Zheng Zhou, Bo Zhu, Haibin Wang, Honghua Zhong
We study the effect of the interplay between parity-time ($\mathcal{PT}$) symmetry and optical lattice (OL) potential on dynamics of quantum droplets (QDs) forming in a binary bosonic condensate trapped in a dual-core system. It is found that the stability of symmetric QDs in such non-Hermitian system depends critically on the competition of gain and loss $γ$, inter-core coupling $κ$, and OL potential. In the absence of OL potential, the $\mathcal{PT}$-symmetric QDs are unstable against symmetry-breaking perturbations with the increase of the total condensate norm $N$, and they retrieve the stability at larger $N$, in the weakly-coupled regime. As expected, the stable region of the $\mathcal{PT}$-symmetric QDs shrinks when $γ$ increases, i.e., the $\mathcal{PT}$ symmetry is prone to break the stability of QDs. There is a critical value of $κ$ beyond which the $\mathcal{PT}$-symmetric QDs are entirely stable in the unbroken $\mathcal{PT}$-symmetric phase. In the presence of OL potential, the $\mathcal{PT}$-symmetric on-site QDs are still stable for relatively small and large values of $N$. Nevertheless, it is demonstrated that the OL potential can assist stabilization of $\mathcal{PT}$-symmetric on-site QDs for some moderate values of $N$. On the other hand, it is worth noting that the relatively small $\mathcal{PT}$-symmetric off-site QDs are unstable, and only the relatively large ones are stable. Furthermore, collisions between stable $\mathcal{PT}$-symmetric QDs are considered too. It is revealed that the slowly moving $\mathcal{PT}$-symmetric QDs tend to merge into breathers, while the fast-moving ones display quasi-elastic collision and suffer fragmentation for small and large values of $N$, respectively.
Pak-Yeung Chan, Bo Zhu
We establish a dichotomy on the curvature decay for four dimensional complete noncompact non Ricci flat steady gradient Ricci soliton with linear curvature decay and proper potential function. A similar dichotomy is also shown in higher dimensions under the additional assumption that the Ricci curvature is nonnegative outside a compact subset.
Bo Zhu, Honghua Zhong, Jun Jia, Fuqiu Ye, Libin Fu
Manipulating the global $PT$ symmetry of a non-Hermitian composite system is a rather significative and challenging task. Here, we investigate Floquet control of global $PT$ symmetry in 2D arrays of quadrimer waveguides with transverse periodic structure along $x$-axis and longitudinal periodic modulation along $z$-axis. For unmodulated case with inhomogeneous inter- and intra- quadrimer coupling strength $κ_1\neqκ$, in addition to conventional global $PT$-symmetric phase and $PT$-symmetry-breaking phase, we find that there is exotic phase where global $PT$ symmetry is broken under open boundary condition, whereas it still is unbroken under periodical boundary condition. The boundary of phase is analytically given as $κ_1\geqκ+\sqrt{2}$ and $1\leqγ\leq2$, where there exists a pair of zero-energy edge states with purely imaginary energy eigenvalues localized at the left boundary, whereas other $4N-2$ eigenvalues are real. Especially, the domain of the exotic phase can be manipulated narrow and even disappeared by tuning modulation parameter. More interestingly, whether or not the array has initial global $PT$ symmetry, periodic modulation not only can restore the broken global $PT$ symmetry, but also can control it by tuning modulation amplitude. Therefore, the global property of transverse periodic structure of such a 2D array can be manipulated by only tuning modulation amplitude of longitudinal periodic modulation.
Jinmin Wang, Zhizhang Xie, Guoliang Yu, Bo Zhu
We prove a quantitative upper bound on the filling radius of complete, spin manifolds with uniformly positive scalar curvature using the quantitative operator $K$-theory and index theory.
