Manipulative resonant nonlinear optics: nonlinear interference effects and quantum control of nonlinearity, dispersion, transparency and inversionless amplification in an extended strongly-absorbing inhomogeneously-broadened medium

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by  A. K. Popov

Manipulative resonant nonlinear optics: nonlinear interference effects and quantum
control of nonlinearity, dispersion, transparency and inversionless amplification in an
extended strongly-absorbing inhomogeneously-broadened medium
A. K. Popov,1,2,∗ S. A. Myslivets,2 and Thomas F. George3,†
1Department of Physics & Astronomy and Department of Chemistry,
University of Wisconsin-Stevens Point, Stevens Point, WI 54481, USA
2Institute of Physics of the Russian Academy of Sciences, 660036 Krasnoyarsk, Russia
3Department of Chemistry & Biochemistry and Department of Physics & Astronomy,
University of Missouri-St. Louis, St. Louis, MO 63121, USA
(Dated: July 29, 2004)
Specific features of nonlinear interference processes at quantum transitions in near- and fully-
resonant optically-dense Doppler-broadened medium are studied. The feasibility of overcoming of
5
the fundamental limitation on a velocity-interval of resonantly coupled molecules imposed by the
0
Doppler effect is shown based on quantum coherence. This increases the efficiency of nonlinear-
0
optical processes in atomic and molecular gases that possess the most narrow and strongest res-
2
onances. The possibility of all-optical switching of the medium to opaque or, alternatively, to

n absolutely transparent, or even to strongly-amplifying states is explored, which is controlled by a
a small variation of two driving radiations. The required intensities of the control fields are shown
J to be typical for cw lasers. These effects are associated with four-wave mixing accompanied by

1 Stokesgainandbytheirinterferenceinfully-andnear-resonantoptically-densefar-from-degenerate
double-Lambda medium. Optimum conditions for inversionless amplification of short-wavelength

1 radiation above the oscillation threshold at the expense of the longer-wavelength control fields, as
v well as for Raman gain of the generated idle infrared radiation, are investigated. The outcomes
1 are illustrated with numerical simulations applied to sodium dimer vapor. Similar schemes can be
0 realized in doped solids and in fiberoptics.
0
1 PACSnumbers: 32.80.Qk,42.50.Gy,42.50.Hz,42.50.Ct,42.79.Ta
0
5
0 I. INTRODUCTION limitations and thus substantially increase the efficiency
/
h of the nonlinear optical processes.
p Coherence is an important concept in physics. Co-
- herent oscillations may interfere to give rise to counter-
t In the early stages of the laser era four decades ago,
n intuitive effects. A common example is a standing light
the major problem was extracting coherent optical elec-
a wave:
tromagnetic radiation from incoherently-excited materi-
u
q als. The contemporary trend is the inverse, i.e., control y(t) =y (t)+y (t)=asin(ωt−kz)+asin(ωt+kz)
1 2
: of physical properties of materials by coherence injected
v =2acos(ωt)cos(kz).
with lasers. In such a context, this paper is aimed at
i
X studying the opportunities of manipulating optical and
Byoverlappingtwocoherentlightfluxes,onecanproduce
r nonlinear-optical properties of materials without chang- completely-dark spots, instead of achieving doubled illu-
a
ing their composition and structure based on quantum
mination,orviceversa,onecanachieveatwofoldincrease
coherence and interference. The phenomena of reso-
of the oscillation amplitude, which in turn yields a four-
nance, Doppler shift of a frequency, coherenceand inter-
foldincreaseintheillumination. Theconceptofinterfer-
ference are fundamental concepts of physics. Resonance
ence,whichisameasureofcoherence,isappliedtooscil-
enablesone togreatlyenhanceprocessesrelatedtooscil-
lations of any nature. Similar effects can be anticipated
lations. Thestrongestresonancesinopticsareattributed
inregardtooscillationsofboundintra-atomicandmolec-
to quantum transitions in free atoms and molecules.
ular charges. Since induced periodically-accelerated and
However, one of the fundamental limitations of optical
decelerated motion of the atomic electrons is responsi-
physics is that only a small fraction of molecules can be
ble for optical process, the question is how to manage
resonantly coupled with coherent radiation concurrently
destructive interference and completely stop induced os-
in an inhomogeneously-broadened medium. This paper
cillations of an optical electron even in the presence of
shows that coherent control enables one to remove such
a resonant electromagnetic field. This would mean the
decouplingoflightandresonantmedia,vanishedabsorp-
tion and dispersion. Then, if destructive interference is
turnedintoconstructive,thatwouldprovideforstrongly-
∗Electronic address: [email protected];
enhanced coupling.
URL:http://www.kirensky.ru/popov
†Electronic address: [email protected]; The basic idea behind such opportunities can be pre-
URL:http://www.umsl.edu/chancellor sented as follows. The oscillating dipole moment of the
Typeset by REVTEX
2
unityvolume,P(t),isadrivingterminMaxwell’selectro- therelativedirectionofpropagationofthecoupledwaves.
