Skip to main content
SpringerLink
Log in

Nonlinear Quantum Mechanics

  • Conference paper
  • First Online:
Coupled Mathematical Models for Physical and Biological Nanoscale Systems and Their Applications (BIRS-16w5069 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 232))

  • 675 Accesses

Abstract

We show that the nonlinear bifurcations found by simulations in single quantum wells in the terahertz regime [5, 6, 25] also occur in semiconductor superlattices (SSLs) in the gigahertz range [2, 3, 11]. The only exception is the second Hopf bifurcation to quasi-periodic orbits on a torus. The advantage of experiments on SSLs in the gigahertz range is that the experiments can be conducted at room temperature and a chaotic oscillator due to the random dressing of a period two-orbit has already been measured [31, 32]. We determine [42] that the route to chaos for SSLs in the sequential tunneling regime is the period doubling cascade. Shorter (10-period) superlattices are observed to exhibit faster oscillations compared with longer (50-period) ones. Two plateaus are observed as functions of the voltage bias, and intrinsically chaotic dynamics on the second plateau are possible only for shorter SSLs, while the dynamics in the first plateau contain intrinsic chaos only for longer (N > 50) SSLs [21].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Alvaro, M. Carretero, L. Bonilla, Noise-enhanced spontaneous chaos in semiconductor superlattices at room temperature. EPL (Europhys. Lett.) 107, 37002 (2014)

    Article  Google Scholar 

  2. A. Amann, J. Schlesner, A. Wacker, E. Schöll, Chaotic front dynamics in semiconductor superlattices. Phys. Rev. B 65, 193313 (2002)

    Article  Google Scholar 

  3. T. Ando, A.B. Fowler, F. Stern, Electronic properties of two-dimensional systems. Rev. Mod. Phys. 54, 437 (1982)

    Article  Google Scholar 

  4. S. Asmussen, P.W. Glynn. Stochastic Simulation: Algorithms and Analysis, vol. 57 (Springer Science & Business Media, 2007)

    Google Scholar 

  5. A.A. Batista, P.I. Tamborenea, B. Birnir, M. Sherwin, D.S. Citrin, Nonlinear dynamics in far-infrared driven quantum-well intersubband transitions. Phys. Rev. B 66, 195325 (2002)

    Article  Google Scholar 

  6. A.A. Batista, B. Birnir, P.I. Tamborenea, D.S. Citrin, Period-doubling and Hopf bifurcations in far-infrared driven quantum well intersubband transitions. Phys. Rev. B 68, 035307 (2003)

    Article  Google Scholar 

  7. B. Birnir, Chaotic perturbations of KdV equations: I. rational solutions. Phys. D Nonlinear Phenom. 18, 464 (1986)

    Article  MathSciNet  Google Scholar 

  8. B. Birnir, R. Grauer, An explicit description of the global attractor of the damped and driven sine-Gordon equation. Commun. Math. Phys. 162, 539 (1994)

    Article  MathSciNet  Google Scholar 

  9. B. Birnir, Basic Attractors and Control (To be published by Springer, New York, 2017)

    Google Scholar 

  10. K. Binder, D.W. Heermann, Monte Carlo Simulation in Statistical Physics: An Introduction (2002)

    Google Scholar 

  11. L.L. Bonilla, J. Galán, J. Cuesta, F.C. Martínez, J.M. Molera, Dynamics of electric field domains and oscillations of the photocurrent in a simple superlattice model. Phys. Rev. B 50, 8644 (1994)

    Article  Google Scholar 

  12. L.L. Bonilla, G. Platero, D. Sánchez, Microscopic derivation of transport coefficients and boundary conditions in discrete drift-diffusion models of weakly coupled superlattices. Phys. Rev. B 62, 2786 (2000)

    Article  Google Scholar 

  13. L.L. Bonilla, Theory of nonlinear charge transport, wave propagation and self-oscillations in semiconductor superlattices. J. Phys. Condens. Matter 14, R341 (2002)

    Article  Google Scholar 

  14. L.L. Bonilla, H.T. Grahn, Non-linear dynamics of semiconductor superlattices. Rep. Prog. Phys. 68, 577 (2005)

    Article  Google Scholar 

  15. L.L. Bonilla, S.W. Teitsworth, Nonlinear Wave Methods for Charge Transport (Wiley VCH, Weinheim, 2009)

    Google Scholar 

  16. L.L. Bonilla, R. Escobedo, G. Dell’Acqua, Voltage switching and domain relocation in semiconductor superlattices. Phys. Rev. B 73, 115341 (2006)

