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Routes to chaos in the Maxwell-Bloch equations
Volume 53, number 2
OPTICS COMMUNICATIONS
15 February 1985
ROUTES TO CHAOS IN THE MAXWELL-BLOCH EQUATIONS P.W. MILONNI * and J.R. ACKERHALT Theoretical Division (T-12), Los Alamos National Laboratory, Los Alamos, NM 87545, USA
and Mei-Li SHIH Department o f Physics, University o f Arkansas, Fayetteville, A R 72701, USA
Received 22 August 1984 Revised manuscript received 6 December 1984
We consider the single-mode MaxweU-Blochequations describing a Doppler-broadening ring laser. For different parameter ranges this system undergoes period-doubling, intermittency, and two-frequency routes to chaos as a single parameter (e.g., detuning) is varied, as has been observed in recent experiments with He-Ne and He-Xe lasers.
The Maxwell-Bloch equations describe the resonant interaction o f light with matter [ 1 ]. In various forms they describe a large variety o f interesting nonlinear phenomena, such as soliton propagation (self-induced transparency) [ 1 - 3 ] , photon echoes [1], and optical bistability [4]. They also form the basis of the semiclassical theory o f the laser [5 ]. In this letter we present evidence showing that under certain conditions the Maxwell-Bloch system is deterministically chaotic, and displays "universal" routes to chaos that are of much current interest in dynamical systems theory. Besides adding to the rich variety of behavior already known for the Maxwell-Bloch equations, these results are o f interest in connection with recent observations o f universal routes to chaos in He-Ne and He-Xe lasers [6,7]. We believe this is the first instance where a physically relevant set of equations for a realistic choice o f parameters is both tractable and exhibits all three "universal" routes to chaos. In fluid flow all three o f these routes to chaos have also been observed; however, the dynamical system o f equations is practically intractable. We consider Maxwell-Bloch equations under the * Permanent address: Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA. 0 030-4018/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
simplifying assumption that the spatial dependence o f the atomic and field variables may be ignored. The resuiting equations describe, within certain approximations (e.g., slowly-varying amplitudes, plane-wave field) a unidirectional ring laser: Jc = --( A -- D - k v ) y - / i x ,
(1)
f: = - - ( A _ ~ -- k o ) x - BY + ~ ( z 2 - Z l ) ,
(2)
1
z~2 = R 2 - 72z2 - ~ y ,
(3)
i 1 = R 1 - 71Zl + ½ a y ,
(4)
21rNd 2 o3
~2 = -7cS2 + ~
J
dv W ( v ) y ( v , t)
(5)
_oo
~ 2 = - 2zrNd2 •
h
fo. d o W ( v ) x ( v , t ) .
(6)
--oo
Here ~2 = de/h is the Rabi frequency [1 ] : z 2 and z 1 are the upper- and lower-state occupation probabilities;x and y are the in-phase and in-quadrature (with the field) parts of the expectation value o f the induced dipole m o m e n t ; k v is the Doppler shift for an atom with velocity v along the optical axis, with W(v) the 133
Volume 53, number 2
O4
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OPTICS COMMUNICATIONS
04
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15 February 1985
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Fig. 1. Period-doubling route to chaos with varying detuning. (a) A = #, period 1; (b) £x = 40, period 2; (c) £x = 4.60, period 4; (d) A = 6~, period 16; (e) ZX= 8#, chaos.
Volume 53, number 2
OPTICS COMMUNICATIONS
one-dimensional Maxwell-Boltzmann distribution; A is the detuning o f the field from the center o f the atomic l i n e ; d and co are the transition dipole moment and (angular) f r e q u e n c y ; N is the number density o f active atoms;/3, 7 1 , 7 2 , R 1 and R 2 are damping and pumping rates associated with the transition, and in particular/3 = 2rr6v0, where 6v 0 is the homogeneous linewidth (HWHM) o f the transition;Tc is the field loss rate, with the electric field given by E ( z , t) = e(t) exp [if(t)] exp [i(kz -
~ot)]
.
