Topological invariants play a key role in the characterization of topological states. Due to the existence of exceptional points, it is a great challenge to detect topological invariants in both Hermitian and non-Hermitian systems via a unified approach. Here, we put forward a dynamic winding number, the winding of realistic observables in long-time average, for exploring band topology in generic two-band models. We build a concrete relation between dynamic winding numbers and equilibrium topological invariants. In one dimension, the dynamical winding number directly gives the conventional winding number. In two dimension, the Chern number relates to the weighted sum of dynamic winding numbers of all phase singularity points, in which the weight is +1 for the north pole and −1 for the south pole. This work opens a new avenue to detecting topological aspects of both Hermitian and non-Hermitian systems via general dynamic evolution.
We study the Floquet edge states in arrays of periodically curved optical waveguides described by the modulated Su-Schrieffer-Heeger model. Beyond the bulk-edge correspondence, our study explores the interplay between band topology and periodic modulations. By analysing the quasienergy spectra and Zak phase, we reveal that, although topological and non-topological edge states can exist for the same parameters, they can not appear in the same spectral gap. In the highfrequency limit, we find analytically all boundaries between the different phases and study the coexistence of topological and non-topological edge states. In contrast to unmodulated systems, the edge states appear due to either band topology or modulation-induced defects. This means that periodic modulations may not only tune the parametric regions with nontrivial topology, but may also support novel edge states. may coexist, how to distinguish between topological and non-topological edge states.In this work, we study the Floquet edge states (FESs) in arrays of periodically curved optical waveguides described by a periodically modulated Su-Schrieffer-Heeger (SSH) model [33]. We analyse, for the first time to our knowledge, the interplay between band topology and periodic modulations, and describe the coexistence of both topological and non-topological edge states supported by the same parameters. Our results show that, for a specific gap, the Zak phase Z Gm is either 0 or π, so that the topological edge states appear only in the gap of Z Gm = π. Through controlling both modulation frequency and amplitude, we may drive the system from non-topological to topological regime, and vice versa. We demonstrate analytically that periodic modulations induce a virtual defect at the boundary, being the key mechanism for the formation of non-topological edge states.The paper is organized as follows. In Sec. II, we introduce our physical model and derive its coupled-mode equations. In Sec. III, we calculate the quasi-energy spectra under OBC. In Sec. IV, by employing the multi-scale perturbation analysis, we give the effective coupled-mode equations and demonstrate the periodic modulations can induce virtual defects at boundaries. The FESs include defect-free surface states and Shockley-like surface states, which induced by virtual defects and the alternating strong and weak couplings between waveguides, respectively. In Sec. V, we analytically obtain the asymptotic phase boundary and numerically give the phase digram of appear FESs, respectively. We explore the topological nature of all FESs via calculate the bulk topological invariant Zak phase. We find that Shockley-like surface
Adiabatic quantum pumping in one-dimensional lattices is extended by adding a tilted potential to probe better topologically nontrivial bands. This extension leads to almost perfectly quantized pumping for an arbitrary initial state selected in a band of interest, including Bloch states. In this approach, the time variable offers not only a synthetic dimension as in the case of the Thouless pumping, but it assists also in the uniform sampling of all momenta due to the Bloch oscillations induced by the tilt. The quantized drift of Bloch oscillations is determined by a one-dimensional time integral of the Berry curvature, which is effectively an integer multiple of the topological Chern number in the Thouless pumping. Our study offers a straightforward approach to yield quantized pumping, and it is useful for probing topological phase transitions.
Controlling the balanced gain and loss in a PT -symmetric system is a rather challenging task. Utilizing Floquet theory, we explore the constructive role of periodic modulation in controlling the gain and loss of a PT -symmetric optical coupler. It is found that the gain and loss of the system can be manipulated by applying a periodic modulation. Further, such an original non-Hermitian system can even be modulated into an effective Hermitian system derived by the high-frequency Floquet method. Therefore, compared with other PT symmetry control schemes, our protocol can modulate the unbroken PT -symmetric range to a wider parameter region. Our results provide a promising approach for controlling the gain and loss of a realistic system.
We investigate how the driving field affects the bound states in the one-dimensional two-particle Bose–Hubbard model with an impurity. In the high-frequency regime, compared with the static lattice [Phys. Rev. Lett. 109 (2012) 116405], a new type of Floquet bound state can be obtained even for a weak particle–particle interaction by tuning the driving amplitude. Moreover, the localization degree of the Floquet bound molecular state can be adjusted by tuning the driving frequency, and even the Floquet bound molecular state can be changed into the Floquet extended state when the driving frequency is below a critical value. Our results provide an efficient way to manipulate bound states in the many-body systems.
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