Dynamic Equations and Nonlinear Dynamics of Cascade Two-Photon Laser

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1 Commun. Theor. Phys. (Beiing, China) 45 (6) pp c International Acaemic Publishers Vol. 45, No. 6, June 5, 6 Dynamic Equations an Nonlinear Dynamics of Cascae Two-Photon Laser XIE Xia,,, HUANG Hong-Bin, QIAN Feng, ZHANG Ya-Jun, YANG Peng, an QI Guan-Xiao Department of Physics, Southeast University, Naning 96, China Department of Physics, Nantong University, Nantong 67, China (Receive April 3, 5; Revise June 7, 5) Abstract We erive equations an stuy nonlinear ynamics of cascae two-photon laser, in which the electromagnetic fiel in the cavity is riven by coherently prepare three-level atoms an classical fiel inecte into the cavity. The ynamic equations of such a system are erive by using the technique of quantum Langevin operators, an then are stuie numerically uner ifferent riving conitions. The results show that uner certain conitions the cascae twophoton laser can generate chaotic, perio oubling, perioic, stable an bistable states. Chaos can be inhibite by atomic populations, atomic coherences, an inecte classical fiel. In aition, no chaos occurs in optical bistability. PACS numbers: 5.45.Ac, 5.45.Gg, 4.55.Ah, 4.65.Pc Key wors: chaos, cascae two-photon laser, atomic coherence, inecte fiel Introuction Lorenz Haken equations, which were obtaine by Lorenz for the escription of convection flows of fluis, [] an by Haken for the escription of one-photon laser ynamics, [] are a famous moel in the stuy of chaotic ynamics. Strange attractor an perio oubling bifurcation are two main ynamic phenomena of the Lorenz Haken equations. [3] We recently generalize these equations to account for atomic coherence an inecte classical fiel, [4] an foun that the atomic coherence an inecte classical fiel can inhibit the chaos. In the stuy of fiel squeezing, the two-photon laser was propose, [5] but its chaotic ynamics has not been reveale so far. In the present paper, we use the formalism of Langevin operators with the consieration of atomic coherences an inecte classical fiel to erive the ynamic equations for the cascae two-photon laser (CTPL). From the ynamic equations, we emonstrate that the Lorenz-like strange attractor, an perioic oubling bifurcation also eist in the CTPL. As in one-photon lasers, [4] the atomic populations, atomic coherences, an the inecte classical fiel can also be use to control the chaos. Further more, we show that the CTPL ynamic equations can be transforme to Lorenz-like equations for onephoton laser. [4] In Sec., we erive the Langevin operator equations for the two-photon laser fiel an atomic variables, an then convert the operator Langevin equations into c- number stochastic ifferential equations. Finally, we give the imensionless nonlinear ynamic equations for CTPL without aiabatic approimation. In Sec. 3, we numerically stuy the ynamics of the equations by calculating the Lyapunov eponent an its time evolution. The effects of