Many biological, culinary, and engineering processes lead to the coencapsulation of several soft particles within a liquid interface. In these situations the particles are bound together by the capillary forces that deform them and influence their biological or rheological properties. Here, we introduce an experimental approach to encapsulate a controlled number of soft beads within aqueous droplets in oil. These droplet-encapsulated gels are manipulated in a deformable microfluidic device to merge them and modify the liquid fraction. In the dry limit the contact surface between the hydrogels is found to be determined by the elastocapillary number $E_c$, with the contact radius following a $E_c^{1/3}$ dependence, indicating that the deformation increases for soft or small particles. When multiple beads are coencapsulated within a single droplet they can be arranged into linear or three-dimensional aggregates that remain at a local energy minimum.

Magic states are a scarce resource for two-dimensional qubit stabilizer codes. Magic state cultivation was recently proposed to reduce the cost of magic state preparation by measuring the transversal Clifford operator of the color code. Cultivation achieves $\sim {10}^{-9}$ logical error rates for the $d=5$ color code, with substantially lower space-time overhead than magic state distillation. However, due to the $\mathcal{O}(d)$ depth of the Clifford measurement circuit, magic state cultivation becomes impractical for $d>5$. Here, we perform logical $X{S}^{†}$ measurements on the color code by gauging a transversal Clifford gate, resulting in a constant-depth logical measurement circuit. We employ repeated gauging measurements with postselection rather than performing error correction on the Clifford stabilizer code that emerges during the gauging protocol, thus gaining simplicity at the cost of scalability. Our protocol requires a regular square grid connectivity and yields logical error rates comparable to magic state cultivation. The $d=7$ version of our protocol gives access to the ${10}^{-12}$ logical error rate regime at 0.05% physical error rate while retaining more than 1% of the shots after the equivalent of the cultivation stage.

The observation of the associated production of an $\mathrm{\Upsilon }(1\mathrm{S})$ meson with a $Z$ boson and a measurement of the ratio of its fiducial cross section to the fiducial cross section of the $Z$ boson are presented. Both the $\mathrm{\Upsilon }(1\mathrm{S})$ meson and the $Z$ boson are identified via decays into a pair of opposite-sign muons. The analysis is based on proton-proton ($pp$) collision data at $\sqrt{s}=13\mathrm{TeV}$ collected with the CMS detector in 2016–2018 and corresponding to an integrated luminosity of $138{\mathrm{fb}}^{-1}$. Using the production of the $Z$ boson decaying into four muons as a normalization channel, the ratio of the fiducial cross sections $\sigma [pp\to Z+\mathrm{\Upsilon }(1\mathrm{S})]\mathcal{B}(Z\to {\mu }^{+}{\mu }^{-})\mathcal{B}[\mathrm{\Upsilon }(1\mathrm{S})\to {\mu }^{+}{\mu }^{-}]$ to $\sigma (pp\to Z)\mathcal{B}(Z\to 4\mu )$ is measured to be ${\mathcal{R}}_{Z+\mathrm{\Upsilon }(1\mathrm{S})}=[21.1\pm 5.5(\mathrm{stat})\pm 0.6(\mathrm{syst})]\times {10}^{-3}$, where stat and syst denote the statistical and systematic uncertainties, respectively. The result is used to extract the effective double-parton scattering cross section ${\sigma }_{\mathrm{eff}}={13.0}_{-3.4}^{+7.7}\mathrm{mb}$. In addition, for the first time, ${\sigma }_{\mathrm{eff}}$ is measured in bins of the transverse momentum of the $\mathrm{\Upsilon }(1\mathrm{S})$ meson or of the $Z$ boson.

