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Structured light, where complex optical fields are tailored in all their degrees of freedom, has become highly topical of late, advanced by a sophisticated toolkit comprising both linear and nonlinear optics. Removing undesired structure from light is far less developed, leveraging mostly on inverting the distortion, e.g., with adaptive optics or the inverse transmission matrix of a complex channel, both requiring that the distortion is fully characterised through appropriate measurement. Here we show that distortions in spatially structured light can be corrected through difference frequency generation in a nonlinear crystal without any need for the distortion to be known. We demonstrate the versatility of our approach by using a wide range of aberrations and structured light modes, including higher-order orbital angular momentum (OAM) beams, showing excellent recovery of the original undistorted field. To highlight the efficacy of this process, we deploy the system in a prepare-and-measure communications link with OAM, showing minimal crosstalk even when the transmission channel is highly aberrated, and outline how the approach could be extended to alternative experimental modalities and nonlinear processes. Our demonstration of light correcting light without the need for measurement opens a new approach to measurement-free error correction for classical and quantum structured light, with direct applications in imaging, sensing and communication, Comment: 13 pages, 7 figures

Quantum interference is a central resource in many quantum-enhanced tasks, from computation to communication protocols. While it usually occurs between identical input photons, quantum interference can be enabled by projecting the quantum state onto ambiguous properties that render the photons indistinguishable, a process known as a quantum erasing. Structured light, on the other hand, is another hallmark of photonics: it is achieved by manipulating the degrees of freedom of light at the most basic level and enables a multitude of applications in both classical and quantum regimes. By combining these ideas, here we design and experimentally demonstrate a simple and robust scheme that tailors quantum interference to engineer photonic states with spatially structured coalescence along the transverse profile, a type of quantum mode with no classical counterpart. To achieve this, we locally tune the distinguishability of a photon pair via spatial structuring of their polarisation, creating a structured quantum eraser. We believe these spatially-engineered multi-photon quantum states may be of significance in fields such as quantum metrology, microscopy, and communications.

In this paper, we study the regularity of weak solutions for a class of nonlinear Choquard equations driven by the $p$-Laplacian operator. We also establish a Pohozaev type identity.

In this paper, we show the existence of a nontrivial weak solution for a nonlinear problem involving the fractional $p$-Laplacian operator and a Berestycki-Lions type nonlinearity. This solution satisfies a Pohozaev identity. Moreover, we prove that any sufficiently smooth solution fulfills the Pohozaev identity., Comment: to appear in Bulletin of the London Mathematical Society

Exciton-polaritons are hybrid elementary excitations of light and matter that, thanks to their nonlinear properties, enable a plethora of physical phenomena ranging from room temperature condensation to superfluidity. While polaritons are usually exploited in high density regime, evidence of quantum correlations at the level of few excitations has been recently reported, thus suggesting the possibility of using these systems for quantum information purposes. Here we show that integrated circuits of propagating single polaritons can be arranged to build deterministic quantum logic gates in which the two-particle interaction energy plays a crucial role. Besides showing their prospective potential for photonic quantum computation, we also show that these systems can be exploited for metrology purposes, as for instance to precisely measure the magnitude of the polariton-polariton interaction at the two-body level. In general, our results introduce a novel paradigm for the development of practical quantum polaritonic devices, in which the effective interaction between single polaritonic qubits may provide a unique tool for future quantum technologies.

Accurately measuring mechanical displacements is essential for a vast portion of current technologies. Several optical techniques accomplish this task, allowing for non-contact sensing even below the diffraction limit. Here we introduce an optical encoding technique, dubbed "linear photonic gears", that enables ultra-sensitive measurements of transverse displacements by mapping these into polarization rotations of a laser beam. In ordinary ambient conditions, we measure the relative shift between two objects with a resolution of 400 pm. We argue that a resolution of 50 pm should be achievable with existing state-of-the-art technologies. Our single-optical-path scheme is intrinsically stable and it could be implemented as a compact sensor, using integrated optics. We anticipate it may have a strong impact on both research and industry.

