol/cm3). Such a density choice will allow us in particular to numerically reduce the

reflections of the incoming electric field on the “boundaries” of the gas. The TDSE is approximated by afinite difference Crank-Nicolson scheme in time, and the Laplace operator is approximated using a 3-pointsstencil. Such a scheme, allows to preserve the ℓ2-norm and is a second order scheme. The Yee scheme isused to solve the Maxwell equations [39] that consists in a finite difference scheme where the electric andmagnetic fields are computed on two spatial and temporal staggered grids. Under a CFL condition this isa stable and order two scheme. At this point it is important to recall that to be valid the macroscopicalMaxwell equations have to be applied on a sufficiently large domain (as it is obtained by a spatial average onmicroscopical Maxwell equations). Typically if we denote by ∆z′M the Maxwell space step in the z′-direction,

we should have: ∆z′M × n1/30 >> 1 but also have ∆z′M < λmin, with λmin corresponding to the highest

harmonics created during the HHG process. In each Maxwell cell, a large number of molecules is contained,but in practice we solve only one TDSE to determine a local polarization P. The simulation has then somemultiple scales as ∆z′M >> ∆yS where ∆yS is the Schrodinger grid space step in the y-direction. We alsohave the condition ∆tM >> ∆tS where ∆tS , ∆tM are respectively the Schrodinger and Maxwell solverstime steps. We then set

N = E[∆tM

∆tS

].

Now denoting by S the Schrodinger operator and by B the Maxwell operator, the global coupled numericalscheme is written as the following splitting scheme (here written at order 1, but is computed at order 2 to

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preserve the Maxwell and Schrodinger solvers order), from time tn to time tn+1:

En+1 ← e∆tMBn

En, ∀r′ ∈ ΩM , solving the Maxwell equations,

ψn+1 ← e[∆tM−(N−1)∆tS ]Sn

e(N−1)∆tSSn

ψn, solving the TDSE in each “Maxwell cell”,

Pn+1 ← ψn+1, by definition of the polarization.

3.1 Time Dependent Schrodinger Equation

The TDSE can be summarized as,

i∂tψr′(r, R, t) = (A+B + Cr

′)ψr′(r, R, t).

with

A = −1

2r + Vc, B = − 1

4mpR + Vi, Cr

′ = r ·Er′(t).

The domain we work on is then given by ΩS× [0, R∞]. The space steps are given by ∆r = (∆xS ,∆yS ,∆zS)T

and ∆R and in the following ∆tS will denote the time step of the numerical scheme for solving the TDSE.

3.1.1 Initial data and numerical scheme

The first step consists in computing the initial data of the TDSE given by the smallest eigevenvalue andassociated eigenvector of the free Hamiltonian A+B; that corresponds to the initial energy state. This is donehere using a splitting scheme. Indeed, even if it is possible in theory to compute directly the eigenelementsof the operator A + B in multidimensions this constitutes a sparse eigenvalue problem that can becomecomputationally very costly in terms of storage and algorithmic complexity. That is why we propose thefollowing two-step process.We first solve:

Aφ(r, R) = λ(R)φ(r, R),

that gives a set of eigenvalues λ(R) and associated eigenvectors φ(r, R) corresponding to R fixed (Born-Oppenheimer), to the smallest eigenvalues of A. This is numerically obtained by a simple three pointscheme to approximate the Laplace operator and by the ARPACK (www.arpack.org) Arnoldi method tocompute the smallest eigenvalue and associated eigenvector of the associated sparse matrix.In a second time, we compute:

[B + λ(R)Id

]ξ(R) = µξ(R)

using again a three point scheme for the Laplacian operator and an Arnoldi method to compute the smallesteigenvalue µ and associated eigenvector ξ.Finally the initial data corresponding to an approximation of the initial state wave function for the consideredmolecule is given by:

ψ(r, R) = φ(r, R)ξ(R).

in the sense that it approximates ψ(r, R) obtained from the calculation of the smallest eigenvalue and asso-ciated eigenvector of A + B. This approximation can be easily justified neglecting the term φ(R)Bψ(r, R)due to the mass ratio between nuclea and electrons.

Setting ψn an approximation of ψ(·, ·, tn) the Crank-Nicolson scheme writes:

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Figure 5: Initial data: probability |ψ|2(R = R0)

iψn+1/2 − ψn

∆tS= −rψ

n+1/2

4+

(Vc

2+ r ·En+1/2

)ψn+1/2 − rψ

n

4+

(Vc

2+ r ·En

)ψn,

iψn+1 − ψn+1/2

∆tS= −Rψ

n+1

2mp+Vi

2ψn+1 − Rψ

n+1/2

2mp+Vi

2ψn+1/2.

This approach is very classical and allows in particular to preserve, when the electric field is zero, the ℓ2−norm of ψ and is of order 2. Numerically, the resolution of the sparse linear system coming from the semi-implicit Crank-Nicolson scheme is given by a LU-preconditionned GMRES iterative method (see [36]). Thestorage is a compressed row storage (C.R.S.).

3.1.2 Parallelization of the TDSE beyond Born-Oppenheimer approximation

The operator splitting proposed above induces a naturel parallelization. Indeed, the “electronic” step:

n+1/2rj ,k − ψn

rj ,k

∆tS= −rψ

n+1/2rj ,k

4+

(Vc

2+ r ·En+1/2

n+1/2rj ,k −

rψnrj ,k

4+

(Vc

2+ r ·Enψn

rj ,k

)ψn

rj ,k,

has to be computed for all k ∈ −M, · · · ,M (that is for all R in [0, R∞]), but the differential Laplaceoperator is independent of R (then of k). So that we have to solve M := 2M + 1 independent sparse linearsystems. These numerical resolutions are done using a Krylov iterative method with initial data dependingon k.

Pψn+1k = Qψn

k , k ∈ −M, · · · ,M, ∀rj

where P and Q are some matrices of MN ,N (C). We then sample −M, · · · ,M on the nodes of the parallelcomputer. In term of algorithmic complexity each linear system preconditionned resolution is of orderO(N 3/2). As the number of resolutions is equal to the number of cellsM in the R−grid the total algorithmiccomplexity at each iteration is given by O(MN 3/2). By independence of the linear systems, and for a numberof processors equal to p, each processor computes O(MN 3/2/p) operations.Considering now the parallelization of the nuclear step: k ∈ −M, · · · ,M

iψn+1

rj ,k − ψn+1/2rj ,k

∆tS= −Rψ

n+1rj ,k

8mp+Vi

2ψn+1

rj ,k −Rψ

n+1/2rj ,k

8mp+Vi

n+1/2rj ,k , ∀rj

the processing is different due to the fact that applying the previous parallelization would be very costly in3D (as the number of a cells is of order N 3). We propose an approach based on the fact that numerically

9

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Proc 1 Proc 2 Proc 3 Proc 4

Resolution for Resolution for Resolution for Resolution for

k=−M/2+1,...,0 k=0,1,...,M/2 k=M/2+1,...,M−1, Mk=−M,−M+1,..k=−M,−M+1,..,−M/2

Figure 6: Sampling

the number of cells in the R−grid is much smaller than in the electronic grid. The numerical nuclear stepcan be summarized by the compact form:

Hψn+1rj

= Kψnrj,

where H and K are matrices of MM,M(C), where M is the number of grid-points in the R-grid. We thenpropose to compute and store the constant operator H−1K. This operation has to be done only one time.We are led to compute N matrix-vector products to deduce the global wavefunction. The natural paral-lelization then consists in the well-known parallelization of the matrix-vector product H−1K × ψn+1. Theuse of this technique implies an adapted numerical code design that is not detailled here.

In 7, we represent the computational time with respect to the number of processors. The simulations usea computer of 576 nodes Dell 1425SC (1152 CPU Intel Xeon, 3.6 GHz) 8 GB RAM / node with infinibandnetwork 4X. The baby-benchmark consists in a one-dimensional laser-molecule interaction with 100 pointsin the “electronic”-grid, 100 points in the “nuclear”-grid.

