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Superpositions of macroscopically distinct quantum states, introduced in Schrödinger's famous Gedankenexperiment, are an epitome of quantum ‘strangeness’ and a natural tool for determining the validity limits of quantum physics. The optical incarnation of Schrödinger's cat (SC)—the superposition of two opposite-amplitude coherent states—is also the backbone of continuous-variable quantum information processing. However, the existing preparation methods limit the amplitudes of the component coherent states, which curtails the state's usefulness for fundamental and practical applications. Here, we convert a pair of negative squeezed SC states of amplitude 1.15 to a single positive SC state of amplitude 1.85 with a success probability of ∼0.2. The protocol consists in bringing the initial states into interference on a beamsplitter and a subsequent heralding quadrature measurement in one of the output channels. Our technique can be realized iteratively, so arbitrarily high amplitudes can, in principle, be reached. The amplitude of a Schrödinger's cat (SC) state — superposed coherent state — is increased using a homodyne measurement. A pair of negative SC states with amplitude of 1.15 is probabilistically converted to a single positive SC state with amplitude of 1.85.

We study the existence of positive solutions for the nonlinear Schrödinger equation with the fractional Laplacian \\begin{gather*}(-\\Delta)^{\\alpha}u+u=f(x,u)\\text{~in~}\\mathbb{R}^{N},\\\u>0\\text{~in~}\\mathbb{R}^{N},\\qquad\\lim_{|x|\\rightarrow\\infty}u(x)=0.\\end{gather*} Furthermore, we analyse the regularity, decay and symmetry properties of these solutions.

In this paper, a novel compact operator is derived for the approximation of the Riesz derivative with order $\\alpha\\in(1,2].$ The compact operator is proved with fourth-order accuracy. Combining the compact operator in space discretization, a linearized difference scheme is proposed for a two-dimensional nonlinear space fractional Schrodinger equation. It is proved that the difference scheme is uniquely solvable, stable, and convergent with order $O(\\tau arrow up +h pound sterling )$, where $\\tau$ is the time step size, $h=\\max\\{h_1,h_2\\}$, and $h_1,\\,h_2$ are space grid sizes in the $x$ direction and the $y$ direction, respectively. Based on the linearized difference scheme, a compact alternating direction implicit scheme is presented and analyzed. Numerical results demonstrate that the compact operator does not bring in extra computational cost but improves the accuracy of the scheme greatly.

Over the past decade, the Anderson localization of light and a wide variety of associated phenomena have come to the forefront of research. Numerous investigations have been made into the underlying physics of how disorder affects transport in a crystalline lattice incorporating disorder. The physics involved relies on the analogy between the paraxial equation for electromagnetic waves and the Schrödinger equation describing quantum phenomena. Experiments have revealed how wavefunctions evolve during the localization process, and have led to discoveries of new physics that are universal to wave systems incorporating disorder. This Review summarizes the phenomena associated with the transverse localization of light, with an emphasis on the history, new ideas and future exploration of the field. The Anderson localization of light within disordered media has become a topic of great interest in recent years. Here the characterization of the effect and its related phenomena are reviewed, with a discussion on the role that nonlinearity and quantum correlated photons can play.

General high-order rogue waves in the nonlinear Schrödinger equation are derived by the bilinear method. These rogue waves are given in terms of determinants whose matrix elements have simple algebraic expressions. It is shown that the general N-th order rogue waves contain N−1 free irreducible complex parameters. In addition, the specific rogue waves obtained by Akhmediev et al. (Akhmediev et al. 2009 Phys. Rev. E 80, 026601 (doi:10.1103/PhysRevE.80.026601)) correspond to special choices of these free parameters, and they have the highest peak amplitudes among all rogue waves of the same order. If other values of these free parameters are taken, however, these general rogue waves can exhibit other solution dynamics such as arrays of fundamental rogue waves arising at different times and spatial positions and forming interesting patterns.

The distinctive non-classical features of quantum physics were first discussed in the seminal paper 1 by A. Einstein, B. Podolsky and N. Rosen (EPR) in 1935. In his immediate response 2 , E. Schrödinger introduced the notion of entanglement, now seen as the essential resource in quantum information 3 , 4 , 5 as well as in quantum metrology 6 , 7 , 8 . Furthermore, he showed that at the core of the EPR argument is a phenomenon that he called steering. In contrast to entanglement and violations of Bell's inequalities, steering implies a direction between the parties involved. Recent theoretical works have precisely defined this property, but the question arose as to whether there are bipartite states showing steering only in one direction 9 , 10 . Here, we present an experimental realization of two entangled Gaussian modes of light that in fact shows the steering effect in one direction but not in the other. The generated one-way steering gives a new insight into quantum physics and may open a new field of applications in quantum information. Recent theory predicts that Einstein–Podolsky–Rosen arguments enable an effect in which one party can steer the other but not the converse. Researchers have now demonstrated this one-way steering effect with two entangled Gaussian modes of light, potentially opening up a new field of applications in quantum information.

