Study of the Double Nonlinear Quantum Resonances in Diatomic Molecules

ABSTRACT

We study the quantum dynamics of diatomic molecule driven by a circularly polarized resonant electric field. We look for a quantum effect due to classical chaos appearing due to the overlapping of nonlinear reso-nances associated to the vibrational and rotational motion. We solve the Schrödinger equation associated with the wave function expanded in term of proper stationary states, |n> |lm> (vibrational angular momentum states). Looking for quantum-classic correspondence, we consider the Liouville dynamics in the two dimensional phase space defined by the coherent-like state of vibrational states. We consider the rela-tionship between the overlapping of the classical resonances and the mixing of the quantum states, and it is found some similarities when the quantum dynamics is pictured in terms of number and phase operators.

We study the quantum dynamics of diatomic molecule driven by a circularly polarized resonant electric field. We look for a quantum effect due to classical chaos appearing due to the overlapping of nonlinear reso-nances associated to the vibrational and rotational motion. We solve the Schrödinger equation associated with the wave function expanded in term of proper stationary states, |n> |lm> (vibrational angular momentum states). Looking for quantum-classic correspondence, we consider the Liouville dynamics in the two dimensional phase space defined by the coherent-like state of vibrational states. We consider the rela-tionship between the overlapping of the classical resonances and the mixing of the quantum states, and it is found some similarities when the quantum dynamics is pictured in terms of number and phase operators.

Cite this paper

nullG. López and J. Zanudo, "Study of the Double Nonlinear Quantum Resonances in Diatomic Molecules,"*Journal of Modern Physics*, Vol. 2 No. 6, 2011, pp. 472-480. doi: 10.4236/jmp.2011.26057.

nullG. López and J. Zanudo, "Study of the Double Nonlinear Quantum Resonances in Diatomic Molecules,"

References

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[2] A. Messiah, Quantum Mechanics I, North Holland,John Wiley & Sons, Inc., New York, London, 29, (1964).

[3] A.J. Lichtenberg and M.A. Liberman, Regular and Stochastic Motion, Springer-Verlag, Berlin, (1983).

[4] G. Casati, B.V. Chirikov, D.L. Shepelyansky, and I. Guarnery, Relevance of classical chaos in quantum mechanics: The hydrogen atom in monochromatic field Phys. Rep., 154, 77, (1983).

[5] P. Lobastie, M.C. Bordas, B. Tribollet, and M. Boyer, Stroboscopic Effect between Electronic and Nuclear Motion in Highly Excited Molecular Rydberg Sates Phys. Rev. Lett., 52, 1681, (1984).

[6] J. Chevaleyre, C. Bordas, M. Boyer, and P. Labastie, Stark Multiplets in Molecular Rydberg States Phys. Rev. Lett., 57, 3027, (1986).

[7] C. Bordas, P.F. Brevet, M. Boyer, J. Chevaleyre, P. Labastie, and J.P. Perrot, electric-field-hinhered vibrational autoionization in molecular Rydberg states Phys. Rev. Lett., 60, 917, (1988).

[8] M. Lombardi, P. Labastie, M.C. Bordas, and M. Boyer, Molecular Rydberg states: Clasical chaos and its correspondence in quantum mechanics, J. Chem. Phys., 89, 3479, (1988).

[9] M. Lombardi and T.H. Seligman, Universal and nonuniversal statistical properties of levels and intensities for chaotic Rydberg molecules Phys. Rev. A, 47, 3571, (1993).

[10] J.J. Kay, S.L. Coy, V.S. Petrovi , B.M. Wong, and R.W. Field, Separation of long-range and short-range interactions in Rydberg states of diatomic molecules J. Chem. Phys., 128, 194301, (2008).

[11] D. Sugny, L. Bomble, T. Ribeyre, O. Dulieu, and M. Desouter-Lecomte, Rotovibrational controlled-NOT gates using optimized stimulated Raman adiabatic passage techniques and optimal control theory Phys. Rev. A, 80, 042325, (2009).