To conclude this section, here is a detail of the communications between processors for the construction of

200

400

600

800

1000

1200

1400

1600

1800

0 2 4 6 8 10 12 14 16

"CPU time"

Figure 7: Speed-up: 1→ 16 processors

the initial data. To simplify the presentation let us suppose that we use only two processors.

1. The processor 1 (root) first computes φR1(r, R) and λR1

(R) solutions of the eigenvalue problem forthe electronic Hamiltonian, for R ∈ [0, R1[, 0 < R1 < R∞. The second processor (slave) computesφR2

(r, R) and λR2(R) solution of the eigenvalue problem for the electronic Hamiltonian, for R ∈

[R1, R∞].

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2. Then λR2(R) is sent by the slave processor to the root processor in order to compute the solution of

the eigenvalue problem of the nuclear Hamiltonian. This then gives ξ(R) for all R ∈ [0, R∞].

3. Then ξ(R) is sent to the slave in order to construct φR2(r, R)ξ(R) for R ∈ [R1, R∞]. In the same time,

the root processor gets φR1(r, R)ξ(R) for R ∈ [0, R1[.

4. Then the two processors compute the square integral of their wavefunction part that gives I21 and I2

2

respectively for the root and slave processor.

5. Finally the value I2 is sent to the root processor to compute the total integral I =√I21 + I2

2 and theroot processor normalizes its part of the wavefunction: φR1

(r, R)ξ(R)/I,

6. and sends to the slave processor I for normalization: φR2(r, R)ξ(R)/I.

At the end, the total normalized wavefunction φ(r, R)ξ(R)/I is then given by

(φR1

(r, R)ξ(R)/I, φR2(r, R)ξ(R)/I

),

3.2 Boundary conditions for the TDSE

Let us now discuss the boundary conditions problem for the TDSE. The initial wavefunction (initial state)support is located in the Schrodinger computational domain center. Then the laser interacts with a moleculeexpanding the wavefunction by ionization whose support that can become very large. Numerically thismeans that it is necessary to discretize the Schrodinger equation in a very large domain. To overcome thiswell-known problem in numerical scattering theory, we have to find an adapted method. The usual idea tocircumvent this difficulty is to reduce the computational domain and to impose some particular numericalboundary conditions on the reduced domain. It is well-known that imposing Dirichlet or Neumann boundaryconditions leads to important numerical oscillations and reflections on the boundary of the domain interactingwith the “physical” waves inside the domain. Absorbing boundary conditions are also used in order to absorbthe numerical spurious reflections. Even if this kind of methods allows effectively to reduce the spuriousreflections, they are often empirical (see for instance [17] in this framework), as some parameters haveto be adapted for each numerical situation. Moreover the spurious reflections are made to vanish but apart of the wavefunction can also be partially absorbed by the absorber. In particular the ℓ2-norm of thewavefunction can hardly be preserved. Ideally we would like to impose particular boundary conditions suchthat the solution of the whole space problem restricted to the compact domain is equal to the solution onthe compact domain (i.e. without spurious reflections).

3.2.1 One-dimensional boundary conditions

To be more precise let us consider the following simplified one-dimensional model without laser:

(i∂tu+u+ V (x) · u

)(x, t) = 0,

u(x, 0) = u0(x), u0 ∈ H1(R).

We suppose that the supports of u0 and V are strictly included in a compact set Ω. One then considers thedomain Ω× [0, T ] with Ω ⊂ Ω and one denotes by Γ the boundary of Ω. One then looks for v solution of

(i∂tv +v + V (x) · v

)(x, t) = 0, x ∈ Ω,

B(x, ∂x, ∂t)v(x, t) = 0, x ∈ Γ,

v(x, 0) = u0(x), x ∈ Ω, u0 ∈ H1(R).

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Γu defined on R

v defined on Ω

Ω

u = vΩ

3

Figure 8: Dirichlet-Neumann conditions

such that

u|Ω×[0,T ]= v. (10)

The main problem consists then in finding an adequate (pseudo-)differential boundary operator B on Γ suchthat (10) occurs. As it well-known these conditions, called Dirichlet-Neumann, are nonlocal in time (and inspace in multidimension). To be more precise we can rewrite this system as follows:

(i∂tv +v + V v)(x, t) = 0, x ∈ Ω,

∂nv = Dv, x ∈ Γ,

v(x, 0) = u0(x), u0 ∈ H1(R).

with n the outward normal of Γ and ∂n is the trace operator on Γ. And D is a first order nonlocal pseudo-

−Γ Γ

Γnn−Γ Molecular domain

Figure 9: Outward normals

differential operator. One then has to solve a TDSE outside Ω. To do this one uses the Laplace transformin time, the Sommerfeld radiation condition and the solution of the second order complex ODE. One canfinally obtain:

v = −eiπ/4

∫ t

0

∂nv(xΓ, τ )√π(t− τ )

dτ on Γ. (11)

So that one has the following set of continuous equations:

(i∂tv +v + V · v)(x, t) = 0, x ∈ Ω,

v(x, t) = −eiπ/4∫ t

0

∂xv(xΓ, τ )√π(t− τ )

dτ, x ∈ Γ,

v(x, t) = eiπ/4∫ t

0

∂xv(−xΓ, τ )√π(t− τ )

dτ, x ∈ −Γ.

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This approach has been previously described in particular in [6], and some numerical issues can be found in[7], [8]. As unfortunately these conditions are nonlocal in time, some numerical methods have been devotedto find effective numerical approximations. At each iteration we can for instance decompose the integral(11) as the sum of a local part and historical part as proposed in [27]. Thus at time tn+1:

v(xΓ, tn+1) = −eiπ/4

∫ tn+1

0

∂xv(xΓ, τ )√π(tn+1 − τ )

dτ.

is decomposed into

v(xΓ, tn+1) = −eiπ/4

∫ tn

0

∂xv(xΓ, τ )√π(tn+1 − τ )

dτ − eiπ/4

∫ tn+1

tn

∂xv(xΓ, τ )√π(tn+1 − τ )

dτ.

One can then compute the historical part eiπ/4∫ tn

0

∂xv(xΓ, τ )√π(tn+1 − τ )

dτ using an explicit approximation and to

implicit the local part eiπ/4∫ tn+1

tn

∂xv(xΓ, τ )√π(tn+1 − τ )

dτ . Such an approach is known to reduce drastically the

artificial reflections on the boundaries (see again [27]).

Now we present some ideas related to the boundary conditions for the TDSE coupled with a laser. Asthe laser operator does not depend on the inter-nuclear distance R we suppose it here constant, so that theequation we consider here simply writes:

i∂tψ(y, t) =(− 1

2y + V (y) + yE(t)

)ψ(y, t). (12)

It is important to note that the term V (y)+ yE(t) has not a compact support, so that it is no more possibleto solve “easily” the free potential TDSE outside a bounded domain as done above. Indeed, outside a givendomain containing the support of the Coulomb potential, the equation writes:

i∂tψ(y, t) =(− 1

2y + yE(t)

)ψ(y, t).

Using as above a Laplace transform in time would lead to a convolution product between the Laplacetransforms of E and ψ. We then propose not to solve exactly this equation but to give an approximatecondition based on a splitting operator. With the known solution on the boundaries at time tn we searchfor it at time tn+1 splitting the equation (12):

i∂tψ(y, t) = yE(t)ψ(y, t), x ∈ Γ, ]tn, tn+1/2],

i∂tψ(y, t) =(− 1

2y + V (y)

)ψ(y, t), x ∈ Γ, ]tn+1/2, tn+1].

The first equation provides the following solution:

ψ(xΓ, tn+1/2) = e−ixΓ

R

tn+1

tn E(s)dsψ(xΓ, tn).

To approximate the second one we consider the solution at time tn+1 when the laser is null:

ψ(xΓ, tn+1) = −eiπ/4

∫ tn+1

0

∂xψ(xΓ, τ )√π(tn+1 − τ )

dτ.

We then decompose this integral in two parts corresponding to the local and historical parts.

