We investigate quantum dynamics of a quantum walker on a finite bipartite non-Hermitian lattice, in which the particle can leak out with certain rate whenever it visits one of the two sublattices. Quantum walker initially located on one of the non-leaky sites will finally totally disappear after a length of evolution time and the distribution of decay probability on each unit cell is obtained. In one regime, the resultant distribution shows an expected decreasing behavior as the distance from the initial site increases. However, in the other regime, we find that the resultant distribution of local decay probability is very counterintuitive, in which a relatively high population of decay probability appears on the edge unit cell which is the farthest from the starting point of the quantum walker. We then analyze the energy spectrum of the non-Hermitian lattice with pure loss, and find that the intriguing behavior of the resultant decay probability distribution is intimately related to the existence and specific property of edge states, which are topologically protected and can be well predicted by the non-Bloch winding number. The exotic dynamics may be observed experimentally with arrays of coupled resonator optical waveguides.
Loss induced generalized measurements have been introduced years ago as a mean to implement generalized quantum measurements (POVM). Here the original idea is extended to a complete equivalence of lossy evolution and a certain widely used class of POVM. This class includes POVM used for unambiguous state discrimination and entanglement concentration. One implication of this equivalence is that unambiguous state discrimination schemes based on PT-symmetric and non-Hermitian Hamiltonians have the same performance as those of standard POVM. After discussing several key points of this equivalence …
The dynamical quantum phase transitions (DQPTs) and the associated winding numbers have been extensively studied in the context Hermitian system. We consider the non-Hermitian analogue of $p$-wave superconductor, supporting Hermitian gapless phase with complex hopping, in presence of on-site or superconducting loss term. This allows us to investigate the effect of non-Hermitian gapless phases on the DQPTs in addition to the Hermitian gapless phases. Our findings indicate that contour analysis of the underlying Hamiltonian, enclosing the origin and/or exceptional points, can predict the occurre…
Experimental control over ultracold quantum gases has made it possible to investigate low-dimensional systems of both bosonic and fermionic atoms. In closed 1D systems there are a lot of similarities in the dynamics of local quantities for spinless fermions and strongly interacting "hard-core" bosons, which on a lattice can be formalised via a Jordan-Wigner transformation. In this study, we analyse the similarities and differences for spinless fermions and hard-core bosons on a lattice in the presence of particle loss. The removal of a single fermion causes differences in local quantities comp…
Pseudo-Hermitian operators can be used in modeling electromagnetic wave propagation in stationary lossless media. We extend this method to a class of non-dispersive anisotropic media that may display loss or gain. We explore three concrete models to demonstrate the utility of our general results and reveal the physical meaning of pseudo-Hermiticity and quasi-Hermiticity of the relevant wave operator. In particular, we consider a uniaxial model where this operator is not diagonalizable. This implies left-handedness of the medium in the sense that only clockwise circularly polarized plane-wave s…
The spectral singularity have been extensively studied over the last one and half decade for different non-Hermitian potentials in non-Hermitian quantum mechanics. The nature of spectral singularities have not been studied for the case of quaternionic potential. In the present work we perform an analytical study for the scattering from a quaternionic point interaction represented by delta function. New features of spectral singularities are observed which are different than the case of complex (non-quaternionic) point interaction. Most notable difference is the occurrence of spectral singulari…
We report on the experimental realization and detection of dynamical currents in a spin-textured lattice in momentum space. Collective tunneling is implemented via cavity-assisted Raman scattering of photons by a spinor Bose-Einstein condensate into an optical cavity. The photon field inducing the tunneling processes is subject to cavity dissipation, resulting in effective directional dynamics in a non-Hermitian setting. We observe that the individual tunneling events are superradiant in nature and locally resolve them in the lattice by performing real-time, frequency-resolved measurements of …
Discrete-time quantum walks are known to exhibit exotic topological states and phases. Physical realization of quantum walks in a noisy environment may destroy these phases. We investigate the behavior of topological states in quantum walks in the presence of a lossy environment. The environmental effects in the quantum walk dynamics are addressed using the non-Hermitian Hamiltonian approach. We show that the topological phases of the quantum walks are robust against moderate losses. The topological order in one-dimensional split-step quantum walk persists as long as the Hamiltonian is $\mathc…
Reflectionless defects in Hermitian tight-binding lattices, synthesized by the intertwining operator technique of supersymmetric quantum mechanics, are generally not invisible and time-of-flight measurements could reveal the existence of the defects. Here it is shown that, in a certain class of non-Hermitian tight-binding lattices with complex hopping amplitudes, defects in the lattice can appear fully invisible to an outside observer. The synthesized non-Hermitian lattices with invisible defects possess a real-valued energy spectrum, however they lack of parity-time (PT) symmetry, which does …
We show that a non-Hermitian operator with a tridiagonal matrix representation in a finite-dimensional vector space is similar to an Hermitian operator. The required condition is sufficient and simple examples show that it is not necessary. We derive quite general features of the eigenvalues and eigenvectors for a somewhat particular case.
