Local field effects and multimode stimulated light scattering in a two-level medium

0.68 MB·English

by  Vl. K. Roerich

Local field effects and multimode stimulated light scattering in a two-level medium
Vl. K. Roerich, A. A. Panteleev, and M. G. Gladush
FSUE SRC RF TRINITI, Federal State unitary Enterprise State Research Center
of Russian Federation Troitsk Institute of Innovation and Fusion Researches,
TRINITI, Troitsk, Moscow region, Russia 142190∗
Scattering of a short (much shorter then the spontaneous lifetime) laser pulse has been consid-
ered in a dense resonant medium subject to local field effects. The system was studied in the limit
of Hartree-Fock approximation for Bogolubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of
equationsforreduceddensityoperators. Aclosedsetofequationsforatomicandfielddensityoper-
atorswasderivedtodescribestimulatedscatteringofradiation. Numericallyaswellastheoretically
the capability of multicomponent spectrum with maximums multiple to Rabi frequency has been
demonstrated. Relative line intensities in thespectrum were found in the limit of low density.
4
0 PACSnumbers: 42.65.Pc,42.50.Fx,05.30.-d,32.50.+d
0
2
I. INTRODUCTION. theoretically[6,7,8,9]andobservedinexperimentswith
n
a impurities embedded into crystalline matrices or doped
J glasses [10, 11]. It has been demonstrated that local
Thepropertiesofscatteredlightwhichonecanobserve
4 fieldeffectscannotablyaffectthespontaneousdecayrate
under action of a strong laser field have been the area of
1 [12, 13, 14]. Also, for these case the strict condition for-
intensive research in quantum optics. Wigner-Weisskopf bidding realization of such bistability mechanism cased

1 [1] were the first to demonstrate that when a two-level
by collisional broadening was eliminated [15, 16].
v system is excited with a weak monochromatic field its
Another area of studies is linked to collective spon-
9 fluorescence spectrum was determined by Rayleigh scat-
taneous decay. In addition to higher decay rate [4] in-
7
tering. In this case the criterionfor the weak pump field
0 teratomic interactioncan substantially modify the spon-
wastheratioofRabifrequencytoeitherthespontaneous
1 taneous spectrum in comparison with the single atom
decay rate or detuning from resonance. In strong fields
0 case[17, 18]. Interactionbetween the atoms givesriseto
4 the stimulated (Rayleigh) component is accompanied by additional resonances in absorption and emission spec-
0 spontaneous radiation displaying two additional maxi-
tra which is explained by possibility for simultaneous
/ mums in the spectrum building the well known Mollow
h excitation of different atoms and excitation exchange
triplet [2]. It must be noted that for stationary pump
p [19, 20, 21, 22, 23, 24, 25, 26, 27, 28].
- modes in the strong field limit the coherent component
t of radiation is reciprocally proportional to the square of Additional lines including satellites multiple to Rabi
n frequency in superradiantspectra was first reported and
a Rabifrequencyandis,therefore,negligible. These result
discussedin[17, 18, 19]. However,intensities ofsuchad-
u havefoundnumerousexcellentexperimentalverifications
q [3]. ditional sidebands were shown to be utterly small, i.e.,
: 10−6 to the central peak. It must be noted that there
v The general picture is much more complicated when ∼are some other difficulties found in connection to condi-
i
X such a process is studied in dense media. This is mainly tions for the superradiantmode. This fact puts the limi-
r due to strong interatomic interactions giving rise to col- tationsonthevalueofthetransverserelaxationconstant
a lectivebehaviorofatomsexposedtoexternallaserfields. [29] γ Nγ, where γ is the radiative relaxation rate.