ψ(xΓ, tn+1) = −eiπ/4

∫ tn

0

∂xψ(xΓ, τ )√π(tn+1 − τ )

dτ − eiπ/4

∫ tn+1

tn

∂xψ(xΓ, τ )√π(tn+1 − τ )

dτ.

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The historical part depends on ψ at time tn and is then known at time tn+1. For the local part we useψ(xΓ, t

n+1/2) in order to compute ψ(xΓ, tn+1). More precisely and following some approximations proposed

by Greengard in [27], the historical part is approximated by:

−eiπ/4

∫ tn

0

∂xψ(xΓ, τ )√π(tn+1 − τ )

∼ −eiπ/4M∑

j=1

wjcj(n),

where(cj(n)

)j

and (wj)j are sequences described in [27]. The local part is then approximated a by Gauss-

Legendre formula,

eiπ/4

∫ tn+1

tn

∂xψ(xΓ, τ )√π(tn+1 − τ )

dτ ∼ −eiπ/4

√∆tn+1

π

(αψ

n+1/2x,Γ + (2− α)ψn

x,Γ

), 0 6 α 6 2.

With such an approach we then have the following boundary conditions. If J denotes the last cell index:J∆x = xΓ ∈ Γ.

ψn+1J = −eiπ/4

( ∑Nc

j=1 wjcj(n) +

√∆tn+1

π

(αψ

n+1/2J − ψn+1/2

J−1

∆xe−ixΓ

R

tn+1

tn E(s)ds + (2− α)ψn

J − ψnJ−1

∆x

)),

cj(n+ 1) = es2j∆tn

cj(n) +∆tn

2

(e−s2

j∆tn+1 ψnJ − ψn

J−1

∆x+ e−s2

j (∆tn+∆tn+1)ψn+1

J − ψn+1J−1

∆x

).

With then,

ψn+1/2J = e−ixΓ

R

tn+1

tn E(s)dsψnJ , ψ

n+1/2J−1 = e−ixΓ

R

tn+1

tn E(s)dsψnJ−1.

Note that by induction, the historical part depends also on the laser. A symmetric approach is proposedin −xΓ. These 2 discrete equations in −J and J close the Crank-Nicolson system. Then the hermi-tian structure of the sparse matrix is lost so that a GMRES method is used to solve the linear system.Our strong assumption consists then in searching for a numerical boundary condition that behaves like∫ t

0∂yψ(x, τ)/

√π(t− τ )dτ . It is straightforward to apply the technique presented above in the velocity

gauge as we do in the following numerical example.To illustrate this technique we propose a simple benchmark on the TDSE for fixed R, in the velocity gauge(4). We suppose that the Coulomb potential is equal to zero and the electron function solutions are Volkovstates [17], [14].

(i∂tu+ ∂xxu+ iA(t)∂xu(x, t) = 0, x ∈ R, t > 0,

u(x, 0) = u0(x), u0 ∈ H1(R).

The benchmark we propose is as follows. The space grid τ contains 100 cells with a space step equal to 0.1.The convective part is approximated using an upwind scheme and the diffusive one is approximated using aCrank-Nicolson scheme. The computational domain is divided in 3 equal regions. At each internal frontier(located in x1 = −12.5 and x2 = 12.5), we impose Dirichlet-Neumann boundary conditions combined withDirichlet boundary conditions for the convection operator. The laser field is stated in fig. 10. Note that theexact solution always remains inside τ ; we can then apply Dirichlet boundary conditions at the extremalboundaries located in x = −25 and x = 25. We observe a good behavior of the numerical solution withrespect to the exact one (fig. 10) despite the coarsity of the grid: after 185 iterations (2 laser cycles), inℓ∞-norm the relative error between the exact and approximate solution is 6%.

3.2.2 Extension to cubic domains

As discussed in the one-dimensional case above, the treatment of the boundary conditions is crucial in orderto limit the computational complexity. In practice adequate boundary conditions allow to limit the size of

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10

20

30

40

50

60

0 50 100 150 200 250 300−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

7

ZONE 1 ZONE 2 ZONE 3

Dirichlet−Neumann + Laser boundary conditions

−25 −20 −15 −10 −5 0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Initial dataNumerical solutionExact solution

Figure 10: Laser field - Comparison between the exact and numerical solutions with coupled Dirichlet-Neumann conditions

the computational box for the TDSE. Thus instead of discretizing R3, we can reduce the domain of compu-tation to a “small” box and then can reduce the algorithmic complexity. The Dirichlet-Neumann boundaryconditions proposed are a possible solution. However a direct application of the method seen above willintroduce some numerical instabilities.

We next consider the computational domain in a cubic box. The Crank-Nicolson scheme for the TDSErequires boundary conditions to be imposed at the cube faces. These faces are planar it is then trivial toextend the modified Dirichlet-Neumann conditions seen above. We denote by Γ1, Γ2, Γ3, Γ4, Γ5, Γ6 the sixfaces of the domain, located in x = xΓ1

, x = xΓ2, y = yΓ3

, y = yΓ4, z = zΓ5

, z = zΓ6. Then, supposing that

the propagation is one-dimensonal (in z′) and polarized in y the boundary conditions are:

−∂nΓ1u(xΓ1

, y, z, t) = eiπ/4∫ t

0

∂xu(xΓ1, y, z, τ)

√π(t− τ )

dτ, (x, y, z) ∈ Γ1, t > 0,

−∂nΓ2u(xΓ2

, y, z, t) = eiπ/4∫ t

0

∂xu(xΓ2, y, z, τ)

√π(t− τ )

dτ, (x, y, z) ∈ Γ2, t > 0,

−∂nΓ3u(x, yΓ3

, z, t) = eiπ/4∫ t

0

∂yu(x, yΓ3, z, τ)

√π(t− τ )

dτ, (x, y, z) ∈ Γ3, t > 0 + laser conditions,

−∂nΓ4u(x, yΓ4

, z, t) = eiπ/4∫ t

0

∂yu(x, yΓ4, z, τ)

√π(t− τ )

dτ, (x, y, z) ∈ Γ4, t > 0 + laser conditions,

−∂nΓ5u(x, y, zΓ5

, t) = eiπ/4∫ t

0

∂zu(x, y, zΓ5, τ )

√π(t− τ )

dτ, (x, y, z) ∈ Γ5, t > 0,

−∂nΓ6u(x, y, zΓ6

, t) = eiπ/4∫ t

0

∂zu(x, y, zΓ6, τ )

√π(t− τ )

dτ, (x, y, z) ∈ Γ6, t > 0.

The coupling with the laser field is done the same way as for 1D. We extend therefore the idea developpedin 1D on Γ3, Γ4. Let us consider the conditions on Γ3 at time tn:

u(x, yΓ3, z, tn+1) = eiπ/4

∫ tn+1

0

∂zu(x, yΓ3, z, τ)

√π(t− τ )

dτ, (x, z) ∈ Γ3.

Numerically:

u(x, yΓ3, z, tn+1) = eiπ/4

∫ tn

0

∂zu(x, yΓ3, z, τ)

√π(t− τ )

dτ + eiπ/4

∫ tn+1

tn

∂zu(x, yΓ3, z, τ)

√π(t− τ )

dτ. (13)

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For the historic part we use the previous computed solutions on the boundary and depending on the laser.In this goal we then compute the numerical solution of

ut + yEz′(t)u = 0, (x, y, z) ∈ Γ3, t ∈ [tn, tn+1[.

Between [tn, tn+1[ we have

u(x, yΓ3, z, tn)e−i

R

tn+1

tn E(s)ds denoted by un+1/2Γ3

(x, z).

We then use this solution in order to approximate the local part in (13) using the same approach as above:

eiπ/4

∫ tn+1

tn

∂zu(x, yΓ3, z, τ)

√π(tn+1 − τ )

dτ ∼ −eiπ/4

√∆tn+1

π

(4

3u

n+1/2Γ3

(x, z) +2

3un

z (x, yΓ3, z)

), ∀(x, z) ∈ Γ3

However the computational domain is singular and these conditions are no more valid on the plane inter-sections and corners of the domain. We propose a more adequate way to treat the boundary conditions inmultidimensions. In order to limit the effect of the singularities we propose to work on a “more regular”grid, a circular cylinder.

