Critical slowing down with bistable higher harmonics

Optics Communications 311 (2013) 389–396

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Critical slowing down with bistable higher harmonics Yasser A. Sharaby a,1, S.S. Hassan b,c, A. Joshi d,n,2 a

Suez University, Faculty of Applied Sciences, Physics Department, Suez, Egypt University of Bahrain, College of Science, Department of Mathematics, P.O. Box 32038 Bahrain School Of Computing, Maths&Digital Technology, Manchester Metropolitan University, Manchester M1 5GD, UK d Department of Physics, Eastern Illinois University, Charleston, IL61920, USA b c

art ic l e i nf o

a b s t r a c t

Article history: Received 3 March 2013 Received in revised form 27 July 2013 Accepted 15 August 2013 Available online 30 August 2013

Switching response in an optical bistable model of two-level atoms in a ring cavity is investigated outside the rotating wave approximation (RWA) in the high- and low-Q cavity cases. Analytical and numerical investigations of the non-autonomous model Bloch equations, up to first Fourier harmonics, show that the switching time in response to linear perturbation of the incident field at the critical points of the bistable curves is significantly affected by the atomic and cavity detuning parameters. The faster oscillatory behavior outside the RWA reflects itself in the additional ultra-low output (first harmonic) field component, which has reversed bistable feature in both the high-Q and low-Q cases. Irregular oscillations with increased atomic detuning are showed only in the lower bistable branch of the first harmonic field. Irregularity of the oscillations is due to the interference of the oscillations of the higher frequency terms with the atomic dispersive polarization in the high-Q case, and the Rabi oscillations in the low-Q case. & 2013 Elsevier B.V. All rights reserved.

Keywords: Optical bistability Critical slowing down Rotating wave approximation Switching time

1. Introduction Optical bistable devices are of great importance for all optical information processing applications [1]. For the simple model of optical bistability with two-level atomic systems [2] placed in an optical cavity, various studies have been the subject of interest, because of its potential applications in all optical switches, memories, transistors and logic circuits [3–5]. These studies show that one can control the bistable threshold intensity and the hysteresis loop via many approaches, such as field induced transparency [6], phase fluctuations [7,8], squeezed state field [9–11], spontaneously generated coherence [12–15], atomic cooperative parameter [16] and the intensity of the microwave field [17]. Effects of the squeezed vacuum field on the optical bistable behavior were studied in both Fabry-Perot and ring cavity configurations filled with two-level atomic systems [10,18–22]. In addition, transverse field variations and inhomogeneous broadening effects have been examined in the ring cavity case in the presence

n

Correspondence author. Tel.: þ 1 2175815950. E-mail addresses: [email protected] (Y.A. Sharaby), [email protected] (S.S. Hassan), [email protected], [email protected] (A. Joshi). 1 Alternative address: Salman bin Abdulaziz University, College of Arts and Sciences, Mathematics Department, Wadi Addwasir, Saudi Arabia. 2 Present address: IQSE, Texas A & M University, College Station, TX 77843, USA. 0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.08.041

of squeezed vacuum field [23]. Very recently, dispersive switching effects that describe the relationship between the output field and atomic detuning for different OB and mesoscopic systems have been investigated in [24]. The forementioned works [1–24] were based on rotating wave approximation (RWA) which neglects the faster oscillatory terms (terms that oscillate at twice the driving field frequency), and usually valid in the near resonance and weak field coupling limits (cf. [25]). Specifically, with non-dissipative systems, RWA is valid for Ω oog(Ω ¼Rabi frequency of the driving field, g is the coupling constant), while with dissipative systems it is valid for γ oo ω (γ ¼ characteristic decay constant, ω is the transition (or carrier) frequency). Beyond the RWA regime, with non-dissipative systems the coupling constant (g) is of order or greater than the Rabi frequency (Ω) and with dissipative systems the transition frequency (ω) is not that small relative to the atomic damping constant (γ ). Beyond the RWA, mesoscopic multistable systems have displayed novel knotted hysteresis cycles (butterfly bistability) in the additional first harmonic component of the output field ([26] and references therein). Recently, we have investigated the optical bistable system consisting of homogeneously broadened two-level atoms placed inside a ring cavity outside the RWA where reversed (clockwise) and butterfly (closed loop) hysteresis structures [27] were predicted for the first harmonic output field component. Inclusion of atomic inhomogeneous broadening and transverse field features could further control the first harmonic field component to exhibit a one- or two-way switching processes [28].

