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In this reply, we include new experimental results and verify that the observed non-linearity in rippled-MoS$_2$ (leading to mobility kink) is an intrinsic property of a disordered system, rather than contact effects (invasive probes) or other device issues. Noting that Peng Wu's hypothesis is based on a highly ordered ideal system, transfer curves are expected to be linear, and the carrier density is assumed be constant. Wu's model is therefore oversimplified for disordered systems and neglects carrier-density dependent scattering physics. Thus, it is fundamentally incompatible with our rippled-MoS$_2$, and leads to the wrong conclusion.

Controlling light at the nanoscale by exploiting ultra-confined polaritons - hybrid light and matter waves - in various van der Waals (vdW) materials empowers unique opportunities for many nanophotonic on-chip technologies. So far, mainstream approaches have relied interfacial techniques (e.g., refractive optics, meta-optics and moire engineering) to manipulate polariton wavefront. Here, we propose that orbital angular momentum (OAM) of incident light could offer a new degree of freedom to structure vdW polaritons. With vortex excitations, we observed a new class of accelerating polariton waves - Airy-like hyperbolic phonon polaritons (PhPs) in high-symmetry orthorhombic vdW crystal {\alpha}-MoO3. In analogous to the well-known Airy beams in free space, such Airy-like PhPs also exhibit self-accelerating, nonspreading and self-healing characteristics. Interestingly, the helical phase gradient of vortex beam leads to asymmetry excitation of polaritons, as a result, the Airy-like PhPs possess asymmetric propagation feature even with a symmetric mode, analogous to the asymmetry hyperbolic shear polaritons in low-symmetry crystals. Our finding highlights the potential of OAM to manipulate polaritons in vdW materials, which could be further extended into a variety of applications such as active structured polaritonic devices.

Singularities arise in diverse disciplines and play a key role in both exploring fundamental laws of physics and making highly-sensitive sensors. Higher-order (>3) singularities, with further improved performance, however, usually require exquisite tuning of multiple (>3) coupled degrees of freedom or nonlinear control, thus severely limiting their applications in practice. Here we propose theoretically and confirm using mechanics experiments that, cubic singularities can be realized in a coupled binary system without any nonlinearity, only by observing the phase tomography of the driven response. By steering the cubic phase-tomographic singularities in an electrostatically-tunable micromechanical system, enhanced cubic-root response to frequency perturbation and voltage-controlled nonreciprocity are demonstrated. Our work opens up a new phase-tomographic method for interacted-system research and sheds new light on building and engineering advanced singular devices with simple and well-controllable elements, with a wide range of applications including precision metrology, portable nonreciprocal devices, and on-chip mechanical computing.

Planar lenses with optimized but disordered structures can focus light beyond the diffraction limit. However, these disordered structures have inevitably destroyed wide-field imaging capability, limiting their applications in microscopy. Here we introduce information entropy S to evaluate the disorder of an objective chip by using the probability of its structural deviation from standard Fresnel zone plates. Inspired by the theory of entropy change, we predict an equilibrium point S0=0.5 to balance wide-field imaging (theoretically evaluated by the Strehl ratio) and subdiffraction-limit focusing. To verify this, a NA=0.9 objective chip with a record-long focal length of 1 mm is designed with S=0.535, which is the nearest to the equilibrium point among all reported planar lenses. Consequently, our fabricated chip can focus light with subdiffraction-limit size of 0.44{\lambda} and image fine details with spatial frequencies up to 4000 lp/mm in experiment. These unprecedented performances enable ultracompact reflective confocal microscopy for superresolution imaging., Comment: 42 pages, 4 figures

Matters Arising in Nature on "Evidence for a single layer van der Waals multiferroic". The search for two dimensional multiferroic materials is an exciting yet challenging endeavor. Recently, Song reported the exciting discovery of type II multiferroic order in an antiferromagnetic (AFM) NiI2 single layer and concluded the ferroelectric (FE) polarization induced by a helical magnetic order1. Their finding was presented from all optical experimental evidence, based on the methods of second harmonic generation (SHG) and linear dichroism (LD). However, the all optical characterizations cannot serve as unanimous evidence of the FE existence, particularly in conjunction with magnetic orders in a single layer NiI2 multiferroic. We have designed and built a Magneto-Optical-Electric Joint-measurement Scanning Imaging system (MOEJSI) for identification of two-dimensional vdW multiferroic., Comment: 7 pages

Metal halide perovskite quantum dots (PQDs) have emerged as promising materials due to their exceptional photoluminescence (PL) properties. A wide range of applications could benefit from adjustable luminescence properties, while preserving the physical and chemical properties of the PQDs. Therefore, post-synthesis engineering has gained attention recently, involving the use of ion-exchange or external stimuli, such as extreme pressure, magnetic and electric fields. Nevertheless, these methods typically suffer from spectrum broadening, intensity quenching or yield multiple bands. Alternatively, photonic antennas can modify the radiative decay channel of perovskites via the Purcell effect, with the largest wavelength shift being 8 nm to date, at an expense of 5-fold intensity loss. Here, we present an optical nanoantenna array with polarization-controlled quasi-bound-states-in-the-continuum (q-BIC) resonances, which can engineer and shift the photoluminescence wavelength over a ~39 nm range and confers a 21-fold emission enhancement of FAPbI3 perovskite QDs. The spectrum is engineered in a non-invasive manner via lithographically defined antennas and the pump laser polarization at ambient conditions. Our research provides a path towards advanced optoelectronic devices, such as spectrally tailored quantum emitters and lasers., Comment: 39 pages, 4 figures in the main text and 10 figures in the supporting information