(7)
Casperson [8] considered essentially the same system in b o t h the traveling- and standing-wave cases, and showed that the "bad-cavity" (7c >/3) instability is more easily realizable when inhomogeneous broadening is dominant * 1. He gave numerical and experimental evidence that a cw-pumped, Doppler-broadened laser could have a pulsating output, and he suggested that in some cases the output might be chaotic. Our computations support this conjecture. The recent experiments o f Abraham [7], Arecchi [ 11 ], and Weiss [6], and their collaborators, has shown that lasers can indeed operate in classifiably chaotic ways. Three wellcharacterized routes to chaos in dissipative systems the Feigenbaum (period doubling), Ruelle-Takens (two-frequency), and Pomeau-Manneville (intermittency) scenarios - have been nearly observed in their experiments with gas lasers [6,7] ,2 We have observed chaotic behavior for a considerable range o f parameters. Here we consider a few exampies, always assuming x = y = z I = z 2 = 0 initially, with having some small initial value in order to start the laser in the semiclassical approach based on (1)--(6); in a fully quantum-mechanical approach this initial "noise" is due to spontaneous emission. Typically ~ 5 0 velocity groups or more are needed to accurately simulate the continuous velocity distribution appearing in (5) and (6), so that our dynamical system on a computer consists o f ~102 ordinary differential equations. The accuracy o f the simulation is tested b y checking that the results are not altered by increasing the number o f velocity groups beyond a chosen value. The computations are also checked by decreasing the step
15 February 1985
size in a (fourth-order) Runge-Kutta integrator, or by using a different algorithm (e.g., a predictor-corrector routine). Additional numerical examples will be presented elsewhere. Fig. 1a shows the intracavity intensity as a function o f time, after initial transients have died out. We have taken/3 = 50 MHz, 71 = 1.8/3, 72 = 0.057/3, 7c = 2.1/3, 2rcNd2co/h = 3.6 X 1021 s - 2 , and assumed a Doppler width (FWHM) 6v 0 = 110 MHz. R 1 and R 2 have been taken to be 0 and 5 X 10 - 6 t , respectively, giving a line-center small-signal gain o f 0.09 cm -1 . In fig. l a the detuning A =/3, and we observe regular pulsations o f the intensity at a frequency ~ 2 2 MHz. As the detuning is increased we observe successive period doublings and eventually chaos (figs. l b - e ) . This is the "universal" period-doubling route to chaos, as exemplified by the mapping Xn+ 1 = r x n (1 - Xn) with increasing parameter r ( 0 ~< r ~< 4) [ 13 ]. In the case o f fig. 1 the parameter that is varied is A, but we have also observed the period-doubling route to chaos by varying other parameters (e.g., R 2 or 7) [14]. In other cases we observe the development of chaos via increasingly frequent bursts o f chaotic oscillations. An example o f such a chaotic burst is shown in fig. 2. Such an intermittency route to chaos was observed experimentally by Weiss et al. [6] and was also inferred from power spectra by Gioggia and Abraham [7]. In some instances we have found long-lived metastable
e4 120 E
°
80
E 2~
c-
40
C
0 0
2
4
6
Time(microsec)
:1:1The Haken-Lorenz instability applies directly to the case of homogeneous broadening; see Haken [9]. 42 Chaos in a bad-cavity unidirectional ring laser has also been reported in ref. [12].
Fig. 2. A chaotic burst computed for the parameters of fig. 1, except that A = 0 and R 2 = 1.2875 X 10-s/3. As R 2 is increased the chaotic bursts become more frequent, and eventually the motion is fully chaotic. 135
OPTICS COMMUNICATIONS
Volume 53, number 2 i
i
o
15 February 1985
have been extended or clarified [ 15,16]. Furthermore the appearance of deterministic chaos in the MaxwellBloch equations with inhomogeneous broadening has been studied independently by Bandy et al. [17]. Research at Arkansas was supported by NSF grant PHY-8308040. We are also grateful to N.B. Abraham, G. Mayer-Kress and D.K. Umberger for helpful remarks.