2
The pioneer consideration of this issue was presented in Itallma≪kesitverycomplicatedtoobservethe sidebands
work[4]by Dicke who demonstratedthat in anoptically experimentally.
dense medium of N atoms (nλ3 1, where n is the One should note that in the references cited above
≫
density of atoms and λ=λ/2π λ is the wavelength)col- therewassteadystatespectraanalysisonlysothecontri-
lective spontaneous decaywith its intensity proportional bution of stimulated scattering (Rayleigh component) in
to N2 was possible. resonance fluorescence has not been considered. On one
One of the most studied range of phenomena in this hand it could have been conditioned by its smallness,
aspect is linked to the interaction of closely positioned but on the other hand in the steady state it is scattered
atomic systems through their radiation field and com- ”elastically”. However, as we demonstrated in [30] in
bines a number of well investigated cooperative effects transient spectra the profile of stimulated scattering is
[5]. Thistypeofphenomenaincludeslocal-fieldeffectsor different from delta function and its frequency is not the
the near dipole-dipole interaction, a striking sequence of same as that of the pump field. Besides, its intensity is
whichisintrinsicopticalbistabilitypredictedandstudied no small in comparison with that of spontaneous radia-
tion. Indensemedia,similartosuperradiation,intensity
isproportionaltothesquareofatomicdensityandunder
certainconditionsmaydominateoverspontaneousinten-
∗Correspondingauthore-mail: [email protected] sity especially when the superradiant mode is not likely
2
to arise because of collisions. the frequency of a photon with the wave vector k and
The aim of this work is to study transient spectra of polarization α.
stimulated scattering in a dense medium excited by a Operator V describes interaction of atom
af
shortlaserpulse. InthelimitofHartree-Fockapproxima- l with a mode of the quantized field, where
trieodnucaepdpldieednsfitoyr oBpBeGraKtoYrs-h[i3e1r]arwcehywoilfl erqecueaitvieonesqufoartiothnes gl(kα)= 2πωkeikrl(~µ21ǫ(kα)) defines the coupling
¯hW
of motion for atomic and field density operators which constant,rhere ~µ is the dipole transition operator,
21
imply local field corrections giving rise to nonlinearities ǫ(k ) is a unitary polarization vector, and k ǫ(k )=0,
α α
ofthesystem. Expressionsdescribingintensityspectraof α = 1,2 W is the quantization volume. O·perator V
a
scattered light will be also obtained. Using the method describes interaction of an atom with the laser field
of successive approximations we will conduct theoretical R(t,r) = ~µ21EL(t,r), where E (t,r) is the laser field
analysisofscatteredlightincasewhenthemediumisex- − 2h¯ L
cited by a short laser pulse of constant intensity profile. strength.
In conclusion we will make a comparison of the analyti- Besides,itis necessarytointroduceanoperatorwhich
calsolutionwiththeexactsolutionobtainednumerically describesradiativeabsorptionbythecavitywallsrelated
and study the spectral properties in a wide range of pa- to either operation of a detector device or whatever else
rameters. irretrievable loss of radiation. Allowing for the losses
This work is structured as follows. In Sec. II we brings an additional term to the Hamiltonian (3) hav-
present the approximations to be used and derive the ing the form kKk[ρ] and describing interaction of the
equations of motion for the atomic and field density op- quantizedfieldwiththermostatatzerotemperature[32]:
P
erators. InSec. IIIweapplythemethodofsuccessiveap-
proximationsto obtainanalytical solutions for the equa- Kk[ρ]=−iηk/2(aˆ+kαaˆkαρ−2aˆkαρaˆ+kα +ρaˆ+kαaˆkα), (4)
tions and discuss their properties. In Sec. IV the spec-
trum of scattered light is studied numerically. A com- where ηk determines the mode losses.
