Effects of Atomic Coherence and Injected Classical Field on Chaotic Dynamics of Non-degenerate Cascade Two-Photon Lasers

Commun. Theor. Phys. Beijing China) 48 2007) pp. 288 294 c International Academic Publishers Vol. 48 No. 2 August 15 2007 Effects of Atomic Coherence and Injected Classical Field on Chaotic Dynamics of Non-degenerate Cascade Two-Photon Lasers ZHANG Ya-Jun 12 HUANG Hong-Bin 1 YANG Peng 1 and XIE Xia 1 1 Department of Physics Southeast University Nanjing 210096 China 2 Nanjing Institute of Astronomical Optics and Technology Nanjing 210042 China Received August 15 2006) Abstract Based on the cascade two-photon laser dynamic equation derived with the technique of quantum Langevin operators with the considerations of coherently prepared three-level atoms and the classical field injected into the cavity we numerically study the effects of atomic coherence and classical field on the chaotic dynamics of a two-photon laser. Lyapunov exponent and bifurcation diagram calculations show that the Lorenz chaos and hyperchaos can be induced or inhibited by the atomic coherence and the classical field via crisis or Hopf bifurcations. PACS numbers: 05.45.Gg 42.50.Gy 42.60.Mi Key words: nondegenerate cascade two-photon laser chaos bifurcation injected classical field atomic coherence 1 Introduction Atomic coherence plays an important role in the interaction between photon and atoms. [1 4] Many remarkable phenomena caused by atomic coherence are revealed such as lasing without population inversion [5] electromagnetically induced transparency [67] slow and stopped light [89] quantum computation and quantum information processing [10] and inhibition of Lorenz strange attractor. [11] Manipulation of atomic coherence now becomes an important technique to control quantum and even classical processes. Studies of the relations between atomic coherences and classical dynamics will provide deep insight in the fundamental physics. Although the dynamics of nondegenerate cascade twophoton laser was studied in the absence of atomic coherences and injected classical field [12] the dynamics of the system in the presence of atomic coherences and classical field has not been studied so far. In this paper we give the dynamic equation of the cascade nondegenerate two-photon laser derived with the technique of quantum Langevin operators with the considerations of atomic coherences and injected classical field and numerically study its dynamic characteristics caused by atomic coherences and injected classical field. The results show richer dynamic phenomena than that of degenerate cascade twophoton laser. [13] 2 Model and Equations We now briefly describe the derivation of the twophoton laser dynamic equation. The initial coherently prepared cascade three-level atoms and the classical field are injected into the doubly resonant cavity [5] where the atoms interact with two nondegenerate modes of the field and the cavity fields are driven by the injected classical field. The atomic level configuration is shown in Fig. 1. Fig. 1 Atomic level configuration for cascade twophoton lasers. The atoms are initially prepared in a coherent superposition of states 1 2 and 3. The Hamiltonian of such a system under the rotatingwave approximation is given by = 1) H = Ω 1 a 1 a 1 + Ω 2 a 2 a 2 + 3 ω l l l j j l=1 [ + θt t j )g 1 a 1 2 1 j + g 2 a 2 3 2 j ) j + iε d1 e iω d 1 t a 1 + iε d 2 e iω d 2 t a 2 + H.c. ]. 