Signature of chaos in the semi quantum behavior of a classically regular triple well heterostructure

ABSTRACT

We analyze the phenomenon of semiquantum chaos in the classically regular triple well model from classical to quantum. His dynamics is very rich because it provides areas of regular be-havior, chaotic ones and multiple quantum tun-neling depending on the energy of the system as the Planck’s constant varies from 0 to 1. The Time Dependent Variational Principle TDVP using generalized Gaussian trial wave function, which, in many-body theory leads to the Hartree Fock Approximation TDHF, is added to the tech-niques of Gaussian effective potentials and both are used to study the system. The extended classical system with fluctuation variables non- linearly coupled to the average variables exhibit energy dependent transitions between regular behavior and semi quantum chaos monitored by bifurcation diagram together with some numerical indicators.

We analyze the phenomenon of semiquantum chaos in the classically regular triple well model from classical to quantum. His dynamics is very rich because it provides areas of regular be-havior, chaotic ones and multiple quantum tun-neling depending on the energy of the system as the Planck’s constant varies from 0 to 1. The Time Dependent Variational Principle TDVP using generalized Gaussian trial wave function, which, in many-body theory leads to the Hartree Fock Approximation TDHF, is added to the tech-niques of Gaussian effective potentials and both are used to study the system. The extended classical system with fluctuation variables non- linearly coupled to the average variables exhibit energy dependent transitions between regular behavior and semi quantum chaos monitored by bifurcation diagram together with some numerical indicators.

Cite this paper

Lekeufack, T. , Yamgoue, S. and Kofane, T. (2010) Signature of chaos in the semi quantum behavior of a classically regular triple well heterostructure.*Natural Science*, **2**, 145-154. doi: 10.4236/ns.2010.23024.

Lekeufack, T. , Yamgoue, S. and Kofane, T. (2010) Signature of chaos in the semi quantum behavior of a classically regular triple well heterostructure.

References

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[2] Ringot, J., Lecocq, Y., Garreau, J.C. and Szriftgiser, P. (1999) Generation of phase-coherent laser beams for Raman spectroscopy and cooling by direct current modulation of a diode laser.European Physical Journal D, 7, 285; Ringot, J., Szriftgiser, P., Garreau, J.C., and Delande, D. (2000) Physical Review Letters, 85, 2741.

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[5] Berry, M.V. (1989) Quantum chaology, not quantum chaos.Physica Scripta, 40, 335.

[6] Ozorio De Almeida, A. (1988) Hamiltonian systems: Chaos and quantization. Cambridge University Press, Cambridge.

[7] Gutzwiller, M.C. (1990) Chaos in classical and quantum mechanics. Springer Verlag, New York.

[8] Reichl, L.E. (1992) The transition to chaos in conservative classical systems: Quantum manifestations. Springer Verlag, New York.

[9] Sengupta, S. and Chattaraj, P.K. (1996) Physics Letters A, 215, 119.

[10] Kluger, Y., Eisenberg, J.M., Svetisky, B., Cooper, F. and Mottola, E. (1991) Physical Review Letters, 67, 2427.

[11] Pradhan, T. and Khare, A.V. (1973) American Journal of Physics, 41, 59.

[12] Jona Lasinio, G., Presilla, C. and Capasso, F. (1992). Physical Review Letters, 68, 2269.

[13] Leo, K., Shah, J., Gobel, E.O., Damen, T.C., Schmitt S., Ring, Schafer, W. and Kholer, K. (1991) Physical Review Letters, 66, 201.

[14] Elze, H.T. (1995) Quantum decoherence, entropy and thermalization in strong interactions at high energy. Nuclear Physics B, 436, 213; Nuclear Physics B, 39, 169.

[15] Jackiw, R. and Kerman, A. (1978) Physics Letters A, 71, 158.

[16] Heller, E.J. (1975) Calculations and mathematical techniques in atomic and molecular physics. Journal of Chemical Physics, 62, 1544; Heller, E.J. and Sundberg, R.L. (1985) Chaotic behavior in quantum systems. Plenum Press, New York, 255.

[17] Pattanayak, A.K. and Schieve, W.C. (1994) Physical Review Letters, 72, 2855;(1992) Physical Review Letters,46, 1821, Proceedings from workshop in honor of Sundarshan., E.G.G. Gleeson, A.M., Ed., (world scientific,Singapore, in press).

[18] Cooper, F., Pi, S.Y. and Stancioff, P.N. (1986) Physical Review D, 34, 3831.

[19] Kovner, A. and Roseinstein, B. (1983) Physical Review D, 39, 2332.

[20] Littlejohn, R.G. (1988) Physical Review Letters, 61, 2159.

[21] Tannoudji, C.C., Diu, B. and Laloe, F. (1988) Quantum mechanics, Ed., Masson.

