Bistable dynamics beyond rotating wave approximation

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Journal of Nonlinear Optical Physics & MaterialsVol. 23, No. 2 (2014) 1450019 (19 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218863514500192

Bistable dynamics beyond rotating wave approximation

Y. A. Sharaby∗,†,‖, S. Lynch‡,∗∗, A. Joshi§,†† and S. S. Hassan¶,‡‡∗Faculty of Applied Sciences, Physics Department,

Suez University, Suez, Egypt

†Mathematics Department, College of Arts and Sciences,Salman bin Abdulaziz University,

Wadi Addwasir, Saudi Arabia

‡School of Computing,Mathematics & Digital Technology,Manchester Metropolitan University,John Dalton Building, Chester St.,

Manchester M1 5GD, UK

§IQSE,Texas A&M University, College Station,

TX 77843, USA

¶Department of Mathematics, College of Science,University of Bahrain,

P. O. Box 32038, Bahrain‖Yasser [email protected]

∗∗[email protected]††[email protected]

‡‡[email protected]

Received 11 January 2014

In this paper, we investigate the nonlinear dynamical behavior of dispersive opticalbistability (OB) for a homogeneously broadened two-level atomic medium interactingwith a single mode of the ring cavity without invoking the rotating wave approximation(RWA). The periodic oscillations (self-pulsing) and chaos of the unstable state of the OBcurve is affected by the counter rotating terms through the appearance of spikes duringits periods. Further, the bifurcation with atomic detuning, within and outside the RWA,

shows that the OB system can be converted from a chaotic system to self-pulsing systemand vice-versa.

Keywords: Optical bistability; bifurcation; rotating wave approximation; self-pulsing;chaos.

‖Corresponding author.

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http://dx.doi.org/10.1142/S0218863514500192

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Y. A. Sharaby et al.

1. Introduction

Optical bistability (OB) phenomenon has a vital role in nonlinear optics, and it hasbeen well studied both experimentally and theoretically,1–19 because of its poten-tial applications in high speed all optical signal processing, optical switching andall optical computing. Optically bistable elements, those based upon the generationof two stable field states for the same input field, can be used as logic gates, mem-ory devices, switches, differential amplifiers, components in optical computers andcommunication systems.5,20,21 Many promising technological applications of thebistability phenomenon in all optical telecommunication networks, optical comput-ing using nonlinear semiconductor lasers and laser amplifiers have been recentlyreviewed.22 Further, studies of the OB system can be beneficial for preparing novelquantum computational devices, quantum information processing circuits and net-works realizing new vistas of generating entanglement within the context of cooper-ative behavior of the system.23 Optical chaos generated in nonlinear optical systemsis applicable for communication of information at high data rates24 and useful inapplications of random number generation.25,26

In earlier works7,27,28 a linear stability analysis for the absorptive optical bista-bility in a ring cavity showed that under certain conditions, part of the upper branchof the hysteresis curve is unstable. In this situation, the system can operate as atransformer of continuous wave light into pulsed light, where a self-pulsing oscil-lation occurs with a period equal to the cavity round-trip time. This analysis wasgeneralized in the dispersive OB case29 (see Refs. 14, 15 and references therein).Another mechanism of instability was theoretically studied by Ikeda et al.,30–33

where periodic instabilities and chaotic behavior could occur in optical bistabletwo-level system in a ring cavity with time delay effects, when the propagation timeof the radiation field from the output- to the input-end is longer than the atomicdecay time. This instability can lead to period doubling and chaos as reported3

for Fabry–Perot cavity containing sodium atoms in the absorptive OB case. Sin-gle mode instabilities ranging from gain assisted laser systems34 to passive two-level optical bistable systems have been investigated35 and revisited with moredetails.36–38 Effects of the squeezed vacuum on the dynamic behavior of two-levelOB systems have been discussed in Ref. 39. Recently, novel instability of OB sys-tems of very thin material (of thickness much smaller than the wavelength)40 hasbeen discussed for a pure absorptive OB case in a Fabry–Perot cavity sustainingmulti-longitudinal modes. In that work, the analysis of OB was carried out withthe change of the position of the thin absorber in the cavity.

