Gain without population inversion in V-type systems driven by a frequency-modulated field

20 Pages·0.37 MB·English

by  Harshawardhan Wanare

Gain without population inversion in V-type systems driven by a
frequency-modulated field
HarshawardhanWanare
Department of Physics, Indian Institute of Technology, Kanpur 208016, Uttar Pradesh, India
(February 2, 2008)
We obtain gain of the probe field at multiple frequencies in a closed three-level V-type system
using frequency modulated (FM) pump field. There is no associated population inversion among
2 theatomicstatesoftheprobetransition. Wedescribeboththesteady-stateandtransientdynamics
0
ofthissystem. Undersuitableconditions,thesystem exhibitslargegain simultaneously at seriesof
0
frequencies far removed from resonance. Moreover, the system can be tailored to exhibit multiple
2
frequency regimes where the probe experiences anomalous dispersion accompanied by negligible

n gain-absorption overalarge bandwidth,adesirable feature forobtaining superluminal propagation
a of pulses with negligible distortion.
J

4 42.50.Hz,42.50.Gy,32.80-t
1

]
s
c I. INTRODUCTION
i
t
p
Inthelastdecade,alotofremarkableeffectsofatomiccoherencehavebeenobservedinvariousmultilevelatoms[1].
o
Oneoftheseeffectsislasingwithoutinversion(LWI)[2]whichhasprospectsinshort-wavelengthlasing. Conventional
.
s lasersbasedonpopulationinversionbecomeimpracticalinthesewavelengthregimesdueto the ω3 dependence ofthe
c
Einstein A coefficient. Several experiments have provided unequivocal evidence of LWI; starting from amplification
i
s withoutinversionintransient[3]theninthesteadystateregime[4],leadingultimatelytoLWI[5]. IntheLWIschemes
y
anexternaldrivingfieldalonganearbytransitiongeneratesatomiccoherencewhichcontributestogainandalleviates
h
p the population inversioncondition. In this paper we show another significant contribution that becomes operative in
[ presence of strong modulation and results in gain at multiple frequencies. The conventionalLWI based laser systems

sufferfromamajorlimitationduetothesmallgaintheyexhibitincomparisontothepopulationinversionbasedlaser
1
systems. However,this difficulty is largelyovercomeinthis FMfield drivensystembecause under suitable conditions
v
9 one can obtain gain enhancement of (∼ 103%) with respect to the conventional LWI schemes (monochromatic field
2 driven systems).
0 TheFMfieldinteractingwithatwo-levelsystemhasbeenintenselystudiedearlier[6–10]andmorerecently[11–15].
1 The FM pump field produces a large number of sidebands [13] which lead to periodic modulation of the absorption
0 coefficient and the fluorescence signal [10]. Analytical solutions have been obtained for the weak modulation case
2
in [8]. The periodic FM field forms the basis of ultrasensitive absorption spectroscopy [9,11] where the resonant
0
information is made to ride on the modulation frequency or its harmonics thus overcoming the predominantly low
/
s frequency noise of lasers. The FM field provides many independent parameters which can be sensitively controlled
c
and thus result in novel effects like the trapping of population [15], in multilevel systems suppression of a series of
i
s resonances occurs in the Autler-Townes spectra [16]. The trapping phenomenon due to the periodic FM field [17] or
y
amplitude dependent phase modulation [20]can be exploited to achieve robusttransfer of population across multiple
h
p levels. The systemsdrivenby amplitude modulatedfieldhavealsobeenwidely studied: the two-levelsystem[18]and
: the three-level system [19] both exhibit multiple resonances resulting from the modulation. We exploit the multiple
v
resonances generated by the FM field and then tailor various parameters to obtain the desired gain features. To our
i
X knowledge studies have not been undertaken to obtain gain in multilevel systems using FM fields, particularly in the
r strong modulation regime, we undertake this study in this paper.
a The organization of the paper is as follows: in section II we obtain the density matrix equations that govern the
dynamics of the V-type system pumped by a FM field on one transition and is probed by a monochromatic field on
the neighboring transition. We present the analysis of the results in section III, where we begin with the description
ofthe steadystate responseofthe systemandgoontodiscussthetransientdynamicsandfinallydiscussthe physical
basis of the gain obtained. In the steady state analysis we first deal with the off-resonant case where the central
frequency of the FM field is detuned from atomic resonance followed by the on-resonance case. The contribution of
atomic coherence between the two excited states of the V-type system is known to be responsible for inversionless
probegaininsystemspumped byamonochromaticfield[21,22]. Here,wedescribe anothercontributionwhichcomes
into being purely due to the modulation and plays a critical role in obtaining gain. We also describe the Floquet
analysis which sheds light on the striking change that occurs in the probe spectrum at specific values of the index
1
of modulation. We utilize some of these features to obtain anomalous dispersion for the probe field accompanied by
negligible absorption-gainin any desired frequency regime. This is an attractive feature for obtaining distortion free
superluminal propagation [23]. Next, we describe the transient dynamics and try to identify the dominant nonlinear
processes involved. The intricate dynamics of the time evolution is presented. We describe in detail the physical
mechanism which is based on spontaneous emission assisted nonlinear optical process that we believe dominates the
gain process. We present our conclusions in section IV.