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Such hysteresis shapes [27,28] have been shown in different systems within the RWA theoretically [29–33] and experimentally [34]. The smallness of the first harmonic output field component according to our model [27, 28] suggests the physical possibility to generate pico-switching optical bistability intensity desired for ultra-low output intensity operation in optical signal processing [35]. Furthermore, inverted mushroom diagram which is consisting of joined clockwise and anti-clockwise hysteresis cycles has been obtained for optical bistable system outside RWA was comprising of two different species of two- level atoms placed in single ring cavity [36]. Similar behavior [36] has also been obtained for double cavity system containing three-level atomic systems with various controllable nonlinear dynamics within RWA [37]. The interaction of the double cavity fields with the homogeneously broadened three-level atomic medium occurs such that the two fields experience feedback via the two independent ring cavities [37]. The transient behavior in optical bistability is very important for technical purposes, where the variation of the switching time as a function of the system parameters can have a physical significance. In optical bistability, it is well known that the switching time goes to infinity when the bistable system is close to the critical point (critical slowing down, CSD). This effect was examined earlier within RWA for the absorptive optical bistable system in the normal vacuum (NV) case [38] and later in the presence of the squeezed vacuum (SV) field [22]. An alternative slowing down effect was discussed by Erneux and Mandel [39] based on a non-linear evolution of the system on both sides of the critical point without adiabatic elemination scheme as in [38]. Also, effect of periodic perturbation of the input field amplitude in dispersive OB system near a critical point shows that CSD can be utilized for dynamical stabilization [40]. In this paper we extend our studies in [22,27] to investigate the critical slowing down phenomenon for optical bistable device that operates with homogeneously broadened two-level atomic system in a ring cavity outside the RWA. A main result is that, the fast oscillatory terms outside the RWA regime brings irregular oscillations in the lower branch of the reversed bistable curve of the first harmonic output field, due to interference with dispersive polarization or Rabi oscillations. We examine the switching response of the optical bistable device within the plane wave approximation in the high-Q and low-Q cavity cases. The paper is organized as follows: A review of our model is presented in Section 2. The switching response of the optical bistability in both high- and low-Q cavity cases are examined in sections 3 and 4 respectively. Summary and conclusion are given in Section 5.

2. Model review Consider a single mode ring cavity containing a homogeneously broadened two-level atomic medium of length L. The c-number model Maxwell–Bloch equations, in the plane wave and mean field approximations, and in a rotating frame at ωL , outside the RWA, have the dimensionless form [27] (outline of derivation is given in Appendix (A)), h i pffiffiffi dx ¼ κ Yð1 þ iθÞx þ 2 2Cr ; dt

ð1aÞ

 n d γ γ d r  ¼  ð1 þ iδÞr  þ pffiffiffir 3 ðx þ xn eiηt Þ ¼ rþ ; dt dt 2 2

ð1bÞ

i d γ γ h r 3 ¼  ð1 þ 2r 3 Þ pffiffiffi r þ ðx þ xn eiηt Þ þ c:c: : dt 2 2 2

ð1cÞ

The notations in Eq. (1) are as follows: r 7 are the mean values of the quadrature atomic polarization components, r 3 is the mean value of the atomic inversion, γ is the A- coefficient of Einestein and δ ¼ 2ðωo ωL Þ=γ is the normalized atomic detuning with respect to γ , ω0 is the atomic transition frequency, ωL is the input field frequency, and η ¼ 2ωL . The quantities x and Y are the normalized output and input field amplitudes, respectively, θ ¼ ðωc ωL Þ=κ is the normalized cavity detuning with ωc ¼ cavity mode frequency, and κ ¼ cavity decay constant and C ¼ g 2 =ðγ κ Þ is the cooperative parameter, where g is the coupling between the cavity field and the atoms and (c.c.) stands for the complex conjugate. The terms containing e 7 iηt ð ¼ e 7 2iωL t Þ in Eqs. (1b) and (1c) represent the effect of interaction of the cavity field with the atoms outside the RWA. Within RWA these terms are discarded and equations (1) yield the well-known input-output field steady state equation [41, 42], " ! !# 2C 2C δ Y ¼ xo 1 þ θ  þi ; ð2Þ 2 2 1 þ δ þ jxo j2 1 þ δ þ jxo j2 where, xo is the fundamental output field amplitude within the RWA. The solutions for the atomic Bloch components r 8 ;3 ðtÞ and the cavity field x according to equations (1) contain all harmonics of frequency 2ωL n (n ¼ integer), due to the presence of the harmonic coefficients e 7 iηt . Following our analysis in [26,27] the field and atomic variables are decomposed up to first harmonic (n ¼ 71) as follows, iη t r 8 ;3 ðtÞ ¼ r o8 ;3 ðtÞ þ r þ þ r 8 ;3 ðtÞeiηt ; 8 ;3 ðtÞe