parison with the analytic solution is carried out while In order to obtain equations describing the spectrum
thepropertiesofobtainedspectraarestudiedforawider of scattered light we will use the BBGKY-hierarchy in
range of parameters. In the summary we list the basic the framework of Hartree-Fock approximation apply-
results. The appendix contains the explicit derivation ing the well-known cluster expansion for reduced den-
scheme for the local field operator. sity operators [31]. While carrying out the derivation
steps we will neglect the spontaneous radiation and col-
lisional relaxation considering the laser pulse duration
II. BASIC EQUATIONS sufficiently small. However, we take into account the
factthatthe resonantmediumcanbe denseenough,i.e.,
eraWteetwwioll-lceovnelsiadteorminstweriathctaiolnasoefrafimeldedrieusmonoafnntotnod1egen2- fne(crt)sλt32o1/b8eπ3st≫ron1g.toHleerte,thne(rl)ociaslt(hceoodpeenrsaittyivea)ndfieλld21eifs-
atomic transition. In the dipole and rotative wave→ap- the transition wavelength. We will further neglect spon-
proximations the Hamiltonian of the system, describing taneousscatteringasitsintensityislowifcomparedwith
the interaction of N atoms and laser field, in units 1/¯h thatofthesimulatedscatteringbeingproportionaltothe
has the form: squared density [30].
We assume that initially the atoms are not perturbed
N N which makes it possible to represent the atom-field den-
H = (H +V )+ H + V , (1)
a a f af sity operator as the product of atomic and field parts:
l=1 k l=1 k
X X XX
(2) ρ= ρa0(ρa11 =1) |0kαih0kα|. (5)
where YN Ykα
Vaf V=Hai=(g=li((∆kRα2(1)t||,22rii)hh122||a,ˆk1αH−fRg=l∗((kνt,αkraˆ))∗+k|1α1aˆihk22α|),aˆ.+kα), (3) Ihnietrharechvyiewtakoefsatphperofoxrimma1t:ion taken above the BBGKY-
a | ih |− | ih | dρa ′′
In (3) H describes the Hamiltonian of an unperturbed i dt −[Ha+Va,ρa]−[Vaf′′,ρaρf′′]f =0,
atomic saystem, ∆21 = ω21 − ωL is the detuning from iddρtf −[Hfr,ρf]−Kk[ρf]−[Va′′f,ρa′′ρf]a′′ =0, (6)
creasrorniearnfcree,qωue21ncbyeionfgathlaesetrrapnusiltsieo,njfr,eqju,ejnc=y1a,n2daωrLe tthhee ρa(0)=ρa0, ρf(0)=|0kαih0kα|,
| i h |
projection operators for respective atomic states.
The next term H describes the Hamiltonian of quan-
f
tized radiation field, where aˆ+kα,aˆkα are formation and 1 Unless otherwise stated the initial conditions for the systems
annihilation operators respectively; νk =ωk ωL, ωk is consideredremainunchanged.
−
3
where ρa,ρ are atomic and field single particle density Summationoveratomicandfieldvariables(seeAppendix
f
operators respectively; [.]f,[.]a denote tracing operation A) in [V˜p ,ρaρ(2)]f yields:
af f
over field and atomic variables with [.] being usual com-
mutatorbrackets. TheBBGKY-hierarchyinthisapprox- [V˜p ,ρaρ(2)]f =[Vp,ρa],
imation is comprised of a Bloch equation for two-level af f a
3n(r)Gλ3 (12)
atoms interacting with laser field complemented with Vp = i 21γ/2 2 1ρa +H.c..
the term [V ,ρaρ ]f which is responsible for the feed- a − 8π2 | ih | 21
af f
back to the atom from the scattered light and the field
where γ is the radiative relaxation constant (see Ap-
equationwritteninShrodingerrepresentation. Here,the
pendix);andGisthegeometricalfactorbeingafunction
term [V ,ρaρ ]a is the quantum representation of field
af f of selected geometry of the sample, its dimensions, and
induced atomic polarization.
polarization of scattered light [33]. In its structure and
Inordertofindasolutionforthesystem(6)wewilluse
naturetheoperator(12)isanalogoustotheLorentzfield
thetransformationwhichalreadyusedinpapers[30]. At
or the local field correction operator obtained in works
first, however, it is expedient to change to the wave pic-
[33] or using other methods and approaches.