The pursuit of programmable light sources with high coherence and spectral purity hinges on establishing a precise set of phase relationships in light-matter interactions. Here, we demonstrate that the quadratic dispersion of a freely propagating electron wave packet serves as a programmable quantum medium. Prepared in a coherent momentum-state ladder via a single laser interaction, the electron subsequently undergoes deterministic phase evolution during free propagation—an intrinsic process that compiles its quantum state into two distinct output channels. This mechanism, quantified by a quantum bunching factor, enables (i) Talbot-resonant bunching, where the electron density self-structures into subcycle combs with tunable harmonic selectivity, and (ii) coherent phase transfer of the programmed quadratic phase to light, generating nonclassical photon states such as multicomponent Schrödinger cat states via energy-resolved electron readout. This quadratic-phase programming provides a drift-section platform for Talbot-selected coherent short-wavelength emission and nonclassical light synthesis based on electron self-interference.

Nonlinearity at the single-photon level is important in both fundamental research and applications of quantum optics. Strong single-photon nonlinearity is usually hard to achieve with a single atom coupled to a cavity mode. Unfortunately, stronger nonlinearity cannot be achieved by coupling more atoms straightforwardly because of the gradual homogeneity of the system energy levels. Here we propose two proposals to realize deeper photon blockade with the increase of the atom number with the help of quantum Zeno dynamics. The single-photon blockade that manifests single-photon nonlinearity is achieved with high single-photon efficiency and purity. We developed a hybrid method by partitioning the system states into two subsets with one computed via a master equation and the other via a non-Hermitian Hamiltonian. This method significantly reduces computational resource requirements and enables simulations of systems with up to thousands of atoms. The results show that our proposals are robust against the inhomogeneity of atom frequency and coupling strength, which is favorable in experiments.

Liquid interfaces appear throughout nature and engineering and are typically contaminated by surface active agents (surfactants), which are characterized by a wide range of solubility. We demonstrate that solubility enhances by an order of magnitude surfactant-induced flow in air-liquid-air films, in contrast to previously studied geometries where solubility dampens the flow. The enhancement is described by a single parameter comparing the depletion length to the film thickness. Our experiments are well described by an asymptotic theory of the Navier-Stokes equations, including surfactant kinetics.

The pressure-induced phase transformation of semiconductors, particularly silicon (Si), has long been a fundamental problem in condensed matter physics. Decades of research have predominantly focused on external factors such as temperature, load, and unloading rate, leaving the role of inherent electronic properties fundamentally unresolved. This Letter addresses this gap by demonstrating that hole doping (i.e., hole concentration) is a critical governing factor in the pressure-induced phase transformation. We experimentally establish, for the first time, that high hole-doping redirects the phase transformation by facilitating the amorphization of Si. Our simulations powered by carrier-dependent interatomic potentials reveal that the three-coordinated Si atoms are dynamically stabilized by the localized holes during the decompression from $\beta$-tin (Si-II) phase, irreversibly driving the collapse of long-range order in Si to facilitate amorphization. Furthermore, we experimentally observe the hole-facilitated amorphization in Ge and SiC, while GaAs shows a somewhat weaker yet consistent trend. Our Letter unveils the hidden role of holes from mere charge carriers to fundamental mediators of structural stability under high pressure, establishing a new perspective on the interplay between electronic and structural properties in semiconductors.

Scanning tunneling microscopy measurements exploiting the natural superstructure modulation of the cuprate superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ (Bi-2212) have revealed a possible correlation between the Cu-apical-O distance ${\delta }_{\mathrm{api}}$ and the superconducting order parameter $m_{\mathrm{SC}}$, as reported recently by O’Mahony et al. [On the electron pairing mechanism of copper-oxide high temperature superconductivity, Proc. Natl. Acad. Sci. U.S.A. 119, e2207449119 (2022)]. These observations were interpreted as evidence for a direct link between superconductivity and the charge-transfer gap, and more broadly revived the long-standing question of the role of apical oxygens in cuprate superconductivity. Using a combination of density-functional theory and cluster dynamical mean-field theory, we compute from first principles the variations of $m_{\mathrm{SC}}$ induced solely by apical oxygen displacement in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CuO}}_{6+\delta }$, Bi-2212, and ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$. The quantitative agreement between our calculations and experiments allows us to unambiguously attribute the observed variations of $m_{\mathrm{SC}}$ to changes in ${\delta }_{\mathrm{api}}$. We demonstrate, however, that these variations of $m_{\mathrm{SC}}$ originate predominantly from changes in the effective hole-doping of the ${\mathrm{CuO}}_2$ planes, with negligible effect on the charge-transfer gap. The modest magnitude of the $m_{\mathrm{SC}}$ modulation induced by apical-oxygen displacement alone warrants caution in interpreting correlations between $T_c$ and ${\delta }_{\mathrm{api}}$ inferred from comparisons across different cuprate compounds. Our work demonstrates that the present ab initio framework can quantitatively resolve the influence of specific structural degrees of freedom on superconductivity in correlated oxides.