In this paper we study the multiplicity and concentration of positive solutions for the following $(p, q)$-Laplacian problem: \begin{equation*} \left\{ \begin{array}{ll} -\Delta_{p} u -\Delta_{q} u +V(\varepsilon x) \left(|u|^{p-2}u + |u|^{q-2}u\right) = f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in W^{1, p}(\mathbb{R}^{N})\cap W^{1, q}(\mathbb{R}^{N}), \quad u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $1< p

In this paper, we are concerned with the following fractional relativistic Schr\"odinger equation with critical growth: \begin{equation*} \left\{ \begin{array}{ll} (-\Delta+m^{2})^{s}u + V(\varepsilon x) u= f(u)+u^{2^{*}_{s}-1} \mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 \, \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $m>0$, $N> 2s$, $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical exponent, $(-\Delta+m^{2})^{s}$ is the fractional relativistic Schr\"odinger operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a superlinear continuous nonlinearity with subcritical growth at infinity. Under suitable assumptions on the potential $V$, we construct a family of positive solutions $u_{\varepsilon}\in H^{s}(\mathbb{R}^{N})$, with exponential decay, which concentrates around a local minimum of $V$ as $\varepsilon\rightarrow 0$., Comment: This is the final version (several modifications have been made with respect to the previous versions)

We study the multiplicity and concentration of complex valued solutions for a fractional magnetic Schr\"odinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik-Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems., Comment: arXiv admin note: text overlap with arXiv:2003.06155

We investigate the nonlinear optical response of suspended 1D photonic crystal nanocavities fabricated on a silicon nitride chip. Strong thermo-optical nonlinearities are demonstrated for input powers as low as $2\,\mu\text{W}$ and a self-sustained pulsing regime is shown to emerge with periodicity of several seconds. As the input power and laser wavelength are varied the temporal patterns change in period, duty cycle and shape. This dynamics is attributed to the multiple timescale competition between thermo-optical and thermo-optomechanical effects and closely resembles the relaxation oscillations states found in mathematical models of neuronal activity. We introduce a simplified model that reproduces all the experimental observations and allows us to explain them in terms of the properties of a 1D critical manifold which governs the slow evolution of the system., Comment: 9 pages, 6 figures

Entanglement in high-dimensional quantum systems, where one or more degrees of freedom of light are involved, offers increased information capacities and enables new quantum protocols. Here, we demonstrate a functional source of high-dimensional, noise-resilient hyperentangled states encoded in time-frequency and vector-vortex structured modes, which in turn carry single-particle entanglement between polarisation and orbital angular momentum. Pairing nonlinearity-engineered parametric downconversion in an interferometric scheme with spin-to-orbital-angular-momentum conversion, we generate highly entangled photon pairs at telecom wavelength that we characterise via two-photon interference and quantum state tomography, achieving near-unity visibilities and fidelities. While hyperentanglement has been demonstrated before in photonic qubits, this is the first instance of such a rich entanglement structure involving spectrally and spatially structured light, where three different forms of entanglement coexist in the same biphoton state., Comment: 5 pages, 4 figures, Supplement: 6 pages, 2 figures

Concentrating light within subwavelength spatial regions is a central topic in nanophotonics. In this letter, we introduce a general principle for the subwavelength localization and enhancement of electric fields in high-index resonators, based on the flux conservation of the displacement current density. We apply this design rule to a ring resonator by locally squeezing its section: since the flux is conserved, the electric field is necessarily enhanced to compensate the reduction of the section area. The introduced principle may constitute an important step toward the control of the displacement current density at the nanoscale, guiding the design of the topology and the geometry of complex dielectric structures., Comment: comments/feedback welcome!

