How do we detect topological phases from a single ground state? Entanglement entropy and spectrum have long been the standard tools -- but the reduced density matrix (RDM) itself contains far more information. We show that the RDM of a 2D topologically ordered system, expressed at the entanglement cut, realizes a 1D mixed-state phase. For the $\mathbb{Z}_2$ toric code phase, it is a 1D $\mathbb{Z}_2$ strong-to-weak spontaneous symmetry breaking (SW-SSB) phase, where deconfinement of anyons manifests as the short-range correlation of both $\mathbb{Z}_2$ charge and $\mathbb{Z}_2$ domain-wall in the RDM. The bulk $e$-$m$ duality translates into a Kramers--Wannier self-duality of the SW-SSB phase. Extending the framework to gapped $\mathbb{Z}_2$ spin liquids, the global spin-rotation symmetry manifests as an additional weak symmetry for the 1D RDM. Spin-$\frac{1}{2}$ spinons result in a cusp on the disorder parameter of spin-rotation at $\theta=\pi$, providing a direct, ground-state signature of symmetry fractionalization. We verify this prediction analytically using the matrix product density operator formalism and numerically for the kagome-lattice resonating valence bond state. The proposed observable requires only a single ground-state wavefunction, making it amenable to quantum simulation platforms.

We report the discovery of a previously unrecognized electrostatic hose instability in electron-beam-sustained plasmas, driven by the coupling between the electron beam centroid and the plasma generated via the beam-impact ionization. Unlike the conventional hose instability of relativistic beams propagating in underdense plasmas, this instability requires only ionization-capable electron beams readily produced by common emission processes and sheath acceleration, indicating broad relevance across various discharges. A linear theory is developed to predict the hosing frequency and growth rate, and particle-in-cell/Monte Carlo simulations confirm both the onset of instability and the theoretical predictions.

Spin-orbit torque (SOT) enables the electrical manipulation of the magnetization with high speed and low energy consumption for magnetic random-access memory (MRAM) applications. Previous studies of short-pulse SOT switching have mainly focused on the nanosecond regime, whereas reports employing picosecond pulses remain scarce and have largely relied on field-assisted switching using bulky, high-power laser systems, limiting prospects for chip-level integration. Here, we introduce an all-electrical on-chip nanoplasma pulse generator capable of producing pulses as short as 6.4 ps, enabling ultrafast picosecond field-free SOT switching in magnetic trilayers. We show that reducing the pulse width lowers the writing energy by 2-3 orders of magnitude, with ultrafast Joule heating assistance playing an essential role in the enhanced efficiency of the picosecond regime. Our demonstration of ultrafast, all-electrical, and field-free SOT switching establishes the nanoplasma pulse generator as an on-chip platform for ultrafast spintronic studies, with promise for high-speed, energy-efficient, and scalable SOT-MRAM technologies.

We consider Schrödinger operators in $\ell^2(\mathbb{Z})$ with real-valued potentials in $\ell^\infty(\mathbb{Z})$ and show that the generic spectral type is pure point. More specifically, we show that for a generic bounded potential, the essential spectrum of the associated Schrödinger operator is a Cantor set and has zero weight with respect to all spectral measures.

We investigate the influence of asymmetric magnetic fields on the atmospheric circulation of hot Jupiters based on a Shallow-Water Magnetohydrodynamic (SWMHD) model. The Shallow-Water Hydrodynamic (SWHD) models predict eastward equatorial jets and hotspot offsets, while some observations have revealed westward hotspots, suggesting that magnetic fields may play an important role. We incorporate asymmetric magnetic fields between hemispheres, and analyze their effects through linear perturbation analysis and numerical calculations. Our results indicate that strong magnetic fields play a dominant role in momentum transport. Asymmetric magnetic field configurations lead to hemispheric temperature contrasts, with the dayside temperature maxima in the hemisphere of stronger magnetic field located closer to the equator. With the magnetic field fixed in one hemisphere, the equatorial hotspots shift westward then eastward as the other hemisphere's field strengthens, exhibiting a pronounced westward offset only at moderate field strengths and weak hemispheric asymmetry. These findings highlight the significance of magnetic field geometry in explaining observed atmospheric dynamics and hotspot variability in hot Jupiters.

