Belief propagation is a well-studied algorithm for approximating local marginals of multivariate probability distribution over complex networks, while tensor network states are powerful tools for quantum and classical many-body problems. Building on a recent connection between the belief propagation algorithm and the problem of tensor network contraction, we propose a block belief propagation algorithm for contracting two-dimensional tensor networks and approximating the ground state of $2D$ systems. The advantages of our method are three-fold: 1) the same algorithm works for both finite and infinite systems; 2) it allows natural and efficient parallelization; 3) given its flexibility it would allow to deal with different unit cells. As applications, we use our algorithm to study the $2D$ Heisenberg and transverse Ising models, and show that the accuracy of the method is comparable to state-of-the-art results.

Structured waves are ubiquitous for all areas of wave physics, both classical and quantum, where the wavefields are inhomogeneous and cannot be approximated by a single plane wave. Even the interference of two plane waves, or a single inhomogeneous (evanescent) wave, provides a number of nontrivial phenomena and additional functionalities as compared to a single plane wave. Complex wavefields with inhomogeneities in the amplitude, phase, and polarization, including topological structures and singularities, underpin modern nanooptics and photonics, yet they are equally important, e.g., for quantum matter waves, acoustics, water waves, etc. Structured waves are crucial in optical and electron microscopy, wave propagation and scattering, imaging, communications, quantum optics, topological and non-Hermitian wave systems, quantum condensed-matter systems, optomechanics, plasmonics and metamaterials, optical and acoustic manipulation, and so forth. This Roadmap is written collectively by prominent researchers and aims to survey the role of structured waves in various areas of wave physics. Providing background, current research, and anticipating future developments, it will be of interest to a wide cross-disciplinary audience.

We present magneto-optical studies of a self-assembled semiconductor quantum dot, concentrating specifically on the case in which the dot is doubly positively charged, studying this way the confined hole - hole exchange interaction. A simple harmonic potential model, which we extend to capture the influence of an externally applied magnetic field in Faraday configuration fully describe the observed polarization sensitive magneto-photoluminscence spectra. We deduce the effective composition of the quantum dot from its measured electronic g-factor. Using this value we determine the dot effective permittivity and quantitatively describe various measured excitonic transitions, their measured Zeeman splittings and their diamagnetic shifts. In particular, the model quantitatively accounts for an observed pronounced negative diamagnetic shift, which provides a direct measure for the hole-hole exchange interaction and its dependence on the externally applied magnetic field strength.

For over 80 years of research, the conventional description of free-electron radiation phenomena, such as Cherenkov radiation, has remained unchanged: classical three-dimensional electromagnetic waves. Interestingly, in reduced dimensionality, the properties of free-electron radiation are predicted to fundamentally change. Here, we present the first observation of Cherenkov surface waves, wherein free electrons emit narrow-bandwidth photonic quasiparticles propagating in two dimensions. The low dimensionality and narrow bandwidth of the effect enable us to identify quantized emission events through electron energy loss spectroscopy. Our results support the recent theoretical prediction that free electrons do not always emit classical light and can instead become entangled with the photons they emit. The two-dimensional Cherenkov interaction achieves quantum coupling strengths over 2 orders of magnitude larger than ever reported, reaching the single-electron–single-photon interaction regime for the first time with free electrons. Our findings pave the way to previously unexplored phenomena in free-electron quantum optics, facilitating bright, free-electron-based quantum emitters of heralded Fock states.

Long-range coherence and correlations between electrons in solids are the cornerstones for developing future quantum materials and devices. In 1954, Dicke described correlated spontaneous emission from closely packed quantum emitters, forming the theoretical basis of superradiance and superfluorescence. Since then, it has remained an open challenge to observe such phenomena with nanometer spatial resolution, precisely the important scale at which the collective correlations occur. Here, we report the first instance of free-electron-driven superfluorescence - superfluorescent cathodoluminescence - enabling us to excite and observe correlations at nanometer spatial scales. To exemplify this concept in our experiments, superlattices of lead halide perovskite quantum dots are excited by focused pulses of multiple free electrons. The electrons trigger superfluorescence: collective ultrafast emission observed at rates faster than both the lifetime and decoherence time. By controlling the area illuminated by the electron beam, we create a transition from a non-correlated spontaneous emission to a correlated superfluorescent emission. The observed signatures of superfluorescence are a reduction of the intrinsic emitter lifetime, a narrower linewidth, and a distinct redshift. We develop the theory of superfluorescent cathodoluminescence, which matches the results and highlights the unique features of electron-driven versus light-driven superfluorescence. Our observation and theory introduce a novel way to characterize coherence and correlations in quantum materials with nanometer spatial resolution, a key for future engineering of quantum devices.

