Two-level atoms and solitons

ISSN 1062�8738, Bulletin of the Russian Academy of Sciences. Physics, 2013, Vol. 77, No. 12, pp. 1407–1411. © Allerton Press, Inc., 2013.Original Russian Text © S.V. Sazonov, 2013, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2013, Vol. 77, No. 12, pp. 1713–1718.

1407

INTRODUCTION

A two�level atom is the simplest quantum modelused in many problems of light–matter interaction.The model became especially popular in the 1960safter the invention of lasers—sources of coherent lightradiation. If the light’s frequency ω is close to fre�quency ω0 of the transition between any two quantumlevels in an atom (resonance), we can confine our�selves to considering these two levels.

A soliton is a solitary wave packet that regains itsshape after interaction with other solitons. Mathemat�ically, this object is a partial differential solution to anonlinear equation (or system of equations). It is par�ticularly important that the equation be nonlinear, andthus the standard superposition principle typical ofsolutions to linear equations in mathematical physicsdoes not hold here. In addition, the equation underconsideration is integrable; i.e., we can find a solutionto the corresponding Cauchy problem using analyticalapproaches. The elastic interaction between solitons isdue to the integrability of the equation in question.Apart from nonlinearity, dispersion is also required forthe existence of a soliton. In terms of interactionbetween the light field and atoms, this means thatthere is a time delay in the medium’s response to thefield effect. It is the mutual cancellation of nonlinear�ity and dispersion that results in the formation of asoliton.

The first theoretical work on solitons was publishedas far back as 1895 [1] and dealt with solitary waves inshallow water. That was the first time that theKorteweg–de Vries (KdV) nonlinear partial differen�tial equation was presented.

The invention of Q�switched and mode�lockedlasers in the 1960s made it possible to generate lightpulses with duration of several nanoseconds (andlater several picoseconds), thus making it feasible toobserve optical solitons.

pτ

OPTICAL ENVELOPE SOLITONS

It is interesting that two books completely unre�lated to one another yet both having to do with solitonswere published in 1967.

A scheme for integrating the KdV equation wasproposed in [2], a theoretical work. The scheme waslater termed the inverse scattering method (ISM) andextended to a great number of other nonlinear partialdifferential equations.

During the experiment in [3], a nanosecond�dura�tion optical soliton that was a manifestation of the self�induced transparence (SIT) effect, was observed forthe first time. The SIT effect means that upon exceed�ing a particular intensity threshold, a light pulse prop�agating in an absorbing resonant medium remains vir�tually unattenuated but is slowed greatly.

The corresponding system of wave and materialequations is referred to as the Maxwell–Bloch (MB)system. It has the form

(1)

Here is the complex Rabi frequency of apulse propagating along the z axis, d is the dipolemoment of the quantum transition under consider�ation; is the Planck constant; is thedetuning of the pulse field from the resonance uponthe quantum transition; ε is the complex slowly vary�ing envelope (SVE) of pulse electric field E, related toone another by the formula

(2)

(with the abbreviation c.c. standing for complex con�jugation); k is the wave number; R is the complexenvelope of the nonstationary atomic dipole momentinduced by the pulse field; W is the difference in pop�ulation between the excited and the ground state of the

( )* *

24 ,

,

.2

mn d ni Rz c t c

R i R i Wt

W i R Rt

∂ψ ∂ψ π ω+ = −

∂ ∂

∂= Δ + ψ

∂

∂= ψ − ψ

∂

�

2dψ = ε ћ

� 0Δ = ω − ω

( )[ ] c.с.,expE i t kz= ε ω − +

Two�Level Atoms and SolitonsS. V. Sazonov

National Research Center Kurchatov Institute, Moscow, 123182 Russiae�mail: [email protected]

Abstract—Integrable equations that produce a wide range of various solitons arising from interactionsbetween two�level atoms with optical pulses are briefly reviewed.

DOI: 10.3103/S1062873813120101

1408

BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 77 No. 12 2013

SAZONOV

atom (inversion); n is the concentration of two�levelatoms; c is the speed of light in a vacuum; and is therefractive index of the nonresonant matrix.