II. MODEL AND CALCULATION
We consider a closed three-level V-type system (Fig. 1) wherein, one of the transitions is coupled to a FM pump
field and the other transitionis coupled to a probe field as well as an incoherent(broadband) pump. The field at the
atom is given as
E=E e−i[ω1t+Φ(t)]+E e−iω2t+c.c.; (1)
1 2
Φ(t)= Msin(Ωt),
where E (E ) is the amplitude of the pump (probe) field, the FM field is sinusoidally modulated about the central
1 2
frequency ω and the modulation is characterized by two independent parameters - the frequency of modulation Ω
1
and the index of modulation M. The FM field couples the |1i↔|3i transitionand the monochromaticprobe field at
ω couples the |2i↔|3i transition.
2
The total Hamiltonian of the system is
H =h¯ω |1ih1|+h¯ω |2ih2|−d·E, (2)
13 23
where, d=d |1ih3|+d |2ih3|+c.c. The first two terms in the Hamiltonian correspondto the unperturbed atomic
13 23
system where the energies are measured from the ground state |3i, and the last term is the interaction term in the
dipole approximation. The semi-classical density matrix equation is
dρ i
= [H,ρ]−γ (|1ih1|ρ−2ρ |3ih3|+ρ|1ih1|)−(γ +Λ)(|2ih2|ρ−2ρ |3ih3|+ρ|2ih2|)
1 11 2 22
dt ¯h
−Λ(|3ih3|ρ−2ρ |2ih2|+ρ|3ih3|), (3)
33
where,2γ (2γ )isthe rateofspontaneousemissionfromthe level|1i(|2i)to|3i,2Λisthe rateofincoherentpumping
1 2
on the |2i↔|3i transition.
We transform the equation of motion (3) into a frame rotating with the instantaneous frequency of the field by
using the following relations
ρ˜ =ρ , i=1,2,3,
ii ii
ρ˜ =ρ ei[ω1t+Φ(t)],
13 13
ρ˜ =ρ eiω2t,
23 23
ρ˜ =ρ ei[(ω1−ω2)t+Φ(t)],
12 12
and undertake the rotating-wave approximation by neglecting the counter-rotating terms at nearly twice the optical
frequency, like e±2i[ω1t+Φ(t)] and e±2iω2t. This approximation is valid if |dΦ(t)/dt| ≪ ω1 which is valid for typical
frequency modulation in the optical regime. Due to the coupling to the FM field the slowly varying density matrix
(ρ˜) equations involve time-dependent detuning factors which go as dΦ(t)/dt.
We intend to obtainexact solutions ofthe density matrixequationforarbitrarystrengthofthe fields andarbitrary
values of the index of modulation. For this purpose we use the Fourier decomposition
∞
ρ˜ = ρ(n)e−inΩt; i,j =1,2,3, (4)
ij ij
n=X−∞
which involves integral multiples of the frequency of modulation Ω.
On substituting the expression (4) in the equation of evolutions of the slowly varying density matrix (ρ˜ ) and
ij
equating various powers of Ω, we obtain the following infinite set of equations where n is an integer that varies from
−∞ to ∞. The equations that governthe dynamics of the system are
2
dρ(n)
11 =−(2γ −inΩ)ρ(n)+iG (ρ(n)−ρ(n)),
dt 1 11 1 31 13
dρ(n)
22 =−(2γ +2Λ−inΩ)ρ(n)+2Λρ(n)+iG (ρ(n)−ρ(n)),
dt 2 22 33 2 32 23
dρ(n)
33 =−(2Λ−inΩ)ρ(n)+(2Λ+2γ )ρ(n)+2γ ρ(n)−iG (ρ(n)−ρ(n))−iG (ρ(n)−ρ(n)),
dt 33 2 22 1 11 1 31 13 2 32 23
dρ(1n2) =−(γ +γ +Λ−i(∆ −∆ )−inΩ)ρ(n)+iG ρ(n)−iG ρ(n)+ iMΩ(ρ(n+1)+ρ(n−1)),
dt 1 2 1 2 12 1 32 2 13 2 12 12
dρ(1n3) =−(γ +Λ−i∆ −inΩ)ρ(n)+iG (ρ(n)−ρ(n))−iG ρ(n)+ iMΩ(ρ(n+1)+ρ(n−1)),
dt 1 1 13 1 33 11 2 12 2 13 13
dρ(n)
23 =−(γ +2Λ−i∆ −inΩ)ρ(n)+iG (ρ(n)−ρ(n))−iG ρ(n), (5)
dt 2 2 23 2 33 22 1 21
where ∆ = ω −ω is the detuning of the central frequency of the FM field from the atomic resonance on the
1 13 1
|1i ↔ |3i transition, the probe field detuning is ∆ = ω −ω on the |2i ↔ |3i transition. The strength of the
2 23 2
atom-field coupling is given by the Rabi frequency 2G = 2d ·E /¯h on the |ii ↔ |3i transition, for i = 1 and 2
i i3 i
denoting the pump and probe Rabi frequencies, respectively. The six equations in Eqs. (5) and three equations of
motion for ρ(n), ρ(n) and ρ(n) form a set of nine equations for each n. We note that the set of equations for n are
21 31 32
coupled to set for n±1. The closure of the above system requires that ρ(n)+ρ(n)+ρ(n) =δ .
