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Self-pulsing and chaos of a two-photon optical bistable model in a ring cavity
Chaos, Solitons and Fractals 28 (2006) 590–600 www.elsevier.com/locate/chaos
Self-pulsing and chaos of a two-photon optical bistable model in a ring cavity S.M.A. Maize Physics Department, Faculty of Science, Menoufia University, Shebin El-Kom, Egypt Accepted 19 April 2005
Communicated by Prof. G. Iovane
Abstract Stability analysis and numerical investigation of the periodic and chaotic behavior are presented for two-photon optical bistable system where two sorts of atom are placed in a ring cavity. The effect of the cooperative parameter ÔCÕ on the state of the system is examined and a wealth of new dynamic characteristics was observed including: period doubling sequence to chaos, complex bistability and strange attractors evolved from torus bifurcation sequence. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Developments in nonlinear dynamics, in particular ideas about deterministic chaos and pattern formation have had implications in the understanding of complex behavior in nonlinear optical systems [1]. It has been demonstrated that optical chaos is not bad for all applications; it may be ideally suited for communicating information at high data rates [2]. Moreover, it is believed that chaos can be employed for a variety of applications, many of which are yet to be discovered. A recent work about stability analysis and chaos for one photon optical bistable (OB) model [3] in contact with a squeezed vacuum reservoir [4] is carried out. The study [3,4] generalizes earlier work on the same model in normal vacuum (see [5] and references therein). On the other hand, a two-photon OB model that utilizes two sorts of atom in a ring cavity has, been studied by Thompson and Hermann [6]. The main feature of the steady state input-output is the possibility to get tristability where five solutions exist for each input field value. They illustrate that the realistic model can be a mixture of two metal vapours (like Potassium has an 8d level lying 1.67 cm1 below the 16d level of rubidium at around 33 180 cm1). Here, in the light of recent advances in fundamental knowledge and techniques this system [6] is revisited on a higher level of computational technique and analysis of the results. A wealth of new dynamic features is discovered which makes the results of earlier investigation [7] along the usual analytical treatment of linear stability analysis [3,7] not more than a scratch on the surface of the problem. The description of the system of interest is given in the following section.
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.100
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2. The model Maxwell–Bloch equations The system of interest here consists of a ring cavity containing atoms with doubly degenerate working level. This degeneracy may be achieved by choosing a mixture of two different sorts of atom [6], as shown in Fig. 1. We assume that cross-linking of the states is forbidden by selection rules and that therefore the intermediate states for the two types of transition form disjoint sets. We associate a Bloch vector ÔrÕ with a–b transition and ÔsÕ with c–d transition. These vectors turn at different rates because the two-photon matrix elements Kab and Kcd differ in value. The equations of motion of ÔrÕ and ÔsÕ can be written as [6] drþ 2ijk ab j 2 k bb k aa þ 2 E r3 i ¼ jE j D rþ c? rþ h dt h dr3 ijk ab j 2 2 rþ Eþ r E ck ðr3 þ 1Þ ¼ h dt ð1Þ dsþ 2ijk cd j 2 k dd k cc þ 2 E s3 i ¼ jE j D0 sþ f? c? sþ h dt h ds3 ijk cd j 2 2 sþ Eþ s E fk ck ðr3 þ 1Þ ¼ h dt The electric field is chosen to be in the form E ¼ Eþ eiðxtkzÞ þ E eiðxtkzÞ
ð2Þ
Self-consistently the polarization ÔPÕ takes the same form as the field ÔEÕ in (2) where the field and polarization amplitudes and phases are slowly varying in ÔtÕ and ÔzÕ [6], so MaxwellÕs equation reduces to oEþ 1 oE x þ þ ¼i P cv ot 2e0 cv oz
ð3Þ
where [6,7] 1. kaa, kbb, kab, kcc, kdd and kcd have the common definition for the two-photon transition k aa ¼
2 X 2 xja laj ; h j x2ja x2
k bb ¼
2 X 2 xjb lbj ; h j x2jb x2
k ab ¼
1 X lja lbj ; etc. h j x þ xjb
2. the atomic detunings D = xba 2x and D 0 = xdc 2x 3. ck, c?, fkck, f?c? are the longitudinal and transverse decay rates appropriate to an electron from the jai and jci state respectively. We define the weights of the two sorts of atom expressed by the Bloch vectors ÔrÕ and ÔsÕ to be ÔW1Õ and ÔW2Õ respectively. The expression of the polarization and hence the positive frequency component ÔP+Õ can be substituted in MaxwellÕs equation (3) to give " sffiffiffiffiffi !# sffiffiffiffiffiffiffiffiffiffi oEþ 1 oEþ x N W1 c? þ W2 f? c? þ jk ab j s3 E þ s E r3 E þ r E þ þ ¼i d1 q d2 ð4Þ cv ot 2e0 cv V oz W1þW2 ck W1 fk ck
Fig. 1. The optical ring cavity and the energy level scheme for the two-photon two sorts of atom mixture.
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where q = jkcdj/jkabj, the stark term d1 ¼ kbb2jkkabaaj
qffiffiffiffi ck c?
and d2 has a similar definition with respect to c–d transition.