Recent observations of 55 Cancri e suggest an atmosphere rich in CO or CO$_2$ (Hu et al. 2024); other observations indicate the planet's eclipse depth is highly variable (e.g. Patel et al. 2024). So far, these observations have only been interpreted using 1D models without self-consistent heat redistribution, as the planet's extreme temperatures make it inaccessible to most 3D models. Here we perform cloud-free GCM simulations of 55 Cancri e using custom correlated-$k$ coefficients developed from the ExoMol database. Our best-fit simulations match the JWST spectra from Hu et al. (2024) well, favoring an atmosphere that is both thick ($\ge$ 10 bar) and CO$_2$-rich ($>1\%$ CO$_2$ volume mixing ratio), while ruling out thin ($<$ 10 bar) and pure-CO/CO$_2$-poor atmosphere, which were previously proposed based on 1D models (Hu et al. 2024; Zilinskas et al. 2025). We also find large-scale atmospheric dynamics, i.e. weather, is insufficient to explain the observed variability. A thick, CO$_2$-rich atmosphere implies that 55 Cancri e likely formed with significantly more volatiles than Earth and Venus. In addition, a thick atmosphere makes it unlikely that the planet's variability is caused by transient outgassing (Heng 2023), favoring other variability mechanisms (e.g. clouds). Our work provides model constraints for upcoming JWST observations of 55 Cancri e, and highlights the importance of interpreting thermal emission observations with self-consistent 3D models.

SN 1006 is a historical Type Ia supernova remnant that exhibits non-thermal emission ranging from radio to multi-TeV $\gamma$-rays. Most of this emission (particularly X-rays and $\gamma$-rays) is concentrated in polar caps aligned with the ambient magnetic field, which makes it an ideal laboratory for studying cosmic ray (CR) acceleration at different shock obliquities and the hadronic/leptonic nature of the $\gamma$-ray emission. We model SN 1006's morphology, multi-wavelength spectrum, and radial profile using a self-consistent multi-zone kinetic model of particle acceleration that accounts for: CR-driven shock modification, magnetic field amplification, drift in magnetic fluctuations, and temporal dynamics including adiabatic and synchrotron losses. Our model can reproduce both the observed spectral and spatial properties, with the exception of the radio profile that we argue requires 3D hydrodynamic effects to replicate. We find that quasi-parallel regions (where the shock normal aligns with the ambient magnetic field) exhibit very prominent CR acceleration ($\sim$20% efficiency), while quasi-perpendicular regions exhibit efficiencies below 1%, consistent with the results of kinetic simulations. We also find that electrons are responsible for the majority of the $\gamma$-ray emission from SN 1006 (i.e., it is a leptonic source), with the exception of the northwest region due to an encounter with a dense cloud.

The development of simple and scalable fabrication strategies for cost-effective electrodes is crucial to advance water splitting in alkaline water electrolyzers (AWEs). Here, we present a coating-annealing method to conformally coat a MoSx catalyst layer onto a porous Ni foam (NF) substrate. By controlling the annealing process, the composition of the MoSx layer could be tuned from MoS2 to MoS3 and its catalytic performance for hydrogen evolution reaction (HER) in alkaline media was optimized. The MoS3@NF synthesized by this method achieved industrially relevant HER current densities of 200 mA/cm2 at a low overpotential of 246 mV, maintaining stable operation for over 240 h. The MoS3@NF cathode, combined with a stainless steel anode, enabled an alkaline water electrolyzer (AWE) cell to operate steadily at 1.96 V and 1 A/cm2 for 1000 h. This performance surpasses that of most of the previously reported water electrolyzers employing MoSx-based cathodes. Our work demonstrates the potential of MoS3 (with its abundant edge-sulfur atoms serving as active sites) as a high-performance cathode material for industrial AWEs.

Melting and solidification of eutectic systems is a classical topic in physical metallurgy, yet the mechanisms at nanoscale are less investigated, due to experimental limitations in spatiotemporal resolution. The advent of \textit{in situ} STEM heating with MEMS technology has recently enabled investigation of eutectic behavior as a function of temperature, time and electrical resistivity. Using this methodology, we investigate the evolution of a nanocrystalline hypoeutectic Al--Cu alloy. Melting initiated in the hotter central region and propagated outward, with grain boundaries acting as preferred sites for eutectic liquid formation via Cu enrichment. The Al$_2$Cu phase melted prior to complete matrix melting. Liquid-state Cu redistribution over a distance of \SI{258}{\micro\metre} -- several orders of magnitude beyond solid-state diffusion limits -- resulted in Al-rich rim accumulations and Cu enrichment at the outermost edge of the observed chip region. These observations are discussed in the context of classical predictions for melting of eutectic systems.

