The bias-variance trade-off is a central concept in supervised learning. In classical statistics, increasing the complexity of a model (e.g., number of parameters) reduces bias but also increases variance. Until recently, it was commonly believed that optimal performance is achieved at intermediate model complexities which strike a balance between bias and variance. Modern deep learning methods flout this dogma, achieving state-of-the-art performance using “overparameterized models” where the number of fit parameters is large enough to perfectly fit the training data. As a result, understanding bias and variance in overparameterized models has emerged as a fundamental problem in machine learning. Here, we use methods from statistical physics to derive analytic expressions for bias and variance in two minimal models of overparameterization (linear regression and two-layer neural networks with nonlinear data distributions), allowing us to disentangle properties stemming from the model architecture and random sampling of data. In both models, increasing the number of fit parameters leads to a phase transition where the training error goes to zero and the test error diverges as a result of the variance (while the bias remains finite). Beyond this threshold, the test error of the two-layer neural network decreases due to a monotonic decrease in both the bias and variance as opposed to the classical bias-variance trade-off. We also show that in contrast with classical intuition, overparameterized models can overfit even in the absence of noise and exhibit bias even if the student and teacher models match. We synthesize these results to construct a holistic understanding of generalization error and the bias-variance trade-off in overparameterized models and relate our results to random matrix theory.

The nature of dark matter remains obscure in spite of decades of experimental efforts. The mass of dark matter candidates can span a wide range, and its coupling with the Standard Model sector remains uncertain. All these unknowns make the detection of dark matter extremely challenging. Ultralight dark matter, with $m\sim {10}^{-22}$ eV, is proposed to reconcile the disagreements between observations and predictions from simulations of small-scale structures in the cold dark matter paradigm while remaining consistent with other observations. Because of its large de Broglie wavelength and large local occupation number within galaxies, ultralight dark matter behaves like a coherently oscillating background field with an oscillating frequency dependent on its mass. If the dark matter particle is a spin-1 dark photon, such as the $U{\left(1\right)}_B$ or $U{\left(1\right)}_{B-L}$ gauge boson, it can induce an external oscillating force and lead to displacements of test masses. Such an effect would be observable in the form of periodic variations in the arrival times of radio pulses from highly stable millisecond pulsars. In this study, we search for evidence of ultralight dark photon dark matter (DPDM) using 14-year high-precision observations of 26 pulsars collected with the Parkes Pulsar Timing Array. While no statistically significant signal is found, we place constraints on coupling constants for the $U{\left(1\right)}_B$ and $U{\left(1\right)}_{B-L}$ DPDM. Compared with other experiments, the limits on the dimensionless coupling constant $\epsilon$ achieved in our study are improved by up to two orders of magnitude when the dark photon mass is smaller than $3\times {10}^{-22}$ eV (${10}^{-22}$ eV) for the $U{\left(1\right)}_B$ ($U{\left(1\right)}_{B-L}$) scenario.

Applying the strong pinning formalism to the mixed state of a type-II superconductor, we study the effect of thermal fluctuations (or creep) on the penetration of an ac magnetic field as quantified by the so-called Campbell length ${\lambda }_{\mathrm{C}}$. Within strong pinning theory, vortices get pinned by individual defects, with the jumps in the pinning energy $\left(\mathrm{\Delta }{e}_{\mathrm{pin}}\right)$ and force $\left(\mathrm{\Delta }{f}_{\mathrm{pin}}\right)$ between bistable pinned and free states quantifying the pinning process. We find that the evolution of the Campbell length ${\lambda }_{\mathrm{C}}\left(t\right)$ as a function of time $t$ is the result of two competing effects, the change in the force jumps $\mathrm{\Delta }{f}_{\mathrm{pin}}\left(t\right)$ and a change in the trapping area $S_{\mathrm{trap}}\left(t\right)$ of vortices; the latter describes the area around the defect where a nearby vortex gets and remains trapped. Contrary to naive expectation, we find that during the decay of the critical state in a zero-field cooled (ZFC) experiment, the Campbell length ${\lambda }_{\mathrm{C}}\left(t\right)$ is usually nonmonotonic, first decreasing with time $t$ and then increasing for long waiting times. Field cooled (FC) experiments exhibit hysteretic effects in ${\lambda }_{\mathrm{C}}$; relaxation then turns out to be predominantly monotonic, but its magnitude and direction depend on the specific phase of the cooling-heating cycle. Furthermore, when approaching equilibrium, the Campbell length relaxes to a finite value, different from the persistent current, which vanishes at long waiting times $t$, e.g., above the irreversibility line. Finally, measuring the Campbell length ${\lambda }_{\mathrm{C}}\left(t\right)$ for different states, zero-field cooled, field cooled, and relaxed, as a function of different waiting times $t$ and temperatures $T$, allows to spectroscopyse the pinning potential of the defects.