atomic populations, atomic coherences, an inecte classical fiel on the chaotic ynamics are reveale by the Lyapunov eponents. Finally, in Sec. 4, we summarize our results. Dynamic Equations Fig. Energy-level iagram for two-photon lasers which are initially prepare in a coherent superposition of levels, an 3. We consier cascae three-level atoms interacting with a single-moe of raiation fiel in a laser cavity (Fig. ). The top level 3 an the bottom level are of the same parity, which is opposite to that of the mile level. The energy of level l (l =,, 3) is hω l. The atomic transitions an 3 interact with the same moe of the cavity fiel with moe frequency ω. The atoms are initially prepare in a coherent superposition of the three levels,, an 3, which are inecte into a laser cavity. The cavity moe is riven by the inecte classical fiel ε e iωt. The Hamiltonian of such a system in the rotating-wave approimation is given by ( h = ) H = ωa a + 3 ω l l l + g θ(t t ) l= The proect partially supporte by Natural Science Founation of Jiangsu Province of China uner Grant No. BK56 Corresponing author, ieia@seu.eu.cn

2 No. 6 Dynamic Equations an Nonlinear Dynamics of Cascae Two-Photon Laser 43 [a( 3 + ) + a ( 3 + )] + i(ε a e iω t ε a e iω t ), () where a an a are the creation an annihilation operators for the electromagnetic fiel, while l l (l, l =,, 3) is the atomic operator. The parameter g enotes the coupling constant between atoms an fiel, an θ(t t ) is the step function escribing the interaction of the -th atom with the fiel at its inection time t. Using the quantum Langevin equation (i /t)x = [X, H] + f X, we can erive from Eq. () the following quantum Langenvin equations for the fiel an the atomic operators: t a = ( iω κ)a ig θ(t t )( 3 + ) + ε e iωt + f a, t = [ i(ω ω ) γ ] igθ(t t )a( ) igθ(t t )a 3 + f, t 3 = [ i(ω 3 ω ) γ 3 ] 3 igθ(t t )a( 3 3 ) + igθ(t t )a 3 + f 3, t 3 = [ i(ω 3 ω ) γ 3 ] 3 igθ(t t )a( 3 ) + f 3, t = igθ(t t )a igθ(t t )a γ + f, t = igθ(t t )a( 3 ) igθ(t t )a ( 3 ) γ + f, t 3 3 = igθ(t t )a 3 + igθ(t t )a 3 γ f 3 3. () Here γ i an γi (i, =,, 3) are the transverse an longituinal relaation rates of an atom, respectively. The operators f in Eq. () are the Langevin noise operators which arise through the interaction with a heat bath. These operators are specifie through their first an secon moments. The normally orere noise correlation functions for f can be foun in Ref. [6], which are not given here. We now eliminate the quickly time-varying part in Eq. () by moving into a frame rotating with laser frequency ν, a = A e iνt, = r e iνt, 3 = 3 r e iνt, an 3 = 3 r e iνt. Net, we efine the macroscopic atomic operators: M = i θ(t t ) r, M = i θ(t t ) 3 r, M 3 = i θ(t t ) 3 r, N = θ(t t )( ), N 3 = θ(t t )( 3 3 ), N 3 = θ(t t )( 3 3 ). (3) The Langevin equations for A, M i, an N i can be obtaine by ifferential Eq. (3) an using Eq. (): [7] Ȧ = ( i κ)a + g(m + M ) + ε e i t + F A, M = irρ + ( i γ )M + gan iga M 3 + F M, M = irρ 3 + ( i 3 γ 3 )M + gan 3 + iga