1) Here a i and a i i = 1 2) are the creation and annihilation operators for the two cavity field modes with frequency The project partially supported by the Natural Science Foundation of Jiangsu Province of China under Grant No. BK2005062 E-mail: yjzhang@niaot.ac.cn E-mail: hongbinh@seu.edu.cn

No. 2 Effects of Atomic Coherence and Injected Classical Field on Chaotic Dynamics of Non-degenerate 289 Ω i and ω l l = 1 2 3) is the energy of atomic level l. The parameter g i i = 1 2) denotes the coupling constant between atoms and field θt t j ) is the step function and accounts for the fact that every atom starts the interaction with the field at its injection time t j. The cavity mode a i is driven by the injected classical field ε di with frequency ω di. Defining A 1 = [ iω 1 ν 1 ) k 1 ]A 1 ig 1 θ j 1 2 j + ε 1 + f A1 A 2 = [ iω 2 ν 2 ) k 2 ]A 2 ig 2 θ j 2 3 j + ε 2 + f A2 t 1 1 j = γ 1 1 j ig 1 θ j [ A 1 2 1 j + A 1 1 2 j ] + f 11 j j a 1 = A 1 e iν1t a 2 = A 2 e iν2t 1 2 = 1 2 e iν1t 2 3 = 2 3 e iν2t 1 3 = 1 3 e iν1+ν2)t 2) and using Eq. 1) we can derive the following Heisenberg equations of motion for the field and atomic operators: t 2 2 j = γ 2 2 j ig 1 θ j [A 1 2 1 j A 1 1 2 j ig 2 θ j [ A 2 3 2 j + A 2 2 3 j ] + f 22 t 3 3 j = γ 3 3 j ig 2 θ j [A 2 3 2 j A 2 2 3 j ] + f 33 t 1 2 j = [ iω 2 ω 1 ν 1 ) γ ] 1 2 j ig 1 θ j [ 1 1 j 2 2 j ]A 1 ig 2 θ j 1 3 j A 2 f 12 t 2 3 j = [ iω 3 ω 2 ν 2 ) γ ] 2 3 j ig 2 θ j [ 2 2 j 3 3 j ]A 2 + ig 1 θ j 1 3 j A 1 f 23 t 1 3 j = [ iω 31 ν 1 ν 2 ) γ ] 1 3 j + ig 1 A 1 2 3 j ig 2 θ j A 2 1 2 j + f 13. 3) Here γ γ ) is the longitudinal transverse) decay rate the same for all levels) k i is the cavity decay rate and f i is the Langevin noise operators θ j θt t j ). Defining the macroscopic atomic operators: M 1 = i j θt t j ) 1 2 j M 2 = i j θt t j ) 2 3 j M 3 = i j θt t j ) 1 3 j N 1 = j θ j 1 1 j N 2 = j θ j 2 2 j N 3 = j θ j 3 3 j and setting ω d1 = ν 1 ω d2 = ν 2 ν i is laser frequency) we get the following equations from Eq. 3): Ȧ 1 = i 1 k 1 )A 1 + g 1 M 1 + ε 1 + F A1 Ȧ 2 = i 2 k 2 )A 2 + g 2 M 2 + ε 2 + F A2 Ṅ 1 = Rρ 11 γ N 1 + g 1 A 1 M 1 + A 1 M 1 ) + F N 1 Ṅ 2 = Rρ 22 γ N 2 g 1 A 1 M 1 + A 1 M 1 ) + g 2A 2 M 2 + A 2 M 2 ) + F N 2 Ṅ 3 = Rρ 33 γ N 3 g 2 A 2 M 2 + A 2 M 2 ) + F N 3 Ṁ 1 = irρ 21 + i 21 γ )M 1 + g 1 N 2 N 1 )A 1 ig 2 A 2 M 3 + F M1 Ṁ 2 = irρ 32 + i 32 γ )M 2 + g 2 N 3 N 2 )A 2 + ig 1 A 1 M 3 + F M2 Ṁ 3 = irρ 31 + i 31 γ )M 3 + ig 1 A 1 M 2 ig 2 A 2 M 1 + F M3 Ṅ 21 = γ [N 21 N 210 ] 2g 1 A 1 M 1 + M 1 A 1) + g 2 A 2 M 2 + A 2 M 2 ) + F 21 Ṅ 32 = γ [N 32 N 320 ] 2g 2 A 2 M 2 + M 2 A 2) + g 1 A 1 M 1 + A 1 M 1 ) + F 32 