Multi-Directional Periodic Driving of a Two-Level System beyond Floquet formalism*

This talk introduces an exact expression for the response of a two-level quantum system subject to arbitrary periodic driving. Determining the dynamics of a two-level system driven by an arbitrary periodic waveform necessitates the use of numerical methods such as Floquet theory. In particular, one seeks to represent the time-dependent Hamiltonian by an extended space Floquet Hamiltonian given by an infinite matrix requiring truncation. However, truncation may lead to various issues such as loss of information concerning interference effects or removing the ability to predict novel effects generated by the drive. To alleviate this issue, we analytically investigate the dynamics of a two-level quantum system with a modulation given by arbitrary multi-directional, multi-harmonic driving. To consolidate the driving to a single term, we introduce the generalized Bessel function (GBF) allowing for analytical expressions using perturbative methods. Beyond perturbative regimes, the ⋆-resolvent formalism is advantageously used to determine an exact series solution for the unitary evolution generated from a compact two-time kernel expression. The resulting series solution is expanded in a non-harmonic Fourier series basis given by the divided difference of complex exponentials with coefficients corresponding to products of the GBFs.