Derivation of a modified Korteweg–de Vries model for few-optical-cycles soliton propagation from a general Hamiltonian

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Derivation of a modified Korteweg–de Vries model for few-optical-cycles soliton propagation from a general

Hamiltonian

Houria Triki, Hervé Leblond, Dumitru Mihalache

To cite this version:

Houria Triki, Hervé Leblond, Dumitru Mihalache. Derivation of a modified Korteweg–de Vries model

for few-optical-cycles soliton propagation from a general Hamiltonian. Optics Communications, Else-

vier, 2012, 285 (13–14), pp.3179 - 3186. �10.1016/j.optcom.2012.02.045�. �hal-03192911�

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Derivation of a modi ﬁ ed Korteweg – de Vries model for few-optical-cycles soliton propagation from a general Hamiltonian

H. Triki

^b

, H. Leblond

^a,

⁎ , D. Mihalache

^a,c,d

aLUNAM Université, Université d'Angers, Laboratoire de Photonique d'Angers, EA 4464, 2 Bd. Lavoisier, 49045 Angers Cedex 01, France

bRadiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P. O. Box 12, 23000 Annaba, Algeria

cHoria Hulubei National Institute for Physics and Nuclear Engineering (IFIN-HH), 30 Reactorului, Magurele-Bucharest, 077125, Romania

dAcademy of Romanian Scientists, 54 Splaiul Independentei, Bucharest 050094, Romania

a b s t r a c t a r t i c l e i n f o

Article history:

Received 11 October 2011

Received in revised form 18 February 2012 Accepted 20 February 2012

Available online 3 March 2012 Keywords:

Few-cycle pulses Few-cycle solitons

Modiﬁed Korteweg–de Vries equation mKdV equation

Propagation of few-cycles optical pulses in a centrosymmetric nonlinear optical Kerr (cubic) type material described by a general Hamiltonian of multilevel atoms is considered. Assuming that all transition frequencies of the nonlinear medium are well above the typical wave frequency, we use a long-wave approximation to derive an approximate evolution model of modiﬁed Korteweg–de Vries type. The model derived by rigorous application of the reductive perturbation formalism allows one the adequate description of propagation of ultrashort (few-cycles long) solitons.

1. Introduction

Since 1999, few-optical cycle pulses [1

–

4] with durations of only a few periods of the optical radiation have become the primary compo- nents in many important problems of dynamics of nonlinear optical waves. In particular, these ultrashort (femtosecond) pulses [5,6]

ﬁ

nd applications in a wide variety of research areas, such as light

–

matter interactions at high

ﬁ

eld intensities, high-order harmonic generation, extreme nonlinear optics [7], and attosecond physics [8,9]. Notably, the theoretical modeling used to correctly describe the dynamics of such pulses in nonlinear optical media has also been developed in parallel to these incentive experimental studies.

It should be said that, the search for new ideas or even new mathe- matical concepts is of great interest as it is helpful to better under- stand the ultrashort pulse propagation in nonlinear optical media and the formation of robust few-optical-cycle solitons.

The continuing experimental progress in the study of wave dy- namics of few-cycle pulses (FCPs) in nonlinear optical media has paved the way for the development of new theoretical approaches to model their propagation in a lot of physical settings. Three classes of main dynamical models for FCPs have been put forward in the past years: (i) the quantum approach [10

–

14], (ii) the re

ﬁ

nements within the framework of slowly varying envelope approximation (SVEA) of the nonlinear Schrödinger-type envelope equations [15

–

24], and

(iii) the non-SVEA models [25

–

43]. The propagation of FCPs in Kerr media can be described beyond the SVEA by using the modi

ﬁ

ed Korteweg

–

de Vries (mKdV) [31

–

33], sine-Gordon (sG) [34

–

36], or mKdV

–

sG equations [37

–

41]. The mKdV and sG equations are completely integrable by means of the inverse scattering transform method [44,45], whereas the mKdV

–

sG equation is completely inte- grable only if some condition between its coef

ﬁ

cients is satis

ﬁ

ed [46,47].

Other relevant works on few-cycle pulses deal with propagation and interaction of extremely short electromagnetic pulses in quadratic nonlinear media [48

–

51], the study of few-cycle light bullets created by femtosecond

ﬁ

laments [52], the investigation of ultrashort spatiotem- poral optical solitons in quadratic nonlinear media [53], the ultrashort spatiotemporal optical pulse propagation in cubic (Kerr-like) media without the use of the slowly varying envelope approximation [54,55], the possibility of generating few-cycle dissipative optical solitons [56,57], generation of unipolar pulses from nonunipolar optical pulses in a quadratic nonlinear medium [58], and the existence of guided optical solitons of femtosecond duration and nanoscopic mode area, that is, femtosecond nanometer-sized optical solitons [59].

We also mention recent studies of ultrafast pulse propagation in mode-locked laser cavities in the few femtosecond pulse regime and the derivation of a master mode-locking equation for ultrashort pulses [60]. A relevant recent theoretical work presents a class of few-cycle elliptically polarized solitary waves in isotropic Kerr media, propose a method of producing multisolitons with different polarization states, and study their binary-collision dynamics [61].

Robust circularly polarized few-optical-cycle solitons in Kerr media

⁎Corresponding author. Tel.: + 33 2 41 73 54 31; fax: + 33 2 41 73 52 16.

E-mail address:herve.leblond@univ-angers.fr(H. Leblond).

doi:10.1016/j.optcom.2012.02.045

Contents lists available atSciVerse ScienceDirect

Optics Communications

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / o p t c o m

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in both long-wave and short-wave approximation regimes were also studied in recent works [62

–

64].

