While Hermiticity lies at the heart of quantum mechanics, recent experimental advances in controlling dissipation have brought about unprecedented flexibility in engineering non-Hermitian Hamiltonians in open classical and quantum systems. Examples include parity-time-symmetric optical systems with gain and loss, dissipative Bose-Einstein condensates, exciton-polariton systems and biological networks. A particular interest centers on the topological properties of non-Hermitian systems, which exhibit unique phases with no Hermitian counterparts. However, no systematic understanding in analogy with the periodic table of topological insulators and superconductors has been achieved. In this paper, we develop a coherent framework of topological phases of non-Hermitian systems. After elucidating the physical meaning and the mathematical definition of non-Hermitian topological phases, we start with one-dimensional lattices, which exhibit topological phases with no Hermitian counterparts and are found to be characterized by an integer topological winding number even with no symmetry constraint, reminiscent of the quantum Hall insulator in Hermitian systems. A system with a nonzero winding number, which is experimentally measurable from the wave-packet dynamics, is shown to be robust against disorder, a phenomenon observed in the Hatano-Nelson model with asymmetric hopping amplitudes. We also unveil a novel bulk-edge correspondence that features an infinite number of (quasi-)edge modes. We then apply the K-theory to systematically classify all the non-Hermitian topological phases in the Altland-Zirnbauer (AZ) classes in all dimensions. The obtained periodic table unifies time-reversal and particle-hole symmetries, leading to highly nontrivial predictions such as the absence of non-Hermitian topological phases in two dimensions. We provide concrete examples for all the nontrivial non-Hermitian AZ classes in zero and one dimensions. In particular, we identify a Z2 topological index for arbitrary quantum channels (CPTP maps). Our work lays the cornerstone for a unified understanding of the role of topology in non-Hermitian systems. *

Main TextIn monolayer transition metal dichalcogenides (TMDs), there is a valley pseudospin 1/2 which describes the two inequivalent but energy degenerate band edges (the ±K valleys) at the corners of the hexagonal Brillouin zone 1 . With broken inversion symmetry, electrons in the two valleys can have finite orbital contributions to their magnetic moments which are equal in magnitude but opposite in sign by time reversal symmetry. This orbital magnetic moment is thus linked to the valley pseudospin in the same way that the bare magnetic moment ( S) is linked to the real spin S, where is the Bohr magneton and is the Lande -factor. The orbital magnetic moment in turn has two parts: a contribution from the parent atomic orbitals, and a "valley magnetic moment" contribution from the lattice structure 1 (Fig. 1a, [18][19][20][21][22][23] , are subject to a momentum-dependent gauge field arising from electron-hole exchange, or valley-orbit coupling, which at zero magnetic field is predicted to result in massless and massive dispersion respectively within the light cone 24 . This implies the possibility of controlling excitonic valley pseudospin via the Zeeman effect in an external magnetic field.Our measurements of polarization-resolved photoluminescence (PL) in a perpendicular magnetic field are performed on mechanically exfoliated WSe2 monolayers. We have obtained consistent results from many samples. The data presented here are all taken from one sample at a temperature of 30 K. In order to resolve the splitting between the +K and -K valley excitons, which is significantly smaller than the exciton linewidth (~10 meV), we both excite and detect with a single helicity of light. In this way we address one valley at a time, and the splitting can be determined by comparing the peak positions for different polarizations. The splitting in the applied magnetic field breaks the valley degeneracy, enabling control of the valley polarization. To investigate this we measure the degree of PL polarization for both helicities of incident circular polarization. Figure 2a shows PL for σ -excitation with σ -(red) and σ + (orange) detection at a field of -7 T. The suppression of the σ + signal relative to the co-polarized σ -peak is a signature of optically pumped valley polarization 7-10 . The degree of exciton valley polarization is clearly larger for σ + excitation than for − (Fig. 2b). On the other hand, when the magnetic field is reversed to +7 T (Figs. 2c and d) the polarization becomes larger for − . This observation implies that, while the sign of the valley polarization is determined by the helicity of the excitation laser, its magnitude depends on the relationship between the helicity and the magnetic field direction. Figure 2e shows the degree of PL polarization for both σ + (blue) and σ -(red) excitation as a function of B between -7 T and +7 T for the neutral exciton peak. It is linear in B with a negative (positive) slope. This "X" pattern implies that the valley Zeeman splitting induces an asymmetry in the intervalle...

