Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Variational Adaptive Gaussian Decomposition: Scalable Quadrature-Free Time-Sliced Thawed Gaussian Dynamics

Rahul Sharma, Amartya Bose

TL;DR

A quadrature-free variational framework for Gaussian wave packet decomposition is introduced, reformulating it as an optimization problem in which the parameters of Gaussian wave packets are chosen to maximize the overlap with the time-evolving wave function.

Abstract

Time-slicing has emerged as a strategy for incorporating semiclassical propagation into real-time path integral formulation and recovering full quantum dynamics. A central step is the decomposition of a time-evolved wave function into a superposition of Gaussian wave packets (GWPs). Here we introduce a quadrature-free variational framework for GWP decomposition, reformulating it as an optimization problem in which the GWP parameters are chosen to maximize the overlap with the time-evolving wave function. An autoencoder-decoder neural network is used for this optimization, with the representation being adaptively reoptimized during propagation. Each wave packet in this decomposition represents a localized patch of the underlying semiclassical manifold, while retaining full correlations between all degrees of freedom. This variational adaptive Gaussian decomposition (VAGD) approach yields a compact Gaussian expansion, providing a scalable route to time-sliced semiclassical quantum dynamics. While general, applying VAGD to facilitate time-slicing of thawed Gaussian approximation (TGA) allows a route to improving the semiclassical treatment to the full quantum mechanical result in a systematic manner.

Variational Adaptive Gaussian Decomposition: Scalable Quadrature-Free Time-Sliced Thawed Gaussian Dynamics

TL;DR

A quadrature-free variational framework for Gaussian wave packet decomposition is introduced, reformulating it as an optimization problem in which the parameters of Gaussian wave packets are chosen to maximize the overlap with the time-evolving wave function.

Abstract

Time-slicing has emerged as a strategy for incorporating semiclassical propagation into real-time path integral formulation and recovering full quantum dynamics. A central step is the decomposition of a time-evolved wave function into a superposition of Gaussian wave packets (GWPs). Here we introduce a quadrature-free variational framework for GWP decomposition, reformulating it as an optimization problem in which the GWP parameters are chosen to maximize the overlap with the time-evolving wave function. An autoencoder-decoder neural network is used for this optimization, with the representation being adaptively reoptimized during propagation. Each wave packet in this decomposition represents a localized patch of the underlying semiclassical manifold, while retaining full correlations between all degrees of freedom. This variational adaptive Gaussian decomposition (VAGD) approach yields a compact Gaussian expansion, providing a scalable route to time-sliced semiclassical quantum dynamics. While general, applying VAGD to facilitate time-slicing of thawed Gaussian approximation (TGA) allows a route to improving the semiclassical treatment to the full quantum mechanical result in a systematic manner.
Paper Structure (14 sections, 14 equations, 11 figures)

This paper contains 14 sections, 14 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Time-Sliced Thawed Gaussian
  4. Variational Adaptive Gaussian Decomposition (VAGD)
  5. Encoding of Parameters
  6. Numerical Stabilization Strategies
  7. Numerical Illustrations
  8. Morse Potential
  9. 1D Explorations
  10. Multidimensional Explorations
  11. Double Well Potentials
  12. One-Dimensional Problem
  13. Tunneling in a Two-Dimensional Double-Well
  14. Conclusion

Figures (11)

  • Figure 1: Schematic of auto-encoder decoder setup which acts as the basis transformation.
  • Figure 2: Breakdown of TGA with anharmonicity of 1D Morse potential.
  • Figure 3: Overlap of the VAGD-TGA method with the SOFT wave function for different maximum number of Gaussians.
  • Figure 4: Overlap of VAGD-TGA wave function with SOFT solution (solid, left $y$-axis) and number of VAGD trajectories (dashed, right $y$-axis) of 1D-Morse oscillator for different values of $\chi$.
  • Figure 5: Overlap of TGA solutions with the SOFT solution for multidimensional Morse models as a function of dimensionality.
  • ...and 6 more figures