We study proximity-induced unconventional superconductivity in a twisted graphene/NbSe$_2$ van der Waals heterostructure using the Bogoliubov-de Gennes formalism. The normal-state parameters of proximitized graphene are extracted from ab initio calculations at a twist angle of $23.4^\circ$, which reduces the common symmetry of the heterostructure to $\mathbf{C}_3$. We construct symmetry-allowed superconducting gap functions of the graphene layer according to the irreducible representations of the $\mathbf{C}_3$ group, containing singlet and triplet pairing channels and their mixtures. Computing the topological invariants as a function of the mixing parameters, we find a rich phase diagram of chiral topological superconducting phases, characterized by nonzero Chern numbers $C\in\{-4,-2,2,4\}$. While the nature of the superconducting order parameter of NbSe$_2$ remains debated, the formation of the van der Waals heterostructure and the related symmetry reduction can alter the relative stability of competing pairing channels, potentially stabilizing a chiral component that is proximity-induced into graphene and triggers the topological phases identified here, making the twisted graphene/NbSe$_2$ heterostructure a promising platform for chiral topological superconductivity detectable via quasiparticle interference imaging and transport measurements.

In this chapter we review some results describing the high-frequency eigenmodes of the Laplacian on compact manifolds, or Euclidean domains, for which the geodesic flow is chaotic. We focus on the macroscopic distribution of these eigenmodes, which is described by the concept of semiclassical measure. The main result on the question is the Quantum Ergodicity theorem, originally due to Schnirelman. We provide the detailed proof of this theorem, including the adjustments necessary to treat the case of manifolds with boundary. We also discuss the Quantum Unique Ergodicity conjecture, and some progress towards this conjecture for strongly chaotic (Anosov) systems. In particular, we describe the constraints on admissible semiclassical measures, in terms of their Kolmogorov-Sinai entropy, as well as more recent delocalization results.

We present a one-month-ahead conditional probabilistic framework for wind-power forecasting at ten-minute resolution. Monthly Weibull shape and scale parameters are estimated from serially dependent SCADA wind-speed data, corrected through a Godambe covariance, and forecast by a heteroskedastic Kalman filter on a bivariate VAR(1) state-space model. Conditional on the MMSE forecasted Weibull invariant law, we construct and compare three positive wind-speed SDE models: an Ornstein-Uhlenbeck-Weibull transform, a Fokker-Planck drift-first diffusion, and a Fokker-Planck diffusion-first model. The simulated wind-speed ensembles are mapped to power through a calibrated XGBoost power curve. Applied to January 2021 data from a Senvion MM92 turbine at Kelmarsh Wind Farm, the three SDE formulations are statistically indistinguishable in probabilistic accuracy, with mean CRPS values between 1.569 and 1.575 m/s. The diffusion-first model is therefore preferred on computational grounds, reducing runtime by about a factor of seven relative to the OU-Weibull model. In the power domain, the Wasserstein distance between simulated and observed distributions is 26.1-27.6 kW, below $1.4\%$ of rated capacity, while the monthly energy-yield bias is about $-7.3\%$ for the examined month. Exceedance-probability errors remain below 1.6 percentage points over the 0-1500 kW range and about 2.2 percentage points near rated power. These quantities provide decision-relevant probabilistic inputs for downstream operational problems, rather than completed reserve, storage, market, or fatigue-optimization decisions. Full marginalisation over the Kalman predictive law of the Weibull parameters is left as a natural extension.