The orbital angular momentum (OAM) of light holds great promise for applications in optical communication, super-resolution imaging, and high-dimensional quantum computing. However, the spatio-temporal coherence of the light source has been essential for generating OAM beams, as incoherent ambient light would result in polychromatic and obscured OAM beams in the visible spectrum. Here, we extend the applications of OAM to ambient lighting conditions. By miniaturizing spiral phase plates and integrating them with structural color filters, we achieve spatio-temporal coherence using only an incoherent white light source. These optical elements act as building blocks that encode both color and OAM information in the form of colorful optical vortices. Thus, pairs of transparent substrates that contain matching positions of these vortices constitute a reciprocal optical lock and key system. Due to the multiple helical eigenstates of OAM, the pairwise coupling can be further extended to form a one-to-many matching and validation scheme. Generating and decoding colorful optical vortices with broadband white light could find potential applications in anti-counterfeiting, optical metrology, high-capacity optical encryption, and on-chip 3D photonic devices.

The intrinsic limitation of the material nonlinearity inevitably results in the poor mode purity, conversion efficiency and real-time reconfigurability of the generated harmonic waves, both in optics and acoustics. Rotational Doppler effect provides us an intuitive paradigm to shifting the frequency in a linear system, which needs to be facilitated by a spiraling phase change upon the wave propagation. Here we numerically and experimentally present a rotating linear vortex metasurface and achieve close-to-unity mode purity (above 95%) and conversion efficiency (above 65%) in audible sound frequency as low as 3000 Hz. The topological charge of the transmitted sound is almost immune from the rotational speed and transmissivity, demonstrating the mechanical robustness and stability in adjusting the high-performance frequency conversion in situ. These features enable us to cascade multiple vortex metasurfaces to further enlarge and diversify the extent of sound frequency conversion, which are experimentally verified. Our strategy takes a step further towards the freewheeling sound manipulation at acoustic frequency domain, and may have far-researching impacts in various acoustic communications, signal processing, and contactless detection.

Spectral/angular emissivity $e$ and absorptivity ${\alpha}$ of an object are widely believed to be identical by Kirchhoff's law of thermal radiation in reciprocal systems, but this introduces an intrinsic and inevitable energy loss for energy conversion and harvesting devices. So far, experimental evidences of breaking this well-known balance are still absent, and previous theoretical proposals are restricted to narrow single-band nonreciprocal radiation. Here we observe for the first time, to our knowledge, the violation of Kirchhoff's law using ultrathin ($0.6$ has been experimentally demonstrated under a moderate external magnetic field. Moreover, based on magnetized ENZ building blocks supporting asymmetrically radiative Berreman and surface ENZ modes, we show versatile shaping of nonreciprocal thermal radiation: single-band, dual-band, and broadband nonreciprocal emission spectra at different wavebands. Our findings of breaking Kirchhoff's law will advance the conventional understanding of emission and absorption processes of natural objects, and lay a solid foundation for more comprehensive studies in designing various nonreciprocal thermal emitters. The reported recipe of diversely shaping nonreciprocal emission will also breed new possibilities in renovating next-generation nonreciprocal energy devices in the areas of solar cells, thermophotovoltaic, radiative cooling, etc., Comment: 21 pages, 4 figures

Interlayer electronic coupling in two-dimensional (2D) materials enables tunable and emergent properties by stacking engineering. However, it also brings significant evolution of electronic structures and attenuation of excitonic effects in 2D semiconductors as exemplified by quickly degrading excitonic photoluminescence and optical nonlinearities in transition metal dichalcogenides when monolayers are stacked into van der Waals structures. Here we report a novel van der Waals crystal, niobium oxide dichloride, featuring a vanishing interlayer electronic coupling and scalable second harmonic generation intensity of up to three orders higher than that of exciton-resonant monolayer WS2. Importantly, the strong second-order nonlinearity enables correlated parametric photon pair generation, via a spontaneous parametric down-conversion (SPDC) process, in flakes as thin as ~46 nm. To our knowledge, this is the first SPDC source unambiguously demonstrated in 2D layered materials, and the thinnest SPDC source ever reported. Our work opens an avenue towards developing van der Waals material-based ultracompact on-chip SPDC sources, and high-performance photon modulators in both classical and quantum optical technologies.

Polarization (P), angular index (l), and radius index (p) are three independent degrees of freedom (DoFs) of vector vortex beams, which have been widely used in optical communications, quantum optics, information processing, etc. Although the sorting of one DoF can be achieved efficiently, it is still a great challenge to sort all these DoFs simultaneously in a compact and efficient way. Here, we propose a beam sorter to deal with all these three DoFs simultaneously by using a diffractive deep neural network (D$^2$NN) and experimentally demonstrated the robust sorting of 120 Laguerre-Gaussian (LG) modes using a compact D$^2$NN formed by one spatial light modulator and one mirror only. The proposed beam sorter demonstrates the great potential of D$^2$NN in optical field manipulation and will benefit the diverse applications of vector vortex beams.

Quantized bulk quadrupole moment has unveiled a nontrivial boundary state, exhibiting lower-dimensional topological edge states and simultaneously hosting the in-gap corner modes of zero dimension. All state-of-the-art strategies for topological thermal metamaterials have so far failed to observe such higher-order hierarchical features, since the absence of quantized bulk quadrupole moments in thermal diffusion fundamentally forbids the possible expansions of band topology, unlike its photonic counterpart. Here, we report a recipe of creating quantized bulk quadrupole moments in diffusion, and observe the quadrupole topological phases in non-Hermitian thermal systems. The experiments demonstrate that both the real- and imaginary-valued bands showcase the hierarchical hallmarks of bulk, gapped edge, and in-gap corner states, in stark contrast to the higher-order states only observed on real-valued bands in classic wave fields. Our findings open up unique possibilities for diffusive manipulations and establish an unexplored playground for multipolar topological physics.