    Article  Google Scholar 

  17. L.L. Bonilla, M. Alvaro, M. Carretero, Chaos-based true random number generators. J. Math. Ind. 7, 1 (2016)

    Article  MathSciNet  Google Scholar 

  18. P. Collet, J. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Modern Birkhäuser Classics (Birkhäuser Boston, 2009)

    Chapter  Google Scholar 

  19. K. Craig et al., Undressing a collective intersubband excitation in quantum well. Phys. Rev. Lett. 76, 2382 (1996)

    Article  Google Scholar 

  20. G. Dell’Acqua, L.L. Bonilla, R. Escobedo, Hopf Bifurcation in a Superlattice Model, in Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering (2006)

    Google Scholar 

  21. J. Essen, M. Ruiz-Garcia, I. Jenkins, M. Carretero, L.L. Bonilla, B. Birnir, Parameter dependence of high-frequency nonlinear oscillations and intrinsic chaos in short GaAs/(Al,Ga)As superlattices (2017) (Submitted)

    Google Scholar 

  22. J. Essen, Simulations of open ballistic quantum systems and semiclassical semiconductor superlattices. Ph.D. thesis, University of California, Santa Barbara, 2017

    Google Scholar 

  23. R.G. Gallager, Principles of Digital Communication (Cambridge University Press, Cambridge, UK, 2008)

    Book  Google Scholar 

  24. B. Galdrikian, M. Sherwin, B. Birnir, Self-consistent Floquet states for periodically driven quantum wells. Phys. Rev. B 49, 13744 (1994)

    Article  Google Scholar 

  25. B. Galdrikian, B. Birnir, Period Doubling and Strange Attractors in Quantum Wells. Phys. Rev. Lett. 76, 3308 (1996)

    Article  Google Scholar 

  26. R. Graham, Squeezing and frequency changes in harmonic oscillations. J. Mod. Optics 34, 873 (1987)

    Article  MathSciNet  Google Scholar 

  27. R. Grauer, B. Birnir, The center manifold and bifurcations of damped and driven sine-Gordon breathers. Phys. D Nonlinear Phenom. 56, 165 (1992)

    Article  MathSciNet  Google Scholar 

  28. J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Springer, New York, 1983)

    Book  Google Scholar 

  29. J.N. Heyman et al., Resonant harmonic generation and dynamics screening in a double quantum well. Phys. Rev. Lett. 72, 2183 (1994)

    Article  Google Scholar 

  30. J.N. Heyman et al., Temperature and intensity dependence of intersubband relaxation rates from photovolgate and absorption. Phys. Rev. Lett. 74, 2682 (1995)

    Article  Google Scholar 

  31. Y. Huang, W. Li, W. Ma, H. Qin, Y. Zhang, Experimental observation of spontaneous chaotic current oscillations in GaAs/Al0.45Ga0.55As superlattices at room temperature. Chin. Sci. Bull. 57, 2070 (2012)

    Article  Google Scholar 

  32. Y. Huang, W. Li, W. Ma, H. Qin, H.T. Grahn, Y. Zhang, Spontaneous quasi-periodic current self-oscillations in a weakly coupled GaAs/(Al, Ga)As superlattice at room temperature. Appl. Phys. Lett. 102, 242107 (2013)

    Article  Google Scholar 

  33. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, M. Rosenbluth, An optical ultrafast random bit generator. Nat. Photonics 4, 58 (2010)

    Article  Google Scholar 

  34. J. Kastrup, R. Hey, K.H. Ploog, H.T. Grahn, L.L. Bonilla, M. Kindelan, M. Moscoso, A. Wacker, J. Galán, Electrically tunable GHz oscillations in doped GaAs-AlAs superlattices. Phys. Rev B. 55, 2476 (1997)

    Article  Google Scholar 

  35. E.W. Laedke, K.H. Spatschek, On localized solutions in nonlinear faraday resonance. J. Fluid Mech. 223, 589–601 (1991)

    Article  MathSciNet  Google Scholar 

  36. K. Leo, P.H. Bolivar, F. Brüggemann, R. Schwedler, K. Klaus, Köhler, Observation of Bloch oscillations in a semiconductor superlattice. Solid State Commun. 10, 943–946 (1992)