Γ

ΓΓΓ

Γ

Γ

6

X

Y

Z

1

2

3 4

5

Figure 11: Cubic domain

3.2.3 Using real 2D Dirichlet-Neumann boundary conditions - Application to 3D domains

In this section we examine some possible solutions to improve the boundary conditions treatment in multiD.The numerical computation of these boundary conditions are in progress. As seen above the singularitiesof the 3D domain (cubic box) lead to construct truly multi-dimensional Dirichlet-Neumann boundary con-ditions. In this goal let us recall some important results. Considering the 2D problem (with r = (x, y)here).

i∂tu = −(r + V (r)

)u, r ∈ R2,

u(r, 0) = u0(r).

To determine the Dirichlet-Neumann boundary conditions we have to solve outside a compact domain Ωcontaining the supports of u0 and V , the equation:

i∂u = −ru, r ∈ ΩC .

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The solution is obtained using the Laplace transform in space (see [38]) and the Sommerfeld radiationcondition that ensures unicity and positivity of the energy flux and is nonlocal in space and in time. Thatis the condition on each point of the boundary depends on all the boundary points at all previous times.It is then computationally very complex to apply directly such a method. It is necessary to establish anappropriate numerical method effective in space and time or at least in time or in space. With this aim,we propose to use the approach of Antoine and Besse as a starting point. They proved that the Dirichlet-Neumann conditions (nonlocal in space and time) can be approximated (to order 3) by some differentialoperators. That is the spatial nonlocality is replaced by local differential operators even if the formulationremains nonlocal in time. In particular they proved in 2D the following theorem (see [7]):

Theorem 3.1 [Antoine & Besse, ’01] The TDSE with artificial boundary conditions of Dirichlet-Neumann

type of order m/2 with m ∈ 1, · · · , 4 is defined by the initial boundary valyue problem:

i∂tu(r, t) +ru(r, t) = 0, (r, t) ∈ Ω× R∗+,

∂nu(r, t) + Tm/2u(r, t) = 0, (r, t) ∈ Ω× R∗+,

u(r, 0) = u0(r), r ∈ Ω.

where Tm/2 are pseudo-differential in time and differential in space operators given by:

T1/2u = e−iπ/4∂1/2t u, ∂Ω× R∗

+

T1u = T1/2u+κ

2u, ∂Ω× R

∗+

T3/2u = T1u− eiπ/4(κ2

8+

1

2∂Ω

)I1/2t u, ∂Ω× R

∗+,

T2u = T3/2u+ i(κ3

8+

1

2∂s(κ∂s) +

∂κ

8

)Itu, ∂Ω× R

∗+

.

The operator∂Ω is the Laplace-Beltrami operator and κ is the local curvature of ∂Ω. The pseudo-differential

convolution I1/2t and ∂

1/2t are defined by:

∂1/2t u(t) =

1√t

d

dt

∫ t

0

u(y)√t− ydy, I

1/2t u(t) =

1√π

d

dt

∫ t

0

u(y)√t− ydy.

These conditions in the twodimensional case allow to reduce drastically the spurious reflexions on ∂Ω (see[8]). The extension of these formulas to the 3D case are of course possible and are based on the Laplacetransform in time of the TDSE, the resolution of the obtained Helmoltz equation under the Sommerfeldradiation condition and then by inverse Laplace transform. We then propose to combine these boundaryconditions with the electric field operator. In order to use the 2D approach, we propose to work in cylindricalcoordinates for the 3D problem .

We suppose for the presentation that the nuclei are fixed (Born-Oppenheimer approximation) and the po-tential are set to zero. This assumption is only imposed for the sake of simplicity. This has no consequenceon the method for the global problem with potentials and with moving nuclei. In cylindrical coordinates ourequations write:

(i∂t +r +

1

r∂r +

1

r2θ +z

)u(r, θ, z, t) = 0, (r, θ, z, t) ∈ R+ × [0, 2π[×R× R∗

+,

u(r, θ, z, 0) = u0(r, θ, z), (r, θ, z) ∈ R+ × [0, 2π[×R.

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The spatial variables are then θ ∈ [0, 2π], r ∈]0, Rmax[ and z ∈ [−zmax, zmax]. We denote by Γθ and Γz

and Γ−z the following surfaces:

Γθ =(r, θ, z), such that r = Rmax, θ ∈ [0, 2π[, z ∈ [−zmax, zmax]

,

Γz =(r, θ, z), such that r ∈]0, Rmax], θ ∈ [0, 2π[, z = zmax

,

Γ−z =(r, θ, z), such that r ∈]0, Rmax], θ ∈ [0, 2π[, z = −zmax

.

Finally Γ = Γθ ∪Γ−z ∪Γz. The finite difference discretization will be discussed later. The natural Dirichlet-Neumann boundary conditions are then given by extension of the previous formulas by:

−∂nΓ−zv(r, θ, z, t) = eiπ/4

∫ t

0

∂xu(r, θ, zΓ−z, τ )

√π(t− τ )

dτ, (r, θ, z) ∈ Γ−z,

−∂nΓzv(r, θ, z, t) = eiπ/4

∫ t

0

∂xu(r, θ, zΓz, τ )

√π(t− τ )

dτ, (r, θ, z) ∈ Γz,

−∂nΓθv(r = Rmax, θ, z, t) + Tm/2u(r = Rmax, θ, z, t) = 0, (r, θ, z) ∈ Γθ.

In the particular case of a circular cylinder the Laplace-Beltrami operator Γθis given by θ.

Suppose now that a laser polarized in z independant of θ interacts with the molecule. Then for (r, θ, z, R, t) ∈R

+ × [0, 2π[×R× R+ × R

∗+:

(i∂t +r +R +

1

r∂r +

1

r2θ +

1

R+z + Vc(r, θ, z, R) + Vi(R) + zE(t)

)u(r, θ, z, R, t) = 0,

u(r, θ, z, R, 0) = u0(r, θ, z, R).

We suppose that the nuclei are fixed (Born-Oppenheimer approximation). This also corresponds to theelectronic step in the global scheme with moving nuclei:

(i∂t +r +

1

r∂r +

1

r2θ +z + Vc(r, θ, z) + zE(t)

)u(r, θ, z, t) = 0, (r, θ, z, t) ∈ R+ × [0, 2π[×R× R∗

+,

u(r, θ, z, 0) = u0(r, θ, z), (r, θ, z) ∈ R+ × [0, 2π[×R.

Then, the coupling of the Dirichlet-Neuman boundary conditions is direct using the previous study. Indeedthe TDSE with Dirichlet-Neumann boundary conditions on Γθ can be written as:

(i∂t +r +

1

r∂r +

1

r2θ + Vc(r, θ, z)

)u(r, θ, z, t) = 0, (r, θ, z, t) ∈ [0, Rmax[×[0, 2π[×R× R

∗+,

−∂nΓθu(r = Rmax, θ, z, t) + Tm/2u(r = Rmax, θ, z, t) = 0, (r, θ, z) ∈ Γθ, t > 0.

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As[r,z

]=

[θ,z

]= 0. So that we solve:

(i∂t +r +

1

r∂r +

1

r2θ + Vc(z, θ, z)

)u(r, θ, z, t) = 0, (r, θ, z, t) ∈ [0, Rmax[×[0, 2π[×R× R∗

+

−∂nΓθu(r = Rmax, θ, z, t) + Tm/2(r = Rmax, θ, z)u(r, θ, z, t) = 0, (r, θ, z) ∈ Γθ, m > 1, t > 0,

(i∂t +z + Vc(z, θ, z) + yE(t)

)u(r, θ, z, t) = 0, (r, θ, z, t) ∈ [0, Rmax[×[0, 2π[×R× R∗

+

−∂nΓ−zu(r, θ, z, t) = eiπ/4

∫ t

0

∂xu(r, θ, zΓ−z, τ )

√π(t− τ )

dτ, (r, θ, z) ∈ Γ−z, t > 0 + laser conditions,

−∂nΓzu(r, θ, z, t) = eiπ/4

∫ t

0

∂xu(r, θ, zΓz, τ )

√π(t− τ )

dτ, (r, θ, z) ∈ Γz, t > 0 + laser conditions.