[12] A. Ruiz, J.P. Palao, and E.J. Heller, Nearly resonant multidimensional systems under a transient perturbative interaction Phys. Rev. E, 80, 066606, (2009).

[13] B.V. Chirikov, A universal instability of many-dimensional oscillator systems Phys. Rep. 52, 263, (1979).

[14] é. V. Shuryak, Nonlinear resonances in quantum systems Sov. Phys. JEPT, 44, 1070, (1976).

[15] R.P. Parson, Vibrational adiabaticity and infrared multiphoton dynamics J. Chem. Phys. 88, 3655, (1987).

[16] P.S. Dardi and K. Gray Classical and quantum mechanics studies of HF in an intense laser field J. Chem. Phys. 77, 1345, (1982).

[17] G.P. Berman and A.R. Kolovsky, Quantum chaos in a diatomic molecule interacting with a resonant field Sov. Phys. JEPT 68, 898, (1989).

[18] G.P. Berman and A.R. Kolovsky, Quantum chaos in interactions of multilevel quantum systems with a coherent radiation field Sov. Phys. Usp. 35, 303, (1992).

[19] P. M. Morse, Diatomic Molecules According to the Wave Mechanics II. Vibrational Lavels Phys. Rev. 34, 57 (1929).

[20] G. P. Berman, E. N. Bulgakov and D. D. Holm, Nonlinear resonance and dynamical chaos in a diatomic molecule driven by a resonant RF field Phys. Rev. A 52, 3074 (1995).

[21] Same as (13).

[22] L. Susskind and J. Glogower, Quantum mechanical phase and time operator, Physics (Long Island City, N.Y.) 1, 49-61 (1964).

[23] A. Lahiri, G. Ghosh, and T.K. Kar, Action-Angle variables in quantum mechanics Phys. Lett. A, 4-5, 239. (1998).

[24] P. Carruthers and M.M. Nieto, Phase and Angle Variables in Quantum Mechanics Rev. Mod. Phys 40, 411. (1968).

[25] G. J. Milburn, Quantum and classical Liouville dynamics of the anharmonic oscillator Phys. Rev. A 33, 674 (1986).

[26] M. Burton, IC-443-The interaction of a supernova remnant with a molecular cloud Roy. Ast. Soc., 28, 269. (1987).

[27] R. Chevalier, Supernova Remnants in Molecular-Clouds Astrophysical Journal 511, 798 (1999).

[28] E. Wigner, On the quantum Correction For Thermodynamic Equilibrium Phys. Rev., 40, 749 (1932).

[29] K. Husimi, Some Formal Properties of the Density Matrix Proc. Phys. Math. Soc. Japan 22, 264 (1940).

[30] R. J. Glauber, Coherent and Incoherent States of the Radiation Field Phys. Rev. 131, 2766 (1963).

[31] E.C.G. Sudarshan, Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams Phys. Rev. Lett., 10, 277 (1963).

[1] Linda E, Reichl, The Transition to Chaos, Springer-Verlag, New York, Inc., (2004).

[2] A. Messiah, Quantum Mechanics I, North Holland,John Wiley & Sons, Inc., New York, London, 29, (1964).

[3] A.J. Lichtenberg and M.A. Liberman, Regular and Stochastic Motion, Springer-Verlag, Berlin, (1983).

[4] G. Casati, B.V. Chirikov, D.L. Shepelyansky, and I. Guarnery, Relevance of classical chaos in quantum mechanics: The hydrogen atom in monochromatic field Phys. Rep., 154, 77, (1983).

[5] P. Lobastie, M.C. Bordas, B. Tribollet, and M. Boyer, Stroboscopic Effect between Electronic and Nuclear Motion in Highly Excited Molecular Rydberg Sates Phys. Rev. Lett., 52, 1681, (1984).

[6] J. Chevaleyre, C. Bordas, M. Boyer, and P. Labastie, Stark Multiplets in Molecular Rydberg States Phys. Rev. Lett., 57, 3027, (1986).

