SShape preserving atomic pulse amplifier
Nilamoni Daloi ∗ and Tarak Nath Dey † Department of Physics, Indian Institute of Technology Guwahati, Guwahati 781039, Assam, India (Dated: February 3, 2021)Propagation of a weak probe pulse through a Λ system in a resonant gain configuration is inves-tigated. We employ the control field intensity that permits the amplification of probe pulse duringpropagation, without instability at two photon resonance. Posterior to amplification, a broadenedprobe pulse is obtained, which retains its initial pulse shape and travels at the speed of light invacuum, without experiencing any delay, absorption, and dispersion. The salient feature of thistechnique lies in the fact that in addition to preserving the initial pulse shape, it also ensures stablepulse propagation after amplification. It also works for arbitrary probe pulse shapes.
I. INTRODUCTION
Light pulse generation, reshaping and shape preserv-ing propagation has received considerable attention inmultilevel atomic systems due to its potential applica-tions in optical communication and information sciences[1–3]. Population inversion [4] and atomic coherencesare two main constituents for the pulse generationand reshaping [5]. Coherently controlled light-matterinteraction can produce desired atomic coherences andpopulation among the various level systems. A varietyof techniques based on stimulated Raman adiabaticprocesses (STIRAP)[6], electromagnetically inducedtransparency (EIT)[7], coherent population trapping(CPT)[8], and saturated absorption [9] has been adoptedto control the dynamics of population and coherenceseffectively. The induced coherence among the lower levelstates of a molecular system [10] or atomic system [11]are the central issue of producing pulse radiations. Vari-ous techniques have been proposed for the generation ofoptical pulses [12, 13].Self induced transparency (SIT) is a prominentexample for shape preserving ultrashort optical pulsespropagating through a resonant medium [14, 15]. Thephysical explanation of SIT comes from re-radiation ofatoms from the excited state of the two level system inthe presence of pulse power which is beyond some criticalvalue [15]. This lossless shape preserving propagationcan be extended to two pulses in an otherwise opaquethree level medium which are referred to as simultons[16]. Remarkably, the two pulses may copropagate ascomplementary pulse shapes in a three level Λ-systemwhich is dictated by the input envelope shapes [17]. Anextensive study on co-existence of stable propagation ofsech and tanh pulse envelopes have been investigated inthree- and multilevel systems [18–22]. The three-levelΛ configuration also offers the temporal cloning of anarbitrary shaped strong pulse envelope on to a weakfield [23]. Recently the predictions of simultons [16] ∗ [email protected] † [email protected] have been experimental verified in a V-type thermalatomic system referred to as a quasisimulton [24]. Inthese studies, radiative and non-radiative decay is nottaken into consideration due to the short duration ofthe pulse. However the influence of various atomicrelaxation processes plays a crucial role on spectralbandwidth of the generated pulse and the propagationvelocity [25]. The weak probe pulse along with strongcontrol field propagate as matched pulses through anabsorptive medium [26]. All of these studies is mostlyrequires optical pulses to be of some very specific inputforms or definite energy in order to demonstrate shapepreserving propagation throughout the length of themedium [27].Media which exhibit huge absorption, limits anyrealistic application based on pulse generation andshape preserving propagation. The absorptive mediumaccompanied by its inherent pulse broadening, distortionas well as suppression of output transmission, practicalstops its usage [28]. Hence coherent manipulation of ab-sorption of the medium is essential for arbitrary shapedoptical-pulse propagation with controllable width andgain [29, 30]. Much attention has been paid to make anopaque medium transparent for supporting propagationof pulses without changing their initial profiles [31–33].Nonetheless, most of the studies failed to support thelong distance propagation of arbitrary shaped opticalpulse. In this work we investigate the propagation of anarbitrary shaped probe pulse through the three level Λsystem in presence of a strong continuous control field.Both the probe and control field experience absorptionduring initial length of propagation. However, theprobe pulse is progressively amplified and retains itsinitial shape. The population transfer from groundstate to excited state due to control field is the mainreason behind the continuous energy transfer to theprobe field. With suitable choice of parameters, theprobe pulse retains its initial shape and propagateswithout any delay, distortion and absorption, after theamplification process. Conventionally the gain systemmanifests an instability due to the interplay betweennon-linearity and anomalous dispersion [34, 35], never-theless the proposed system is immune from instability. a r X i v : . [ phy s i c s . op ti c s ] F e b The transmitted probe pulse display broadening is incontrast to the gain associated with pulse narrowing [30].The paper is organized as follows. Sec. II containsthe theoretical formulation of the relevant level system.Sec. III contains the results of numerical simulationsalong with detailed explanations. Sec. IV contains thesummary and conclusion.