So far, most of theoretical investigations concern only FCPs propagating in a nonlinear optical medium which is described by a Hamiltonian related to two-level atoms. However, to the best of our knowledge, studies based on very general Hamiltonians describing the wave dynamics of such ultrashort pulses have not yet been reported. It is, nevertheless, an interesting and potentially useful de- scription if FCPs can form and can be robust when we consider this more general physical setting. The aim of this paper is to investigate such a multilevel system in the framework of the reductive perturba- tion method (multiscale analysis). This fact allows one to extend the existing studies to a more general physical situation. Thus in this work we give a detailed mathematical derivation of the modi

ﬁ

ed Korteweg

–

de Vries equation for a general Hamiltonian. We assume that the absorption spectrum of the nonlinear medium does not ex- tend below some cutoff frequency, and that the typical frequency of the FCP is much less than the latter. In other words, we assume that the transparency range of the medium is very large, and consider only the frequencies located in the ultraviolet spectral domain and further. The effect of the infrared transitions, which yield a sine- Gordon model in the case of two-level atoms, will be considered in a further study.

The present study can be considered from two different points of view: (i) a nonlinear cubic medium which has no transition in the infrared is actually described, and (ii) the most general medium con- taining two kinds of transitions, both in the infrared and in the ultra- violet domains, is expected to be described by a more general modi

ﬁ

ed Korteweg

–

de Vries

–

sine-Gordon equation. However, we give here the detailed mathematical derivation for the modi

ﬁ

ed Kor- teweg

–

de Vries part of the most general model.

This paper is organized as follows. In the next section we present in detail a derivation of the modi

ﬁ

ed-Korteweg

–

de Vries equation as a rigorous formal asymptotics of the Maxwell

–

Bloch equations for the most general Hamiltonian for multilevel atoms. This evolution equation describes the propagation of ultrashort (a few femtosecond long) optical solitons in the so-called long-wave regime.

In Section 3 we analyze both analytically and numerically the modi-

ﬁ

ed Korteweg

–

de Vries breather, which is the prototype of few- optical-cycle solitons in such cubic nonlinear optical media. Finally, in Section 4 we present our conclusions.

2. The derivation of a modiﬁed Korteweg–de Vries model from a general Hamiltonian

We consider a set of multilevel atoms described by a very general Hamiltonian:

H

₀

¼

ℏ

ω

1

0

⋯

0 0 ω

2 ⋯ ⋯

⋯ ⋯ ⋯ ⋯

0

⋯ ⋯

ω

N

0 BB

@

1 CC

A:

ð1Þ

The evolution of the density matrix ρ is governed by the Schrödinger

–

von Neumann equation

i

ℏ∂t

ρ ¼ ½ H

;

ρ

;

ð2Þ

in which the total Hamiltonian

H ¼ H

₀−^→

μ

⋅^→

E ð3Þ

includes a term accounting for the coupling between the electric

ﬁ

eld

^→

E and the atoms through a dipolar momentum operator

^→

μ .

The evolution of the electric

ﬁ

eld is governed by Maxwell

–

Helmholtz wave equation

∂²z^→

E ¼ 1

c

²∂²t ^→

E þ 1 ε

0

P

→

;

ð4Þ

in which the polarization density P

^→

expresses as P

→

¼ NTr ρ μ

^→

:

ð5Þ

We consider a cubic (Kerr) nonlinearity, i.e. we assume that the material is centrosymmetric, so that the second order susceptibility χ

⁽²⁾

vanishes. For the sake of simplicity, we will assume a linearly po- larized wave. Then, only the component of

^→

E along the direction of polarization, and the corresponding component of μ

^→

will be involved.

Hence, we suppose that the operator μ

^→

¼ μ

^→

e

x

, where

^→

e

x

is the unitary vector along the x-axis. Also, we have

^→

E ¼ E e

^→x

, and

^→

P ¼ P e

^→x

. Note that E

→

and μ

^→

in Eq. (3) are replaced with scalar quantities E and μ , in which the matrix

μ ¼ ð μ

nm

Þ

_ð_n_;_m_Þ∈_½₁_;_N;

ð6Þ is Hermitian, i.e. μ

mn

= μ

nm∗

, where the star denotes the complex con- jugate. Due to centrosymmetry, since μ

mn

are the matrix elements of an odd operator, the matrix μ is off-diagonal.

We assume that the characteristic pulse frequency ω

w

has the same order of magnitude as the inverse of the pulse duration 1/t

_w

, and is very small with respect to any resonance frequency

Ω

nm

¼ ω

n−ωm

ð7Þ

in the atomic spectrum, i.e.

1

=

t

_w

˜ ^ω

^ω^bbΩ^nm

^{for all} ⁿ

^;

^m

^:

^ð8Þ

This motivates the use of the long-wave approximation [65]. We thus introduce the slow variables

τ ¼ ε t

−

z V

;

ð9Þ

ζ ¼ ε

³

z

;

ð10Þ

so that we obtain

∂t

¼ ε∂

τ;

ð11Þ

∂z

¼

−

ε

V

∂_τ

þ ε

³∂_ζ:

ð12Þ

The scaling (Eqs. (9)

–

(10)) clearly assumes unidirectional propa- gation. Accurate studies [56,58] proved that this assumption may lead to erroneous results in non-homogeneous media, since some re

ﬂ

ections which are negligible for long pulses are not for few-cycle pulses. This must be taken into account in the interpretation of our re- sults, the input wave should indeed be the pulse after it has entered the homogeneous nonlinear medium, which may appreciably differ from the incident pulse.

The

ﬁ

eld E, the polarization density P and the density matrix ρ are expanded in a power series of some small parameter ε as

E ¼

∑

p≥1

ε

^p

E

_p;

ð13Þ

P ¼

∑

p≥1

ε

^p

P

_p;

ð14Þ

(4)

ρ ¼

∑

p≥0

ε

^p

ρ

^{ð Þ}^p:

ð15Þ

As t

→−∞

, ρ

⁽⁰⁾

is assumed to be the density matrix at thermal equilibrium, while E

p

, P

p

, and ρ

^(p)

vanish, for any p

≥

1. 2.1. Order 0

The Schrödinger

–

von Neumann equation (2) at order ε

⁰

is 0 ¼

h

H

₀;

ρ

^{ð Þ}⁰i

;

ð16Þ

and H

₀;

ρ

½

ð Þ

_nm

¼

ℏΩnm

ρ

nm:

ð17Þ

We assume that each level is non-degenerated, and hence Ω

nm≠

0 if n

≠

m. Then Eq. (16) yields ρ

nm(0)

= 0 for n

≠

m (i, j = 1, 2,.., N).