The necessity of quantising the gravitational field is still subject to an open debate. In this paper we compare the approach of quantum gravity, with that of a fundamentally semi-classical theory of gravity, in the weak-field non-relativistic limit. We show that, while in the former case the Schrödinger equation stays linear, in the latter case one ends up with the so-called Schrödinger-Newton equation, which involves a nonlinear, non-local gravitational contribution. We further discuss that the Schrödinger-Newton equation does not describe the collapse of the wave-function, although it was initially proposed for exactly this purpose. Together with the standard collapse postulate, fundamentally semi-classical gravity gives rise to superluminal signalling. A consistent fundamentally semi-classical theory of gravity can therefore only be achieved together with a suitable prescription of the wave-function collapse. We further discuss, how collapse models avoid such superluminal signalling and compare the nonlinearities appearing in these models with those in the Schrödinger-Newton equation.

It is well known that graphene deposited on hexagonal boron nitride produces moiré patterns in scanning tunnelling microscopy images. The interaction that produces this pattern also produces a commensurate periodic potential that generates a set of Dirac points that are different from those of the graphene lattice itself. The Schrödinger equation dictates that the propagation of nearly free electrons through a weak periodic potential results in the opening of bandgaps near points of the reciprocal lattice known as Brillouin zone boundaries 1 . However, in the case of massless Dirac fermions, it has been predicted that the chirality of the charge carriers prevents the opening of a bandgap and instead new Dirac points appear in the electronic structure of the material 2 , 3 . Graphene on hexagonal boron nitride exhibits a rotation-dependent moiré pattern 4 , 5 . Here, we show experimentally and theoretically that this moiré pattern acts as a weak periodic potential and thereby leads to the emergence of a new set of Dirac points at an energy determined by its wavelength. The new massless Dirac fermions generated at these superlattice Dirac points are characterized by a significantly reduced Fermi velocity. Furthermore, the local density of states near these Dirac cones exhibits hexagonal modulation due to the influence of the periodic potential.

We take a new look at the relation between the optimal transport problem and the Schrödinger bridge problem from a stochastic control perspective. Our aim is to highlight new connections between the two that are richer and deeper than those previously described in the literature. We begin with an elementary derivation of the Benamou–Brenier fluid dynamic version of the optimal transport problem and provide, in parallel, a new fluid dynamic version of the Schrödinger bridge problem. We observe that the latter establishes an important connection with optimal transport without zero-noise limits and solves a question posed by Eric Carlen in 2006. Indeed, the two variational problems differ by a Fisher information functional . We motivate and consider a generalization of optimal mass transport in the form of a (fluid dynamic) problem of optimal transport with prior . This can be seen as the zero-noise limit of Schrödinger bridges when the prior is any Markovian evolution. We finally specialize to the Gaussian case and derive an explicit computational theory based on matrix Riccati differential equations. A numerical example involving Brownian particles is also provided.

Atomic matter waves provide a controllable platform for studying the behaviour of solitons. In a lithium condensate, a characterization of the dynamics of collisions between solitons reveals a dependence on their relative phases. Solitons are localized wave disturbances that propagate without changing shape, a result of a nonlinear interaction that compensates for wave packet dispersion. Individual solitons may collide, but a defining feature is that they pass through one another and emerge from the collision unaltered in shape, amplitude, or velocity, but with a new trajectory reflecting a discontinuous jump. This remarkable property is mathematically a consequence of the underlying integrability of the one-dimensional (1D) equations, such as the nonlinear Schrödinger equation, that describe solitons in a variety of wave contexts, including matter waves 1 , 2 . Here we explore the nature of soliton collisions using Bose–Einstein condensates of atoms with attractive interactions confined to a quasi-1D waveguide. Using real-time imaging, we show that a collision between solitons is a complex event that differs markedly depending on the relative phase between the solitons. By controlling the strength of the nonlinearity we shed light on these fundamental features of soliton collisional dynamics, and explore the implications of collisions in the proximity of the crossover between one and three dimensions where the loss of integrability may precipitate catastrophic collapse.