Krylov Complexity of Optical Hamiltonians

TL;DR

This paper explores Krylov complexity in various quantum optical models with group symmetries, analyzing how complexity evolves under different driving regimes and resonances, using group-theoretic methods and numerical algorithms.

Contribution

It introduces a group-theoretic approach to compute Krylov complexity in optical Hamiltonians with $SU(2)$, $SU(1,1)$, and $SU(3)$ symmetries, providing new insights into their dynamical complexity.

Findings

Krylov complexity exhibits resonance-dependent behavior.

Group decompositions simplify complexity analysis.

Lanczos algorithm reveals complexity dynamics in $SU(3)$ systems.

Abstract

In this work, we investigate the Krylov complexity in quantum optical systems subject to time--dependent classical external fields. We focus on various interacting quantum optical models, including a collection of two--level atoms, photonic systems and the quenched oscillator. These models have Hamiltonians which are linear in the generators of $SU(2)$, $H(1)$ (Heisenberg--Weyl) and $SU(1,1)$ group symmetries allowing for a straightforward identification of the Krylov basis. We analyze the behaviour of complexity for these systems in different regimes of the driven field, focusing primarily on resonances. This is achieved via the Gauss decomposition of the unitary evolution operators for the group symmetries. Additionally, we also investigate the Krylov complexity in a three--level $SU(3)$ atomic system using the Lanczos algorithm, revealing the underlying complexity dynamics. Throughout we have exploited the the relevant group structures to simplify our explorations.