While memory effects emerge from systems of wildly varying lengthscales and timescales, the reduction of a complex system with many interacting elements into one simple enough to be understood without also losing the complex behavior continues to be a challenge. Here, we investigate how bistable cylindrical origamis provide such a reduction via tunably interactive memory behaviors. We base our investigation on folded sheets of Kresling patterns that function as two-state memory units. By linking several units, each with a selected activation energy, we construct a one-dimensional material that exhibits return-point memory. After a comprehensive experimental analysis of the relation between the geometry of the pattern and the mechanical response for a single bit, we study the memory of a bellows composed of four bits arranged in series. Since these bits are decoupled, the system reduces to the Preisach model, and we can drive the bellows to any of its 16 allowable states by following a prescribed sequence of compression and extension. We show how to reasonably discriminate between states by measuring the system's total height and stiffness near equilibrium. Furthermore, we establish the existence of geometrically disallowed defective stable configurations that expand the configuration space to 64 states with a more complex transition pattern. Using empirical considerations of the mechanics, we analyze the hierarchical structure of the corresponding diagram, which includes Garden of Eden states and subgraphs. We highlight two irreversible transformations, namely shifting and erasure of defects, leading to memory behaviors reminiscent of those observed with more complex glassy systems.

Topological valley-Hall edge states have been realized in a variety of photonic structures across the electromagnetic spectrum because they can be easily engineered by breaking certain lattice symmetries. However, the valley-Chern numbers that characterize the topological phase are usually fixed by the symmetry-breaking perturbation and therefore the valley-Hall edge states are forced to propagate in a fixed direction. Here we consider a kagome metasurface comprised of a subwavelength array of dipole emitters/antennas, and we unveil that one can modify the geometrical and topological properties of the polaritons by structuring the local photonic environment. As a proof of concept, we show that one can induce topological transitions via accidental Dirac points by embedding the metasurface inside a cavity waveguide. Varying the cavity width modifies the nature of the dipole-dipole interactions which enables one to manipulate the Berry curvature and invert the valley-Chern numbers without inverting the symmetry-breaking perturbation. Consequently, we demonstrate that one can switch the chirality of the polariton valley-Hall edge states by varying only the cavity width. This alternative approach to engineering topological transitions via structuring the photonic environment could also have implications for other topological phases such as photonic higher-order topological insulators.

A single-spin qubit placed near the surface of a conductor acquires an additional contribution to its $1/T_1$ relaxation rate due to magnetic noise created by electric current fluctuations in the material. We analyze this technique as a wireless probe of superconductivity in atomically thin two-dimensional materials. At temperatures $T\lesssim T_c$, the dominant contribution to the qubit relaxation rate is due to transverse electric current fluctuations arising from quasiparticle excitations. We demonstrate that this method enables detection of metal-to-superconductor transitions, as well as investigation of the symmetry of the superconducting gap function, through the noise scaling with temperature. We show that scaling of the noise with sample-probe distance provides a window into the nonlocal quasistatic conductivity of superconductors, both clean and disordered. At low temperatures the quasiparticle fluctuations get suppressed, yet the noise can be substantial due to resonant contributions from collective longitudinal modes, such as plasmons in monolayers and Josephson plasmons in bilayers. Potential experimental implications are discussed.

We show that the nonlinear stochastic dynamics of a measurement-feedback-based coherent Ising machine (MFB-CIM) in the presence of quantum noise can be exploited to sample degenerate ground and low-energy spin configurations of the Ising model. We formulate a general discrete-time Gaussian-state model of the MFB-CIM, which faithfully captures the nonlinear dynamics present at and above system threshold. This model overcomes the limitations of both mean-field models, which neglect quantum noise, and continuous-time models, which assume long photon lifetimes. Numerical simulations of our model show that when the MFB-CIM is operated in a quantum-noise-dominated regime with short photon lifetimes (i.e., low cavity finesse), homodyne monitoring of the system can efficiently produce samples of low-energy Ising spin configurations, requiring many fewer roundtrips to sample than suggested by established high-finesse, continuous-time models. We find that sampling performance is robust to, or even improved by, turning off or altogether reversing the sign of the parametric drive, but performance is critically reduced in the absence of optical nonlinearity. For the class of MAX-CUT problems with binary-signed edge weights, the number of roundtrips sufficient to fully sample all spin configurations up to the first-excited Ising energy, including all degeneracies, scales with the problem size $N$ as $1.{08}^N$. At $N=100$ with a few dozen (median $\sim 20$) such desired configurations per instance, we have found median sufficient sampling times of $6\times {10}^6$ roundtrips; in an experimental implementation of an MFB-CIM with a 10 GHz repetition rate, this corresponds to a wall-clock sampling time of 60 ms.