dynamicequationsthatdeterminestheopticalproperties The solution is convenient for further numerical analysis
of the materials. It is proportional to the number den- andexperiments. Itisshownthatinhomogeneousbroad-
sityofatomsN andtotheinducedatomicdipolemoment ening of Raman transitions, which is inherent for far-
d(t). If, besides the probe radiation E (t), an auxiliary from-degenerate schemes, creates important features on
p
strong driving field E (t) is turned on [E (t) = E quantum interference in resonant schemes. The feasibil-
d p,d p,d
exp(iω t)], opticalelectronsbecome involvedinnonlin- ityofovercomingwiththe aidofthe coherenceprocesses
p,d
ear oscillations: of the seemingly fundamental limitation imposed by the
Doppler effect on the velocity-interval of the molecules
P(t)=χEp(t)+χ(3)|Ed(t)|2Ep(t). (1) concurrently involved in resonant nonlinear coupling is
shown. Thus, a major part of Maxwell’s velocity dis-
This equation displays the superposition of two compo-
tribution can be involved in the concurrent interaction
nentsoscillatingwiththesamefrequency. Inthe vicinity
thatdramaticallyincreasesthecross-sectionofnonlinear-
of a resonance, the real and imaginary parts of the sus-
optical processes in warm gases. As an application, the
ceptibilities can be commensurate. Therefore, one can
conversion of the easily-achievable long-wavelength gain
control the amplitudes and relative phase of the inter-
to a higher frequency interval is explored. It is shown
fering oscillations (i.e., constructive and destructive in-
that the most favorable are the optically-dense samples,
terference) by varying the intensity and detuning from
wheretheinhomogeneityofthefieldintensitiesalongthe
the resonance of the driving field. In quantum terms,
medium imposes other important features that must be
this means that the coupling of light and matter and,
accounted for as well. The optimum conditions for the
therefore,opticalpropertiesofthematerials,canbecon-
realization of such processes are investigated, and the
trolledthroughjudiciousmanipulationoftheinterference
potentialof quantum switching,amplification andlasing
ofone-photonandmultiphotonquantumpathways. Con-
withoutpopulationinversionbasedoncoherentcoupling
sequently, this enables one to selectively deposit energy
at inhomogeneously-broadened transitions in optically-
in certain quantum states or, alternatively, to convert
dense spatially inhomogeneous media is demonstrated
energy incoherently deposited in specific quantum states
with the aid of numerical simulations. Another impor-
into coherentlight,while decouplingsuchradiationfrom
tant outcome investigated in this paper is the feasibility
the absorbing atoms at resonant lower levels. As such,
ofmanipulation ofthe resonantmedium, from becoming
spectralstructuresinthe transparencyandrefractivein-
almost entirely optically opaque to strongly amplifying
dex canbe formedandmodified with the aidofthe aux-
via a completely transparent state by only a small vari-
iliary control resonant or quasi-resonantradiation.
ation of the intensity or frequency of one of the lower-
Coherent quantum control of optical processes has
frequency control fields and of a large gain for an idler
proven to be a powerful tool to manipulate refraction,
infrared radiation. It is shown that the required condi-
absorption,transparency,gainandconversionof electro-
tions can be fulfilled, e.g., in the framework of exper-
magnetic radiation (for a review see, e.g., [1, 2, 3, 4,
iments similar to [19] with sodium dimer vapor. The
5, 6, 7, 8]). Among the recent achievements are the
important features of both the signaland idler radiation
slowing down of the light group speed to a few m/s,
inherent for such a scheme include entangled states and
highly-efficient frequency up-conversion, squeezed quan-
suppression of quantum noise.
tumstate lightsourcesandopticalswitchesforquantum
information processing on this basis [9, 10, 11, 12, 13, This paper is organized into seven major sections.
14, 15], and judicious control and feasibilities of near Section II describes a principal experimental scheme
100% population transfer between the quantum energy aimed at manipulating optical transparency of initially
levels [16]. Much interest has been shown in the physics strongly-absorbing media, and provides a set of equa-
and diverse practical schemes of lasing without popula- tions for coupled electromagnetic waves and their solu-
tion inversion (see, e.g.,[1, 2, 3, 4, 5, 6, 7, 17]). Large tion for several limiting cases. A corresponding theo-
enhancementoffully-resonantfour-wavemixing through retical model and the underlying physics are discussed.
quantumcontrolviacontinuumstateshasbeenproposed Section III gives the density matrix equations. Their so-
in [18]. lutions related to the problems under consideration are
The present paper considers optical processes con- giveninAppendix A. Thephysicalprinciples ofmanipu-
trolled through ”injected” quantum coherence, which is lating the local absorption(amplification) and refractive
followedby destructive and constructive nonlinear inter- indices,aswellastheprinciplesofeliminatinginhomoge-
ference effects. The near- and fully-resonant far-from- neous broadening of nonlinear resonances, are described
degeneratedouble-Lambdaschemeisinvestigated,where in Sec. IV. Results of numerical experiments towards
inhomogeneousbroadeningofallcoupled(includingmul- the outlinedeffectsarepresentedinSec. V. The empha-
tiphoton) transitions is accounted for. Analytical solu- sis is placedonopticalswitching andamplificationwith-
tions for density-matrix elements (level populations and out population inversion in an optically-dense medium
coherences)arefoundthataccountforvariousrelaxation assisted by Stokes gain, as well as on the optimal con-
channels,incoherentexcitationofthecoupledlevels,and ditions to achieve the effects. Section VI outlines earlier
Doppler shifts of the resonances. These shifts depend on experiments on quantum control based on constructive
3
and destructive interference and is to show a great vari- opportunities for manipulating local optical parameters.
etyofapplicationsofsimilarschemesofquantumcontrol. But three coupled fields, a probe and two driving fields,
These include up-conversion of infrared, and generation may generate radiation in the vicinity of the transition
of short-wavelength VUV radiation, selective detection gnthroughfour-wavemixing(FWM)processesω +ω =
4 2
of relaxation processes and weak electric and magnetic ω +ω . This idler radiation may actually not get de-
1 3
dc fields, as well as selective population of energy levels. pleted but rather substantially enhanced, since the driv-
SectionVIIsummarizesthemainoutcomesofthepaper. ing field E can easily produce a large Stokes gain. This
1
gain, in turn, can be controlled with the driving field
E . These processes dramatically change the propaga-
3
II. RESONANT FOUR-WAVE MIXING tion properties of the probe field, because three fields,
ASSISTED BY STOKES GAIN enhanced Stokes radiation and two driving fields at ω ,
2
ω and ω , contribute back to the probe field through
1 3
Consider a simple experiment designed to explore the the FWM process. Thus, doubled FWM assisted by
manipulation of optical properties through quantum in- Stokes amplification processes opens extra channels for
terference based on the experimental setup described in thetransferofenergyfromthedrivingfieldstotheprobe
a number of publications on resonantnonlinear optics in oneand,consequently,allowscontrolofthetransparency
sodium vapor (e.g. [19]). Two-atom sodium molecules of the medium for the probe radiation. Such a process
(dimers), Na , possess an electronic-vibration-rotation is usuallycalledopticalparametricamplification(OPA).