Any quantum computation consists of a sequence of unitary evolutions described by a finite set of Hamiltonians. For the case where this set consists of only products of Pauli operators, known as Pauli strings, we provide a necessary and sufficient condition for it to generate $\mathfrak{su}(2^n)$, i.e., to be universal for quantum computation on $n$ qubits. When combining Pauli strings with a general Hamiltonian, we show a sufficient (and in certain circumstances even necessary) condition for universality based on the Pauli-basis expansion of the Hamiltonian. As an application of these results, we prove two corollaries: (i) a necessary and sufficient condition for the universality of a general Hamiltonian given arbitrary single-qubit control on all qubits, and (ii) the universality of an XYZ Heisenberg Hamiltonian with local control of just two adjacent qubits.

In this work, we systematically study the gravitational lensing properties and observational signatures of compact boson stars. Unlike black holes, the photon effective potential of a compact boson star develops a nearly flat region, whose width increases with the compactness of the star. This flat structure significantly broadens the range of impact parameters that can produce large-angle deflections, leading to noticeably wider lensing rings of all orders. Photons constituting these rings traverse more complex paths, rendering the resulting images more sensitive to the spatial distribution of the accretion flow. Ray tracing results show that, compared to black hole models, the image topology and visibility amplitudes of compact boson stars exhibit a stronger dependence on the accretion flow structure. These results highlight qualitative differences in the observational properties of compact boson stars and black holes.

A specific heat treatment at 300 C, named medium-temperature baking (Mid-T baking) is applied to superconducting radio-frequency (SRF) accelerating cavities to improve the quality factor (Qo) at medium accelerating fields (10 to 20 MV/m). This treatment is very successful when done properly as it can reduce by almost a factor of two the power dissipations in this field range. However, surface contamination can lead to the degradation of Qo instead. Plasma-based surface treatment provides an effective approach to eliminate contaminants from the Niobium surface. In this study an insitu plasma cleaning process with argon containing 10 % O2 was performed to remove hydrocarbons from Niobium surface. The treatment was applied before and after a heat treatment at 500 C under ultra-high vacuum conditions (Mid-T baking). Changes in chemical speciation and oxide layer alteration induced by plasma processing were analyzed using insitu X-ray photoelectron spectroscopy (XPS) and exsitu scanning electron microscopy (SEM). The results show that plasma treatment modifies the composition of Niobium oxides, converting a Nb2O5 layer into NbO2. Furthermore, a plasma treatment before Mid-T baking helps reduce unstable oxides such as NbxO and significantly increases the proportion of metallic Niobium at the surface. The Niobium sample treated by plasma prior to Mid-T baking showed a 53 % reduction in carbide formation. Moreover, the C1s component attributed to NbC bonds shifts toward lower binding energy, indicating the formation of a more metallic NbC phase. Whereas without plasma treatment, the higher binding energy component observed after Mid T baking is consistent with Nb2C.

Elastically accommodated grain-boundary sliding (EAGBS) is a plausible source of upper-mantle seismic attenuation and dispersion, yet classical theory predicts a localized Debye-like peak that is absent or only weakly expressed in dry olivine experiments. Here we test whether microstructural heterogeneity can explain this discrepancy using 2-D finite-element simulations on periodic Voronoi tessellations. We find that irregular grain geometry changes the baseline EAGBS response relative to the regular hexagonal benchmark, but increasing grain-size variance alone produces only modest changes in modulus and peak height, with little spectral broadening. In contrast, a broad distribution of grain-boundary viscosities progressively suppresses and broadens the Debye-like loss peak into a weak background spanning a wide frequency interval. This broadening arises from the superposition of many localized relaxation processes with distinct characteristic timescales and motivates a reduced-order 0-D description of the aggregate response. These results suggest that the absence of a pronounced EAGBS peak in dry olivine does not necessarily imply the absence of EAGBS mechanism itself. If grain boundaries sample a sufficiently broad viscosity distribution, the macroscopic EAGBS contribution may appear experimentally only as part of a broad attenuation background, while still remaining relevant for upper-mantle seismic attenuation and velocity dispersion.

Non-Hermitian systems can evade Anderson localization and exhibit delocalized states even in one dimension. Here, we show that such non-Hermitian delocalized states under periodic boundary conditions (PBC) are intrinsically critical, realizing the universality class of one-dimensional random Dirac fermions. By linking spectral winding to topological Anderson transitions via Hermitization, we demonstrate that the delocalized PBC states exhibit a Dirac-type criticality with universal algebraic correlations. In contrast to Hermitian systems, where this criticality occurs only at fine-tuned transition points, it emerges generically in non-Hermitian systems as a consequence of spectral topology. These results identify a universal mechanism by which non-Hermiticity promotes criticality, providing a unified description of non-Hermitian delocalization in one dimension.