2jk ab j ffiffiffiffiffiffiffi Eþ to Maxwell–Bloch equations (1) and (4), we get By introducing the normalized field nþ ¼ hp ck c? þ 2 pffiffiffiffiffiffiffiffiffi 2 drþ ¼ i ck c? n r3 i d1 c? n D rþ c? rþ dt pffiffiffiffiffiffiffiffiffi ck c? þ2 dr3 ¼i rþ n r n2 ck ðr3 þ 1Þ dt 2 ! pffiffiffiffiffiffiffiffiffi 2 dsþ f? c? þ 2 0 ¼ iq ck c? n s3 i qd2 n D sþ f? c? sþ dt fk ck pffiffiffiffiffiffiffiffiffi ck c? 2 2 ds3 ¼ iq sþ nþ s n fk ck ðs3 þ 1Þ dt 2 " sffiffiffiffiffi !# sffiffiffiffiffiffiffiffiffiffi onþ 1 onþ x N W1 c? þ W2 f? c? þ jk ab j s3 n þ s n r3 n þ r n þ d1 q d2 þ ¼i cv ot 2e0 cv V oz W1þW2 ck W1 fk ck
ð5Þ
3. The self-pulsing within the mean field limit Here we consider the case of the fast relaxation of the transverse Bloch components, i.e., the transversal relaxation time is considered to be the shortest among all characteristic times in the problem (c?, f?c? ck, fkck, k, . . .). Consequently, the Bloch components r+ and s+ take their stationary values. After this adiabatic elimination of r+ and s+, we integrate the field equation in (5) with respect to z within the mean fieldRlimit (valid for T 1 where T is the transmisL sivity of the ring cavity mirror) and the field takes its average value L1 0 EðzÞdz [8]. And use the boundary conditions of the ring cavity [3–7] Eð0; t0 Þ ¼ TY þ REðL; t0 Þ expðihT Þ and
ET ðtÞ EðL; tÞ ¼ pffiffiffiffi ¼ X T
where h is the normalized cavity detuning, the set of Eqs. (5) is reduced to the following set of differential equations: " ! # ck dr3 jX j4 ¼ þ 1 r3 þ 1 dtn j 1 þ X21 " ! # fk ck ds3 jX j4 ¼ þ 1 s3 þ 1 dtn j B þ X22 " ! ! # pffiffiffi 2 du c? A f? c? BjX j jX j2 jX j2 X1 jX j2 X 2 p ffiffiffi r ¼ Y u þ hv þ 2C u þ v þ vd þ þ v þ v d u 1 3 2 s3 dtn ck fk ck 1 þ X21 1 þ X21 B þ X22 B þ X22 B " ! ! # pffiffiffi 2 BjX j dv c1 A f? c? jX j2 jX j2 X1 jX j2 X 2 r3 þ pffiffiffi v ¼ v hu þ 2C v u ud1 u u d2 s3 ck fk ck dtn 1 þ X21 1 þ X21 B þ X22 B þ X22 B
ð6Þ
pffiffiffiffi pffiffiffiffi where Y (the input field) ¼ nI = T , X (the output field) ¼ nþ ðL; t0 Þ ¼ nT = T ¼ u iv, tn (the normalized time) = jt 0 , j (the cavity decay time) = cvT/Lt, Lt is the length of the ring cavity, t 0 = t + (Lt L)z/(cvL); L is the length of medium inside the cavity. qffiffiffiffi c The cooperative parameter C ¼ 2TL 2N0 cxv V jk ab j c?k ; A ¼ fk WW 21 and B ¼ fk f? q2 . The set of equation (6) represents lumped system, which can be solved numerically and computationally by the following technique.
4. Numerical tools and presentation techniques The numerical investigation of the dynamic characteristics of the model is carried out using principles of bifurcation theory coupled with continuation techniques. The continuity diagrams are obtained using the software AUTO [9]. This package is able to perform both steady state (static) and dynamic bifurcation analysis, including the determination of
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the entire periodic solution branches. AUTO also computes the Floquet multipliers [10] along periodic solution branches and therefore determines the stability of the periodic orbit [11]. Because of periodicity there is always a Floquet multiplier equal to +1. When all Floquet multipliers lie inside the unit circle, the periodic solution is asymptotically stable. Complex Floquet multipliers indicate that the periodic solution behaves like a focus, while real Floquet multipliers denote node or saddle-like behavior. Periodic attractors can lose their stability by a number of ways. The most important ones are period doubling and torus bifurcation. A Floquet multiplier leaving the unit circle through -1 indicates a period doubling bifurcation, while passage of the complex Floquet multipliers out of the unit circle indicates that the periodic orbit bifurcates to invariant torus. The DGEAR subroutine (1986) with automatic step size to ensure accuracy for stiff differential equations is used for numerical simulations of point as well as periodic attractors and accuracy up to 1015 is attainable [12].