We analyze the coherent transport of a single photon, which propagates in a one-dimensional coupled-resonator waveguide and is scattered by a controllable two-level system located inside one of the resonators of this waveguide. Our approach, which uses discrete coordinates, unifies low and high energy effective theories for single-photon scattering. We show that the controllable two-level system can behave as a quantum switch for the coherent transport of a single photon. This study may inspire new electro-optical single-photon quantum devices. We also suggest an experimental setup based on superconducting transmission line resonators and qubits.

No abstract

In monolayer group-VI transition metal dichalcogenides, charge carriers have spin and valley degrees of freedom, both associated with magnetic moments. On the other hand, the layer degree of freedom in multilayers is associated with electrical polarization. Here we show that transition metal dichalcogenide bilayers offer an unprecedented platform to realize a strong coupling between the spin, valley and layer pseudospin of holes. Such coupling gives rise to the spin Hall effect and spin-dependent selection rule for optical transitions in inversion symmetric bilayer and leads to a variety of magnetoelectric effects permitting quantum manipulation of these electronic degrees of freedom. Oscillating electric and magnetic fields can both drive the hole spin resonance where the two fields have valley-dependent interference, making an interplay between the spin and valley as information carriers possible for potential valley-spintronic applications. We show how to realize quantum gates on the spin qubit controlled by the valley bit.

We report the observation of anomalously robust valley polarization and valley coherence in bilayer WS 2 . The polarization of the photoluminescence from bilayer WS 2 follows that of the excitation source with both circular and linear polarization, and remains even at room temperature. The near-unity circular polarization of the luminescence reveals the coupling of spin, layer, and valley degree of freedom in bilayer system, and the linearly polarized photoluminescence manifests quantum coherence between the two inequivalent band extrema in momentum space, namely, the valley quantum coherence in atomically thin bilayer WS 2 . This observation provides insight into quantum manipulation in atomically thin semiconductors.valleytronics | spin-valley coupling | valley quantum control T ungsten sulfide WS 2 , part of the family of group VI transition metal dichalcogenides (TMDCs), is a layered compound with buckled hexagonal lattice. As WS 2 thins to atomically thin layers, WS 2 films undergo a transition from indirect gap in bulk form to direct gap at monolayer level with the band edge located at energy-degenerate valleys (K, K′) at the corners of the Brillouin zone (1-3). Like the case of its sister compound, monolayer MoS 2 , the valley degree of freedom of monolayer WS 2 could be presumably addressed through nonzero but contrasting Berry curvatures and orbital magnetic moments that arise from the lack of spatial inversion symmetry at monolayers (3, 4). The valley polarization could be realized by the control of the polarization of optical field through valley-selective interband optical selection rules at K and K′ valleys as illustrated in Fig. 1A (4-6). In monolayer WS 2 , both the top of the valence bands and the bottom of the conduction bands are constructed primarily by the d orbits of tungsten atoms, which are remarkably shaped by spin-orbit coupling (SOC). The giant spin-orbit coupling splits the valence bands around the K (K′) valley by 0.4 eV, and the conduction band is nearly spin degenerated (7). As a result of time-reversal symmetry, the spin splitting has opposite signs at the K and K′ valleys. Namely, the Kramer's doublet jK↑i and jK′↓i is separated from the other doublet jK′↑i and jK↓i by the SOC splitting of 0.4 eV. The spin and valley are strongly coupled at K (K′) valleys, and this coupling significantly suppresses spin and valley relaxations as both spin and valley indices have to be changed simultaneously.In addition to the spin and valley degrees of freedom, in bilayer WS 2 there exists an extra index: layer polarization that indicates the carriers' location, either up-layer or down-layer. Bilayer WS 2 follows the Bernal packing order and the spatial inversion symmetry is recovered: each layer is 180°in plane rotation of the other with the tungsten atoms of a given layer sitting exactly on top of the S atoms of the other layer. The layer rotation symmetry switches K and K′ valleys, but leaves the spin unchanged, which results in a sign change for the spin-valley coupling from layer to layer (Fi...