Self-pulsing and chaos are exhibited by the three-level atoms when they areinteracting with multiple fields,41,42 probe field experiencing feedback,43 singlemode field in ring cavity environment along with squeezed vacuum field,44 andunder two-photon atomic transition.45,46 Recently, the authors of Ref. 47 investi-gated a novel type of instability different from earlier instabilities of OB30–33,48–51

in which the occurrence of self-pulsing does not require finite detuning of either the

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bistable field coupled to the atom or the cavity detuning,52 and it occurs withoutinvolving coupling to multiple modes of the cavity field. Further, the instabilityexamination in the lower cooperative branch for three ∧- and V-atomic systemsinside a double-cavity53 showed that it is due to the leakage of population fromthe coherently prepared lower atomic levels. This leakage occurs via spontaneousemission into the ground state leading to instability in the lower cooperative branch.Such instability occurs only in the Λ-type system wherein incoherent decay cou-pling of the pair of ground states is essential to obtain instabilities in the cooperativebranch.

In all the previous theoretical studies of OB,1–53 the usual rotating wave approx-imation (RWA) has been adopted, which amounts to neglecting highly oscilla-tory terms in the model Maxwell–Bloch equations. Effects of such terms on thebistable behavior for two-level atomic systems in an optical ring cavity were recentlystudied by us for the homogeneously and nonhomogeneously broadened atomicmedium.54,55 It was shown that54 the first harmonic output field component, gener-ated outside the RWA, exhibits reversed (clockwise) or closed (butterfly) loop bista-bility, simultaneously with the usual (anti-clockwise) bistable output fundamentalfield component. Thus, transforming opposite coding information simultaneously insuch a system is a physical possibility. Also, the order of the first harmonic outputfield is small, O(λ2) (λ = γ/ωL; γ = the atomic damping constant, ωL = circularfrequency of the injected laser field) and sensitive to the damping ratio (γ/κ); where(κ) is the cavity damping constant. As argued in Ref. 54, such weak output signalsmay be detected via phase sensitive detection techniques which have been used indetecting squeezed light. In addition, when effects of atomic linewidth are takeninto consideration,55 the output field component outside the RWA, which exhibitsthe usual anti-clockwise bistable behavior in the absorptive case, switches in thedispersive case to a closed loop (switching-down process) with Lorentzian linewidthor to reversed (clockwise) bistable behavior with Gaussian linewidth.

Enhancement of the parameter (λ) can be achieved in strong coupling regimes(λ > 1) as reported recently in experiments with superconductor qubits,56 solid-state semi-conductors57 and nano-mechanical resonators.58 Recently, some theoret-ical investigations were carried out on the entanglement of nondissipative stronglycoupled two-qubit system interacting with a single mode quantized field outsidethe RWA (see Ref. 59 and references therein). Also, there are some earlier studiesoutside the RWA on dissipative quantum systems interacting with squeezed vac-uum baths60–62 and nondissipative mesoscopic multistable63,64 and pulsed-drivenqubit65,66 systems. Outside the RWA, critical slowing down behavior has been inves-tigated recently67 in which near the critical point of the incident field, the systemexhibits switching to the lower branch of the first harmonic output field with irreg-ular oscillations in both high and low quality cavity cases.

In this paper, we further extend the work reported in Ref. 54 to investigate thedynamical behavior of homogeneously broadened two-level atomic systems placed

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in an optical ring cavity, outside the RWA. We show that the dynamics of thesystem and chaos are sensitively dependent on the input field amplitude and theatomic-field detuning parameter.

The paper is organized as follows: The nonautonomous model of Maxwell–Blochequations outside the RWA with analytical treatment via Fourier series decompo-sition up to the first generated harmonic component of the output field is reviewedin Sec. 2. Nonlinear dynamics and bifurcation behavior are computationally inves-tigated in Sec. 3. A summary is presented in Sec. 4.