ture by means of the substitution ρf = e−iHfrtρ(f1)eiHfrt. It is obvious that with the local field term and tak-
As the result, the system (6) now reads ing account for the fact that for the chosen initial con-
ditions Kk[ρ(2)] 0 one can write ρ(2) 0k 0k
dρa f ≡ f ≡ | ih |
i dt −[Ha+Va,ρa]−[V˜af,ρaρ(f1)]f =0, [V˜af,ρaρ(f2)]f ≡ 0. Thus, the system (13) can, indeed,
(1) be reduced to a single equation, which is
dρ (7)
i dft −Kk[ρ(f1)]−[V˜af,ρaρ(f1)]a =0, dρa
V˜af =igl(kα)|2ih1|aˆkαe−iνkt+H.c.. i dt −[Ha+Va+Vap,ρa]=0, (13)
Let us write out explicitly the operator describing the Changingbacktothepicturewestartedwithonourway
induced atomic polarization: from the equations (6) to (13) and using the properties
ofthecoherentstateoperators,thefielddensityoperator
[V˜ ,ρaρ(1)]a =[P ,ρ(1)]= for the scattered field one can represent in the following
af f f f
N (8) form:
m i[gm(kα)ρa12aˆkαe−iνkt+H.c.,ρ(f1)]. ρf =(e−iνkaˆ+kαaˆkαtD(βk(t))0k 0k D+(βk(t))eiνkaˆ+kαaˆkαt),
X | ih |
ρf =(D(βk(t)e−iνkt)0k 0k D+(βk(t)e−iνkt)).
It follows from (8) for operator P that the exponent | ih |
f
exp(−iPf) is nothing but a coherent state. Allowing for Now, substituting βk(t)e−iνkt = βk′(t) we can finally
the mode losses via Kk[ρ(1)], the operator which elimi- write the system of equations describing the scatered
f
natesthetermslinearinaˆ+kα,aˆkα fromthe fieldequation light:
is expressed as follows [32]:
dρa
i [H +V +Vp,ρa]=0,
ddβtke=xp−(−η2ikLβfk)(−t)=Negxmp((−kαiφ)∗(ρt)a2)1Dei(νβktk(=t))ζ,k(t), (9) ddβtk′ =−dt(iν−k+ηak/2)aβk′ −aNm gm(kα)∗ρa21, (14)
dφk = Xmi(ζkβk∗ ζk∗βk)/2. ρf =|βk′(t)ihβk′X(t)|.
dt − −
Furthermore, one can easily find from (14) the average
Letusnotethatthefollowingordinaryoperatorrelations value of observed spectral intensity of scattered light:
for coherent state operators are valid D(βk(t)) [29, 32]:
DD++((ββkk((tt))))aaˆˆ+kkααDD((ββkk((tt))))==((aaˆˆ+kkαα ++ββkk∗((tt))))., (10) =h¯Iωkk(htβ)=k′(th¯=)ω|k−η[k−i¯hKiωKkk[k|ρβ[fρk′]f(aˆ]t+kaˆ)α|+k2aαˆ.kaˆαk|αβ]nk′(=t)i= (15)
Using (10) while performing a transformation so that where [.]n is the trace over number of photons.