The dynamical and topological properties of non-Hermitian systems have attracted great attention in recent years. In this work, we establish an intrinsic connection between two classes of intriguing phenomena -- topological phases and dynamical quantum phase transitions (DQPTs) -- in non-Hermitian systems. Focusing on one-dimensional models with chiral symmetry, we find DQPTs following the quench from a trivial to a non-Hermitian topological phase. Moreover, the number of critical momenta and critical time periods of the DQPTs are found to be directly related to the topological invariants of t…
The spectral and transport properties of a non-Hermitian tight-binding lattice with unidirectional hopping are theoretically investigated in three different geometrical settings. It is shown that, while for the infinitely-extended (open) and for the ring lattice geometries the spectrum is complex, lattice truncation makes the spectrum real. However, an exceptional point of order equal to the number of lattice sites emerges. When a homogeneous dc force is applied to the lattice, in all cases an equally-spaced real Wannier-Stark ladder spectrum is obtained, corresponding to periodic oscillatory …
In contrast to a Hermitian system, in which zero modes are usually degenerate and localized edge state in the thermodynamic limit, the zero mode of a finite non-Hermitian system can be a single nontrivial long-range entangled state at the exceptional point (EP). In this work, we demonstrate this feature with a concrete example based on exact solutions. Numerical simulations show that the entangled state can be generated through the dynamic process with a high fidelity.
We explore a way of finding the link between a non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of Bethe Ansatz solutions for a class of non-Hermitian Hamiltonians and the scattering problems for the corresponding Hermitian Hamiltonians. It is shown that a scattering state of an arbitrary Hermitian lattice embedded in a chain as the scattering center shares the same wave function with the corresponding non-Hermitian tight binding lattice, which consists of the Hermitian lattice with two additional on-site complex potentials, no matter the non-Hermitian is broken PT symmetry…
We propose a method to generate a real-energy flat band in a two-dimensional (2D) non-Hermitian Lieb lattice. The coincidence of the flat band eigenstate in both real and momentum spaces is essential for the proposed flat band, which is flexible at the appropriate match between the synthetic magnetic flux and non-Hermiticity. The proposed method is not limited to the 2D non-Hermitian Lieb lattice, and is applied to the 2D non-Hermitian Tasaki's decorated square lattice, dice lattice, and kagome lattice. Our findings make a step forward for the flat band engineering in 2D non-Hermitian optical …
While classic quantum chaos originated from the idea to set into context nonlinear physics and Hermitian quantum mechanics, non-Hermitian models have enhanced the field in recent years. At the same time, low-dimensional effective matrix models have proven to be a powerful tool in accessing the physical properties of a system in a semiquantitative manner. Here, we focus on two realizations of non-Hermitian physics in mesoscopic systems. First, we consider spiral optical microcavities in which the asymmetric scattering between whispering gallery modes induces the non-Hermitian behaviour. Second,…
We study a general class of PT-symmetric tridiagonal Hamiltonians with purely imaginary interaction terms in the quasi-hermitian representation of quantum mechanics. Our general Hamiltonian encompasses many previously studied lattice models as special cases. We provide numerical results regarding domains of observability and exceptional points, and discuss the possibility of explicit construction of general metric operators (which in turn determine all the physical Hilbert spaces). The condition of computational simplicity for the metrics motivates the introduction of certain one-parametric sp…