2. Model Review

We consider a single mode ring cavity, which contains a homogeneously broad-ened two-level atomic medium of length L. The c-number model Maxwell–Blochequations, in the plane wave and mean field approximations, and in a rotatingframe rotating at frequency ωL, outside the RWA, have the following dimensionlessform54,67:

dxdt

= κ[Y − (1 + iθ)x + 2√

2Cr−], (2.1a)

ddt

r− = −γ⊥2

(1 + iδ)r− +γ⊥√

2r3(x + x∗eiηt) =

(ddt

r+

)∗, (2.1b)

ddt

r3 = −γll

2(1 + 2r3) − γll

2√

2[r+(x + x∗eiηt) + c.c.]. (2.1c)

The notations in Eqs. (2.1a)–(2.1c) are as follows: r± are the mean values of thequadrature atomic polarization components, r3 is the mean value of the atomicinversion, γ⊥ is the transverse relaxation rate of the atomic polarization, γll is thelongitudinal relaxation of the atomic inversion and δ = 2(ωo − ωL)/γ⊥ is the nor-malized atomic detuning, ω0 is the atomic transition frequency, ωL is the input fieldfrequency, and the parameter η = 2ωL. The quantities x and Y are the normalizedoutput and input field amplitudes, respectively, θ = (ωc − ωL)/κ is the normal-ized cavity detuning, ωc = cavity mode frequency, κ = cavity decay constant andC = g2/(γ⊥κ) is the cooperative parameter, where g is the coupling between thecavity field and the atoms. In Eq. (2.1c), (c.c.) stands for complex conjugate.

The terms containing e±iηt(= e±2iωLt) in Eqs. (2.1b) and (2.1c) are arisingdue to interaction of the cavity field with the atoms outside the RWA.67 Withinthe RWA, these terms are discarded and Eqs. (2.1a)–(2.1c) yield the well-knowninput-output field steady state equation:6–8

Y = xo

[(1 +

2C

1 + δ2 + |xo|2)

+ i

(θ − 2Cδ

1 + δ2 + |xo|2)]

, (2.2)

where xo is the fundamental output field amplitude within the RWA.

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The solution for the atomic Bloch components r∓,3 and the cavity field x accord-ing to Eqs. (2.1a)–(2.1c) contain all harmonics of frequency 2ωLn (n = integer),due to the presence of the harmonic coefficients e±iηt. Following our analysis inRefs. 54, 63 and 64 the field and atomic variables are decomposed up to first har-monic (n = ±1) as follows:

r∓,3(t) = ro∓,3(t) + r+∓,3(t)e

iηt + r−∓,3(t)e−iηt,

x(t) = xo(t) + x+(t)eiηt + x−(t)e−iηt,(2.3)

where xo and ro∓,3 are the fundamental field and atomic variables components,

respectively, within the RWA, and the corresponding components x∓, r±∓,3 are thosegenerated outside the RWA. Substituting Eq. (2.3) into (2.1a)–(2.1c) and comparingthe coefficients of e±inηt (n = 0, 1), we reach the following system of ordinary dif-ferential equations (ODEs) for the atom and field harmonic amplitude components:

ddt

ro−(t) = −γ⊥

2(1 + iδ)ro

−(t) +γ⊥√

2[ro

3(t)(xo(t) + x∗+(t))

+ r−3 (t)(x∗o(t) + x+(t)) + r+

3 (t)x−(t)]

=(

d

dtro+(t)

)∗, (2.4a)

ddt

r+−(t) = −γ⊥

2

(1 + iδ +

2iη

γ

)r+−(t) +

γ⊥√2[ro

3(t)(x∗o(t) + x+(t))

+ r+3 (t)(xo(t) + x∗

+(t)) + r−3 (t)x∗−(t)]

=(

d

dtr−+(t)

)∗, (2.4b)

ddt

r−−(t) = −γ⊥2

(1 + iδ − 2iη

γ

)r−−(t) +

γ⊥√2

× [ro3(t)x−(t) + r−3 (t)(xo(t) + x∗

+(t))]

=(

d

dtr++(t)

)∗, (2.4c)

ddt

ro3(t) = −γll

2− γllr

o3(t) −

γll

2√

2[(ro

+(t) + r+−(t))(xo(t) + x∗

+(t))

+ r++(t)x−(t) + c.c.], (2.4d)

ddt

r+3 (t) = −γll

(1 +

iη

γ

)r+3 (t) − γll

2√

2[(ro

+(t) + r+−(t))(x∗

o(t) + x+(t))

