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Microscopic quantum generalization of classical Liénard oscillators

Srijan Bhattacharyya, Arnab Ghosh, and Deb Shankar Ray
Phys. Rev. E 103, 012118 – Published 19 January 2021
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  • INTRODUCTION
  • QUANTUM ANALOG OF THE CLASSICAL LIÉNARD…
  • C-NUMBER DESCRIPTION OF THE QUANTUM…
  • CONSTRUCTION OF THE QUANTUM LIÉNARD…
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Based on a system-reservoir model and an appropriate choice of nonlinear coupling, we have explored the microscopic quantum generalization of classical Liénard systems. Making use of oscillator coherent states and canonical thermal distributions of the associated c numbers, we have derived the quantum Langevin equation of the reduced system which admits single or multiple limit cycles. It has been shown that detailed balance in the form of the fluctuation-dissipation relation preserves the dynamical stability of the attractors even in the case of vacuum excitation. The quantum versions of Rayleigh, van der Pol, and several other variants of Liénard oscillators are derived as special cases in our theoretical scheme within a mean-field description.

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    • Received 4 August 2020
    • Accepted 28 December 2020

    DOI:https://doi.org/10.1103/PhysRevE.103.012118

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Fluctuations & noiseNonequilibrium statistical mechanicsQuantum statistical mechanics
    1. Techniques
    Langevin equationStatistical methods
    Nonlinear DynamicsStatistical Physics & Thermodynamics

    Authors & Affiliations

    Srijan Bhattacharyya1, Arnab Ghosh1,*, and Deb Shankar Ray2

    • 1Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh 208016, India
    • 2Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India

    • *arnab@iitk.ac.in

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    Vol. 103, Iss. 1 — January 2021

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    Images

    • Figure 1
      Figure 1

      The phase space trajectories of the (a) classical and the (b) quantum Liénard oscillators [Eq. (32)] with f(x)=εx2 are plotted in the absence (a) and the presence (b) of intrinsic noise. The average trajectory is shown (blue) in (b) for internal noise. (c) The quantum Liénard oscillator in the presence of external noise: the lack of detailed balance destroys the limit cycle oscillation. We set ε=0.01 and KT=50 (arbitrary units).

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    • Figure 2
      Figure 2

      The phase space trajectories of a (a) classical [Eq. (26)] and a (b) quantum van der Pol oscillator [Eq. (40)] are plotted for ε=2 and KT=0. It is shown that quantum fluctuations can retain the general shape of the limit cycle oscillation even at absolute zero (arbitrary units). The average trajectory is shown (blue) in (b) closely resembles the classical one. (c) In the absence of detailed balance, an external noise completely destroys the limit cycle.

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    • Figure 3
      Figure 3

      (a) The phase space trajectories of a (a) classical [Eq. (27)] and a (b) quantum Rayleigh oscillator [Eq. (43)] are plotted for the parameters ε=1 and KT=10 (arbitrary units). The statistical average over noisy trajectories in (b) is identical to its classical counterpart. (c) An external noise breaks the stable limit cycle.

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    • Figure 4
      Figure 4

      The phase space trajectories of the (a) classical and the (b) quantum Liénard systems with f(x)=0.005*x6−0.144*x4+x2−1 [16] are plotted for ε=0.01. In both cases we have two stable (blue line) and one unstable (green dotted line) limit cycles. Pure quantum fluctuations at T=0 may lead to noisy but stable oscillations even for the two stable limit cycles (arbitrary units).

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    • Figure 5
      Figure 5

      The phase space trajectories of the (a) classical and the (b) quantum versions of a generalized Rayleigh oscillator for polynomial ẋ6+ẋ4−ẋ2−1 are plotted for ε=1.5. We have put KT=20 for plotting the quantum case (arbitrary units).

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