5. Results and discussion The continuity diagram for the two-photon ring-cavity model (the input–output field) is shown in Fig. 2 for a set of model parameters: D1 ¼ 60;
D2 ¼ 70;
h ¼ 80;
C ¼ 110;
A ¼ 3;
B ¼ 81;
ck =j ¼ 1:5;
f k ck =j ¼ 2:25;
d1 ¼ d2 ¼ 0
This diagram show two stable static branches (solid line) connected by three unstable branches (dashed line) through four limit points. Moreover, this diagram contains two Hopf bifurcation points [13], which are denoted by the squares on the curve. The presence of Hopf points indicates necessarily the existence of periodic behavior of the system. The Hopf point arises when a pair of complex eigenvalues of the Jacobean of the model crosses the imaginary axis transversal. It is noticed that the two Hopf bifurcation points are almost degenerate Hopf where the transversality condition of the definitions of the Hopf bifurcation point is violated (the conditions for the crossing of the imaginary axis of the eigenvalues) that is d Re½kðY Þ ¼0 dY Y0 which means that the derivative of the real part of the eigenvalue with respect to the control parameter Y is equal to zero at this Hopf point. The continuity diagram also presents periodic behavior which loses its stability and starts a period doubling bifurcation scenario at Y = 16.9564 (this scenario will be analyzed later). This periodic branch terminates homoclinically [13] almost at Y = 7.76 as shown in Fig. 3 where the period goes to infinity almost at this value of Y. The two-parameter continuation technique, which we can get by using the AUTO package, will help in exploring the system under consideration. The importance of the two-parameter continuation comes from its ability to give the answer of some arising questions like: where does the periodic branch start? . . . which points are degenerate Hopf points? It is important to know the answer of the last question because it is known that we can often find rich dynamics close to the degenerate Hopf points. Figs. 4a and b represent the two parameter continuation diagram between the cooperation parameter C and the input field Y. The figure shows the loci of the Hopf bifurcation points (solid line) and the loci of
Fig. 2. Bifurcation diagram X versus Y where (- - -) represents the unstable static branch, (—) represents the stable static branch, (h) represents the Hopf bifurcation points; the periodic branch is the unstable positive slop branch .
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Fig. 3. The period versus the input field Y.
Fig. 4. (a) Two-parameter continuation diagram C versus Y. (b) Enlargement of the first part of (a).
static limit point (dashed line). Fig. 4a answers the first question and determines the starting point of the periodic solution of Fig. 2 at Y = 428.02 (the base case where C = 110). Fig. 4b is an enlargement of the first part of the curves in Fig. 4a where we can see two kinds of degeneracy. The first one at the coincidence of Hopf points loci to the static limit points loci and the second at the intersection of the loci of Hopf points and the loci of the static limit point at C = 50.5. We are going to study the two types of degeneracy by choosing deferent values of C. 5.1. The base case C = 110 In all the results presented the input field Y is used as the control parameter (bifurcation parameter) and all presentation techniques discussed before are utilized to investigate the rich dynamic behavior of the system. Fig. 5 shows the continuity diagram at C = 110. Three Hopf bifurcation points characterize this diagram. Two of them are almost degenerate Hopf. The third Hopf at Y = 427.986 is the starting gate to periodic solution. The periodic branch which is emerged from this point loses its stability at Y = 16.9564 through period doubling and also shows chaotic behavior. This bifurcation to chaos is investigated by using one-dimensional Poincare bifurcation diagram as shown in Fig. 6. The following characteristics are evident: (i) With decreasing Y the periodic branch with P1 (single period) which originated from the third Hopf bifurcation point goes through the sequence of period doubling (P1, P2, P4, P8, . . .) leading to chaos. This chaotic behavior goes under a number of bifurcations from chaos to periodicity and vice versa. (ii) The chaotic behavior vanishes where P4 is observed and bifurcate to P8 (the four bulbs which are shown in the figure). (iii) The periodic branch terminates homoclinically at Y = 6.95.
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Fig. 5. Bifurcation diagram at C = 110.
Fig. 6. 1D Poincare bifurcation diagram at C = 110.
5.1.1. Tristability Regarding to Fig. 2 we have more than one solution at the same value of the bifurcation parameter. Fig. 7 illustrates the phase plane at Y = 5 where two stationary non-equilibrium states coexist in the same state space. Fig. 8 shows another kind of bistability where one stationary non-equilibrium state coexist with period one attractor (P1) at Y = 20. Figs. 9a and b represents the complex bistable solution where two stationary non-equilibrium states coexist with a chaotic solution. Fig. 9a interprets how complex the trajectory approaches the two stationary points. Here we can see two reasons for the sensitivity to the initial conditions. One of them is due to the existence of more than one solution sharing the state space whereas the second one is due to the existence of the chaotic behavior. Fig. 10 shows the dynamic characteristics of a chaotic attractor at Y = 11.5. Fig. 10a represents the time trace. Fig. 10b shows the phase plane (u versus v). Fig. 10c shows another phase plane (u versus X). Fig. 10d shows 2D Poincare map. 5.2. The cooperative parameter C 110 Here we take the two cases of C = 130 and 150. Figs. 11a shows the continuity diagram at C = 150 performed by AUTO software, which indicates period doubling at Y = 19.661. Fig. 11b shows the one dimensional Poincare map for Y 2 (20, 8.3). It is clear that the figure is characterized by incomplete period doubling sequence and no more than period P4 is present in this region. The sequence is P1, P2, P4, P2, and P1. The periodic branch terminates homoclinically almost at Y = 8.3. When the cooperative parameter C decreases to 130 two banded chaos, in the region Y 2 (18.1069, 7.5906), are observed and this chaos bifurcate by period halving scenario to P1 attractor which terminate also homoclinically at Y = 6.728 (As shown in Fig. 12).
