Adiabatic Floquet representation of atomic systems

Phillip Weinberg, Marin Bukov, Luca D’Alessio +3 more

Anatoli Polkovnikov

Jesús Hernández‐Saz

T. T. Nguyen‐Dang, Evgueni Sinelnikov, Arne Keller +1 more

A perturbative series is derived for the systematic construction of a dynamical invariant (Lewis invariant) for a time-dependent Hamiltonian which is characterized by a time-scale parameter $\ensuremath{\tau}$, as appears in the usual formulation of the adiabatic theorem. The derivations make efficient use of the quantum averaging method, and the perturbative series obtained permits the construction of the invariant in successive orders in $ϵ\ensuremath{\sim}1/\ensuremath{\tau}$, corresponding to high-order adiabatic approximations. The series can be considered and resummed analytically to all orders, yielding an exact invariant for an harmonic oscillator linearly driven by an external field. For a nondegenerate two-level system, approximate invariants have been obtained up to the fifth order and illustrate how the adiabatic approximations of increasing orders successively approach the exact dynamics. The construct is applicable also to Floquet Hamiltonians and, in this context, it furnishes high-order adiabatic representations for the time evolution of Floquet states associated with a material system in an aperiodic laser pulse. We illustrate in particular how an exact Floquet dynamical invariant, defining exact adiabatic transports of Floquet states, is obtained for a laser-driven harmonic vibrational mode.

Masaya Nakagawa, Robert-Jan Slager, Sho Higashikawa +1 more

A universal feature of topological insulators is that they cannot be adiabatically connected to an atomic limit, where individual lattice sites are completely decoupled. This property is intimately related to a topological obstruction to constructing a localized Wannier function from Bloch states of an insulator. Here, we generalize this characterization of topological phases toward periodically driven systems. We show that nontrivial connectivity of hybrid Wannier centers in momentum space and time can characterize various types of topology in periodically driven systems, which include Floquet topological insulators, anomalous Floquet topological insulators with micromotion-induced boundary states, and gapless Floquet states realized with topological Floquet operators. In particular, nontrivial time dependence of hybrid Wannier centers indicates impossibility of continuous deformation of a driven system into an undriven insulator, and a topological Floquet operator implies an obstruction to constructing a generalized Wannier function which is localized in real and frequency spaces. Our results pave a way to a unified understanding of topological states in periodically driven systems as a topological obstruction in Floquet states.

Gayanath Fernando, R. Matthias Geilhufe