E "6 £2-
References
--5 i
0
2O
4O
Frequency(mhz) Fig. 3. Power spectrum of the intensity for the parameters of fig. 1, except that now A = 2.2#, 7C = 2.8/3, and R2 = 1.01 X 10-5 /3. The peaks in the spectrum are linear combinations of two basic frequencies, the smaller one corresponding to a slow modulation of higher-frequency pulsations. As 3'c is reduced the broadband component of the spectrum grows with the development of chaos. chaos followed by a transition to a constant intensity. Chaos can also occur after the development of two initially incommensurate frequencies (and their harmonics) in the dynamics. Fig. 3 is an example of such a spectrum. After the onset of the two frequencies the spectrum quickly becomes broadband (a signature of chaos) as the control parameter is varied. From a purely theoretical perspective it is remarkable that the ubiquitous system [ 1 - 6 ] displays all this rich and chaotic behavior that is of so much current interest. At the present time a detailed comparison of these computations with experimental results for He-Ne and He-Xe lasers is complicated by the fact that some of the rates, particularly/3, are not very accurately k n o w n ; the particular route to chaos followed in our numerical experiments can depend sensitively on/~. However, our results show that standard semiclassical laser theory predicts the same routes to chaos that have been observed so far in single-mode chaotic lasers. Finally, we should point out that several papers have recently appeared in which the earlier stability analyses
136
[ 1] See L. Allen and J.H. Eberly, Optical resonance and twolevel atoms (Wiley, N.Y., 1975). [2] S.L. McCall and E.L. Hahn, Phys. Rev. 183 (1969) 457. [3] G.L. Lamb, Elements of soliton theory (Wiley, N.Y., 1980). [4] E. Abraham and S.D. Smith, Rep. Prog. Phys. 45 (1982) 815. [5] See M. Sargent III, M.O. Scully and W.E. Lamb Jr., Laser physics (Addison-Wesley,Reading, 1974). [6] C.O. Weiss and H. King, Optics Comm. 44 (1982) 59; C.O. Weiss,A. Godone and A. Olafsson, Phys. Rev. A28 (1983) 892. [7] R.S. Gioggia and N.B. Abraham, Phys. Rev. Lett. 51 (1983) 650; see also N.B. Abraham, T. Chyba, M. Coleman, R.S. Gioggia, N.J. Halas, L.M. Hoffer, S.-N. Liu, M. Maeda and J.C. Wesson, in: Laser physics, eds. J.D. Harvey and D.F. Walls (Springer-Verlag,Berlin, 1983) pp. 107-131. [8] L.W. Casperson, IEEE J. Quantum Electron. QE-14 (1978) 756; Phys. Rev. A21 (1980) 911. [9] H. Haken, Phys. Lett. 53A (1975) 77. [10] L.W. Casperson, Laser physics, eds. J.D. Harvey and D.F. Walls (Springer-Verlag,Berlin, 1983) p. 88. [ 11] F.T. Arecchi, R. Meucci, G. Puccioni and J. Tredicce, Phys. Rev. Lett. 49 (1982) 1217. [12] L.E. Urbach, S.-N. Liu and N.B. Abraham, in: Coherence and quantum optics V, eds. L. Mandel and E. Wolf (Plenum, N.Y., 1984). [ 13] M.J. Feigenbaum, J. Stat. Phys. 19 (1978) 25 ; 21 (1979) 669. [14] M.-L. Shih, P.W. Milonni and J.R. Ackerhalt, J. Opt. Soc. Am. B, to be published, January, 1985. [ 15 ] P. Mandel, Optics Comm. 44 (1983) 400; 45 (1983) 269. [ 16 ] L.A. Lugiato, L.M. Narducci, D.K. Bandy and N.B. Abraham, Optics Comm. 46 (1983) 115. [17] D.K. Bandy, L.M. Narducci, L.A. Lugiato and N.B. Abraham, J. Opt. Soc. Am. B, to be published, January, 1985.