The SVE approximation allowed the initial sec�ond�order wave equation to be reduced to the firstorder (see the first equation in (1)) by ignoring the sec�ond derivatives of the envelopes.

The MB system (1) is completely integrable andtherefore has solutions in the form of solitons thatelastically interact with one another [4].

In the case of exact resonance ( ), the Rabi fre�quency of the pulse assumes real values and system (1)becomes the so�called sine–Gordon (SG) equation

(3)

which also turned out to be integrable by ISM [4]. Here

and

where is the initial difference in population betweenthe quantum levels of the two�level atoms ( atthermodynamic equilibrium).

Soliton solution (3) describes an SVE pulse propa�gating in a resonant absorbing medium. It has the form

(4)

where the velocity of the soliton υ is connected to itsduration τp by the relation

(5)

It is evident that the soliton velocity and amplitudeincrease with diminishing duration. The second termon the right�hand side of (5) corresponds to the greatslowing down of propagation relative the speed of lightin a vacuum. According to the experimental data, thevelocity of an SVE soliton in a resonant medium canbe two to four orders of magnitude slower than thespeed of light. This slowing down is due to the solitonneeding a great deal of time to excite the atoms with itsleading edge and regain the spent energy with its trail�ing edge.

Let the quasi�resonance condition

(6)which corresponds to the relatively weak interactionbetween the optical pulse and matter, be fulfilled. In thiscase, Crisp’s expansion [5] in the small parameter μ

(7)

(8)

is valid.

mn

0ω = ω

2

sin ,z∂ θ

= −α θ∂ ∂τ

',dτ

−∞

θ = ψ τ∫ ,mt n z cτ = −

24 ,d n Wc

∞

π ωα = −

�

W∞

0W∞<

sech2 ,p p

t z⎛ ⎞− υψ = ⎜ ⎟τ τ⎝ ⎠

.21 mp

n

c= + ατ

υ

( )1

1,p−

μ = Δ τ �

2 2

3 4

2 3

3 2 4 3

3

2 2

,

W W WR i

t

W W Wi i

t t t

∞ ∞ ∞

∞ ∞ ∞

∂ψ= − ψ + ψ ψ − ψ

Δ ∂Δ Δ

∂ψ ∂ ψ ∂ ψ+ + −

Δ ∂ Δ ∂ Δ ∂

( )**

2

2 31 .

2 2iW W

t t∞

⎡ ⎤ψ ∂ψ ∂ψ= − + ψ − ψ⎢ ⎥∂ ∂Δ Δ⎣ ⎦

Inserting (7) into the right�hand side of (1), weobtain a closed equation for the complex envelope. Ifin (7) we confine ourselves to the lowest degrees ofnonlinearity and dispersion and keep only the first,second, fourth, and fifth terms on the right�hand side,we arrive at the nonlinear Schrödinger equation(NLSE), which is also integrable by ISM [3]:

(9)

where the linear

group velocity is defined by the expression =

and

(10)

Parameter is the group velocity dispersion(GVD) coefficient. In our case, it is equal to nonlin�earity coefficient β. Note, however, that NLSE gener�ally describes the propagation of a light pulse in a non�resonant medium with weak nonlinearity and disper�sion. If the medium is different from the system oftwo�level atoms,

Soliton solution (9) has the form

(11)

The coefficient before z in the imaginary exponentdefines a nonlinear addition to the inverse phasevelocity proportional to the square of the solitonamplitude. In other words, it is a nonlinear addition tothe refractive index of the medium.

At the same time, as is evident from (11), the groupvelocity of the NLSE soliton does not depend oneither its amplitude or duration and is equal to the lin�ear group velocity It is for this reason we proposethat NLSE solitons be used in fiber�optic communica�tion systems. Indeed, it is by virtue of this property thatthe propagation sequence of solitons with differentamplitudes and durations does not change: they leavethe medium in the same sequence in which they wereformed upon entering it, and the encoded informationdoes not change. NLSE solitons are therefore referredto as fundamental solitons [6].