In this paper, we study the topological properties of the non-Hermitian Su-Schrieffer-Heeger (SSH) lattice by periodically introducing onsite imaginary potentials in the manner of ($iγ_1$, $-iγ_2$, $-iγ_1$, $iγ_2$) where $γ_1$ and $γ_2$ are the imaginary potential strengths. Results show that by changing the lattice to a tetramerized non-Hermitian system, such imaginary potentials induce the nontrivial transition of the topological properties of the SSH system. First, the topologically-nontrivial region is extended, followed by the non-Hermitian spontaneous breaking of the anti-$\cal PT$ symme…
We propose an ultracold-atom setting where a fermionic superfluidity with attractive s-wave interaction is uploaded in a non-Hermitian Lieb optical lattice. The existence of a real-energy flat band solution is revealed. We show that the interplay between the skin effect and flat-band localization leads to exotic localization properties. We develop a multiband mean-field description of this system and use both order parameters and superfluid weight to describe the phase transition. A relation between the superfluid weight and non-Hermitian quantum metric of the quantum states manifold is built.…
Exceptional points (EPs) are degeneracies of non-Hermitian operators where, in addition to the eigenvalues, corresponding eigenmodes become degenerate. Classical and quantum photonic systems with EPs have attracted tremendous attention due to their unusual properties, topological features, and an enhanced sensitivity that depends on the order of the EP, i.e. the number of degenerate eigenmodes. Yet, experimentally engineering higher-order EPs in classical or quantum domains remains an open challenge due to the stringent symmetry constraints that are required for the coalescence of multiple eig…
The Comment by Xiong et al. (arXiv:1610.06275) criticizing my Letter [Phys. Rev. Lett. 116, 133903 (2016)] was rejected by Physical Review Letters. In this Reply, I show that all their claims are wrong.
Quantum walks often provide telling insights about the structure of the system on which they are performed. In PT-symmetric and lossy dimer lattices, the topological properties of the band structure manifest themselves in the quantization of the mean displacement of such a walker. We investigate the fragile aspects of a topological transition in these two dimer models. We find that the transition is sensitive to the initial state of the walker on the Bloch sphere, and the resultant mean displacement has a robust topological component and a quasiclassical component. In PT symmetric dimer lattic…
Tongji University（同济大学）；The Chinese University of Hong Kong（香港中文大学）；Jiangsu University（江苏大学）；Institute of Acoustics, Chinese Academy of Sciences（中国科学院声学研究所）
We study how unique features of non-Hermitian lattice systems can be harnessed to improve Hamiltonian parameter estimation in a fully quantum setting. While the so-called non-Hermitian skin effect does not provide any distinct advantage, alternate effects yield dramatic enhancements. We show that certain asymmetric non-Hermitian tight-binding models with a $\mathbb{Z}_2$ symmetry yield a pronounced sensing advantage: the quantum Fisher information per photon increases exponentially with system size. We find that these advantages persist in regimes where non-Markovian and non-perturbative effec…
The non-Hermitian systems with the non-Hermitian skin effect (NHSE) are very sensitive to the imposed boundary conditions and lattice size, which leads to size-dependent non-Hermitian skin effects. Here, we report the experimental observation of NHSE with different boundary conditions and different lattice size in a unidirectional hopping model based on a circuit platform. The circuit admittance spectra and corresponding eigenstates are very sensitive to the presence of the boundary. Meanwhile, our experimental results show how the lattice size and boundary terms together affect the strength o…
We show that the bulk-boundary correspondence for topological insulators can be modified in the presence of non-Hermiticity. We consider a one-dimensional tight-binding model with gain and loss as well as long-range hopping. The system is described by a non-Hermitian Hamiltonian that encircles an exceptional point in momentum space. The winding number has a fractional value of 1/2. There is only one dynamically stable zero-energy edge state due to the defectiveness of the Hamiltonian. This edge state is robust to disorder due to protection by a chiral symmetry. We also discuss experimental rea…