ð3aÞ

xðtÞ ¼ xo ðtÞ þ x þ ðtÞeiηt þx ðtÞeiηt ;

ð3bÞ

are the fundamental field and atomic variables where, xo and components, respectively, within the RWA and the corresponding components x 8 , r 7 8 ;3 are the generated ones outside the RWA. Note that in Eq. (3b) the expansion of the cavity field into three field components xo; 7 ðtÞ implies that they circulate the cavity with equal speed outside the medium as well as inside the medium. This is not a bad assumption since the speed of the electromagnetic field outside the medium is independent of the frequency of the field. Within the active medium the three components have different speeds due to dispersion which is not significant for small medium length. Using Eqs. (3a,3b) into Eqs. (1a–1c) and comparing the coefficients of e 7 inηt (n ¼ 0,1) we reach the following system of ODE for various atom and field harmonic amplitude components, r o8 ;3

d o γ γ  r ðtÞ ¼  ð1 þ iδÞ r o ðtÞ þ pffiffiffi r o3 ðtÞðxo ðtÞ þ xnþ ðtÞÞ dt  2 2  n  d o  n r þ ðtÞ ; þ r 3 ðtÞðxo ðtÞ þ x þ ðtÞÞ þ r 3 þ ðtÞx ðtÞ ¼ dt   d þ γ 2iη þ γ  r  ðtÞ ¼  1 þ iδ þ r ðtÞ þ pffiffiffi r o3 ðtÞðxno ðtÞ þ x þ ðtÞÞ dt γ  2 2  n  d  þ n  r þ ðtÞ ; þ r 3 ðtÞðxo ðtÞ þ x þ ðtÞÞ þr 3 ðtÞxn ðtÞ ¼ dt

ð4aÞ

ð4bÞ

   d  γ 2iη  γ  n r  ðtÞ ¼  1 þiδ r ðtÞ þ pffiffiffi r o3 ðtÞx ðtÞ þ r  3 ðtÞðxo ðtÞ þ x þ ðtÞÞ dt 2 γ  2  n d r þ þ ðtÞ ; ð4cÞ ¼ dt  d o γ γ  r ðtÞ ¼  γ r o3 ðtÞ pffiffiffi ðr oþ ðtÞ þ r þ ðtÞÞðxo ðtÞ þ xnþ ðtÞÞ þ r þ þ ðtÞx ðtÞ þ c:c: 2 dt 3 2 2

ð4dÞ

Y.A. Sharaby et al. / Optics Communications 311 (2013) 389–396

  d þ iη þ γ  r ðtÞ pffiffiffi ðr oþ ðtÞ þ r þ ðtÞÞðxno ðtÞ þ x þ ðtÞÞ þ r þ r 3 ðtÞ ¼ γ 1 þ þ ðtÞðxo ðtÞ γ 3 dt 2 2

 þ xnþ ðtÞÞ þ r þ ðtÞxn ðtÞ þ r o ðtÞxn ðtÞ ¼



n d  r 3 ðtÞ ; dt

h i pffiffiffi dxo ¼ κ Yð1 þ iθÞxo ðtÞ þ 2 2Cro ðtÞ ; dt    pffiffiffi dx þ iη ¼ κ 1 þ iθ þ x þ ðtÞ2 2Crþ ðtÞ ; dt κ dx ¼ κ dt



  pffiffiffi iη x ðtÞ2 2Cr 1 þ iθ  ðtÞ :

κ

ð4eÞ

ð4f Þ ð4gÞ

ð4hÞ

Our analytical and numerical results [27,28] show that the amplitude of the additional first harmonic component x þ , is of 2 smaller order, Οðλ Þ with λ ¼ γ =2ωL compared with the fundamental field component amplitude xo and to the same order of approximation the field component x ¼ 0 and hence the dipole component r   ¼ 0. Consequently, Eqs. (4a–h) reduce to the following smaller set of ODE,  n d o γ γ d o r  ðtÞ ¼  ð1 þ iδÞr o ðtÞ þ pffiffiffir o3 ðtÞxo ðtÞ ¼ r þ ðtÞ ; ð5aÞ dt dt 2 2