Phonon lasers, exploiting coherent amplifications of phonons, have been a cornerstone for exploring nonlinear phononics, imaging nanomaterial structures, and operating phononic devices. Very recently, by levitating a nanosphere in an optical tweezer, a single-mode phonon laser governed by dispersive optomechanical coupling has been demonstrated, assisted by alternating mechanical nonlinear cooling and linear heating. Such levitated optomechanical (LOM) devices, with minimal noises in high vacuum, can allow flexible control of large-mass objects without any internal discrete energy levels. However, untill now, it is still elusive to realize phonon lasing with levitated microscale objects, due to much stronger optical scattering losses. Here, by employing a Yb3+-doped active system, we report the first experiment on nonlinear multi-frequency phonon lasers with a micro-size sphere governed instead by dissipative LOM coupling. In this work, active gain plays a key role since not only 3-order enhancement can be achieved for the amplitude of the fundamental-mode phonon lasing, compared with the passive device, but also nonlinear mechanical harmonics can emerge spontaneously above the lasing threshold. Furthermore, for the first time, coherent correlations of phonons are observed for both the fundamental mode and its harmonics. Our work drives the field of LOM technology into a new regime where it becomes promising to engineer collective motional properties of typical micro-size objects, such as atmospheric particulates and living cells, for a wide range of applications in e.g., acoustic sensing, gravimetry, and inertial navigation., Comment: Accepted for Publication in Nature Physics

Cavity optomechanical (COM) sensors, as powerful tools for measuring ultraweak forces or searching for dark matter, have been implemented to date mainly using linear COM couplings. Here, quantum force sensing is explored by using a quadratic COM system which is free of bistability and allows accurate measurement of mechanical energy. We find that this system can be optimized to achieve a force sensitivity $7$ orders of magnitude higher than any conventional linear COM sensor, with experimentally accessible parameters. Further integrating a quadratic COM system with a squeezing medium can lead to another $3$ orders enhancement, well below the standard quantum limit and reaching the zeptonewton level. This opens new prospects of making and using quantum nonlinear COM sensors in fundamental physics experiments and in a wide range of applications requiring extreme sensitivity., Comment: 9+10 pages,5+5 figures

As originally introduced, the EPR phenomenon was the ability of one party (Alice) to steer, by her choice between two measurement settings, the quantum system of another party (Bob) into two distinct ensembles of pure states. As later formalized as a quantum information task, EPR-steering can be shown even when the distinct ensembles comprise mixed states, provided they are pure enough and different enough. Consider the scenario where Alice and Bob each have a qubit and Alice performs dichotomic projective measurements. In this case, the states in the ensembles to which she can steer form the surface of an ellipsoid ${\cal E}$ in Bob's Bloch ball. Further, let the steering ellipsoid ${\cal E}$ have nonzero volume. It has previously been shown that if Alice's first measurement setting yields an ensemble comprising two pure states, then this, plus any one other measurement setting, will demonstrate EPR-steering. Here we consider what one can say if the ensemble from Alice's first setting contains only one pure state $\mathsf{p}\in{\cal E}$, occurring with probability $p_\mathsf{p}$. Using projective geometry, we derive the necessary and sufficient condition analytically for Alice to be able to demonstrate EPR-steering of Bob's state using this and some second setting, when the two ensembles from these lie in a given plane. Based on this, we show that, for a given ${\cal E}$, if $p_\mathsf{p}$ is high enough [$p_{\sf p} > p_{\rm max}^{{\cal E}} \in [0,1)$] then any distinct second setting by Alice is sufficient to demonstrate EPR-steering. Similarly we derive a $p_{\rm min}^{{\cal E}}$ such that $p_\mathsf{p}>p_{\rm min}^{{\cal E}}$ is necessary for Alice to demonstrate EPR-steering using only the first setting and some other setting. Moreover, the expressions we derive are tight; for spherical steering ellipsoids, the bounds coincide: $p_{\rm max}^{{\cal E}} = p_{\rm min}^{{\cal E}}$., Comment: 30 pages, 9 figures

Metasurfaces are planar structures that can manipulate the amplitude, phase and polarization (APP) of light at subwavelength scale. Although various functionalities have been proposed based on metasurface, a most general optical control, i.e., independent complex-amplitude (amplitude and phase) control of arbitrary two orthogonal states of polarizations, has not yet been realized. Such level of optical control can not only cover the various functionalities realized previously, but also enable new functionalities that are not feasible before. Here, we propose a single-layer dielectric metasurface to realize this goal and experimentally demonstrate several advanced functionalities, such as two independent full-color printing images under arbitrary elliptically orthogonal polarizations and dual sets of printing-hologram integrations. Our work opens the way for a wide range of applications in advanced image display, information encoding, and polarization optics., 23 pages, 4 figures

Colour changes can be achieved by straining photonic crystals or gratings embedded in stretchable materials. However, the multiple repeat units and the need for a volumetric assembly of nanostructures limit the density of information content. Inspired by surface reliefs on oracle bones and music records as means of information archival, here we endow surface-relief elastomers with multiple sets of information that are accessible by mechanical straining along in-plane axes. Distinct from Bragg diffraction effects from periodic structures, we report trenches that generate colour due to variations in trench depth, enabling individual trench segments to support a single colour. Using 3D printed cuboids, we replicated trenches of varying geometric parameters in elastomers. These parameters determine the initial colour (or lack thereof), the response to capillary forces, and the appearance when strained along or across the trenches. Strain induces modulation in trench depth or the opening and closure of a trench, resulting in surface reliefs with up to six distinct states, and an initially featureless surface that reveals two distinct images when stretched along different axes. The highly reversible structural colours are promising in optical data archival, anti-counterfeiting, and strain-sensing applications., Comment: Repeated informatin