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Fig. 7. Phase plane at Y = 5 and C = 110 for the two steady state bistable case.
Fig. 8. Phase plane at Y = 20 and C = 110 for the case of stable steady state coexist with stable limit cycle.
Fig. 9. Phase planes of bistable case where two stable steady state are coexist with chaotic attractor. (i)The trajectory approaches the two stable steady state only. (ii) The same as (a) but includes the chaotic attractor.
5.3. The cooperative parameter C 110 At C = 90 the continuity diagram is shown in Fig. 13a where we can see the periodic branch has originated from the highest Hopf point (the Hopf point which correspond to the highest value of the control parameter Y). This periodic
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Fig. 10. The dynamic characteristic of the chaotic attractor at Y = 11.5 and C = 110: (a) time trace, (b) phase plane ÔvÕ versus ÔuÕ, (c) phase plane ÔXÕ versus ÔuÕ and (d) 2D Poincare map.
Fig. 11. (a) Bifurcation diagram at C = 150. (b) 1D Poincare map of (a).
branch loses its stability at Y = 16.7337 and goes under a number of bifurcations from chaos to periodicity and vice versa until it terminates homoclinically at almost Y = 10.47 (as shown in Fig. 13b). The Poincare bifurcation diagram is characterized by a boundary crises at Y = 14.25, where the chaotic attractor changes its shape due to the collision by an unstable periodic attractor [14]. When C decreases to 40 (Fig. 14), which is very close to the second type of degenerate Hopf points (the intersection between the loci of Hopf points and the loci of the static limit point), it is clear that the bifurcation scenario is totally different from the previous cases where other strange attractors are observed. The periodic branch originated from the highest Hopf point loses its stability at Y = 36.7195. Fig. 14b shows the Poincare
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Fig. 12. (a) Bifurcation diagram at C = 130. (b) 1D Poincare map of (a).
Fig. 13. (a) Bifurcation diagram at C = 90. (b) 1D Poincare map of (a).
Fig. 14. (a) Bifurcation diagram at C = 40. (b) 1D Poincare map of (a).
bifurcation diagram for Y 2 (33.4, 36.8). It is clear that torus bifurcation scenario is observed. Figs. 15 show the dynamic characteristics to the strange toroidal attractor: Fig. 15a represents the time trace, Fig. 15b is the phase plane (u versus X), Fig. 15c is another phase plane (u versus v) and Fig. 15d gives the 2D Poincare map. This strange attractor shows periodic behavior where the trajectory revisits fixed points in the state space as shown in Fig. 15d (the separate points in the figure).
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Fig. 15. The dynamic characteristic of a strange attractor at Y = 36.3 and C = 40: (a) time trace, (b) phase plane ÔXÕ versus ÔuÕ, (c) phase plane ÔvÕ versus ÔuÕ and (d) 2D Poincare map.
6. Conclusion In this work we considered a two-photon optical bistable system where two sorts of atom are placed in a ring cavity. Within a base set of the values of the parameter space an X–Y bifurcation diagram has been obtained. More exploration of the system under investigation using the two-parameter continuation technique uncovers more richness of the system dynamics where two types of degeneracy are observed. We studied the dynamic behavior of the system for different values of the cooperative parameter ÔCÕ considering the two types of degeneracy. At C = 110, the system dynamics is characterized by the existence of 3-Hopf points. Complex bistability regions are observed: • Region of two steady states. • Region of two steady states and a periodic attractor. • Region of two steady states and a chaotic attractor. The bifurcation scenario originated by period doubling sequence and terminated homoclinically. As the value of the cooperative parameter C increases the chaotic region shrinks (e.g. C = 150 where we can see incomplete period doubling characterizes the dynamics of the system). By decreasing C, e.g. C = 90, crises interrupt the period doubling bifurcation scenario which indicates dangerous boundaries. Further decrease of C, e.g. C = 40, shows another type of strange attractor and the bifurcation scenario is totally different from previous cases where we can find torus bifurcation scenario.
References [1] Luther GG, Alber MS. Nonlinear Sci Today 1997:1; Gauthier DJ. Nonlinear Sci Today 1994;4:2–11.