Qiaochu Ma, Jinmin Wang, Guoliang Yu, Bo Zhu
In this paper, we prove an upper bound on the $\widehat{A}$ genus of a smooth, closed, spin Riemannian manifold using its scalar curvature lower bound, Neumann isoperimetric constant, and volume. The proof of this result relies on spectral analysis of the Dirac operator. We also construct an example to show that the Neumann isoperimetric constant in this bound is necessary. Our result partially answers a question of Gromov on bounding characteristic numbers using scalar curvature lower bound.
Yi Wu, Bo Zhu, Shufang Hu, Zheng Zhou, Honghua Zhong
Controlling the balanced gain and loss in a PT-symmetric system is a rather challenging task. Utilizing Floquet theory, we explore the constructive role of periodic modulation in controlling the gain and loss of a PT-symmetric optical coupler. It is found that the gain and loss of the system can be manipulated by applying a periodic modulation. Further, such an original non-Hermitian system can even be modulated into an effective Hermitian system derived by the high-frequency Floquet method. Therefore, compared with other PT symmetry control schemes, our protocol can modulate the unbroken PT-symmetric range to a wider parameter region. Our results provide a promising approach for controlling the gain and loss of a realistic system.
Bo Zhu, Tianlong Ma
The strong matching preclusion number of a graph, introduced by Park and Ihm in 2011, is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, the fractional strong matching preclusion number of a graph is the minimum number of edges and vertices whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we obtain the fractional strong matching preclusion number for Cartesian product graphs. As an application, the fractional strong matching preclusion number for torus networks is obtained.
Bo Zhu, Jeremiah Z. Liu, Bruce R. Rosen, Matthew S. Rosen
Image reconstruction plays a critical role in the implementation of all contemporary imaging modalities across the physical and life sciences including optical, MRI, CT, PET, and radio astronomy. During an image acquisition, the sensor encodes an intermediate representation of an object in the sensor domain, which is subsequently reconstructed into an image by an inversion of the encoding function. Image reconstruction is challenging because analytic knowledge of the inverse transform may not exist a priori, especially in the presence of sensor non-idealities and noise. Thus, the standard reconstruction approach involves approximating the inverse function with multiple ad hoc stages in a signal processing chain whose composition depends on the details of each acquisition strategy, and often requires expert parameter tuning to optimize reconstruction performance. We present here a unified framework for image reconstruction, AUtomated TransfOrm by Manifold APproximation (AUTOMAP), which recasts image reconstruction as a data-driven, supervised learning task that allows a mapping between sensor and image domain to emerge from an appropriate corpus of training data. We implement AUTOMAP with a deep neural network and exhibit its flexibility in learning reconstruction transforms for a variety of MRI acquisition strategies, using the same network architecture and hyperparameters. We further demonstrate its efficiency in sparsely representing transforms along low-dimensional manifolds, resulting in superior immunity to noise and reconstruction artifacts compared with conventional handcrafted reconstruction methods. In addition to improving the reconstruction performance of existing acquisition methodologies, we anticipate accelerating the discovery of new acquisition strategies across modalities as the burden of reconstruction becomes lifted by AUTOMAP and learned-reconstruction approaches.
Bo Zhu, Zhi Tan, Huilin Gong, Honghua Zhong, Xin-You Lü, Xiaoguang Wang
Fractional quantized response appears to be a distinctive characteristic in interacting topological systems. Here, we discover a novel phenomenon of tilt-induced fractional quantize drift in non-interacting system constructed by a time-modulated superlattice subjected to a external time-independent gradient potential. Depending on the tilt strength, Rabi oscillations between adjacent lowest enegy bands caused by Landau-Zener tunneling, can induce that the one-cycle-averaged drift displacement is fraction, which is relate to the ratio of the sum of Chern numbers of multiple bands to the number of energy bands involved in Landau Zener tunneling. As representative examples, we construct fractional (1/3, 1/2) quantize drift only via adjusting period of lattice. The numerical simulations allow us to consider a realistic setup amenable of an experimental realization. Our findings will expand the research implications of both fractional quantize response and topological materials.