We study the reduction of the supercurrent by a gate electrode in a purely metallic superconductor–normal metal–superconductor Josephson junction by performing, on the same device, a detailed investigation of the gate-dependent switching probability together with the local tunneling spectroscopy of the normal metal. We demonstrate that high energy electrons leaking from the gate trigger the reduction of the critical current which is accompanied by an important broadening of the switching histograms. The switching rates are well described by an activation formula including an additional term accounting for the injection of rare high energy electrons from the gate. The rate of electrons obtained from the fit remarkably coincides with the independently measured leakage current. Concomitantly, a negligible elevation of the local temperature in the junction is found by tunneling spectroscopy which excludes stationary heating induced by the leakage current as a possible explanation of the reduction of the critical current. This incompatibility is resolved by the fact that phase dynamics and thermalization effects occur at different timescales.

We discuss the stochastic trajectories of single molecules in a phase-separated liquid, when a dense and a dilute phase coexist. Starting from a continuum theory of macroscopic phase separation we derive a stochastic Langevin equation for molecular trajectories that takes into account thermal fluctuations. We find that molecular trajectories can be described as diffusion with drift in an effective potential, which has a steep gradient at phase boundaries. We discuss how the physics of phase coexistence affects the statistics of molecular trajectories and in particular the statistics of displacements of molecules crossing a phase boundary. At thermodynamic equilibrium detailed balance imposes that the distributions of displacements crossing the phase boundary from the dense or from the dilute phase are the same. Our theory can be used to infer key phase separation parameters from the statistics of single-molecule trajectories. For simple Brownian motion, there is no drift in the presence of a concentration gradient. We show that interactions in the fluid give rise to an average drift velocity in concentration gradients. Interestingly, under non-equilibrium conditions, single molecules tend to drift uphill the concentration gradient. Thus, our work bridges between single-molecule dynamics and collective dynamics at macroscopic scales and provides a framework to study single-molecule dynamics in phase-separating systems.

We explore topological edge states in periodically driven nonlinear systems. Based on a self-consistency method adjusted to periodically driven systems, we obtain stationary states associated with topological phases unique to Floquet systems. In addition, we study the linear stability of these edge states and reveal that Floquet stationary edge states experience a sort of transition between two regions, I and II, in which lifetimes of these edge states are extremely long and short, respectively. We characterize the transitions in lifetimes by Krein signatures, or equivalently the pseudo-Hermiticity breaking, and clarify that the transitions between regions I and II are signified by collisions of edge-dominant eigenstates of Floquet operators for fluctuations. We also analyze the effects of random potentials and clarify that lifetimes of various stationary edge states are equalized due to the randomness-induced mixing of edge- and bulk-dominant eigenstates. This intriguing phenomenon originating from a competition between nonlinearity and randomness results in random potentials prolonging lifetimes in region II and vice versa in region I. These changes of lifetimes induced by nonlinear and/or random effects should be detectable in experiments by preparing initial states akin to edge states.

We propose a cooling scheme for realizing single-photon sideband cooling on particles trapped in a state-dependent optical potential. We develop a master rate equation from an ab initio model and find that in experimentally feasible conditions it is possible to reduce the average occupation number of the vibrational levels in one dimension by more than $90\%$ by applying a frequency sweep on the cooling laser that sequentially cools all the motional states. Notably, this cooling scheme works also when a particle experiences a deeper trap in its internal ground state than in its excited state, a condition for which conventional single-photon sideband cooling does not work. In our analysis, we consider two cases: a two-level particle confined in an optical tweezer, and Li atoms confined in an optical lattice, and we find conditions for efficient cooling in both cases. The results from the model are confirmed by a full quantum Monte Carlo simulation of the system Hamiltonian. Our findings provide an alternative cooling scheme that can be applied in principle to any particle, e.g., atoms, molecules, or ions, confined in a state-dependent optical potential.