In one-dimensional $p$-wave superconductors with short-range interactions, topologically protected Majorana modes emerge, whose mass decays exponentially with system size, as first shown by Kitaev. In this Letter, we extend this prototypical model by including power law long-range interactions within a self-consistent framework, leading to the self-consistent long-range Kitaev chain (seco-LRKC). In this model, the gap matrix acquires a rich structure where short-range superconducting correlations coexist with long-range correlations that are exponentially localized at both chain edges simultaneously. As a direct consequence, the topological edge modes hybridize even if their wave function overlap vanishes, and the edge mode mass inherits the asymptotic scaling of the interaction. In contrast to models with imposed power law pairing, where massive Dirac modes emerge for exponents $\nu <d$, we analytically motivate and numerically demonstrate that, in the fully self-consistent model, algebraic edge mode decay with system size persists for all interaction exponents $\nu >0$, despite exponential wave function localization. While the edge mode remains massless in the thermodynamic limit, finite-size corrections can be experimentally relevant in mesoscopic systems with effective long-range interactions that decay sufficiently slowly. Our findings have direct implications for quantum simulations of ultracold microwave-shielded dipolar molecules in one-dimensional optical lattices.

Andreev reflection converts an incoming fermion into an outgoing hole and is usually tied to a superconducting interface. We show that an analogous charge-conjugating boundary condition emerges from a purely lattice duality defect. Starting from a Majorana representation of the transverse-field Ising chain, we construct a folded lattice model in which a boundary Majorana impurity implements a one-site translation of a staggered Majorana chain. In the continuum, this translation becomes a chiral fermion-parity defect: it flips the sign of the only left-moving Majorana mode while leaving the right-moving mode unchanged. When the two Majorana modes are recombined into a complex fermion in the folded geometry, this sign flip becomes the Andreev-like boundary condition. Our lattice formulation gives a microscopic interpretation of the Emery-Kivelson boundary of the two-channel Kondo problem and of Maldacena-Ludwig monopole scattering, while identifying the boundary as the interface between a Kitaev-chain symmetry-protected topological phase and a gapless chain. The same Majorana translation defect also provides a lattice realization of an axial $\mathrm{U}(1{)}_A$-symmetric charge-flip boundary.

Electronic flat bands have localized Wannier-like orbitals as zero modes. In the Lieb or kagome models, the localized orbitals satisfy a topological condition that entails two noncontractible loop eigenstates along the $x\text{or}y$ axis in real space and one topological band touching point with other bands in momentum space. In these topological flat bands, the Bloch state at the touching point is ill defined and so is any topological invariant for the entire band. We propose a new topological condition in which the loop states in different directions are linearly dependent. Its satisfaction removes the singularity at the band touching point and enforces nontrivial, well-defined topological invariants. Enforcing the new condition, we obtain topological-topological flat (${\mathrm{top}}^2$-flat) bands in 2D and 3D with nontrivial invariants including the Chern number, the ${\mathbb{Z}}_2$ invariants, and the topological-crystalline invariants. Under small, generic interactions, ${\mathrm{top}}^2$-flat bands flow to correlated topological insulators with a symmetric mass term, and specially designed interacting models can have ${\mathrm{top}}^2$-flat bands as exact zero modes.