The Schiff moment of a nucleus is a symmetry-violating nuclear moment that indicates new physics beyond the Standard Model. We place the limit, $|\mathscr{S}({}^{153}$Eu)$| < 1.7 \times 10^{-8}$ $e\,$fm$^3$ (95\% confidence), on the Schiff moment of the $^{153}$Eu nucleus, using nuclear spin resonances in two ensembles of oppositely-polarized $^{153}$Eu$^{3+}$ ions in a Y${}_2$SiO${}_5$ crystal. This measurement using octupolar nuclei in a mm-scale crystal constrains new physics at the TeV energy-scale.

The combinatorial design space of multilayer perovskite solar cells is vast, yet exhaustive experimental or computational screening of all possible material combinations remains impractical. Here, we integrate SCAPS-1D device simulations with machine learning to systematically explore 125 device architectures constructed from five electron transport layers (ETL), five absorbers (including lead-free double perovskites), and five hole transport layers (HTL). A representative subset of configurations is used to train a machine learning (ML) model, which predicts the power conversion efficiency (PCE) of the remaining unexplored structures. Leave-One-Group-Out cross-validation yields a Spearman rank correlation, demonstrating reliable ranking capability. SHAP (SHapley Additive exPlanations) analysis reveals that the HTL band gap, absorber band gap, and ETL electron affinity are the most influential descriptors, providing physical insights into interfacial recombination and charge extraction. The machine learning model identifies several high-performance configurations that are subsequently verified by full SCAPS-1D simulations. Among them, the device FTO/TiO$_2$/Cs$_2$AgBiBr$_6$/NiO/Ag achieves a PCE of 28.85%, and the ML-suggested structure FTO/SnO$_2$/Cs$_2$AgInBr$_6$/NiO/Ag exhibits 28.62%, outperforming a closely related literature architecture by approximately 4% absolute. Notably, eight of the top-11 structures employ the lead-free double perovskite Cs$_2$AgInBr$_6$. This work demonstrates that a physics-based, data-driven workflow combining SCAPS-1D, ML, and SHAP can accelerate the discovery of high-efficiency, environmentally friendly perovskite solar cells while providing transparent design rules. The approach is generalizable to other multilayer optoelectronic systems.

We solve exactly the relativistic finite-difference equation for the quantum three-dimensional ring-shaped Quesne oscillator potential. Our investigation is based on a finite-difference version of relativistic quantum mechanics. So-called relativistic configurational r-space is a key concept here. We show that the radial wavefunctions and angular wavefunctions are expressed through the continuous dual Hahn polynomials and Jacobi polynomials, respectively. A discrete energy spectrum has been found. The radial wave functions and energy spectrum have the correct nonrelativistic limit. We also build a dynamical symmetry group SU (1, 1) for the radial part of the equation of motion, which allows us to find the energy spectrum purely algebraically.

In quantum mechanics, a measurement applied to a state in general produces some amount of intrinsic randomness. This is not only a fundamental feature of the theory, but is also at the basis of any quantum process to generate random numbers. The simplest of such processes consists of a single, fully charaterized, measurement acting on a single, fully characterized, state. Unfortunately, no general method to estimate the intrinsic randomness produced in such setups is known. In this work, we address this issue by presenting a semidefinite programming formulation of the maximum probability with which an adversary, Eve, can guess the outcomes of characterized but untrusted prepare-and-measure setups. We then present several applications of this construction. First, we apply our method to a variety of specific setups, allowing us both to benchmark the approach and, more importantly, to determine the exact amount of certifiable randomness in scenarios where only upper bounds were previously available. Then, we show that the presence of entanglement between the device preparing the state and the measurement strictly increases Eve's predictive power, already in the most elementary setup of a binary measurement acting on a qubit state.