Bo Zhu, Yongguan Ke, Wenjie Liu, Zheng Zhou, Honghua Zhong
We study the Floquet-surface bound states embedded in the continuum (BICs) and bound states out the continuum (BOCs)in a resonantly driven 1D tilted defect-free lattice. In contrast to fragile single-particle BICs assisted by specially tailored potentials, we find that Floquet-surface BICs, stable against structural perturbations, can exist in a wide range of parameter space. By using a multiple-time-scale asymptotic analysis in the high-frequency limit, the appearance of Floquet-surface bound states can be analytically explained by effective Tamm-type defects at boundaries induced by the resonance between the periodic driving and tilt. The phase boundary of existing Floquet-surface states is also analytically given. Based on the repulsion effect of surface states, we propose to detect transition points and measure the number of Floquet-surface bound states by quantum walk. Our work opens a new door to experimental realization of BICs in quantum system.
Jinmin Wang, Zhizhang Xie, Bo Zhu, Xingyu Zhu
We investigate the influence of uniformly positive scalar curvature on the size of a non-collapsed Ricci limit space coming from a sequence of $n$-manifolds with non-negative Ricci curvature and uniformly positive scalar curvature. We prove that such a limit space splits at most $n-2$ lines or $\mathbb{R}$-factors. When this maximal splitting occurs, we obtain a uniform upper bound on the diameter of the non-splitting factor. Moreover, we obtain a volume gap estimate and a volume growth order estimate of geodesic balls on such manifolds.
Bo Zhu, Wei Yu, Hugh H. T. Liu
Proportional-Integral-Differential (PID) control is widely used in industrial control systems. However, up to now there are at least two open problems related with PID control. One is to have a comprehensive understanding of its robustness with respect to model uncertainties and disturbances. The other is to build intuitive, explicit and mathematically provable guidelines for PID gain tuning. In this paper, we introduce a simple nonlinear mapping to determine PID gains from three auxiliary parameters. By the mapping, PID control is shown to be equivalent to a new PD control (serving as a nominal control) plus an uncertainty and disturbance compensator (to recover the nominal performance). Then PID control can be understood, designed and tuned in a Two-Degree-of-Freedom (2-DoF) control framework. We discuss some basic properties of the mapping, including the existence, uniqueness and invertibility. Taking as an example the PID control applied to a general uncertain second-order plant, we prove by the singular perturbation theory that the closed-loop steady-state and transient performance depends explicitly on one auxiliary parameter which can be viewed as the virtual singular perturbation parameter (SPP) of PID control. All the three PID gains are monotonically decreasing functions of the SPP, indicating that the smaller the SPP is, the higher the PID gains are, and the better the robustness of PID control is. Simulation and experimental examples are provided to demonstrate the properties of the mapping as well as the effectiveness of the mapping based PID gain turning.
Simone Cecchini, Jinmin Wang, Zhizhang Xie, Bo Zhu
Let $(M,g)$ be a closed connected oriented (possibly non-spin) smooth four-dimensional manifold with scalar curvature bounded below by $n(n-1)$. In this paper, we prove that if $f$ is a smooth map of non-zero degree from $(M, g)$ to the unit four-sphere, then $f$ is an isometry. Following ideas of Gromov, we use $μ$-bubbles and a version with coefficients of the rigidity of the three-sphere to rule out the case of strict inequality. Our proof of rigidity is based on the harmonic map heat flow coupled with the Ricci flow.
Jinmin Wang, Zhichao Wang, Bo Zhu
We prove a scalar-mean rigidity theorem for compact Riemannian manifolds with boundary in dimension less than five by developing a dimension reduction argument for mean curvature, which extends Schoen-Yau's dimension reduction argument for scalar curvature. As a corollary, we prove the sharp spherical radius rigidity theorem and best NNSC fill-in in terms of the mean curvature. Moreover, we prove a Lipschitz Listing type scalar-mean rigidity theorem for these dimensions.