From the diminishing sound of a plucked guitar string to the quick fading of ripples on water, wave attenuation in lossy media is a ubiquitous phenomenon. Here, we unveil a hidden connection between this familiar effect and a time-domain non-Hermitian topology. We show that a broad class of velocity-dependent dissipation, such as everyday air damping, can essentially break temporal reciprocity by introducing an imaginary gauge field along the time axis. This gauge field opens a point gap in momentum bands and stimulates a temporal analog of the non-Hermitian skin effect, underpinning the topological origin of loss-induced wave attenuation. Despite its simplicity and ubiquity, this temporal nonreciprocity serves as a powerful knob for sculpting both the steady-state and dynamic responses of emergent time crystals, from braiding momentum bands in the frequency Brillouin zone to reshaping interface states in the temporal domain. Moreover, the inherent causality of time grants unprecedented flexibility in controlling boundary conditions and system size—features prohibited in conventional spatial models—as we experimentally demonstrate using a simple mechanical platform.

Real-space singularities govern a broad spectrum of wave phenomena, yet they remain largely unexplored in elastic wave systems. Here, we report hybrid topological singularities that emerge on the surfaces of finite-sized solids due to the full vectorial character of elastic waves. These textures fuse spin-field singularities with displacement-field singularities and exhibit unique nonpairwise topological charge dynamics. Moreover, a subset of these singularities imprint dislocation lines onto the radiated acoustic field, generating robust acoustic vortices in free space from an otherwise achiral source and structure. Our results establish elastic waves as a powerful platform for engineering real-space singularities and open avenues for singular phononics and the exploration of rich topological defects in elastic media.

We explore synchronization transitions in a generalized Kuramoto model with $n$-dimensional vector oscillators on complete graphs (CG) and on $d$-dimensional lattices. On the CG, we analytically derive universal critical exponents $(\beta ,\overline{\nu })=(1/2,5/2)$ for all even $n$. Through the Botet-Jullien-Pfeuty relation, the CG exponent $\overline{\nu }=5/2$ predicts an upper critical dimension $d_u=5$ for the corresponding lattice models. For locally coupled systems with $d<d_u$, we identify frequency entrainment for arbitrary $n$ using a newly developed entrainment connectivity matrix, and characterize the transitions with two complementary order parameters. Using spin-wave theory together with numerical simulations, we obtain critical exponents across the $(n,d)$ parameter space. These results suggest a family of $n$-independent but $d$-dependent universality classes, offering a unified perspective on how symmetry and dimensionality shape synchronization phenomena.

Transitions from reversible to irreversible or fluctuating states above a critical density and shear amplitude have been extensively studied in nonthermal cyclically sheared suspensions and amorphous solids. Here, we propose that the same type of reversible to irreversible transition occurs for a system of particles with competing short-range attraction and long-range repulsion, which can form crystals, stripes, and bubbles as the ratio of attraction to repulsion varies. By oscillating the strength of the attractive part of the potential, we find that the system can organize into either time-periodic states consisting of nondiffusive complex closed orbits, or into a diffusive fluctuating state. A critical point separates these states as a function of the maximum strength of the attraction, oscillation frequency, and particle density. We also find a nonmonotonic behavior of the reversible state as a function of the strength of the attraction and the oscillation frequency.

Shadow estimation is a powerful framework for predicting quantum properties from randomized measurements. While 3-design protocols achieve optimal worst-case performance, the minimal number of measurement bases required for such optimality has remained open. Here we prove that $\mathrm{\Theta }(d^2)$ measurement bases are both necessary and sufficient for worst-case optimal shadow estimation for a $d$-dimensional system and construct an explicit basis family. In stark contrast, any state 2-design already suffices for average-case optimality: the mean squared shadow norm of normalized observables is bounded by a universal constant, and we prove strong concentration for Haar-random states, yielding constant sample complexity for generic pure-state fidelity estimation. Easily implementable 2-designs—from mutually unbiased bases, cyclic measurements, or shallow $\mathcal{O}(\log n)$-depth circuits on $n$ qubits—enable optimal average-case protocols with remarkably simple measurement strategies. Our results establish a fundamental complexity separation: worst-case estimation requires $\mathrm{\Theta }(d^2)$ bases, whereas average-case performance requires only $\mathrm{\Theta }(d)$ bases, with broad implications for quantum information theory and near-term experiments.