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Damped Oscillations01:07

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In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
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If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about...
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An idealized LC circuit of zero resistance can oscillate without any source of emf by shifting the energy stored in the circuit between the electric and magnetic fields. In such an LC circuit, if the capacitor contains a charge q before the switch is closed, then all the energy of the circuit is initially stored in the electric field of the capacitor. This energy is given by
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Consider designing an oscillator circuit, a crucial component in various electronic devices and systems. The objective is to create an oscillator circuit with specific characteristics: a damped natural frequency of 4 kHz and a damping factor of 4 radians per second. To accomplish this, a parallel RLC circuit is employed, known for its ability to sustain oscillations at a resonant frequency. In this case, the damping factor is pivotal in achieving the desired performance.
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An RLC circuit combines a resistor, inductor, and capacitor, connected in a series or parallel combination.
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When an oscillator is forced with a periodic driving force, the motion may seem chaotic. The motions of such oscillators are known as transients. After the transients die out, the oscillator reaches a steady state, where the motion is periodic, and the displacement is determined.
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Theory for Dissipative Time Crystals in Coupled Parametric Oscillators.

Stuart Yi-Thomas1, Jay D Sau1

  • 1University of Maryland, College Park, Joint Quantum Institute, Condensed Matter Theory Center and , Department of Physics, Maryland 20742-4111, USA.

Physical Review Letters
|January 29, 2025
PubMed
Summary

Discrete time crystals, novel phases of matter, were modeled using parametrically driven coupled oscillators. This classical system demonstrates robust discrete time crystal phases, even with symmetry breaking.

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Area of Science:

  • Condensed Matter Physics
  • Quantum Mechanics
  • Nonlinear Dynamics

Background:

  • Discrete time crystals represent a novel phase of matter breaking discrete time translational symmetry in periodically driven systems.
  • Understanding discrete time crystals is crucial for advancing quantum technologies and fundamental physics.
  • Previous research has primarily focused on quantum systems, leaving classical analogues less explored.

Purpose of the Study:

  • To propose and investigate a classical system of weakly nonlinear parametrically driven coupled oscillators as a test bed for discrete time crystals.
  • To demonstrate that period-doubling instabilities in this classical system lead to discrete time crystal phases.
  • To establish the robustness of these discrete time crystal phases under various approximations and conditions.

Main Methods:

  • Modeling a classical system of weakly nonlinear parametrically driven coupled oscillators.
  • Analyzing the system's behavior in a limit approaching Langevin dynamics in a symmetry-breaking potential.
  • Employing numerical simulations with Glauber dynamics approximation to study phase stability under Ising symmetry breaking.
  • Utilizing field-theoretic arguments to assess robustness against approximations like the semiclassical limit for dissipative quantum systems.

Main Results:

  • The proposed classical system effectively models period-doubling instabilities relevant to Josephson junction arrays and semiconductor lasers.
  • Numerical simulations confirm the existence of a discrete time crystal phase in the classical system, even with Ising symmetry breaking.
  • Field-theoretic analysis demonstrates the robustness of these findings to various approximations, including the semiclassical limit.

Conclusions:

  • A classical system of parametrically driven coupled oscillators serves as a viable and insightful test bed for studying discrete time crystals.
  • The identified discrete time crystal phase exhibits robustness, suggesting broad applicability and potential for experimental realization.
  • This work bridges the gap between classical and quantum descriptions of time crystals, offering new avenues for research.