The switching time of the OB device can be obtained by solving Eqs. (6a,b), numerically, in the vicinity of the incident field Y. This is carried out by replacing Y in Eq. (6a) by Y c þ β with β o 1 is a small (real) perturbation and Y c is the critical (switching-on) value of the incident field given analytically in the case of θ ¼ 0, C 4 1 by [22], Yc ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðC þ 1Þ2 þ δ

ð7Þ

which reduces to Y c ¼ C þ 1 at δ ¼ 0 [2,38]. Otherwise, for the off resonance case (δ; θ a 0) Y c is obtained numerically. In the resonant case (δ ¼ 0, θ ¼ 0), C ¼ 20, λ ¼ 106 and ν ¼ 108 (suitable for high-Q cavity), the initial conditions used in solving equations (6) are taken in the vicinity of Y c ,viz., xo ð0Þ ¼ 1, x þ ð0Þ ¼ 1  1013 . By increasing the value of the parameter β , the time delay (τ ¼ κ t) for the fundamental field component xo ðτÞ , within the RWA, to switch-up

25 20 15

x

xo

25

10

 n   d þ γ 2iη þ γ d  r  ðtÞ ¼  1 þ iδ þ r þ ðtÞ ; r  ðtÞ þ pffiffiffir o3 ðtÞxno ðtÞ ¼ dt dt γ 2 2

ð5bÞ

 d o γ γ  r ðtÞ ¼  γ r o3 ðtÞ pffiffiffi r oþ ðtÞxo ðtÞ þ c:c: ; dt 3 2 2 2

ð5cÞ

h i pffiffiffi dxo ¼ κ Yð1 þ iθÞxo ðtÞ þ 2 2Cro ðtÞ ; dt    pffiffiffi dx þ iη ¼ κ 1 þ iθ þ x þ ðtÞ2 2Crþ ðtÞ : κ dt

391

15

5

5 0

0

0

10

20

0

5

15

30

25

Y

40

50

ð5dÞ

4. 10 13 ð5eÞ

3. 10 13

Next, we investigate the time response (switching time) of the optical bistable device outside the RWA according to Eqs. (5a–e) when it jumps to the upper (lower) state from lower (upper) state by perturbing incident field (Y) in the vicinity of its critical point for different hysteresis cycles in the high-Q and low-Q cavity cases. In this work, the enhancement of the switching (fast response) in optical bistable device is achieved by increasing the intensity of the incident field via small perturbation β o 1. In our previous study [22] we have controlled the switching time within the RWA via injecting the bistable system with a squeezed vacuum field.

x

2. 10 13

x

1.2 10

13

4. 10

14

0

0 5

15

Y

25

1. 10 13 0

4. 10

13

3. 10

13

2. 10

13

1. 10

13

0

10

0

10

20

30

40

50

3. High-Q cavity case In this case where, the life time of photons inside the cavity (κ ) is much greater than the atomic lifetime (γ 1 ) and hence the atomic variables can be eliminated adiabatically from Eqs. (5a)–(5c). Accordingly, the time-dependent Eqs. (5d, 5e) for the normalized output field components xo; þ ðτÞ are of the forms, 1

dxo 2Cxo ðτÞð1iδÞ ¼ Yð1 þ iθÞxo ðτÞ ; 2 dτ ð1 þ δ þ jxo ðτÞj2 Þ

ð6aÞ

  2 dx þ iη 2Cxo n ðτÞð1 þ δ Þ ¼  1 þ iθ þ x þ ðτÞ : 2 dτ κ ð1 þ iðδ þ 2=λÞÞð1 þ δ þ jxo ðτÞj2 Þ

ð6bÞ

x

0

With τ ¼ κ t, λ ¼ γ =η and ν ¼ κ =η. Note that, the time-dependent behavior and hence the switching response of the first harmonic output field component x þ ðτÞ, outside RWA, depending on two further parameters, namely, λ and ν.