2
spectrum in the near-IR and visible ranges that can However,asshownbelow,inresonance,onecannotthink
be coveredwithcommercialfrequency-tunablecw lasers. about the entire process as a sequence of conventional
Figure 1 depicts a relevant partial scheme of energy lev- Raman and FWM processes. This would be misleading
in predicting and interpreting the relevant experiments.
Indeed, the distinguishing differences in the frequency-
m
correlation properties of single- and multiphoton radi-
ation processes dramatically change in strong resonant
fields. Thisleadstoavarietyofimportanteffectsandap-
g plications that are discussed below. Thus, the the effect
under consideration should be viewed as amplification
without inversion (AWI) rather than conventional OPA
accompanied by absorption and Stokes gain. The quan-
ω ω ω ω
tuminterference involved in resonantschemes plays such
4 1 2 3
a crucial role that thinking about the conversion process
n under investigation in terms of the Manley-Rowe photon
conservation law would be misleading [20].
l Thepropagationoffourcoupledopticalwavesinares-
onant medium is described by the set of coupled equa-
tions for the waves (E /2) exp[i(k z − ω t)] +c.c. (i =
FIG.1: Opticalswitching at ω4 through quantumcontrolby i i i
1...4) traveling in an optically-thick medium:
two fields at ω1 and ω3 assisted byidler radiation at ω2.
dE (z)/dz =iσ E +iσ˜ E E E∗ , (2)
4,2 4,2 4,2 4,2 1 3 2,4
elsandtransitions. Similarschemesaretypicalforsolids dE (z)/dz =iσ E +iσ˜ E E E∗ . (3)
dopedby rareearths(like Er3+ andPr3+). Assume that 1,3 1,3 1,3 1,3 4 2 3,1
the sample is probed with a sufficiently-weak (and thus Here, ω +ω =ω +ω , k are wave numbers in a vac-
4 2 1 3 j
non-perturbative) optical field in the vicinity of the blue uum,σ =−2πk χ =δk +iα /2,whereα andδk are
j j j j j j j
transitionml. Thentherefractiveandabsorptionindices the intensity-dependent absorption indices and disper-
inthecorrespondingfrequencyintervalcanbecontrolled sion parts of the wave numbers, σ˜ = −2πk χ˜ (etc.) is
4 4 4
with the driving fields coupledto the lowerorupper lev- aFWMcross-couplingparameter,andχ andχ˜ arethe
j j
els. Resonantdrivingfieldsinduceback-and-forthtransi- corresponding nonlinear susceptibilities dressed by the
tions with the frequency G =E d /2~ betweenlevels j driving fields. Amplification or absorption of any of the
i i jk
andk,whereG (calledtheRabifrequency)isthemagni- coupledradiationsinfluences the propagationfeatures of
i
tudeofthecorrespondinginteractionHamiltonianscaled the others. In the following sections, we will account for
to Planck’s constant. Modulation of the probability am- possible depletion of the fundamental beams along the
plitudesleadstoasplittingoftheresonanceattheprobe medium due to absorption. However, qualitative feature
transitionandtointerferenceoftheone-andtwo-photon ofamplificationofthe probe beamandgenerationofthe
quantum pathways controlled by the driving radiation. idler radiation are seen within the approximation that a
Thismayexhibititselfasoddcoherenceandinterference changeofthedrivingradiationsE alongthemediumis
1,3
structuresintheabsorptionandrefractionspectraofthe neglected. This is possible in some cases, e.g., at the ex-
probe radiation. pense of the saturation effects. Then the system (2)-(3)
Twoindependently-variedfields providefor evenmore reduces to two coupled standard equations of nonlinear
4
optics for E and E , whereas the medium parameters are homogeneous along z. The solution is standard:
4 2
α β γ∗
E∗ =exp − 2 z−βz E∗ cosh(Rz)+ sinh(Rz) −i 2E sinh(Rz) ,
2 (cid:16) 2 (cid:17)(cid:26) 20(cid:20) R (cid:21) R 40 (cid:27)
α β γ
E =exp − 4 z+βz E cosh(Rz)− sinh(Rz) +i 4E∗ sinh(Rz) . (4)
4 (cid:16) 2 (cid:17)(cid:26) 40(cid:20) R (cid:21) R 20 (cid:27)
Here, R= β2+γ2, β =[(α −α )/2+i∆k]/2, ∆k = where P is the polarization of the medium oscillating
4 2
δk1+δk3−δpk2−δk4,γ2 =γ2∗γ4,γ4,2 =χ4,2E1E3,andE2∗0 with the frequency ω4, N is the number density of
and E are input values (at z = 0). The first terms in molecules, d is the electric dipole moment of the tran-
40 ml
the curvedbracketsindicate OPA,and the second terms sition, and ρ is the density matrix element. Other po-
lm
the processes of FWM. larizations are determined in the same way. The density
If ∆k = δk +δk −δk −δk = 0, the input values matrixequationsforamixtureofpurequantummechan-
1 3 2 4
I ≡ |E |2 at z = 0 are I =0 and I 6= 0, and the ical ensembles in the interaction representation can be
4,2 4,2 20 40
absorption (gain) rate substantially exceeds that of the written in a general form as
nonlinear optical conversion obtained at the exit of the
L ρ =q −i[V,ρ] +γ ρ ,
medium of length L: nn nn n nn mn mm
L ρ =L ρ =−i[V,ρ] ,L =d/dt +Γ , (8)
lm lm 4 4 lm ij ij
I4/I40 = exp(−α4l/2)+(γ2/(2β)2) V =G ·exp{i[Ω t−kz]}, G =−E ·d /2~,
lm lm 4 lm 4 lm
× (cid:12)(cid:12)[exp(g2L/2)−exp(−α4L/2)]|2. (5) where Ω = ω ,−ω is the frequency detuning from
4 4 ml
the corresponding resonance; Γ - homogeneous half-
Alternatively, if I =0,I 6=0, mn
40 20
widthsofthecorrespondingtransition(inthecollisionless
η4 =I4/I20 = (|γ4|2/(2β)2) regime Γmn = (Γm + Γn)/2); Γn = jγnj - inverse
× |exp(g L/2)−exp(−α L/2)|2. (6) lifetimes of levels; γmn - rate of relaxatPion from level m
2 4 to n; and q = w r - rate of incoherent excitation
n j nj j
Here, γ2 = γ2∗γ4, γ4,2 = χ4,2E1E3, β = (α4 − α2)/4, to state n fromPunderlying levels. The equations for the
(|γ2|/β2 ≪ 1), and g ≡ −α . From (5) and (6), other elements are written in the same way.
