The classical mechanics of a system of two nonlinearly coupled oscillators driven by an oscillating electric field is studied. The presence of quasiperiodic and chaotic motion in the unforced system is shown to influence the nature of energy absorption. Two essentially different types of behavior are observed. In the first, energy is exchanged in a multiply periodic manner between the system and the forcing field. In the second regime, the energy exchange is erratic and a statistical analysis of a family of trajectories shows the role of the chaotic motion in the unforced system in the dissociation process. A theory for rate of photodissociation is presented and results are compared with those obtained from an ensemble of exact classical trajectories.
REFERENCES
1.
See, for example,
N.
Bloembergen
and E.
Yablonoviteh
, Phys. Today
31
, 23
(1978
);M. F. Goodman, J. Stone, and E. Thiele, in Multiple Photon Excitation and Dissociation of Polyatomic Molecules, Topics in Current Physics, edited by C. D. Cantrell (Springer, New York, 1980).
2.
See, for example, (a) R. V. Ambartzumian and V. S. Letokhov, Chemical and Biochemical Applications of Lasers, edited by C. B. Moore (Academic, New York, 1977), Vol. 2;
(b) D. Bomse, R. L. Woodin, and J. Beauchamp, in Advances in Laser Chemistry, edited by A. H. Zewail (Springer, New York, 1978);
(c)
Aa.
Sudbo/
, P. A.
Schulz
, E. R.
Grant
, Y. R.
Shen
, and Y. T.
Lee
, J. Chem. Phys.
70
, 912
(1978
);Aa.
Sudbo/
, P. A.
Schulz
, E. R.
Grant
, Y. R.
Shen
, and Y. T.
Lee
, 69
, 2312
(1978
); , J. Chem. Phys.
(d)
P. A.
Hackett
, C.
Willis
, and M.
Gauthier
, J. Chem. Phys.
71
, 2682
(1979
);(e)
P.
Kolodner
, C.
Winterfield
, and E.
Yablonoviteh
, Opt. Commun.
20
, 119
(1977
).3.
(b) N. Bloembergen, C. Cantrell, and D. Larsen, in Tunable Lasers and Applications, edited by A. Mooradian, T. Jaeger, and P. Stokseth (Springer, New York, 1976);
(e)
E. R.
Grant
, P. A.
Schulz
, Aa. S.
Sudbo/
, Y. R.
Shen
, and Y. T.
Lee
, Phys. Rev. Lett.
40
, 115
(1978
);(f)
J. G.
Black
, E.
Yablonoviteh
, N.
Bloembergen
, and S.
Mukamel
, Phys. Rev. Lett.
38
, 1131
(1977
);4.
also, W. E. Lamb, paper presented at Conference on Laser Chemistry, Steamboat Springs, Colorado, 1976.
5.
(b)
D.
Noid
, M. L.
Koszykowski
, R. A.
Marcus
, and J. D.
McDonald
, Chem. Phys. Lett.
51
, 540
(1977
);6.
P. R. Sethna, in Nonlinear Differential Equations and Nonlinear Mechanics, edited by J. P. LaSalle and S. Lefschetz (Academic, New York, 1963). Equation (4.5) was obtained for the case by assuming (see Ref. 20). In an exact solution of Eq. (4.5), contains an additional term that is periodic in t and has a time average of zero. For simplicity, this term was neglected.
7.
See, for example, H. T. Davis, Introduction to Nonlinear Differential and Integral Equations (Dover, New York, 1954);
W. J. Cunningham, Introduction to Nonlinear Analysis (McGraw‐Hill, New York, 1958);
A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, New York, 1979).
8.
(a) V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978);
(b) J. Ford, in Fundamental Problems in Statistical Mechanics, edited by E. D. G. Cohen (Elsevier, New York, 1975);
(c) See Appendix 8 in Ref. 8(a).
9.
D. W.
Noid
, M. L.
Koszykowski
, and R. A.
Marcus
, J. Chem. Phys.
71
, 2864
(1979
), and references therein.10.
(a) See P. Brumer, Adv. Chem. Phys. (in press) for a review;
(b) See M. Tabor, Adv. Chem. Phys. (in press) for a review.
11.
(a)
S.
Chapman
, B.
Garrett
, and W.
Miller
, J. Chem. Phys.
64
, 502
(1976
);12.
The quantum energy levels were determined using standard matrix diagonalization programs.
13.
Integrations of Eq. (3.2) were performed on a VAX 11/780 computer using Fortran programs.
14.
N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations (Hindustan, New Delhi, 1961).
15.
16.
(b) G. E. O. Giacaglia, Perturbation Methods in Nonlinear Systems (Springer, New York, 1972);
(c) The autonomous system has a near 2:1 resonance, and so the near‐resonant algorithm in (a) was used. The η term, which contributes relatively little to was neglected, and to obtain Eq. (4.15) the in Eq. (49) of Ref. 16(a) was set equal to π for simplicity. (See Ref. 20).
17.
In a preliminary report [R. Ramaswamy and R. A. Marcus, in Classical, Semiclassical, and Quantum Mechanical Problems in Mathematics, Chemistry, and Physics, edited by K. Gustavson and W. P. Reinhardt (Plenum, New York, in press)] an agreement was reported for a single numerical and perturbation calculated amplitude and period in the quasi‐periodic regime, but was fortuitous.
18.
R. A. Marcus, D. W. Noid, and M. L. Koszykowski, in Advances in Laser Chemistry, edited by A. Zewail (Springer, New York, 1978);
D. W.
Noid
, M. L.
Koszykowski
, and R. A.
Marcus
, J. Chem. Phys.
67
, 404
(1977
);19.
G.
Benettin
, L.
Galgani
, and J. M.
Strelcyn
, Phys. Rev. A
14
, 2338
(1976
).20.
Simple stability analysis [W. J. Cunningham, Introduction to Nonlinear Analysis (McGraw‐Hill, New York, 1958)] at the equilibrium points for the resonant angle θ and its conjugate momentum [which are and p in Ref. 16(a)] and are the variables and in the KBM‐type analysis shows that the equilibrium points at which or are of neutral stability. In both of the perturbation treatments in Sec. IV, θ was replaced for simplicity by one of its equilibrium values. The value chosen for the Lie transform treatment was that which gave the smallest imaginary eigenvalue. In the KBM case the eigenvalues were complex conjugates for 0 and π and the value used was an ad hoc one which gave best numerical results.
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© 1981 American Institute of Physics.
1981
American Institute of Physics
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