11 22 33 n,0
We obtain exact non-perturbative solutions of the above equations in both the transient as well as steady-state
regimes. Thetransientsolutionsareobtainedbytakingalargesetofclosed(2N+1)×9firstordercoupleddifferential
equations where the harmonic index n varies from −N to N, and numerically integrating them using fourth-order
Runge-Kuttaroutine. Needlesstosay,wehavecheckedtheconvergenceofthesolutionsbyincreasingN anddecreasing
the step-size of integration. We obtain the steady-state solutions by setting the left hand side time derivatives to
zero in Eqs. (5) and obtain tri-diagonal recurrence relations which are solved using the infinite continued fraction
technique [24,10].
III. RESULTS AND DISCUSSION
A. Steady-state response
Wepresentthesteady-stateresponseofthesystemdescribedinsectionII.Weconsidertwodifferentcases,firstone
is the off-resonantcase whereinthe centralfrequency of the FM fieldis detuned fromthe atomic transition(∆ 6=0),
1
and the second case where the central frequency of the FM field is on resonance with the atomic transition. The
second case permits a comparison with the usual system pumped by a monochromatic field instead of the FM field
[21]. In all our calculations the relevant frequency/time variables are appropriately normalized with the spontaneous
emission decay on the pump transition, namely γ .
1
In presence of the FM field on |1i↔ |3i transition, the probe field on the |3i↔|2i transition exhibits two combs
of absorption peaks which are slightly displaced from each other. Each comb has frequencies that are separated by
∼±ΩandtherelativeshiftbetweenthecombsdependsonM,∆ andG . Theoverallspectrumappearsasaseriesof
1 1
double peak structures displaced by ∼±Ω. The width of the resonances depend on the incoherent processes like the
decaysandthe incoherentpump. Fig. 2ashowsatypicalabsorptionspectrumofthe probefieldmodifiedby astrong
FM field on the neighboring transition. In our notation, absorptionof the probe field occurs when Im(ρ(0))<0. The
32
two displaced comb like resonances result from probing the two linearly independent set of Floquet states resulting
from the time periodic nature of the FM field-atom interaction. We discuss these aspects in detail below.
Inpresenceofthe incoherentpumpΛonecomboffrequenciesexperiencesgain(Im(ρ(0))>0correspondsto probe
32
gain). The two set of combs continue to remain displaced and one of them is flipped over in the opposite direction
exhibiting gain as seen in Fig. 2b.
Intermsofthedensitymatrixequationsthesteadystatecontributionstotheprobegain/absorptioncanberesolved
into three different terms, namely,
i G (ρ(0)−ρ(0)) (γ +γ +Λ−i(∆ −∆ ))
F = 2 22 33 1 2 1 2 ,
1 (γ +γ +Λ−i(∆ −∆ ))(γ +2Λ+i∆ )+G2
1 2 1 2 2 2 1
3
G G ρ(0)
F = 2 1 13 ,
2 (γ +γ +Λ−i(∆ −∆ ))(γ +2Λ+i∆ )+G2
1 2 1 2 2 2 1
MΩ G (ρ(1)+ρ(−1))
F =− × 1 12 12 ,
3 2 (γ +γ +Λ−i(∆ −∆ ))(γ +2Λ+i∆ )+G2
1 2 1 2 2 2 1
ρ(0) = F + F + F . (6)
32 1 2 3
The first term F contains the contribution of the population difference between the energy levels |2i and |3i (when
1
M =0itcorrespondstotheconventionalpopulationinversionterm),the nexttwotermscorrespondtothecoherence
term in the conventionalmodulation free (M =0) scheme [21,22]. Note that all the three components have the same
denominator. The second term F provides the probe response to the dynamics at the central FM frequency ω , or,
2 1
to the n = 0 contribution on the |1i ↔ |3i transition. The third term F , we call the modulation term, provides
3
the contributions arising purely due to modulation. The contributions from the higher order terms, namely |n| ≥ 1,
are mainly channeled through the F term. Note that the F term provides the coupling of n = 0 response to the
3 3
n = ±1 term and this coupling is proportional to G MΩ, moreover, the coupling is independent of the probe Rabi
1
frequency G . Similar expressions can be written for each n and the coupling from higher order terms n±1 always
2
occursthroughG MΩandisindependentofG . Noapproximationsaremadeinwritingoutthe variouscomponents
1 2
in Eq. (6), all the individual factors contained therein, like ρ(0),ρ(0),ρ(0) and ρ(±1), are calculated exactly and thus
22 33 13 12
contain all the higher order contributions.
