Gibbs phenomenon and the emergence of the steady-state in quantum transport

TL;DR

This paper links the formation of steady states in quantum transport simulations to the Gibbs phenomenon, explaining how individual and total steady states develop rapidly despite oscillations, with implications for nanoscale and ultracold atomic systems.

Contribution

It reveals that steady state emergence in quantum transport is a manifestation of the Gibbs phenomenon, providing a new understanding of the dynamics in explicit reservoir simulations.

Findings

Steady state formation is related to the Gibbs phenomenon.

Individual particles develop steady states at different rates depending on energy.

The total steady state is reached rapidly within a timescale of π/W.

Abstract

Simulations are increasingly employing explicit reservoirs - internal, finite regions - to drive electronic or particle transport. This naturally occurs in simulations of transport via ultracold atomic gases. Whether the simulation is numerical or physical, these approaches rely on the rapid development of the steady state. We demonstrate that steady state formation is a manifestation of the Gibbs phenomenon well-known in signal processing and in truncated discrete Fourier expansions. Each particle separately develops into an individual steady state due to the spreading of its wave packet in energy. The rise to the steady state for an individual particle depends on the particle energy - and thus can be slow - and ringing oscillations appear due to filtering of the response through the electronic bandwidth. However, the rise to the total steady state - the one from all particles - is rapid, with timescale $\pi/W$, where $W$ is the bandwidth. Ringing oscillations are now also filtered through the bias window, and they decay with a higher power. The Gibbs constant - the overshoot of the first ring - can appear in the simulation error. These results shed light on the formation of the steady state and support the practical use of explicit reservoirs to simulate transport at the nanoscale or using ultracold atomic lattices.