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[2] Gauthier DJ. Science 1998;279:1156; Van Wiggeren GD, Roy R. Science 1998;279:1198; Corcoran E. Sci Amer 1991;264(August):82. [3] Batarfi HA, Hassan SS, Saunders R, Bullough RK. Eur Phys J D 2000;8:417–29. [4] Hassan SS, Batarfi HA, Saunders R, Bullough RK. Eur Phys J D 2000;8:403–16. [5] Lugiato LA, Narducci LM. In: Dalibard J, Raimond JM, Zinn-Justin J, editors. Fundamental systems in quantum optics. Amsterdam: Elsevier Science Publishers; 1992. p. 942–1047. [6] Thompson BV, Hermann JA. Phys Lett 1981;83A:376; Thompson BV, Hermann JA. In: Knight PL, editor. Quant Elec & Elec Opt. John Wiley and Sons Ltd; 1983. p. 209–12. [7] Maize SMA. PhD Thesis , University of Manchester, UK, 1986. [8] Bonifacio R, Lugiato LA. Lett Nuovo Cimento 1978;21:505. [9] Doedel EJ, Kernevez JP. AUTO: software for continuation and bifurcation problems in ordinary differential equations. Pasadena, CA: California Institute of Technology; 1986. [10] Drazin PG. Nonlinear system. Cambridge University Press; 1992. pp.160. [11] Guckenheimer J, Holmes P. Nonlinear oscillation, dynamical systems and bifurcation of vector fields. New York: Springer; 1983. [12] IMSL/PC-Library, 1986. [13] Arecchi FT, Harrison RG. Instability and chaos in quantum optics. Berlin Heidelberg: Springer-Verlag; 1987. [14] Grebogi C, Ott E, Yorke JA. Physica 1983;7D:181.
Self-pulsing and chaos of a two-photon optical bistable model in a ring cavity S.M.A. Maize Physics Department, Faculty of Science, Menoufia University, Shebin El-Kom, Egypt Accepted 19 April 2005
Communicated by Prof. G. Iovane
Abstract Stability analysis and numerical investigation of the periodic and chaotic behavior are presented for two-photon optical bistable system where two sorts of atom are placed in a ring cavity. The effect of the cooperative parameter ÔCÕ on the state of the system is examined and a wealth of new dynamic characteristics was observed including: period doubling sequence to chaos, complex bistability and strange attractors evolved from torus bifurcation sequence. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Developments in nonlinear dynamics, in particular ideas about deterministic chaos and pattern formation have had implications in the understanding of complex behavior in nonlinear optical systems [1]. It has been demonstrated that optical chaos is not bad for all applications; it may be ideally suited for communicating information at high data rates [2]. Moreover, it is believed that chaos can be employed for a variety of applications, many of which are yet to be discovered. A recent work about stability analysis and chaos for one photon optical bistable (OB) model [3] in contact with a squeezed vacuum reservoir [4] is carried out. The study [3,4] generalizes earlier work on the same model in normal vacuum (see [5] and references therein). On the other hand, a two-photon OB model that utilizes two sorts of atom in a ring cavity has, been studied by Thompson and Hermann [6]. The main feature of the steady state input-output is the possibility to get tristability where five solutions exist for each input field value. They illustrate that the realistic model can be a mixture of two metal vapours (like Potassium has an 8d level lying 1.67 cm1 below the 16d level of rubidium at around 33 180 cm1). Here, in the light of recent advances in fundamental knowledge and techniques this system [6] is revisited on a higher level of computational technique and analysis of the results. A wealth of new dynamic features is discovered which makes the results of earlier investigation [7] along the usual analytical treatment of linear stability analysis [3,7] not more than a scratch on the surface of the problem. The description of the system of interest is given in the following section.
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.100
S.M.A. Maize / Chaos, Solitons and Fractals 28 (2006) 590–600
591
2. The model Maxwell–Bloch equations The system of interest here consists of a ring cavity containing atoms with doubly degenerate working level. This degeneracy may be achieved by choosing a mixture of two different sorts of atom [6], as shown in Fig. 1. We assume that cross-linking of the states is forbidden by selection rules and that therefore the intermediate states for the two types of transition form disjoint sets. We associate a Bloch vector ÔrÕ with a–b transition and ÔsÕ with c–d transition. These vectors turn at different rates because the two-photon matrix elements Kab and Kcd differ in value. The equations of motion of ÔrÕ and ÔsÕ can be written as [6] drþ 2ijk ab j 2 k bb k aa þ 2 E r3 i ¼ jE j D rþ c? rþ h dt h dr3 ijk ab j 2 2 rþ Eþ r E ck ðr3 þ 1Þ ¼ h dt ð1Þ dsþ 2ijk cd j 2 k dd k cc þ 2 E s3 i ¼ jE j D0 sþ f? c? sþ h dt h ds3 ijk cd j 2 2 sþ Eþ s E fk ck ðr3 þ 1Þ ¼ h dt The electric field is chosen to be in the form E ¼ Eþ eiðxtkzÞ þ E eiðxtkzÞ
ð2Þ
Self-consistently the polarization ÔPÕ takes the same form as the field ÔEÕ in (2) where the field and polarization amplitudes and phases are slowly varying in ÔtÕ and ÔzÕ [6], so MaxwellÕs equation reduces to oEþ 1 oE x þ þ ¼i P cv ot 2e0 cv oz
ð3Þ
where [6,7] 1. kaa, kbb, kab, kcc, kdd and kcd have the common definition for the two-photon transition k aa ¼
2 X 2 xja laj ; h j x2ja x2
k bb ¼
2 X 2 xjb lbj ; h j x2jb x2
k ab ¼
1 X lja lbj ; etc. h j x þ xjb
2. the atomic detunings D = xba 2x and D 0 = xdc 2x 3. ck, c?, fkck, f?c? are the longitudinal and transverse decay rates appropriate to an electron from the jai and jci state respectively. We define the weights of the two sorts of atom expressed by the Bloch vectors ÔrÕ and ÔsÕ to be ÔW1Õ and ÔW2Õ respectively. The expression of the polarization and hence the positive frequency component ÔP+Õ can be substituted in MaxwellÕs equation (3) to give " sffiffiffiffiffi !# sffiffiffiffiffiffiffiffiffiffi oEþ 1 oEþ x N W1 c? þ W2 f? c? þ jk ab j s3 E þ s E r3 E þ r E þ þ ¼i d1 q d2 ð4Þ cv ot 2e0 cv V oz W1þW2 ck W1 fk ck
Fig. 1. The optical ring cavity and the energy level scheme for the two-photon two sorts of atom mixture.