The major contribution from the first two terms (F and F ) for a weak probe occur at frequencies
1 2
∆ 1
∆ → 1 ± ∆2+4G2, (7)
2 2 2q 1 1
which is similar to the usual monochromatic pump case. The third term F provides the higher order contributions
3
which become significantin presence of strong modulationdue to the coupling proportionalto G MΩ, as seen in Eq.
1
(6). FortheparameterschosenforFig. 2,thecontributionfromtheF termdwarfstheF andF contributionsatall
3 1 2
the frequencies other thanthose givenin Eq. (7), see Fig. 2c. Thus, inthe strong modulationregimethe modulation
term dominates over the population inversion and the coherence terms of the usual monochromatic pump case.
In order to understand the two comb of resonances one can obtain the Floquet spectrum resulting from the FM
field coupling the atomic transition. Moreover, it has been demonstrated that in presence of strong modulation and
under certain conditions the Autler-Townes response exhibits simultaneous suppression of semi-infinite number of
resonances [16]. The conditions under which this occurs can also be obtained by analyzing the Floquet-spectrum.
Following the approach developed by Shirley [25] to obtain the Floquet states, we consider exactly the effect of the
strong FM field on the atomic transitionand neglect the weak effects of dissipation and the weak probe field because
G ,Ω≫γ ,γ ,Λ,G . In this limit it is advantageousto look at the dressedstates |Ψ i and |Ψ i (dressedby the FM
1 1 2 2 1 3
field) instead of the bare atomic states |1i and |3i. The Schr¨odinger equation of the evolution of the corresponding
dressed state amplitudes ψ and ψ is
1 3
d ψ ∆ −MΩcos(Ωt) −G ψ
i 1 = 1 1 1 . (8)
dt(cid:18)ψ3 (cid:19) (cid:18) −G1 0 (cid:19)(cid:18)ψ3 (cid:19)
As the Hamiltonian in Eq. (8) depends on time periodically, with a period 2π/Ω, one can obtain two linearly
independent solutions of Eq. (8) in the following form
∞
ψ±(t)=e−iλ±t χ±,ne−inΩt (i=1 and 3), (9)
i i
n=X−∞
where the index + and − distinguish the two solutions. On substituting expression (9) into Eq. (8) we obtain an
infinite set of recursion relations for χn, i=1 and 3
i
−nΩ+∆ −G χn MΩ 1 0 χn+1 χn−1 χn
(cid:18) −G1 1 −nΩ1 (cid:19)(cid:18)χ1n3 (cid:19)− 2 (cid:18)0 0(cid:19)(cid:20)(cid:18)χ1n3+1 (cid:19)+(cid:18)χ31n−1 (cid:19)(cid:21)=λ(cid:18)χ1n3 (cid:19), (10)
wherein χn is coupled to its nearestneighbors χn±1, for brevity we have dropped the ± superscript. These recursion
i i
relations can be written as an infinite matrix eigenvalue problem with the following infinite dimensionalHamiltonian
(H )
f
4
.. : : : : : ↓ : : : ..
.. −MΩ 0 −(n−1)Ω+∆ −G −MΩ 0 0 0 0 ..
 2 1 1 2 
.. 0 0 −G −(n−1)Ω 0 0 0 0 0 ..
1
 .. 0 0 −MΩ 0 −nΩ+∆ −G −MΩ 0 0 .. 
 2 1 1 2 . (11)
 → 0 0 0 0 −G −nΩ 0 0 0 .. 
 1 
 .. 0 0 0 0 −MΩ 0 −(n+1)Ω+∆ −G −MΩ .. 
 2 1 1 2 
 .. 0 0 0 0 0 0 −G −(n+1)Ω 0 .. 
 1 
 .. : : : : : : : : : .. 
 
Here, we have ordered the elements of the Floquet Hamiltonian H such that i runs over the states 1 and 3 before a
f
change in the harmonic index n whichtakes integer values from−∞ to ∞. The eigenvalues ofthe H matrix are the
f
quasienergies associatedwith various levels of the dressed states. The structure of the matrix (11) results in periodic
eigenvalues, and each dressed state has an infinite ladder of quasienergy levels separated by ±Ω. In other words the
combofresonancescorrespondto probing the two linearlyindependent set ofdressedstates + and−, whoseenergies
given by λ±+nΩ, where n takes integer values from −∞ to ∞. Moreover, two levels belonging to different dressed
states complement each other, i.e., λ+−λ− =∆1 modulo(Ω).
