The missing quantum number of the Floquet states

TL;DR

This paper introduces an overlooked quantum number, the average energy, into Floquet theory for periodically driven quantum systems, improving the stability and completeness of Floquet state calculations.

Contribution

It identifies the average energy as a crucial quantum number, resolving quasi-energy degeneracy issues and enabling a variational approach for Floquet states.

Findings

The average energy quantum number provides a stable ordering of Floquet states.

Inclusion of average energy resolves degeneracy issues in the continuum limit.

A variational method for Floquet states is proposed.

Abstract

We reformulate the Floquet theory for periodically driven quantum systems following a perfect analogy with the proof of Bloch theorem. We observe that the current standard method for calculating the Floquet eigenstates by the quasi-energy alone is incomplete and unstable, and pinpoint an overlooked quantum number, the average energy. This new quantum number resolves many shortcomings of the Floquet method stemming from the quasi-energy degeneracy issues, particularly in the continuum limit. Using the average energy quantum number we get properties similar to those of the static energy, including a unique lower-bounded ordering of the Floquet states, from which we define a ground state, and a variational method for calculating the Floquet states. This is a first step towards reformulating Floquet first-principles methods, that have long been thought to be incompatible due to the limitations of the quasi-energy.