2 2
it follows that at relatively small lengths, the FWM It is necessary to distinguish the open and closed
coupling may even increase the depletion rate of the energy-level configurations. In the open case (where the
probe radiation, depending on the signs of Imγ and lowest level is not the ground state), the rate of inco-
4,2
Reγ . In order to achieve amplification, large optical herent excitation to various levels by an external source
4,2
lengths L and significant Stokes gain on the transition essentially does not depend on the rate of induced tran-
gn (exp(g L/2)≫|(2β)2/γ2|), as well as effective FWM sitions between the considered levels. In the closed case
2
bothatω andω ,arerequired. Asseenfrom(4),(5)and (where the lowest level is the ground state), the excita-
2 4
(6),the evolutionofthe coupledfieldsalongthe medium tionrate to differentlevels andvelocitiesdepends onthe
strongly depends on whether the Stokes field is injected value and velocity distribution at other levels, which are
atthe entrance to the medium, andif so,onthe value of dependentontheintensityofthedrivingfields. Foropen
the ratio of the intensities of the input probe and Stokes configurations, qi are primarily determined by the pop-
fields. ulation of the ground state and do not depend on the
driving fields. The Doppler shifts of the resonances k v
i
canbeaccountedforinthefinalformulasbysubstituting
III. DENSITY MATRIX EQUATION AND Ωi for Ω′i =Ωi−kiv (v is the atomic velocity).
MACROSCOPIC PARAMETERS OF THE In a steady-state regime, the solution of a set of
MEDIUM density-matrix equations can be cast in the following
form:
The macroscopic parameters in Eqs. (4) and (5) are
ρ =r , ρ =r ·exp(iΩ t), ρ =r ·exp(iΩ t),
convenientfor calculationswith the density matrix tech- ii i lg 1 1 nm 3 3
nique, which allows one to account for various relax- ρng =r2·exp(iΩ1t)+r˜2·exp[i(Ω1+Ω3−Ω4)t],
ation and incoherent excitation processes. The power- ρ =r ·exp(iΩ t)+r˜ ·exp[i(Ω −Ω +Ω )t],
lm 4 4 4 1 2 3
dependent susceptibility χ , responsible for absorption
4 ρln =r12·exp[i(Ω1−Ω2)t]+r43·exp[i(Ω4−Ω3)t].
and refraction, can be found as
The density matrix amplitudes r determine the absorp-
i
P(ω )=χ E , P(ω )=Nρ d +c.c., (7) tion/gain and refraction indexes, and r˜ determine the
4 4 4 4 lm ml i
5
four-wave mixing driving nonlinear polarizations. Then At Γ → ∞ and Γ → ∞, the interference disap-
gm ln
the problem reduces to the set of algebraic equations pears. Consequently, the effect of the driving fields re-
ducestoachangeinthelevelpopulations(opticalpump-
P r =iG ∆r −iG r∗ +ir∗ G , ing). Indeed, the two-photon coherences r and r
2 2 2 2 3 32 12 1 41 12
d r˜ =−iG r∗ +ir∗ G , are the source of quantum nonlinear interference effects
2 2 3 41 43 1
(NIE),whichleadtoAWI,Autler-Townessplittingofthe
P r =i[G ∆r −G r +r G ],
4 4 4 4 1 41 43 3 resonances and to related effects of electromagnetically-
d r˜ =−iG r +ir G
4 4 1 32 12 3 inducedtransparency,coherentpopulationtrapping,and
P r =−iG∗r +ir∗G , to corresponding changes in the refraction index. From
41 41 1 4 1 4
P43r43 =−iG4r3∗+ir4G∗3, (12) or (A1) at G3 =0, it follows that
P32r32 =−iG∗2r3+ir2∗G3, χ4 Γ4 ∆r4−g1∆r1
= ·
P12r12 =−iG1r2∗+ir1G∗2, χ04 P4 ∆n4(1+g4)
P1r1 =iG1∆r1, P3r3 =iG3∆r3, (9) Γ4 ∆r4P41−(∆r1|G1|2/P1)
= · , (14)
∆n P P +|G |2
4 41 4 1
Γmrm =−2Re{iG∗3r3}+qm, χ2 = Γ2 · ∆r2−g3∆r1
Γ r =−2Re{iG∗r }+γ r +γ r +q , χ02 P2 ∆n2(1+g2)
n n 3 3 gn g mn m n
Γgrg =−2Re{iG∗1r1}+qg, = ∆Γn2 · ∆r2P1P∗2∗−P(∆+r1|G|G1|2|2/P1∗). (15)
Γ r =−2Re{iG∗r }+γ r +γ r +q, (10) 2 12 2 1
l l 1 1 gl g ml m l
Both equations are of similar structures, and we will
where G = −E d /2~, G = −E d /2~, G =
1 1 lg 2 2 gn 3 briefly review major NIE using the example of Eq. (15).