20

30

40

50

Fig. 1. (a) The transient (fundemental) output field component xo ðτÞ , within RWA versus the normalized time τ ¼ κt in the high-Q limit (ν ¼ 108 ) for C ¼ 20, θ ¼ 0; δ ¼ 0 and different values of the input field perturbation Y ¼ Y C þ β, Y C ¼ 21; β ¼ 0:03(——), β ¼ 0:04(       ) and β ¼ 0:12(    ). Inset shows the steady state curve (jxo jvs. Y). (b) As (a) but for the (first harmonic) output field 6 component x þ ðτÞ , outside RWA for λ ¼ 10 and β ¼ 0:03. Inset shows the steady state curve ( x þ vs. Y). (c) As (b) but for β ¼ 0:12.

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to the upper branch is reduced [38] as shown in Fig. 1(a). The same is also true for the first harmonic field component x þ ðτÞ as depicted in Fig. 1(b, c), generated outside the RWA but the switching occurs in the reverse way, i.e., switching-down, followed by dense oscillations over a short period of time. Variation the atomic detuning ( δ ) has a noticeable effect on the reversed steady state OB curve of x þ (Fig. 2a), while the fundamental component jxo j is hardly affected (see inset of Fig. 2a). For a fixed perturbation β ¼ 0:03 and increasing value of detuning ( δ ) the switching time increases significantly for both components jxo j, x þ (Fig. 2b–2d). The oscillation occurring in the lower branch of x þ ðτÞ at different time locations depends on

x

2. 10

13

1.5 10

13

1. 10

13

5. 10

14

the sign of δ and is due to the contribution of the atomic dispersive polarization field (Fig. 2d). Also, when δ Z 1, the sign of δ affects only the switching down location of the component x þ ðτÞ , Fig. 2(c, d). In the case of detuned cavity ( θ ¼ 4) the effect of increasing the atomic detuning ( δ ) on the steady state OB curves for both components jxo j, x þ is to decrease the switching-on and -off points (Fig. 3a), with a closed loop formation at δ ¼ 1 for the field component x þ . For non zero values of the same sign of (δ; θ) the switching time slightly increases (Fig. 3b–3d). Note, for non-zero values of (δ; θ) such that δθ o0 the bistable behavior tends to diminish in conformity with the analysis in [41], for both

4. 10

13

2.7 10

13

x 1.3 10

0

0

5

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0 0

25

20

13

20

4. 10

13

2.7 10

13

10

15

xo

30

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50

20

30

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50

1

10

x 1.3 10

5 0

20

0

10

20

30

40

13

0 0

50

10

Fig. 2. (a) The steady state output field component x þ against the incident field Y for fixed C ¼ 20, θ ¼ 0 and different values of the absolute detuning parameter jδj; atomic 0(——), 0.5(       ) and 1(    ). Inset shows the corresponding field component jxo j against Y. (b) The transient output field component xo ðτÞ versus τ in the high-Q limit for C ¼ 20, θ ¼ 0, β ¼ 0:03 and different values of the atomic detuning parameterδ; δ ¼ 0(——),δ ¼ 0:5(       ) and δ ¼ 1(    ). (c) The output field component x þ ðτÞ 6 8 against (τ) for the same data as (b), λ ¼ 10 , ν ¼ 10 and for δ ¼ 0:5. (d) As (c) but for δ ¼ 1. Inset shows the same but for δ ¼ 1.

x

1.4 10

13

1. 10

13

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14

2. 10

14

0

4. 10

13

2.5 10

13

1. 10

13

x

0

5

10

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25

0

10

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50

0

10

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30

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50

Y

6

4. 10

13

2.5 10

13

1. 10

13

5 4

xo

x

3 2 1 0

0

5

10

15

20

25

Fig. 3. (a) As Fig. 2(a) but with jθj ¼ 4. (b) As Fig. 2(b) for the field component xo ðτÞ but with jθj ¼ 4. (c) As (2c) but for the field component x þ ðτÞ at δ ¼ 0, θ ¼ 4. Inset shows the same but for θ ¼ 4. (d) As (c) but for δ ¼ 1.

Y.A. Sharaby et al. / Optics Communications 311 (2013) 389–396

components j xo j, x þ . The transient oscillations in both output field components xo ðτÞ , x þ ðτÞ with increasing (δ; θ) (Fig. 3b, 3d) is a manifestation of balancing process between the internal atomic and cavity dispersive fields. Further, for the case δ ¼ 0; θ a 0 (Fig. 3c), the irregular oscillations that appear in the lower branch for θ o 0, δ ¼ 0 turn to a single peak for δ a 0 (Fig. 3d).