Ṅ 31 = γ [N 31 N 310 ] g 2 A 2 M 2 + A 2 M 2 ) g 1A 1 M 1 + A 1 M 1 ) + F 31 4) where i = Ω i ν i 21 = ω 2 ω 1 ν 1 32 = ω 3 ω 2 ν 2 31 = ω 31 ν 1 ν 2 and N ij = N i N j. ρ ij = ρ ij e iθij is the matrix element of the initial atoms and R is the injection rate of atoms. N ij0 = Rρ ii ρ jj )/γ is the initial atomic population inversion. We now transform the operator 4) into corresponding C-number Langevin equations which can be obtained only if we define the correspondence between a product of C-number and a product of operators. [15] Introducing the quantities: N 21 = N 210 1 z ) 1 N 32 = N 320 1 z ) 2 N 31 = N 310 1 z ) 3 M 1 = N 210 y 1 e iϕ1 M 2 = N 320 y 2 e iϕ2 c 1 c 2 c 3 2c 1 2c 2 M 3 = N 310 2c 3 y 3 e iϕ3 A 1 = n 10 x 1 e iθ1 A 2 = n 20 x 2 e iθ2 Y 1 = ε 1 k 1 n10 Y 2 = ε 2 k 2 n20 with n i0 = γ2 4g 2 i b = γ γ σ i = k i γ τ = γ t δ i = i k i δ ij = ij γ

290 ZHANG Ya-Jun HUANG Hong-Bin YANG Peng and XIE Xia Vol. 48 c 1 = N 210g 2 1 k 1 γ = Rg2 1 k 1 γ γ ρ 22 ρ 11 ) = c 10 ρ 22 ρ 11 ) c 2 = N 320g 2 2 k 2 γ = Rg2 2 k 2 γ γ ρ 33 ρ 22 ) = c 20 ρ 33 ρ 22 ) c 3 = N 310g 1 g 2 = Rg 1g 2 ρ 33 ρ 11 ) = c 30 ρ 33 ρ 11 ) k1 k 2 γ k1 k 2 γ γ and setting θ 1 = θ 2 = ϕ 1 = ϕ 2 = ϕ 3 = 0 we have the following C-number equations for dimensionless variables x i y i and z i neglecting the noise forces): ẋ 1 = σ 1 y 1 x 1 + Y 1 ) ẋ 2 = σ 2 y 2 x 2 + Y 2 ) ẏ 1 = 2bc 10 ρ 21 + c 1 x 1 x 1 z 1 y 1 ± 1 2 αβx 2y 3 ẏ 2 = 2bc 20 ρ 32 + c 2 x 2 x 2 z 2 y 2 1 1 2 αβ x 1y 3 ẏ 3 = 2bc 30 ρ 31 y 3 ± 1 2 αβx 1y 2 1 1 2 αβ x 2y 1 ż 1 = bz 1 + x 1 y 1 1 2 α2 β 2 x 2 y 2 ż 2 = bz 2 + x 2 y 2 1 1 2 α 2 β 2 x 1y 1 ż 3 = bz 3 + 1 1 2 αβ x 1y 1 + 1 2 αβx 2y 2 5) where k 2 /k 1 = β 2 g1/g 2 2 2 = α 2 c 10 = α 2 β 2 c 20 and c 30 = αβc 20. In deriving Eqs. 5) we have used the phase-locking conditions: ϕ 1 = θ 1 ϕ 2 = θ 2 δ 1 = δ 2 = δ 21 = δ 32 = δ 31 = 0 θ 21 ϕ 1 = θ 32 ϕ 2 = θ 31 ϕ 3 = π/2 ϕ 3 ϕ 1 θ 2 = ±π/2 and ϕ 3 ϕ 2 θ 1 = ±π/2. Equations 5) are the dynamic equations for the non-degenerate cascade two-photon lasers CTPL). Here x 1 and x 2 are the dimensionless laser fields generated from the atomic transitions 2 1 and 3 2 respectively y 1 y 2 and y 3 are the dimensionless atomic polarizations between atomic levels 1 2 2 3 and 1 3 respectively while z 1 z 2 and z 3 are the dimensionless atomic population inversions between levels 2 1 3 2 and 3 1 respectively. The dimensionless classical field Y i i = 1 2) is used to drive the field x i and the time is scaled as τ=γ t. In deriving Eq. 5) we have assumed that the atom j is initially prepared in a coherent superposition state ψ j 0) = 3 l=1 ρll e iϕ lj l with ρ 11 +ρ 22 +ρ 33 = 1 and ρ lm = l ψ j 0) ψ j 0) m = ρ ll ρ mm exp[iϕ lj ϕ mj )] = ρ lm e iϕ lmj. If the atomic phase ϕ lmj is different for different atom j then we write expiϕ lmj ) = expiϕ lm0 + iδ lmj ) and averaging it over all atoms yields expiϕ lm0 + iδ lmj ) expiϕ lm0 ) exp δlmj 2 /2) = a lm expiϕ lm0 ) 0 a lm 1). Thus the nondiagonal matrix element that is the atomic coherence ρ lm between