In the previous formula the pseudo-differential operators Tm/2 are defined in [7] and theorem (3.1). The

ΓLz

Γ

Γθ nΓθ

H+

H+

z

−z

z

E(t)

Figure 12: 3D domain

discretization we propose is again very classical and is based on the three-point Crank-Nicolson scheme. Inthe following we denote by (ri)i, (θj)j , (zk)k the grid points.

• If r 6= 0:

iψn+1i,j,k = iψn

i,j,k + ∆tM

(1

2rψ

n+1i,j,k +

1

2rψ

ni,j,k +

1

2r2iθψ

n+1i,j,k+

1

2r2iθψ

ni,j,k +

1

2ri∂ψn+1

i,j,k +1

2ri∂ψn

i,j,k +1

2zψ

n+1i,j,k +

1

2zψ

ni,j,k

).

• If r = 0 it is well known that the equation becomes:

(i∂t + 2r +z

)ψ(r, θ, z, t) = 0, (r = 0, θ, z, t) ∈ R

+ × R× [0, 2π[×R∗+,

so that:

iψn+10,j,k = iψn

0,j,k + ∆tM

(rψ

n+10,j,k +rψ

n0,j,k +

1

2zψ

n+10,j,k +

1

2zψ

n0,j,k

)

with obviously

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• If r 6= 0

θψi,j,k

r2i∼ψi,j+1,k − 2ψi,j,k + ψi,j−1,k

r2i ∆θ2, ∀j, i 6= 0,

∂rψi,j,k

ri

∼ψi+1,j,k − ψi−1,j,k

2ri∆r, ∀j, i 6= 0,

rψi,j,k ∼ψi,j,k+1 − 2ψi,j,k + ψi,j,k−1

∆r2, ∀j, k, zψi,j,k ∼

ψi+1,j,k − 2ψi,j,k + ψi−1,j,k

∆z2, ∀i, j

• If r = 0, then the approximation is given by 2rψ(0, zk) ∼ 4ψ(r1, zk)− ψ(0, zk)

∆r2, ∀k.

This corresponds to a free Hamiltonian Dirichlet-Neumann boundary conditions and has to be combinedwith the laser is polarized in the y-direction, the combination consists in coupling the laser field with theboundary conditions on Γz and Γ−z. On Γθ as long as the laser polarizarition remains one-dimensional andpolarized in y we can keep the same conditions than above.

3.3 Maxwell equations

The Maxwell equations are approximated using the Yee scheme [39] with Mur’s absorbing boundary condi-tions [30]. This is a stable order 2 scheme under a CFL condition:

∆tM 6 1/c

√1

∆x2M ′

+1

∆y2M ′

+1

∆z2M ′

. (14)

In vacuum we then use the classical Yee scheme and we add the polarization derivative in time ∂tP, whenthe laser interacts with the gas. An important issue is to ensure that inside the gas the divergence conditionon D = E + 4πP/c, ∇ ·D = 0 is effectively verified at the discrete level.

Proposition 3.1 The condition1 ∇h · Dnh = 0 is numerically verified by the modified Yee scheme for all

n ∈ N provided that ∇h ·D0h = 0.

Proof. It is well-known that the Yee scheme in the vacuum leads to

∇h ·En+1h = ∇h ·En

h.

This allows to prove that the Yee scheme verifies ∇hEnh = 0 provided that ∇hE

0h = 0. In the non-

homogeneous case, the same calculations lead us to the following discrete equation:

∇h ·En+1h + 4π∆tM∇h

(∂tP

n+1h

)= ∇h ·En

h.

That is suppose P regular enough in time and space (see next section):

∇h ·En+1h + 4π∆tM∂t

(∇hP

n+1h

)= ∇h ·En

h.

Now using the approximation of the partial derivative in time we have used:

∂t

(∇hP

n+1h

)=∇hP

n+1h −∇hP

nh

∆tM,

then

∇h ·En+1h + 4π∆tM

(∇hPn+1h −∇hP

nh

∆tM

)= ∇h ·En

h.

1Eh, Dh, ∇h denote the discrete versions of E, D and ∇.

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So that:

∇h ·En+1h + 4π∇hP

n+1h = ∇h ·En

h + 4π∇hPnh.

Then, provided that ∇h ·D0h = 0, it is true for all n.

Typically the kind of initial data we consider is as fig. 13. And obviously we have:

Theorem 3.2 The global scheme is an order 2 scheme, stable under the CFL condition (14).

Figure 13: Initial electric field - x′, y′-plan (z′ is fixed) - λ ∼ 800nm and ∼ 6 cycles

3.4 Polarization computation

The polarization computation is central in this work as it allows to couple the Maxwell equations and TDSE.As said above the polarization is deduced from the TDSE by (7). In practice in each Maxwell-cell we willsolve numerically one single TDSE and we will deduce the local polarization for a physical volume of size∆x′M ×∆y′M ×∆z′M and containing n(r′)∆x′M ×∆y′M ×∆z′M molecules. For a one-dimensional gas prop-

E(z’,t)

z’

: H2+ molecule

y

Figure 14: Maxwell and Schrodinger scales (1D)

agation with a 100 cycles laser with a wavelength of 800nm interacting with molecules, we would have tosolve numerically at each iteration 8000(80×100) TDSE, for a Maxwell cell size equal to ∆z′M = 10nm (oneTDSE per Maxwell cell). As noticed above, the non-Born-Oppenheimer TDSE we have to solve is 4D andis then costly in CPU time. We then propose a technique based on a simple Taylor expansion that allows to

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reduce the numerical cost, supposing the polarization smooth in space. As discussed above, this assumptionseems a priori valid when the electric field is smooth enough and its wavelength encompasses a large numberof molecules (see fig. 14).

3.4.1 Decreasing the polarization computation cost

We make a partition of the domain ΩM = ∪i=1,··· ,I−1[ω′

i, ω′

i+1[ with I a small integer. We suppose that eachsub-domain ωi contains a sufficiently large number of Maxwell cells. For each sub-domain ωi, we choosea reference cell r′i,0 (located for example in the center of the sub-domain). For this cell we compute thecorresponding TDSE which in this framework is written as,

ω1 ω ω ω

ωωωω

2 43

5 6 7 8

Maxwell grid

ω iSub−domains:

H2+ molecules

Z’

Y’

H2+ molecules of reference

Figure 15: Decomposition of the domain

i∂tψri,0′(r, R, t) =(− 1

2r + V (r, R)− 1

mpR +

1

R+ r ·E

r′

i,0(t)

r′

i,0(r, R, t).

and the corresponding polarization:

P(r′i,0, t) = −n(r′i,0)

∫ψ

r′

i,0(r, R, t) · r · ψ∗

r′

i,0(r, R, t)drdR. (15)

For every other cell located in r′ ∈ ωi we have:

P(r′, t) = P(r′i,0, t) +∇r′P(r′i,0, t) · (r′ − r′i,0) +O

((r′ − r′i,0)

2).

Then we deduce the value of the polarization for every cell of the sub-domain ωi. It is necessary to compute∇r

′P(r′i,0, t). With this aim, by derivation in r′ of (15), we obtain

∇r′P(r′i,0, t) = −n(r′i,0)

∫∇r

′ψr′

i,0(r, R, t) · r · ψ∗

r′

i,0(r, R, t)drdR−

n(r′i,0)∫ψ

r′

i,0(r, R, t) · r · ∇r

′ψ∗r′

i,0(r, R, t)drdR−

(∇r′n)(r′i,0)

∫ψ

r′

i,0(r, R, t) · r · ψ∗

r′

i,0(r, R, t)drdR.