It has been shown in detail [16] that when
λ+mΩ=0 (12)
for some integer m the determinant of the characteristic equation of the above eigenvalue problem factorizes into
determinants of two semi-infinite blocks
det[Hf −Iλ]λ=−mΩ = −G21 det[Ha−Iλ] det[Hb−Iλ] = 0. (13)
The two semi-infinite blocks are such that the harmonic index n in H and H runs from −∞ to m−1 and m+1
a b
to ∞, respectively. The quasienergies are ambiguous within integer multiple of Ω because replacing λ by λ+kΩ,
where k is an integer, leaves the Eq. (13) unchanged. The above factorization results from the structure of the
Floquet Hamiltonian - if the integer m happens to be n in matrix (11) then the row and column marked by small
horizontal and vertical arrows, respectively, in the matrix Eq. (11) contains only one non-trivial element namely
−G , corresponding to the χm term. Now if χm = 0 then the half-infinite system of equations is closed resulting in
1 3 3
the above factorization, eq. (13). To determine bounded non-trivial solutions of the characteristic equation (13) one
requires that one of the following conditions is satisfied, either
det[H +I mΩ]=0 or det[H +I mΩ]=0. (14)
a b
The above conditions results in
χn =χn =0 for all n<m or n>m (15)
1 3
depending onwhich ofthe conditions in Eq. (14)is satisfied. The variousharmonics ofthe dressedstates χn and the
i
correspondingquasienergiesλ are determined numerically with the dimensions of the Floquet Hamiltonian matrix to
be 2(N +1) where N is a large positive integer and the harmonic index n in eq. (11) varies from −N to N.
We calculate the variation of the quasienergies as a function of the modulation index M for a fixed value of the
pump field coupling strength, its detuning and the modulation frequency. In Fig. 3a the corresponding parameters
are G =20γ , ∆ =20γ , and Ω=30γ . Henceforth, we drop the harmonic superscript from the density matrix
1 1 1 1 1 (0)
variables as we will deal with only the zeroth order atomic response at the frequencies ω and ω unless otherwise
1 2
specified. As seen in Fig. 3a, the quasienergy level plotted with a solid line crosses the zero energy periodically at
M = 2.915, 4.06, 6.26, 7.35 and so on. At these values of M the condition (12) is satisfied. The probe spectrum
exhibits only one set of absorption peaks separated by Ω, instead of the usual set of double peaks, as Eq. (15) is
satisfied. Fig. 3b highlights this feature for M =2.915 which is the first zero crossingof the quasienergy level in Fig.
3a. Thereisonlysinglepeakedstructurefor∆ >0,whereasthespectrafor∆ <0continuestoexhibitthetwosetof
2 2
peaks. InFig. 3btheproberesponseisshownforvariousvaluesoftheincoherentpumpΛ/γ =0,0.2,0.4,0.6,0.8and
1
1.0. In the absence of the incoherentpump the probe absorptionspectra for ∆ <0 contains two comb of absorption
2
dips, whereas for ∆ > 0 it contains a single comb of absorption dips separated by ∼ Ω. As the incoherent pump
2
rate is increased one set of absorption dips transform into gain peaks. The gain is maximum at about Λ/γ = 0.5.
1
We have observed that for parameters where the semi-infinite resonances are suppressed, the surviving single comb
of resonances almost always exhibit gain in presence of the incoherent pump.
Astheprobespectrumshowswellseparatedgainpeaks(for∆ >0)the intermediateregionaccordsthe possibility
2
of anomalous dispersion. In these frequency regimes the probe beam can experience superluminal velocity due to
5
anomalous dispersion [23]. Regions of anomalous dispersion (d Re(ρ )/dω < 0) accompanied by a flat region of
32 2
nearly zero absorption (Im(ρ ) ≈ 0) is particularly desirable in achieving superluminal propagation. In Fig. 4, the
32
regionbetweenasetofgainpeaksisexpandedandonecanseetheanomalousdispersionaccompaniedbyaflatregion
ofnegligiblegain/absorption. ThesuperluminalpropagationinRef.[23]isachievedinbetweenRamangainlines. The
anomalous region between the gain lines, in their system, is quite limited in terms of the available bandwidth for the
probe pulse. This isbecause the two fields responsibleforthe Ramangaincannotbe too far detunedfromresonance.
The trade off between achieving sufficiently large gainso as to obtain sharp change in dispersion in between the gain
lines, and having the gainlines separatedfar enough so as to obtain minimal gainat the line center severely restricts
the bandwidth in their system. Moreover, Doppler broadening further reduces the desirable anomalous region. In
mostother schemestoo one lacksthe controloverthe separationbetweencloselyspaceddoubletexhibiting inversion,
as the doublets arise from either hyperfine splitting or isotope shift [26]. This limitation is overcome in our system
because the bandwidth depends on the frequency of modulation andone can achieve a few hundred GHz modulation
frequency at optical frequencies. Another advantage of this system is availability of multiple periodically separated
gain peaks so that one could choose to operate the probe far off resonance and thus the possibility of decreasing
the noise arising from spontaneous emission [27]. Hence, our system in principle accordsmore flexibility in obtaining
anomalousdispersionregionofdesiredfrequencybandwidthinadesiredfrequencyregime. Thepossibilityoftailoring
dispersion/absorption-gain characteristics by controlling different parameters of the modulation and the incoherent
pumptoovercomerestrictionsduetoDopplerbroadeningandpowerbroadeningareaddedadvantagesofthissystem.