If we consider all of the terms in expansion (7) andperform the phase transformation

(11)

222

2,

2

kQ Qi Q Qz

∂ ∂= − − β

∂ ∂τ

,gt zτ = − υ

2

2 34 ,d nk W

c∞

π ω= β = −

Δ�

gυ1

gυ

2

21 4 ,m

d nn Wc

∞

⎛ ⎞π ω−⎜ ⎟Δ⎝ ⎠�

( )exp .Q i zα= ψ

Δ

2k

2 .k ≠ β

exp sech2 22

1 .2

g

p pp

t zk kQ i z

⎛ ⎞ ⎛ ⎞− υ= ⎜ ⎟ ⎜ ⎟τ β ττ ⎝ ⎠⎝ ⎠

.gυ

( )29exp3 27

Q i zΔ α⎡ ⎤ψ = τ −⎢ ⎥⎣ ⎦Δ


TWO�LEVEL ATOMS AND SOLITONS 1409

we arrive at the modified Korteweg–de Vries (MKdV)equation

(12)

where and –

Soliton solution (12) takes the form of (4), whereits velocity is

It is well known [3] that this equation is also inte�grable and has soliton solutions.

Let there now be both resonant and nonresonanttwo�level atoms in the medium as, e.g., in a mixture oftwo isotopes of a chemical element when there is anisotopic shift of the quantum levels [7]. Combining (3)and (9), we obtain the Konno–Kameyama–Sanuki(KKS) equation

(13)

This equation is integrable only if the ratio betweenthe coefficients of the third and fourth terms on theleft�hand side is 3/2 [8]. This ratio is seen to arise inthe optical problem under consideration in a naturalway without any limits imposed on the mediumparameters. The terms mentioned above describerespectively the nonlinearity and the dispersion of theresponse of the quasi�resonant atoms.

The soliton solution of KKS equation (13) coincideswith (4), but its velocity is related to the duration as

The solitons considered above are envelope soli�tons, i.e., they contain a great many optical oscilla�tions. Nanosecond and picosecond pulses containaround a million and a thousand oscillations, respec�tively. They are thus quasi�monochromatic solitons.

Producing light pulses of increasingly shorter dura�tions under laboratory conditions is one line in thedevelopment of modern laser physics. It is safe to saythat femtosecond optics is now a well�established lineof research [9]. We shall discuss in more detail how itapplies to two�level atoms in the next section.

FEW�CYCLE PULSES

Pulses with durations of a few femtoseconds havearound one light oscillation period and are normallyreferred to as few�cycle pulses (FCPs) [10]. The spec�trum width of the FCP is comparable to the centralfrequency of its spectrum. It is therefore a wide�bandpulse; i.e., it is impossible to distinguish any of the car�rier frequencies in it. The standard ISM approxima�tion of nanosecond and picosecond optics thereforecannot be applied here. We must write and solve the

32

33 0,2

Q Q QQz T T

∂ ∂ ∂− β − β =

∂ ∂ ∂

4α

β =Δ

T t=

22 .3

mnz

cα⎛ ⎞+⎜ ⎟

⎝ ⎠Δ

2

2 21 1 2 1 .

3 pc

⎛ ⎞α α= + − ⎜ ⎟υ ΔτΔ Δ ⎝ ⎠

( )22 2 4

2 43sin 0.2Z T T T T

∂ θ ∂θ ∂ θ ∂ θ+ α θ − β − β =

∂ ∂ ∂ ∂ ∂

2

2 21 2 .

3m

p

p

n

c

βα= + + ατ −

υ Δ τ

corresponding equations for the field and the dipolemoment themselves, rather than for their envelopes.

The first attempt to abandon the ISM approxima�tion was made when an alternative approach todescribing SIT was devised in [11].

Dispensing with the historic description, wepresent a system of the corresponding material andwave equations in the form

(14)

Here E is the electric field of the pulse,U is the atomic dipole moment induced by the pulsefield, W is the difference in population between thequantum levels, V determines the rapidity of dipolemoment variation, and parameter D characterizes theasymmetry of an atomic electron shell when there isno pulse field. The last is referred to as the constantdipole moment (CDM) of the atom.