Surface phonon polaritons (SPhPs) in polar dielectrics offer new opportunities for infrared nanophotonics due to sub-diffraction confinement with low optical losses. Though the polaritonic field confinement can be significantly improved by modifying the dielectric environment, it is challenging to break the fundamental limits in photon confinement and propagation behavior of SPhP modes. In particular, as SPhPs inherently propagate isotropically in these bulk polar dielectrics, how to collectively realize ultra-large field confinement, in-plane hyperbolicity and unidirectional propagation remains elusive. Here, we report an approach to solve the aforementioned issues of bulk polar dielectric's SPhPs at one go by constructing a heterostructural interface between biaxial van der Waals material (e.g., MoO3) and bulk polar dielectric (e.g., SiC, AlN, and GaN). Due to anisotropy-oriented mode couplings at the interface, the hybridized SPhPs with a large confinement factor (>100) show in-plane hyperbolicity that has been switched to the orthogonal direction as compared to that in natural MoO3. More interestingly, this proof of concept allows steerable, angle-dependent and unidirectional polariton excitation by suspending MoO3 on patterned SiC air cavities. Our finding exemplifies a generalizable framework to manipulate the flow of nano-light and engineer unusual polaritonic responses in many other hybrid systems consisting of van der Waals materials and bulk polar dielectrics.

While structural colors are ubiquitous in nature, saturated reds are mysteriously absent. Hence, a longstanding problem is in fabricating nanostructured surfaces that exhibit reflectance approaching the theoretical limit. This limit is termed the Schrodinger red and demands sharp spectral transitions from "stopband" to a high reflectance "passband" with total suppression of higher-order resonances at blue and green wavelengths. Current approaches based on metallic or dielectric nanoantennas are insufficient to simultaneously meet these conditions. Here, for the 1st time, we designed and fabricated tall Si nanoantenna arrays on quartz substrate to support two partially overlapping y polarized quasi bound-state-in-the-continuum (q-BIC) modes in the red wavelengths with sharp spectral edges. These structures produce possibly the most saturated and brightest reds with ~80% reflectance, exceeding the red vertex in sRGB and even the cadmium red pigment. We employed a gradient descent algorithm with structures supporting q BIC as the starting point. Although the current design is polarization dependent, the proposed paradigm has enabled us to achieve the elusive structural red and the design principle could be generalized to Schrodinger's pixels of other colors. The design is suitable for scale up using other nanofabrication techniques for larger area applications, such as red pixels in displays, decorative coatings, and miniaturized spectrometers with high wavelength selectivity., Comment: 40 pages, 4 figures in the main text, 15 figures in the supplementary information

Various unusual behaviors of artificial materials are governed by their topological properties, among which the edge state at the boundary of a photonic or phononic lattice has been captivated as a popular notion. However, this remarkable bulk-boundary correspondence and the related phenomena are missing in thermal materials. One reason is that heat diffusion is described in a non-Hermitian framework because of its dissipative nature. The other is that the relevant temperature field is mostly composed of modes that extend over wide ranges, making it difficult to be rendered within the tight-binding theory as commonly employed in wave physics. Here, we overcome the above challenges and perform systematic studies on heat diffusion in thermal lattices. Based on a continuum model, we introduce a state vector to link the Zak phase with the existence of the edge state, and thereby analytically prove the thermal bulk-boundary correspondence. We experimentally demonstrate the predicted edge states with a topologically protected and localized heat dissipation capacity. Our finding sets up a solid foundation to explore the topology in novel heat transfer manipulations.

In a previous work (Li et al. Science 364, 170) [1], we proposed a heat transfer system that preserves the anti-parity-time (APT) symmetry, and observe the rest-to-motion phase transition during the symmetry breaking. Recently, it was suggested (Zhao et al. arXiv:1906.08431) [2] that the behaviours of the system can be understood using scaling analysis based on the P\'eclet and Nusselt numbers (Pe and Nu). It was further proposed that there exists a third regime in the phase diagram in addition to the symmetric and symmetry broken phases. Although we appreciate the proposal to characterize the contributions of coupling, diffusion, and advection with dimensionless numbers, here we show that they do not help to predict or interpret the behaviours of the APT system. The dimensionless numbers do not provide enough details about the system to conclude that there is a motionless phase, a phase transition, to find the critical point, or to give the correct phase diagram with only two regimes.

Polaritons, coupled excitations of photons and dipolar matter excitations, can propagate along anisotropic metasurfaces with either hyperbolic or elliptical dispersion. At the transition from hyperbolic to elliptical dispersion (corresponding to a topological transition), various intriguing phenomena are found, such as an enhancement of the photonic density of states, polariton canalization and hyperlensing. Here we investigate theoretically and experimentally the topological transition and the polaritonic coupling of deeply subwavelength elements in a uniaxial infrared-phononic metasurface, a grating of hexagonal boron nitride (hBN) nanoribbons. By hyperspectral infrared nanoimaging, we observe, for the first time, a synthetic transverse optical phonon resonance (that is, the strong collective near-field coupling of the nanoribbons) in the middle of the hBN Reststrahlen band, yielding a topological transition from hyperbolic to elliptical dispersion. We further visualize and characterize the spatial evolution of a deeply subwavelength canalization mode near the transition frequency, which is a collimated polariton that is the basis for hyperlensing and diffraction-less propagation. Our results provide fundamental insights into the role of polaritonic near-field coupling in metasurfaces for creating topological transitions and polariton canalization.