A flexible control of wave scattering in complex media is of relevance in different areas of classical and quantum physics. Recently, a great interest has been devoted to scattering engineering in non-Hermitian systems, with the prediction and demonstration of new classes of non-Hermitian potentials with unique scattering properties, such as transparent and invisibile potentials or one-way reflectionless potentials. Such potentials have been found for both continuous and discrete (lattice) systems. However, wave scattering in lattice systems displays some distinct features arising from the dis…
Many features of Bloch oscillations in one-dimensional quantum lattices with a static force can be described by quasiclassical considerations for example by means of the acceleration theorem, at least for Hermitian systems. Here the quasiclassical approach is extended to non-Hermitian lattices, which are of increasing interest. The analysis is based on a generalised non-Hermitian phase space dynamics developed recently. Applications to a single-band tight-binding system demonstrate that many features of the quantum dynamics can be understood from this classical description qualitatively and ev…
Unidirectional and robust transport is generally observed at the edge of two- or three-dimensional quantum Hall and topological insulator systems. A hallmark of these systems is topological protection, i.e. the existence of propagative edge states that cannot be scattered by imperfections or disorder in the system. A different and less explored form of robust transport arises in non-Hermitian systems in the presence of an {\it imaginary} gauge field. As compared to topologically-protected transport in quantum Hall and topological insulator systems, robust non-Hermitian transport can be observe…
Topologically engineered optical materials support robust light transport. Herein, the investigated non-Hermitian lattice is trimerized and inhomogeneously coupled using uniform intracell coupling. The topological properties of the coupled waveguide lattice are evaluated, the PT-symmetric phase of a PT-symmetric lattice can have different topologies; the edge states depend on the lattice size, boundary configuration, and competition between the coupling and degree of non-Hermiticity. The topologically nontrivial region extends in the presence of periodic gain and loss. The nonzero geometric ph…
The search of topological states in non-Hermitian systems has gained a strong momentum over the last two years climbing to the level of an emergent research front. In this Perspective we give an overview with a focus in connecting this topic to others like Floquet systems. Furthermore, using a simple scattering picture we discuss an interpretation of concepts like the Hamiltonian's defectiveness, i.e. the lack of a full basis of eigenstates, crucial in many discussions of topological phases of non-Hermitian Hamiltonians.
Topological phenomena in non-Hermitian systems have recently become a subject of great interest in the photonics and condensed-matter communities. In particular, the possibility of observing topologically-protected edge states in non-Hermitian lattices has sparked an intensive search for systems where this kind of states are sustained. Here, we present the first study on the emergence of topological edge states in two-dimensional Haldane lattices exhibiting balanced gain and loss. In line with recent studies on other Chern insulator models, we show that edge states can be observed in the so-ca…
We show how generic non-Hermitian tight-binding lattice models can be realized in an unconditional, quantum-mechanically consistent manner by constructing an appropriate open quantum system. We focus on the quantum steady states of such models for both fermionic and bosonic systems. Surprisingly, key features and spatial structures in the steady state cannot be simply understood from the non-Hermitian Hamiltonian alone. Using the 1D Hatano-Nelson model as a paradigmatic example, we show that the steady state has a marked sensitivity to boundary conditions. This dependence however is qualitativ…
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 transit…