−E d /2~, G = −E d /2~, P = Γ + iΩ ,
3 nm 4 4 ml 1 lg 1 The effects of the driving field G contributing to the
1
P = Γ + iΩ , P = Γ + iΩ , P = Γ + iΩ ,
2 ng 2 3 nm 3 4 lm 4 changeofabsorptionandrefractionindicescanbeclassi-
P12 = Γln + i(Ω1 − Ω2), P43 = Γln + i(Ω4 − Ω3), fiedas: (i)achangeofthepopulations∆r (|G |2),domi-
4 1
P =Γ +i(Ω −Ω ), P =Γ +i(Ω −Ω ), d =
32 gm 3 2 41 gm 4 1 2 nantatΓ Γ →∞,and(ii)asplittingoftheresonance
41 gm
Γ +i(Ω +Ω −Ω ), d =Γ +i(Ω −Ω +Ω ),Ω =
ng 1 3 4 4 lm 1 2 3 1 (seethedenominatorin(14),dominantat∆r =0. With
1
ω −ω , Ω =ω −ω , Ω =ω −ω , Ω =ω −ω ,
1 lg 3 3 mn 2 2 gn 4 4 ml anincreaseinthedetuningΩ ,oneofthe componentsof
1
∆r = r −r , ∆r = r −r , ∆r = r −r , ∆r =
1 l g 2 n g 3 n m 4 Autler-Townes splitting determines the ac-Stark shift of
r −r .
l m the resonance.
For a closed scheme, Eq. (10), r must be replaced by
l As for any interference, NIE cause only redistribu-
tion absorption and amplification over the frequency in-
terval (line shape) but do not change the frequency-
r =1−r −r −r . (11)
l n g m integrated absorption (amplification). The latter is de-
terminedbythepopulationchangeonly: (χ /χ0)dΩ =
Thesolutionoftheseequationsasappliedtotheproblem 4 4 4
πΓ4∆r4/∆n4. The term in (14) associaRted with rm =
under consideration is given in Appendix A.
n describes the line shape (probability) of pure emis-
m
sion, where all the rest (at r = 0) describe pure ab-
m
sorption. Indeed, aswasfirstemphasizedin[21,23],this
IV. NONLINEAR INTERFERENCE EFFECTS difference determines NIE in absorption, emission and
IN ABSORPTION AND AMPLIFICATION
amplificationspectrawhichenabletheentireelimination
SPECTRA
of absorption and appearance of transparency at unequal
populations of the levels at the probed transition and to
A. Nonlinear interference effects in three-level amplification without population inversion in some fre-
schemes
quency interval(s) at the expense of enhanced absorption
in other intervals. As seen from (14), at Ω = Ω = 0,
4 1
Considertwosimplesubcases,G3 =0(V-scheme)and the corresponding requirements to achieve transparency
G1 =0 (Λ-scheme). From (9), it follows that and then amplification without a change of sign of ∆r4
are
P r =i[G ∆r −G r ],
4 4 4 4 1 41
P41r41 =−i[G∗1r4−r1∗G4], (12) ∆r1|G1|2/ΓgmΓlg ≥∆r4. (16)
Therefore, owing to NIE, population inversion between
P2r2 =i[G2∆r2+r1∗2G1], the initial and final bare states is not required in order
P r =−i[G r∗−r G∗]. (13) to achieve AWI in this case. The requirements for trans-
12 12 1 2 1 2
parency at unequal populations of levels at the probed
The structure of (12) and (13) is the same as that of resonant transition and that for amplification without
(1): they present the interference of two oscillations. population inversion for Ne transitions was investigated
6
in [24] (see also [25]) during the course of early studies for molecules at different velocities due to the Doppler
of NIE with He-Ne lasers. Then lasing without popula- shifts. This is not the case in near-degenerate schemes.
tion inversion was experimentally realized in [27] in gas The interference of elementary quantum pathways, ac-
dischargeatthe transition2s –2p (λ = 1.15µ) ofNe in counting for Maxwell’s velocity distribution and satu-
2 4
the presence of the driving field of another He-Ne laser ration effects, results in a nontrivial dependence of the
resonanttotheadjacenttransition2s –2p (λ=1.52µ). macroscopic parameters on the intensities of the driving
2 1
It is misleading to think about resonant processes in Λ fields and on the frequency detunings from the centers
and V schemes as conventional Raman processes. The oftheinhomogeneously-broadenedresonances. However,
amplificationandabsorptionacquirethe featuresoftwo- this limitation that seems fundamental can be overcome
photonprocesses,while the driving field is detuned from by a judicious compensation of Doppler shifts with ac-
resonanceandinterferencebetweenone-andtwo-photon Stark shifts.