ρ(1) =exp( iL )(t)ρ(2)exp(iL )(t)thesystem(7)comes
f − f f f
to the form
III. SCATTERED LIGHT SPECTRUM OF A
idρa [H +V ,ρa] [V˜ +V˜p ,ρaρ(2)]f =0, DOT SAMPLE EXCITED BY A SHORT LASER
dt − a a − af af f PULSE
dρ(2) (11)
f (2)
i dt −Kk[ρf ]=0, Inthis sectionwewillstudythe propertiesofthe scat-
V˜apf =igl(kα)(|2ih1|βk(t)e−iνkt+H.c.. tered light spectrum for a dot sample excited by a short
4
laser pulse. The main emphasis in our analysis falls on its existence in the following analysis. Now, using the
the effects induced by the local field correction. We will following relation [34]:
assume that the collection of atoms is embedded in a
∞
sample with dimensions small enough to let us drop all
cos(Bcos(u))= ( 1)jJ (B)ei2ju,
spatial dependencies in equations (14). Besides, for the − |2j|
sakeofsimplicity we considera pulse ofrectangularpro- ∞j=X−∞ (23)
file, i.e., R(t) = R at 0 < t < T with pulse duration sin(Bcos(u))= ( 1)jJ (B)ei(2j+1)u,
|2j+1|
as long as T, and take the strict resonance condition −
j=−∞
X
when the excitation frequency is exactly the transition
frequency providing ∆ =0. the integral (22) can be presented as
21
Thesolutionforthisnonlinearsetofequations(14)we
will find using the method of successive approximations β′(t)=Me−iεkU Ueeiεku ∞ ( 1)jeijuY ,
andbeaccuratetothefirstorderapproximationonly. In × − j
Z0 j=−∞
case of exactresonancethe zero approximation(no local X
Y = J eiB+J (B)e−iB , (24)
field) for the equation has the form: 2j+1 |2j| |2j+2|
Y2j = J|2j(cid:0)|+1(B)eiB J|2j|+3(B)e−i(cid:17)B ,
−
dρa(0) Y =Y∗, Y =2J (B)cos(B).
i [V ,ρa(0)]=0, (16) −2(cid:0)j 2j 0 1 (cid:1)
a
dt −
Performing integration we get
(17)
∞
and its solution is found easily: β′(t)=Me−iεkU ( 1)jS(j)Y ,
j
−
j=−∞ (25)
ρa(0)(t)= 21(cid:18)1−−sicnos(2(2RRt)t) 1+sinco(s2(R2tR)t)(cid:19). (18) S(m)= ei(iε(kXε+km+)Ume −) 1.
In the first order approximation the equation is Substituting(25)and(22)into(15)anddroppingallnon
resonant cross terms we can write the spectrum of scat-
dρa(1) tered light in the form:
i [V +Vp(ρa(0))]=0, (19)
dt − a a
∞
(20) Ik(t)=ηk¯hωk βk′(t)2 =N2G2ke−ηkt1 S(j)2 Yj 2
| | | | | |
j=−∞
which is as well integrated: X
S(m)2 = 1+e−ηkT −2e−ηkT/2cos((νk+2mR)T),(26)
1 1 cos(V(t)) sin(V(t)) | | (νk+2mR)2+ηk2/4
ρa(1)(t)= 2(cid:18) −−sin((V(t)) 1+cos((V(t))(cid:19), (21) Gk =ηk¯hωk2¯hπWωk||µ21||2,
where t<T t =0, t>T t =t T. Let us now
1 1
V(t)=2Rt B(1 cos(2Rt)), see this expre→ssion in various m→arginal c−ases.
− −
1) ηkT 1. In this limit the contribution from the
where B = 3n(r)Gλ321 γ . Substituting results of (21) terms cont≫aining e−ηkT is negligible over long times, so
8π2 4R (25) takes the form:
′
in (14) for β (t) and neglecting the signalretardationwe
get ∞ Y 2
Ik(t)=N2G2ke−ηkt1" (νk+2j|Rj)|2+ηk2/4#. (27)
β′(t)=Me−iεkU Ueeiεkusin(u+B(1 cos(u)))du+ j=X−∞
"Z0 − The spectrum represents a collection of lines multiple to
U Rabifrequencywith line widths ofthe orderofηk2/4and
ZUe eiεkusin(Ue+B(1−cos(Ue)))du# (22) isnittieenssaitsiefsupnrcotipoonrstioofnaBl taore|Ysjh|2o.wLninine wFIidGth1s faonrdseinvteerna-l
2Rτ =u, 2Rt=U, U =2RT,
e central lines. It must be noted that such analysis is rea-
N 2πωk νk iηk/2 sonableifcarriedoutatthevalueofparameterB 1as
M = µ21 , − =εk, ≪
2R ¯hW || || 2R theresultofthissectionwereobtainedusingthe method
r
ofsuccessiveapproximationsinB. Asitfollowsfromthe
whereN isthenumberdensityinthesample. Thesecond
picture the line intensities decrease sharply with larger
partintheexpression(22)describesradiationofresidual
valuesofj which,initsturn,followsfromtheasymptotic
polarizationinducedduringpulsepropagation. Ithasthe j
1 eB
nature of ordinary Rayleigh scattering. It is of no par- expansion for Bessel functions: J (B)
j
ticular interestforthe presentstudy so we willdisregard ∼ √2πj 2j !