II. THEORETICAL FORMULATIONA. Level system
FIG. 1: (Color online) Schematic diagram of a three level Λ-system. Here, | (cid:105) is the excited state, | (cid:105) is an intermediatemeta stable state and | (cid:105) the ground state with energy setto zero. The atomic transitions | (cid:105) ↔ | (cid:105) and | (cid:105) ↔ | (cid:105) arecoupled by a weak probe field (cid:126)E p , with frequency ω p , and astrong control field (cid:126)E c , with frequency ω c , respectively. Thespontaneous emission decay rate of | (cid:105) to | j (cid:105) ( j ∈ ,
3) transi-tion is denoted by γ j . The detunings, and Rabi frequenciesof the fields are denoted by ∆ i , and Ω i respectively ( i ∈ p, c represents probe and control field, respectively). A Λ-system as shown in Fig. 1 can meet the desired cri-teria for achieving gain by considering that all populationis kept at ground state | (cid:105) initially. Unlike the absorptionbased EIT configuration, in Fig. 1, the ground state | (cid:105) and excited state | (cid:105) are coupled by a strong control field (cid:126)E c and the metastable state | (cid:105) and | (cid:105) are coupled by aweak probe field (cid:126)E p . This configuration leads to popula-tion transfer from | (cid:105) to | (cid:105) which spontaneously decaysto | (cid:105) . The spontaneous decay from | (cid:105) to | (cid:105) makes pro-vision for the probe field being resonantly enhanced bystimulated emission in presence of the control field, caus-ing probe amplification. The probe and control fields areconsidered to be propagating along z direction and aredefined as: (cid:126)E j ( r, t ) = ˆ e j E j ( r, t ) e − i ( ω j t − (cid:126)k j .(cid:126)r ) + c.c. , (1)where ˆ e j are the unit polarization vectors, E j ( r, t ) are theslowly varying envelope functions, ω j are the field carrier frequencies and (cid:126)k j = k j ˆ z are the wave vectors of thefields. The index j ∈ p, c represents probe and controlfields, respectively.The time dependent Hamiltonian of the system, underthe electric dipole approximation is given as: H = H + H I , (2a) H = ¯ h ( ω | (cid:105) (cid:104) | + ω | (cid:105) (cid:104) | ) , (2b) H I = | (cid:105) (cid:104) | (cid:126) d . ˆ e p E p e − i ( ω p t − k p z ) , + | (cid:105) (cid:104) | (cid:126) d . ˆ e c E c e − i ( ω c t − k c z ) + h.c. , (2c)where ω j ( j ∈ ,
2) denotes the resonance frequency of | j (cid:105) ↔ | (cid:105) transition and (cid:126) d j = (cid:104) j | ˆ d | (cid:105) are matrix el-ements of the dipole moment operator ˆ d , representingthe induced dipole moments, corresponding to | j (cid:105) ↔ | (cid:105) transition. To write the Hamiltonian in a time indepen-dent form, the following unitary transformation is used: U = e − i V t , (3a) V = ω c | (cid:105) (cid:104) | + ( ω c − ω p ) | (cid:105) (cid:104) | . (3b)The effective Hamiltonian obeying the Schr¨odinger equa-tion in the transformed basis is given as: H = U † . H . U − i ¯ h U † . ∂ U ∂t , (4)which under RWA (Rotating wave approximation) gives H ¯ h = − ∆ c | (cid:105) (cid:104) | − (∆ c − ∆ p ) | (cid:105) (cid:104) |− Ω p | (cid:105) (cid:104) | − Ω c | (cid:105) (cid:104) | + h.c. . (5)The single photon detunings for the probe and controlfields are defined as:∆ p = ω p − ω , ∆ c = ω c − ω , (6)and the Rabi frequencies of probe and control fields arewritten as:Ω p = (cid:126) d . ˆ e p E p ¯ h e ik p z , Ω c = (cid:126) d . ˆ e c E c ¯ h e ik c z . (7)The dynamics of atomic state populations and coher-ences is governed by the following Liouville equation: ∂ρ∂t = − i ¯ h [ H , ρ ] + L ρ . (8)The Liouville operator L ρ , describes all incoherent pro-cesses and can be expressed as: L ρ = − (cid:88) i =1 3 (cid:88) j =1 ,j (cid:54) = i γ ij | i (cid:105) (cid:104) i | ρ − | j (cid:105) (cid:104) j | ρ ii + ρ | i (cid:105) (cid:104) i | ) , (9)where γ ij represent the radiative decay rates from excitedstates | i (cid:105) to ground states | j (cid:105) .The equations of motion for atomic state populationsand coherences of the three level Λ-system are then givenas: ˙ ρ = − Γ ρ + i (Ω p ρ − ρ Ω ∗ p ) + i (Ω c ρ − Ω ∗ c ρ ) , (10a)˙ ρ = − (cid:18) Γ2 − i ∆ p (cid:19) ρ + i Ω c ρ + i Ω p ( ρ − ρ ) , (10b)˙ ρ = − (cid:18) Γ2 − i ∆ c (cid:19) ρ + i Ω c ( ρ − ρ ) + i Ω p ρ , (10c)˙ ρ = γ ρ + i (Ω ∗ p ρ − ρ Ω p ) , (10d)˙ ρ = − (cid:20) γ − i (∆ c − ∆ p ) (cid:21) ρ + i (Ω ∗ p ρ − Ω c ρ ) , (10e)˙ ρ = γ ρ + i ( ρ Ω ∗ c − Ω c ρ ) , (10f) ρ ∗ ij = ρ ji , (10g) with initial conditions ρ ( z,