Then to compute ∇r′P(r′i,0, t) we compute ∇r

′ψr′

i,0(r, R, t) and then we solve the following system obtained

by derivation in r′ of the TDSE,

i∂t

(∇r

′ψr′

i,0(r, R, t)

)=

(− 1

2r + V (r, R)− 1

4mpR +

1

R+ r ·E

r′

i,0(t)

)∇r

′ψr′

i,0(r, R, t)+

i(∇r

′Er′

i,0(t)

)· r · ψ

r′

i,0(r, R, t).

(16)

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The numerical scheme to solve the previous system is simply deduced from our Crank-Nicolson scheme:

(∇r

′ψr′

i,0

)n+1=

(∇r

′ψr′

i,0

)n+1

2+

(∇r

′ψr′

i,0

)n

2− i

2Hn+1

(∇r

′ψr′

i,0

)n+1 − i

2Hn∇r

′ψnr′

i,0+ i

(∇r

′Er′

i,0

)n+1 · r · ψn+1r′

i,0. (17)

In (17) the quantity ψn+1r′

i,0is supposed to have been previously computed by the numerical scheme for the

TDSE with Hn given by:

Hn =(− 1

2y + V (r, R)− 1

mpR +

1

R+ r ·En

r′

i,0

)(1, 1, 1)T , n > 0.

Finally, for each sub-domain ωi we compute ψr′

i,0(r, R, t) and∇r

′ψr′

i,0(r, R, t) to deduce linearly from P(r′i,0, t)

the polarization for each cell located in r′ of this sub-domain. The error due to this process is naturally hereat order one. It is possible to increase the order using a higher order Taylor expansion. In this case, it wouldbe necessary to compute ∇2

r′r′ψr

i,0(r, R, t) obtained by double derivation in r′ of the TDSE. We can then

deduce easily the following result:

Proposition 3.2 Supposing the polarization and the molecular density smooth enough, the approximation

proposed above in each r′ ∈ ωi, is of order 1 or 2 depending on the Taylor expansion order.

In practice we can expect in some cases, a strong reduction of the number of TDSE to solve. Typically thesub-domains size will depend on the transmitted electric field harmonics, then on the intensity and frequencyof the incoming laser. The higher the harmonics will be, the smaller the size of the sub-domains will bechosen.Taking into account the previous method to reduce the computational cost of the polarization, that is thenumber of TDSE to solve, we have the following temporal scheme from tn to tn+1, denoting by I − 1 thenumber of sub-domains, with I << number of cells in the Maxwell grid:

En+1 ← En, ∀r′ ∈ Ω,

ψn+1r′

i,0← ψn, for r′i,0 ∈ ωi, ∀i ∈ [1, I],

Pn+1r′

i,0← ψn+1

r′

i,0for r′i,0 ∈ ωi, ∀i ∈ [1, I],

∇r′ψn+1

r′

i,0← ψn+1

r′

i,0and eq. (16), for r′i,0 ∈ ωi, ∀i ∈ [1, I],

∇r′Pn+1

r′

i,0← ∇r

′ψn+1r′

i,0for r′i,0 ∈ ωi, ∀i ∈ [1, I],

Pn+1r′ ← Pn+1

r′

i,0, (∇r

′n)(r′i,0) and ∇r′P n+1

r′

i,0, ∀r′ ∈ ωi, and ∀i ∈ [1, I].

Note that from another point of view it would be interesting to investigate a coupling of the polarizationsolution with a local (in each Maxwell-cell) slowly varying envelope approximation (see section 2.).

3.4.2 Parallelization of the polarization computation in the Born-Oppenheimer approxima-

tion

Many approaches are possible to parallelize the Maxwell-TDSE. One of the most effective one is as follows.To simplify the presentation let us suppose that the Maxwell-grid possesses N cells and that we solve NTDSE (one by Maxwell-cell) with a code running on N processors. We also suppose in this paragraph thatthe Maxwell-domain is not “very large” so that it is possible to solve the Maxwell equations on one singleprocessor. At each temporal Maxwell iteration, each processor solves one single Schodinger equation, and

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computes the corresponding polarization. Then it sends it to the root processor. Then with the polarizationon the whole domain the root processor can solve the non-homogeneous Maxwell equations (recall that inour configuration (small Maxwell-domain) the CPU cost of the Maxwell equations computation is negligiblecompared to the TDSE computation cost). Then the root processor sends to the slaves the updated electricfield (this process is summarized in fig. 16). Such a simple parallelization allows us to obtain a very goodspeed-up. Note also that the data are distributed between the computer nodes allowing to consider a largenumber of molecules with sufficiently large spatial grids. Fig. 17 represents for a given mesh: in abscissa

processor2processor1 processor3 processor4

1 2 3 4

processor1

Slaves send the polarization to the root processor (processor 1)

processor1 processor2 processor3 processor4

Root processor sends the updated electric field to slaves

Resolution of the Maxwell equations by the root processor

TIME

n n n n

n+1 n+1 n+1 n+1

n+1

Resolution of the Schrodinger equations then the polarizationP(r , t ) P(r ,t ) P(r ,t ) P(r ,t )

E(r,t )

P(r,t ) P(r,t ) P(r,t ) P(r,t )

Figure 16: Parallelization for 4 TDSE

the number of TDSE (4, 16, 64, 256 TDSE) to solve also equal to the number of processors (one equationby processor) and in ordinate the real time for the 3D code to solve the corresponding Maxwell-TDSE. Theideal configuration would be “real time=constant”, corresponding to a speed-up equal to the number ofprocessors. The coupling of our proposed reduced computation of the polarization with this parallelization

1.5 2 2.5 3 3.5 4 4.5 5 5.5

7.8

7.85

7.9

7.95

8

8.05

8.1

8.15

x 104

Number of processors = number of TDSE (log)

Rea

l com

puta

tiona

l tim

e (in

sec

ond)

4 TDSE

256 TDSE

Figure 17: Computation of the 3D Maxwell-TDSE with one TDSE per processor. In ordinate the real time(in seconds) necessary for the numerical computation.

allows us to extend simulations to a very large number of molecules. Note also that the parallelization isalmost totally independent of the way we solve numerically the Schrodinger and Maxwell equations.

We present next another approach used in particular for large 3D Maxwell-domains. In this case, the par-allelization we propose is also based on a domain decomposition of the Maxwell domain. Let suppose thatwe have N processors and N2 TDSE to solve. We make a partition of the Maxwell domain in N1 = N −N2

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sub-domains. In practice, N1 << N2 as the TDSE are more costly than the Maxwell ones. Suppose for

instance that the gas is totally located in the Nth sub-domain (N 6 N1). The computational steps become:

• the Maxwell equations are solved with processors 1, · · · , N1 on each sub-domain with message passingat the sub-domains interfaces.

• the processor N sends its electric field to the molecules composing the gas that is to processors N1 +1, · · · , N.

• independently, the N2 TDSE are solved and give the associated polarization.

• each processor N1 + 1, · · · , N sends to the Nth processor its computed polarization.

Resolution of the Maxwell equations by processors 0,1,2

Processor 3

Processor 1 Processor 2

Processor 3 Processor 4 Processor 5

Processor 4 Processor 5

Processor 6 Processor 7

Processor 7Processor 6

then polarization by processors 3,4,5,6,7Resolution of the Schordinger equations

TIME

n n

n+1 n+1 n+1 n+1 n+1

1 2 3 4 5

5 4 3 2 1

n+1E(r,t ) E(r,t ) E(r,t )

Processor 0

subdomain 1 subdomain 2 subdomain 3

Processors 3,4,5,6,7 send the updated polarization to processor 2

Processor 2 sends the updated electric field to processors 3,4,5,6,7

P(r ,t ) P(r ,t ) P(r ,t ) P(r ,t )P(r ,t )n n n

n+1 n+1

P(r ,t ) P(r ,t ) P(r ,t ) P(r ,t )P(r ,t )

Schroedinger resolutionMaxwell resolution

Figure 18: Another parallelization approach for 5 TDSE and 8 processors

The temporal process consists in two sub-iterations: computing the Maxwell equations onN1 processors (firsttemporal sub-iteration) and then computing the TDSE on N2 processors (second temporal sub-iteration).See figs. 18 and 19. Note also that such a decomposition allows us to distribute the memory betweenthe nodes of the parallel-computer. This parallelization has been used for the largest computations (seenumerical results section).