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where q = jkcdj/jkabj, the stark term d1 ¼ kbb2jkkabaaj
qffiffiffiffi ck c?
and d2 has a similar definition with respect to c–d transition.
2jk ab j ffiffiffiffiffiffiffi Eþ to Maxwell–Bloch equations (1) and (4), we get By introducing the normalized field nþ ¼ hp ck c? þ 2 pffiffiffiffiffiffiffiffiffi 2 drþ ¼ i ck c? n r3 i d1 c? n D rþ c? rþ dt pffiffiffiffiffiffiffiffiffi ck c? þ2 dr3 ¼i rþ n r n2 ck ðr3 þ 1Þ dt 2 ! pffiffiffiffiffiffiffiffiffi 2 dsþ f? c? þ 2 0 ¼ iq ck c? n s3 i qd2 n D sþ f? c? sþ dt fk ck pffiffiffiffiffiffiffiffiffi ck c? 2 2 ds3 ¼ iq sþ nþ s n fk ck ðs3 þ 1Þ dt 2 " sffiffiffiffiffi !# sffiffiffiffiffiffiffiffiffiffi onþ 1 onþ x N W1 c? þ W2 f? c? þ jk ab j s3 n þ s n r3 n þ r n þ d1 q d2 þ ¼i cv ot 2e0 cv V oz W1þW2 ck W1 fk ck
ð5Þ
3. The self-pulsing within the mean field limit Here we consider the case of the fast relaxation of the transverse Bloch components, i.e., the transversal relaxation time is considered to be the shortest among all characteristic times in the problem (c?, f?c? ck, fkck, k, . . .). Consequently, the Bloch components r+ and s+ take their stationary values. After this adiabatic elimination of r+ and s+, we integrate the field equation in (5) with respect to z within the mean fieldRlimit (valid for T 1 where T is the transmisL sivity of the ring cavity mirror) and the field takes its average value L1 0 EðzÞdz [8]. And use the boundary conditions of the ring cavity [3–7] Eð0; t0 Þ ¼ TY þ REðL; t0 Þ expðihT Þ and
ET ðtÞ EðL; tÞ ¼ pffiffiffiffi ¼ X T
where h is the normalized cavity detuning, the set of Eqs. (5) is reduced to the following set of differential equations: " ! # ck dr3 jX j4 ¼ þ 1 r3 þ 1 dtn j 1 þ X21 " ! # fk ck ds3 jX j4 ¼ þ 1 s3 þ 1 dtn j B þ X22 " ! ! # pffiffiffi 2 du c? A f? c? BjX j jX j2 jX j2 X1 jX j2 X 2 p ffiffiffi r ¼ Y u þ hv þ 2C u þ v þ vd þ þ v þ v d u 1 3 2 s3 dtn ck fk ck 1 þ X21 1 þ X21 B þ X22 B þ X22 B " ! ! # pffiffiffi 2 BjX j dv c1 A f? c? jX j2 jX j2 X1 jX j2 X 2 r3 þ pffiffiffi v ¼ v hu þ 2C v u ud1 u u d2 s3 ck fk ck dtn 1 þ X21 1 þ X21 B þ X22 B þ X22 B
ð6Þ
pffiffiffiffi pffiffiffiffi where Y (the input field) ¼ nI = T , X (the output field) ¼ nþ ðL; t0 Þ ¼ nT = T ¼ u iv, tn (the normalized time) = jt 0 , j (the cavity decay time) = cvT/Lt, Lt is the length of the ring cavity, t 0 = t + (Lt L)z/(cvL); L is the length of medium inside the cavity. qffiffiffiffi c The cooperative parameter C ¼ 2TL 2N0 cxv V jk ab j c?k ; A ¼ fk WW 21 and B ¼ fk f? q2 . The set of equation (6) represents lumped system, which can be solved numerically and computationally by the following technique.
4. Numerical tools and presentation techniques The numerical investigation of the dynamic characteristics of the model is carried out using principles of bifurcation theory coupled with continuation techniques. The continuity diagrams are obtained using the software AUTO [9]. This package is able to perform both steady state (static) and dynamic bifurcation analysis, including the determination of
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the entire periodic solution branches. AUTO also computes the Floquet multipliers [10] along periodic solution branches and therefore determines the stability of the periodic orbit [11]. Because of periodicity there is always a Floquet multiplier equal to +1. When all Floquet multipliers lie inside the unit circle, the periodic solution is asymptotically stable. Complex Floquet multipliers indicate that the periodic solution behaves like a focus, while real Floquet multipliers denote node or saddle-like behavior. Periodic attractors can lose their stability by a number of ways. The most important ones are period doubling and torus bifurcation. A Floquet multiplier leaving the unit circle through -1 indicates a period doubling bifurcation, while passage of the complex Floquet multipliers out of the unit circle indicates that the periodic orbit bifurcates to invariant torus. The DGEAR subroutine (1986) with automatic step size to ensure accuracy for stiff differential equations is used for numerical simulations of point as well as periodic attractors and accuracy up to 1015 is attainable [12].