M 3 + F M, M 3 = irρ 3 + ( i 3 γ 3 )M 3 iga(m M ) + F M3, N = γ ( N N ) g[a(m M ) + A (M M )] + F N, N 3 = γ 3 ( N 3 N 3 ) g[a(m M ) + A (M M )] + F N3, N 3 = γ 3 ( N 3 N 3 ) g[a(m + M ) + A (M + M )] + F N3, (4) where = ω ν, = ω ν, = ω ω ν, 3 = ω 3 ω ν, 3 = ω 3 ω ν, Ni = N i (γ /γi )N, an N i = (R/γ i )(ρ ii ρ ) with R the average time-inepenent atomic inection rate an ρ i (i, =,, 3) the initial atomic ensity matri element satisfying ρ + ρ + ρ 33 =. F is the noise operator. Now we transform the operator equations into the corresponing c-number equations. [7] Here we choose the normal orering A, M, M, M 3, N, N 3, N 3, M, M, M 3, A an erive the c-number Langevin equations for A, M i, N i : A = ( i κ)a + g(m + M ) + ε e i t + F A,

3 44 XIE Xia, HUANG Hong-Bin, QIAN Feng, ZHANG Ya-Jun, YANG Peng, an QI Guan-Xiao Vol. 45 Ṁ = irρ + ( i γ )M + gan iga M 3 + F M, Ṁ = irρ 3 + ( i 3 γ 3 )M + gan 3 + iga M 3 + F M, Ṁ 3 = irρ 3 + ( i 3 γ 3 )M 3 iga(m M ) + F M3, N = γ ( N N ) g[a(m M ) + A (M M )] + F N, N 3 = γ 3 ( N 3 N 3 ) g[a(m M ) + A (M M )] + F N3, N 3 = γ 3 ( N 3 N 3 ) g[a(m + M ) + A (M + M )] + F N3. (5) In orer to numerically stuy Eq. (5), we transform the variables A, M i, N i into imensionless variables by efining (setting γ i = γ an γ i = γ ) A = n e iϕ, M = N y e iϕy, M = N 3 y e iϕy, M3 = N 3 y 3 e iϕy 3, c c c 3 ( N = N z ) (, N 3 = N 3 z ) (, N 3 = N 3 z ) 3 (6) c c c 3 with n = γ /(g), c = (g /κγ )N = c (ρ ρ ), c = (g /κγ )N 3 = c (ρ 33 ρ ), c 3 = c + c = (g /κγ )N 3 = c (ρ 33 ρ ), c = Rg /(κγ γ ), an N i = (R/γ )(ρ ii ρ ). Neglecting the Langevin noise force, equation (5) now becomes ẋ = σ[ + y cos(ϕ y ϕ ) + y cos(ϕ y ϕ ) + Y cos(ϕ ϕ )], ϕ = σ[ δ + y sin(ϕ y ϕ ) + y sin(ϕ y ϕ ) + Y sin(ϕ ϕ )], ẏ = bc ρ sin(ϕ ϕ y ) y + (c z ) cos(ϕ ϕ y ) y 3 sin(ϕ + ϕ y ϕ y3 ), y ϕ y = bc ρ cos(ϕ ϕ y ) δ y + (c z ) sin(ϕ ϕ y ) y 3 cos(ϕ + ϕ y ϕ y3 ), ẏ = bc ρ 3 sin(ϕ 3 ϕ y ) y + (c z ) cos(ϕ ϕ y ) + y 3 sin(ϕ + ϕ y ϕ y3 ), y ϕ y = bc ρ 3 cos(ϕ 3 ϕ y ) δ 3 y + (c z ) sin(ϕ ϕ y ) + y 3 cos(ϕ + ϕ y ϕ y3 ), ẏ 3 = bc ρ 3 sin(ϕ 3 ϕ y3 ) y 3 + [y sin(ϕ + ϕ y ϕ y3 ) y sin(ϕ + ϕ y ϕ y3 )], y 3 ϕ y3 = bc ρ 3 cos(ϕ 3 ϕ y3 ) δ 3 y 3 [y cos(ϕ + ϕ y ϕ y3 ) y cos(ϕ + ϕ y ϕ y3 )], ż = bz + y cos(ϕ ϕ y ) y cos(ϕ ϕ y ), ż = bz + y cos(ϕ ϕ y ) y cos(ϕ ϕ y ), ż 3 = bz 3 + y cos(ϕ ϕ y ) + y cos(ϕ ϕ y ), (7) where σ = κ/γ, b = γ /γ, δ = /κ, δ i = i /γ, ρ i = ρ i e iϕi, an Y = ε e iϕ /(κ n ). The ot now represents erivation with respect to = γ t. Setting all the time erivations of phase equal to zero, we obtain the phase-locking conitions: ϕ = ϕ = ϕ y = ϕ y, ϕ ϕ y = ϕ 3 ϕ y = ϕ 3 ϕ y3 = π/, ϕ + ϕ y ϕ y3 = ±π/, an δ = δ i =. Equation (7) now becomes ẋ = σ( + y + y + Y ), ẏ = bc ρ y + c z y 3, ẏ = bc ρ 3 y + c z ± y 3, ẏ 3 = bc ρ 3 y 3 ± (y y ), ż = bz + y y, ż = bz + y y, ż 3 = bz 3 + (y + y ). (8) Here the up signs correspon to ϕ + ϕ y ϕ y3 = π/, an the lower signs correspon to ϕ + ϕ y ϕ y3 = π/. It is clear that equations (8) are two nonlinear couple Lorenz Haken equations. 3 Dynamics of Cascae Two-Photon Laser Before numerically stuying the ynamics of Eq. (8), we first iscuss the stationary solutions of Eq. (8). For simplicity, we first set Y = an ρ = ρ 3 = ρ 3 =, which results in three stationary stable solutions:

4 No. 6 Dynamic Equations an Nonlinear Dynamics of Cascae Two-Photon Laser 45 X (, y, y, y 3, z, z, z 3 ) = (,,,,,, ) for c 3, an X ± = (±, ±y, ±y, y 3, z, z, z 3 ) with = b(c 3 ), y = [ c c ] + b(c3 ), + (3 + b)(c 3 ) y = [ c c ] b(c3 ), y 3 = b(c c )(c 3 ) + (3 + b)(c 3 ) + (3 + b)(c 3 ), z = [ 3(c c ) ] + (c 3 ), z = [ 3(c c ) ] (c 3 ), z 3 = c 3, (9) + (3 + b)(c 3 ) + (3 + b)(c 3 ) for c 3 c 3r (c 3r can be etermine by numerical simulation). That is, whatever the initial conitions are, the traectory of the system in phase space efine by, y, y, y 3, z, z, z 3 eventually approaches the trivial solution X if c 3, while the traectory will approach an be stable at X + or X if c 3 c 3r. Bifurcation iagram of laser intensity as a function of parameter c an the corresponing maimal Lyapunov eponents are calculate an shown in Fig.. In this figure, hopf bifurcation occurs at c A 9.5, the nontrivial stationary solutions X ± lose their stabilities, an the traectory ehibits strange behaviors. For c A < c < c B 466, the strange attractor is the main ynamic behavior, an perioic winows are foun in this parameter region in 356 < c < 393 an some other small regions. When ecreasing c from on, for c > c C 488, the attractor is a simple close curve. At c = c C, perio oubling bifurcation occurs, an at c = c B, the system enters the chaotic region. If Y an ρ + ρ 3 =, we have the bistable solution for ρ > ρ 33, [8] an laser solution for ρ 33 > ρ : Y = + + ( /b) [c (ρ ρ 33 ) bc ( ρ + ρ 3 )]. () We will numerically show that there is no chaos in bistable solution. All the parameters in Eq. (8) can affect the ynamics of the system. Figure 3 gives the maimal Lyapunov eponent (MLE) vs. ifferent parameters σ, b, c for ρ =., ρ =.3, ρ 33 =, Y =, an ρ = ρ 3 = ρ 3 =. As we know, the nonlinear ynamics of the system is etermine by MLE: if MLE >, the cascae two-photon laser system will be in the chaotic states, while MLE < correspons to the stable evolution of the system. In the following numerical simulation we choose σ =.453 an b =.778, an stuy the effects of the atomic population ρ ii, the atomic coherences, an the inecte classical fiel on the chaotic ynamics of the system. Fig. Etrema of laser intensity as a function of the pumping parameter c. This figure shows us the bifurcation iagram for σ =.453, b =.778, Y =, ρ =.3, ρ =., ρ 33 =, an ρ i =. Maimal Lyapunov eponents as a function of c for the same laser parameters as in, which has a goo agreement with the bifurcation iagram in representing critical c value for ifferent ynamics..8.4 b c 3.5 (c) 5 c σ σ.5 b Fig. 3 MLE vs. σ an b for c =, ρ =., ρ =.3, ρ 33 =, Y =, an a i =. MLE vs. σ an c for b =.778, ρ =., ρ =.3, ρ 33 =, Y =, an a i =. (c) MLE vs. b an c for σ =.453, ρ =., ρ =.3, ρ 33 =, Y =, an a i =.