We now discuss in detail the resonant case where the central frequency of the FM field is on resonance with the
|1i↔|3itransitionandcontrastourresultswiththetraditionalmonochromatic(M =0)case. Inthemonochromatic
pumpcase,gainarisespurelyfromthecoherencebetweentheexcitedstates|1iand|2iwhicharenotdipoleconnected,
and, has been shown to occur without inversionin any state basis [21]. We calculate the steady-state response of the
system in presence of modulation and Fig. 5a depicts the gain obtained as a function of the index of modulation M.
At M = 0 one obtains the gain typical of the monochromatic case, whereas for strong modulation (M away from
zero)one finds much largergain. The Im(ρ ) at M =10.17is 3.75×10−5 in comparisonto 3.05×10−6 atM =0, a
32
∼103% increment overthe monochromatic case. It should be borne in mind that the dynamics in presence of strong
modulationis suchthatsignificantcontributionsoccur fromthe numerousside-bandresonancesleadingto suchlarge
gain.
The absorption peaks (sharp dips in Fig. 5a) occur at those values of M for which J (M)= 0, note that we have
o
chosenΩ>>γ . Thiseffect is expectedonphysicalgroundsas onecanseefromthe spectralcontentofthe FMfield,
1
namely
+∞
eiMsin(Ωt) = J (M)eipΩt, (16)
p
p=X−∞
where J (M) is the Bessel function of integer order p. Whenever the resonant central frequency is absent on the
p
|1i↔|3itransitionthe populationin level|3idominantly experiencesabsorptiononthe|3i↔|2itransition,andthe
largevalueofΩalsoensuresthatspectralcomponentsat±Ωwiththe weightofJ±1(M)arefarremovedinfrequency
from the resonantline center. In between these peaks of absorptionone obtains gain. The fact that one obtains gain
only when J (M)6=0 points toward the spontaneous emission assisted nonlinear optical process responsible for gain
0
which will be discussed in detail at the end of this section.
In terms of the density matrix the major contribution to the gain is from the modulation term F of Eq. (6) as
3
seen in Fig. 5b. For M = 0 the contribution from the inversion term F is negative while the positive coherence
1
term F completely offsets this and is responsible for gain. The F term is zero at M =0 and starts increasing as M
2 3
getslarger. For strongmodulationtheF termlargelyovershadowsallothercontributionsandis responsibleforgain
3
in between the zeros of J (M). The smaller dips in the F component, that occur in between the the zeros of the
0 3
J (M), result from the next higher contribution arising from ω ±Ω. These set of dips that occur when F is large
0 1 3
and positive are due to the zeros of the the Bessel J1(M). The weights of the contribution at ω1±Ω are J±1(M),
these contributions are absent when M takes values such that J (M) = 0. It should be noted that even though the
1
contribution from ±Ω disappears at these values of M, the contribution from higher multiples of Ω continues to be
significantresultinginlargeF termandthe resultinggain. The steadystate populationdistributionisshowninFig.
3
5c. AtM =0,ρ <ρ <ρ andthis resultsingainwithoutinversioninthebarestatebasisaswellasnoinversion
22 11 33
for the Raman process. For M =3.7 where the gain is 1.6×10−5 about ∼400%more than the monochromatic case,
the relative population distribution is the same, i.e., ρ < ρ < ρ ; there is no inversion in the bare state basis
22 11 33
as well as no inversion for the stimulated Raman scattering (SRS) process. The two photon SRS inversion condition
requiresthatρ >ρ . ForverylargevaluesofM, notshownin figure,eventhoughthere is no inversionin the bare
22 11
state basis (ρ <ρ ) there is Raman inversionfor the SRS process namely the |2i→|3i→|1i process, because ρ
22 33 22
6
becomes larger than ρ . Note that in all our discussions the feature of the gain being inversionless is between the
11
bare atomic states |1i, |2i and |3i.
B. Transient dynamics
We discuss the time evolution of atomic polarization and population distribution in the FM field pumped V-type
systemandhowitapproachesthesteadystatevalues. DuetothepresenceofastrongFMfieldthetransientresponse
ofthe systemisverycomplexmakingit difficultto identify the physicalmechanismresponsibleforgain. We consider
in detail one case in which one can identify the dominant process and is comparable to the monochromatic pump
case. We will also present a general case wherein the dynamics is more complex.
We choose the on-resonance case where ∆ =∆ =0, the modulation frequency is large (Ω=80γ ) and the index
1 2 1
ofmodulationis chosensuchthatit is a zeroofthe firstorderBesselfunction, i.e., J (M)=0 for M =10.1734. This
1
particularchoiceissuchthatthe FMfieldspectrumhasthe centralpeakatω withweightJ (M), nopeakatω ±Ω
1 0 1
as J (M) = 0, the next peak in frequency is far removed and is at ω ±2Ω with weight J (M), and other peaks at
1 1 2
ω ±nΩ with n > 2 and weight J (M) are further removed from resonance. Choice of large Ω and the appropriate
1 n
M ensures that the effects of ω ±nΩ for n 6= 0 are minimal and the population and atomic polarization dynamics
1
appear to be somewhat similar to the monochromatic pump case.