In this paper we study the following class of fractional relativistic Schr\"odinger equations: \begin{equation*} \left\{ \begin{array}{ll} (-\Delta+m^{2})^{s}u + V(\varepsilon x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $m>0$, $N> 2s$, $(-\Delta+m^{2})^{s}$ is the fractional relativistic Schr\"odinger operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential satisfying a local condition, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous subcritical nonlinearity. By using a variant of the extension method and a penalization technique, we first prove that, for $\varepsilon>0$ small enough, the above problem admits a weak solution $u_{\varepsilon}$ which concentrates around a local minimum point of $V$ as $\varepsilon\rightarrow 0$. We also show that $u_{\varepsilon}$ has an exponential decay at infinity by constructing a suitable comparison function and by performing some refined estimates. Secondly, by combining the generalized Nehari manifold method and Ljusternik-Schnirelman theory, we relate the number of positive solutions with the topology of the set where the potential $V$ attains its minimum value.

In this paper we study the following class of fractional Kirchhoff problems: \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}M(\varepsilon^{2s-N}[u]^{2}_{s})(-\Delta)^{s}u + V(x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $N\geq 2$, $(-\Delta)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive continuous function, $M: [0, \infty)\rightarrow \mathbb{R}$ is a Kirchhoff function satisfying suitable conditions and $f:\mathbb{R}\rightarrow \mathbb{R}$ fulfills Berestycki-Lions type assumptions of subcritical or critical type. Using suitable variational arguments, we prove the existence of a family of positive solutions $(u_{\varepsilon})$ which concentrates at a local minimum of $V$ as $\varepsilon\rightarrow 0$.

In this work we consider the following class of fractional $p\&q$ Laplacian problems \begin{equation*} (-\Delta)_{p}^{s}u+ (-\Delta)_{q}^{s}u + V(\varepsilon x) (|u|^{p-2}u + |u|^{q-2}u)= f(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $1< p

We prove the existence of infinitely many nontrivial weak periodic solutions for a class of fractional Kirchhoff problems driven by a relativistic Schr\"odinger operator with periodic boundary conditions and involving different types of nonlinearities.

In this paper we consider the following class of fractional Kirchhoff equations with critical growth: \begin{equation*} \left\{ \begin{array}{ll} \left(\varepsilon^{2s}a+\varepsilon^{4s-3}b\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s}u+V(x)u=f(u)+|u|^{2^{*}_{s}-2}u \quad &\mbox{ in } \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \quad u>0 &\mbox{ in } \mathbb{R}^{3}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b>0$ are constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-\Delta)^{s}$ is the fractional Laplacian operator, $V$ is a positive continuous potential and $f$ is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we prove the existence of a family of positive solutions $u_{\varepsilon}$ which concentrates around a local minimum of $V$ as $\varepsilon\rightarrow 0$., Comment: arXiv admin note: text overlap with arXiv:1810.04561

We deal with a class of fractional magnetic Schr\"odinger equations in the whole line with exponential critical growth. Under a local condition on the potential, we use penalization methods and Ljusternik-Schnirelmann category theory to investigate the existence, multiplicity and concentration of nontrivial weak solutions., Comment: arXiv admin note: text overlap with arXiv:1807.07444, arXiv:1801.00199, arXiv:1808.09295

We investigate the existence, multiplicity and concentration of nontrivial solutions for the following fractional magnetic Kirchhoff equation with critical growth: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3} [u]_{A/\varepsilon}^{2}\right)(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u+|u|^{\2-2}u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon$ is a small positive parameter, $a, b>0$ are fixed constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential verifying the global condition due to Rabinowitz \cite{Rab}, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ subcritical nonlinearity. Due to the presence of the magnetic field and the critical growth of the nonlinearity, several difficulties arise in the study of our problem and a careful analysis will be needed. The main results presented here are established by using minimax methods, concentration compactness principle of Lions \cite{Lions}, a fractional Kato's type inequality and the Ljusternik-Schnirelmann theory of critical points., Comment: arXiv admin note: text overlap with arXiv:1808.09295

In this paper we deal with the following fractional Choquard equation \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{sp}(-\Delta)^{s}_{p} u + V(x)|u|^{p-2}u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u) \mbox{ in } \mathbb{R}^{N},\\ u\in W^{s,p}(\R^{N}), \quad u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $p\in (1, \infty)$, $N>sp$, $(-\Delta)^{s}_{p}$ is the fractional $p$-Laplacian, $V$ is a positive continuous potential, $0<\mu0$ small enough.