2.2. Order 1

The Schrödinger

–

von Neumann equation (2) at order ε

¹

is i

ℏ∂τ

ρ

^{ð Þ}⁰

¼ ½ H

₀;

ρ

1

−

E

₁h

μ; ρ

^{ð Þ}⁰i

:

ð18Þ

Since μ; ρ

½ ð Þ

_nm

¼

X^N

ν¼1

μ

nν

ρ

νm−ρnν

μ

_νm

ð Þ

;

ð19Þ

and using Eq. (17), we obtain the equation

i

ℏ∂τ

ρ

^{ð Þ}nm⁰

¼

ℏΩnm

ρ

^{ð Þ}nm¹−

E

₁X^N

ν¼1

μ

nν

ρ

^{ð Þ}ν⁰m−ρ^{ð Þ}n⁰ν

μ

νm

;

ð20Þ

(recall that Ω

nm

= ω

n−

ω

m

).

Since ρ

⁽⁰⁾

is diagonal, Eq. (20) reduces to

ρ

^{ð Þ}nm¹

¼ μ

nm

E

₁

ρ

^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

ℏΩnm

ð21Þ for n

≠

m, while the diagonal components ρ

nn0

are constant.

The polarization density is given by Eq. (5), i.e., at order ε

¹

, by P

₁

¼ N

∑

nm

ρ

^{ð Þ}nm¹

μ

mn:

ð22Þ

Since μ is off-diagonal, the sum in Eq. (22) extends over n

≠

m only.

Reporting Eq. (21) into Eq. (22), and using the fact that μ is Hermitian, we get

P

₁

¼ NE

₁∑

n≠m

μ

nm

2

ρ

ℏΩnm :

ð23Þ

The Maxwell

–

Helmholtz wave equation (4) at leading order ε

³

is 1

V

²∂²τ

E

1

¼ 1

c

²∂²τ

E

1

þ P

₁

ε

0

:

ð24Þ

Reporting Eq. (23) into Eq. (24) yields 1

V

²∂²_τ

E

₁

¼ 1

c

²∂²_τ

E

₁

þ NE

₁

ε

0ℏ ∑

n≠m

μ

nm

2

ρ

Ω

nm

2 4

3

5:

ð25Þ

Eq. (25) admits a nonzero solution if V=c/n

0

, in which the refractive index n

0

is

n

₀

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ N

ε

0ℏ∑

n≠m

μ

nm

2

ρ

Ω

nm

vu ut

:

ð26Þ

From Eq. (26), we can deduce the linear susceptibility χ

⁽¹⁾

, which is related to the refractive index through n

0

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ χ

^{ð Þ}¹ p

, as

χ

^{ð Þ}¹

¼ N ε

0ℏ∑

n≠m

μ

nm

2

ρ

Ω

nm :

ð27Þ

Eq. (27) is worth being compared to the known expression of the lin- ear susceptibility, which is given in Ref. [66] (Eq. 3.5.15, p. 163):

χ

^{ð Þ}nm¹

ω

p

¼ N ε

0ℏ∑

nm

ρ

μ

ⁱmn

μ

^jnm

Ω

nm−ωp−

i γ

nm:

ð28Þ Here we restrict to a linear polarization, hence χ

⁽¹⁾

= χ

xx(1)

and μ

mn

= μ

mnx

. We neglect the damping, i.e. γ

nm

= 0, and due to the long-wave approximation, the susceptibility must be evaluated as ω

p

tends to zero. Then Eq. (27) coincides with Eq. (28).

2.3. Order 2

The Schrödinger

–

von Neumann equation (2) at order ε

²

is

i

ℏ∂_τ

ρ

^{ð Þ}¹

¼

h

H

₀;

ρ

^{ð Þ}²i

−

E

₂h

μ; ρ

^{ð Þ}⁰i

−

E

₁h

μ; ρ

^{ð Þ}¹i

:

ð29Þ

Inserting Eq. (17) into Eq. (29), we get the equation

i

ℏ∂τ

ρ

^{ð Þ}nm¹

¼

ℏΩnm

ρ

^{ð Þ}nm²−

E

₂X^N

ν¼1

μ

nν

ρ

^{ð Þ}νm⁰−ρ^{ð Þ}nν⁰

μ

νm

−

E

₁X^N

ν¼1

μ

nν

ρ

^{ð Þ}νm¹−ρ^{ð Þ}nν¹

μ

νm

:

ð30Þ

Since ρ

⁽⁰⁾

is diagonal, Eq. (30) reduces to

i

ℏ∂τ

ρ

^{ð Þ}nm¹

¼

ℏΩnm

ρ

^{ð Þ}nm²−

E

₂

μ

nm

ρ

−

E

₁X^N

ν¼1

μ

nν

ρ

μ

_νm

:

ð31Þ For n=m, by using Eq. (21), Eq. (31) becomes

i

ℏ∂_τ

ρ

^{ð Þ}nn¹

¼

−

E

²₁∑

ν≠n

μ

nν

μ

_νn

ρ

^{ð Þ}nn⁰−ρ^{ð Þ}νν⁰

ℏΩ_νn −

μ

_νn

μ

nν

ρ

^{ð Þ}νν⁰−ρ^{ð Þ}nn⁰

ℏΩnν

2 4

3 5:

ð32Þ

The right-hand-side of Eq. (32) is zero, and hence ρ

nn(1)

= 0 for all n.