Non-Hermitian systems, with symmetric or antisymmetric Hamiltonians under the parity-time ($\mathcal{PT}$) operations, can have entirely real eigenvalues. This fact has led to surprising discoveries such as loss-induced lasing and topological energy transfer. A merit of anti-$\mathcal{PT}$ systems is free of gain, but in recent efforts on making anti-$\mathcal{PT}$ devices, nonlinearity is still required. Here, counterintuitively, we show how to achieve anti-$\mathcal{PT}$ symmetry and its spontaneous breaking in a linear device by spinning a lossy resonator. Compared with a Hermitian spinning device, significantly enhanced optical isolation and ultrasensitive nanoparticle sensing are achievable in the anti-$\mathcal{PT}$-broken phase. In a broader view, our work provides a new tool to study anti-$\mathcal{PT}$ physics, with such a wide range of applications as anti-$\mathcal{PT}$ lasers, anti-$\mathcal{PT}$ gyroscopes, and anti-$\mathcal{PT}$ topological photonics or optomechanics., 10+14 pages, 3+6 figures, 2 tables, to be published in Nano Letters

The Zeeman effect offers unique opportunities for magnetic manipulation of the spin degree of freedom (DOF). Recently, valley Zeeman splitting, referring to the lifting of valley degeneracy, has been demonstrated in two-dimensional transition metal dichalcogenides (TMDs) at liquid helium temperature. However, to realize the practical applications of valley pseudospins, the valley DOF must be controllable by a magnetic field at room temperature, which remains a significant challenge. Magnetic doping in TMDs can enhance the Zeeman splitting, however, to achieve this experimentally is not easy. Here, we report unambiguous magnetic manipulation of valley Zeeman splitting at 300 K (g = -6.4) and 10 K (g = -11) in a CVD-grown Fe-doped MoS2 monolayer; the effective g factor can be tuned to -20.7 by increasing the Fe dopant concentration, which represents an approximately fivefold enhancement as compared to undoped MoS2. Our measurements and calculations reveal that the enhanced splitting and geff factors are due to the Heisenberg exchange interaction of the localized magnetic moments (Fe 3d electrons) with MoS2 through the d-orbital hybridization.

Twisted two-dimensional bi-layers offer exquisite control on the electronic bandstructure through the interlayer rotation and coupling, enabling magic-angle flat-band superconductivity and moir\'e excitons. Here, we demonstrate how analogous principles, combined with large anisotropy, enable extreme control and manipulation of the photonic dispersion of phonon polaritons (PhPs) in van der Waals (vdW) bi-layers. We experimentally observe tunable topological transitions from open (hyperbolic) to closed (elliptic) dispersion contours in twisted bi-layered {\alpha}-MoO3 at photonic magic angles, induced by polariton hybridization and robustly controlled by a topological quantity. At these transitions the bilayer dispersion flattens, exhibiting low-loss tunable polariton canalization and diffractionless propagation with resolution below {\lambda}0/40. Our findings extend twistronics and moir\'e physics to nanophotonics and polaritonics, with great potential for nano-imaging, nanoscale light propagation, energy transfer and quantum applications., Comment: Nature, in press

Recent advances in twistronics of low-dimensional materials, such as bilayer graphene and transition-metal dichalcogenides, have enabled a plethora of unusual phenomena associated with moir�� physics. However, several of these effects require demanding manipulation of superlattices at the atomic scale, such as the careful control of rotation angle between two closely spaced atomic lattices. Here, we study moir�� hyperbolic plasmons in pairs of hyperbolic metasurfaces (HMTSs), unveiling analogous phenomena at the mesoscopic scale. HMTSs are known to support confined surface waves collimated towards specific directions determined by the metasurface dispersion. By rotating two evanescently coupled HMTSs with respect to one another, we unveil rich dispersion engineering, topological transitions at magic angles, broadband field canalization, and plasmon spin-Hall phenomena. These findings open remarkable opportunities to advance metasurface optics, enriching it with moir�� physics and twistronic concepts., 17 pages, 4 figures

Although TMDC monolayers offer giant optical nonlinearity within few-angstrom thickness, it is still elusive to modulate and engineer the wavefront of nonlinear emissions. The grain size of high-quality monolayers also restricts possibilities of imparting inhomogeneous or gradient profiles of phases and amplitudes by classical light sources. Plasmonic nanosieves can support huge field enhancement and precise nonlinear phase control with hundred-nanometer pixel-level resolution, however they suffer from intrinsically weak nonlinear susceptibility. In nature, epoxy represents a strong glue whose magic adhesion comes from the bonding of two intrinsic loose and nonsticky resins. Inspired by the nature of epoxy, we report a multifunctional and powerful nonlinear interfaces via integrating transition-metal dichalcogenide (TMDC) monolayers with linear plasmonic nanosieves, which cannot be accessed by either constituents. We experimentally demonstrate orbital angular momentum (OAM) generation, beam steering, versatile polarization control and holograms and of SH emission, with the proposed nonlinear interfaces. In addition, an effective second-harmonic (SH) nonlinearity \c{hi}(2) of ~25 nm/V is obtained for such systems. This designer platform synergizes the complementary advantages of TMDC monolayer, plasmonic nanosieves, geometric phases, and field enhancement, paving a distinct avenue toward designer, multi-functional, and ultra-compact nonlinear optical devices.