This paper is concerned with the multiplicity and concentration behavior of nontrivial solutions for the following fractional Kirchhoff equation in presence of a magnetic field: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3} [u]_{A/\varepsilon}^{2}\right)(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b>0$ are constants, $s\in (\frac{3}{4}, 1)$, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential having a local minimum and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ subcritical nonlinearity. Applying penalization techniques and Ljusternik-Schnirelman theory, we relate the number of nontrivial solutions with the topology of the set where the potential $V$ attains its minimum., Comment: arXiv admin note: text overlap with arXiv:1807.07444, arXiv:1808.01925

We consider the following fractional Schr\"odinger-Poisson type equation with magnetic fields \begin{equation*} \varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (\frac{3}{4}, 1)$, $t\in (0,1)$, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous electric potential and $f:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a continuous function with subcritical growth. By using suitable variational methods, we show the existence of families of nontrivial solutions concentrating around local minima of the potential $V(x)$ as $\varepsilon\rightarrow 0$., Comment: arXiv admin note: substantial text overlap with arXiv:1807.07444; text overlap with arXiv:1801.00199, arXiv:1807.06861

We deal with the following fractional Schr\"odinger-Poisson equation with magnetic field \begin{equation} \varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u+|u|^{2^{*}_{s}-2}u \quad \mbox{in} \mathbb{R}^{3}, \nonumber \end{equation} where $\varepsilon>0$ is a small parameter, $s\in (\frac{3}{4}, 1)$, $t\in (0,1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a subcritical nonlinearity. Under a local condition on the potential $V$, we study multiplicity and concentration of nontrivial solutions as $\varepsilon\rightarrow 0$. In particular, we relate the number of nontrivial solutions with the topology of the set where the potential $V$ attains its minimum., Comment: arXiv admin note: text overlap with arXiv:1801.00199, Potential Analysis (2018)

We consider the following nonlinear fractional Choquard equation $$ \varepsilon^{2s}(-\Delta)^{s}_{A/\varepsilon} u + V(x)u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $0<\mu<2s$, $N\geq 3$, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is a smooth magnetic potential, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive potential with a local minimum and $f$ is a continuous nonlinearity with subcritical growth. By using variational methods we prove the existence and concentration of nontrivial solutions for $\varepsilon>0$ small enough., Comment: arXiv admin note: text overlap with arXiv:1801.00199

This paper is devoted to the study of fractional Schr\"odinger-Poisson type equations with magnetic field of the type \begin{equation*} \varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon>0$ is a parameter, $s,t\in (0, 1)$ are such that $2s+2t>3$, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a continuous electric potential and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ subcritical nonlinear term. Using variational methods, we obtain the existence, multiplicity and concentration of nontrivial solutions for $\varepsilon>0$ small enough.

Quantum teleportation establishes a correspondence between an entangled state shared by two separate par- ties that can communicate classically and the presence of a quantum channel connecting the two parties. The standard benchmark for quantum teleportation, based on the average fidelity between the input and output states, indicates that some entangled states do not lead to channels which can be certified to be quantum. It was re- cently shown that if one considers a finer-tuned witness, then all entangled states can be certified to produce a non-classical teleportation channel. Here we experimentally demonstrate a complete characterization of a new family of such witnesses, of the type proposed in Phys. Rev. Lett. 119, 110501 (2017) under different con- ditions of noise. Furthermore, we show non-classical teleportation using quantum states that can not achieve average teleportation fidelity above the classical limit. Our results have fundamental implications in quantum information protocols and may also lead to new applications and quality certification of quantum technologies., Comment: 5 pages + Supp. Materials