0) = ρ ( z,
0) = 0 , and ρ ( z,
0) = 1 . Assuming that the system is closed, the total populationremains conserved; i.e. , ρ + ρ + ρ = 1. In Eq. (10),the overdots stand for time derivatives and “ ∗ ” denotescomplex conjugate. The decoherence rate of ρ is de-noted by γ and the total decay rate of excited state | (cid:105) is written as Γ = γ + γ . The decay rate of excitedstate | (cid:105) , to states | (cid:105) and | (cid:105) are assumed to be equal;i.e., γ = γ = γ . B. Propagation equations
In order to explore the effects of populations and co-herences on the propagation dynamics of probe pulsethrough the gain medium, the study of Maxwells equa-tion is inevitable. Under the slowly varying envelope ap-proximation, the propagation equations for the probe andcontrol fields can be expressed as (cid:18) ∂∂z + 1 c ∂∂t (cid:19) Ω p ( z, t ) = η p ρ ( z, t ) , (11a) (cid:18) ∂∂z + 1 c ∂∂t (cid:19) Ω c ( z, t ) = η c ρ ( z, t ) , (11b)where η i ( i ∈ p, c ) are called the coupling constants ofthe respective fields. For simplicity, the resonance fre-quencies of | (cid:105) → | j (cid:105) ( j ∈ ,
3) transitions are assumedto be equal; i.e. , ω = ω . This gives η p = η c = η with η = 3 N λ γ/ π , where N is the number of atoms per unit volume inside the medium and λ is the wave-length of the fields. To facilitate numerical integration ofEq. (11), a frame moving at the speed of light in vacuum c , is used. The necessary coordinate transformations forthat are τ = t − z/c , and ζ = z . This allows for theround bracketed terms of Eq. (11) to be replaced bypartial derivatives with respect to the single independentvariable ζ . III. NUMERICAL RESULTS
In this section, the numerical results acquired fromcoupled Maxwell-Bloch equations are presented. For nu-merical simulation, we first consider spatiotemporal evo-lution of a Gaussian probe pulse in the presence of acontinuous control field. At the entrance of the medium,the Gaussian probe pulse and the continuous control fieldare defined as:Ω p (0 , τ ) = Ω p exp (cid:20) − ( τ − τ ) σ (cid:21) , (12a)Ω c (0 , τ ) = Ω c , (12b)where, Ω p , Ω c are the amplitudes of probe and controlfields respectively and σ , τ are the initial pulse widthand delay of the probe pulse, respectively. The results arecurated in Fig. 2, where the temporal profiles of probeand control field magnitudes are plotted at different prop-agation lengths. Time, and propagation length are madedimensionless as T = γτ × − and Z = ηζ/γ × − respectively throughout the paper.Both the probe pulse and the continuous control fieldundergo observable reshaping during propagation. InFig. 2(a), from 0 ≤ Z ≤ .
4, the probe pulse experiencesgroup delay, absorption and indiscernible broadeningwith increasing propagation length. From 2 . ≤ Z ≤ . Z ≥ .
2, a broadened and amplified Gaus-sian probe pulse is obtained, which travels at the speedof light in vacuum, without any delay, absorption anddispersion.In Fig. 2(b), the control field undergoes absorption atits leading end as it propagates though the medium, giv-ing it an appearance of experiencing delay at the leadingend. . . . . . T . . . . . . . . . . Z | Ω p / Ω p | . . . . T . . . . . . . . . . Z | Ω c / Ω c | (a)(b) FIG. 2: (Color online) (a) Temporal profile of probe pulse atdifferent propagation lengths Z . (b) Temporal profile of con-trol field corresponding to the propagation lengths mentionedin Fig. 2(a). Parameters used are: Ω p = 0 . γ , Ω c = 4 γ , τ = 200 /γ , σ = 15 /γ and ∆ p = ∆ c = 0, γ = 0 . γ .Field magnitudes are normalized by their respective field mag-nitudes at Z = 0. Axes are not made to scale for visibility. A detailed explanation of the results in Fig. 2 areprovided in the following sections.