5 46 XIE Xia, HUANG Hong-Bin, QIAN Feng, ZHANG Ya-Jun, YANG Peng, an QI Guan-Xiao Vol. 45 (i) Effects of initial atomic populations on ynamics In orer to reveal the effect of the initial atomic population ρ ii on the chaotic ynamics of the CTPL, we o not consier the effects of the initial atomic coherences an the inecte classical fiel here, an set ρ = ρ 3 = ρ 3 = an Y =. From Eq. (8) we see that the ynamic evolution of the system can be controlle by c = c (ρ ρ ) an c = c (ρ 33 ρ ). Figure 4 gives the numerical result of the epenence of the MLE on the initial atomic populations ρ an ρ (noting that ρ + ρ + ρ 33 = ) at parameter c =. It is clear that figure 4 is ivie into si regions by si lines: l : ρ = ρ (ρ 33 = ), an l i : ρ = a i ρ (ρ 33 ρ = a i ) with a =, a.8, a 3.464, a 4.4, an a We will stuy the ynamic behaviors in each region ivie by these lines. The region between l an l, in which ρ > ρ 33 correspons to the bistable state of the system, no chaos occurs in this region. The region < ρ 33 ρ <.79 between l an l correspons to the stable an metastable chaos states as ρ 33 ρ increases from to.79. An the states in this region will eventually evolve to the state X for < ρ 33 ρ <.5, an X + or X for.5 < ρ 33 ρ <.79. The region.79 < ρ 33 ρ < 9 between l an l 5 correspons to the Lorenz-like strange chaotic states as shown in Fig. 5. The Lyapunov imension of this strange attractor is.7473, which is larger than that of Lorenz attractor (.54). Aitionally, not only strange attractors eist in this whole region. In fact, there eist many small internals, calle perioic winows in Fig. 4, in which the attractor is perioic. For instance, the region 36 < ρ 33 ρ < 89 between l 3 an l 4 is the largest perioic winow. Figure 6 shows the T perio behavior in this perioic winow ρ..3 ρ ρ ρ Fig. 4 MLE vs. ρ an ρ for σ =.453, b =.778, c =, Y =, an a i = y y y 3 3 Fig. 5 Time series an traectories in phase space for c =, ρ =.3, ρ =., ρ 33 =, Y =, an ρ i =. The two focuses X ± are (±5.754, ±.8, ±4.944,.84, , , 59). MLE =.46 an the Lyapunov imension is y 5 Fig. 6 The same as in Fig. 5, but for ρ =.38, ρ =, ρ 33 =.7. The square of the -proection as a function of time an -y proection of traectory in a perioic winow. The perio is T an MLE =.. As ρ 33 ρ ecreases from.73 to 9, the perio oubling occurs: T T 4T 8T.... Figure 7 correspons to the T an T behaviors of the system. We note that the lines l i are ifferent for ifferent c. Figure 8 gives the MLE for ifferent c an ρ.