levels l and m in Eqs. 5) can be written as ρ lm = a lm ρll ρ mm 0 a lm 1). For simplicity we write Eqs. 5) in the form of Ẋ = / t x 1 x 2 y 1 y 2 y 3 z 1 z 2 z 3 ) T = F X) and set α = β = 1. It is interesting to note that equation 5) is composed of two nonlinear coupled Lorenz Haken equations. The nine stationary solutions C i = x 1 x 2 y 1 y 2 y 3 z 1 z 2 z 3 ) of Eqs. 5) for Y i = ρ lm = 0 are given by C 0 = 0 0 0 0 0 0 0 0) C 1± = ± bc 1 1) 0 ± bc 1 1) 0 0 c 1 1) 1 2 c 1 1) 1 ) 2 c 1 1) C 2± = 0 ± bc 2 1) 0 ± bc 2 1) 0 1 2 c 2 1) c 2 1) 1 ) 2 c 2 1) C 3± = ±d e ±d e 0 c 1 1) c 2 1) c 3 2)) C 4± = ±d e ±d e 0 c 1 1) c 2 1) c 3 2)) with d = 2bc 1 + c 3 3)/3 e = 2bc 2 + c 3 3)/3. The linear stabilities of these solutions C i can be analyzed by calculating the eigenvalues of the Jacobian matrix DF C i ). The eight eigenvalues of DF C 0 ) are given by λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 λ 8 ) = 1 b b b 3/2 + 1/2) 1 + 8c 1 3/2 1/2) 1 + 8c 1 1 + c 2 1 c2 ). It is obvious that all eigenvalues λ i < 0 if 0 c i < 1 i = 1 2) in this case C 0 is stable. The solution C 0 becomes unstable for c i ) i = 1 2) > 1 which bifurcates via a pitchfork bifurcation to C 1± at c 1 =1 and c 2 < 1 or to C 2± at c 2 = 1 and c 1 < 1. Bifurcation of C 0 to C 3± or C 4± is possible only for both c 1 = 1 and c 2 = 1. In this case both C 1± and C 2± are unstable. C 1± becomes unstable at Hopf bifurcation point c 1 = σ 1 σ 1 + b + 3)/σ 2 b 1) if σ 1 > b + 1 which is the same as that in Lorenz Haken model. [15] C 1± can also transit to C 3± or C 4± via pitchfork bifurcation at c 1 = 1 + 4bc 2 1)/b 3bχ) with χ = g 2 σ 1 /g 1 σ 2. The cases are similar for C 2± except for the changes 1 2 χ 1/χ). The two field solutions C 3± and C 4± become unstable via Hopf bifurcations. We have not studied the effects of the pumping parameter c 1 and c 2 or the linear stabilities of those stationary solutions C i. The detailed discussions of the bifurcations of C i± i = 1 2 3 4) as c i increases can be found in Ref. [12]. Here we mainly study the effects of the injected classical field Y i and ρ ij on the two-photon laser dynamics. Before the following discussions we should note that

No. 2 Effects of Atomic Coherence and Injected Classical Field on Chaotic Dynamics of Non-degenerate 291 equations 5) are symmetric with respect to the change x i y i z i ) x i y i z i ) i = 1 2) for Y i = ρ ij = 0 so for every solution x i t) y i t) z i t)) the symmetric solution x i t) y i t) z i t)) also exists. 3 Lyapunov Exponents and Bifurcation In this section we mainly focus on the effects of atomic populations atomic coherences and injected classical field on the dynamics. In order to distinguish different dynamic states of the system we give the maximum Lyapunov exponent λ 1 MLE) versus ρ 11 and ρ 22 for Y i =0 and ρ ij i j) = 0 See Fig. 2). λ 1 > 0 corresponds to the chaotic motion of the system and λ 1 < 0 corresponds to the stable motion. In this simulation we have taken the upper sign in Eqs. 5) and set α = β = 1). Figure 2 can be divided into six regions 1) 2) 3) 4) 5) and 6) by lines ρ 11 = 1 ρ 22 ρ 11 = ρ 22 ρ 11 = 1 ρ 22 )/2 and ρ 11 = 1 2ρ 22. Different regions correspond to different atomic populations: 1) ρ 11 > ρ 33 > ρ 22 ; 2) ρ 22 > ρ 33 > ρ 11 ; 3) ρ 33 > ρ 11 > ρ 22 ; 4) ρ 22 > ρ 11 > ρ 33 ; 5) ρ 33 > ρ 22 > ρ 11 ; and 6) ρ 11 > ρ 22 > ρ 33. Since the pump parameters c 1 = c 20 ρ 22 ρ 11 ) and c 2 = c 20 ρ 33 ρ 22 ) affect the chaotic dynamics of the Lorenz oscillator 1 x 1 y 1 z 1 ) and Lorenz oscillator 2 x 2 y 2 z 2 ) respectively from Eqs. 5) and Fig. 2 we easily see that there exist domains in regions 1) and 3) in which x 1 y 1 y 3 ) = 0 0 0) while x 2 y 2 z 1 z 2 and z 3 are in chaotic states. Fig. 3 The MLE λ 1 top curves) and the second larger Lyapunov exponent λ 2 lower curves) for different c 20 with σ 1 = 2 σ 2 = 1 b = 0.1 Y i = 0 and a ij = 0. a) λ 1 > 0 and λ 2 0 for ρ 11 = 0.1 ρ 22 = 0.3 and ρ 33 = 0.6 the system is now in hyperchaotic state; b) λ 1 > 0 and λ 2 0 for ρ 11 = 0.2 ρ 22 = 0.5 and ρ 33 = 0.3 the system is now in Lorenz chaotic state with x 2 = 0; c) λ 1 > 0 and λ 2 0 for ρ 11 = 0.6 ρ 22 = 0.1 and ρ 33 = 0.3 the system is now in Lorenz chaotic state with x 1 = 0. Fig. 2 The MLE λ 1 vs. ρ 11 and ρ 22 for σ 1 = 2 σ 2 = 1 b = 0.1 c 2 = 300 Y i = 0 and ρ ij i j) = 0. Similarly there exist domains in regions 2) and 4) where x 2 y 2 y 3 ) = 0 0 0) while x 1 y 1 z 1 z 2 and z 3 are in chaotic states. In the chaotic domain of region 5) all variables are in chaotic states and which are hyperchaotic state because the chaotic attractors have two positive Lyapunov exponents while in regions 1) 4) the system has only one positive Lyapunov exponent See Fig. 3). The trajectory of the hyperchaos two-field chaos) in phase space spirals around one or two saddle focus points and the transition from one to another motion will occur when the parameters are changed while the one-field chaos is the usual Lorenz strange attractor. [15] In region 6) the system is in the steady state C 0. The non-chaotic states in regions 1) 5) are either stationary state metastable chaotic state or quasiperiodic state. The coexistence of stable state and chaotic state in two-photon lasers is an interesting phenomenon. The saddle focus points in this case are either C 1± or C 2± and the attractor is usual Lorenz strange attractor in x i y i z i ) phase space because the single Lorenz equation is now decoupled from the nonlinear coupled Eqs. 5). The situation changes if the classical field Y i and the atomic coherence ρ ij = a ij ρii ρ jj 0 a ij 1) exist. Figure 4 shows