We deal with the following nonlinear problem involving fractional $p\&q$ Laplacians: \begin{equation*} (-\Delta)^{s}_{p}u+(-\Delta)^{s}_{q}u+|u|^{p-2}u+|u|^{q-2}u=\lambda h(x) f(u)+|u|^{q^{*}_{s}-2}u \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $10$ is a parameter, $h$ is a nontrivial bounded perturbation and $f$ is a superlinear continuous function with subcritical growth. Using suitable variational arguments and concentration-compactness lemma, we prove the existence of a nontrivial non-negative solution for $\lambda$ sufficiently large.

We consider some nonlinear fractional Schr\"odinger equations with magnetic field and involving continuous nonlinearities having subcritical, critical or supercritical growth. Under a local condition on the potential, we use minimax methods to investigate the existence and concentration of nontrivial weak solutions., Comment: arXiv admin note: text overlap with arXiv:1807.07444

We consider the following class of fractional parametric problems \begin{equation*} \left\{ \begin{array}{ll} (-\Delta_{Dir})^{s} u= f(x, u)+t\varphi_{1}+h &\mbox{ in } \Omega\\ u=0 &\mbox{ on } \partial \Omega, \end{array} \right. \end{equation*} where $\Omega\subset \mathbb{R}^{N}$ is a smooth bounded domain, $s\in (0, 1)$, $N> 2s$, $(-\Delta_{Dir})^{s}$ is the fractional Dirichlet Laplacian, $f: \bar{\Omega} \times \mathbb{R} \rightarrow \mathbb{R}$ is a locally Lipschitz nonlinearity having linear or superlinear growth and satisfying Ambrosetti-Prodi type assumptions, $t\in \mathbb{R}$, $\varphi_{1}$ is the first eigenfunction of the Laplacian with homogenous boundary conditions, and $h:\Omega\rightarrow \mathbb{R}$ is a bounded function. Using variational methods, we prove that there exists a $t_{0}\in \mathbb{R}$ such that the above problem admits at least two distinct solutions for any $t\leq t_{0}$. We also discuss the existence of solutions for a fractional periodic Ambrosetti-Prodi type problem.

In this paper we investigate the existence of multiple solutions for the following two fractional problems \begin{equation*} \left\{\begin{array}{ll} (-\Delta_{\Omega})^{s} u-\lambda u= f(x, u) &\mbox{in} \Omega \\ u=0 &\mbox{in} \partial \Omega \end{array} \right. \end{equation*} and \begin{equation*} \left\{\begin{array}{ll} (-\Delta_{\mathbb{R}^{N}})^{s} u-\lambda u= f(x, u) &\mbox{in} \Omega \\ u=0 &\mbox{in} \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{equation*} where $s\in (0,1)$, $N>2s$, $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$, and $f:\bar{\Omega}\times \mathbb{R}\rightarrow \mathbb{R}$ is a superlinear continuous function which does not satisfy the well-known Ambrosetti-Rabinowitz condition. Here $(-\Delta_{\Omega})^{s}$ is the spectral Laplacian and $(-\Delta_{\mathbb{R}^{N}})^{s}$ is the fractional Laplacian in $\mathbb{R}^{N}$. By applying variational theorems of mixed type due to Marino and Saccon and Linking Theorem, we prove the existence of multiple solutions for the above problems., Comment: Adv. Nonlinear Stud. (2018)