Due to the CDM, the electric field of the pulseinteracting with two�level atoms not only excites aquantum transition, thereby changing the levels’ pop�ulations (see the penultimate equation in (14)) butalso dynamically shifts the frequency of this transition(see the first two equations in (14)).

Here the wave equation is reduced from the secondto the first order using the unidirectional propagation(UP) approximation [11]. In this approximation, onlya wave propagating in the z direction is consideredwhile a wave reflected from induced nonlinear inho�mogeneities is ignored. This is possible when the con�centration of two�level atoms is sufficiently low, asexpressed by the inequality

(15)

When there is no CDM (i.e., when D = 0), sys�tem (14), (15) is referred to as a reduced Maxwell–Bloch system (RMB). This is the RMB that was inves�tigated in [11]. It was also shown there that the systemwas integrable via ISM and generated breather�typesolutions

(16)

(17)

(18)

( )0 ,2

U D Vt d

∂= − ω − Ω

∂

( )0 ,2

V D U Wt d

∂= ω − Ω + Ω

∂

,W Vt

∂= −Ω

∂

208

.mn d nV

z c t c

π ω∂Ω ∂Ω+ =

∂ ∂ �

2 ,dEΩ = ћ

2

0

8 1.d nπ

ω��

( )( )sech

4

1arctan sin ,gph

p p

t

t zt z

∂Ω =∂

⎧ ⎡ ⎛ ⎞ ⎤⎫− υ× ω − υ⎨ ⎬⎜ ⎟⎢ ⎥

ωτ τ⎩ ⎣ ⎝ ⎠ ⎦⎭

( )( )

2 2 202

22 2 2 2 20

11 ,1 4

pmp

g p p

n

c

ω + ω τ += + ατ

υ ⎡ ⎤ω − ω τ + + ω τ⎣ ⎦

( )( )

2 2 202

22 2 2 2 20

11 ,1 4

pmp

ph p p

n

c

ω − ω τ −= + ατ

υ ⎡ ⎤ω − ω τ + + ω τ⎣ ⎦

1410


SAZONOV

where duration and central frequency ω of thebreather spectrum are considered free parameters.

Since phase velocity and group velocity aredifferent, the shape of the breather changes and recursperiodically as it propagates.

If Eqs. (16)–(18) produce the envelope soli�tons of the MB system where Ω = 2ψcos[ω(t – z/υph)]with ψ defined by (4) and the group and phase veloci�

ties defined as + and

When Δ = 0, the latter expression for the groupvelocity is transformed, as expected, into (5).

If the solution to (16)–(18) corresponds toan optical soliton of the FCP type.

The approach used in [11] therefore allows us notonly to describe resonant envelope solitons in a systemof two�level atoms but also to find FCP�type solutionsin the same system.

One disadvantage of the RMB is that it uses anapproximation of a medium with a low concentrationof two�level atoms (15). As a result, the velocity of thepulses that are solutions to the RMB cannot be muchdifferent from the linear velocity c/nm. At the sametime, as was mentioned above, the velocity of the SITsolitons can be hundreds to thousands of times lowerthan the linear velocity.

The authors of [12] proposed using the spectraloverlap (SO) approximation

(19)

instead of an approximation with a low concentra�tion of resonant atoms. The idea underlying thisapproximation is that the spectral width of a pulse is

i.e., the pulse spectrum overlaps thefrequency of the quantum transition under consider�ation and the excitation of atoms can thus be quitehigh.

It was shown in [12] that under condition (19), thepulse dynamics obeys SG equation (3). Now, however,the truncated area of the pulse is governed by its elec�

tric field rather than the field envelope: =

In [12], the authors also proposed using the opticaltransparency (OT) approximation

(20)

which corresponds to relatively weak interactionbetween the pulse field and matter, and showed that inthis case pulse propagation was described by an MKdVin the form of (12) for the electric field but not for itsenvelope.

Note that the SG and MKdV, like the RMB, havebreather�type solutions [13] that become NLSE�type

pτ

phυ gυ

1,pωτ �

1 m

g

n

c=

υ ( )

2

21

p

p

ατ

+ Δτ

1 .m

ph

n

c=

υ

~ 1,pωτ

20( ) 1.pω τ �

0~ 1 ,pδω τ > ω

'dτ

−∞

θ = Ω τ∫'(2 ) .d Ed

τ

−∞

τ∫�

20( ) 1,pω τ �

nonresonant envelope solitons when anddescribe the propagation of FCPs when .