5
IV. NUMERICAL MODELING
1,0
Intheprevioussectionweobtainedexplicitexpressions
0,9
for intensity spectra of scattered radiation in case when
0,8
parameter B 1 which corresponds to the small den-
0,7 |Y|2 sity of atoms.≪This limitation was due to the method
1
0,6 |Y0|2 ofsuccessiveapproximationsselectedforouranalysis. In
0,5 |Y2|2 caseoflargedensityvaluesthismethodisnotapplicable.
|Y|2
0,4 |Y3|2 Thus, we have to solve the system of nonlinear system
4
0,3 equations (14) using numerical methods.
0,2 In order to estimate the range of parameters where
0,1 multicomponentspectracouldbeexpectedweintroduced
0,0 operators describing radiative and collisional relaxation.
0,0 0,2 0,4 0,6 0,8 1,0
B Γ[ρa]= iγ/2(2 2ρa+ρa 2 2 21 2ρa 2 1),
Γ−[ρa]=| ihiγ| (1 2|ρaih+|−2 |1iρha|). | ih | (32)
s − 2 | ih | 12 | ih | 21
Substituting these damping terms into the equations
FIG. 1: Relative intensities of spectral lines. gives
dρa
i [H +V +Vp,ρa] Γ[ρa] Γ [ρa]=0,
2th)eηkexTpr≪ess1i.onInfotrhitsheliminittenitsiitsyvsapliedcttroaptuotceh−aηnkgTe∼it t1oin ddtdβtk−′ =−a(iνka+ηka/2)βk′ −− N gm−(kαs)∗ρa21, (33)
m
Ik(t)=N2G2ke−ηkt1 ∞ 2Yj 2V(j), Ik(t)=ηk¯hωk|βk′X(t)|2.
| |
j=−∞ (28) Also, we present the time integrated intensity spectra to
X
V(m)= 1−cos((νk+2mR)T). make experimental verificationof obtained results possi-
(νk+2mR)2+ηk2/4 ble if such is to take place.
∞
Over long times the spectrum of scattered light is now
IkT = Ik(t)dt. (34)
as well a collection of lines multiple to Rabi frequency
with intensities proportional to Y 2. However, unlike Z0
j
the oppositecasethey aremodula|ted| withthe frequency In ris. 2 one can see the intensity spectra of scattered
inverse to the pulse duration time. This modulation the lightforalongcomparedtothemodedumpingtimepulse
linewidthsareoftheorderofinversepulsedurationtime. 3 , ηkT 1, for different values of parameter B. For
≫
For the centralpeak it is seen fromthe following expres- small values of B in ris. 2(a) ot is seen that the ratios of
sion 2: line intensities generally meet the results reportedin the
previoussection. Thecentralpeakwithνk =0istheonly
1 cos(νkT) ∞ ( 1)i−1(νkT)2i/(2i)! exception. This fact is explainedby existence ofresidual
ν−2+η2/4 = i=1 −ν2+η2/4 (29) radiationafterthepulse hasalreadypassedthemedium.