The rings, spots and stripes found on some butterflies, Pachyrhynchus weevils, and many chameleons are notable examples of natural organisms employing photonic crystals to produce colorful patterns. Despite advances in nanotechnology, we still lack the ability to print arbitrary colors and shapes in all three dimensions at this microscopic length scale. Commercial nanoscale 3D printers based on two-photon polymerization are incapable of patterning photonic crystal structures with the requisite ~300 nm lattice constant to achieve photonic stopbands/ bandgaps in the visible spectrum and generate colors. Here, we introduce a means to produce 3D-printed photonic crystals with a 5x reduction in lattice constants (periodicity as small as 280 nm), achieving sub-100-nm features with a full range of colors. The reliability of this process enables us to engineer the bandstructures of woodpile photonic crystals that match experiments, showing that observed colors can be attributed to either slow light modes or stopbands. With these lattice structures as 3D color volumetric elements (voxels), we printed 3D microscopic scale objects, including the first multi-color microscopic model of the Eiffel Tower measuring only 39-microns tall with a color pixel size of 1.45 microns. The technology to print 3D structures in color at the microscopic scale promises the direct patterning and integration of spectrally selective devices, such as photonic crystal-based color filters, onto free-form optical elements and curved surfaces.

The uncertainty relation, as one of the fundamental principles of quantum physics, captures the incompatibility of noncommuting observables in the preparation of quantum states. In this work, we derive two strong and universal uncertainty relations for $N(N\ge2)$ observables with discrete and bounded spectra, one in multiplicative form and the other in additive form. To verify their validity, for illustration, we implement in the spin-1/2 system an experiment with single-photon measurement. The experimental results exhibit the validity and robustness of these uncertainty relations, and indicate the existence of stringent lower bounds., Comment: 9 pages, 5 figures. Slightly extended

Taking advantage of coherent light beams, we experimentally investigate the variancebased uncertainty relations and the optimal majorization uncertainty relation for the two-dimensional quantum mechanical system.Different from most of the experiments which devoted to record each individual quantum, we examine the uncertainty relations by measuring an ensemble of photons with two polarization degree of freedom characterized by the Stokes parameters which allow us to determine the polarization density matrix with high precision. The optimality of the recently proposed direct-sum majorization uncertainty relation is verified by measuring the Lorenz curves. Results show that the Lorenz curve method represents a faithful verification of the majorization uncertainty relation and the uncertainty relation is indeed an ensemble property of quantum system., Comment: 7 pages, 7 figures

Rotation in micro/nanoscale provides extensive applications in mechanical actuation$^{1, 2}$, cargo delivery$^{3, 4}$, and biomolecule manipulation$^{5, 6}$. Light can be used to induce a mechanical rotation remotely, instantly and precisely$^{7-13}$, where liquid throughout serves as a must-have enabler to suspend objects and remove impact of adhesion. Achieving light-driven motion in non-liquid environments faces formidable challenges, since micro-sized objects experience strong adhesion and intend to be stuck to contact surfaces. Adhesion force for a usual micron-sized object could reach a high value$^{14, 15}$ (nN - {\mu}N) which is several orders of magnitude higher than both its gravity (~ pN) and typical value of optical force (~ pN) in experiments$^{16}$. Here, in air and vacuum, we show counter-intuitive adhesion-assisted rotary locomotion of a micron-sized metal nanoplate with ~30 nm-thickness, revolving around a microfiber. This locomotor is powered by pulsed light guided into the fiber, as a coordinated consequence of photothermally induced surface acoustic wave on the nanoplate and favorable configuration of plate-fiber geometry. The locomotor crawls stepwise with sub-nanometer locomotion resolution actuated by designed light pulses. Furthermore, we can control the rotation velocity and step resolution by varying the repetition rate and pulse power, respectively. A light-actuated micromirror scanning with 0.001{\deg} resolution is then demonstrated based on this rotary locomotor. It unfolds unprecedented application potential for integrated micro-opto-electromechanical systems, outer-space all-optical precision mechanics and controls, laser scanning for miniature lidar systems, etc., Comment: 13 pages, 5 figures

Bloch oscillations (BOs) refer to a periodically oscillatory motion of particle in lattice systems driven by a constant force. By temporally modulating acoustic waveguides, BOs can be generalized from spatial to frequency domain, opening new possibilities for spectrum manipulations. The modulation can induce mode transitions in the waveguide band and form an artificial frequency lattice, with the mismatched wave vector during transitions acting as a constant force that drives frequency Bloch oscillations (FBOs). Furthermore, the modulation phase accompanying transitions serves as a gauge potential that controls the initial oscillation phase, providing an additional degree of freedom to tailor FBOs. We report that multiple FBOs with judiciously designed oscillation phases can be further cascaded to realize acoustic spectrum self-imaging, unidirectional transduction and bandwidth engineering. The study proposes the concept of FBOs in acoustic systems and functionalizes its cascade configurations for advanced control of sound spectrum. This paradigm may find versatile applications in underwater secure communication, voice encryption and signal processing., Comment: 4 figures

The uncertainty relation is a distinguishing feature of quantum theory, characterizing the incompatibility of noncommuting observables in the preparation of quantum states. Recently, many uncertainty relations were proposed with improved lower bounds and were deemed capable of incorporating multiple observables. Here we report an experimental verification of seven uncertainty relations of this type with single-photon measurements. The results, while confirming these uncertainty relations, show as well the relative stringency of various uncertainty lower bounds., slightly extended. To appear in PRA

We use metasurfaces to enable acoustic orbital angular momentum (a-OAM) based multiplexing in real-time, postprocess-free and sensor-scanning-free fashions to improve the bandwidth of acoustic communication, with intrinsic compatibility and expandability to cooperate with other multiplexing technologies. The mechanism relied on encoding information onto twisted beams is numerically and experimentally demonstrated by realizing the real-time picture transfer, which differs from existing static data transfer by encoding data onto OAM states. Our study can boost the capacity of acoustic communication links and offer potential to revolutionize relevant fields.

