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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