4. Low-Q cavity case In this case, κ 1 oo γ 1 and hence the field variables can be eliminated adiabatically from Eq. (5d, 5e) to give, pffiffiffi Y þ2 2Cr o ; ð8aÞ xo ¼ 1 þiθ 30

xo

xþ ¼

pffiffiffi 2 2Crþ : 1 þ iðθ þ η=κ Þ

pffiffiffi   Y þ 2 2Cr oþ ðτ′Þ d þ 1 2iη þ 1 r  ðτ′Þ ¼  1 þ iδ þ ; r  ðτ′Þ þ pffiffiffir o3 ðτ′Þ dτ′ 2 γ 1iθ 2  n d  r ðτ′Þ ¼ dτ ′ þ 35

20

30

15

25

15 10

0

10

20

30

40

5

50

0 6.

4.

10

10

9

x

4. 10

8

2. 10

8

0.06

0

0 10

40

45

50

9

0 0

6.

10

9

4.

10

9

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x

30

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60

2.

10

9

50

10

9

10

9

0.12

8

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0 0

1.2

10

8

8.

10

9

4.

10

9

0

x 2.

20

9

4. 10 4.

10

x

0 6.

0

9

x 2.

ð9bÞ

20

xo

5 0

ð8bÞ

Subsitiuting Eq. (8a) for xo in Eqs. (5a–5c) we get the system dynamics governed by the following equations for the polarization 0 0 0 components r o ðτ Þ, r þ ðτ Þ and the inversion component r o3 ðτ Þ 0 where τ ¼ γ t, pffiffiffi d o 1 1 Y þ2 2Cr o ðτ′Þ r  ðτ′Þ ¼  ð1 þ iδÞr o ðτ′Þ þ pffiffiffir o3 ðτ ′Þ ; dτ′ 2 1 þ iθ 2  n d o r ðτ′Þ ð9aÞ ¼ dτ ′ þ

25

10

393

0 20

10

20

30

25

40

30

x

50

0 0 Fig. 4. (a) The transient output field component xo ðτ Þ versus the time τ in the low-Q limit (ν ¼ 104 ) for C ¼ 20, θ ¼ 0; δ ¼ 0 and different values of the input field perturbation Y ¼ Y C þ β, Y C ¼ 21; β ¼ 0:06(——), 0:08(       ) and β¼ 0 0 β ¼ 0:12(    ). (b), (c) The output field component x þ ðτ Þ against (τ ) for the 6 same data as (a), λ ¼ 10 and β ¼ 0:06; 0:12, respectively. Insets show zooming of the curves for time t A ½40; 50 and for time t A ½20; 30.

0

50

60

70

0 Fig. 5. (a) As Fig. 4(a) but with jδj ¼ 1:4. (b) The output field component x þ ðτ Þ 0 6 against (τ ) for the same data as (a) with λ ¼ 10 and β ¼ 0:06. Inset shows the same but for δ ¼ 1:4. (c) As (b) but for β ¼ 0:12.

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" # pffiffiffi d o 1 o 1 Y þ 2 2Cr o ðτ′Þ o r ðτ′Þ ¼  r 3 ðτ′Þ pffiffiffi r þ ðτ′Þ þ c:c: : dτ ′ 3 2 1 þ iθ 2 2

ð9cÞ

Eqs. (9a–9c) are numerically solved for r o ðτ′Þ, r þ ðτ′Þ with initial conditions in the vicinity of Y c and then the output field components xo; þ ðτ′Þ are evaluated from Eqs. (8a,8b). In the resonant case (δ ¼ 0, θ ¼ 0), the optical bistable device with low-Q cavity (ν ¼ 104 , Fig. 4), switches slower than that for the high-Q cavity (ν ¼ 108 , Fig. 1a and b) within and outside the RWA. Further, the system oscillates strongly in the upper (lower) branch of the fundamental (first harmonic) component of the output field xo ðx þ Þ. This strong Rabi oscillations are simply damped for the xo ðτ′Þ component (Fig. 4(a)) within the RWA, but

40

30

xo

20

10

0

6.