6 No. 6 Dynamic Equations an Nonlinear Dynamics of Cascae Two-Photon Laser y 4 4 y c Fig. 7 Time evolution of the variable an the -y proection of traectory for perio oubling. Here ρ =.3, ρ =., ρ 33 =, Y =, ρ i =, an c = 38 for with the perio being T an MLE =.78. c = 7 for with perio T an MLE= ρ Fig. 8 MLE vs. ρ an c, ρ =., Y =, an a i =. (ii) Effects of atomic coherences on ynamics In orer to stuy the effects of atomic coherence ρ i for fie atomic population, we write ρ i = a i ρii ρ with a i. The complete, partial, an zero atomic coherence between level i an level correspons to a i =, < a i <, an a i =, respectively a ρ.8..4 (c) a.4 a ρ.4..4 ρ.8.4 Fig. 9 MLE vs. ρ an a 3 for c =, Y =, ρ =., an a = a 3 = ; MLE vs. ρ an a for c =, Y =, ρ =., an a 3 = a 3 = ; (c) MLE vs. ρ an a 3 for c =, Y =, ρ =., an a = a 3 =. Figure 9 gives the relations between MLE an a i. Figures 9 an 9 show that the atomic coherences ρ an ρ 3 have strong effects on the chaotic ynamics. The strange chaotic attractor can be inhibite by ρ or ρ 3 for certain ρ. In this case, the stationary stable solution of the laser fiel is always more than zero, that is >. The reason for this is that the fiel is riven by atomic coherences. The system unergoes the strange chaotic states an metastable states, an eventually to stationary stable states with increasing a or a 3. Figure 9(c) shows that the atomic coherence ρ 3 has no effect on the chaotic ynamics. The region of atomic population in which chaos occurs ecreases with increasing a, a 3. Figure gives the relation between MLE an atomic population for complete atomic coherences a = a 3 = a 3 =. Compare with Fig. 4, the chaotic region is greatly ecrease.

7 48 XIE Xia, HUANG Hong-Bin, QIAN Feng, ZHANG Ya-Jun, YANG Peng, an QI Guan-Xiao Vol ρ ρ Fig. MLE vs. ρ an ρ for a = a 3 = a 3 =, other variables are the same as those in Fig. 4. Y ρ Fig. MLE vs. ρ an Y for σ =.453, b =.778, c =, ρ =., an a i = (iii) Effect of inecte classical fiel on ynamics The classical fiel Y inecte into the cavity also has strong effect on the strange attractor. The state will go from strange chaotic attractor to stable state with increasing Y. Figure ehibits the epenence of MLE on Y for ρ =. an a i =. 4 Conclusion We have erive the ynamic equations of the cascae egenerate two-photon laser by using the quantum Langenvin equation. These ynamic equations have both laser an optical bistability solutions. Lorenz-like strange attractor eists in the solution of the equations with larger Lyapunov imension than that in usual Lorenz equations. Holf bifurcation an perio oubling bifurcation are foun as the roas to chaos. The ynamics has epenence on initial atomic populations, atomic coherences, an inecte classical fiel. The point to be stresse is that the chaos can be inhibite by those factors. Another phenomenon shoul be note is that there is no chaos in the optical bistability. References [] E.N. Lorenz, J. Atmos. Sci. (963) 3. [] H. Haken, Phys. Lett. A 53 (975) 77. [3] C.O. Weiss an R. Vilaseca, Dynamics of Lasers, VCH, Weinheim, Germany (99). [4] X.L. Deng, H.Q. Ma, B.D. Chen, an H.B. Huang, Phys. Lett. A 9 () 77. [5] H.P. Yuan, Phys. Rev. A 3 (976) 6; M.O. Scully, K. Wókiewicz, M.S. Zubairy, J. Bergou, N. Lu, an J. Meyer Vehn, Phys. Rev. Lett. 6 (988) 83; N. Lu an S.Y. Zhu, Phys. Rev. A 4 (989) [6] M. Sargent III, M.O. Scully, an W.E. Lamb, Laser Physics, Aision-Wesley, Reaing, MA (974); M. La, in Statistical Physics, Phase Transition an Superconuctivity, es. M. Chritien, E.P. Gross, an S. Deeser, Goron an Breach, New York (968), Vol. II, p. 45. [7] C. Benkert an M.O. Scully, Phys. Rev. A 4 (99) 87; C. Benkert, M.O. Scully, J. Bergou, L. Daviovich, M. Hillery, an M. Orszag, Phys. Rev. A 4 (99) 756. [8] H.B. Huang, J.L. Song, J. Wei, X.L. Zhang, an H. Lu, Chin. J. Lasers B 7 (998) 33.