+ r++(t)(xo(t) + x∗

+(t)) + r−+(t)x∗−(t) + ro

−(t)x∗−(t)]

=(

d

dtr−3 (t)

)∗, (2.4e)

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dxo

dt= κ[Y − (1 + iθ)xo(t) + 2

√2C ro

−(t)], (2.4f)

dx+

dt= −κ

[(1 + iθ +

iη

κ

)x+(t) − 2

√2Cr+

−(t)]

, (2.4g)

dx−dt

= −κ

[(1 + iθ − iη

κ

)x−(t) − 2

√2Cr−−(t)

]. (2.4h)

Our analytical and numerical results54,55 show that the amplitude of the additionalfirst harmonic component x+ is of smaller order, O(λ2) with λ = γ/2ωL, comparedwith the fundamental field component amplitude xo. To the same order of approx-imation, the field component x− = 0, and hence the dipole component r−− = 0.Consequently, Eqs. (2.4a)–(2.4h) reduce to the following smaller set of ODEs:

ddt

ro−(t) = −γ⊥

2(1 + iδ)ro

−(t) +γ⊥√

2ro3(t)xo(t) =

(ddt

ro+(t)

)∗, (2.5a)

ddt

r+−(t) = −γ⊥

2

(1 + iδ +

2iη

γ⊥

)r+−(t) +

γ⊥√2ro3(t)x

∗o(t) =

(ddt

r−+(t))∗

, (2.5b)

ddt

ro3(t) = −γll

2− γllr

o3(t) −

γll

2√

2[ro

+(t)xo(t) + c.c.], (2.5c)

dxo

dt= κ[Y − (1 + iθ)xo(t) + 2

√2Cro

−(t)], (2.5d)

dx+

dt= −κ

[(1 + iθ +

iη

κ

)x+(t) − 2

√2Cr+

−(t)]

. (2.5e)

Optical instabilities most likely35–38 arise when the atomic system interacts with(large) multimode field leading to chaos and self-pulsing or with intense single cavitymode field that saturates the collection of atoms in the absorptive OB.

When the system in the vicinity of the unstable region of the upper branch ofOB curve, either it transfers to the lower branch or exhibits either periodic (self-pulsing) oscillatory behavior or non-periodic chaotic behavior. The analytical studyof instability14,15 for absorptive and dispersive bistability shows that the instabilityof the upper branch may exist if at least one of the off-resonant modes are unstable.Here, we investigate the nonlinear dynamical behavior of the OB system, Eq. (2.4),within and outside the RWA in the dispersive OB case when the atomic polarizationvariables are eliminated adiabatically. In this adiabatic case, the atomic relaxationtime for the atomic polarization components is shorter than the other characteristictimes (γ⊥〉〉γll, κ, κθ, κC) of the system. Hence by substituting in Eqs. (2.5a)–(2.5e)for ro,±

± by their stationary values,54 and then expressing the normalized outputfields as xo = uo + ivo and x+ = u+ + iv+, we obtain the following system of

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ODEs:

κ′ dro3(τ)dτ

= −12(1 + 2ro

3(τ)) − ro3(τ)

(1 + δ2)(u2

o(τ) + v2o(τ)), (2.6a)

duo(τ)dτ

= Y − uo(τ) + θvo(τ) + 4Cro3(τ)

(δvo(τ) + uo(τ))1 + δ2

, (2.6b)

dvo(τ)dτ

= −vo(τ) − θuo(τ) + 4Cro3(τ)

(−δuo(τ) + vo(τ))1 + δ2

, (2.6c)

du+(τ)dτ

= −u+(τ) +(

θ +1ν

)v+(τ) − 4Cro

3(τ)

((δ +

2λ

)vo(τ) − uo(τ)

)

1 +(

δ +2λ

)2 ,

(2.6d)

dv+(τ)dτ

= −v+(τ) −(

θ +1ν

)u+(τ) − 4Cro

3(τ)

((δ +

2λ

)uo(τ) + vo(τ)

)

1 +(

δ +2λ

)2 .