OPTICS COMMUNICATIONS
15 February 1985
ROUTES TO CHAOS IN THE MAXWELL-BLOCH EQUATIONS P.W. MILONNI * and J.R. ACKERHALT Theoretical Division (T-12), Los Alamos National Laboratory, Los Alamos, NM 87545, USA
and Mei-Li SHIH Department o f Physics, University o f Arkansas, Fayetteville, A R 72701, USA
Received 22 August 1984 Revised manuscript received 6 December 1984
We consider the single-mode MaxweU-Blochequations describing a Doppler-broadening ring laser. For different parameter ranges this system undergoes period-doubling, intermittency, and two-frequency routes to chaos as a single parameter (e.g., detuning) is varied, as has been observed in recent experiments with He-Ne and He-Xe lasers.
The Maxwell-Bloch equations describe the resonant interaction o f light with matter [ 1 ]. In various forms they describe a large variety o f interesting nonlinear phenomena, such as soliton propagation (self-induced transparency) [ 1 - 3 ] , photon echoes [1], and optical bistability [4]. They also form the basis of the semiclassical theory o f the laser [5 ]. In this letter we present evidence showing that under certain conditions the Maxwell-Bloch system is deterministically chaotic, and displays "universal" routes to chaos that are of much current interest in dynamical systems theory. Besides adding to the rich variety of behavior already known for the Maxwell-Bloch equations, these results are o f interest in connection with recent observations o f universal routes to chaos in He-Ne and He-Xe lasers [6,7]. We believe this is the first instance where a physically relevant set of equations for a realistic choice o f parameters is both tractable and exhibits all three "universal" routes to chaos. In fluid flow all three o f these routes to chaos have also been observed; however, the dynamical system o f equations is practically intractable. We consider Maxwell-Bloch equations under the * Permanent address: Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA. 0 030-4018/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
simplifying assumption that the spatial dependence o f the atomic and field variables may be ignored. The resuiting equations describe, within certain approximations (e.g., slowly-varying amplitudes, plane-wave field) a unidirectional ring laser: Jc = --( A -- D - k v ) y - / i x ,
(1)
f: = - - ( A _ ~ -- k o ) x - BY + ~ ( z 2 - Z l ) ,
(2)
1
z~2 = R 2 - 72z2 - ~ y ,
(3)
i 1 = R 1 - 71Zl + ½ a y ,
(4)
21rNd 2 o3
~2 = -7cS2 + ~
J
dv W ( v ) y ( v , t)
(5)
_oo
~ 2 = - 2zrNd2 •
h
fo. d o W ( v ) x ( v , t ) .
(6)
--oo
Here ~2 = de/h is the Rabi frequency [1 ] : z 2 and z 1 are the upper- and lower-state occupation probabilities;x and y are the in-phase and in-quadrature (with the field) parts of the expectation value o f the induced dipole m o m e n t ; k v is the Doppler shift for an atom with velocity v along the optical axis, with W(v) the 133
Volume 53, number 2
O4
E
OPTICS COMMUNICATIONS
04
80
15 February 1985
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Fig. 1. Period-doubling route to chaos with varying detuning. (a) A = #, period 1; (b) £x = 40, period 2; (c) £x = 4.60, period 4; (d) A = 6~, period 16; (e) ZX= 8#, chaos.
Volume 53, number 2
OPTICS COMMUNICATIONS
one-dimensional Maxwell-Boltzmann distribution; A is the detuning o f the field from the center o f the atomic l i n e ; d and co are the transition dipole moment and (angular) f r e q u e n c y ; N is the number density o f active atoms;/3, 7 1 , 7 2 , R 1 and R 2 are damping and pumping rates associated with the transition, and in particular/3 = 2rr6v0, where 6v 0 is the homogeneous linewidth (HWHM) o f the transition;Tc is the field loss rate, with the electric field given by E ( z , t) = e(t) exp [if(t)] exp [i(kz -
~ot)]
.