The H1 photonic crystal cavity supports two degenerate dipole modes of orthogonal linear polarization which could give rise to circularly polarized fields when driven with a $\pi$/2 phase difference. However, fabrication errors tend to break the symmetry of the cavity, which lifts the degeneracy of the modes, rendering the cavity unsuitable for supporting circular polarization. We demonstrate numerically a scheme that induces chirality in the cavity modes, thereby achieving a cavity that supports intrinsic circular polarization. By selectively modifying two air holes around the cavity, the dipole modes could interact via asymmetric coherent backscattering, which is a non-Hermitian process. With suitable air hole parameters, the cavity modes approach the exceptional point, coalescing in frequencies and linewidths as well as giving rise to significant circular polarization close to unity. The handedness of the chirality can be selected depending on the choice of the modified air holes. Our results highlight the prospect of using the H1 photonic crystal cavity for chiral light-matter coupling in applications such as valleytronics, spin-photon interfaces, and the generation of single photons with well-defined spins.

The interplay between charge-density-wave (CDW) order and superconductivity (SC) in the kagome metal $\mathrm{Rb}{\mathrm{V}}_3{\mathrm{Sb}}_5$ is studied by tracking the evolutions of their transition temperatures $T^{*}$ and $T_{\mathrm{c}}$ as a function of pressure ($P$) via measurements of resistivity and magnetic susceptibility under various hydrostatic pressures up to ∼5 GPa. It is found that the CDW order at $T^{*}$ experiences a subtle modification at $P_{\mathrm{c}1}\approx 1.5\mathrm{GPa}$ before it is completely suppressed around $P_{\mathrm{c}2}\approx 2.4\mathrm{GPa}$. Accordingly, the superconducting transition $T_{\mathrm{c}}\left(P\right)$ exhibits a shallow M-shaped double superconducting dome with two extrema of $T_{\mathrm{c}}^{\mathrm{onset}}\approx 4.4$ and 3.9 K around $P_{\mathrm{c}1}$ and $P_{\mathrm{c}2}$, respectively, leading to a fourfold enhancement of $T_{\mathrm{c}}$ with respect to that at ambient pressure. The constructed $T-P$ phase diagram of $\mathrm{Rb}{\mathrm{V}}_3{\mathrm{Sb}}_5$ resembles that of $\mathrm{Cs}{\mathrm{V}}_3{\mathrm{Sb}}_5$ and shares similar features to many other unconventional superconducting systems with intertwined competing electronic orders. The strong competition between CDW and SC is also evidenced by the broad superconducting transition width in the coexistent region. Our results shed more light on the intriguing physics involving intertwined electronic orders in this topological kagome metal family.

Nitrogen-vacancy (NV) centers in diamond have emerged as promising room-temperature quantum sensors for probing condensed matter phenomena ranging from spin liquids, two-dimensional (2D) magnetic materials, and magnons to hydrodynamic flow of current. Here we propose and demonstrate that the nitrogen-vacancy center in diamond can be used as a quantum sensor for detecting the photonic spin density, the spatial distribution of light's spin angular momentum. We exploit a single spin qubit on an atomic force microscope tip to probe the spinning field of an incident Gaussian light beam. The spinning field of light induces an effective static magnetic field in the single spin qubit probe. We perform room-temperature sensing using Bloch sphere operations driven by a microwave field (XY8 protocol). This nanoscale quantum magnetometer can measure the local polarization of light in ultra-sub-wavelength volumes. We also put forth a rigorous theory of the experimentally measured phase change using the NV center Hamiltonian and perturbation theory involving only virtual photon transitions. The direct detection of the photonic spin density at the nanoscale using NV centers in diamond opens interesting quantum metrological avenues for studying exotic phases of photons, nanoscale properties of structured light as well as future on-chip applications in spin quantum electrodynamics.

