We study the on-resonance Spin-1/2 Double Kicked Rotor, a periodically driven quantum system that hosts topological phases. Motivated by experimental constraints, we analyze the effects of open and periodic boundary conditions in contrast to the idealized case of infinite momentum space. As a bulk probe for topological invariants, we focus on the Mean Chiral Displacement (MCD) and show that it exhibits a pronounced sensitivity to boundary conditions, which can be traced to the dynamics in momentum space. Under open boundaries, states that would otherwise extend freely become localized at the edges of the finite momentum space, forming quasienergy edge states. While the bulk response measured by the MCD is strongly affected once the evolving wave packet reaches the boundaries, the persistence of these edge states still reflects the bulk-edge correspondence and provides reliable signatures of topological transitions.
Witnessing and tracking topological phase transitions induced by interactions with the environment is a crucial challenge. Among the various experimental approaches to detect topological properties, the Mean Chiral Displacement (MCD) has emerged as a powerful bulk probe in one-dimensional chiral systems, allowing the extraction of the topological invariant from single-particle dynamics. Here we study the dynamics of a single particle in a one-dimensional Su-Schrieffer-Heeger chain coupled to multiple cavity modes via inter-cell hopping terms, focusing on the out-of-equilibrium behavior of the MCD. We show that, whenever the frequency is larger than the static hopping amplitudes, the coupling induces a discontinuous jump in the MCD, already at small times, signaling that such a coupling also leaves a signature in the survival edge probability when the dynamics are initialized at one of the two edges. For frequencies comparable to the static hopping amplitudes, topological order competes with dissipative effects, which makes the MCD behave smoothly, retaining information about the driven-dissipative topology.
We study the topological properties of one-dimensional discrete-time quantum walks with Fibonacci quasiperiodic modulation. Spectral analysis under open boundary conditions reveals isolated edge modes that coexist at both zero and
π energies within bulk gaps. Using the mean chiral displacement (MCD) as a dynamical bulk probe, we obtain a fractal, butterfly-like phase diagram, indicating a nontrivial bulk topology. However, since the MCD involves wave-packet averaging, its direct correspondence with boundary-localized states remains ambiguous. To resolve this issue, we determine the integer topological invariant by employing the Schur function formalism, a scattering-based approach defined on the unit disk. This analysis identifies a topological phase with winding number
W=2 inside the ``wings'' of the butterfly diagram, demonstrating that the coexistence of zero- and
π-energy edge modes is an intrinsic bulk property. Moreover, we find that the topological index crucially depends on the phason degree of freedom: the winding number can take values of
W=0,
2, and even
4 depending on the local surface termination. This behavior originates from the surface impedance, which yields an effective winding number ranges from complete masking by reflective boundaries (
W=0) to enhancement through surface resonances (
W=4). Finally, we show that although individual surface terminations lead to anisotropic phase diagrams, their ensemble average restores the overall geometric structure observed in the MCD phase diagram. Our results establish a complete bulk-edge correspondence in quasiperiodic quantum walks and provide a guiding principle for surface topological engineering, where edge transport can be controlled purely through boundary manipulation.
We present a concrete theoretical proposal for detecting topological phase transitions in double kicked atom-optics kicked rotors with internal spin-1/2 degree of freedom. The implementation utilizes a kicked Bose-Einstein condensate evolving in one-dimensional momentum space. To reduce the influence of atom loss and phase decoherence, we aim to keep experimental durations short while maintaining a resonant experimental protocol. Experimental limitations induced by phase noise, quasimomentum distributions, symmetries, and the ac-Stark shift are considered. Our results thus suggest a feasible and optimized procedure for observing topological phase transitions in quantum kicked rotors.
Discrete-time quantum walks have been shown to simulate all known topological phases in one and two dimensions. Being periodically driven quantum systems, their topological description, however, is more complex than that of closed Hamiltonian systems. We map out the topological phases of the particle-hole symmetric one-dimensional discrete-time quantum walk. We find that there is no chiral symmetry in this system: its topology arises from the particle-hole symmetry alone. We calculate the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological invariant in a simple way that is consistent with a general definition for one-dimensional periodically driven quantum systems. These results allow for a transparent interpretation of the edge states on a finite lattice via the the bulk-boundary correspondence. We find that the bulk Floquet operator does not contain all the information needed for the topological invariant. As an illustration to this statement, we show that in the split-step quantum walk, the edges between two bulks with the same Floquet operator can host topologically protected edge states.
Topologically protected edge states exactly at topological phase boundaries challenge the conventional belief that topological states must be associated with a bulk energy gap. Because periodically driven (Floquet) systems host unusually intricate topological phase boundaries, topological edge states can be prolific at such Floquet quantum criticality. Working on a class of chiral-symmetric, Floquet-driven Majorana fermion chains, we analytically and computationally show that the precise boundaries between different Floquet topological gapped phases can accommodate topological edge modes, including the so-called critical Majorana π modes. We also identify a general bulk-edge correspondence formula to predict and understand the emergence of topological edge modes at Floquet quantum criticality. Of direct interest to quantum simulation experiments, our work opens an avenue for exploring topological physics intrinsic to nonequilibrium phase boundaries. The authors develop a theoretical framework to characterize the topology and edge states at the nonequilibrium critical points of periodically driven (Floquet) quantum systems. The rule of bulk-edge correspondence applicable to both gapped and gapless Floquet topological phases in 1D chiral symmetric driven systems is proposed.
Novel topological properties that arose in the periodically driven system are unique, in which there are two kinds of quasienergy gaps, the zero quasienergy gap and the
π quasienergy gap. The corresponding edge modes would traverse either the zero quasienergy gap or the
π quasienergy gap, or traverse both two quasienergy gaps. And the characterization of these two kinds of edge modes might not be the same. However, in this paper, we find that both the zero edge modes and the
π edge modes in the Floquet Chern insulators can be characterized by the same topological invariant, where the corresponding Dirac mass term is periodically kicked. Particularly, we take the periodically kicked Qi-Wu-Zhang model as an illustrative example. In this model, the topology is characterized by the Floquet Chern number
CF, and there are six different topological phases in total, denoted as
CF={−10,−2,−1π,1π,2,10}. Furthermore, we find that the Floquet operator associated with the periodically kicked Qi-Wu-Zhang model reduces to a Dirac Hamiltonian in the low-energy limit. Then, the phase diagram is uncovered by examining the topology of this Dirac Hamiltonian. Additionally, we explore the orders of topological phase transitions in the context of Floquet stationary states by analyzing the von Neumann entropy of these states. Our work provides further insights into the topological phases in periodically driven systems.