Bo Zhu, Pietro Govoni, Yajun Mao, Chiara Mariotti, Weimin Wu
WW scattering is an important process to study electroweak symmetry breaking in the Standard Model at the LHC, in which the Higgs mechanism or other new physics processes must intervene to preserve the unitarity of the process below 1 TeV. This channel is expected to be one of the most sensitive to determine whether the Higgs boson exists. In this paper, the final state with two same sign Ws is studied, with a simulated sample corresponding to the integrated luminosity of 60 fb$^{-1}$ in pp collision at $\sqrt{s}=$10 TeV. Two observables, the invariant mass of $μμ$ from W decays and the azimuthal angle difference between the two $μ$s, are utilized to distinguish the Higgs boson existence scenario from the Higgs boson absence scenario. A good signal significance for the two cases can be achieved. If we define the separation power of the analysis as the distance, in the log-likelihood plane, of pseudo-experiments outcomes in the two cases, with the total statistics expected from the ATLAS and CMS experiments at the nominal centre-of-mass energy of 14 TeV, the separation power will be at the level of 4 $σ$.
Bo Zhu
In this paper, we study the interplay of geometry and positive scalar curvature on a complete, non-compact manifold with non-negative Ricci curvature. In three-dimensional manifold, we prove a minimal volume growth, an estimate of integral of scalar curvature and width. In higher dimensional manifold, we obtain a volume growth with a stronger condition.
Bo Zhu, Shi Hu, Honghua Zhong, Yongguan Ke
We propose to measure band topology via quantized drift of Bloch oscillations in a two-dimensional Harper-Hofstadter lattice subjected to tilted fields in both directions. When the difference between the two tilted fields is large, Bloch oscillations uniformly sample all momenta, and hence the displacement in each direction tends to be quantized at multiples of the overall period, regardless of any momentum of initial state. The quantized displacement is related to a reduced Chern number defined as a line integral of Berry curvature in each direction, providing an almost perfect measurement of Chern number. Our scheme can apply to detect Chern number and topological phase transitions not only for the energy-separable band, but also for energy-inseparable bands which cannot be achieved by conventional Thouless pumping or integer quantum Hall effect.
Zhu Bo, Rang Liu, Ming Li, Qian Liu
The recently emerged symbol-level precoding (SLP) technique has been regarded as a promising solution in multi-user wireless communication systems, since it can convert harmful multi-user interference (MUI) into beneficial signals for enhancing system performance. However, the tremendous computational complexity of conventional symbol-level precoding designs severely hinders the practical implementations. In order to tackle this difficulty, we propose a novel deep learning (DL) based approach to efficiently design the symbol-level precoders. Particularly, in this correspondence, we consider a multi-user multi-input single-output (MU-MISO) downlink system. An efficient precoding neural network (EPNN) is introduced to optimize the symbol-level precoders for maximizing the minimum quality-of-service (QoS) of all users under the power constraint. Simulation results demonstrate that the proposed EPNN based SLP design can dramatically reduce the computing time at the price of slight performance loss compared with the conventional convex optimization based SLP design.
Jinmin Wang, Bo Zhu
We establish a sharp upper bound for the bottom spectrum of the Beltrami Laplacian on universal covers of closed Riemannian manifolds with scalar curvature lower bound. Moreover, we prove a scalar curvature rigidity theorem when this bound is achieved. Additionally, we prove a net characterization of scalar curvature for general complete noncompact Riemannian manifolds.
Bo Zhu, Xingyu Zhu
In this note, we prove that positive scalar curvature can pass to three dimensional Ricci limit spaces of non-negative Ricci curvature when it splits off a line. As a corollary, we obtain an optimal Bonnet-Myers type upper bound. Moreover, we obtain a similar statement in all dimensions for Alexandrov spaces of non-negative curvature.