5. Results and discussion The continuity diagram for the two-photon ring-cavity model (the input–output field) is shown in Fig. 2 for a set of model parameters: D1 ¼ 60;
D2 ¼ 70;
h ¼ 80;
C ¼ 110;
A ¼ 3;
B ¼ 81;
ck =j ¼ 1:5;
f k ck =j ¼ 2:25;
d1 ¼ d2 ¼ 0
This diagram show two stable static branches (solid line) connected by three unstable branches (dashed line) through four limit points. Moreover, this diagram contains two Hopf bifurcation points [13], which are denoted by the squares on the curve. The presence of Hopf points indicates necessarily the existence of periodic behavior of the system. The Hopf point arises when a pair of complex eigenvalues of the Jacobean of the model crosses the imaginary axis transversal. It is noticed that the two Hopf bifurcation points are almost degenerate Hopf where the transversality condition of the definitions of the Hopf bifurcation point is violated (the conditions for the crossing of the imaginary axis of the eigenvalues) that is d Re½kðY Þ ¼0 dY Y0 which means that the derivative of the real part of the eigenvalue with respect to the control parameter Y is equal to zero at this Hopf point. The continuity diagram also presents periodic behavior which loses its stability and starts a period doubling bifurcation scenario at Y = 16.9564 (this scenario will be analyzed later). This periodic branch terminates homoclinically [13] almost at Y = 7.76 as shown in Fig. 3 where the period goes to infinity almost at this value of Y. The two-parameter continuation technique, which we can get by using the AUTO package, will help in exploring the system under consideration. The importance of the two-parameter continuation comes from its ability to give the answer of some arising questions like: where does the periodic branch start? . . . which points are degenerate Hopf points? It is important to know the answer of the last question because it is known that we can often find rich dynamics close to the degenerate Hopf points. Figs. 4a and b represent the two parameter continuation diagram between the cooperation parameter C and the input field Y. The figure shows the loci of the Hopf bifurcation points (solid line) and the loci of
Fig. 2. Bifurcation diagram X versus Y where (- - -) represents the unstable static branch, (—) represents the stable static branch, (h) represents the Hopf bifurcation points; the periodic branch is the unstable positive slop branch .
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Fig. 3. The period versus the input field Y.
Fig. 4. (a) Two-parameter continuation diagram C versus Y. (b) Enlargement of the first part of (a).
static limit point (dashed line). Fig. 4a answers the first question and determines the starting point of the periodic solution of Fig. 2 at Y = 428.02 (the base case where C = 110). Fig. 4b is an enlargement of the first part of the curves in Fig. 4a where we can see two kinds of degeneracy. The first one at the coincidence of Hopf points loci to the static limit points loci and the second at the intersection of the loci of Hopf points and the loci of the static limit point at C = 50.5. We are going to study the two types of degeneracy by choosing deferent values of C. 5.1. The base case C = 110 In all the results presented the input field Y is used as the control parameter (bifurcation parameter) and all presentation techniques discussed before are utilized to investigate the rich dynamic behavior of the system. Fig. 5 shows the continuity diagram at C = 110. Three Hopf bifurcation points characterize this diagram. Two of them are almost degenerate Hopf. The third Hopf at Y = 427.986 is the starting gate to periodic solution. The periodic branch which is emerged from this point loses its stability at Y = 16.9564 through period doubling and also shows chaotic behavior. This bifurcation to chaos is investigated by using one-dimensional Poincare bifurcation diagram as shown in Fig. 6. The following characteristics are evident: (i) With decreasing Y the periodic branch with P1 (single period) which originated from the third Hopf bifurcation point goes through the sequence of period doubling (P1, P2, P4, P8, . . .) leading to chaos. This chaotic behavior goes under a number of bifurcations from chaos to periodicity and vice versa. (ii) The chaotic behavior vanishes where P4 is observed and bifurcate to P8 (the four bulbs which are shown in the figure). (iii) The periodic branch terminates homoclinically at Y = 6.95.
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Fig. 5. Bifurcation diagram at C = 110.
Fig. 6. 1D Poincare bifurcation diagram at C = 110.