k k P k k With increasing value of B the relative intensity of the
central line grows substantially. The relative intensities
forlargevaluesofνk ηk theexpressiontakestheform:
≫ of the side bands decrease while shifting to the center of
the spectral range, see FIG 2(b),(c). Moreover, one can
1ν−2c+osη(2ν/k4T) ≈T2/2!−νk2T4/4!+O(νk4), (30) see that peaks multiple to 3 and 4 Rabifrequencies arise
k k shifted significantly to the center. At extremely large
densities B 1 all components merge leaving the only
from which it follows that the line width value is of the ∼
central peak (FIG 2(d)).
order of 1/T. Let us note that the intensity in the line
The short pulse scattering, ηkT 1, is demonstrated
center is always different from null. Indeed, at νk → 0 in FIG 3 for different values of den≪sity. In FIG 3(a),(b)
we have:
some additional peaks are well seen. These peaks arise
duetomodulationsofthekind(30)for6πand10πpulses
1+e−ηkT −2e−ηkT/2cos(νkT) =T2+O(T4) (31) respectively. The general behavior of spectrum is just
νk2+ηk2/4 like the one displayed in the previous case, that is, the
2 similarlyforothercomponents 3 Thiscaseisexaminedascase1)intheSec. III
6
1.2E+006
(a)
1.0E+006
8.0E+005
IT
6.0E+005
k
k
4.0E+005
2.0E+005
0.0E+000
-120 -60 n 0 60 120
k k
3.5E+006
(b)
2.8E+006
2.1E+006
IT
k
k 1.4E+006
7.0E+005
0.0E+000
-150 -100 -50 n0 50 100 150
k k
(c) 1.0E+007
8.0E+006
IT6.0E+006
k k
4.0E+006
2.0E+006
0.0E+000
-150 -100 -50 n0 50 100 150
k k
(d) 5.0E+007
4.0E+007
3.0E+007
IT
k k
2.0E+007
1.0E+007
0.0E+000
-150 -100 -50 n0 50 100 150
k k
FIG.2: IntensityspectrumofthescatteredlightIk andtimeintegratedspectrumIkT,valuesofIk,IkT arepresentedinrelative
units and proportional to N2. ∆ = 0,R = 10π,γ = 0.01, ηk = 5, T = 1.9 B = 0.32, 0.64, 0.85, 1.2 for pictures (a,b,c,d)
respectively.
7
8E+004
(a)
6E+004
5E+004
k IT
k
3E+004
2E+004
0E+000
-150 -100 -50 n 0 50 100 150
k
k
(b) 3E+005
2E+005
k
IT
k
1E+005
0E+000
k -150 -100 -50 n 0 50 100 150
k
FIG. 3: Same as in FIG2. ∆=0,R=10π,γ =0.01, ηk =0.1, picture (a) T =0.3, B=0.28, picture(b) T =0.5, B =0.36
peaks shift toward the central line at B 1 and finally excited by a short rectangular laser pulse. A possibility
∼
merge. The calculationcarriedoutfor differentvaluesof wasdemonstratedto generatescatteredlightatfrequen-
γ,γ showed that with shorter relaxation time the spec- cies multiple to Rabi frequency which is contributed by
2
tral picture is ”blurred”. At γ,γ 2/T the additional transientlocalfield. Theratiooflineintensitieshasbeen
2
∼
peaks induced by modulation practically vanish. How- determined. Wealsodemonstratedthatadditionalsatel-
ever, the peaks multiple to 4R remain observable up lites can occur in the spectrum due to modulation Rabi
±
until γ T 8 (FIG 4). We examined the spectra as components as a result of finite pulse length.
2
∼
functions of detuning and found that the spectral pat- Inthefinalpartofthisworkwecarriedoutthenumer-
tern is very sensitive to the value of detuning. Even for ical analysis of spectra. It was demonstrated that in the
∆ 0.1R the spectrum is distorted showing increased limit of small density the theoretical analysis well meets
∼
intensities of the lines lying close to the excitation fre- the results of calculations. However, for large densities,
quency. At ∆ 0.8R the multimode structure is no i.e., B > 0.3 this theoretical approach becomes inappli-
∼
longer observed. The light is scattered at the frequency cable. As the density is increased one can observe a sig-
of the pulse. nificant shift of the satellites to the central line in came
cases accompanied with generation of additional bands
multiple to three and four Rabi frequencies.
