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

TL;DR

This paper presents an exact analytical method for calculating the response of a two-level quantum system under arbitrary periodic driving, surpassing traditional Floquet approaches by avoiding matrix truncation and capturing all interference effects.

Contribution

The authors introduce a novel $ ext{ extsterling}$-resolvent formalism with the path-sum theorem to derive an exact series solution for the transition probabilities in periodically driven two-level systems.

Findings

Provides an exact series solution for transition probabilities.

Eliminates the need for matrix truncation in Floquet analysis.

Offers a compact kernel expression capturing all drive information.

Abstract

In this manuscript, we introduce an exact expression for the response of a semi-classical two-level quantum system subject to arbitrary periodic driving. Determining the transition probabilities of a two-level system driven by an arbitrary periodic waveform necessitates numerical calculations through methods such as Floquet theory, requiring the truncation of an infinite matrix. However, such truncation can lead to a loss of significant interference information, hindering quantum sensors or introducing artifacts in quantum control. To alleviate this issue, we use the $\star$-resolvent formalism with the path-sum theorem to determine the exact series solution to Schr\"odinger's equation, therefore providing the exact transition probability. The resulting series solution is generated from a compact kernel expression containing all of the information of the periodic drive and then expanded in a non-harmonic Fourier series basis given by the divided difference of complex exponentials with coefficients corresponding to products of generalized Bessel functions. The present method provides an analytical formulation for quantum sensors and control applications.