4 Numerical examples

In this part we present some results obtained using the numerical and parallel approaches introduced above.We consider Ti:sapphire pulses with intensity I ∼ 1014W/cm2 at 800nm interacting with a H+

2 gas.

We first present two results obtained when solving the Maxwell-TDSE in 1D and 3D for fixed nuclei withone-dimensional propagation. That is we neglect here the transversal effects of the gas on the electric field.The first one fig. 20 represents for a 1D computation, the transmitted electric field harmonics for a prop-agation in a H+

2 media of length equal from 0.4nm (corresponding to 1 TDSE (molecule)), to 12.8nm (32TDSE (molecules)). The Schrodinger equations are solved on a mesh containing 2000 points with a spacestep equal to 0.2a.u. Note that we obtain a quadratic scaling as expected (see [34] for instance) of the lowelectric harmonic intensities as a function of number of molecules from 1 to 32, when we increase the domainlength. The second numerical result fig. 21 is the harmonic intensities of the transmitted electric field fora 3D computation, with a propagation in a H+

2 media of length respectively equal to 15nm (correspondingto 4 TDSE), 60nm (16 TDSE), 240nm (64 TDSE). The Schrodinger equations are solved on a cartesian

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VACUUM

GAS

Processor 0

Processor 1

Processor 2

(M.E. solved by processors 0 −>4)

(S.E. solved by processors 5 −>N)

Processor 4

Processor 3

Figure 19: 3D Maxwell-Schrodinger parallelization

1e-20

1e-15

1e-10

1e-05

1

100000"4 Schroedinger equations"

"16 Schroedinger equations""1 Schroedinger equation"

"Initial electric field""32 Schroedinger equations"

Figure 20: 1D electric harmonic intensities for 0.4nm (1 molecule), 1.6nm (4 molecules), 6.4nm (16 molecules)and 12.8nm (32 molecules).

mesh (101×101×501 points) with a space size equal to 0.2a.u. The one-dimensional Maxwell equations aresolve using space steps equal to 70a.u. The code ran for ∼ 22 hours respectively on 4, 16 and 64 processors(see www.ccs.usherbrooke.ca/mammouth). In this case, the molecules are located on a single line and thepropagation is 1D, that is we here solve a 1D version of the Maxwell equations. Again we remark that thecomputation gives us a quadratic scaling as a function of number of molecules for the low harmonics withrespect to the propagation length.

The next benchmark is the computation of our 3D Maxwell-Schrodinger model with the following datas: theMaxwell domain is a cubic box [0, 2.25×105]3 (in atomic units) with a uniform grid ∆x′M = ∆y′M = ∆z′M =750a.u. (allowing to consider rigorously at least the 20 first transmitted harmonics). The TDSE have been

26

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1e-20

1e-15

1e-10

1e-05

1

100000

0 20 40 60 80 100

"Initial electric field""240nm (64 TDSE)"

"60nm (16 TDSE)""15nm (4 TDSE)"

Figure 21: 3D, H+2 harmonics for 15nm (4 molecules), 60nm (16 molecules), 240nm (64 molecules).

solved using the Crank-Nicolson scheme presented above with grid steps of length ∆xS = ∆yS = ∆zS =0.2a.u. The initial state for the molecules has been computed exactly in the Born-Oppenheimer approxima-tion with R = 3.2a.u. At such distance and λ = 800nm, there is a three-photon ionization resonance for the1σg − 1σu excitation [17]. The initial electric field is presented in fig. 13. In this benchmark we have solved3 × 3 × 56 = 504 TDSE representing the H+

2 molecules located in the middle of the Maxwell domain, in abox of size 2250 × 2250 × 42000 (in atomic units). The total number of processors used in this numericalcomputation is 511. More precisely 504 processors have been used for the TDSE computation and 7 forthe Maxwell equations computation, following the process presented in fig. 18. The real computation timeis 1167 minutes and the physical computational one is 632a.u that is ∼ 15fs. In fig. 22 we represent theelectric field solution propagated in the vacuum and in the final electric field after interaction with gas. Wealso illustrate the modification of the transmitted pulses by substracting the vacuum pulse at the end of thepropagation fig. 23.Next, using thre same datas as above with 5× 5× 10 TDSE (250 molecules), the electric field substractionis given by 24, 25. We expect with this kind of simulations to have a better understanding of filamentation(see [24]) at the microscopical scale. This point will be more deeply studied in a forthcoming paper.

The next benchmark consists in comparing the effect of the molecular density on the transmitted electric

Figure 22: Electric field for a 3D-Maxwell-Schodinger propagation in the vacuum

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Figure 23: Difference between the electric field for a 3D-Maxwell-Schodinger propagation in the gas and inthe vacuum (504 TDSE)

Figure 24: Difference between the electric field for a 3D-Maxwell-Schodinger propagation in the gas and inthe vacuum (250 TDSE)

Figure 25: Difference (250 TDSE) - other cut

field. A circular electric field (fig. 26) propagates through aH+2 -gas of volume equal to 3750×3750×7500a.u.3.

The number of Schrodinger equations to solve is equal to 5 × 5 × 10 = 250 with the same numerical and

28

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physical datas than above (laser frequency, Maxwell domain, R = 3.2a.u., etc). The three cases we considerare for molecular densities equal to n0 = 10−2, n0 = 10−3, n0 = 10−4 (figs. 27,28, 29). We then observeas excepted an amplification of the difference between the transmitted electric field in the vacuum and inthe gas when the molecular density increases (see forthcoming papers for physical study of this phenomenonand consequences on filamentation).

Figure 26: Transmitted Electric field in the vacuum

Figure 27: Difference between the electric field for a 3D-Maxwell-Schodinger propagation in the gas and inthe vacuum (250 TDSE) and n0 = 10−4

We conclude this part on a remark on a numerical phenomenon noticed in [23]. The authors observed forhigh harmonics some minima in the harmonics spectra for H+

2 molecules interacting with very short pulses(few cycles). In our simulations we also observe that these minima that seem not to vanish when the laserpropagates in the gas on a length corresponding to 64 Schrodinger equations that is 0.250µm, figs 30, 31.

5 Conclusion

In this paper, we have presented a Maxwell-Schrodinger model for time dependent laser-matter interactionand some aspects of its numerical computation in 1D and 3D. In particular, we have proposed some methodsto reduce the algorithmic complexity of the numerical problem due to the multiple scales (polarizationcomputation for instance) but also due to the boundary conditions due to dissociative ionization of molecules.This has allowed to us to develop a numerical code for a Maxwell-TDSE system of equations which guaranties

29

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Figure 28: Difference between the electric field for a 3D-Maxwell-Schodinger propagation in the gas and inthe vacuum (250 TDSE) and n0 = 10−3

Figure 29: Difference between the electric field for a 3D-Maxwell-Schodinger propagation in the gas and inthe vacuum (250 TDSE) and n0 = 10−2

1e-35

1e-30

1e-25

1e-20

1e-15

1e-10

1e-05

1

100000

1e+10"1 S.E."

"64 S.E."

Figure 30: One-dimensional Maxwell-Schrodinger. Electric field harmonics

30

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1e-35

1e-30

1e-25

1e-20

1e-15

1e-10

1e-05"1 S.E."

"64 S.E."

Figure 31: One-dimensional Maxwell-Schrodinger. Electric field harmonics. Zoom

scalability of intensities of harmonics as a function of the number of molecules in 1D and 3D, and thestructure of the propagated pulses. Numerical simulations were obtained so far only for fixed R. the non-born-Oppenheimer TDSE for H+

2 has been applied previously to single molecules only [17]. Computationswith our Maxwell-Schrodinger code via equations (5), (7), (6) require Peta-Byte supercomputers. Amongthe important points to investigate note the following one. For a 10 Torr pressure, and supposing anideal molecular equi-distribution, the distance between two molecules is on order of 12nm. For a 800nm-laser,the 67th harmonic wavelength is approximatively equal to 12nm. This is raising new issues about theapplicability of TDSE and macroscopic Maxwell equations.

