Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering

Abstract

We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged Hamiltonian. These classes cover systems, such as the Kapitza pendulum, the Harper-Hofstadter model of neutral atoms in a magnetic field, the Haldane Floquet Chern insulator and others. In all setups considered, we discuss both the infinite-frequency limit and the leading finite-frequency corrections to the Floquet Hamiltonian. We provide a short overview of Floquet theory focusing on the gauge structure associated with the choice of stroboscopic frame and the differences between stroboscopic and non-stroboscopic dynamics. In the latter case one has to work with dressed operators representing observables and a dressed density matrix. We also comment on the application of Floquet Theory to systems described by static Hamiltonians with well-separated energy scales and, in particular, discuss parallels between the inverse-frequency expansion and the Schrieffer-Wolff transformation extending the latter to driven systems. PACS

Schrieffer-Wolff Transformation for Periodically Driven Systems: Strongly Correlated Systems with Artificial Gauge Fields

“…In the following, we work in the limit J 0 U, Ω and assume that the amplitude of the periodic modulation also scales with Ω [40].…”

“…Using the HFE to perform the SWT offers a few advantages: (i) the SW generator comes naturally out of the calculation, (ii) one can systematically compute higherorder corrections [33][34][35][36][37][38]40], and (iii) the HFE allows for obtaining not only the effective Hamiltonian but also the kick operator, which keeps track of the mixing between orbitals and describes the intra-period dynamics [34,40]. This is important for identifying the fast timescale associated with the large frequency U in dynamical measurements [41], and expressing observables through creation and annihilation operators dressed by orbital mixing [40].…”

Floquet engineering from long-range to short-range interactions

“…In this Rapid Communication, we show how to use Floquet engineering [16,17] to reshape a long-range interaction into a short-range one. Although we focus on making the interaction as short range as possible, our approach can be used to engineer other interaction profiles.…”

“…We let the gradient strength scale with in order to get a nontrivial Floquet Hamiltonian in the limit of large [16,17]. The gradient can be generated experimentally by a magnetic field [22,23] or ac Stark shift [24,25].…”

“…The advantages of these setups are that the interaction strength is large and that the atoms do not have to be very cold. However, these setups have long-range interactions, so it would be beneficial to somehow remove the longrange tail while otherwise preserving the magnitude of the interactions.In this Rapid Communication, we show how to use Floquet engineering [16,17] to reshape a long-range interaction into a short-range one. Although we focus on making the interaction as short range as possible, our approach can be used to engineer other interaction profiles.…”