Communication: Gibbs phenomenon and the emergence of the steady-state in quantum transport
Michael P. Zwolak
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 the emergence of the steady state 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 its wave-like nature. 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.