1. Absorption of control
In Fig. 3, at the position of control field’s leading endon time axis, there occurs a population transfer from | (cid:105) to | (cid:105) , with an intermediary transient population transferto | (cid:105) . This population jump from | (cid:105) to | (cid:105) as shown inFig. 3 (2 nd row), requires a certain amount of controlfield energy. Thus, for every infinitesimal increment in Z , energy corresponding to a tiny portion of the controlfield’s total area gets absorbed in the medium. Therefore,at any given Z , a chunk of control field area shows upmissing from the leading end as shown in Fig. 3 (1 st row). | Ω c / Ω c | t (a) Z = 2 . t (b) Z = 2 . t (c) Z = 3 . ρ
11 00.51 ρ ρ . . T . . . . FIG. 3: (Color online) Temporal profile of control field andpopulation distribution are plotted column wise at propaga-tion lengths Z = 2 .
3, 2 .
9, and 3 . t (cid:48) represents the position of control field’sleading end on time axis. Control field magnitude normaliza-tion and parameters used are same as Fig. 2. Figures pertain-ing to a particular row and column have a common verticaland horizontal axis respectively. The control field undergoes linear absorption with in-creasing propagation length Z , as shown in Fig. 4, wherethe graph between the position of control field’s leadingend ( t (cid:48) ) on time axis vs Z gives a straight line.00.20.4 0 1 2 3 4 t Z FIG. 4: Position of control field’s leading end t (cid:48) , on time axisvs propagation length Z . The slope of the graph β ≈ . Z . Parametersused are same as Fig. 2. The formula for the slope of the graph in Fig. 4 isderived as follows. The energy density of control fieldis given as |E c | / π . Therefore, if A is the area of crosssection of the control laser beam, then the energy passingthrough the transverse plane at any given z , for a timeinterval t (cid:48) /γ is (cid:90) t (cid:48) /γ cA |E c | π dτ, (13)which is also the amount of control field energy that getsabsorbed after propagating for a distance z , inside themedium. Again, due to control field absorption, approx-imately half of the population momentarily jumps fromground state | (cid:105) to excited state | (cid:105) as marked in Fig.3 (2 nd row). Therefore, when each atom in a homoge-neous medium with atomic number density N , absorbsone control field photon of energy ¯ hω , then the totalenergy absorbed by the medium is E ≈ ¯ hω N Az. (14)Equating Eqs. (13) and (14) gives t (cid:48) ≈ | Ω c /γ | Z . (15)Therefore, the theoretical slope of t (cid:48) vs Z plot is β t =2 / | Ω c /γ | . The slopes obtained numerically ( β n ) for dif-ferent control magnitudes are in good agreement withthe theoretical ones as shown in Table I. Where, the ra-tio β t /β n ≈ TABLE I: Table showing comparison between β n and β t | Ω c /γ | β n β t β t /β n . .
023 0.219 0 .
22 1 .
004 0.123 0 .
125 1 .
025 0.077 0 .
08 1 .
046 0.053 0 .
056 1 .
067 0.039 0 .
041 1 .
2. Probe delay, dispersion and absorption
The probe pulse travels incontrovertibly under the in-fluence of control field from 0 ≤ Z ≤ . st row). The presence of control field creates a pop-ulation distribution of ρ = ρ = 0 , ρ = 1 as shownin Fig. 5 (2 nd , 3 rd , 4 th row), which serves as an absorb-ing, dispersive medium for the probe pulse. Hence, theprobe undergoes noticeable absorption and delay within0 ≤ Z ≤ .
4. The probe absorption coefficient α (cid:48) , can becalculated from the imaginary part of the probe suscep-tibility χ , of the medium. The probe susceptibility χ , is written in terms of the atomic coherence as: χ = η (cid:48) Ω p ρ (cid:18) η (cid:48) = 3 N λ γ π (cid:19) , (16a) ρ = − Ω p (cid:2) | Ω p | + γ ( γ − i ∆ p ) (cid:3) ( γ − i ∆ p ) [ γ (∆ p + iγ ) + i | Ω p | ] + iγ | Ω c | , (16b)where ρ , in Eq. (16b), comes from the steady statesolutions of Eq. (10) with ∆ c = 0, under the weak probe(Ω p (cid:28) Ω c ) approximation. The steady state boundarycondition ρ = ρ = 0, and ρ = 1 is used for thederivation of Eq. (16b). The probe absorption coeffi-cient is defined as α (cid:48) = 2 πω Im[ χ ] /c , which is made di-mensionless as α ( d ) = γα (cid:48) /η . At two photon resonance(∆ p = ∆ c = 0), α ( d ) = 1 − | Ω c | γγ + | Ω c | + | Ω p | . (17)For the parameters in Fig. 2, α ( d ) ≈ . × − , which isclose to the value obtained numerically, 6 . × − . . . . . . . | Ω i / Ω i | Probe (a) Z = 0 Control (b) Z = 1 . Z = 2 . ρ ρ ρ T FIG. 5: (Color online) (1 st row) Probe ( i = p , crisscross) andcontrol ( i = c , solid) field magnitude vs time T , are plottedcolumn wise for Z = 0, 1 .