The population of Compton-thick (CTK) AGN represents a critical yet elusive phase in the growth of supermassive black holes. Constraining their abundance and evolution at high z is essential for understanding both SMBH growth and the origin of the cosmic X-ray background. We investigate the X-ray luminosity function (XLF) of CTK AGN at z > 3 using one of the largest available samples of X-ray-selected AGN at high z, containing 811 sources from XMM-Newton XXL-N and Chandra CCLS and CDF-S/N surveys. We first selected a subsample of ten high-probability CTK candidates, identified through x-ray spectral fitting. Their multiwavelength properties are examined through SED modelling to assess the robustness of their CTK classification. For most sources, the inferred X-ray luminosities appear overestimated when compared with their IR luminosities. After updating the NH posteriors with IR-informed priors, only three sources remain consistent with the CTK regime. To compute the XLF for the entire CTK AGN population, we used 24 microns photometry to estimate IR luminosities and update the X-ray posteriors for all the remaining sources. Incorporating IR priors systematically reduces the inferred CTK number densities, yielding a more conservative and physically consistent estimate of the XLF. We find that CTK AGN constitute 17 per cent of the total AGN population at 3 < z < 6, consistent with results at lower z. Our analysis reveals no statistically significant evolution in the CTK fraction up to z about 6, suggesting that the most heavily obscured accretion phase remains a persistent component of black hole growth throughout cosmic history. While the overall obscured AGN fraction (NH > 1e23 cm-2) increases toward higher redshifts, the stable CTK fraction supports the interpretation that, at these epochs, the interstellar medium in typical host galaxies cannot produce CTK levels of obscuration.

Mirrors that preserve the handedness of optical radiation upon reflection are an essential building block for the design of numerous resonant nanophotonic structures with capabilities for enantiomeric discrimination. Ordinary metallic and Bragg dielectric mirrors are not suitable in these context since they flip handedness of electromagnetic field upon reflection around normal incidence. While there has been considerable progress in the development of such reflecting structures, this research area remains largely unexplored. Here, we present a detailed numerical and theoretical analysis of the potential of low-symmetry periodic lattices composed of high-symmetry non-chiral meta-atoms for resonant reflection with handedness preservation (HP). Using full-wave numerical simulations, we analyze a family of rhombic and monoclinic (oblique) lattices of circular dielectric disks and/or holes, and in each identify the regime of near-perfect HP reflection. We study the robustness of these structures to geometric deviations, material losses, and incidence angle. Finally, we describe the resonant HP response of these structures using the coupled-mode theory formalism.

Particulate contaminants decrease the power output of solar panels, the transparency of windows, and are detrimental to microelectronics, where even a single particle can induce a short circuit. Despite significant research on particle adhesion and self-cleaning, it remains unclear when and how a drop can remove a particle from a surface, thus efficiently cleaning the surface. Here, by combining lattice Boltzmann simulations and confocal microscopy experiments, we show that at least six different scenarios arise from the complex interplay between capillary and friction forces when a drop collides with a particle. Notably, the capillary force plays a dual role in particle removal: while its tangential component always drives removal, its normal component can also hinder it. By introducing a dimensionless capillary capture parameter, we can predict particle removal across a wide range of particle and surface properties. These results provide quantitative design principles for easy-to-clean surfaces that minimize water and chemical usage.

The harmonic oscillator is considered within generalized frameworks using complex and quaternionic numbers. The classical oscillator is considered in terms of a complex position function, and quantum oscillators are examined in terms of complex wave functions, and in terms of quaternionic wave functions as well. Both of the quantum solutions are obtained within the real Hilbert space formalism. The results reveal the complex and quaternionic descriptions as suitable frameworks for non-stationary processes, including damped oscillations, forced oscillations, and additionally self-interacting processes that cannot be appropriately described otherwise.

