[Paper Review] On the Convergence of Time Splitting Methods for Quantum Dynamics in the Semiclassical Regime

François Golse, Shi Jin+1 more|arXiv (Cornell University)|Jun 8, 2019

Numerical methods for differential equationsMathematics24 references21 citations

TL;DR

This paper establishes uniform-in-Planck-constant error estimates for time splitting methods—Lie-Trotter and Strang—for the von Neumann equation in the semiclassical regime. By employing a pseudo-metric analogous to the Wasserstein distance of order 2 between quantum density operators and classical phase-space densities, it proves that numerical convergence is independent of $̂\hbar$, enabling $̂\Delta t = O(1)$ time steps even as $\hbar \to 0$. The key result is explicit error bounds that remain valid uniformly in $\hbar$, resolving a long-standing challenge in semiclassical quantum dynamics simulations.

ABSTRACT

By using the pseudo-metric introduced in [F. Golse, T. Paul: Archive for Rational Mech. Anal. 223 (2017) 57-94], which is an analogue of the Wasserstein distance of exponent $2$ between a quantum density operator and a classical (phase-space) density, we prove that the convergence of time splitting algorithms for the von Neumann equation of quantum dynamics is uniform in the Planck constant $\\hbar$. We obtain explicit uniform in $\\hbar$ error estimates for the first order Lie-Trotter, and the second order Strang splitting methods.

Motivation & Objective

• To address the computational challenge of resolving small de Broglie wavelengths ($\sim \hbar$) in quantum dynamics simulations.
• To rigorously justify the use of $O(1)$ time steps in time-splitting spectral methods, which had previously only been justified at a formal level.
• To establish error estimates that are uniform in the Planck constant $\hbar$, enabling reliable numerical simulation in the semiclassical regime.
• To bridge the gap between quantum dynamics and classical limits by using a phase-space metric that captures the semiclassical behavior of mixed quantum states.

Proposed method

• The authors employ a pseudo-metric inspired by the Wasserstein distance of order 2, defined between a quantum density operator and a classical phase-space probability density.
• The metric is applied to compare the Wigner transform of the quantum state obtained via time splitting with the classical transport solution.
• The analysis relies on the Wigner transform to connect quantum evolution to classical Liouville dynamics in phase space.
• Error estimates are derived using trace-class and Hilbert-Schmidt operator norms, combined with coupling techniques between quantum and classical measures.
• The method leverages known operator norm bounds for exponential splitting schemes (Lie-Trotter and Strang) and applies them to the von Neumann equation for mixed states.
• A key technical tool is the use of coherent states $|q,p\rangle$ as a basis for constructing couplings between quantum and classical distributions.

Experimental results

Research questions

• RQ1 Can time splitting methods for the von Neumann equation achieve convergence with time steps independent of $\hbar$ in the semiclassical regime?
• RQ2 Is there a suitable metric that allows for uniform-in-$\hbar$ error estimates in quantum dynamics simulations?
• RQ3 How do the Lie-Trotter and Strang splitting schemes perform in terms of error when $\hbar \to 0$?
• RQ4 Can the Wigner transform framework be used to rigorously justify $O(1)$ time steps in time-splitting spectral methods?
• RQ5 What is the dependence of the error on $\hbar$, $\Delta t$, and initial data in the semiclassical limit?

Key findings

• The first-order Lie-Trotter splitting method yields a uniform-in-$\hbar$ error bound of order $\Delta t + \min(\frac{\Delta t}{\hbar}, \sqrt{\hbar})$, which simplifies to $\Delta t + \Delta t^{1/3}$ when $\hbar = \Delta t^{4/3}$.
• The second-order Strang splitting method achieves a uniform-in-$\hbar$ error bound of order $\Delta t + \min(\frac{\Delta t^2}{\hbar}, \sqrt{\hbar})$, leading to $\Delta t + \Delta t^{2/3}$ when $\hbar = \Delta t^{4/3}$.
• The error estimates are uniform in $\hbar$, meaning that the convergence rate does not deteriorate as $\hbar \to 0$, which is crucial for semiclassical simulations.
• The pseudo-metric used is a quantum-classical analogue of the Wasserstein distance of order 2, enabling comparison between quantum density operators and classical phase-space densities.
• The analysis shows that standard $L^2$ or Sobolev norms would yield errors scaling as $O((\Delta t / \hbar)^m)$, which precludes $O(1)$ time steps, but the chosen metric avoids this issue.
• The results validate the empirical success of time-splitting spectral methods in the semiclassical regime and provide a rigorous foundation for their use with $\Delta t = O(1)$.