Let us now return to system (14) at In thiscase, it is referred to as the CDM RMB. As was shownin [14], it is integrable by ISM and thus produces soli�ton solutions.

Let us consider a few specific cases correspondingto the approximations that produce new integrableequations and systems.

Let OT condition (20) be fulfilled. For a mediumwith strongly anisotropic atoms, we then obtain from(14) the KdV equation for the electric field of the pulsewhen D/d � 1:

(21)

where and

The KdV equation was mentioned above in con�nection with the propagation of waves in shallowwater. Here it describes thepropagation of an opticalpulse in a system of anisotropic two�level atoms.

Problems involving optical methods for generatingterahertz electromagnetic radiation have recentlybecome very important [15]. Since the terahertz rangeis appreciably lower on the frequency scale than theoptical range, the field in system (14) can be repre�sented as the additive sum of an optical envelope pulseand field of an FCP�type terahertz pulse. Applyingthe SVE approximation to an optical field and the OTand UP approximations to a terahertz field, we arriveat a system with the form [16]

(22)

where Q is related to ψ by transformation (10),

ΩT = DET/ћ, μ = k2 =

and g =

System (22) was derived using the condition of themost effective terahertz pulse generation, i.e., that theoptical group velocity and the terahertz phase velocitybe equal.

This system, referred to as the Yajima–Oikawa(YO) system, was analyzed in [17], where its integra�bility by the ISM was demonstrated.

The soliton solution to system (22) takes the form

(23)

1pωτ �

~ 1pωτ

0.D ≠

3

30,q

z∂Ω ∂Ω ∂ Ω

+ Ω − β =∂ ∂τ ∂τ

2

20

8 d nW Dqdc

∞π

= −

ω�

2

30

8.

d nW

c∞

πβ = −

ω�

TE

22

2,

2T

kQ Qi g Qz

∂ ∂= − + Ω

∂ ∂τ

( )2 ,T Qz

∂Ω ∂= −μ

∂ ∂τ

2

0

4,

d nW Dc d

∞π

−ω Δ�

203

8,

d n W

c∞

π ω−

Δ�2

02

2.

d n W Ddc

∞π ω

−Δ�

( )[ ]sech02 2 exp ,p p

t zdQ i zD

⎛ ⎞− υω δ= − δτ − κ ⎜ ⎟Δτ Δ τ⎝ ⎠

sech2

24 ,T

pp

t zdD

⎛ ⎞− υΩ = − ⎜ ⎟

τΔτ ⎝ ⎠


TWO�LEVEL ATOMS AND SOLITONS 1411

where velocity υ of the synchronous propagation ofoptical and terahertz pulses is defined by the expres�sion

The quantity δ determines the shift of the opticalpulse’s carrier frequency to the red region as it gener�ates a terahertz signal. Note that the value of this shiftis proportional to the intensity of the optical pulseitself. This shift can be treated as resulting from thescattering of the optical pulse by its generated tera�hertz signal. Note that this shift was observed in theexperiments [18]. Here it appears in the soliton solu�tion of the YO system.

In [19] we analyzed the interaction between a sys�tem of anisotropic two�level atoms and a laser pulsefield when the excited state is degenerated with respectto a projection of the angular momentum onto thepreferred axis. In this case, the field is of vector char�acter and has an ordinary and an extraordinary component. When propagation is normal to theanisotropy axis, system (14) is modified such that theordinary component induces a quantum transitionand the extraordinary component shifts the frequencyof the transition [19].

Within SO approximation (19), we then obtain forthe ordinary component a modified SG (MSG) equa�tion in the form

(24)

where = and

and the extraordinary component is related to theordinary one by the expression

where Ωe = DEe/ћ.Equation (24) becomes the SG equation at τc ~ D = 0

obtained in [12]. The MSG equation in the form of(24) is also integrable by ISM. Its soliton solution hasthe form

(25)

where and velocity of propagation υ

is defined by expression (5).A detailed analysis of solution (25) is given in [19].