292 ZHANG Ya-Jun HUANG Hong-Bin YANG Peng and XIE Xia Vol. 48 the MLE λ 1 versus Y i and a ij for ρ 11 = 0.4 ρ 22 = 0.1 and ρ 33 = 0.5 in region 3). The field x 2 is now in chaotic state while x 1 = 0 for Y i = 0 and a ij = 0. Figure 4c) shows that λ 1 becomes negative for a 32 0.13 that is the x 2 chaos is inhibited by atomic coherence between levels 3 and 2 and now x 1 = 0 and x 2 = 3.46. to those in region 3) discussed above except that a 21 can only generate x 1 chaos but cannot inhibit the hyperchaos. Similar case occurs for a 31 and a 32 in regions 2) and 4). The hyperchaos in region 5) is also sensitive to Y i and a ij. Small values of these dynamic parameters can inhibit the hyperchaotic state to periodic state and then to stationary state when Y i and a ij become larger See Figs. 5a) 5c)). Fig. 4 The MLE λ 1 versus different dynamic parameters Y i and a ij for ρ 11 = 0.4 ρ 22 = 0.1 ρ 33 = 0.5 b = 0.1 σ 1 = 2 σ 2 = 1 and c 20 = 300. The point to be stressed is that the generation of x 1 chaotic state is very sensitive to Y 1 a 21 and a 31 very small value 10 9 ) of these parameters can generate x 1 chaos and thus the Lorenz chaotic state becomes hyperchaotic state. Figure 4a) shows that both x 1 and x 2 chaos can be simultaneously inhibited to steady state for certain values of a 31 and or) a 21. For example x 1 = 1.07 and x 2 = 3.47 for a 31 0.96 and a 21 = 0 or x 1 = 0.33 and x 2 = 3.46 for a 21 0.9 and a 31 = 0. With increasing a 21 and a 31 the hyperchaos state evolves from hyperchaotic state through metastable chaotic states to stationary state. The similar case occurs for increasing Y 2. But with increasing Y 1 system evolves from hyperchaotic state 0 Y 1 < 4.9) through periodic state 4.9 Y 1 < 17.5) and metastable chaotic state to stationary state x 2 = 0 and x 1 = 16.97 Y 1 17.5). The effects of Y i and a ij on the dynamics of fields x 1 and x 2 in region 1) are similar Fig. 5 The MLE λ 1 and bifurcation versus different dynamic parameters Y i and a ij for ρ 11 = 0.2 ρ 22 = 0.3 ρ 33 = 0.5 σ 1 = 2 σ 2 = 1 b = 0.05 and c 20 = 300. a) λ 1 for different Y i with a ij = 0; b) λ 1 for different a 21 and a 32 with a 31 = 0 and Y i = 0; a ) Corresponding bifurcation of a) for different Y 2; c) λ 1 vs. a 31 with Y i = 0 and a 21 = a 32 = 0. It is interesting to note that in region 6) x 1 = x 2 = 0 for Y i = ρ ij = 0. This state C 0 ) goes to hyperchaotic state both x 1 and x 2 are in chaotic states) via cascade Hopf bifurcations with increasing the atomic coherence ρ 31. The

No. 2 Effects of Atomic Coherence and Injected Classical Field on Chaotic Dynamics of Non-degenerate 293 Hopf bifurcation and the MLE λ 1 describing this process are shown in Fig. 6a) and Fig. 6a ) respectively. In this case the atomic coherence ρ 32 and ρ 21 cannot induce or inhibit the hyperchaos. Fig. 6 The MLE λ 1 and bifurcation versus different dynamic parameters Y i and a ij for ρ 11 = 0.5 ρ 22 = 0.3 ρ 33 = 0.2 b = 3.3 σ 1 = 2 σ 2 = 1 and c 20 = 200. a) Bifurcation for different ρ 31; b) Bifurcation for different Y 1 with a 31 = 0.932; a ) Corresponding MLE λ 1 of a). b ) MLE λ 1 vs. Y i corresponding b). Fig. 7 a) Bifurcations for different Y i with ρ 11 = 1/6 ρ 22 = 1/3 ρ 33 = 1/2 σ 1 = σ 2 = 1.4253 b = 0.2778 and c 1 = c 2 = 25; b) The same as a) but for different ρ 31. This atomic coherence induced hyperchaos which occurs in the region 0.932 a 31 1 for ρ 11 = 0.5 ρ 22 = 0.3 and ρ 33 = 0.2 can also