In closed quantum systems, dynamical phase transitions are identified by the nonanalytic behavior of the return probability as a function of time. In this work, we study the nonunitary dynamics following quenches across exceptional points in a non-Hermitian lattice realizable by optical resonators. Dynamical quantum phase transitions with topological signatures are found when an isolated exceptional point is crossed during the quench. A winding number defined by a real, noncyclic geometric phase is introduced, whose value features quantized jumps at critical times of these phase transitions and remains constant elsewhere, playing the role of a topological order parameter. This work provides a simple framework to study dynamical and topological quantum phase transitions in non-Hermitian systems.

Topological states of matter in non-Hermitian systems have attracted a lot of attention due to their intriguing dynamical and transport properties. In this paper, we propose a periodically driven non-Hermitian lattice model in one dimension, which features rich Floquet topological phases. The topological phase diagram of the model is derived analytically. Each of its non-Hermitian Floquet topological phases is characterized by a pair of integer winding numbers, counting the number of real zero- and $\pi$-quasienergy edge states at the boundaries of the lattice. Non-Hermiticity-induced Floquet topological phases with unlimited winding numbers are found, which allow arbitrarily many real zero- and $\pi$-quasienergy edge states to appear in the complex quasienergy bulk gaps in a well-controlled manner. We further suggest probing the topological winding numbers of the system by dynamically imaging the stroboscopic spin textures of its bulk states.

Second-order topological insulators (SOTI) exhibit protected gapless boundary states at their hinges or corners. In this paper, we propose a generic means to construct SOTIs in static and Floquet systems by coupling one-dimensional topological insulator wires along a second dimension through dimerized hopping amplitudes. The Hamiltonian of such SOTIs admits a Kronecker sum structure, making it possible for obtaining its features by analyzing two constituent one-dimensional lattice Hamiltonians defined separately in two orthogonal dimensions. The resulting topological corner states do not rely on any delicate spatial symmetries, but are solely protected by the chiral symmetry of the system. We further utilize our idea to construct Floquet SOTIs, whose number of topological corner states is arbitrarily tunable via changing the hopping amplitudes of the system. Finally, we propose to detect the topological invariants of static and Floquet SOTIs constructed with our approach in experiments by measuring the mean chiral displacements of wavepackets.

The double kicked rotor model is a physically realizable extension of the paradigmatic kicked rotor model in the study of quantum chaos. Even before the concept of Floquet topological phases became widely known, the discovery of the Hofstadter butterfly spectrum in the double kicked rotor model [J. Wang and J. Gong, Phys. Rev. A 77, 031405 (2008)] already suggested the importance of periodic driving to the generation of Floquet topological matter. In this work, we explore Floquet topological phases of a double kicked rotor with an extra spin-$1/2$ degree of freedom. The latter has been experimentally engineered in a quantum kicked rotor recently by loading ${}^{87}\mathrm{Rb}$ condensates into a periodically pulsed optical lattice. Theoretically, we found that under the on-resonance condition, the spin-$1/2$ double kicked rotor admits rich topological phases due to the interplay between its external and internal degrees of freedom. Each of these topological phases is characterized by a pair of winding numbers, whose combination predicts the number of topologically protected zero and $\pi$-quasienergy edge states in the system. Topological phases with arbitrarily large winding numbers can be easily found by tuning the kicking strength. We discuss an experimental proposal to realize this model in kicked ${}^{87}\mathrm{Rb}$ condensates, and suggest detecting its topological invariants by measuring the mean chiral displacement in momentum space.

Majorana edge modes are candidate elements of topological quantum computing. In this paper, we propose a Floquet engineering approach to generate multiple non-Hermitian Majorana zero and $\pi$ modes at the edges of a one-dimensional topological superconductor. Focusing on a Kitaev chain with periodically kicked superconducting pairings and gain/losses in the chemical potential or nearest-neighbor hopping terms, we found rich non-Hermitian Floquet topological superconducting phases, which are originated from the interplay between drivings and non-Hermitian effects. Each of the phases is characterized by a pair of topological winding numbers, which can in principle take arbitrarily large integer values thanks to the applied driving fields. Under the open boundary condition, these winding numbers also predict the number of degenerate Majorana edge modes with quasienergies zero and $\pi$. Our findings thus expand the family of Floquet topological phases in non-Hermitian settings, with potential applications in realizing environmentally robust Floquet topological quantum computations.

Floquet states of periodically driven systems could exhibit rich topological properties. Many of them are absent in their static counterparts. One such example is the chiral edge states in anomalous Floquet topological insulators, whose description requires a different topological invariant and type of bulk-edge correspondence. In this work, we propose a prototypical quenched lattice model, whose two Floquet bands could exchange their Chern numbers periodically and alternatively via touching at quasienergies 0 and $\pi$ under the change of a single system parameter. This process in principle allows the generation of as many Floquet chiral edge states as possible in a highly controllable manner. The quantized transmission of these edge states are extracted from the Floquet scattering matrix of the system. The flexibility in controlling the number of topological edge channels provided by our scheme could serve as a starting point for the engineering of robust Floquet transport devices.