Consequently, taking into account the fact that μ is off-diagonal, we get from Eq. (31)

ρ

^{ð Þ}nm²

¼ i

Ω

nm∂τ

ρ

^{ð Þ}nm¹

þ E

₂

ℏΩnm

μ

nm

ρ

þ E

1

ℏΩnm ∑

ν≠n;m

μ

nν

ρ

μ

νm

;

ð33Þ

(5)

for n

≠

m, i.e., inserting Eq. (21) into Eq. (33) we get,

ρ

^{ð Þ}nm²

¼ i μ

nm

ρ

ℏΩ²nm

∂τ

E

1

þ E

₂

μ

nm

ρ

ℏΩnm

−

E

²₁ ℏ²

Ω

nm

ν≠∑n;m

μ

nν

μ

_νm

ρ

Ω

nν −

ρ

^{ð Þ}mm⁰ −ρ^{ð Þ}νν⁰

Ω

νm

2 4

3 5:

ð34Þ

The polarization density at order ε

²

is thus

P

2

¼ iN

ℏ∂τ

E

1 ∑

n;m;n≠m

μ

nm

μ

mn

ρ

Ω

²nm

þ NE

₂

ℏ ∑

n;m;n≠m

μ

nm

μ

mn

ρ

Ω

nm

þ NE

²₁ ℏ² ∑

m

;

n

;

ν

m

≠

n

≠ν

½

^ρ

^{ð Þ}^mm⁰ ^−ρ^{ð Þ}^νν⁰

^μ

^mn

^Ω

^nm

^μ

ⁿ

^Ω

^ν

^μ

^ν^m^ν^m

−

ρ

^{ð Þ}_νν⁰−ρ^{ð Þ}nn⁰

μ

mn

μ

_νm

μ

nν

Ω

nm

Ω

nν

^:

ð35Þ

The second order nonlinear polarization is thus given by

P

^NL₂

¼ ε

0

χ

^{ð Þ}²

ð Þ E

₁ ²;

ð36Þ

where the second order susceptibility χ

⁽²⁾

is given by

χ

^{ð Þ}²

¼ N ε

0ℏ²

∑

m

;

n

;

ν m

≠

n

≠ν

ρ

μ

mn

μ

_nν

μ

_νm

Ω

nm

Ω

νm −

ρ

μ

mn

μ

_νm

μ

_nν

Ω

nm

Ω

nν

:

ð37Þ The expression of χ

⁽²⁾

can be found in Ref. [66] (Eq. 3.6.14 p. 173), as

χ

^{ð Þ}ijk²

ω

p

þ ω

q;

ω

q;

ω

p

¼ N 2

0ℏ²

∑

mnν

f

^ρ

^{ð Þ}^mm⁰ ^−ρ^{ð Þ}^νν⁰

½

^Ω

^nm^−ω^p^−ω^q⁻

^μ

ⁱ^mn

ⁱ ^γ ^μ

^nm^jⁿ^ν

^μ

^k^ν

^Ω

^m^νm^−ω^p⁻

ⁱ ^γ

^νm

þ μ

ⁱmn

μ

^knν

μ

^j_νm

Ω

nm−ωp−ωq−

i γ

nm

Ω

_νm−ωq−

i γ

_νm

−

ρ

½

^Ω

^nm^−ω^p^−ω^q

^μ

⁻ⁱ^mn

ⁱ ^γ ^μ

^nm^j^ν^m

^μ

^kⁿ^ν

^Ω

ⁿ^ν^−ω^p⁻

ⁱ ^γ

ⁿ^ν

þ μ

ⁱmn

μ

^k_νm

μ

^jnν

Ω

nm−ωp−ωq−

i γ

nm

Ω

nν−ωq−

i γ

nν

g

^:

ð38Þ

According to the chosen polarization, we should have

χ

^{ð Þ}²

¼ χ

^{ð Þ}xxx²

ð 2 ω; ω; ω Þ

;

ð39Þ evaluated as ω tends to zero, and neglecting the damping ( γ

nm

= 0). It is seen that Eqs. (36) and (38) coincide under these conditions, taking into account the fact that μ is off-diagonal. In the present study, we

assume a centrosymmetric medium, and hence χ

⁽²⁾

= 0. The coef

ﬁ

- cient of

∂τ

E

1

in Eq. (35) is

A ¼ iN

∑

nm

μ

nm

μ

mn

ρ

ℏΩ²nm

:

ð40Þ

Permuting the dummy subscripts m and n changes the sign of the term in the latter sum, hence A= 0.

The Maxwell

–

Helmholtz wave equation (4) at order ε

⁴

is

1 V

²∂²τ

E

₂

¼ 1

c

²∂²τ

E

₂

þ P

2

ε

0

;

ð41Þ

which is automatically satis

ﬁ

ed if the expression (Eq. (26)) of the re- fractive index is taken into account.

2.4. Order 3

The Schrödinger

–

von Neumann equation (2) at order ε

³

is

i

ℏ∂_τ

ρ

^{ð Þ}²

¼

h

H

₀;

ρ

^{ð Þ}³i

−

E

₃h

μ; ρ

^{ð Þ}⁰i

−

E

₂h

μ; ρ

^{ð Þ}¹i

−

E

₁h

μ; ρ

^{ð Þ}²i

:

ð42Þ

Using Eqs. (17) and (19) into Eq. (42), we obtain the equation

i

ℏ∂_τ

ρ

^{ð Þ}nm²

¼

ℏΩnm

ρ

^{ð Þ}nm³−

E

₃X^N

ν¼1

μ

nν

ρ

^{ð Þ}_ν⁰m−ρ^{ð Þ}n⁰ν

μ

_νm

−

E

₂X^N

ν¼1

μ

nν

ρ

^{ð Þ}_ν¹m−ρ^{ð Þ}n¹ν

μ

_νm

−

E

₁X^N

ν¼1

μ

nν

ρ

^{ð Þ}_ν²m−ρ^{ð Þ}n²ν

μ

_νm

:

ð43Þ

A

ﬁ

rst step is to compute the diagonal terms ρ

nn(2)

. For m= n, Eq. (43) becomes

i

ℏ∂τ

ρ

^{ð Þ}nn²

¼

−

S

_1n

E

₂−

S

_2n

E

₁;

ð44Þ

where we have set S

_jn

¼

∑

ν≠n

μ

nν

ρ

^{ð Þ}νn^j−ρ^{ð Þ}nν^j

μ

_νn

;