7, and 2 . ρ = ρ = 0,and ρ = 1. Normalization of field magnitudes and parame-ters used are same as Fig. 2. Figures connected to a particularrow and column have a common vertical and horizontal axesrespectively. It is well known that, after propagating for a distance z inside an absorbing, dispersive medium, a Gaussian pulseof the form: Ω p (0 , t ) = Ω p exp (cid:20) − t σ (cid:21) , (18)gets modified as:Ω p ( z, t ) = Ω p (cid:115) σ σ − ik (cid:48)(cid:48) z exp (cid:20) − ( t − k (cid:48) z ) σ − ik (cid:48)(cid:48) z ) − α (cid:48) z (cid:21) , (19)where k (cid:48) = ∂k ( ω ) ∂ω (cid:12)(cid:12) ω and k (cid:48)(cid:48) = ∂ k ( ω ) ∂ω (cid:12)(cid:12) ω with k ( ω ) and ω being the propagation vector and carrier frequencyof the field respectively. Equation (19) shows that theGaussian pulse gets delayed by a time τ d = k (cid:48) z and thepulse width gets modified as σ ( z ) = σ − ik (cid:48)(cid:48) z , aftertraveling a distance z inside any dispersive medium. Theparameters k (cid:48) and k (cid:48)(cid:48) are written in terms of the probesusceptibility χ , as: k (cid:48) = (cid:32) πω ∂χ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω (cid:33) /c, k (cid:48)(cid:48) = 2 πω c ∂ χ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω , (20)which are made dimensionless as κ = γ k (cid:48) /η and κ = γ k (cid:48)(cid:48) /η respectively. Substituting Eqs. [(16a), (16b)] inEq. (20) gives: κ = γ | Ω c | − ( γγ + | Ω p | ) ( γγ + | Ω c | + | Ω p | ) , (21a) κ =2 i { γ | Ω c | [ γ (2 γ + γ ) + 2 | Ω p | ] − ( γγ + | Ω p | ) } / ( γγ + | Ω c | + | Ω p | ) . (21b)For the parameters in Fig. 2, Eq. (21a) gives κ =0 . . κ = 7 . × − i , for parameters inFig. 2, is very small to cause any significant change inthe pulse width σ ( z ) = (cid:112) σ − ik (cid:48)(cid:48) z . This is why in Fig.5 (1 st row), from 0 ≤ Z ≤ . κ can be perceived as the rate at whichthe probe pulse proceeds along time axis with increasing Z . The aforementioned value of κ = 0 . β = 0 .
125 (see Table I).Therefore, the control field’s leading end moves forwardalong time axis at a greater rate than the probe pulse asevident from Fig. 5 (1 st row).In Fig. 5 (3 rd column), beyond Z ≥ .
4, controlfield’s leading end begins to overlap with the probe pulse,thereby initiating the process of pulse reshaping and am-plification, which continues till the control field’s leadingend completely overtakes the probe pulse at Z = 4 .
3. Probe reshaping and amplification
As seen earlier in Fig. 3 (2 nd row), there occurs amomentary population transfer from the ground state | (cid:105) to excited state | (cid:105) at the position of control field’sleading end. This leads to a positive peak in ( ρ − ρ )causing population inversion in the | (cid:105) ↔ | (cid:105) channel[see Figs. 6(a), (b)]. Hence, in presence of the probefield there is stimulated emission of probe photons in the | (cid:105) ↔ | (cid:105) channel causing probe amplification. This isillustrated in Fig. 6, where the temporal profiles of probeand control field magnitudes along with ( ρ − ρ ) areplotted column wise at different propagation lengths. (a)
01 01 0500.5 0.3 0.5 0.3 0.5 0.3 0.5 | Ω i / Ω i | Control Z = 2 . Probe Z = 2 . Z = 3 ρ − ρ T (b)
015 015 01500.5 0.3 0.5 0.3 0.5 0.3 0.5 | Ω i / Ω i | Z = 3 . Z = 3 . Z = 4 . ρ − ρ T FIG. 6: (Color online) [1 st row of (a), (b)] Temporal profilesof probe ( i = p , crisscross) and control ( i = c , solid) fieldmagnitude are plotted column wise at different propagationlengths Z . [2 nd row of (a), (b)] ( ρ − ρ ) vs time T , atthe corresponding propagation lengths. Plot illustrates howthe positive peak of ( ρ − ρ ) at the position of controlfield’s leading end causes probe amplification at every ( ζ, τ ).Normalization of field magnitude and the parameters usedare same as Fig. 2, except here the normalized control fieldmagnitude is upscaled in a way to accommodate the probepulse inside it, indicating Ω p (cid:28) Ω c . In both Figs. 6(a),(b), plots connected to a particular column have a commontime axis. The vertical axis for ( ρ − ρ ) plots remain samealong a row while the plots for field magnitudes have differentvertical axis along a row. In Fig. 6, with increasing Z , both the positive peakof ( ρ − ρ ) and control field’s leading end, proceedcollinearly at a greater rate than the probe pulse alongtime axis. At Z = 2 .