(7)
Casperson [8] considered essentially the same system in b o t h the traveling- and standing-wave cases, and showed that the "bad-cavity" (7c >/3) instability is more easily realizable when inhomogeneous broadening is dominant * 1. He gave numerical and experimental evidence that a cw-pumped, Doppler-broadened laser could have a pulsating output, and he suggested that in some cases the output might be chaotic. Our computations support this conjecture. The recent experiments o f Abraham [7], Arecchi [ 11 ], and Weiss [6], and their collaborators, has shown that lasers can indeed operate in classifiably chaotic ways. Three wellcharacterized routes to chaos in dissipative systems the Feigenbaum (period doubling), Ruelle-Takens (two-frequency), and Pomeau-Manneville (intermittency) scenarios - have been nearly observed in their experiments with gas lasers [6,7] ,2 We have observed chaotic behavior for a considerable range o f parameters. Here we consider a few exampies, always assuming x = y = z I = z 2 = 0 initially, with having some small initial value in order to start the laser in the semiclassical approach based on (1)--(6); in a fully quantum-mechanical approach this initial "noise" is due to spontaneous emission. Typically ~ 5 0 velocity groups or more are needed to accurately simulate the continuous velocity distribution appearing in (5) and (6), so that our dynamical system on a computer consists o f ~102 ordinary differential equations. The accuracy o f the simulation is tested b y checking that the results are not altered by increasing the number o f velocity groups beyond a chosen value. The computations are also checked by decreasing the step
15 February 1985
size in a (fourth-order) Runge-Kutta integrator, or by using a different algorithm (e.g., a predictor-corrector routine). Additional numerical examples will be presented elsewhere. Fig. 1a shows the intracavity intensity as a function o f time, after initial transients have died out. We have taken/3 = 50 MHz, 71 = 1.8/3, 72 = 0.057/3, 7c = 2.1/3, 2rcNd2co/h = 3.6 X 1021 s - 2 , and assumed a Doppler width (FWHM) 6v 0 = 110 MHz. R 1 and R 2 have been taken to be 0 and 5 X 10 - 6 t , respectively, giving a line-center small-signal gain o f 0.09 cm -1 . In fig. l a the detuning A =/3, and we observe regular pulsations o f the intensity at a frequency ~ 2 2 MHz. As the detuning is increased we observe successive period doublings and eventually chaos (figs. l b - e ) . This is the "universal" period-doubling route to chaos, as exemplified by the mapping Xn+ 1 = r x n (1 - Xn) with increasing parameter r ( 0 ~< r ~< 4) [ 13 ]. In the case o f fig. 1 the parameter that is varied is A, but we have also observed the period-doubling route to chaos by varying other parameters (e.g., R 2 or 7) [14]. In other cases we observe the development of chaos via increasingly frequent bursts o f chaotic oscillations. An example o f such a chaotic burst is shown in fig. 2. Such an intermittency route to chaos was observed experimentally by Weiss et al. [6] and was also inferred from power spectra by Gioggia and Abraham [7]. In some instances we have found long-lived metastable
e4 120 E
°
80
E 2~
c-
40
C
0 0
2
4
6
Time(microsec)
:1:1The Haken-Lorenz instability applies directly to the case of homogeneous broadening; see Haken [9]. 42 Chaos in a bad-cavity unidirectional ring laser has also been reported in ref. [12].
Fig. 2. A chaotic burst computed for the parameters of fig. 1, except that A = 0 and R 2 = 1.2875 X 10-s/3. As R 2 is increased the chaotic bursts become more frequent, and eventually the motion is fully chaotic. 135
OPTICS COMMUNICATIONS
Volume 53, number 2 i
i
o
15 February 1985
have been extended or clarified [ 15,16]. Furthermore the appearance of deterministic chaos in the MaxwellBloch equations with inhomogeneous broadening has been studied independently by Bandy et al. [17]. Research at Arkansas was supported by NSF grant PHY-8308040. We are also grateful to N.B. Abraham, G. Mayer-Kress and D.K. Umberger for helpful remarks.