We deal with the existence of $2\pi$-periodic solutions to the following non-local critical problem \begin{equation*} \left\{\begin{array}{ll} [(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=W(x)|u|^{2^{*}_{s}-2}u+ f(x, u) &\mbox{in} (-\pi,\pi)^{N} \\ u(x+2\pi e_{i})=u(x) &\mbox{for all} x \in \mathbb{R}^{N}, \quad i=1, \dots, N, \end{array} \right. \end{equation*} where $s\in (0,1)$, $N \geq 4s$, $m\geq 0$, $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent, $W(x)$ is a positive continuous function, and $f(x, u)$ is a superlinear $2\pi$-periodic (in $x$) continuous function with subcritical growth. When $m>0$, the existence of a nonconstant periodic solution is obtained by applying the Linking Theorem, after transforming the above non-local problem into a degenerate elliptic problem in the half-cylinder $(-\pi,\pi)^{N}\times (0, \infty)$, with a nonlinear Neumann boundary condition, through a suitable variant of the extension method in periodic setting. We also consider the case $m=0$ by using a careful procedure of limit. As far as we know, all these results are new., Comment: Calculus of Variations and Partial Differential Equations (2018)

This paper is devoted to the study of the following fractional Choquard equation $$ \varepsilon^{2s}(-\Delta)^{s} u + V(x)u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u) \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplacian, $V$ is a positive continuous potential with local minimum, $0<\mu<2s$, and $f$ is a superlinear continuous function with subcritical growth. By using the penalization method and the Ljusternik-Schnirelmann theory, we investigate the multiplicity and concentration of positive solutions for the above problem., Comment: arXiv admin note: text overlap with arXiv:1711.03625

By using the penalization method and the Ljusternik-Schnirelmann theory, we investigate the multiplicity of positive solutions of the following fractional Schr\"odinger equation $$ \e^{2s}(-\Delta)^{s} u + V(x)u = f(u) \mbox{ in } \mathbb{R}^{N} $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplacian, $V$ is a positive continuous potential with local minimum, and $f$ is a superlinear function with subcritical growth. We also obtain a multiplicity result when $f(u)=|u|^{q-2}u+\lambda |u|^{r-2}u$ with $20$, by combining a truncation argument and a Moser-type iteration.

Structured photons are nowadays an interesting resource in classical and quantum optics due to the richness of properties they show under propagation, focusing and in their interaction with matter. Vectorial modes of light in particular, a class of modes where the polarization varies across the beam profile, have already been used in several areas ranging from microscopy to quantum information. One of the key ingredients needed to exploit the full potential of complex light in quantum domain is the control of quantum interference, a crucial resource in fields like quantum communication, sensing and metrology. Here we report a tunable photon-photon interference between vectorial modes of light. We demonstrate how a properly designed spin-orbit device can be used to control quantum interference between vectorial modes of light by simply adjusting the device parameters and no need of interferometric setups. We believe our result can find applications in fundamental research and quantum technologies based on structured light by providing a new tool to control quantum interference in a compact, efficient and robust way.

In this paper we focus our attention on the following nonlinear fractional Schr\"odinger equation with magnetic field \begin{equation*} \varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N\geq 3$, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $A:\mathbb{R}^{N}\rightarrow \mathbb{R}^N$ are continuous potentials and $f:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a subcritical nonlinearity. By applying variational methods and Ljusternick-Schnirelmann theory, we prove existence and multiplicity of solutions for $\varepsilon$ small., Comment: 23 pages

We consider a class of parametric Schr\"odinger equations driven by the fractional $p$-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. By using variational methods and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of positive solutions for small values of the parameter.

In classical physics, properties of the objects exist independently on the context, i.e. whether and how measurements are performed. Quantum physics showed this assumption to be wrong and that Nature is indeed "contextual". Contextuality has been observed in the simplest physical systems such as single particles, and plays fundamental roles in quantum computation advantage. Here, we demonstrate for the first time quantum contextuality in an integrated photonic chip. The chip implements different combinations of measurements on a single photon delocalized on four distinct spatial modes. We show violations of a CHSH-like non-contextuality inequality by 14 standard deviations. This paves the way to compact and portable devices for contextuality-based quantum-powered protocols.