4, the leading end of both probeand control fields coincide on the time axis. From 2 . ≤Z ≤ .
2, the control field’s leading end or in other words,the positive peak of ( ρ − ρ ) grazes through the probepulse on time axis. This causes probe amplification atthe position of the positive peak of ( ρ − ρ ) on timeaxis for every space time coordinate.In Fig. 6(b) (3 rd column), beyond Z ≥ .
2, both thepositive peak of ( ρ − ρ ) and the control field’s leadingend completely overtakes the probe pulse, halting furtheramplification. The probe pulse then travels at the speedof light in vacuum, without any delay, distortion and ab-sorption, whilst retaining its initial Gaussian shape butwith a larger pulse width. This stable propagation of theprobe pulse is explained in the next section.
4. Stable probe propagation . . | Ω i / Ω i | Probe Control(a) ρ (b) ρ (c) ρ T (d) FIG. 7: (Color online) (a) Temporal profiles of probe (criss-cross) and control (solid) field magnitudes at Z = 4 . | (cid:105) , | (cid:105) and | (cid:105) re-spectively. The plot illustrates the population distribution ρ = ρ = 0 , ρ = 1 as seen by the probe after the controlfield’s leading end completely overtakes it. Both probe andcontrol field magnitudes are normalized to unity. Parametersused are same as Fig. 2. All figures have a common time axis. The technique of pulse amplification at hand, ensuresa stable pulse propagation upon completion of amplifi-cation. This is made possible due to the population dis-tributions created by the control field as it propagatesthrough the medium. Figure 7 shows the temporal pro-files of the fields along with population distribution ata position Z = 4 . ≤ T ≤ .
6, the probe pulsesees a population distribution ρ = ρ = 0 , ρ = 1 [see Figs. 7(b), (c), (d)], i.e. , all population gets settled inthe dark state | (cid:105) [7]. Therefore, the general state | ψ (cid:105) ofthe system within the time interval 0 ≤ T ≤ . | ψ (cid:105) = a | (cid:105) + a | (cid:105) + a | (cid:105) = | (cid:105) , (22)where a i ( i = 1 , ,
3) represent the probability amplitudesof state | i (cid:105) . Equation (22) implies a = a = 0 , a = 1.Therefore, from the definition of density matrix elements ρ = a ∗ a = 0. The propagation equation for the probepulse [see Eq. (11a)] then becomes: ∂ Ω p ( ζ, τ ) ∂ζ = iηρ ( ζ, τ )= 0 , (23)which resembles a free space propagation equation. Thusat the end of amplification process, the probe pulse trav-els freely, unattenuated and undistorted at the speed oflight in vacuum, even in the presence of medium.
5. Probe pulse broadening . . | Ω i / Ω i | ControlAmplified portionPortion inside control (a) ρ (b) ρ (c) ρ T (d) FIG. 8: (Color online) (a) Temporal profiles of probe andcontrol field magnitudes at Z = 2 .
8. (b), (c), (d) Populationdistribution of | (cid:105) , | (cid:105) and | (cid:105) respectively. The plot showshow the portion of the probe pulse behind the control field’sleading end [crisscross pattern in (a)] sees a population distri-bution ρ = ρ = 0, ρ = 1. Whereas, the portion of probepulse inside the control field [oblique line pattern in (a)] seesa different population distribution ρ = ρ = 0, ρ = 1.The former behaves like dark state, while the later serves as abright state. Hence the oblique line portion continues to ex-perience delay, absorption and dispersion, while the crisscrossportion undergoes free space propagation. Normalization offield magnitudes and parameters used are same as Fig. 2. Allfigures have a common time axis. Apart from the negligible pulse broadening due to dis-persion in the region 0 ≤ Z ≤ . st row)], theprobe pulse is subjected to a noticeable pulse broadeningduring amplification within 2 . ≤ Z ≤ . st row)]. This section provides a qualitative explanation forsuch a pulse broadening. Figure 8, shows the temporalprofiles of the fields along with population distribution ata position amidst the amplification process. During theamplification process, the probe pulse sort of splits intotwo portions as indicated by the crisscross and obliqueline patterns in Fig. 8(a). The crisscross pattern indi-cates the amplified portion of probe and the oblique linepattern represents the portion of probe which is yet to beamplified. In Figs. 8(b), (c), (d), the oblique line portionsees a population distribution ρ = ρ = 0 , ρ = 1,and the crisscross portion sees a different population dis-tribution ρ = ρ = 0 , ρ = 1. The former servesas an absorbing, dispersive medium, while the later be-haves like dark state, as discussed earlier in Sec. III 2and III 4 respectively. Therefore, the crisscross portionpropagates freely without experiencing any delay, distor-tion and absorption. But the oblique line portion contin-ues to experience noticeable delay, with small absorptionand insignificant broadening due to dispersion. It’s dueto this delay, experienced by the oblique line portion,the probe pulse gets elongated as the control field grazesthrough it along time axis, with increasing propagationlength. The small absorption experienced by the obliqueline portion is compensated by amplification and doesn’tlead to any noticeable shape distortion.
