Robust quantification of spectral transitions in perturbed quantum systems

TL;DR

This paper develops rigorous, time-independent bounds on spectral leakage in perturbed quantum systems, ensuring small leakage over time even with continuous spectra, with practical applications.

Contribution

It introduces novel bounds on quantum spectral leakage that remain valid for all times and accommodate continuous spectra, using Schrieffer-Wolff and Bloch methods.

Findings

Bounded the distance between true and effective dynamics.

Proved leakage remains small for arbitrarily large times.

Applied bounds to practical quantum systems.

Abstract

A quantum system subject to an external perturbation can experience leakage between uncoupled regions of its energy spectrum separated by a gap. To quantify this phenomenon, we present two complementary results. First, we establish time-independent bounds on the distances between the true dynamics and the dynamics generated by block-diagonal effective evolutions constructed via the Schrieffer-Wolff and Bloch methods. Second, we prove that, under the right conditions, this leakage remains small eternally. That is, we derive a time-independent bound on the leakage itself, expressed in terms of the spectral gap of the unperturbed Hamiltonian and the norm of the perturbation, ensuring its validity for arbitrarily large times. Our approach only requires a finite spectral gap, thus accommodating continuous and unbounded spectra. Finally, we apply our bounds to specific systems of practical interest.