5.1.1. Tristability Regarding to Fig. 2 we have more than one solution at the same value of the bifurcation parameter. Fig. 7 illustrates the phase plane at Y = 5 where two stationary non-equilibrium states coexist in the same state space. Fig. 8 shows another kind of bistability where one stationary non-equilibrium state coexist with period one attractor (P1) at Y = 20. Figs. 9a and b represents the complex bistable solution where two stationary non-equilibrium states coexist with a chaotic solution. Fig. 9a interprets how complex the trajectory approaches the two stationary points. Here we can see two reasons for the sensitivity to the initial conditions. One of them is due to the existence of more than one solution sharing the state space whereas the second one is due to the existence of the chaotic behavior. Fig. 10 shows the dynamic characteristics of a chaotic attractor at Y = 11.5. Fig. 10a represents the time trace. Fig. 10b shows the phase plane (u versus v). Fig. 10c shows another phase plane (u versus X). Fig. 10d shows 2D Poincare map. 5.2. The cooperative parameter C 110 Here we take the two cases of C = 130 and 150. Figs. 11a shows the continuity diagram at C = 150 performed by AUTO software, which indicates period doubling at Y = 19.661. Fig. 11b shows the one dimensional Poincare map for Y 2 (20, 8.3). It is clear that the figure is characterized by incomplete period doubling sequence and no more than period P4 is present in this region. The sequence is P1, P2, P4, P2, and P1. The periodic branch terminates homoclinically almost at Y = 8.3. When the cooperative parameter C decreases to 130 two banded chaos, in the region Y 2 (18.1069, 7.5906), are observed and this chaos bifurcate by period halving scenario to P1 attractor which terminate also homoclinically at Y = 6.728 (As shown in Fig. 12).
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Fig. 7. Phase plane at Y = 5 and C = 110 for the two steady state bistable case.
Fig. 8. Phase plane at Y = 20 and C = 110 for the case of stable steady state coexist with stable limit cycle.
Fig. 9. Phase planes of bistable case where two stable steady state are coexist with chaotic attractor. (i)The trajectory approaches the two stable steady state only. (ii) The same as (a) but includes the chaotic attractor.
5.3. The cooperative parameter C 110 At C = 90 the continuity diagram is shown in Fig. 13a where we can see the periodic branch has originated from the highest Hopf point (the Hopf point which correspond to the highest value of the control parameter Y). This periodic
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Fig. 10. The dynamic characteristic of the chaotic attractor at Y = 11.5 and C = 110: (a) time trace, (b) phase plane ÔvÕ versus ÔuÕ, (c) phase plane ÔXÕ versus ÔuÕ and (d) 2D Poincare map.
Fig. 11. (a) Bifurcation diagram at C = 150. (b) 1D Poincare map of (a).
branch loses its stability at Y = 16.7337 and goes under a number of bifurcations from chaos to periodicity and vice versa until it terminates homoclinically at almost Y = 10.47 (as shown in Fig. 13b). The Poincare bifurcation diagram is characterized by a boundary crises at Y = 14.25, where the chaotic attractor changes its shape due to the collision by an unstable periodic attractor [14]. When C decreases to 40 (Fig. 14), which is very close to the second type of degenerate Hopf points (the intersection between the loci of Hopf points and the loci of the static limit point), it is clear that the bifurcation scenario is totally different from the previous cases where other strange attractors are observed. The periodic branch originated from the highest Hopf point loses its stability at Y = 36.7195. Fig. 14b shows the Poincare
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Fig. 12. (a) Bifurcation diagram at C = 130. (b) 1D Poincare map of (a).
Fig. 13. (a) Bifurcation diagram at C = 90. (b) 1D Poincare map of (a).
Fig. 14. (a) Bifurcation diagram at C = 40. (b) 1D Poincare map of (a).
bifurcation diagram for Y 2 (33.4, 36.8). It is clear that torus bifurcation scenario is observed. Figs. 15 show the dynamic characteristics to the strange toroidal attractor: Fig. 15a represents the time trace, Fig. 15b is the phase plane (u versus X), Fig. 15c is another phase plane (u versus v) and Fig. 15d gives the 2D Poincare map. This strange attractor shows periodic behavior where the trajectory revisits fixed points in the state space as shown in Fig. 15d (the separate points in the figure).
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Fig. 15. The dynamic characteristic of a strange attractor at Y = 36.3 and C = 40: (a) time trace, (b) phase plane ÔXÕ versus ÔuÕ, (c) phase plane ÔvÕ versus ÔuÕ and (d) 2D Poincare map.
6. Conclusion In this work we considered a two-photon optical bistable system where two sorts of atom are placed in a ring cavity. Within a base set of the values of the parameter space an X–Y bifurcation diagram has been obtained. More exploration of the system under investigation using the two-parameter continuation technique uncovers more richness of the system dynamics where two types of degeneracy are observed. We studied the dynamic behavior of the system for different values of the cooperative parameter ÔCÕ considering the two types of degeneracy. At C = 110, the system dynamics is characterized by the existence of 3-Hopf points. Complex bistability regions are observed: • Region of two steady states. • Region of two steady states and a periodic attractor. • Region of two steady states and a chaotic attractor. The bifurcation scenario originated by period doubling sequence and terminated homoclinically. As the value of the cooperative parameter C increases the chaotic region shrinks (e.g. C = 150 where we can see incomplete period doubling characterizes the dynamics of the system). By decreasing C, e.g. C = 90, crises interrupt the period doubling bifurcation scenario which indicates dangerous boundaries. Further decrease of C, e.g. C = 40, shows another type of strange attractor and the bifurcation scenario is totally different from previous cases where we can find torus bifurcation scenario.
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