ð45Þ

for j = 1, 2. S

_1n

contains the same expression as the right-hand-side term of Eq. (32) above, and consequently S

1n

= 0. Using the expres- sion (34) of ρ

nν(2)

for n

≠ν

, S

2n

can be expanded as

S

_2n

¼ F

_1n

E

₂

þ F

_2n∂τ

E

₁

þ F

_3n

E

²₁;

ð46Þ where

F

_1n

¼ μ

_νn

μ

nν

ℏ

ρ

^{ð Þ}nn⁰−ρ^{ð Þ}_νν⁰

Ω

νn

þ ρ

Ω

νn

2 4

3

5

¼ 0

;

ð47Þ

as S

1n

= 0 above, and

F

2n

¼

∑

ν≠n

2i μ

nν

μ

νn

ℏΩ²vn

ρ

^{ð Þ}nn⁰−ρ^{ð Þ}vv⁰

;

ð48Þ

(6)

F

_3n

¼

∑^N

ν≠n

f

^ℏΩ

^μ

^nν^νⁿ^l≠n;ν^∑^N

^μ

^ln

^μ

^ν^l²⁴

^ρ

^{ð Þ}ⁿⁿ⁰^ℏΩ^−ρ^ln^{ð Þ}^ll⁰⁻

^ρ

^{ð Þ}^ll⁰^ℏΩ^−ρ^ν^l^{ð Þ}^νν⁰³⁵

−

μ

_νn

ℏΩnν∑^N

l¼1

μ

lν

μ

nl

ρ

^{ð Þ}νν⁰−ρ^{ð Þ}ll⁰

ℏΩlν −

ρ

^{ð Þ}ll⁰−ρ^{ð Þ}nn⁰

ℏΩnl

2 4

3

5

g

^:

^ð49Þ

Then we get ρ

^{ð Þ}nn²

¼ iF

_2n

2

ℏ

E

²₁

þ iF

_3n

ℏ ∫_−∞^τ

E

³₁

d τ: ð50Þ

The off-diagonal terms of ρ

⁽³⁾

can also be computed. Since ρ

⁽⁰⁾

is a diagonal matrix, and μ and ρ

⁽¹⁾

are off-diagonal matrices, Eq. (43) yields, for n

≠

m,

ρ

^{ð Þ}nm³

¼ i

Ω

nm∂_τ

ρ

^{ð Þ}nm²

þ E

3

μ

nm

ℏΩnm

ρ

þ E

2

ℏΩnm ∑

ν≠m;n

μ

_nν

ρ

^{ð Þ}_ν¹m−ρ^{ð Þ}n¹ν

μ

_νm

þ E

₁

ℏΩnm

ν≠∑m;n

μ

nν

ρ

^{ð Þ}_νm²−ρ^{ð Þ}_nν²

μ

_νm

þ E

₁

μ

nm

ℏΩnm

ρ

^{ð Þ}mm² −ρ^{ð Þ}nn²

:

ð51Þ Reporting Eqs. (21), (34) and (51) into Eq. (52), we compute the po- larization density as

P

₃

¼ ε

0

χ

^{ð Þ}¹

E

₃

þ 2 ε

0

χ

^{ð Þ}²

E

₁

E

₂

þ ε

0

χ

^{ð Þ}³

ð Þ E

₁³

þA

∂_τ

E

₂

þ B

∂²_τ

E

₁

þ CE

₁∂_τ

E

₁

þ DE

₁∫_−∞^τ

E

³₁

d τ; ð52Þ in which χ

⁽¹⁾

is given by Eq. (27), and χ

⁽²⁾

is given by Eq. (38). Here χ

⁽³⁾

= χ

R(3)

+ χ

S(3)

, with

χ

^{ð Þ}R³

¼ N ε

0ℏ³∑

nmνl

½ ^μ

^mn

^μ

ⁿ^ν

^Ω ^μ

^lm^nm

^μ ^Ω

^ν^l^ν^m

^ρ ^Ω

^{ð Þ}^mm⁰^lm^−ρ^{ð Þ}^ll⁰⁻

^μ

^mn

^μ

ⁿ^ν

^Ω ^μ

^lm^nm

^μ ^Ω

^ν^l^ν^m

^ρ ^Ω

^{ð Þ}^ll⁰^ν^l^−ρ^{ð Þ}^νν⁰

−

μ

mn

μ

_νm

μ

_lν

μ

nl

ρ

^{ð Þ}_νν⁰−ρ^{ð Þ}_ll⁰

Ω

nm

Ω

_nν

Ω

_lν

þ μ

mn

μ

_νm

μ

_lν

μ

nl

ρ

^{ð Þ}_ll⁰−ρ^{ð Þ}nn⁰

Ω

nm

Ω

_nν

Ω

nl

^;

ð53Þ the sum being extended on all terms for which the denominators do not vanish (the

‘

regular

’

terms, motivating the subscript

‘

R

’

), and χ

^{ð Þ}S³

¼ N

ε

0ℏ ∑

n;m;n≠m

μ

mn

μ

nm

Ω

nm

iF

_2m

2

ℏ −

iF

_2n

2

ℏ

;

ð54Þ

where F

2n

is given by Eq. (48). The coef

ﬁ

cient A is given by Eq. (40),

B ¼

−

N

ℏ ∑

nm

μ

nm

μ

mn

ρ

Ω

³nm

;

ð55Þ

C ¼

−

iN

ℏ²∑

nmν

μ

mn

μ

nν

μ

_νm

ρ

Ω

nm

Ω

²νm

−

μ

mn

μ

nν

μ

_νm

ρ

Ω

nm

Ω

²nν

2 4

3 5

þ 2iN

ℏ² ∑

nmν

μ

mn

μ

_νm

μ

nν

ρ

Ω

²nm

Ω

nν −

μ

mn

μ

_νm

μ

nν

ρ

^{ð Þ}mm⁰ −ρ^{ð Þ}_νν⁰

Ω

²nm

Ω

νm

2 4

3 5;

ð56Þ

and D ¼ N

ℏ ∑

n;m;n≠m

μ

mn

μ

nm

Ω

nm

iF

_3m

2

ℏ −

iF

_3n

2

ℏ

;

ð57Þ

where F

3n

is given by Eq. (49).

