Dynamical quantum phase transitions (DQPTs) are manifested by time-domain nonanalytic behaviors of many-body systems. Introducing a quench is so far understood as a typical scenario to induce DQPTs. In this work, we discover a novel type of DQPTs, termed "Floquet DQPTs", as intrinsic features of systems with periodic time modulation. Floquet DQPTs occur within each period of continuous driving, without the need for any quenches. In particular, in a harmonically driven spin chain model, we find analytically the existence of Floquet DQPTs in and only in a parameter regime hosting a certain nontrivial Floquet topological phase. The Floquet DQPTs are further characterized by a dynamical topological invariant defined as the winding number of the Pancharatnam geometric phase versus quasimomentum. These findings are experimentally demonstrated with a single spin in diamond. This work thus opens a door for future studies of DQPTs in connection with topological matter.DQPTs are often associated with quantum quenches, a protocol in which parameters of a Hamiltonian are suddenly changed [1,2]. A quantum quench across an equilibrium quantum critical point may induce a DQPT. If pre-quench and post-quench systems are in topologically distinct phases, DQPTs may also be characterized by dynamical topological invariants [3][4][5]. As a promising approach to classify quantum states of matter in nonequilibrium situations, DQPTs have been theoretically explored in both closed and open quantum systems at different physical dimensions [2,7,8]. Experimentally, DQPTs have been observed in trapped ions [3,10], cold atoms [11,12], superconducting qubits [13], nanomechanical oscillators [14], and photonic quantum walks [15,16].To date, in most studies of DQPTs, a quantum quench acts as a trigger for initiating nonequilibrium dynamics and then exposing the underlying topological features. However, DQPTs under more general nonequilibrium manipulations are still largely unexplored [17][18][19]. In particular, because the dynamics of systems under time-periodic modulations has led to fascinating discoveries like Floquet topological states [20-24] and discrete time crystals [25][26][27], it is urgent to investigate how DQPTs may occur in such Floquet systems. Along this avenue, there have been scattered studies, but still with the notion that DQPTs are best aroused by a quench to some system parameters [28,29]. Here we introduce a novel class of DQPTs, termed Floquet DQPTs, which can be regarded as intrinsic features of systems with time-periodic modulations. As schematically shown in Floquet DQPT Time Time Rate function Rate function H i H f (a) H(t) (b) Conventional DQPT T T Quench T FIG. 1. Comparison between (a) DQPTs following a quantum quench, and (b) Floquet DQPTs without any quenches.Here Hi and H f denote the Hamiltonians before and after the quench, H(t) denotes a periodically and continuously modulated Hamiltonian. Fig. 1, the Floquet DQPTs we discovered occur within each period of a continuous driving field, without the need for any...

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 unconventional 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 Rb condensates into a periodically pulsed optical lattice. Under the on-resonance condition, the spin-1/2 double kicked rotor admits fruitful 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 0 and π-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 Rb condensates, and suggest to detect its topological invariants by measuring the mean chiral displacement in momentum space.

Topological states of matter in non-Hermitian systems have attracted a lot of attention due to their intriguing dynamical and transport properties. In this study, 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 0-and π-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 0-and π-quasienergy edge states to appear in the complex quasienergy bulk gaps in a wellcontrolled manner. We further suggest to probe the topological winding numbers of the system by dynamically imaging the stroboscopic spin textures of its bulk states.

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Recent discoveries on topological characterization of gapless systems have attracted interest in both theoretical studies and experimental realizations. Examples of such gapless topological phases are Weyl semimetals, which exhibit three-dimensional (3D) Dirac cones (Weyl points), and nodal line semimetals, which are characterized by line nodes (two bands touching along a line). Inspired by our previous discoveries that the kicked Harper model exhibits many fascinating features of Floquet topological phases, in this paper we consider a generalization of the model, where two additional periodic system parameters are introduced into the Hamiltonian to serve as artificial dimensions, so as to simulate a 3D periodically driven system. We observe that by increasing the hopping strength and the kicking strength of the system, many new Floquet band touching points at Floquet quasienergies 0 and π will start to appear. Some of them are Weyl points, while the others form line nodes in the parameter space. By taking open boundary conditions along the physical dimension, edge states analogous to Fermi arcs in static Weyl semimetal systems are observed. Finally, by designing an adiabatic pumping scheme, the chirality of the Floquet-band Weyl points and the π Berry phase around Floquet-band line nodes can be manifested.