pathways vanishes. Indeed, at |Ω | ≈ |Ω | ≫ Γ ,Γ , Consider the principles of such compensation for the
1 4 1 4
|g | ≪ 1,|g | ≪ 1, P ≈ iΩ ,P ≈ iΩ ≈ iΩ , one example of the absorption (gain) index α . In order to
4 1 4 4 1 1 4 2
obtains from (14) the equation for the absorption (am- account for Doppler effects, we must substitute all Ω in
j
plification) index α(Ω ) reduced by its value α0(0) at the above formulas for Ω′ = Ω −k v, where v is the
4 j j j
resonance at G =0: molecularvelocity,andthenperformvelocityintegration
1
over a Maxwell distribution. Corresponding factors P
j
α(Ω4) ≈ Γ24∆r4 −Re Γ4(∆r4g4+∆r1g1) must be substituted for velocity-dependent factors Pj′ =
α0(0) Ω2∆n (cid:26) iΩ ∆n (cid:27) P (Ω′). Let us assume all detunings Ω , except Ω −
4 4 4 4 j j j 1
Γ2∆r Γ Γ |G |2(∆r −∆r ) Ω2,tobemuchgreaterthanthecorrespondingtransition
≈ 4 4 − 4 14 · 1 1 4 Dopplerwidths. ThenallfactorsP′ andP′ ,exceptP′ ,
Ω2∆n Γ2 +(Ω −Ω )2 Ω2∆n j ij 12
4 4 14 4 1 4 4 become independent of v. According to Eq. (15), the
= Γ2lm(rl−rm) dependenceofthedenominatoronvelocityisdetermined
(nl−nm)Ω24 by the factor
− ΓgmΓlm · |G1|2(rm−rg). (17) P1′∗2 +|G1|2/P2′ ≈Γln−i[Ω1−Ω2−(k1−k2)v]
Γ2gm+(Ω4−Ω1)2 Ω24(nl−nm) +[|G1|2/(Γlg+iΩ1)]−i(|G1|/Ω1)2k1v.(18)
The lastterms in (17) describe Raman-likecoupling and Equation (18) indicates a velocity-dependent broaden-
originate both from the nominator and denominator in ing (Re{|G1|2/P2′}) and shift (Im{|G1|2/P2′}) of the res-
Eq. (14). In this case, population inversion between the onance that allows one to compensate for the Doppler
initial and final bare states (rm = nm > rg) is required effect at (|G1|/Ω1)2k1 = k1 − k2, and therefore all
for amplification of the probe field. molecules will be uniformly coupled to the two-photon
In more detail, the nature of NIE in the context resonanceindependent oftheir velocities. Compensation
of interference of quantum pathways and modification of the Doppler shifts with light shifts was proposed in
of the frequency-correlation properties of multiphoton [28,31,32]. Opportunitiesforgreatenhancementofvar-
processes in strong resonant fields was discussed in iousopticalprocesseswiththiseffectwererecentlyinves-
[25, 28, 29]. Numerical examples of AWI, including ef- tigated in more detail in [33, 34, 35, 36, 37]. We shall il-
fectsrelatedtothegrowthoftheamplifiedfield,aregiven lustratethe aboveconsideredeffects withnumericalsim-
in [30] both for open and closed energy level configura- ulationsappliedtotheexperimentalschemedescribedin
tions. Sec. II.
B. Quantum coherence and elimination of V. NUMERICAL SIMULATIONS
inhomogeneous broadening of multiphoton
transitions with light shifts As an example, we shall consider the transitions l −
g−n−m−l (Fig. 1) as those of sodium dimers Na :
2
As outlined above, two-photon coherence r plays a X′Σ+(v” = 0,J” = 45) – A′Σ+(6,45)(λ = 655 nm) –
ln g u 1
keyroleintheprocessesunderinvestigation. Theimpor- X1Σ+(14,45)(λ = 756 nm) – B1Π (5,45)(λ = 532 nm)
g 2 u 3
tant feature of the far-from-frequency-degenerate inter- – X′Σ+(0,45)(λ = 480 nm) from the experiment [19].
g 4
action is that the inhomogeneous (in our case, Doppler) We shall use the experimental relaxation data: γ = 7,
gl
broadening of a two-photon transition is much greater γ = 4, γ = 5, γ = 10, Γ = Γ = 20, Γ = Γ
gn mn ml l n g m
than its homogeneous width, ζ =Γln/(∆Dlg −∆Dng) ≪1, = 120, Γlm = Γnm = Γng = Γlg = 70, Γln = 20, Γgm =
where ∆D and ∆D are the Doppler HWHM of the cor- 120(allin106 s−1). TheDopplerwidthofthetransition
lg ng
respondingtransition. Therefore,onlyaζ fractionofthe (FWHM) at the wavelength λ =480 nm at a tempera-
4
molecules are resonantly involved in the process. Cor- ture of 410◦C used for modelling is approximately equal
respondingly, the magnitude and sign of the multipho- to1.7GHz. ThentheBoltzmannpopulationoflevelnis
ton resonance detunings and, consequently, of the am- 1.4%ofthatoflevell. Allofthis, aswellas the inhomo-
plitude and phase of the lower-statecoherence ρ , differ geneity of the material parameters and phase mismatch
nl
7
∆k, are accounted for in the numerical simulations de- 1
scribed below. 4
l−n 0.02
A. Manipulating absorption and amplification 2)0.5
indices with coherent elimination of Doppler u 0.01
2/
broadening of multiphoton resonances v
−
( 3
p 1
x 0
Figure 2 depicts absorption (upper plots) and Stokes e −0.5 0 0.5
r i 0
amplification(lowerplots)computedatrelativelylowin- ∆
tensities of the driving fields that are set to the corre-
sponding resonances. Plot (a) corresponds to the condi-
2
tions at the the entrance to the medium, and (b) to the
radiationdepletedafterpropagationthroughthemedium −0.5
−0.5 −0.25 0 0.25 0.5
length Z =α40z = 15, where the OPA of the probe field v/u
reaches its maximum (see Fig. 8). As seen from the
plots,thedepletiongivesrisetoadramaticchangeinthe
FIG. 3: Differences of level population vs molecule velocity
spectral properties of the absorptionindex for the probe
projections v onthewavevectors(uis thermalvelocity). l-n
bluefieldandoftheStokesamplificationindex. Theline corresponds to rl−rn, and theother plots to thetransitions
shapes donotresemblethose forconventionRamanpro- as indicated. All parameters are the same as in Fig. 2(b).
cesses. Despite some power-induced depopulation of the
lower level, an increase of α occurs in some frequency
4
interval in the vicinity of the transitionresonance center
at the expense of its decrease in other frequency inter-
vals. Forallthefigureshereandbelow,Ω isdetermined
2
by the equation Ω =Ω +Ω −Ω .
2 1 3 4
1
4
0
0
4
(cid:68)
/(cid:68)i(cid:358)1
(cid:358)2 (a)
2
(cid:358)250 0 250
1
0.5 4
0
4
(cid:68) 0
/(cid:68)i
(cid:358)0.5
(b)
2
FIG.4: Resonancelaser-inducedspectralstructures(highin-
(cid:358)1
(cid:358)250 0 250 tensities of driving fields). Upper plots - absorption of the
(cid:58) (MHz)
4 probe blue radiation, lower - Stokes gain. G1 = 1 GHz,
Ω3 = Ω1 = 0. (a) G3 = 1070 MHz, (b) G3 = 415 MHz.