Due to the centrosymmetry, the coef

ﬁ

cient C is zero. Consider indeed the second order susceptibility χ

xxx(2)

( ω

p

+ ω

q

; ω

q

, ω

p

) given by Eq. (38), for ω

p

= ω

q

= ω , i.e.

χ

^{ð Þ}²

ð 2 ω; ω; ω Þ ¼ N

0ℏ²∑

mnν

½

^ρ

^{ð Þ}^mm⁰ ^−ρ^{ð Þ}^νν⁰

^ð ^Ω

^nm⁻

^μ

^mn

² ^ω ^μ

ⁿ

^Þ

^ν

^ð ^Ω ^μ

^ν^ν^m^m^−ω

^Þ

−

ρ

μ

mn

μ

_νm

μ

nν

Ω

nm−

2 ω

ð Þ ð Ω

nν−ω

Þ

^:

ð58Þ

Taking the derivative with respect to ω , and then the limit as ω tends to zero yields

d

d ω χ

^{ð Þ}²

ð 2 ω; ω; ω Þj

ω¼0

¼ N

0ℏ²∑

mnν

½

^ρ

^{ð Þ}^mm⁰ ^−ρ^{ð Þ}^νν⁰

^μ

^mn

^Ω

^nm

^μ

ⁿ

^Ω

^ν

^μ

²^ν^m^ν^m

−

ρ

μ

mn

μ

_νm

μ

nν

Ω

nm

Ω

²nν

þ 2N

0ℏ²∑

mnν

½

^ρ

^{ð Þ}^mm⁰ ^−ρ^{ð Þ}^νν⁰

^μ

^mn

^Ω

²^nm

^μ

^nν

^Ω ^μ

^ν^m^νm

−

ρ

μ

mn

μ

νm

μ

nν

Ω

²nm

Ω

nν

^:

ð59Þ

It is seen that d

d ω χ

^{ð Þ}²

ð 2 ω; ω; ω Þ j

_ω¼0

^¼

⁻

_ε

₀

^iC

^:

^ð60Þ

The medium being centrosymmetric, χ

⁽²⁾

(2 ω ; ω , ω )

≡

0, and hence C= 0. It has also been seen that A = 0 and χ

⁽²⁾

= 0. The coef

ﬁ

cient D is a nonlinear term of the 4th order. Since the order is even, it must be zero due to centrosymmetry. Notice that this feature cannot be justi-

ﬁ

ed for the expression of the coef

ﬁ

cient itself, as it was the case for the χ

⁽²⁾

coef

ﬁ

cient. Hence D = 0 and

ﬁ

nally the polarization density reduces to

P

₃

¼ ε

0

χ

^{ð Þ}¹

E

₃

þ ε

0

χ

^{ð Þ}³

ð Þ E

₁³

þ B

∂²τ

E

₁:

ð61Þ Consider now the Maxwell

–

Helmholtz wave equation (4) at order ε

⁵

. It is

1 V

²∂²τ

E

3−

2 V

∂ζ∂τ

E

1

¼ 1

c

²∂²τ

E

3

þ P

₃

ε

0

:

ð62Þ

Reporting Eq. (51), and using the expression (26) of the velocity V, the terms involving E

₃

vanish, and Eq. (62) reduces to

∂ζ

E

1

þ γ∂

τ

ð Þ E

1³

þ β∂

³τ

E

1

¼ 0

;

ð63Þ

which is exactly the mKdV equation.

The nonlinear coef

ﬁ

cient is

γ ¼ 1

2n

₀

c χ

^{ð Þ}³;

ð64Þ

with χ

⁽³⁾

given by Eqs. (53)

–

(54). An expression of χ

⁽³⁾

can be found in Ref. [66] (Eq. (3.7.10), p. 182), as

χ

^{ð Þ}kjih³

ω

p

þ ω

q

þ ω

r;

ω

r;

ω

q;

ω

p

¼ P

_I

χ

~^{ð Þ}kjih³

ω

p

þ ω

q

þ ω

r;

ω

r;

ω

q;

ω

p

;

ð65Þ

(7)

in which P

_I

represents an averaging over all permutations of ω

r

, ω

q

and ω

p

, and

χ

~^{ð Þ}kjih³

ω

p

þ ω

q

þ ω

r;

ω

r;

ω

q;

ω

p

¼ N ε

_:ℏ³∑

nmνl

½ Ω

nm−ωp−ω

^ρ

^{ð Þ}^mmq⁰−ω^−ρr^{ð Þ}^ll⁰

^μ

^k^mn

^μ

^jⁿ^ν

^μ

ⁱ^ν^l

^μ

^h^lm

Ω

νm−ωp−ωq

Ω

lm−ωp

−

ρ

^{ð Þ}ll⁰−ρ^{ð Þ}_νν⁰

μ

^kmn

μ

^jnν

μ

ⁱlm

μ

^h_νl

Ω

nm−ωp−ωq−ωr

Ω

νm−ωp−ωq

Ω

νl−ωp

−

ρ

^{ð Þ}_νν⁰−ρ^{ð Þ}ll⁰

μ

^kmn

μ

^j_νm

μ

ⁱnl

μ

^hlν

Ω

nν−ωp−ωq

Ω

lν−ωp

þ ρ

^{ð Þ}ll⁰−ρ^{ð Þ}nn⁰

μ

^kmn

μ

^j_νm

μ

ⁱlν

μ

^hnl

Ω

nν−ωp−ωq

Ω

nl−ωp

^;

ð66Þ

in which we set the relaxation rates γ

nm

to zero.

The present χ

⁽³⁾

should coincide with χ

xxxx(3)

(0; 0, 0, 0), however several terms in the sum (66) are singular as ( ω

r

, ω

q

, ω

p

)

→

(0, 0, 0).

