Related Experiment Video
Updated: Aug 3, 2026
09:23
Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators
Published on: May 30, 2014
Hypersensitivity to perturbations of quantum-chaotic wave-packet dynamics
P G Silvestrov1, J Tworzydło, C W J Beenakker
1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands.
Summary
The Loschmidt echo reveals quantum chaos sensitivity. Initial wave packet decay is double-exponential, transitioning to single-exponential after the Ehrenfest time, reflecting typical quantum dynamics.
Area of Science:
- Quantum chaos
- Quantum dynamics
- Sensitivity to perturbation
Background:
- The Loschmidt echo measures the sensitivity of quantum chaotic dynamics to perturbations.
- Understanding this sensitivity is crucial for characterizing quantum systems.
Purpose of the Study:
- To reexamine the problem of the Loschmidt echo.
- To analyze the decay behavior of wave packets in quantum chaotic systems.
Main Methods:
- Analyzing the overlap squared M(t) of two wave packets evolving under slightly different Hamiltonians.
- Investigating the double-exponential and single-exponential decay regimes.
Main Results:
- The overlap squared M(t) exhibits an initial double-exponential decay: exp(-constant x e(2λ(0)t)).
- The average decay (-)M shows single-exponential behavior: exp(-λ(1)t), with λ(0) as the Lyapunov exponent.
- The phase space volume contributing to (-)M vanishes in the classical limit before the Ehrenfest time (τ(E)).
Conclusions:
- The double-exponential decay characterizes the main part of the phase space initially.
- Single-exponential decay becomes representative of typical initial conditions only after the Ehrenfest time.
- This highlights the time-dependent nature of quantum chaotic system sensitivity.
Related Concept Videos
02:32The de Broglie Wavelength
In the macroscopic world, objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle; it will continue traveling in a straight line unless it collides with another ball, or it is acted on by some other force, such as friction. The ball has a well-defined position and velocity or well-defined momentum, p = mv, which is defined by mass m and velocity v at any given moment. This is the typical...
04:08The Uncertainty Principle
Werner Heisenberg considered the limits of how accurately one can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurate the measurement of the momentum of a particle is known, the less accurate the position at that time is known and vice versa. This is what is now called the Heisenberg uncertainty principle. He mathematically...
02:45The Quantum-Mechanical Model of an Atom
Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra. Schrödinger...
01:02¹H NMR: Interpreting Distorted and Overlapping Signals
Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
As Δν decreases and the signals move closer, the doublets appear increasingly distorted. The intensities of the inner lines increase at the cost of those of the outer lines as the signals are slanted or...
As Δν decreases and the signals move closer, the doublets appear increasingly distorted. The intensities of the inner lines increase at the cost of those of the outer lines as the signals are slanted or...
01:07Interference and Superposition of Waves
When two waves of the same nature occur in the same region simultaneously, they result in interference. Interference of waves implies that the net effect of the waves is the sum of the individual waves' effects. However, it does not imply that the individual waves affect the propagation of other waves.
Interference occurs in mechanical waves, such as sound waves, waves on a string, and surface water waves. Mechanical waves correspond to the physical displacement of particles. Hence,...
Interference occurs in mechanical waves, such as sound waves, waves on a string, and surface water waves. Mechanical waves correspond to the physical displacement of particles. Hence,...
00:53Sound Waves: Interference
Sound waves can be modeled either as longitudinal waves, wherein the molecules of the medium oscillate around an equilibrium position, or as pressure waves. When two identical waves from the same source superimpose on each other, the combination of two crests or two troughs results in amplitude reinforcement known as constructive interference. If two identical waves, that are initially in phase, become out of phase because of different path lengths, the combination of crests with troughs...