10

0

10

20

30

40

are irregularly damped for the x þ ðτ ′Þ component (Fig. 4b and 4c), outside the RWA. The irregularity of the Rabi oscillations for the ðx þ Þ field component can be explained as follows. In this case of low-Q cavity, the first harmonic output field component x þ follows adiabatically the corresponding polarization component r þ (see Eq. (8b)), which oscillates at the higher frequency (δ þ2η=γ ), (Eq. (9b)). In addition to that the interference occurs between this higher frequency and the Rabi oscillations inherent in the fundamental atomic inversion and dipole components r o3; 7 ðτ′Þ- as evident in the second term of Eq. (9b). For non-zero atomic detunings (δ a 0, θ ¼ 0) and larger perturbation (β ), the switching time for the output field component jxo ðτ′Þj increases for both positive and negative values of δ (Fig. 5a). The similar behavior is shown by the component x þ ðτ′Þ but with oscillations in the lower branch are more pronounced for small β, δ 4 0 (Fig. 5b) compared to the larger β, δ o0 (Fig. 5c). For the detuned cavity (θ; δ a0, Fig. 6), the switching times for both field components jxo ðτ′Þj, jx þ ðτ′Þj are much reduced with suppression of Rabi oscillations with larger θ ¼ 2. When δ; θ have opposite signs (i.e., when cavity frequency and the center of the atomic line lie on opposite sides with respect to the of the incident wavelength field), the field output component x þ ðτ′Þ shows reduced oscillations in the lower branch (Fig. 6b), as compared to Figs. (5b, 5c), and for larger δ; θ oscillations are completely suppressed (Fig. 6c). The suppression of oscillations is a result of increased cavity detuning.

5. Summary and conclusion We have investigated the switching response of the optical bistable system consisting of homogeneously broadened two-level

9

4. 10 4.

10

9

10

9

12

x

Κ 0.102 2.

0

x

0

10

20

30

40

50 0 0

6.

10

4. 10 4.

10

20

30

40

50

9

10

9

10

9

12

x

Κ 0.102 2.

x 0

0

20

40

60

80

100

0 Fig. 6. (a) The transient output field component xo ðτ Þ versus the time normalized 0 τ ¼ γt in the low-Q limit (ν ¼ 104 ) for C ¼ 20, jδj ¼ 1, β ¼ 0:06 and different values jθj ¼ 0:5(——) and jθj ¼ 2(    ). (b) The output field of the cavity detuning, 0 0 component x þ ðτ Þ against τ for the same data as (a), λ ¼ 106 and for δ ¼ 1,θ ¼ 0:5. Inset shows the same but for δ ¼ 1,θ ¼ 0:5. (c) As (b) but for θ ¼ 2. Inset shows the same but for δ ¼ 1, θ ¼ 2.

0

0

10

20

30

40

50

Fig. 7. (a) As Fig. 3(d) but with β ¼ 0:06, κ ¼ 0:102. (b) As (a) but in the nonadiabatic (full model) case.

Y.A. Sharaby et al. / Optics Communications 311 (2013) 389–396

atoms placed in a ring cavity and interacting with a single cavity mode outside the RWA. The ultra-low first harmonic output field component x þ outside the RWA and the fundamental componentxo within the RWA switch simultaneously for the same driving field value. Our main results are (i) In high-Q cavity case, the switching time (time delay) for both field components xo ; x þ within and outside the RWA, respectively, decreases with increased perturbation of the input field in the vicinity of its critical (switching-on) value. For non-zero atomic and cavity detunings (δ; θ a0) of the same sign at a fixed perturbation, the switching time increases. The irregular oscillations in the field component x þ due to atomic and cavity dispersive fields show in the lower branch at different time locations as (δ; θ) vary. (ii) In low-Q cavity case, the switching time is longer than that in the high-Q case and the stronger (Rabi) oscillations appear in the lower branch of the x þ component in the absorptive case (δ; θ ¼0). The irregularity in the Rabi oscillations is caused by interference of the higher frequency terms outside the RWA with the (regular) Rabi oscillations inherent in the fundamental atomic variables within the RWA. Finally, we comment on the results obtained here, namely, in the adiabatic limits. Conditions of validity of adiabatic elimination in nonlinear dynamical systems, which depend on selection of system parameters among other criteria, have been investigated at large in [43]. Within the contest of our model we affirm that the adiabatic elimination of atomic or cavity variables (high- and low-Q cavity cases) are not responsible to any overall change in the switching time, nor to the irregular oscillations in the x þ -field component. This has been confirmed through the numerical solution of the full (nonadiabatic) model equations (5), where for low-Q case data we have the same reported results of Section 4 while for high-Q data we obtained the same qualitative and closely quantitative results of Section 3. Some results for high-Q case (κ ¼ 0:102) show that irregular oscillations in the x þ -component are obtained with both adiabatic Eqs. (6a and 6b) and non-adiabatic Eqs. (5a–5e) systems for the same set of system parameters (Fig. 7).