It is straightforwardly seen that the sum of the regular terms exactly coincides with χ

R(3)

as given by Eq. (53). Taking into account the fact that μ is off-diagonal, the singular terms in the sum (66) are:

•

The 1st term for ν =m:

A

mnl

¼

−

K

_mnl

Ω

ω

p

þ ω

q

Ω

lm−ωp

;

ð67Þ

with

K

_mnl

¼ ρ

^{ð Þ}mm⁰ −ρ^{ð Þ}ll⁰

μ

mn

μ

nm

μ

ml

μ

lm:

ð68Þ

•

The 2nd term for ν = m:

B

_mnl

¼ K

_mnl

Ω

ω

p

þ ω

q

Ω

lm

þ ω

p

:

ð69Þ

•

The 3rd term for ν =n:

C

_mnl

¼ K

nml

Ω

ω

p

þ ω

q

Ω

ln−ωp

:

ð70Þ

•

The 4th term for ν =n:

D

_mnl

¼

−

K

_nml

Ω

ω

p

þ ω

q

Ω

ln

þ ω

p

:

ð71Þ

Let us set ( ω

r

, ω

q

, ω

p

) = (

−

ω + δω

r

, ω + δω

q

, ω + δω

p

), in which δω

r

, δω

q

and δω

p

are in

ﬁ

nitesimal quantities. After some calculations, the in

ﬁ

nitesimal quantities simplify and we get

P

I

ð A

mnl

þ B

mnl

Þ ¼

−

K

_mnl

3 ð Ω

nm−ω

Þ

1 Ω

lm

þ ω

ð Þ

²

þ 1

Ω

lm−ω

ð Þ

²

þ 1

Ω

²lm−ω²

" #

:

ð72Þ

It is comparable to the expression of χ

⁽³⁾

found in Ref. [67], but with a slight sign change. As ω

→

0, it yields

P

_I

ð A

_mnl

þ B

_mnl

Þ ¼

−

ρ

^{ð Þ}mm⁰ −ρ^{ð Þ}ll⁰

μ

mn

μ

nm

μ

ml

μ

lm

Ω

nm

Ω

²lm

:

ð73Þ

P

_I

ð C

_mnl

þ D

_mnl

Þ yields the same expression as Eq. (73) with permuted n and m. On the other hand, Eqs. (54) and (48) yield

χ

^{ð Þ}S³

¼

−

2N ε

0ℏ³ ∑

n;m;ν;

n≠m;ν≠m

μ

mn

μ

nm

μ

_νm

μ

mν

Ω

nm

Ω

²νm

ρ

^{ð Þ}mm⁰ −ρ^{ð Þ}_νν⁰

;

ð74Þ

and hence χ

^{ð Þ}S³

¼ N

ε

0ℏ³ ∑ n;m;ν;

n≠m;ν≠m

P

_I

ð A

_mnl

þ B

_mnl

þ C

_mnl

þ D

_mnl

Þ

:

ð75Þ

As a conclusion, taking into account both regular and singular terms, we see that χ

⁽³⁾

exactly coincides with χ

xxxx(3)

(0; 0, 0, 0).

The expression (Eq. (64)) of the nonlinear coef

ﬁ

cient slightly dif- fers from the analogous expression found in Refs. [34,39]. A factor 4 π is due to the system of units (i.e., CGS units versus SI units). In fact, the value of χ

⁽³⁾

given by Eq. (78) in Ref. [67] is smaller by a factor of 1/3. The expression of the nonlinear coef

ﬁ

cient in [34] was based on this value of χ

⁽³⁾

, and this initial error resulted in the erroneous coef

ﬁ

cient of 6 in Eq. (25) in [34] instead of 2 in Eq. (64) above.

The dispersion coef

ﬁ

cient is β ¼ 1

2 ε

0

n

₀

c B

;

ð76Þ

where B is given by Eq. (55). Taking twice the derivative of Eq. (28) with respect to ω

p

, then setting ω

p

= 0, and comparing the result with Eq. (55) shows that

B ¼

−ε0

2 d

²

χ

^{ð Þ}xx¹

ð Þ ω

d ω

²

j

_ω¼0^:

^ð77Þ

The wave vector is k( ω ) =n

0

( ω ) ω /c, and its third derivative for ω = 0 is

d

³

k

d ω

³

j

_ω¼0

^¼ ³ _c ⁿ ^′′

⁰^;

^ð78Þ

where we have set

n ′′

₀

¼ d

²

n

₀

d ω

²

j

_ω¼0^:

^ð79Þ

Taking twice the derivative ofn

0

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ χ

^{ð Þ}¹

p

with respect to ω yields

d

²

n

₀

d ω

²

¼ 1

2n

₀

d

²

χ

^{ð Þ}¹

d ω

² −

1 2n

²₀

d χ

^{ð Þ}¹

d ω

!2

;

ð80Þ

however, d χ

⁽¹⁾

/d ω for ω = 0 is proportional to the coef

ﬁ

cient A given by Eq. (40), hence is zero. Finally, it is found that the dispersion coef-

ﬁ

cient β can be written as

β ¼

−

1 6

d

³

k

d ω

³

j

^ω¼0;

ð81Þ

(8)

which exactly coincides with the expression found in Ref. [34] and generalizes the latter. The equivalent expression

β ¼

−

n ′′

₀

2c

;

ð82Þ

evidences the fact that β is not a third-order dispersion as it could been believed at

ﬁ

rst glance, but accounts in the present approxima- tion for the group velocity dispersion. It also may account for higher order dispersion terms, see Ref. [68].