We address transformer attention on energy-constrained physical substrates. Softmax attention requires exponentiation and global reduction, operations with high energy cost on von Neumann hardware and no natural physical analog. We show that Kuramoto synchronization dynamics (which arise in electrical, mechanical, superconducting, and charge-density-wave oscillator arrays, among other physical systems) implement a well-defined attention operation without either. The resulting mechanism, fixed-query oscillator attention, replaces softmax's arithmetic with the equilibration of a gradient flow on the sphere: queries are learned anchors fixed on the sphere, and free oscillators evolve under Kuramoto-Lohe dynamics until they settle at positions encoding attention weights via cosine similarity. Because the computation is equilibration, it requires no exponentiation; the only global operation is an affine normalization at readout. The fixed point is provably unique and globally attractive from almost every initial condition, a guarantee that holds across every physical realization. Empirically, at the minimal hardware configuration (oscillator dimension $d_{\mathrm{osc}}$ = 2), oscillator attention outperforms softmax on keyword spotting (+1.00 pp) and on subject-verb agreement (+5.27 pp on hard sentences, with zero training failures versus one in five for softmax). On causal language modeling, where softmax retains an advantage, oscillator attention closes the gap as $d_{\mathrm{osc}}$ grows: from +11.09 PPL at $d_{\mathrm{osc}}$ = 2 to +2.98 PPL at $d_{\mathrm{osc}}$ = 32 on WikiText-2, and from +2.39 PPL at $d_{\mathrm{osc}}$ = 2 to +0.57 PPL at $d_{\mathrm{osc}}$ = 32 on TinyStories. The main objective of this work is not to replace softmax in software but to provide a mathematically grounded blueprint for accurate attention on physical substrates.

Attention is the key mechanism underlying in-context learning in transformers, and attention patterns have been observed empirically to emerge abruptly during training. We present a Bayesian theory of feature learning in attention; we then focus on how the copy subcircuit in the first layer of an induction head is learned by analyzing a single-layer softmax attention network trained on a copy task. We derive a closed-form posterior over the attention matrix and reduce it to a low-dimensional order parameter space. This reduction reveals a phase transition in the amount of training data, which we verify using both Bayesian sampling and standard training with Adam. We contrast our results with linear attention and find that softmax attention exhibits a \emph{first-order phase transition} while in linear attention an initial \emph{second-order phase transition} is followed by a smooth, continuous evolution toward the structured attention pattern (\emph{crossover}). Our work provides a first-principles theoretical account of the abrupt emergence of the copy subcircuit, reminiscent of the one observed in training large language models.

Entanglement is a key bottleneck limiting the efficiency of tensor-network and quantum simulations of molecular electronic structures. Here, we systematically assess and extend Clifford disentanglers as a structure-preserving approach to entanglement reduction: they can modify the entanglement structure of qubit wavefunctions while retaining the Pauli-string form of qubit Hamiltonians. To enable a practical search over Clifford transformations, we classify Clifford operators by their action on the Schmidt spectrum across a bipartition, reducing the two- and four-qubit search spaces to 20 and 91392 representatives, respectively. Embedded in an iterative Clifford-augmented matrix product state framework, these transformations reduce the energy errors at fixed bond dimension for the molecular test cases studied and mitigate the dependence on orbital orderings and fermion-to-qubit mappings. We further show that Clifford disentanglers can also benefit quantum simulations such as the shallow-circuit variational quantum eigensolver calculations. Together, these results establish Clifford disentanglers as a useful structure-preserving entanglement-engineering tool for tensor-network and quantum simulations of molecular electronic structure, while also clarifying their correlation dependence and motivating future developments.

This Letter presents a novel analytical framework showing that the dark count probability (PDC) of perimeter-gated single-photon avalanche diodes (pg-SPADs) follows a complementary Gompertz function. Specifically, we show that PDC follows a complementary Gompertz form from which we derive a pixel-specific descriptor, the midpoint perimeter gate voltage, which characterizes a pixel's equiprobable operating point. We further show that a perimeter gate voltage compensation rate may be obtained from this descriptor to offset temperature-induced changes in the pixel's activation function. The proposed framework is experimentally validated using 4,096 pg-SPADs arranged in a 64 x 64 array and manufactured in a 0.35 $\mu$m CMOS process. The devices were characterized at temperatures ranging from -5 $^o$C to 55 $^o$C and perimeter gate voltage magnitudes of 0 to 5 V. The measured results demonstrate deterministic bias control of dark count probability across process and temperature variations.

We develop a Heisenberg-Langevin theory of an exciton mirror accounting for the retardation and the long-range electron-hole exchange. A particular case of a strong transverse magnetic field is analyzed in detail. The optical bistability due to repulsion between the excitons inside the light cone appears to be prone to a modulational instability towards the non-radiative surface polariton modes. Above the corresponding threshold, the pumped 2D exciton gas acts as an optical parametric generator of twin polariton beams. Conversely, below the threshold, the mirror acquires the phase-conjugating properties.