[22] Perez, J.P. and Saint Crieq Chery, N. (1986) Relativity and Quantisation, University Paul Sabatier, Toulouse, Ed., Masson.

[23] Gaudaire, M. (1969) Propriete de la matiere: Onde et Matiere, Colorado Dunod University, Orsay.

[24] Ott, E. (1997) Chaos in dynamical systems. Cambridge University Press, Cambridge.

[25] Carlson, L. and Schieve, W.C. (1989) Physical Review A, 40, 5896.

[26] Stevenson, P. (1984) Physical Review D, 30, 1712; (1985) Physical Review D, 32, 1389.

[27] Toda, M. (1974) Instability of trajectories of the lattice with cubic nonlinearity. Physics Letters A, 48, 335.

[28] Brumer, P. and Duff, J.N. (1976) Journal of Chemical Physics, 65, 3566.

[29] Capasso, F. and Datta, S. (1990) Bandgap and interface engineering for advanced electronic and photonic devices.Physics Today, 43(2), 74.

[30] Kramer, B. (1991) Quantum coherence in mesoscopic systems. Plenum, New York.

[31] Presilla, C., Jona – Lasinio, G.. and Capasso, F. (1991) Nonlinear feedback oscillations in resonant tunneling through double barriers.Physical Review B, 43, 5200

[32] Tchoukuegno, R. and Woafo, P. (2002) Physical D 167, 86; Tchoukuegno, R., Nana, B.R. and Woafo, P. (2002) Physical Review A, 304, 362; (2003) International Journal of Non-Linear Mechanics, 38, 531.

[33] Jing, Z.J., Yang, Z.Y. and Jiang, T. (2006) Bifurcation and chaos in discrete-time predator–prey system.Chaos, Solitons and Fractals, 27, 722.

[34] Sun, Z.K., Xu, W. and Yang, X.L. (2006) Chaos, Solitons and Fractals, 27, 778.

[35] Blum, T.C. and Elze, H.T. (1996) Semiquantum chaos in the double well.Physical Review E, 53, 3123.

[36] Zaslavsky, G.M. (1985) Chaos in dynamic systems. Harwood Academic, Chur, Switzerland.

[37] Wolf, A., Swift, J. B., Swinney, H.L. and Vastano, J.A. (1985) Determining Lyapunov exponents from a time series.Physical Review D, 16, 285.

[38] Melnikov, V.K. (1963) On the stability of a center for time-periodic perturbations. Transactions of the Moscow Mathematical Society, 12, 1.

[39] Gucken, H. J. and Holmes, P. (1983) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer Verlag, Berlin.

[40] Yamgoue, S.B. and Kofane, T.C. (2002) The subharmonic Melnikov theory for damped and driven oscillators revisited .International Journal of Bifurcation and Chaos, 8, 1915; (2003) Chaos, Solitons & Fractals, 17, 155.

[41] Churchill, R.C., Pecelli, G. and Rod, L. (1975) J. Diff. Eqs. 17, 329; (1977) J. Diff. Eqs. 24, 329; Churchill, R.C. and Rod, D.L. (1976) Ibid. 21, 39; (1976) Ibid. 21, 66; (1980) Ibid. 37, 23.

[1] Cooper, F., Dawson, J.F., Meredith, D. and Shepard, H. (1994) Physical Review Letters, 72, 1337; Lichtenberg, A.J. and Lieberman, M.A. (1983) Regular and stochastic motion. Springer, New York.

[2] Ringot, J., Lecocq, Y., Garreau, J.C. and Szriftgiser, P. (1999) Generation of phase-coherent laser beams for Raman spectroscopy and cooling by direct current modulation of a diode laser.European Physical Journal D, 7, 285; Ringot, J., Szriftgiser, P., Garreau, J.C., and Delande, D. (2000) Physical Review Letters, 85, 2741.

[3] Tabor, M. (1989) Chaos and integrability in non linear dynamics. John Wiley and Sons, New York.

[4] Mackey, R.S. and Meiss, J.D. (1987) Strong variation of global-transport properties in chaotic ensembles. Hamilt- onian dynamical systems, Adam Hilger, Bristel.

[5] Berry, M.V. (1989) Quantum chaology, not quantum chaos.Physica Scripta, 40, 335.

[6] Ozorio De Almeida, A. (1988) Hamiltonian systems: Chaos and quantization. Cambridge University Press, Cambridge.

[7] Gutzwiller, M.C. (1990) Chaos in classical and quantum mechanics. Springer Verlag, New York.

[8] Reichl, L.E. (1992) The transition to chaos in conservative classical systems: Quantum manifestations. Springer Verlag, New York.

[9] Sengupta, S. and Chattaraj, P.K. (1996) Physics Letters A, 215, 119.

[10] Kluger, Y., Eisenberg, J.M., Svetisky, B., Cooper, F. and Mottola, E. (1991) Physical Review Letters, 67, 2427.