3. The breather solution of the modiﬁed-Korteweg–de Vries equation

The mKdV equation (63) is completely integrable by means of the inverse scattering transform [69]. The N-soliton solution has been given by Hirota [70]. It is more convenient to write the mKdV equation (63) into the dimensionless form

∂Z

u þ 2

∂T

u

³

þ σ∂

³T

u ¼ 0

;

ð83Þ where σ = ±1, u is a dimensionless electric

ﬁ

eld, and Z and T dimen- sionless space and time variables de

ﬁ

ned relative to the laboratory vari- ables as

u ¼ E

E

₀ ;

Z ¼ z

L

;

T ¼ t

−

z

=

V

t

_w :

ð84Þ

The reference time is thus chosen to be the pulse length t

_w

(in phys- ical units). Recall that the atomic resonance frequencies Ω

nm

have been chosen above as zero order quantities in the perturbative scheme, while t

w

is assumed to be formally large, of order 1/ ε , with respect to the zero order times 1/ Ω

nm

. The characteristic electric

ﬁ

eld and propagation distance are

E

₀

¼ 1 t

_w

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

−

2 σ n

₀

n ′′

₀

χ

^{ð Þ}³ vu

ut ;

ð85Þ

L ¼ 2ct

³_w

−σ

n ′′

0

:

ð86Þ

If χ

⁽³⁾

and n

′′0

have opposite sign, which is typically for χ

⁽³⁾

>0 and anomalous dispersion, then σ =+1, and the mKdV equation (63) is a fo- cusing one. Else, typically for χ

⁽³⁾

>0 and normal dispersion, σ =

−

1,

Eq. (63) is a defocusing one and describes nonlinear dispersion [39]. In the focusing case, the mKdV equation admits real single-soliton solutions, and N-soliton and breather solutions. Integrating the mKdV equation (83) with respect toT, under the assumption that u, i.e. the elec- tric

ﬁ

eld, and its derivatives, vanishes at in

ﬁ

nity, it is seen that the conser- vation law

∂Z∫_−∞^þ∞

udT ¼ 0 ð87Þ is satis

ﬁ

ed. This is the expression in our situation of the general law of the conservation of the electric pulse area, as derived in [56,58]. Due to the Galilean transformation and the scaling (9-10), it is seen from Maxwell equations that the magnetic

ﬁ

eld is

^→

B ¼ uE

0=

V e

^→y;

and that

∫^þ∞_−∞

B

_α

dz

∝∫^þ∞_−∞

udT

;

hence the conservation law of the magnetic pulse area is also satis

ﬁ

ed by the mKdV Eq. (63), since it does not differ from (87).

The two-soliton solution of the mKdV equation is [70]

u ¼ e

^η¹

þ e

^η²

þ

^p_p¹⁻^p²

1þp2

2 e^η¹ 4p²₁

þ

_4p^e^η²2

2

e

^η¹^þη²

1 þ

^e_4p^2η¹2

1

þ

_p ²

1þp2

ð Þ²

e

^η¹^þη²

þ

^e_4p^2η²2 2

þ

^p_p¹⁻^p²

1þp2

4 e^2η1þ2η2

16p²₁p²₂

;

ð88Þ

with

η

j

¼ p

j

T

−

p

³j

Z

−γj;

ð89Þ for j = 1 and 2. The parameters p

1

, p

2

, γ

1

, and γ

2

are arbitrary. If p

2

=p

1∗

, where

∗

denotes the complex conjugate, and γ

2

= γ

1∗

, the ex- plicit solution (88) is an oscillating localized solution, called breather soliton, which actually adequately describes a FCP soliton.

An example of FCP soliton propagation is shown on Fig. 1. The mKdV equation (63) is solved using the exponential time differencing 4th order Runge

–

Kutta scheme [71], for an input data (blue dotted line) of the form

u ¼ Ae

⁻^T²⁼^w²

sin ð ω

0

T þ ϕ Þ

;

ð90Þ with the parameters w= 2.5, A ¼ 1

:

5, ω

0

= 0.6 π , and ϕ = π . The com- putation was run until Z

≃

80. The pulse evolves with very few changes in shape and width, apart from periodic oscillations. We chose a prop- agation distance (Z = 79.72) at which the carrier-envelope phase of the

ﬁ

nal FCP is the same as the initial one, moved to the initial posi- tion and plotted it in Fig. 1 (green thick solid line) for comparison.

A

ﬁ

t with the breather (88) is also shown (dashed red line): it is very close to the numerical result. The values of the parameters which yield the best

ﬁ

t are p

1

= 0.875 + 1.7i, γ

1

= 0.24i (and a small shift in position).

Fig. 1.Propagation of a FCP according to the mKdV equation. Blue dotted line: initial input with Gaussian envelope. Green thick solid line: the FCP soliton observed after some propagation distance (Z= 79.72). Dashed red line:ﬁt of the latter by the analytic breather.

T

Z

-80

-60 -40 -20 0

0 2 4 6 8 10

Fig. 2.Propagation of a FCP according to the mKdV equation:uagainstTandZcomputed with the analytic formula (Eq.(88)); parameters are the same as theﬁt inFig. 1.

(9)

The propagation of the breather, computed from the analytic for- mula (Eq. (88)), is shown in Fig. 2. Note that the same picture can be obtained numerically.

4. Conclusions

In conclusion, in this paper we have studied in detail the propaga- tion of ultrashort pulses containing only few optical cycles, in a cen- trosymmetric nonlinear cubic (Kerr) material described by a general Hamiltonian of multilevel atoms. Assuming that all transition fre- quencies of the material are well above the typical wave frequency, we can use a long-wave approximation to derive an approximate model for nonlinear ultrashort optical pulse propagation in such a medium. The obtained equation is of modi

ﬁ

ed Korteweg

–

de Vries type and allows the adequate description of FCP soliton propagation in a cubic nonlinear optical medium. However, in a real nonlinear opti- cal medium, a broad transparency range is required, and hence all atomic transitions must be far enough from the pulse central frequen- cy. However in such materials not all transitions belong to the ultravi- olet domain. The contribution of the infrared transitions, which are not taken into account in the present work, should be considered in a further study by means of a short-wave approximation applied to the same general quantum model. Therefore a fully realistic model for FCP propagation will be obtained by putting together the results yielded by both long- and short-wave approximations.

Acknowledgments

The work of DM was supported in part by a Senior Chair Grant from the Région Pays de Loire, France. Support from the Romanian Ministry of Education and Research (Project PN-II-ID-PCE-2011-3- 0083) is also acknowledged.

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