In Fig. 6a we show the evolution of the population in presence of the incoherent pump with the initial condition
ρ = 1 and all other ρ = 0 (i, j=1-3). It is seen that at all times ρ < ρ and ρ < ρ , therefore, there
33 ij 22 33 22 11
is no population inversion in the bare state basis and there is no Raman inversion for gain through SRS process,
respectively. Fig. 6b shows the evolution of population in absence of the incoherent pump. The population in level
|2i is negligible because the probability of the population to be excited to level |2i is very small as G << G and
2 1
γ . As expected the population oscillates between level |1i and |3i and in steady state ρ ≈ρ ≈0.5, though ρ is
i 11 33 33
slightly largerthan ρ due to the presence of the FM field and the decays. It is also observedthat in presence of the
11
incoherentpump the population and the atomic coherences reachtheir steady state values earlierin time. Fig. 6c(d)
shows the evolution of gain-absorption for the probe field (FM pump field at ω ) with and without the incoherent
1
pump. We note the following: they both oscillatedominantly with the same frequency and nearly in phase with each
other. Both the probe and the pump field experience gain in the transient regime with or without the incoherent
pump. In presence of the incoherent pump only the probe field experiences gain in the steady state, whereas with
Λ=0 both the fields experience absorption in the steady state.
The probe field experiences gain just after ρ reaches a minimum value favoring the stimulation process from
33
|2i → |3i rather than the |3i → |2i transition. The probe field experiences gain in the interval when dρ /dt > 0
33
and reaches its maximum value at a time when dρ /dt is the steepest. The FM pump field at the ω frequency
33 1
experiences gain just after ρ reaches its maximum value. The contribution to the envelop of the pump gain does
11
not seem to come from SRS process even though the two-photon inversion with ρ > ρ is present. The signature
11 22
for an SRS process would be that the absorption-gain response on the |1i ↔ |3i and |2i ↔ |3i transitions be out of
phase. Inotherwords,emissionononetransitionaccompaniedbyabsorptionontheothertransition. Thisishowever
not present in this case as the envelops oscillate in phase with each other as seen in Figs. 6c and d. We re-emphasize
thatthegaininthe aboveparameterregimeis qualitativelyakintothe monochromaticcaseapartfromthe rapidbut
weak (small amplitude) oscillations due to the FM field on the central pump frequency ω in Fig. 6d.
1
We present in Fig. 7 the temporal evolution of the atomic polarization and the population dynamics when the
central frequency of the FM field is detuned from the atomic resonance. The parameters chosen are identical to that
inFig. 2whichcorrespondstothe steadystatebehavior. Thepopulationinitially isinlevel|3i. Theprobefrequency
is tuned to the gainpeak at ∆ =46.8γ . We observethat the population in levels |1i and |3i oscillate ∼180o out of
2 1
phase with respect to each other. The population in |2i is negligible in absence of the incoherent pump and steadily
increases in presence of the incoherent pump, see Figs. 7a-b. At time τγ ≈ 3 the population ρ becomes larger
1 22
than the population ρ , however, ρ remains smaller than ρ at all times. Again there is no population inversion
11 22 33
betweenthe levels |2iand|3i. After time τγ ≈3,unlike the resonantcaseshownin Fig. 6, the populationρ >ρ
1 22 11
implying inversion for the SRS process for probe gain.
Both the probe and the pump fields experience gain in the transient region with or without the incoherent pump.
It is clear that gain on the central FM frequency ω results just after ρ reaches its maximum value. Apart from
1 11
the high frequency oscillations the envelop of the polarization on the central FM frequency ω closely follows the ρ
1 11
oscillations. These high frequency oscillations riding on the envelop result from higher order contributions of field at
ω ±nΩ for n>0. There is no gain in the steady state at the ω frequency, whereas the probe field experiences gain
1 1
in the steady state.
The frequency of oscillation of the probe field depends on its generalized Rabi frequency and is found to be
7
independent of the incoherent decays and the incoherent pump. A closer look at the probe oscillations in Fig. 7c
shows that periodically there are largerpeaks of gain whichinvariably follow just after ρ reachesa minimum value,
33
pointing to similar process discussed above. Moreover, due to the complexity of the time evolution one cannot rule
out SRS process as some peaks of probe gain do match with pump absorption dips and vice-versa.
