Fig. 2

a A periodically modulated system can be equivalently modelled as a lattice of modes, with the non-zero frequency components of the Hamiltonian providing the coupling between the different frequency “sheets”. b Periodically pumped systems exhibit parametric instabilities at those states which are separated from their time-reversal partner by a modulation frequency \(\Omega = 2\omega_{0}\), where \(\omega_{0}\) is the frequency of the amplified input wave. c These instabilities can be equivalently regarded as the result of the opening of k-gaps, hosting imaginary states which grow or decay exponentially over time. d Acoustic Floquet topological insulator formed by a Kagome lattice featuring a phased time-modulation of the acoustic capacitance \(C\) at its three corner sites, producing an angular bias. e The introduction of the capacitance modulation results in the opening of a topological band-gap. An instance of edge state hosted within the topological gap is shown in panel (d). Adapted from [14]. f A ring resonator is temporally modulated by an electro-optic modulator (EOM), and its clockwise (CW) and counterclockwise (CCW) states are coupled by a pair of non-intersecting waveguides. g, h This setup produces a system with two synthetic dimensions, the Floquet ladder and the angular momentum basis, within a single ring-resonator. Adapted from [38]