[11] Pradhan, T. and Khare, A.V. (1973) American Journal of Physics, 41, 59.

[12] Jona Lasinio, G., Presilla, C. and Capasso, F. (1992). Physical Review Letters, 68, 2269.

[13] Leo, K., Shah, J., Gobel, E.O., Damen, T.C., Schmitt S., Ring, Schafer, W. and Kholer, K. (1991) Physical Review Letters, 66, 201.

[14] Elze, H.T. (1995) Quantum decoherence, entropy and thermalization in strong interactions at high energy. Nuclear Physics B, 436, 213; Nuclear Physics B, 39, 169.

[15] Jackiw, R. and Kerman, A. (1978) Physics Letters A, 71, 158.

[16] Heller, E.J. (1975) Calculations and mathematical techniques in atomic and molecular physics. Journal of Chemical Physics, 62, 1544; Heller, E.J. and Sundberg, R.L. (1985) Chaotic behavior in quantum systems. Plenum Press, New York, 255.

[17] Pattanayak, A.K. and Schieve, W.C. (1994) Physical Review Letters, 72, 2855;(1992) Physical Review Letters,46, 1821, Proceedings from workshop in honor of Sundarshan., E.G.G. Gleeson, A.M., Ed., (world scientific,Singapore, in press).

[18] Cooper, F., Pi, S.Y. and Stancioff, P.N. (1986) Physical Review D, 34, 3831.

[19] Kovner, A. and Roseinstein, B. (1983) Physical Review D, 39, 2332.

[20] Littlejohn, R.G. (1988) Physical Review Letters, 61, 2159.

[21] Tannoudji, C.C., Diu, B. and Laloe, F. (1988) Quantum mechanics, Ed., Masson.

[22] Perez, J.P. and Saint Crieq Chery, N. (1986) Relativity and Quantisation, University Paul Sabatier, Toulouse, Ed., Masson.

[23] Gaudaire, M. (1969) Propriete de la matiere: Onde et Matiere, Colorado Dunod University, Orsay.

[24] Ott, E. (1997) Chaos in dynamical systems. Cambridge University Press, Cambridge.

[25] Carlson, L. and Schieve, W.C. (1989) Physical Review A, 40, 5896.

[26] Stevenson, P. (1984) Physical Review D, 30, 1712; (1985) Physical Review D, 32, 1389.

[27] Toda, M. (1974) Instability of trajectories of the lattice with cubic nonlinearity. Physics Letters A, 48, 335.

[28] Brumer, P. and Duff, J.N. (1976) Journal of Chemical Physics, 65, 3566.

[29] Capasso, F. and Datta, S. (1990) Bandgap and interface engineering for advanced electronic and photonic devices.Physics Today, 43(2), 74.

[30] Kramer, B. (1991) Quantum coherence in mesoscopic systems. Plenum, New York.

[31] Presilla, C., Jona – Lasinio, G.. and Capasso, F. (1991) Nonlinear feedback oscillations in resonant tunneling through double barriers.Physical Review B, 43, 5200

[32] Tchoukuegno, R. and Woafo, P. (2002) Physical D 167, 86; Tchoukuegno, R., Nana, B.R. and Woafo, P. (2002) Physical Review A, 304, 362; (2003) International Journal of Non-Linear Mechanics, 38, 531.

[33] Jing, Z.J., Yang, Z.Y. and Jiang, T. (2006) Bifurcation and chaos in discrete-time predator–prey system.Chaos, Solitons and Fractals, 27, 722.

[34] Sun, Z.K., Xu, W. and Yang, X.L. (2006) Chaos, Solitons and Fractals, 27, 778.

[35] Blum, T.C. and Elze, H.T. (1996) Semiquantum chaos in the double well.Physical Review E, 53, 3123.

[36] Zaslavsky, G.M. (1985) Chaos in dynamic systems. Harwood Academic, Chur, Switzerland.

[37] Wolf, A., Swift, J. B., Swinney, H.L. and Vastano, J.A. (1985) Determining Lyapunov exponents from a time series.Physical Review D, 16, 285.

[38] Melnikov, V.K. (1963) On the stability of a center for time-periodic perturbations. Transactions of the Moscow Mathematical Society, 12, 1.

[39] Gucken, H. J. and Holmes, P. (1983) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer Verlag, Berlin.

[40] Yamgoue, S.B. and Kofane, T.C. (2002) The subharmonic Melnikov theory for damped and driven oscillators revisited .International Journal of Bifurcation and Chaos, 8, 1915; (2003) Chaos, Solitons & Fractals, 17, 155.

[41] Churchill, R.C., Pecelli, G. and Rod, L. (1975) J. Diff. Eqs. 17, 329; (1977) J. Diff. Eqs. 24, 329; Churchill, R.C. and Rod, D.L. (1976) Ibid. 21, 39; (1976) Ibid. 21, 66; (1980) Ibid. 37, 23.