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Microscopic quantum generalization of classical Liénard oscillators.

Srijan Bhattacharyya1, Arnab Ghosh1, Deb Shankar Ray2

  • 1Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh 208016, India.

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This study presents a quantum generalization of classical Liénard systems using a system-reservoir model. The derived quantum Langevin equation reveals stable limit cycles, even with vacuum excitation, preserving dynamical stability.

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Area of Science:

  • Quantum physics
  • Nonlinear dynamics
  • Statistical mechanics

Background:

  • Classical Liénard systems describe nonlinear oscillations.
  • Understanding quantum behavior in such systems is crucial for advanced physics.
  • Previous models lacked a comprehensive quantum framework for Liénard dynamics.

Purpose of the Study:

  • To develop a microscopic quantum generalization of classical Liénard systems.
  • To investigate the quantum Langevin equation and its limit cycle behavior.
  • To analyze the role of fluctuation-dissipation relations in dynamical stability.

Main Methods:

  • Employed a system-reservoir model with nonlinear coupling.
  • Utilized oscillator coherent states and canonical thermal distributions.
  • Derived the quantum Langevin equation for the reduced system.

Main Results:

  • Successfully derived a quantum Langevin equation admitting single or multiple limit cycles.
  • Demonstrated that detailed balance (fluctuation-dissipation relation) ensures attractor stability, even under vacuum excitation.
  • Showcased quantum versions of Rayleigh, van der Pol, and other Liénard oscillators as special cases.

Conclusions:

  • The developed theoretical scheme provides a robust framework for quantum Liénard systems.
  • Quantum fluctuations and dissipation are critical for maintaining the stability of oscillatory attractors.
  • This work bridges classical nonlinear dynamics with quantum mechanics through a mean-field approach.