6. Evaluation of probe pulse width
To check the predictability of the system, a formula forthe probe pulse width at the end of amplification processis derived in terms of experimentally available parame-ters. To simplify the calculation an analogy to a simpleclassical problem is drawn in Fig. 9.In Fig. 9(a), the bars indicated by “A” and “B”move forward along x axis at rates v and v respectively( v > v ). The bars “A” and “B” coincide in an elapsedtime t after traveling a distance A (cid:48) A and B (cid:48) B respec-tively [Fig. 9(b)]. In Fig. 9(c), the trailing end of probepulse and the control field’s leading end move forwardalong time axis at rates β and κ per unit propagationlength Z , respectively ( β > κ ). By analogy the controlfield’s leading end is represented by the bar “A” and,the trailing end of probe pulse is represented by the bar“B”. In Fig. 9(c), Z c denotes the propagation length atwhich both the leading end of probe and control coincideon time axis. In Fig. 9(d), after propagating a distance Z (cid:48) , the control field’s leading end completely overtakesthe probe, leaving behind a broadened, amplified Gaus-sian probe pulse. From observation, the full extent of theprobe pulse T ( Z ) [Fig. 9(d)], after the amplification pro-cess, is simply the time taken by control field’s leadingend to overtake the trailing end of probe pulse on time FIG. 9: (Color online) A diagram illustrating an analogy ofthe problem to a similar classical problem. axis.From analogy, T ( Z ) is equivalent to A (cid:48) A = v t in Fig.9(b). From Fig. 9(b), with simple manipulation A (cid:48) A canbe written as: A (cid:48) A = v L ( v − v ) . (24)From analogy, in Fig. 9, A (cid:48) A ≡ T ( Z ) , v ≡ β, v ≡ κ , L ≡ T ( Z c ) . Therefore, Eq. (24) gives T ( Z ) = βT ( Z c )( β − κ ) , (25)where, with some simple algebra, Z c can be written as: Z c = − B − √ B − C , (26) B = − β − κ ) γτ + κ a β − κ ) (cid:104) a = 2 (cid:112) ) (cid:105) ,C = 4 γ τ − γ σ a β − κ ) . The pulse width σ ( Z ), can be calculated by using therelation (see appendix A): T ( Z ) = 2 (cid:112) ) σ ( Z ) , (27)where σ ( Z ) = σ − iκ Z . Substituting the earlier ob-tained values of κ = 0 . κ = 7 . × − (see Sec.III 2) and β = 0 .
123 (see Table I) in Eqs. [(25) - (27)]gives T ( Z ) = 235 . . β and κ , κ (for Ω p (cid:28) Ω c ) are merely dependent on | Ω c | andindependent of the initial pulse width σ , which makes T ( Z ) [Eq. (25)] independent of σ . Therefore, for aparticular control field intensity, the ratio T ( Z ) /T (0) isexpected to remain constant for any arbitrary σ . Keep- TABLE II: Table shows the ratio of T ( Z ), to the initial fullextent of probe pulse T (0). σ γ (cid:12)(cid:12) Ω p /γ (cid:12)(cid:12) a T (8 . T (0) T (8 . /T (0)15 10 237 .
545 111 .
51 2 . .
522 223 .
02 2 . .
604 334 .
53 2 . .
856 446 .
04 2 . a The values of this column are to be multiplied by 10 − . ing Ω c = 4 γ and area of probe pulse constant, the ra-tios T ( Z ) /T (0), obtained numerically for different val-ues of σ are tabulated in Table II. In Table II, the ra-tio T ( Z ) /T (0) ≈
2, remains constant for different initialprobe pulse width σ , which matches the expected re-sult. Therefore, the formula in Eq. (25) correctly esti-mates the probe pulse width at the end of amplificationprocess, countenancing the correctness of the numericalsimulation.