[7] C. Bordas, P.F. Brevet, M. Boyer, J. Chevaleyre, P. Labastie, and J.P. Perrot, electric-field-hinhered vibrational autoionization in molecular Rydberg states Phys. Rev. Lett., 60, 917, (1988).

[8] M. Lombardi, P. Labastie, M.C. Bordas, and M. Boyer, Molecular Rydberg states: Clasical chaos and its correspondence in quantum mechanics, J. Chem. Phys., 89, 3479, (1988).

[9] M. Lombardi and T.H. Seligman, Universal and nonuniversal statistical properties of levels and intensities for chaotic Rydberg molecules Phys. Rev. A, 47, 3571, (1993).

[10] J.J. Kay, S.L. Coy, V.S. Petrovi , B.M. Wong, and R.W. Field, Separation of long-range and short-range interactions in Rydberg states of diatomic molecules J. Chem. Phys., 128, 194301, (2008).

[11] D. Sugny, L. Bomble, T. Ribeyre, O. Dulieu, and M. Desouter-Lecomte, Rotovibrational controlled-NOT gates using optimized stimulated Raman adiabatic passage techniques and optimal control theory Phys. Rev. A, 80, 042325, (2009).

[12] A. Ruiz, J.P. Palao, and E.J. Heller, Nearly resonant multidimensional systems under a transient perturbative interaction Phys. Rev. E, 80, 066606, (2009).

[13] B.V. Chirikov, A universal instability of many-dimensional oscillator systems Phys. Rep. 52, 263, (1979).

[14] é. V. Shuryak, Nonlinear resonances in quantum systems Sov. Phys. JEPT, 44, 1070, (1976).

[15] R.P. Parson, Vibrational adiabaticity and infrared multiphoton dynamics J. Chem. Phys. 88, 3655, (1987).

[16] P.S. Dardi and K. Gray Classical and quantum mechanics studies of HF in an intense laser field J. Chem. Phys. 77, 1345, (1982).

[17] G.P. Berman and A.R. Kolovsky, Quantum chaos in a diatomic molecule interacting with a resonant field Sov. Phys. JEPT 68, 898, (1989).

[18] G.P. Berman and A.R. Kolovsky, Quantum chaos in interactions of multilevel quantum systems with a coherent radiation field Sov. Phys. Usp. 35, 303, (1992).

[19] P. M. Morse, Diatomic Molecules According to the Wave Mechanics II. Vibrational Lavels Phys. Rev. 34, 57 (1929).

[20] G. P. Berman, E. N. Bulgakov and D. D. Holm, Nonlinear resonance and dynamical chaos in a diatomic molecule driven by a resonant RF field Phys. Rev. A 52, 3074 (1995).

[21] Same as (13).

[22] L. Susskind and J. Glogower, Quantum mechanical phase and time operator, Physics (Long Island City, N.Y.) 1, 49-61 (1964).

[23] A. Lahiri, G. Ghosh, and T.K. Kar, Action-Angle variables in quantum mechanics Phys. Lett. A, 4-5, 239. (1998).

[24] P. Carruthers and M.M. Nieto, Phase and Angle Variables in Quantum Mechanics Rev. Mod. Phys 40, 411. (1968).

[25] G. J. Milburn, Quantum and classical Liouville dynamics of the anharmonic oscillator Phys. Rev. A 33, 674 (1986).

[26] M. Burton, IC-443-The interaction of a supernova remnant with a molecular cloud Roy. Ast. Soc., 28, 269. (1987).

[27] R. Chevalier, Supernova Remnants in Molecular-Clouds Astrophysical Journal 511, 798 (1999).

[28] E. Wigner, On the quantum Correction For Thermodynamic Equilibrium Phys. Rev., 40, 749 (1932).

[29] K. Husimi, Some Formal Properties of the Density Matrix Proc. Phys. Math. Soc. Japan 22, 264 (1940).

[30] R. J. Glauber, Coherent and Incoherent States of the Radiation Field Phys. Rev. 131, 2766 (1963).

[31] E.C.G. Sudarshan, Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams Phys. Rev. Lett., 10, 277 (1963).