V. SUMMARY Authors are grateful to A.N. Starostin for discussions
and remarks received in the course of work. We would
Inthisworkwehavecarriedouttheanalysisofscatter- like to acknowledge the financial support from the Rus-
ingofashortlaserpulseinadensetwo-levelmedium. We sian Foundation for Basic Research No.02-02-17153 and
used the approach based on the BBGKY-hierarchy for No.03-02-06590 and grants of the President of the Rus-
the reduced density operators. In the framework of the sianFederationNo.MK1565.2003.02,No.MD338.2003.02
Hartree-Fock approximation and with use of the prop- and No.NS1257.2003.02.
erties of coherent state operators we derived the coop-
erative field operator in the explicit form and compared
our results with the well-known ones [6]. The transfor-
Appendix A. Cooperative field operator
mations used in this work allowedus to derive the set of
equations describing transient spectra of scattered light.
The equations were analyzed using the method of suc- In this section we will derive the explicit form for the
cessiveapproximationsinthecasewhenthemediumwas cooperative field operator. Our attention is now for the
8
40
30
IT
k k20
10
0
-150 -100 -50 n0 50 100 150
k
k
FIG. 4: Same as in FIG2. ∆=0,R=10π,γ =0.5,γ2 =9,ηk =1, T =1, B=0.74
expression (12): where dΩ is the spatial angle element, k = k, r =
lm
r r . Integralofthekind(A.6)havebeen|co|nsidered
l m
[V˜apf,ρfρa]f = i[gl(kα)|2ih1|βk(t)′− |in −[35] a|nd are representable in the following form:
k (A.1)
g (k )∗X1α 2β′∗(t),ρ ρa],
− l α | ih | k f 2πωk µ21 2
B (x)= || ||
ml
where the equation for βk(t) has the form: ¯hW × (A.7)
sin(x) sin(x) cos(x)
b +c .
ddβtk′ =−(iνk+ηk/2)βk′ − N gm(kα)∗ρa21. (A.2) × x (cid:16) (x)3 − (x)2 (cid:17)!
m
X
(A.3) Transition constants b,c for various polarizations ∆M
are
′
Solving the equation for βk(t) formally one can get:
∆M =0: b=0,c=2,
βk′(t)=− N gm(kα)∗ tρa21(τ)e−i(νk+ηk/2)(t−τ)dτ(A. .4) With use of∆(MA.7=) ±th1e:intebgr=al1(,Ac.=5)−w1a.s examin(eAd.8in)
Xl Z0 [33], where allowing for disregard of signal retardation
they obtained the cooperative field operator:
Nowwesubstitutetheresultin(A.1)andbychangingthe
summationoveratomicandfieldvariablesforintegration
correspondingly over the volume and k come to 3n(r)Gλ3
Vp = i 21γ/2 2 1ρa (t,r )+H.c., (A.9)
[V˜p ,ρ ρ ]f =[R (t,r )2 1 R∗(t,r )1 2,ρ ρa] a − 8π2 | ih | 21 l
af f a l l | ih |− l l | ih | f
t
ρa (Rτ,l(rt,r)le)−=i(νZk+ηZk0/2B)(ltm−τ()kdrτlmdr)×k2dk, (A.5) where γ = 4ω2331|h¯|µc321||2 is the spontaneous decay rate
× 21 m m from the upper state, n(r) is the density of the sample,
andG representsthe geometricalfactor being a function
of selected geometry of the sample, its dimensions, and
Blm(krlm)= gl(kα)gm(kα)∗dΩ, (A.6) polarization of scattered light.
Z α=±1
X
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