Non-perturbative interpretation of the Bloch vector's path beyond rotating wave approximation

TL;DR

This paper provides a non-perturbative analysis of the Bloch vector's complex trajectories in strong coupling regimes, revealing oscillations, cusps, and divergence in curvature beyond the rotating wave approximation.

Contribution

It introduces an analytical and numerical framework to understand the Bloch vector's path beyond RWA, including cusp formation and curvature divergence.

Findings

Rotation speed oscillates between zero and twice the RWA prediction.

Cusps occur at points where the rotation speed vanishes.

Large curvature persists even with quantum fields in the deep quantum regime.

Abstract

The Bloch vector's path of a two-level system exposed to a monochromatic field exhibits, in the regime of strong coupling, complex corkscrew trajectories. By considering the infinitesimal evolution of the two-level system when the field is treated as a classical object, we show that the Bloch vector's rotation speed oscillates between zero and twice the rotation speed predicted by the rotating wave approximation. Cusps appear when the rotation speed vanishes. We prove analytically that in correspondence to cusps the curvature of the Bloch vector's path diverges. On the other hand, numerical data show that the curvature is very large even for a quantum field in the deep quantum regime with mean number of photons $\bar{n}\lesssim 1$. We finally compute numerically the typical error size in a quantum gate when the terms beyond rotating wave approximation are neglected.