Flatbands have become a cornerstone of contemporary condensed-matter physics and photonics. In electronics, flatbands entail comparable energy bandwidth and Coulomb interaction, leading to correlated phenomena such as the fractional quantum Hall effect and recently those in magic-angle systems. In photonics, they enable properties including slow light¹ and lasing². Notably, flatbands support supercollimation—diffractionless wavepacket propagation—in both systems^3,4. Despite these intense parallel efforts, flatbands have never been shown to affect the core interaction between free electrons and photons. Their interaction, pivotal for free-electron lasers⁵, microscopy and spectroscopy^6,7, and particle accelerators^8,9, is, in fact, limited by a dimensionality mismatch between localized electrons and extended photons. Here we reveal theoretically that photonic flatbands can overcome this mismatch and thus remarkably boost their interaction. We design flatband resonances in a silicon-on-insulator photonic crystal slab to control and enhance the associated free-electron radiation by tuning their trajectory and velocity. We observe signatures of flatband enhancement, recording a two-order increase from the conventional diffraction-enabled Smith–Purcell radiation. The enhancement enables polarization shaping of free-electron radiation and characterization of photonic bands through electron-beam measurements. Our results support the use of flatbands as test beds for strong light–electron interaction, particularly relevant for efficient and compact free-electron light sources and accelerators.

Compton scattering is a cornerstone of quantum physics, describing the fundamental electron-photon interaction. Inverse Compton scattering can create attosecond x-ray pulses by high-intensity lasers driving free electrons. So far, in all theory and experiments, the observables of Compton scattering and its generalizations could be described by treating the driving electromagnetic field classically. Motivated by advances in the generation of squeezed light with high intensity, we consider driving the Compton effect with nonclassical light. We develop a framework to describe the nonperturbative interaction of a charged particle with driving fields of an arbitrary quantum light state. We obtain analytical results for the Compton emission spectrum when driven by intense thermal and squeezed vacuum states, showing noticeably broader emission spectra relative to a classical drive, thus reaching higher emission frequencies for the same average intensity. We envision quantum light properties such as squeezing and entanglement as degrees of freedom to control various radiation phenomena.

The last decade has witnessed extensive breakthroughs and significant progress in atomically flat two-dimensional (2D) semiconductor nanoplatelets (NPLs) in terms of synthesis, growth mechanisms, optical and electronic properties and practical applications. Such NPLs have electronic structures similar to those of quantum wells in which excitons are predominantly confined along the vertical direction, while electrons are free to move in the lateral directions, resulting in unique optical properties, such as extremely narrow emission line width, short photoluminescence (PL) lifetime, high gain coefficient, and giant oscillator strength transition (GOST). These unique optical properties make NPLs favorable for high color purity light-emitting applications, in particular in light-emitting diodes (LEDs), backlights for liquid crystal displays (LCDs) and lasers. This review article first introduces the intrinsic characteristics of 2D semiconductor NPLs with atomic flatness. Subsequently, the approaches and mechanisms for the controlled synthesis of atomically flat NPLs are summarized followed by an insight on recent progress in the mediation of core/shell, core/crown and core/crown@shell structures by selective epitaxial growth of passivation layers on different planes of NPLs. Moreover, an overview of the unique optical properties and the associated light-emitting applications is elaborated. Despite great progress in this research field, there are some issues relating to heavy metal elements such as Cd²⁺ in NPLs, and the ambiguous gain mechanisms of NPLs and others are the main obstacles that prevent NPLs from widespread applications. Therefore, a perspective is included at the end of this review article, in which the current challenges in this stimulating research field are discussed and possible solutions to tackle these challenges are proposed.

Recent advances in laser interactions with coherent free electrons have enabled to shape the electron's quantum state. Each electron becomes a superposition of energy levels on an infinite quantized ladder, shown to contain up to thousands of energy levels. We propose to utilize the quantum nature of such laser-driven free electrons as a "synthetic Hilbert space" in which we construct and control qudits (quantum digits). The question that motivates our work is what qudit states can be accessed using electron-laser interactions, and whether it is possible to implement any arbitrary quantum gate. We find how to encode and manipulate free-electron qudit states, focusing on dimensions which are powers of 2, where the qudit represents multiple qubits implemented on the same single electron – algebraically separated, but physically joined. As an example, we prove the possibility to fully control a 4-dimenisonal qudit, and reveal the steps required for full control over any arbitrary dimension. Our work enriches the range of applications of free electrons in microscopy and spectroscopy, offering a new platform for continuous-variable quantum information.

We report on the implementation of degenerate Raman sideband cooling of ${}^{40}K$ atoms. The scheme incorporates a three-dimensional (3D) optical lattice, which confines the atoms and drives the Raman transitions. The optical cooling cycle is closed by two optical pumping beams. The wavelength of the laser beams forming the lattice is close to the $D_2$ atomic transition, while the optical pumping is operated near the $D_1$ transition. With this cooling method, we achieve a temperature of $\sim 1\mu K$ of a cloud with $\sim {10}^7$ atoms. This corresponds to a phase space density of $\ge {10}^{-3}$. Moreover, the fermionic ensemble is spin polarized to conditions which are favorable for subsequent evaporative cooling. We study the dependence of the cooling scheme on several parameters, including the applied magnetic field and the detuning, duration, and intensity profile of the optical pumping beams. Adding this optical cooling stage to current Fermi gas experiments can improve the final conditions and increase the data rate.