A d-dimensional second-order topological insulator (SOTI) can host topologically protected (d−2)dimensional gapless boundary modes. Here we show that a 2D non-Hermitian SOTI can host zero-energy modes at its corners. In contrast to the Hermitian case, these zero-energy modes can be localized only at one corner. A 3D non-Hermitian SOTI is shown to support second-order boundary modes, which are localized not along hinges but anomalously at a corner. The usual bulkcorner (hinge) correspondence in the second-order 2D (3D) non-Hermitian system breaks down. The winding number (Chern number) based on complex wavevectors is used to characterize the second-order topological phases in 2D (3D). A possible experimental situation with ultracold atoms is also discussed. Our work lays the cornerstone for exploring higher-order topological phenomena in non-Hermitian systems.arXiv:1810.04067v3 [cond-mat.mes-hall]

Discrete time crystals are a recently proposed and experimentally observed out-of-equilibrium dynamical phase of Floquet systems, where the stroboscopic evolution of a local observable repeats itself at an integer multiple of the driving period. We address this issue in a driven-dissipative setup, focusing on the modulated open Dicke model, which can be implemented by cavity or circuit QED systems. In the thermodynamic limit, we employ semiclassical approaches and find rich dynamical phases on top of the discrete time-crystalline order. In a deep quantum regime with few qubits, we find clear signatures of a transient discrete time-crystalline behavior, which is absent in the isolated counterpart. We establish a phenomenology of dissipative discrete time crystals by generalizing the Landau theory of phase transitions to Floquet open systems.Introduction.-Phases and phase transitions of matter are key concepts for understanding complex many-body physics [1,2]. Recent experimental developments in various quantum simulators, such as ultracold atoms [3,4], trapped ions [5,6] and superconducting qubits [7,8], motivate us to seek for quantum many-body systems out of equilibrium [9][10][11], such as many-body localized phases [12][13][14][15][16][17] and Floquet topological phases [18][19][20][21][22][23][24][25].In recent years, much effort has been devoted to periodically driven (Floquet) quantum many-body systems that break the discrete time-translation symmetry (TTS) [26]. In contrast to the continuous TTS breaking [27][28][29] that has turned out to be impossible at thermal equilibrium [30,31], the discrete TTS breaking has been theoretically proposed [32][33][34][35][36] and experimentally demonstrated [37,38]. Phases with broken discrete TTS feature discrete time-crystalline (DTC) order characterized by periodic oscillations of physical observables with period nT , where T is the Floquet period and n = 2, 3, · · · . The DTC order is expected to be stabilized by many-body interactions against variations of driving parameters. Note that the system is assumed to be in a localized phase [33,34,36,37] or to have long-range interactions [38][39][40]. Otherwise, the DTC order only exists in a prethermalized regime [41,42] since the system will eventually be heated to a featureless infinite-temperature state due to persistent driving [43][44][45].While remarkable progresses are being made concerning the DTC phase, most studies focus on isolated systems. Indeed, as has been experimentally observed [37,38] and theoretically investigated [46], the DTC order in an open system is usually destroyed by decoherence. On the other hand, it is known that dissipation and decoherence can also serve as resources for quantum tasks such as quantum computation [47] and metrology [48]. From this perspective, it is natural to ask whether the DTC order exists and can even be stabilized in open systems [49]. Such a possibility has actually been pointed out in Ref. [41], but neither a detailed theoretical model nor a concrete experimental imp...