Compact and miniaturized devices with flexible functionalities are always highly demanded in optical integrated systems. Plasmonic nanosieve has been successfully harnessed as an ultrathin flat platform for complex manipulation of light, including holography, vortex generation and non-linear processes. Compared with most of reported single-functional devices, multi-functional nanosieves might find more complex and novel applications across nano-photonics, optics and nanotechnology. Here, we experimentally demonstrate a promising roadmap for nanosieve-based helical devices, which achieves full manipulations of optical vortices, including its generation, hybridization, spatial multiplexing, focusing and non-diffraction propagation etc., by controlling the geometric phase of spin light via over 121 thousands of spatially-rotated nano-sieves. Thanks to such spin-conversion nanosieve helical elements, it is no longer necessary to employ the conventional two-beam interferometric measurement to characterize optical vortices, while the interference can be realized natively without changing any parts of the current setup. The proposed strategy makes the far-field manipulations of optical orbital angular momentum within an ultrathin interface viable and bridges singular optics and integrated optics. In addition, it enables more unique extensibility and flexibility in versatile optical elements than traditional phase-accumulated helical optical devices.

Uncertainty principle is one of the fundamental principles of quantum mechanics. In this work, we derive two uncertainty equalities, which hold for all pairs of incompatible observables. We also obtain an uncertainty relation in weak measurement which captures the limitation on the preparation of pre- and post-selected ensemble and hold for two non-Hermitian operators corresponding to two non-commuting observables., Comment: 4 pages

We demonstrate that a planar and ultrathin binary lens can focus an azimuthally polarized beam with vortical phase (APV) to a subwavelength spot of transverse polarization. The results elaborates that, in the multi-layer medium, this focused spot, which is beyond the Rayleigh diffraction limitation, can be well maintained for several wavelengths after travelling through the dielectric interfaces, which is not attainable by using other vector beams (i.e., radially, linearly and circularly polarized beams) as the illuminating light. This compact optical system can be valuable in data writing and defect identification of wafer or silicon chips, owing to the enhanced polarized focusing through interfaces. It also enables to be highly integrated with traditional microscopy for the far-field super-resolution imaging, surface scanning and detection, and subwavelength focusing, owing to the enhanced focusing performance (reduced width and extended length) as well as the planarized configuration of the ultrathin lens., 8 pages,4 figures

We present two efficient schemes which can simultaneously accomplish hyperentanglement concentration and purification for two-photon four-qubit systems in an unknown partially hyperentangled mixed states. The first can correct errors in the polarization entanglement and extract maximal hyperentanglement in polarization and spatial mode with additional partial frequency entanglement. The second uses additional maximal frequency entanglement to purify and concentrate hyperentanglement in polarization and spatial mode deterministically. Both of the two schemes are only based on existing optical devices and cross-Kerr nonlinearities., 13 pages, 6 figures

Using the phenomenon of extraordinary sound transmission (EST) through artificially constructed density-near-zero (DNZ) materials, we propose an alternative approach to realize a very robust acoustic cloaking, which includes one absorber and one projector connected by an energy channel. The elementary unit cell is made of one single-piece homogeneous copper, but it well maintains both the original planar wavefront and the nearly perfect one-dimensional transmission, in the presence of any inserted objects. Furthermore, the overall cloaked space can be increased, designed, and distributed by arbitrarily assembling additional DNZ cells without any upper limit in the total cloaking volume. We demonstrate acoustic cloaking by different combinations of DNZ cells in free space as well as in acoustic waveguides, which enables any objects inside the cells imperceptible along undistorted sound paths.

The principle of transformation optics has been applied to various wave phenomena (e.g., optics, electromagnetics, acoustics and thermodynamics). Recently, metamaterial devices manipulating dc currents have received increasing attention which usually adopted the analogue of transformation optics using complicated resistor networks to mimic the inhomogeneous and anisotropic conductivities. We propose a distinct and general principle of manipulating dc currents by directly solving electric conduction equations, which only needs to utilize two layers of bulk natural materials. We experimentally demonstrate dc bilayer cloak and fan-shaped concentrator, derived from the generalized account for cloaking sensor. The proposed schemes have been validated as exact devices and this opens a facile way towards complete spatial control of dc currents. The proposed schemes may have vast potentials in various applications not only in dc, but also in other fields of manipulating magnetic field, thermal heat, elastic mechanics, and matter waves.

We theoretically and experimentally demonstrate the focusing of macroscopic 3D darkness surrounded by all light in free space. The object staying in the darkness is similar to staying in an empty light capsule because light just bypasses it by resorting to destructive interference. Its functionality of controlling the direction of energy flux of light macroscopically is fascinating, similar in some sense to the transformation-based cloaking effect. Binary-optical system exhibiting anti-resolution (AR) is designed and fabricated, by which electromagnetic energy flux avoids and bends smoothly around a nearly perfect darkness region. AR remains an unexplored topic hitherto, in contrast to the super-resolution for realizing high spatial resolution. This novel scheme replies on smearing out the PSF and thus poses less stringent limitations upon the object's size and position since the created dark (zero-field) area reach 8 orders of magnitude larger than the square of wavelength in size. It functions very well at arbitrarily polarized beams in three dimensions, which is also frequency-scalable in the whole electromagnetic spectrum., 11 pages, 5 figures

Invisibility has attracted intensive research in various communities, e.g., optics, electromagnetics, acoustics, thermodynamics, etc. However, the most of them have only been experimentally achieved by virtue of simplified approaches due to their inhomogeneous and extreme parameters imposed by transformation-optic method, and usually require challenging realization with metamaterials. In this paper, we demonstrate an advanced bilayer thermal cloak with naturally available materials first time. This scheme, directly from thermal conduction equation, has been validated as an exact cloak rather than a reduced one, and we experimentally confirmed its perfect performance (heat-front maintenance and heat protection) in an actual setup. The proposed scheme may open a new avenue to control the diffusive heat flow in ways inconceivable with phonons.

