Numerical integration of quantum time evolution in a curved manifold

TL;DR

This paper investigates numerical methods for simulating quantum time evolution on curved manifolds, focusing on unitarity preservation and stability of discretized integrators like Crank-Nicolson.

Contribution

It compares a previously proposed unitary algorithm with a generalized non-strictly unitary method for quantum evolution on curved spaces.

Findings

The unitary algorithm integrates the wrong equation.

The generalized algorithm is not strictly unitary at finite time steps.

Both methods' stability and unitarity properties are analyzed.

Abstract

The numerical integration of the Schr\"odinger equation by discretization of time is explored for the curved manifolds arising from finite representations based on evolving basis states. In particular, the unitarity of the evolution is assessed, in the sense of the conservation of mutual scalar products in a set of evolving states, and with them the conservation of orthonormality and particle number. Although the adequately represented equation is known to give rise to unitary evolution in spite of curvature, discretized integrators easily break that conservation, thereby deteriorating their stability. The Crank Nicolson algorithm, which offers unitary evolution in Euclidian spaces independent of time-step size $\mathrm{d}t$, can be generalised to curved manifolds in different ways. Here we compare a previously proposed algorithm that is unitary by construction, albeit integrating the wrong equation, with a faithful generalisation of the algorithm, which is, however, not strictly unitary for finite $\mathrm{d}t$.