Missing quantum number of Floquet states

We reformulate the Floquet theory for periodically driven quantum systems following a perfect analogy with the proof of the Bloch theorem. We observe that the current standard method for calculating the Floquet eigenstates using the quasienergy 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 quasienergy 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, which have long been thought to be incompatible due to the limitations of the quasienergy.

Figure 1

Convergence to the Floquet states using the Lagrange minimization of (a) the average energy [Eq. (40)] and (b) the quasienergy variance [Eq. (53)]. Each line represents a different convergence path starting from the initial states described in Eq. (55), and the final converging solution is duplicated on the rightmost axes. Only the convergent solutions are plotted, where the convergence criteria for both the objective function and the constraints are set to a factor of ${10}^{-12}$.