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Deep Neural Network Emulation of the Quantum-Classical Transition via Learned Wigner Function Dynamics

Kamran Majid

TL;DR

This work tackles the quantum-classical transition by directly learning the time evolution of the Wigner function for Gaussian states in a one-dimensional harmonic oscillator. It trains a deep neural network to map initial state parameters and Planck's constant $\hbar$ to the time-evolved Wigner-function parameters at a fixed time, using an analytically generated dataset and achieving a final loss around $0.0390$. The results show accurate predictions across a broad range of $\hbar$, with predicted phase-space distributions converging toward classical localization as $\hbar$ decreases, demonstrated through uncertainty measures and phase-space plots. This direct phase-space learning approach complements traditional observable-milling methods and offers a scalable framework for exploring the quantum-classical boundary in more complex quantum systems, including potential extensions to anharmonic potentials and decoherence effects.

Abstract

The emergence of classical behavior from quantum mechanics as Planck's constant $\hbar$ approaches zero remains a fundamental challenge in physics [1-3]. This paper introduces a novel approach employing deep neural networks to directly learn the dynamical mapping from initial quantum state parameters (for Gaussian wave packets of the one-dimensional harmonic oscillator) and $\hbar$ to the parameters of the time-evolved Wigner function in phase space [4-6]. A comprehensive dataset of analytically derived time-evolved Wigner functions was generated, and a deep feedforward neural network with an enhanced architecture was successfully trained for this prediction task, achieving a final training loss of ~ 0.0390. The network demonstrates a significant and previously unrealized ability to accurately capture the underlying mapping of the Wigner function dynamics. This allows for a direct emulation of the quantum-classical transition by predicting the evolution of phase-space distributions as $\hbar$ is systematically varied. The implications of these findings for providing a new computational lens on the emergence of classicality are discussed, highlighting the potential of this direct phase-space learning approach for studying fundamental aspects of quantum mechanics. This work presents a significant advancement beyond previous efforts that focused on learning observable mappings [7], offering a direct route via the phase-space representation.

Deep Neural Network Emulation of the Quantum-Classical Transition via Learned Wigner Function Dynamics

TL;DR

This work tackles the quantum-classical transition by directly learning the time evolution of the Wigner function for Gaussian states in a one-dimensional harmonic oscillator. It trains a deep neural network to map initial state parameters and Planck's constant to the time-evolved Wigner-function parameters at a fixed time, using an analytically generated dataset and achieving a final loss around . The results show accurate predictions across a broad range of , with predicted phase-space distributions converging toward classical localization as decreases, demonstrated through uncertainty measures and phase-space plots. This direct phase-space learning approach complements traditional observable-milling methods and offers a scalable framework for exploring the quantum-classical boundary in more complex quantum systems, including potential extensions to anharmonic potentials and decoherence effects.

Abstract

The emergence of classical behavior from quantum mechanics as Planck's constant approaches zero remains a fundamental challenge in physics [1-3]. This paper introduces a novel approach employing deep neural networks to directly learn the dynamical mapping from initial quantum state parameters (for Gaussian wave packets of the one-dimensional harmonic oscillator) and to the parameters of the time-evolved Wigner function in phase space [4-6]. A comprehensive dataset of analytically derived time-evolved Wigner functions was generated, and a deep feedforward neural network with an enhanced architecture was successfully trained for this prediction task, achieving a final training loss of ~ 0.0390. The network demonstrates a significant and previously unrealized ability to accurately capture the underlying mapping of the Wigner function dynamics. This allows for a direct emulation of the quantum-classical transition by predicting the evolution of phase-space distributions as is systematically varied. The implications of these findings for providing a new computational lens on the emergence of classicality are discussed, highlighting the potential of this direct phase-space learning approach for studying fundamental aspects of quantum mechanics. This work presents a significant advancement beyond previous efforts that focused on learning observable mappings [7], offering a direct route via the phase-space representation.

Paper Structure

This paper contains 17 sections, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Theoretical Background: The Wigner Function and the Classical Limit
  3. The Wigner Function
  4. The Classical Limit and the Wigner Function
  5. Gaussian States and Their Wigner Functions
  6. Methodology
  7. Data Generation
  8. Neural Network Architecture and Training
  9. Plot Generation
  10. Uncertainty vs. $\hbar$
  11. Phase Space Convergence
  12. Results
  13. Convergence with $\hbar$
  14. Generalization Performance
  15. Comparison with Analytical Results
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Predicted and analytical time-evolved position uncertainty $\sigma_x(t)$ as a function of $\hbar$ for a fixed initial state, demonstrating the network's accurate capture of the relationship.
  • Figure 2: Predicted Wigner function in phase space at $t=5.0$ for a fixed initial state and varying $\hbar$ values, showcasing the learned convergence towards classical-like distributions.