FIG. 2: Spectral structures induced at low intensity of the Insets - corresponding magnified peaks. ∆ - FWHM of the
resonantdrivingfields. Upperplots-absorptionoftheprobe corresponding peaks.
blue radiation, lower - Stokes gain. Ω1 =Ω3 = 0. (a) G1 =
60 MHz, G3 = 20 MHz; (b) G1 = 16 MHz, G3 = 19 MHz.
distribution of level populations over the velocities pro-
Figure 3 displays a substantial inhomogeneity of the ducedbythedrivingfields. Populationinversionappears
8
1
0.5
0
−0.5
−2 −1 0 1 2 3
Ω (GHz)
4
FIG. 5: Laser-induced spectral structures (high intensities of
driving fields). Upper plots - absorption of the probe blue
radiation, lower -Stokesgain. G1 = 1GHz, G3 = 400 MHz,
Ω1 = 2140 MHz, Ω3 = 400 MHz.
at the Stokes transition ng in a narrow velocity interval,
while no population inversion is produced at the Raman
transition ln. The figure indicates the smallness of the
fraction of molecules resonantly coupled to the driving
fields. This important fact must be taken into account
while considering amplification with and without popu-
lation inversion [30].
Figure 4(a) shows corresponding line shapes at higher
FIG. 6: Compensation of Doppler shifts with light shifts
intensities of the driving fields chosen so that the split-
and sub-Dopplerresonances. Upperplots - absorption of the
ting of coupled levels is nearly equal. This determines probeblueradiation,lower-Stokesgain. Ω1=2140MHz,G1
the coincidence of two out of the four spectral compo- =1GHz. (a) G3 =0, (Ω2 =2140 MHz-Ω4); (b) Ω3 =Ω1,
nentsattheprobeandattheStokestransitions. Plot(b) G3 = 242 MHz (Ω2 = 4280 MHz - Ω4). ∆ is FWHM of
demonstratesfurther substantialchangeofthe spectrum the corresponding peaks. Insets - magnified corresponding
by variation of the intensity of one of the fields and ap- laser-inducedstructures. TheDopplerFWHMoftheRaman
pearanceofnarrowsub-Dopplerlaser-inducedabsorption transition is 170 MHz.
and amplification structures. The position and shape of
thestructurescanbevariedwithinthefrequency-interval
comparable with the Doppler width. the Doppler FWHM of the Ramantransition(170 MHz)
Figure 5 illustrates the opportunities to form a trans- and comparable to the homogeneous FWHM of the Ra-
parency window in any frequency intervalin the vicinity mantransition(6.4MHz). Figure6(b)showsfurtherfea-
of the probe transition. As follows from Figs. 2, 4 and sibilities for producing sub-Doppler structures with two
5, the maxima and minimum in the Stokes gain, as a driving fields.
rule, correspondto the maxima andthe minimum in the Figure 7 displays the sensitivity of the width of the
absorption index. induced structures on the choice of the parameters of
Figure 6 demonstrates the feasibilities of compensat- the driving fields. The width of the most narrow reso-
ing for Doppler shifts with ac-Stark shifts and corre- nance(plot2)isevenlessthanthehomogeneousFWHM
sponding substantial field-induced narrowing of the in- ofthe correspondingtransitionsandcommensurate with
duced resonances. The upper plot in 6(a) presents a that of the most narrow resonance in the system (tran-
red-shiftedandslightly-decreasedDoppler-broadenedab- sition ln). Since all nonlinear resonances possess asym-
sorption resonance, and the lower one the feasibility of metric and non-Lorentzian shapes, the width of such a
theformationofapower-inducedDoppler-freeStokesres- resonance has different properties than those attributed
onance. At G2 ≪ Ω2, the ac-Stark shift of the Stokes to a conventional resonance of a Lorentzian shape. The
1 1
transition is estimated as G2/Ω , and the dressed two- insets prove that the narrowing is related to judicious
1
photonresonanceasΩ =Ω +G2/Ω . Thisisinagood compensation of the Doppler shifts by ac-Stark shifts.
2 1 1 1
agreement with the computed plots. The FWHM of the Plots 2 in the insets indicate that the central portion of
induced resonance is 17 MHz, which is much less than the Maxwell velocity distribution (the majority of atoms)
9
FIG. 8: Laser-induced inversionless amplification, trans-
parency and generated Stokes radiation at low intensities of
fundamental beams (Z = α40z). Ω1 = Ω3 = 0, G10 = 60
MHz, G30 = 20 MHz. (a) probe; (b) generated Stokes idler
radiation.
(GHz) the amplifier are stipulated from the interplay between
absorption,amplificationofgeneratedidlerStokesradia-
tion [Fig. 8(b)], phase-matching, FWM cross-coupling
FIG. 7: Power-induced Doppler-free resonances. Ω1 = Ω3 =
2140MHz,G3 =242MHz. (a)-laser-inducedstructuresin parameters, and variation of all material parameters
absorptionindexα4(Ω4), inset-velocitydistributionα4(v/u) along the medium due to saturation effects accounting
at Ω4 set to the center of the corresponding resonance (u is forthevelocity-dependentNIE.Itisremarkablethatgen-
thermal velocity). Maxwell’s envelope is removed. ∆ - full erated idler Stokes radiation may several hundred times
width at half maximum of the corresponding peaks. (b) - exceed the input probe radiation [Fig. 8(b)]. The in-
same for Stokes amplification index. (a): 1 — G1 = 1500 put Rabi frequencies of the control fields for this graph
MHz(∆ =98 MHz),2— G1 =1000 MHz(∆ = 17.6 MHz), correspond to the focused cw radiations on the order of
3 — G1 = 500 MHz (∆ = 133 MHz), 4 — G1 = 0 (∆ = 232 several tens of mW, i.e., to about one photon per thou-
MHz); (b)–sameparametersasin(a),wheretheFWHMof
sand molecules. The length of the medium is scaled to
thestructures 1, 2 and 3 are 64, 15.8 and 128 MHz.
theabsorptionlengthatω =ω ,withalldrivingfields
4 ml
turned off.