C. Physical mechanism for gain
Although the quantitative results obtained above from solving the density matrix equations provide complete
description of the system under consideration, it still does not provide insight into the underlying nonlinear optical
processes. Deeper understanding can be obtained by systematically summing up individual scattering processes
and identifying the dominant contributions to the gain. We have observed that in the steady state the probe gain
occurs only if γ > γ [28], implying that spontaneous emission plays a crucial role in obtaining inversionless gain
1 2
[29]. In recenttimes irreversiblespontaneousemissionassistednonlinear opticalprocesseshavebeen suggestedwhich
contribute to inversionlessgain(in ladder-typesystems by Sellin et al [30]andinΛ-type systems by Zhu[31]). Along
theselines thephysicalpictureforinversionlessgaininthesteadystateforthe V-typesystemwouldbe thefollowing.
First a stimulated emission process from |2i → |3i of the probe photon ω , followed by absorption from |3i → |1i
2
and finally spontaneous emission from |1i → |3i at the rate of 2γ , see Fig. 8a. Note that the probe experiences
1
significantstimulatedemissiononlyatcertainfrequenciesduetotheFMpumpfield,namelyω ±nΩ. Theseresonant
2
features are absent in the conventional LWI schemes. It should also be noted that in the above process the second
step involving absorption from |3i → |1i depends critically on the availability of the ω photon (for ∆ = 0). In its
1 1
absence,whenthe index ofmodulationM is suchthatJ (M)=0,this nonlinearprocessissuppressedandthe probe
0
experiences only absorption, see Fig. 5a. The competing process (Fig. 8b), of stimulated emission from |1i → |3i
followed by absorption from |3i→|2i and spontaneous emission from |2i→|3i at 2γ rate, is less preferred because
2
of γ >γ . Thus the former process plays a dominant role in inversionless probe gain.
1 2
We have described above the nonlinear optical process responsible for gain when the central FM frequency ω is
1
resonant with the atomic transition. We have also looked at detuned cases and similar nonlinear process seems to
form the basis for obtaining gain. We describe briefly one such detuned case, wherein ∆ is finite and Ω is chosen to
1
be equal to ∆ . It is clear that, in this case,the ω +Ω photon would be requiredin the secondstep of the nonlinear
1 1
process (Fig. 8a) instead of the ω photon. This is further supported by looking at the polarization Im(ρ ) as a
1 32
function of M, and one observes sharp absorption at those values of M for which J (M)=0 (not shown here). This
1
clearly shows that in absence of the ω +Ω photon there would be no spontaneous emission assisted gain process.
1
Hence, the availability of the intermediate photon which would take the population from level |3i to level |1i and the
probe resonant with the periodic sideband response created by the FM field plays a crucial role in obtaining large
gain.
It should be noted that the nonlinear optical process discussed above is assisted by the spontaneous emission and
hence does not necessitate any kind of population inversion. The FM field plays a central role in creating the rich
periodic Floquet structure whose quasienergy levels are probed by the probe field. The on-resonance pump photon
acts as a hub about which the above nonlinear process occurs resulting in gain. We would also like to point out
that if an amplitude modulated field (with modulation frequency α) is used instead of the FM field, then, due to the
availabilityofonly apair ofsidebandphotons(ω ±α)the gainfromthe abovenonlinearprocesswillbe presentonly
1
atapairoffrequencies. Itis the availabilityofω ±nΩphotonsinthe FMfieldthatresultsingainatalargenumber
1
of sideband frequencies far removed from the central frequency.
IV. CONCLUSION
In summary, we have presented an analysis of the occurrence of gain in a V-type system in presence of FM pump
field. We have analyzed both the steady state and the transient dynamics of the light amplification process. In the
steady state analysis we have tried to bring out the significant role played by the modulation term over and above
the coherence term which is dominant in the conventional monochromatic pump schemes. We obtained the Floquet
quasienergies that provide the explanation for the two comb of frequencies in the probe spectrum; furthermore the
quasienergyspectrum wasusedto chooseappropriateindex ofmodulationM suchthatthe resultingprobe spectrum
contains a semi-infinite set of consecutive gain peaks far removed from resonance and could be exploited to obtain
short-wavelengthlasing. Inbetweenthe gainpeaksanomalousdispersionis obtainedaccompaniedbynegligiblegain-
absorption,which could be utilized to obtain distortionfree superluminal pulse propagationin any desired frequency
regime and over a large bandwidth which can be increased by increasing the modulation frequency Ω. The gain
8
obtained in experiments based on LWI schemes is severely limited in its magnitude. The advantage of gain obtained
using the FM field is the significant increment in its magnitude ∼ 103% due to numerous sideband contributions,
and no population inversionbetween the atomic states. It should be noted that the occurrence of gainat frequencies
far removed from resonance could also make this system attractive in terms of the possibility of reduced noise due
to quantum fluctuations, further investigations along these lines would be fruitful. The study of transient dynamics
showed that both the probe and the pump fields experience transient gain. The probe gain is not due to population
inversionintheatomicstatebasis,moreover,incertainregimesonecouldevenruleoutgainviaSRSprocessandtwo-
photon inversion. We have described the physical process, based on spontaneous emission assisted nonlinear optical
process, which is responsible for inversionless gain. This effect is further corroboratedby the spectrum obtained as a
function of the index of modulation in Fig. 5a.
9
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10