    Article  Google Scholar 

  37. W. Li, I. Reidler, Y. Aviad, Y.Y. Huang, H. Song, Y.H. Zhang, M. Rosenbluh, I. Kanter, Fast physical random-number generation based on room-temperature chaotic oscillations in weakly coupled superlattices. Phys. Rev. Lett. 111, 044102 (2013)

    Article  Google Scholar 

  38. C.M. Morris, Interplay of quantum confinement, electron-electron interactions, and terahertz radiation in indium gallium arsenide quantum posts and gallium arsenide quantum wells. Ph.D. thesis, University of California, Santa Barbara, 2011

    Google Scholar 

  39. T.E. Murphy, R. Roy, The world’s fastest dice. Nat. Photonics 2, 714–715 (2008)

    Article  Google Scholar 

  40. M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2000)

    MATH  Google Scholar 

  41. I. Reidler, Y. Aviad, M. Rosenbluh, I. Kanter, Ultrahigh-speed random number generation based on a chaotic semiconductor laser. Phys. Rev. Lett. 103, 024102 (2009)

    Article  Google Scholar 

  42. M. Ruiz-Garcia, J. Essen, M. Carretero, L.L. Bonilla, B. Birnir, Enhancing chaotic behavior at room temperature in GaAs/(Al, Ga)As superlattices. Phys. Rev. B 95, 085204 (2017)

    Article  Google Scholar 

  43. A.K. Saxena, The conduction band structure and deep levels in Ga1-xAlxAs alloys from a high-pressure experiment. J. Phys. C Solid State Phys. 23, 4323 (1980)

    Article  Google Scholar 

  44. M. Sciamanna, K.A. Shore, Physics and applications of laser diode chaos. Nat. Photonics 9, 151–162 (2015)

    Article  Google Scholar 

  45. D.R. Stinson, Cryptography: Theory and Practice, 3rd edn. (CRC Press, Boca Raton, 2006)

    MATH  Google Scholar 

  46. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, P. Davis, Fast physical random bit generation with chaotic semiconductor lasers. Nat. Photonics 2, 728–732 (2008)

    Article  Google Scholar 

  47. J.Q. Wu, D.S. Jiang, B.Q. Sun, Room-temperature microwave oscillation in alas/gaas superlattices. Phys. E Low-dimensional Syst. Nanostruct. 4(2), 137–141 (1999)

    Article  Google Scholar 

  48. Z. Yin, Y. Zhang, M. Ruiz-García, M. Carretero, L.L. Bonilla, K. Biermann, H.T. Grahn, Noise-enhanced chaos in a weakly coupled GaAs/(Al,Ga)As superlattice (unpublished)

    Google Scholar 

  49. M. Zalużny, Influence of the depolarization effect on the nonlinear intersubband absorption spectra of quantum wells. Phys. Rev. B 47, 3995 (1993)

    Article  Google Scholar 

  50. M. Zalużny, Saturation of intersubband absorption and optical rectification in asymmetric quantum wells. J. Appl. Phys. 74, 4716 (1993)

    Article  Google Scholar 

Download references

Acknowledgements

This material is based upon work supported by, or in part by, the U.S. Army Research Laboratory and the U. S. Army Research Office under contract/grant number 444045-22682.

Author information

Authors and Affiliations

  1. Department of Mathematics and CNLS, University of California, Santa Barbara, CA, 93106, USA

    Björn Birnir

  2. University of Iceland, 107, Reykjavík, Iceland

    Björn Birnir

Authors
  1. Björn Birnir
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Björn Birnir .

Editor information

Editors and Affiliations

  1. Department of Materials Science and Engineering and G. Millan Institute, Universidad Carlos III de Madrid, Leganés, Spain

    Luis L. Bonilla

  2. Lyman Laboratory of Physics, Department of Physics, Harvard University, Cambridge, Massachusetts, USA

    Efthimios Kaxiras

  3. The MS2 Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roderick Melnik

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Birnir, B. (2018). Nonlinear Quantum Mechanics. In: Bonilla, L., Kaxiras, E., Melnik, R. (eds) Coupled Mathematical Models for Physical and Biological Nanoscale Systems and Their Applications. BIRS-16w5069 2016. Springer Proceedings in Mathematics & Statistics, vol 232. Springer, Cham. https://doi.org/10.1007/978-3-319-76599-0_1

Download citation

Publish with us

Policies and ethics

Search

Navigation