E "6 £2-
References
--5 i
0
2O
4O
Frequency(mhz) Fig. 3. Power spectrum of the intensity for the parameters of fig. 1, except that now A = 2.2#, 7C = 2.8/3, and R2 = 1.01 X 10-5 /3. The peaks in the spectrum are linear combinations of two basic frequencies, the smaller one corresponding to a slow modulation of higher-frequency pulsations. As 3'c is reduced the broadband component of the spectrum grows with the development of chaos. chaos followed by a transition to a constant intensity. Chaos can also occur after the development of two initially incommensurate frequencies (and their harmonics) in the dynamics. Fig. 3 is an example of such a spectrum. After the onset of the two frequencies the spectrum quickly becomes broadband (a signature of chaos) as the control parameter is varied. From a purely theoretical perspective it is remarkable that the ubiquitous system [ 1 - 6 ] displays all this rich and chaotic behavior that is of so much current interest. At the present time a detailed comparison of these computations with experimental results for He-Ne and He-Xe lasers is complicated by the fact that some of the rates, particularly/3, are not very accurately k n o w n ; the particular route to chaos followed in our numerical experiments can depend sensitively on/~. However, our results show that standard semiclassical laser theory predicts the same routes to chaos that have been observed so far in single-mode chaotic lasers. Finally, we should point out that several papers have recently appeared in which the earlier stability analyses
136
[ 1] See L. Allen and J.H. Eberly, Optical resonance and twolevel atoms (Wiley, N.Y., 1975). [2] S.L. McCall and E.L. Hahn, Phys. Rev. 183 (1969) 457. [3] G.L. Lamb, Elements of soliton theory (Wiley, N.Y., 1980). [4] E. Abraham and S.D. Smith, Rep. Prog. Phys. 45 (1982) 815. [5] See M. Sargent III, M.O. Scully and W.E. Lamb Jr., Laser physics (Addison-Wesley,Reading, 1974). [6] C.O. Weiss and H. King, Optics Comm. 44 (1982) 59; C.O. Weiss,A. Godone and A. Olafsson, Phys. Rev. A28 (1983) 892. [7] R.S. Gioggia and N.B. Abraham, Phys. Rev. Lett. 51 (1983) 650; see also N.B. Abraham, T. Chyba, M. Coleman, R.S. Gioggia, N.J. Halas, L.M. Hoffer, S.-N. Liu, M. Maeda and J.C. Wesson, in: Laser physics, eds. J.D. Harvey and D.F. Walls (Springer-Verlag,Berlin, 1983) pp. 107-131. [8] L.W. Casperson, IEEE J. Quantum Electron. QE-14 (1978) 756; Phys. Rev. A21 (1980) 911. [9] H. Haken, Phys. Lett. 53A (1975) 77. [10] L.W. Casperson, Laser physics, eds. J.D. Harvey and D.F. Walls (Springer-Verlag,Berlin, 1983) p. 88. [ 11] F.T. Arecchi, R. Meucci, G. Puccioni and J. Tredicce, Phys. Rev. Lett. 49 (1982) 1217. [12] L.E. Urbach, S.-N. Liu and N.B. Abraham, in: Coherence and quantum optics V, eds. L. Mandel and E. Wolf (Plenum, N.Y., 1984). [ 13] M.J. Feigenbaum, J. Stat. Phys. 19 (1978) 25 ; 21 (1979) 669. [14] M.-L. Shih, P.W. Milonni and J.R. Ackerhalt, J. Opt. Soc. Am. B, to be published, January, 1985. [ 15 ] P. Mandel, Optics Comm. 44 (1983) 400; 45 (1983) 269. [ 16 ] L.A. Lugiato, L.M. Narducci, D.K. Bandy and N.B. Abraham, Optics Comm. 46 (1983) 115. [17] D.K. Bandy, L.M. Narducci, L.A. Lugiato and N.B. Abraham, J. Opt. Soc. Am. B, to be published, January, 1985.