Numerical approaches to time evolution of complex quantum systems

TL;DR

This paper compares numerical methods for simulating quantum system dynamics, highlighting the efficiency of a Chebyshev polynomial expansion approach over traditional schemes, and demonstrates its application to complex problems like graphene nanoribbons and polaron evolution.

Contribution

It introduces and benchmarks a Chebyshev polynomial-based method for quantum dynamics, showing its advantages over standard techniques and applying it to challenging quantum systems.

Findings

Chebyshev method is faster and more efficient than Crank-Nicholson.

Performance benchmarks show advantages in tunneling and anharmonic potentials.

Applications reveal localization and transmission phenomena in disordered systems.

Abstract

We examine several numerical techniques for the calculation of the dynamics of quantum systems. In particular, we single out an iterative method which is based on expanding the time evolution operator into a finite series of Chebyshev polynomials. The Chebyshev approach benefits from two advantages over the standard time-integration Crank-Nicholson scheme: speedup and efficiency. Potential competitors are semiclassical methods such as the Wigner-Moyal or quantum tomographic approaches. We outline the basic concepts of these techniques and benchmark their performance against the Chebyshev approach by monitoring the time evolution of a Gaussian wave packet in restricted one-dimensional (1D) geometries. Thereby the focus is on tunnelling processes and the motion in anharmonic potentials. Finally we apply the prominent Chebyshev technique to two highly non-trivial problems of current interest: (i) the injection of a particle in a disordered 2D graphene nanoribbon and (ii) the spatiotemporal evolution of polaron states in finite quantum systems. Here, depending on the disorder/electron-phonon coupling strength and the device dimensions, we observe transmission or localisation of the matter wave.