(2.6e)

with τ = γ⊥t, κ′ = κ/γ⊥, λ = γ/η and ν = κ/η.Equations (2.6a)–(2.6e) are treated numerically in the next section using the

standard Runge–Kutta method with suitable initial conditions to display variousdynamical states.

3. Computational Results

Nonlinear systems that display chaos are sensitive to the system parameters andessentially show four modes of operations: chaotic, stable, periodic and quasiperi-odic. Methods of determining the dynamics of such systems rely heavily on math-

ematical packages such as MATLAB r©, Mathematica r© and MapleTM

.68–70 Thesteady state relation that represents the relation between the output field com-ponent outside the RWA, |x+| and the input field Y in the dispersive case(δ = 374, θ = 340) is shown in Fig. 1. The inset shows the bistable behavior ofthe fundamental output field component |xo| within the RWA. Linear stabilityanalysis around the stationary state shows that, points on the upper branch of thebistable behavior are unstable and the associated instability arises with oppositesigns of the detuning parameters (δθ〈0) and multiple periods of unstable pointsoccurs for large cooperativity (C〉300).35 In the usual optical bistable behavior overa certain input range, Y ∈ (0, 2500), as shown in the inset of Fig. 1, the pointsA, B, C, D represent the unstable points in the upper branch of Fig. 1 and corre-spond to the respective Y values as Y = 950, Y = 1225, Y = 1350 and Y = 2000,respectively.

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Fig. 1. The first harmonic output field component |x+| against the incident field Y for C = 70000,κ = 0.25, λ = 10−6 for δ = 374, θ = 340. Inset shows the fundamental output field component|xo| against the incident field Y .

The system of normalized coupled equations (2.6) is integrated numericallyfor fixed values of θ = 340, δ = 374, κ = 0.25. For different Y values, thetime-dependence of the output field components |xo| and |x+|, the phase portrait(Im(|xo,+|), Re(|xo,+|)), 2D Poincare map of the solutions at fixed ro

3 and powerspectrum diagrams of the input intensities are plotted respectively in Figs. 2–5 forfixed parameters of the system. For Y = 2000 [Figs. 2(a) and 2(b)], periodic behav-ior is displayed for both field components (|xo|, |x+|) but with significant spikes forthe output field component out the RWA |x+|. A single period is exhibited in thephase space of both the output field components |xo| and |x+| [Figs. 2(c) and 2(d)],but with weak irregular oscillations occur in the single period of the |x+| component[Fig. 2(d)]. The presence of spikes in the phase portrait is due to the oscillatorynature of the system operating outside the RWA that generates the virtual pho-tons. This single periodic loop appears in the Poincare diagram of the output fieldcomponents (|xo|, |x+|) as a dot [Figs. 2(e) and 2(f)]. The power spectrum showsone main peak for both component |xo| and |x+| outside the RWA [Figs. 2(g) and2(h)].

Decreasing Y to Y = 1350 [Figs. 3(a) and 3(b)], double periods occur for bothfield components |xo| and |x+|, but with weak oscillatory behavior in the |x+|-phase portrait [Figs. 3(c) and 3(d)]. Poincare map that confirms the existence ofperiod doubling is shown by two points [Figs. 3(e) and 3(f)] and two prominentpeaks in the power spectrum [Figs. 3(g) and 3(h)]. At Y = 1225, we have periodquadruplicating (Fig. 4) and finally at Y = 950, the chaotic behavior is exhibited(Fig. 5).

The above results are demonstrated in the bifurcation diagram for the normal-ized output field components within and outside the RWA (|xo| and |x+|) againstthe input field Y for fixed system parameters (δ = 374, θ = 340, C = 70000, λ =10−6, κ = 0.25) as seen in Figs. 6(a) and 6(b)). It is noted that, optical bistable

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(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 2. (a, b) Self-pulse oscillations for both |xo| and |x+| components in the adiabatic caseagainst the time τ = κt for C = 70000, κ = 0.25, λ = 10−6, δ = 374, θ = 340 and Y = 2000.(c, d) The corresponding phase-space representation (Re(xo), Im(xo)) for the fundamental outputfield component and (Re(x+), Im(x+)) for the first harmonic output field component, respectively.