Invisible cloak has long captivated the popular conjecture and attracted intensive research in various communities of wave dynamics, e.g., optics, electromagnetics, acoustics, etc. However, their inhomogeneous and extreme parameters imposed by transformation-optic method will usually require challenging realization with metamaterials, resulting in narrow bandwidth, loss, polarization-dependence, etc. On the contrary, we demonstrate that tunable thermodynamic cloak can be achieved with homogeneous and finite conductivity only employing naturally available materials. The controlled localization of thermal distribution inside the coating layer has been presented, and it shows that an incomplete cloak can function perfectly. Practical realization of such homogeneous thermal cloak has been suggested by using two naturally occurring conductive materials, which provides an unprecedentedly plausible way to flexibly realize flexible thermal cloak and manipulate thermal flow., 12 pages, 4 figures

The creation of wave-dynamic illusion functionality is of great interests to various scientific communities, which can potentially transform an actual perception into the pre-controlled perception, thus empowering unprecedented applications in the advanced-material science, camouflage, cloaking, optical and/or microwave cognition, and defense security, etc. By using the space transformation theory and engineering capability of metamaterials, we propose and realize a functional ghost illusion device, which is capable of creating wave-dynamic virtual ghost images off the original object's position under the illumination of electromagnetic waves. The scattering signature of the object is thus ghosted and perceived as multiple ghost targets with different geometries and compositions. The ghost-illusion material, being inhomogeneous and anisotropic, was realized by thousands of varying unit cells working at non-resonance. The experimental demonstration of the ghost illusion validates our theory of scattering metamorphosis and opens a novel avenue to the wave-dynamic illusion, cognitive deception, manipulate strange light or matter behaviors, and design novel optical and microwave devices., Comment: 19 pages, 6 figures

We manipulate optical Tamm states in graphene-based photonics to achieve and steer large magneto-optical effects. Here we report the presence of a giant Faraday rotation via a single graphene layer of atomic thickness while keeping a high transmission. The Faraday rotation is enhanced across the interface between two photonic crystals due to the presence of an interface mode, which presents a strong electromagnetic field confinement at the location of the graphene sheet. Our proposed scheme opens a promising avenue to realize high performance graphene magneto-optical devices that can be extended to other two-dimensional structures., Comment: 18 pages, 4 figures

We experimentally demonstrate efficient Raman conversion to respective Stokes and anti-Stokes fields in both pulsed and continuous modes with a Rb-87 atomic vapor cell. The conversion efficiency is about 40-50% for the Stokes field and 20-30% for the anti-Stokes field, respectively. This conversion process is realized with feedback of both the Raman pump and the frequency-converted fields (Stokes or anti-Stokes). The experimental setup is very simple and can be applied easily to produce the light source with larger frequency difference using other Raman media. They may have wide applications in nonlinear optics, atomic physics, quantum optics and precise measurement., 4 pages, 5 figure

An optical Fano resonance, which is caused by birefringence control rather than frequency selection, is discovered. Such birefringence-induced Fano resonance comes with fast-switching radiation. The resonance condition $\varepsilon_t< 1/\varepsilon_r$ is revealed and a tiny perturbation in birefringence is found to result in a giant switch in the principal light pole induced near surface plasmon resonance. The loss and size effects upon the Fano resonance have been studied Fano resonance is still pronounced, even if the loss and size of the object increase. The evolutions of the radiation patterns and energy singularities illustrate clearly the sensitive dependence of Fano resonance upon the birefringence.

Phase-preserved optical elevator is an optical device to lift up an entire plane virtually without distortion in light path or phase. Using transformation optics, we have predicted and observed the realization of such a broadband phase-preserved optical elevator, made of a natural homogeneous birefringent crystal without resorting to absorptive and narrowband metamaterials involving time-consuming nano-fabrication. In our demonstration, the optical elevator is designed to lift a sheet upwards, and the phase is verified to be preserved always. The camouflage capability is also demonstrated in the presence of adjacent objects of the same scale at will. The elevating device functions in different surrounding media over the wavelength range of 400-700 nm. Our work opens up prospects for studies of light trapping, solar energy, illusion optics, communication, and imaging.

We propose an analytical scattering theory in spectral domain to model the electromagnetic (EM) fields of a gyrotropic sphere in terms of the eigen-functions and their associated spectral eigenvalues/coefficients in a recursive integral form. Applying the continuous boundary conditions of electromagnetic fields on the surface between the free space and gyrotropic sphere, the spectral coefficients of transmitted fields inside the gyrotropic sphere and the scattered fields in the isotropic host medium can be obtained exactly by expanding spherical vector wave eigenfunctions. Numerical results are provided for some representative cases, which are compared to the results from adaptive integral method (AIM). Good agreement demonstrates the validity of the proposed analytical scattering theory for gyrotropic spheres in spectral domain using Fourier transform., 18 pages, 4 figures

Usually a light beam pushes a particle when the photons act upon it. This is due to that the electric-dipole particle in the paraxial beam is considered. We investigate the scattering forces in non-paraxial gradientless beams and find that the forces can drag certain particles towards the beam source. The major criterion to be carried out to get the attractive force is the strong non-paraxiality of the light beam. The cone angle denoting the non-paraxiality has been investigated to unveil its importance on achieving dragging force. We hope the attractive forces will be very useful in nanoparticle manipulation., Comment: 4 pages, 4 figures