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Analysis of the parametrically periodically driven classical and quantum linear oscillator.

Vladimir Grubelnik1, Marjan Logar1, Marko Robnik2

  • 1Faculty of Electrical Engineering and Computer Science, University of Maribor, Koroška cesta 46, SI-2000 Maribor, Slovenia, European Union.

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Summary
This summary is machine-generated.

This study analyzes the quantum and classical periodically driven linear oscillator (Floquet system). We found that quantum and classical instability regions precisely coincide, offering insights into energy evolution and Floquet system dynamics.

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Area of Science:

  • Quantum mechanics
  • Classical mechanics
  • Nonlinear dynamics
  • Floquet engineering

Background:

  • Periodically driven systems, known as Floquet systems, exhibit complex behaviors.
  • Understanding the interplay between classical and quantum descriptions in these systems is crucial.

Purpose of the Study:

  • To theoretically and computationally investigate the behavior of classical and quantum parametrically driven linear oscillators.
  • To derive analytic formulas for quantum propagators and energy evolution.
  • To analyze system dynamics within stability and instability regions.

Main Methods:

  • Theoretical analysis of a harmonic oscillator frequency.
  • Derivation of analytic formulas for quantum propagator and energy evolution.
  • Computational analysis of system behavior in stability and instability regions.

Main Results:

  • An explicit analytic formula for the quantum propagator was derived in terms of the classical propagator.
  • Exact formulas for the evolution of energy expectation values were obtained.
  • Quantum and classical instability regions were found to coincide exactly.
  • An elliptic-type equation describing exponential energy growth in instability regions was identified.

Conclusions:

  • The derived formulas and findings are applicable to a broad range of linear Floquet systems.
  • The exact coincidence of classical and quantum instability borders validates numerical accuracy.
  • The study provides a strong foundation for a general classical and quantum theory of linear Floquet systems.