In closed quantum systems, a dynamical phase transition is identified by nonanalytic behaviors 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 realized by optical resonators. Dynamical quantum phase transitions with topological signatures are found when an isolated exceptional point is crossed during the quench. A topological 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, mimicking the plateau transitions in quantum Hall effects. This work provides a simple framework to study dynamical and topological responses in non-Hermitian systems.Introduction.-Dynamical quantum phase transitions (DQPTs) are characterized by nonanalytic behavior of physical observables as functions of time [1,2]. These transitions happen in general if the system is ramped through a quantum critical point. As a promising framework to classify quantum dynamics of nonequilibrium many-body systems, DQPTs have been studied intensively in recent years [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The generality and topological feature of DQPTs were demonstrated in both lattice and continuum systems [20], across different spatial dimensions [21][22][23][24][25], and under various dynamical protocols [26][27][28]. The defining features of DQPTs have also been observed in recent experiments [29][30][31].Following the initial proposal, most studies on DQPTs focus on closed quantum systems undergoing unitary time evolution. Efforts have been made to generalize DQPTs to systems prepared in mixed states [32,33]. However, DQPTs in systems with gain and loss, and therefore subject to nonunitary evolution are largely unexplored. One such class of open systems can be descried by a non-Hermitian Hamiltonian. This type of system, realizable in various platforms like photonic lattice [34], phononic media [35], LRC circuits [36] and cold atoms [37,38], has attracted great attention in recent years due to their nontrivial dynamical [39][40][41][42][43][44][45][46][47][48], topological [49][50][51][52][53][54][55][56][57][58][59][60][61][62] and transport properties [63][64][65][66][67][68][69][70]. Many of these features can be traced back to non-Hermitian degeneracy (i.e. exceptional) points mediating gap closing and reopening transitions on the complex plane [71][72][73][74][75][76]. In this work, we explore DQPTs in non-Hermitian systems, with a focus on topological signatures in nonunitary evolution following quenches across exceptional points (EPs).Theory.-We start by summarizing the theoretical framework of DQPTs for systems described by non-Hermitian lattice Hamiltonians. The nonunitary time evolution of the system is governed by a Schrödinger equation i d dt |Ψ(t) = H|Ψ(t) . For concreteness, we present the formalism with a one-dimensional twoband lattice model in mind, while the g...

Downstream analysis of circulating tumor cells (CTCs) has provided new insights into cancer research. In particular, the detection of CTCs, followed by the regulation and monitoring of their intracellular activities, can provide valuable information for comprehensively understanding cancer pathogenesis and progression. However, current CTC detection techniques are rarely capable of in situ regulation and monitoring of the intracellular microenvironments of cancer cells over time. Here, we developed a multifunctional branched nanostraw (BNS)-electroporation platform that could effectively capture CTCs and allow for downstream regulation and monitoring of their intracellular activities in a real-time and in situ manner. The BNSs possessed numerous nanobranches on the outer sidewall of hollow nanotubes, which could be conjugated with specific antibodies to facilitate the effective capture of CTCs. Nanoelectroporation could be applied through the BNSs to nondestructively porate the membranes of the captured cells at a low voltage, allowing the delivery of exogenous biomolecules into the cytosol and the extraction of cytosolic contents through the BNSs without affecting cell viability. The efficient delivery of biomolecules (e.g., small molecule dyes and DNA plasmids) into cancer cells with spatial and temporal control and, conversely, the repeated extraction of intracellular enzymes (e.g., caspase-3) for real-time monitoring were both demonstrated. This technology can provide new opportunities for the comprehensive understanding of cancer cell functions that will facilitate cancer diagnosis and treatment.

Dynamical kicking systems possess rich topological structures. In this work, we study Floquet states of matter in a non-Hermitian extension of double kicked rotor model. Under the on-resonance condition, we find various non-Hermitian Floquet topological phases, with each being characterized by a pair of topological winding numbers. A generalized mean chiral displacement is introduced to detect these winding numbers dynamically in two symmetric time frames. Furthermore, by mapping the system to a periodically quenched lattice model, we obtain the topological edge states and unravel the bulk-edge correspondence of the non-Hermitian double kicked rotor. These results uncover the richness of Floquet topological states in non-Hermitian dynamical kicking systems.