395

þ while the cavity mode operators a^ and a^ are boson operators þ obeying the algebra ½a^ ; a^  ¼ 1, with g, κ o are coupling constants. The first and second terms in (A.1) describe the Hamiltonian of the unperturbed system of atomic system and field, respectively. The third term in (g) is the interaction Hamiltonian of the atomic system with the single cavity field mode outside the RWA. The fourth term in (κ o ) describes the interaction Hamiltonian (within the RWA) of the cavity field mode with the input field (Eo ). The last two terms in (A.1) describe the dissipative processes for both cavity field and atomic system, respectively, and given by   ^ i c þ a^ Γ ^ iþ ; ^ 1 ¼ ∑ a^ þ Γ ðA:3Þ H c i

  ^ k þ s^  Γ ^ kþ ^ 2 ¼ ∑ s^ þ Γ H R R

ðA:4Þ

k

^ ic; Γ ^ i þ ) and (Γ ^ k ;Γ ^ k þ ) are boson reservoir operators for the With (Γ R c R vacuum modes of cavity field and atomic reservoirs, respectively. Note that, according to the model Hamiltonian (A.1) interaction of the cavity field within the active medium is outside the RWA where faster oscillatory terms due to the interacting terms (a^ s^  ) and þ (a^ s^ þ ) are not discarded, while the interaction of cavity field mode with the input field (outside the active medium) is within the RWA. The usual procedure (e.g., [44]) to trace over field and atomic reservoirs in the normal vacuum state leads to the following master equation for the reduced density operator ρ of the system, h i h i ^ ρ þg ða^ þ a^ Þðs^ þ þ s^  Þ; ρ ρ_ ¼ i ωo s^ z þ ωc a^ þ a; h

i h i h i

þ ^ ρa^ þ ^ iωL t þ κ a^ ρ; a^ þ þ a; þ κ o Eo ða^ eiωL t ae

γ

þ ð2s^  ρs^ þ s^ þ s^  ρρs^ þ s^  Þ: 2

ðA:5Þ

where γ ; κ are the atomic and cavity decay constants, respectively. The master equation (A.5) reduces to earlier form [41] in the RWA when phase decay processes in the RWA are ignored.

Defining the c-number expectation values: r 3; 7 ¼ s^ z; 7 , α ¼ a^ and the normalized output x ¼ 2g α=γ 2 and input Y ¼ 2gEo =γ 2 fields and within the semiclassical (decorrelation) approximation we arrive from equation (A.5) to the system of equations (A.1) in rotating frame of frequency ωL and have taken κ o ¼ κ (without loss of generality).

Acknowledgments References Y. A. Sharaby acknowledges the financial support of the Deanship of Scientific Research, Salman bin Abdulaziz University (Project no. 34/h/1432). A. Joshi acknowledges the support of RCSA for this work. S. S. Hassan acknowledges the hospitality of L. Nejad at MMU (July–August, 2013) where the final version of this manuscript was prepared.

Appendix A Here we present an outline of derivation of Eq. (1). The model Hamiltonian of homogeneously broadened two-level atomic system of transition frequency ωo interacting with a single (ring) cavity mode of frequency ωc outside the RWA, which in turn is driven by a coherent input field EðtÞ ¼ Eo eiωL t , of real constant amplitude Eo and frequency ωL is given by (in units of ℏ ¼ 1) (cf [41] in the RWA), ^ ¼ ωo s^ z þ ωc a^ þ a^ þ igða^ þ aÞð ^ s^ þ þ s^  Þ H þ iωL t i ω t ^ 1 þH ^ 2: L ^ ae ÞþH þ iκ o Eo ða^ e

ðA:1Þ

The atomic (density) operators s^ z; 7 are the spin 1/2 Pauli operators obeying the SU(2)-algebra, ½s^ z ; s^ 7  ¼ 7 s^ 7 ; ½s^ þ ; s^   ¼ 2s^ z ;

ðA:2Þ

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