[Paper Review] Échelles de temps pour l'évolution quantique à petite constante de Planck

Thierry Paul|ArXiv.org|Jan 21, 2009

Cold Atom Physics and Bose-Einstein Condensates5 references7 citations

TL;DR

This paper investigates long-time semiclassical approximations of quantum evolution in the limit of small Planck's constant ℏ, identifying three distinct time scales: ℏ⁻¹ for classical-like behavior, log(1/ℏ) for wave packet delocalization in unstable systems, and longer scales where quantum reconstructions—such as wave packet revivals—persist. The key contribution is the identification of these time scales and their connection to classical dynamics, particularly through coherent states and Wigner transforms, revealing non-classical quantum effects even in the semiclassical limit.

ABSTRACT

We present some recent results concerning the long time semiclassical approximation .

Motivation & Objective

• To understand the long-time behavior of quantum systems in the semiclassical limit (ℏ → 0), particularly when quantum evolution extends beyond the standard Ehrenfest time.
• To identify distinct time scales T(ℏ) for which different semiclassical behaviors emerge, especially in systems with stable or unstable classical dynamics.
• To analyze the persistence of quantum effects—such as wave packet revivals and reconstruction—beyond the classical limit, particularly in chaotic or unstable systems.
• To clarify the role of coherent states with non-Gaussian symbols in capturing long-time quantum dynamics, especially when Gaussian states fail.
• To explore the link between quantum indeterminism and classical sensitivity to initial conditions in the long-time semiclassical regime.

Proposed method

• Uses semi-classical analysis and Weyl quantization to study the Schrödinger equation with a small parameter ℏ, focusing on coherent states defined via a Schwartz function a(x).
• Employs the Wigner transform to relate quantum states to classical phase space distributions, enabling comparison between quantum evolution and classical flow.
• Analyzes time evolution of coherent states under Hamiltonian flows, distinguishing between stable (periodic) and unstable (hyperbolic, heteroclinic) classical trajectories.
• Applies microlocal analysis and Egorov-type theorems to study the propagation of quantum observables and the fidelity of semiclassical approximation over time.
• Considers the limit of infinite time by defining a non-deterministic inverse flow via unstable manifolds of fixed points, linking it to quantum delocalization and probabilistic interpretation.
• Uses the concept of wave packet reconstruction to identify the third time scale, where quantum interference leads to recurrence phenomena even as ℏ → 0.

Experimental results

Research questions

• RQ1 What are the distinct time scales T(ℏ) for which different semiclassical limits emerge in quantum evolution?
• RQ2 How does the stability of classical trajectories (stable vs. unstable) affect the long-time behavior of quantum wave packets?
• RQ3 Can quantum reconstructions—such as wave packet revivals—persist in the semiclassical limit, and if so, under what conditions?
• RQ4 What is the role of non-Gaussian coherent states in capturing long-time quantum dynamics beyond the standard Gaussian approximation?
• RQ5 How does the classical sensitivity to initial conditions relate to quantum indeterminism in the long-time semiclassical regime?

Key findings

• Three distinct time scales emerge in the semiclassical limit: T₁(ℏ) = ℏ⁻¹ for classical-like behavior, T₂(ℏ) = log(1/ℏ) for wave packet delocalization in unstable systems, and T₃(ℏ) > log(1/ℏ) for persistent quantum reconstructions.
• In the stable case (e.g., harmonic oscillator), wave packet revivals occur on the T₁(ℏ) = ℏ⁻¹ scale, consistent with periodic quantum evolution.
• In the unstable case (e.g., hyperbolic fixed points or heteroclinic orbits), wave packet delocalization occurs on the T₂(ℏ) = log(1/ℏ) scale, where quantum states spread over the unstable manifold.
• For longer times beyond log(1/ℏ), quantum reconstructions—such as the reappearance of initial wave packets—can still occur, indicating that quantum effects persist in the semiclassical limit.
• The reconstruction of wave packets is not preserved under Gaussian initial states; the reconstructed states have singular symbols in phase space, indicating non-smooth, non-Gaussian structures.
• The non-deterministic inverse flow defined via unstable manifolds of fixed points acquires a quantum meaning through wave packet delocalization and probabilistic interpretation, linking classical chaos to quantum indeterminism.