7. Ancillary results
01 01301 01301 0.5 1 013 0.5 1.5
Complex shape
Input Output sech ( t ) T Square T FIG. 10: (Color online) (left column) Temporal profiles ofprobe (crisscross) and control (solid) field magnitude at Z =0, for different probe pulse shapes. (right column) Corre-sponding temporal profiles of probe and control field mag-nitudes at the end of amplification process. Field magnitudescaling, normalization are same as Fig. 6 and parameters usedare same as Fig. 2, with the additional parameters: σ = 30 /γ , σ = 40 /γ , σ = 45 /γ , τ = 400 /γ , τ = 500 /γ , τ = 500 /γ . Fig-ures in a particular column have a common time axis.
The propagation of a sech , square and a complexshaped pulse, are investigated indiscriminately in pres-ence of a continuous control field. The complex shapedpulse and sech pulse are given as:Ω p (0 , t ) = Ω p (cid:16) e − ( t − τ ) /σ + sech[ − ( t − τ ) /σ ] (cid:17) , Ω p (0 , t ) = Ω p sech [ − ( t − τ ) /σ ] . In presence of a continuous control field, both the com-plex and sech pulses show similar amplification and elon-gation (Fig. 10) as obtained earlier with a Gaussianpulse. The control field also shows similar absorptionas before. The square pulse shows similar elongation butdoesn’t retain it’s square shape at end of amplificationprocess. Therefore, our method can be used to amplifyarbitrary pulse shapes without loss of generality.Next, the propagation of a Gaussian pulse in presenceof a continuous control field is investigated for differentcontrol magnitudes.
01 0201 0701 01501 01501 0.3 0.7 0300 2 5 (a) | Ω c | = 0 . γ Input Output(b) | Ω c | = 1 γ (c) | Ω c | = 2 γ (d) | Ω c | = 4 . γ T (e) | Ω c | = 5 γ T FIG. 11: (Color online) (left column) Temporal profiles ofprobe (crisscross) and control (solid) field magnitudes at Z =0, for different values of | Ω c /γ | . (right column) Correspondingtemporal profiles of probe and control field magnitudes atthe end of amplification process. Field magnitude scaling,normalization and relevant parameters remain same as Fig.6. Figures in a particular column have a common time axis. As explained earlier in Fig. 8, the amplified portionof the probe (crisscross), being uninfluenced by the con-trol, isn’t subjected to pulse broadening due to disper-sion but the portion of probe under the influence ofcontrol (oblique line pattern) continues to experiencebroadening. Therefore, if κ , which is responsible forpulse broadening is large, then the oblique line portion0of probe will experience significant broadening, result-ing in an asymmetric pulse shape posterior to the am-plification process. This is shown in Fig. 11(a), where κ = 31 . i for | Ω c /γ | = 0 .
5, results in an asymmetricoutput pulse shape. Whereas, the output pulse shapefor κ = 4 . × − i corresponding to | Ω c /γ | = 4 . κ decreases with increase in | Ω c | .Therefore, in Fig. 11(a) - (d), as | Ω c /γ | is increased from0 . .
5, the output pulse shape becomes more close tothe input Gaussian shape.In Fig. 11(a) - (e), as | Ω c /γ | is increased from 0 . | Ω c | , beyond acertain value, e.g., | Ω c | = 5 γ in Fig. 11(e), results in un-stable output. This may be attributed to the instabilitiesthat accompanies huge amplification.Therefore, choice of | Ω c | plays a crucial role in decidingthe amount of amplification and the shape of probe pulseat the end of amplification process. A relatively small | Ω c | leads to small amplification and asymmetricities inthe output pulse shape. Again, a large | Ω c | results ininstabilities caused by huge gain. The sweet spot liessomewhere in the middle, as seen in Fig. 11. IV. SUMMARY AND CONCLUSIONS
In conclusion, the propagation of a weak probe pulsethrough the Λ system in a resonant gain configurationis investigated. The gain configuration is different fromthe EIT system in a way that the control and probefields are swapped with one another. This configurationmakes provision for population inversion in the transi-tion coupled by the probe field. Thus, causing probe am-plification through stimulated emission. With a carefulchoice of control field intensity, the probe pulse althoughbroadened, retains its initial shape and travels at thespeed of light in vacuum, without any delay, absorptionand dispersion after the amplification process. With thisscheme, an arbitrary shaped probe pulse can propagatethrough the medium without loss of generality at the endof amplification. Therefore, the proposed model systemcould be useful in optical communication as one of themethods to amplify various signals, for possibly assistinglong distance communication.
ACKNOWLEDGMENTS
T.N.D. gratefully acknowledges funding by the Scienceand Engineering Board (Grant No. CRG/2018/000054).
Appendix A: Relation between T and σ In Fig. 12, the relation between the width σ of a Gaus-sian function ( y = e − x / σ ) and the full extent of theGaussian function T can be approximated by consideringan arbitrarily small value say (10 − ), and then evaluatingthe values of x min and x max , where x min and x max arethe x coordinates of the intersection points of y = 10 − line with the Gaussian function. FIG. 12: Gaussian distribution function
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