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APPENDIX - Proof of theorem 2.1

Proof. In the following we will denote by E the laser operator:

E : (r, R, t) 7→ E(r, R, t) = χ(r, R) ·E(t).

In a first time we regularize the potentials, defining

V εc = − 1

√ε2 + x2 + (y −R/2)2 + z2

− 1√ε2 + x2 + (y +R/2)2 + z2

, V εi = − 1√

ε2 +R2

We then have |V εc | 6 |Vc| and |V ε

i | 6 1/R and ∂tVεc = ∂tV

εi = 0. Then

i∂tψε(r, R, t) =

[− 1

2r + V ε

c (r, R) + χ(r, R) ·E(t) + V εi (R)

]ψε(r, R, t).

As remarked above there exists a unique ψε ∈ C0(0, T ;H1∩H1

)as V ε

i , Vεc , χ(r, R)·E(t) ∈ L∞

(0, T ;C2

b (R4)).

We now search for an estimation in H1 of ψε. First recall that:

‖ψε(t)‖2H1∩H1=

R4

|∇yψε|2 + |∇Rψ

ε|2 +(1 + ‖(r, R)T‖22

)|ψε|2.

Then there exists a constant Mp > 0 (= 1 for instance, as mp > 1) such that:

R4

|∇yψε|2

2+|∇Rψ

ε|2mp

+(1 + ‖(r, R)T ‖22

)|ψε| 6 Mp‖ψε(t)‖2H1∩H1

. (18)

The main difficulty consists in finding a positive constant C such that:

‖ψε(t)‖2H1∩H16 C‖ψ0‖2H1∩H1

(19)

Supposing (19) is true and using a compacity argument, there exists a sequence εn such that

ψεn ∗n→∞ u in L∞

(0, T ;H1 ∩H1

).

We finally get

‖ψ(t)‖2H1∩H16 C‖ψ0‖2H1∩H1

.

That proves the existence of a solution in the set described above.

In order to obtain estimation (19) it is necessary to have an estimation of

d

dt

R4

(1 + ‖(r, R)T‖2

)|ψε|2, and

d

dt

R4

|∇yψε|2

2+|∇Rψ

ε|2mp

.

With this aim and as proposed in [11], we multiply the TDSE by(1+ ‖(r, R)T ‖2

)ψε we integrate on R

4 andwe take the imaginary part:

d

dt

R4

(1 + ‖(r, R)T‖2

)|ψε|2 = Im

R4

∇y

[‖(r, R)T ‖2ψε

]∇yψε

2+∇R

[‖(r, R)T ‖2ψε

]∇Rψε

mp.

We easily obtain by derivation and Cauchy-Schwartz

d

dt

R4

(1 + ‖(r, R)T ‖2

)|ψε|2 6

1

2

R4

‖(r, R)T ‖2|ψε|2 +|∇yψ

ε|22

+|∇Rψ

ε|2mp

.

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The same manner we multiply by ∂tψε we take the real part and we integrate over R4. That is

0 =

R4

Re(− ∂tψ

εyψε

2− ∂tψ

εRψε

mp+ V ε

c ψε∂tψ

ε + V εi ψ

ε∂tψε + Eψε∂tψ

ε). (20)

Then

1

2

∫R4 ∂t

|∇yψε|2

2+ ∂t

|∇Rψε|2

mp=

1

2

∫R4(V

εc + V ε

i + E)∂t|ψε|2

=d

2dt

∫R4(V

εc + V ε

i + E)|ψε|2 − 1

2

∫R4 ∂tE|ψε|2.

Then trivially there exists a constant C2

R4

∂tE|ψε|2 6 C2

∥∥∥∂tE

1 + ‖(r, R)T‖2

∥∥∥L∞

‖ψε‖2H1 .

We then obtain the following estimate:

d

dt

R4

|∇yψε|2

2+|∇Rψ

ε|2mp

61

2

R4

(V εc + V ε

i + E)∂t|ψε|2 +1

2

∥∥∥∂tE

1 + ‖(r, R)T ‖2

∥∥∥L∞

‖ψε‖2H1 . (21)

Then setting:

Eεmp

(t) =d

dt

∫R4

|∇yψε|2

2+|∇Rψ

ε|2mp

+∫

R4(1 + ‖(r, R)T ‖2)|ψε|2,

and because of (20) and (21) we have:

d

dtEε

mp(t) 6

1

2

∫R4 ‖(r, R)T ‖2|ψε|2 +

1

2

∫R4

|∇yψε|2

2+|∇Rψ

ε|2mp

+

d

2dt

∫R4(V

εc + V ε

i + E)|ψε|2 + C2

∥∥∥∂tE

1 + ‖(r, R)T ‖2

∥∥∥L∞

‖Eεmp

(t).

As ψε ∈ H1 and because of (20), there exists a positive constant C3 such that:

d

dtEε

mp(t) 6

d

dt

R4

(V ε

c + V εi + E

)|ψε|2 + C3

[1 +

∥∥∥∂tE

1 + ‖(r, R)T‖2

∥∥∥L∞

].

By integration we have

Eεmp

(t) 6∫

R4

(V ε

c + V εi + E(0)

)|ψε(0)|2+

∫R4

(V ε

c + V εi + E(t)

)|ψε(t)|2+

C3

∫ t

0

[1 +

∥∥∥∂tE

1 + ‖(r, R)T ‖2

∥∥∥L∞

]Eε

mp(s)ds+ Eε

mp(0).

As∫

R4

(V εc + V ε

i )|ψε(t)|2 6

R4

(Vc + Vi)|ψε(t)|2 6

( ∫

R4

(Vc + Vi)2|ψε(t)|2

)1/2(∫

R4

|ψε(t)|2)1/2

.

By definition of Vi and Vc and applying an Hardy inequality, there exists a positive constant C4 such that:

R4

|ψε(t)|2(Vc + Vi)2

6 2

R4

|ψε(t)|2(V 2c + V 2

i ) 6 8

R4

∣∣∇yψε∣∣2 +

∣∣∇Rψε∣∣2 6 D‖∇r,Rψ

ε‖L2(R4).

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So that, by the classical equality ‖ψ0‖2L2 = ‖ψε‖2L2 ,

R4

(V εc + V ε

i )|ψε|2 6C4

2‖∇r,Rψ

ε‖L2(R4) +C4

2‖ψ0‖L2(R4). (22)

Now as E ∈ L∞(0, T ) and χ ∈ C2b (R4)

R4

E(t)|ψε|2 6

∥∥∥E

(1 + ‖(r, R)T ‖22

∥∥∥L∞(0,T ;R4)

‖ψε(t)‖2H1 (23)

and because of (18):

Eεmp

(0) 6 Mp‖ψ0‖2H1∩H1

and by definition of Vc and Vi there exists a positive constant C5 such that∫

R4

(V ε

c + V εi + E(0)

)|ψ0|2 6 C5‖ψ0‖2H1∩H1

. (24)

So that (5), (22), (23), (24) lead to the existence of two positive constants C6 and C7 such that:

‖ψε(t)‖2H1∩H16 C6‖ψ0‖2H1∩H1

+ C7

∫ t

0

[1 +

∥∥∥∂tE

(1 + ‖(r, R)T ‖22

∥∥∥L∞(0,T ;R4)

]‖ψε(s)‖2H1∩H1

ds

We then apply the Gronwall inequality that leads to the existence of a positive constant CI such that:

‖ψε(t)‖2H1∩H16 CI exp

( ∫ t

0

(1 +

∥∥∥∂tE

(1 + ‖(r, R)T ‖22

∥∥∥L∞(0,T ;R4)

)‖ψε(s)‖2H1∩H1

ds)‖ψ0‖2H1∩H1

Now as ∂tE ∈ L1(0, T ), there exists a constant C such that (19) occurs. The uniqueness is based on thesame arguments than [11] again using a Gronwall inequality.

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