CONCLUSIONS

The examples considered above indicate that agreat many soliton equations and systems are pro�duced by a system of two�level atoms interacting withan optical field. The list of integrable models cannot bereduced to the above systems, especially when we aredealing with generalizations of cases where the electric

field is of a vector character [20]. There is also reasonto believe that the simple two�level atom model canlead further to new integrable systems that have solitonsolutions.

ACKNOWLEDGMENTS

The work was supported by the Russian Founda�tion for Basic Research, project no. 13�02�00199a.

REFERENCES

1. Korteweg, D.J. and de Vries, G., Philos. Mag., 1895,vol. 39, p. 422.

2. Gardner, C.S., Green, J.M., Kruskal, M.D., andMiura, R.M., Phys. Rev. Lett., 1967, vol. 19, p. 1095.

3. McCall, S.L. and Hahn, E.L., Phys. Rev. Lett., 1967,vol. 18, p. 908.

4. Lem, J.L., Vvedenie v teoriyu solitonov (Introductioninto Solitons Theory), Moscow: Mir, 1983.

5. Crisp, M.D., Phys. Rev. A, 1973, vol. 8, p. 2128.6. Agrawal, G.P., Nonlinear Fiber Optics, AT&T Bell Lab.,

1989.7. Sazonov, S.V., Bull. Russ. Acad. Sci. Phys., 2007,

vol. 71, no. 1, p. 122.8. Konno, K., Kameyama, W., and Sanuki, H.J., Phys.

Soc. Jpn., 1974, vol. 37, p. 171.9. Akhmanov, S.A., Vysloukh, V.A., and Chirkin, A.S.,

Optika femtosekundnykh lazernykh impul’sov (Optics ofFemtosecond Laser Pulses), Moscow: Nauka, 1988.

10. Brabec, T. and Krausz, F., Rev. Mod. Phys., 2000,vol. 72, p. 545.

11. Caudrey, P.J., Eilbeck, J.C., Gibbon, J.D., andBullough, R.K., J. Phys. A, 1973, vol. 6, p. L53.

12. Belenov, E.M., Nazarkin, A.V., and Ushchapovskii, V.A.,Zh. Eksp. Teor. Fiz., 1991, vol. 100, p. 762.

13. Kosevich, A.M. and Kovalev, A.S., Vvedenie v neli�neinuyu fizicheskuyu mekhaniku (Introduction intoNonlinear Physical Mechanics), Kiev: Naukovadumka, 1989.

14. Agrotis, M., Ercolani, N.M., Glasgow, S.A., andMoloney, J.V., Phys. D, 2000, vol. 138, p. 134.

15. Kryukov, P.G., Femtosekundnye impul’sy (FemtosecondPulses), Moscow: Fizmatlit, 2008.

16. Sazonov, S.V. and Ustinov, N.V., JETP, 2012, vol. 115,no. 5, p. 741.

17. Yajima, N. and Oikawa, M., Prog. Theor. Phys., 1976,vol. 56, p. 1719.

18. Stepanov, A.G., Mel’nikov, A.A., Kompanets, V.O.,and Chekalin, S.V., JETP Lett., 2007, vol. 85, no. 5,p. 227.

19. Sazonov, S.V. and Ustinov, N.V., JETP Lett., 2006,vol. 83, no. 11, p. 483.

20. Sazonov, S.V. and Ustinov, N.V., JETP, 2005, vol. 100,no. 2, p. 256.

Translated by M. Potapov

21 1 .g kυ = υ − δ

oE eE

( )22

21 sin ,cx∂ θ ∂θ

= −α − τ θ∂ ∂τ ∂τ

'odτ

−∞

θ = Ω τ∫ ( ) '2 od E dτ

−∞

τ∫ћ0

,2

cD

dτ =

ω

( )2

2 202 0,

2e e o

Dd

Ω + ω Ω + Ω =

sech2

1 tanh2 ,1 2 tanh

op

q

q q

− χΩ = χ

τ − χ +

,p

t z− υχ =

τ,c

p

qτ

=

τ

Documents

Two-level atoms and solitons