In a good agreement with (5), Fig. 9(a) shows that,
becomes involved in the coupling and simultaneously con-
first, the probe beam is sharply depleted, and a substan-
tributes to the absorption maximum of the narrowed res-
tialopticallengthisrequiredtoachieve transparency, and
onance (compare with Fig. 3). The other wider reso-
then there is significant inversionless gain (main plot).
nances remain inhomogeneously broadened.
The complete transparency is achieved at a length of
about 4. With the same driving fields but with the probe
radiation tunedtotheexactresonance (lowerrightinset),
B. Generation of Stokes radiation, optical amplification is not achievable. The upper inset shows
switching and inversionless amplification of strong depletion of the driving radiation E along the
1
short-wavelength radiation above the oscillation medium at the chosen intensity and detuning from the
threshold in resonant optically-dense media
resonance. The evolution of the amplitudes of the fields
along the medium is determined by the relative phase
Figure 8 displays the results of numerical simulations Ψ=φ −φ +φ −φ +(k −k +k −k ) z =Θ+∆kz.
4 3 2 1 4 3 2 1 z
ofamplificationofshort-wavelengthradiationandofgen- ThelowerleftinsetinFig. 9(a)showssubstantialchange
eration of long-wavelength field controlled by quantum of the relative phase Θ along the medium that is deter-
interference for the conditions corresponding to Fig. 2. mined by the inhomogeneity of the coupled field and,
The optimum parameters and output characteristics of consequently, material parameters. Figure 9(b) presents
10
3300 11 00 (a)
1 3
(cid:237)(cid:237)11
I/II/I4404402200I/II/I1101100000 55 1100 (cid:52)(cid:52)(cid:237)(cid:237)(cid:237)(cid:237)(cid:237)(cid:237)43243200 55 11001155 (i)(4)n/n(cid:39)phph(cid:358)10 24
11 1
1100
0 10 20 Z 30 40
00..55 (cid:358)5
x 10
2
((aa)) 00 (b)
00 55 11001155
00
00 55 1100 1155
1 ZZ (4)nph 2, 4
(i)n/ph0
(cid:39) 1, 3
0.8
1
0.6 (cid:358)2
40 0 10 20 Z 30 40
/I4
I
0.4 0.5
FIG.10: Relativechangeof thephotonnumbersin thecou-
pled fieldsalong themedium at neglected absorption and re-
0.2 fraction. G10 = 60 MHz, G30 = 20 MHz. (a) Ω1 = Ω3 =
040 50 60 70 80 Ω4 = 0, and (b) Ω1 = 1 GHz, Ω3 = −1 GHz, Ω4 = −2.5
(b) G (MHz)
0 10 GHz.
30 35 40 45 50
(cid:58) (MHz)
4
turbation theory. Indeed, Fig. 10(a) displays even qual-
FIG. 9: Propagation of probe radiation (a) and optical itatively incorrect behavior with respect to the Manley-
switching (b) (Z = α40z). Ω1 = Ω3 = 0, G10 = 60 MHz, Rowe relationship (photon conservation law). As detun-
G30 = 20 MHz [(a) and main plot (b)]. (a) Ω4 = 35 MHz ingfromtheresonanceincreases, thedisagreementceases
(main plot) and Ω4 = 0 (lower inset). Upper left inset - de- [Figure 10(b)].
pletion of the fundamental radiation at ω1, upper right inset Figure 11 shows the feasibility of realizing strong am-
- phasefor main (a). (b) Main plot and inset - Z =2, inset
plification of the probe short-wavelength radiation at the
- Ω4 =35 MHz. Minimum transmission is less than 10−3.
expenseofotherlonger-wavelengthdrivingradiations,de-
spite the enhanced absorption index. The importance of
optimization of the conversion parameters based on the
feasibilities of all-optical switching, controlled by quan- processesinvestigatedhere,onthe developedtheoryand
tum interference. Only a small variation of the fre- on the numerical simulation is explicitly seen from com-
quency of the probe field and/or the input intensity of parison of the plots (a) and (c). If so, substantial (sev-
the driving field I10 results in a dramatic change of the eral thousand times) amplification of the shortest wave-
intensity of the transmitted probe radiation. By this, the length radiation becomes feasible. Amplification slightly
sample can be easily driven to the opaque state (less than exceedingunityensuresgenerationinsidetheringcavity.
10−3 of the transparency) or to entire transparency and Hence, Figs. 8, 9 and 11show feasibilities oflasing with-
strong amplification without any change of the level pop- out requirements of population inversion at the resonant
ulations. transition.
Figure 10 displays the change of the number of pho-
tons in each beam along the medium scaled to the num-
ber of photons in the probe field at the entrance to the VI. EARLIER EXPERIMENTS ON RESONANT
medium, assuming σ =0 in Eqs. (2) and (3). The plots NONLINEAR-OPTICAL PROCESSES
i
arecomputedfor the sameinput intensities as Fig. 9(a), CONTROLLED THROUGH QUANTUM
butFig. 10(a)correspondstothefully-resonantcaseand INTERFERENCE
Fig. 10 (b) to far-from-resonance coupling. The figure
proves that the interference of quantum pathways plays Constructiveanddestructivenonlinearinterferenceef-
suchanimportantroleinfully-resonantmulti-wavelength fects,whichplayacrucialroleforquantumswitchingun-
coupling that the process as a whole can not be viewed as derconsiderationinthepresentpaper,areoftenmetand
a sequence of independent conventional elementary one- play an important role in nonlinear spectroscopy and in
photon/two-photon FWM processes, as they were intro- resonant nonlinear optics. Comprehensive experiments
ducedforoff-resonantconditionsintheframeworkofper- on nonlinear interference effects at Doppler-broadened