be inhibited to stable state: x 1 = 28.77 x 2 = 15.855 by Y i via Hopf bifurcations chaos inverse Hopf bifurcations which are shown in Fig. 6b) bifurcation diagram) and Fig. 6b ) MLE for different Y 1 and Y 2 ). The effects of the injected classical field and the atomic coherence on the dynamics can be summarized as follows. The injected classical field Y i can induce the hyperchaos two-field chaos) from the Lorenz chaos one-field chaos) or the stable motion via crisis and inhibit the hyperchaos to stable motion via inverse Hopf bifurcations see Figs. 5a ) 6b) and 7a)). The atomic coherence has the similar effects as that of Y i on the dynamics. However the atomic coherence ρ 31 can induce the hyperchaos from the stable motion via Hopf bifurcations or crisis see Figs. 6a) and 7b)) and the atomic coherences ρ ij can inhibit the hyperchaos via crisis or inverse Hopf bifurcations.

294 ZHANG Ya-Jun HUANG Hong-Bin YANG Peng and XIE Xia Vol. 48 4 Conclusion In conclusion the dynamic equation of nondegenerate cascade two-photon lasers is derived by using the quantum Langevin operator methods which is nonlinear coupled two Lorenz Haken equations. Lyapunov exponent calculation shows that the initial atomic states and the injected classical field have strong effects on the chaotic dynamics. Lorenz chaos and hyperchaos can exist in this system. Bifurcation diagrams show that the atomic coherences and the classical field can induce or inhibit the chaos via Hopf bifurcations or crisis. Especially the generation of hyperchaos is very sensitive to the dynamic parameters Y i and ρ ij a phenomenon cannot occur in degenerate cascade two-photon lasers. [13] References [1] E.N. Lorenz J. Atmos. Sci. 20 1963) 130. [2] R. Ju Y.J. Zhang H.B. Huang and H. Zhao Acta Phys. Sin. 53 2004) 2191 in Chinese). [3] X. Shi and Q.S. Lu Chin. Phys. 14 2005) 1088. [4] Z.G. Zheng Chin. Phys. 10 2001) 703. [5] M.O. Scully and M.S. Zubairy Quantum Optics Cambridge University Press Cambridge 1997). [6] S.E. Harris Phys. Today 507) 1997) 36; E. Arimondo Progress in Optics 35 1996) 257. [7] M.C. Phillios et al. Phys. Rev. Lett. 91 2003) 183602. [8] L.V. Han et al. Nature London) 397 1999) 594; C. Liu et al. Nature London) 409 2001) 490. [9] M. Fleischhauer and M. Lukin Phys. Rev. Lett. 84 2000) 5094; D.F. Phillips et al. Phys. Rev. Lett. 86 2001) 783. [10] M.A. Nielsen and I.L. Chung Quantum Computation and Quantum Information Cambridge University Press Cambridge 2000); C. Macchiavello G.M. Palma and A. Zeilinger Quantum Computation and Quantum Information Theory World Scientific Singapore 2000). [11] X.L. Deng H.Q. Ma B.D. Chen and H.B. Huang Phys. Lett. A 77 2001) 290. [12] G.J. de Valcárcel E. Roldán and R. Vilaseca Phys. Rev. A 45 1992) R2674; Phys. Rev. A 49 1994) 1243. [13] X. Xie H.B. Huang F. Qian. Y.J. Zhang P. Yang and G.X. Qi Commun. Theor. Phys. Beijing China) 45 2006) 1042. [14] C. Benkert and M.O. Scully Phys. Rev. A 42 1990) 2817. [15] C.O. Weiss and R. Vilaseca Dynamics of Lasers VCH Weinheim 1991).