Refuting a Proposed Axiom for Defining the Exact Rotating Wave Approximation

TL;DR

This paper critically examines a proposed axiomatic definition of the effective Hamiltonian for the rotating wave approximation in driven qubits, refuting a specific variational principle through numerical analysis.

Contribution

It challenges the validity of a third proposed axiom for defining the effective Hamiltonian in the rotating wave approximation.

Findings

The third axiom does not hold under numerical tests.

The effective Hamiltonian series may not always converge for arbitrary pulse shapes.

The variational principle is invalid for the effective Hamiltonian definition.

Abstract

For a linearly driven quantum two-level system, or qubit, sets of stroboscropic points along the cycloidal-like trajectory in the rotating frame can be approximated using the exact rotating wave approximation introduced in arXiv:1807.02858. That work introduces an effective Hamiltonian series $\mathcal H_{\text{eff}}$ generating smoothed qubit trajectories; this series has been obtained using a combination of a Magnus expansion and a Taylor series, a Magnus-Taylor expansion. Since, however, this Hamiltonian series is not guaranteed to converge for arbitrary pulse shapes, the same work hypothesizes an axiomatic definition of the effective Hamiltonian. The first two of the proposed axioms define $\mathcal H_{\text{eff}}$ to (i) be analytic and (ii) generate a stroboscopic time evolution. In this work we probe a third axiom---motivated by the smoothed trajectories mentioned above---namely, (iii) a variational principle stating that the integral of the Hamiltonian's positive eigenvalue taken over the full pulse duration is minimized by this $\mathcal H_{\text{eff}}$. We numerically refute the validity of this third axiom via a variational minimization of the said integral.