Completing the Floquet picture: the missing quantum number

The Floquet method in quantum mechanics has become a central tool for calculating periodically driven quantum systems; however, there are known limitations to this method that limit its applicability. The primary issues are the lack of a bounded ordering of the eigenstates and the infinite quasi-energy degeneracy in the continuum, which rule out a Ritz variation principle and efficient first-principles computation methods based on it. Moreover, the Floquet states are considered to be the energy-time counterparts of the spatially periodic Bloch states; however, if we properly follow this analogy, we find that there is no equivalent energy quantum number in the current Floquet formalism based on the quasi-energy.

Here we will show how the Floquet picture can be completed using the average energy¹ as a quantum number, following a perfect analogy with the Bloch systems. This redefines the Floquet eigenstates, having a unique average energy ground state and resolving the previous issues of the current quasi-energy method. With this, we can start adapting conventional first-principles calculation methods for the static ground-state to formulate the equivalent Floquet methods for the average energy Floquet ground state. We will demonstrate this with the Floquet Hartree-Fock calculation.