(e, f) Poincare map for both |xo| and |x+|. (g, h) Power spectrum for both |xo| and |x+|.

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(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 3. Same as (2) but with Y = 1350.

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(a) (b)

(c) (d)

(e) (f)

(g) (h)


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(a) (b)

(c) (d)

(e) (f)

(g) (h)


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device can display chaotic behavior, multi-periodic and single periodic states forboth output field components by increasing the input field Y . In other words, theOB device routes from chaos to self-pulsing by increasing Y . In particular, chaoticbehavior occurs at Y = 950 is transformed to period quadruplicating at Y = 1225,to 2-periods at Y = 1350 and finally to a single period at Y = 2000. Note that,each dot in the bifurcation diagram represents the peak of the oscillation. For fixedY = 1350, the system can be converted from chaotic to aperiodic behavior andagain to chaotic for the output field component |x+| out the RWA, by increasingthe atomic detuning from δ = 360 to 374 or vice-versa (Fig. 7).

For lower atomic detuning (δ = 100), κ = 0.3 and for rest of parameters as inFig. 1, the steady state relation between the output field component outside theRWA, |x+| and the input field Y is shown in Fig. 8, which exhibits closed loopbehavior.54 On the left side of the closed bistable hysteresis loop behavior over acertain Y range, e.g., Y ∈ (0, 5000) (see Fig. 8), the route from self-pulsing to chaosis obtained for both the output field components |xo| and |x+|, as shown in Fig. 9.

(a)

Fig. 6. (a) Bifurcation diagram representing the maximum value of the output field |xo| within

RWA against the input field Y for fixed C = 70, 000, δ = 374 and θ = 340. (b) As (a) but for thefirst harmonic output field component |x+| for κ = 0.25 and λ = 10−6.

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(b)

Fig. 6. (Continued)

Fig. 7. Bifurcation diagram that represent the maximum value of the first harmonic output fieldcomponent |x+| against the atomic detuning δ for fixed values of other parameters as in Fig. 1,but with Y = 1350.

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Fig. 8. Same as Fig. 1, but withκ = 0.3 and δ = 100.

(a) (b)

(c) (d)

Fig. 9. (a–d) Same as (2.4a)–(2.4d), but with κ = 0.3, δ = 100, θ = 340 and Y = 4220.

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These results was obtained earlier in Ref. 29 for the output field component |xo|within the RWA. For example, for Y = 4220, we get period quadruplicating forY = 4220 (Fig. 9).

4. Summary

We have investigated the dynamical behavior of an optical bistable system for homo-geneously broadened two-level atoms interacting with a single mode ring cavity out-side the RWA in the dispersive regime. The system routes from periodic to chaoticbehavior by decreasing the input field. This can be viewed from the bifurcation dia-gram (|xo,+| against Y ) by sweeping the parameter Y forth and back. Outside theRWA, the virtual photons (that results from rapidly oscillating terms of the atom-field interaction) affect the self-pulsing and chaotic behaviors of the unstable stateof the system in the form of spiking. The state of the OB device changing from reg-ular (self-pulsing) to irregular (chaos) oscillations by varying the atomic detuningas seen in the bifurcation diagram of the output field against the atomic detuning.It is noted that, the irregular oscillations that appear in the output field component|x+| outside the RWA vanish for small atomic detuning.

For low atomic detuning parameter, the left side of the closed hysteresis loopof OB exhibits chaos, period quadruplicating, period doubling and single period bydecreasing the magnitude of the incident field amplitude, while by increasing theincident field amplitude (on the right side of the closed loop OB) quasiperiodicity isexhibited by the system in both output field components. These results are similarto that obtained in Ref. 29 for the fundamental output field component |xo| withinRWA. The specific feature outside the RWA is the phase of the time series of theadditional first harmonic field component |x+|. This phase difference is visible inthe phase space portrait as well as in the Poincare map (regarding locations ofdots) while the power spectrum and bifurcation diagram essentially have similartopography.

Acknowledgment

Y. A. Sharaby acknowledges the financial support of the Deanship of Scien-tific Research at Salman bin Abdulaziz University under the research project28/J/1433h.

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