In this paper we deal with the multiplicity and concentration of positive solutions for the following fractional Schr\"odinger-Kirchhoff type equation \begin{equation*} M\left(\frac{1}{\varepsilon^{3-2s}} \iint_{\mathbb{R}^{6}}\frac{|u(x)- u(y)|^{2}}{|x-y|^{3+2s}} dxdy + \frac{1}{\varepsilon^{3}} \int_{\mathbb{R}^{3}} V(x)u^{2} dx\right)[\varepsilon^{2s} (-\Delta)^{s}u+ V(x)u]= f(u) \, \mbox{in} \mathbb{R}^{3} \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (\frac{3}{4}, 1)$, $(-\Delta)^{s}$ is the fractional Laplacian, $M$ is a Kirchhoff function, $V$ is a continuous positive potential and $f$ is a superlinear continuous function with subcritical growth. By using penalization techniques and Ljusternik-Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum., Comment: Mathematical Methods in the Applied Sciences (2017)

In this paper we study the existence, multiplicity and concentration behavior of solutions for the following critical fractional Schr\"odinger system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-\Delta)^{s}u+V(x) u=Q_{u}(u, v)+\frac{1}{2^{*}_{s}}K_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}\varepsilon^{2s} (-\Delta)^{s}u+W(x) v=Q_{v}(u, v)+\frac{1}{2^{*}_{s}}K_{v}(u, v) &\mbox{ in } \mathbb{R}^{N} u, v>0 &\mbox{ in } \R^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplacian operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $W:\mathbb{R}^{N}\rightarrow \mathbb{R}$ are positive H\"older continuous potentials, $Q$ and $K$ are homogeneous $C^{2}$-functions having subcritical and critical growth respectively. We relate the number of solutions with the topology of the set where the potentials $V$ and $W$ attain their minimum values. The proofs rely on the Ljusternik-Schnirelmann theory and variational methods., Comment: arXiv admin note: text overlap with arXiv:1704.00604

The aim of this paper is to investigate the existence, multiplicity and concentration of positive solutions for the following nonlocal system of fractional Schr\"odinger equations \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-\Delta)^{s}u+V(x)u=Q_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}, \varepsilon^{2s} (-\Delta)^{s}v+W(x)v=Q_{v}(u, v) &\mbox{ in } \mathbb{R}^{N}, u, v>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $W:\mathbb{R}^{N}\rightarrow \mathbb{R}$ are positive continuous potentials, $Q$ is a homogeneous $C^{2}$-function with subcritical growth. In order to relate the number of solutions with the topology of the set where the potentials $V$ and $W$ attain their minimum values, we apply penalization techniques, Nehari manifold arguments and Ljusternik-Schnirelmann theory., Comment: arXiv admin note: text overlap with arXiv:1703.04370

In this paper we deal with the following nonlocal systems of fractional Schr\"odinger equations \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-\Delta)^{s}u+V(x)u=Q_{u}(u, v)+\gamma H_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}\\ \varepsilon^{2s} (-\Delta)^{s}v+W(x)v=Q_{v}(u, v)+\gamma H_{v}(u, v) &\mbox{ in } \mathbb{R}^{N} \\ u, v>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$, $s\in (0, 1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $W:\mathbb{R}^{N}\rightarrow \mathbb{R}$ are continuous potentials, $Q$ is a homogeneous $C^{2}$-function with subcritical growth, $\gamma\in \{0, 1\}$ and $H(u, v)=\frac{2}{\alpha+\beta}|u|^{\alpha} |v|^{\beta}$ with $\alpha, \beta\geq 1$ such that $\alpha+\beta=2^{*}_{s}$. We investigate the subcritical case $(\gamma=0)$ and the critical case $(\gamma=1)$, and using Ljusternik-Schnirelmann theory, we relate the number of solutions with the topology of the set where the potentials $V$ and $W$ attain their minimum values.

We deal with the existence of positive solutions for the following fractional Schr\"odinger equation $$ \varepsilon ^{2s} (-\Delta)^{s} u + V(x) u = f(u) \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplacian operator, and $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous positive function. Under the assumptions that the nonlinearity $f$ is either asymptotically linear or superlinear at infinity, we prove the existence of a family of positive solutions which concentrates at a local minimum of $V$ as $\varepsilon$ tends to zero.