In colloidal systems, Brownian motion emerges from the massive separation of time and length scales associated with characteristic dynamics of the solute and solvent constituents. This separation of scales produces several temporal regimes in the colloidal dynamics when combined with the effects of the interaction between the particles, confinement conditions, and state variables, such as density and temperature. Some examples are the short- and long-time regimes in two- and three-dimensional open systems and the diffusive and subdiffusive regimes observed in the single-file (SF) dynamics along a straight line. In this paper, we address the way in which a confining geometry induces new time scales. We report on the dynamics of interacting colloidal particles moving along a circle by combining a heuristic theoretical analysis of the involved scales, Brownian dynamics computer simulations, and video-microscopy experiments with paramagnetic colloids confined to lithographic circular channels subjected to an external magnetic field. The systems display four temporal regimes in the following order: one-dimensional free diffusion, SF subdiffusion, free-cluster rotational diffusion, and the expected saturation due to the confinement. We also report analytical expressions for the mean-square angular displacement and crossover times obtained from scaling arguments, which accurately reproduce both experiments and simulations. Our generic approach can be used to predict the long-time dynamics of many other confined physical systems.

Quantum technologies that rely on photonic qubits require a precise controllability of their properties. For this purpose hybrid approaches are particularly attractive because they offer a large flexibility to address different aspects of the photonic degrees of freedom. When combining photonics with other quantum platforms like phonons, quantum transducers have to be realized that convert between the mechanical and optical domain. Here, we realize this interface between phonons in the form of surface acoustic waves (SAWs) and single photons, mediated by a single semiconductor quantum dot exciton. In this combined theoretical and experimental study, we show that the different sidebands exhibit characteristic blinking dynamics that can be controlled by detuning the laser from the exciton transition. By developing analytical approximations we gain a better understanding of the involved internal dynamics. Our specific SAW approach allows us to reach the ideal frequency range of around 1 GHz that enables simultaneous temporal and spectral phonon sideband resolution close to the combined fundamental time-bandwidth limit.

We investigate a quite strong turbulence in a supercritical fluid near a gas-liquid critical point. Specifically, we consider a case in which the Kolmogorov scale is much smaller than the equilibrium correlation length $\xi$. Although equilibrium critical fluctuations are destroyed by turbulence, $\xi$ still provides a crossover length scale between two types of energy cascade. At scales much larger than $\xi$, the Richardson cascade becomes dominant, whereas at scales much smaller than $\xi$, another type of cascade, which we call the van der Waals cascade, is induced by density fluctuations. Experimental conditions required to observe the van der Waals cascade are also discussed.

The quantum computation of electronic energies can break the curse of dimensionality that plagues many-particle quantum mechanics. It is for this reason that a universal quantum computer has the potential to fundamentally change computational chemistry and materials science, areas in which strong electron correlations present severe hurdles for traditional electronic structure methods. Here we present a state-of-the-art analysis of accurate energy measurements on a quantum computer for computational catalysis, using improved quantum algorithms with more than an order of magnitude improvement over the best previous algorithms. As a prototypical example of local catalytic chemical reactivity we consider the case of a ruthenium catalyst that can bind, activate, and transform carbon dioxide to the high-value chemical methanol. We aim at accurate resource estimates for the quantum computing steps required for assessing the electronic energy of key intermediates and transition states of its catalytic cycle. In particular, we present quantum algorithms for double-factorized representations of the four-index integrals that can significantly reduce the computational cost over previous algorithms, and we discuss the challenges of increasing active space sizes to accurately deal with dynamical correlations. We address the requirements for future quantum hardware in order to make a universal quantum computer a successful and reliable tool for quantum computing enhanced computational materials science and chemistry, and identify open questions for further research.

We use magnetotransport in dual-gated magnetic topological insulator heterostructures to map out a phase diagram of the topological Hall and quantum anomalous Hall effects as a function of the chemical potential (primarily determined by the back gate voltage) and the asymmetric potential (primarily determined by the top gate voltage). A theoretical model that includes both surface states and valence band quantum well states allows the evaluation of the variation of the Dzyaloshinskii-Moriya interaction and carrier density with gate voltages. The qualitative agreement between experiment and theory provides strong evidence for the existence of a topological Hall effect in the system studied, opening up a route for understanding and manipulating chiral magnetic spin textures in real space.

The development of direct probes of entanglement is integral to the rapidly expanding field of complex quantum materials. Here we test the robustness of entangled neutrons as a quantum probe by measuring the Clauser-Horne-Shimony-Holt contextuality witness while varying the beam properties. Specifically, we show that the mode entanglement of the spin and path subsystems of individual neutrons prepared in two different experiments using two different apparatuses persists even after varying the entanglement length, coherence length, and neutron energy difference of the paths. The two independent apparatuses acting as entangler-disentangler pairs are static-field magnetic Wollaston prisms and resonance-field radio-frequency flippers. Our results show that the spatial and energy properties of the neutron beam may be significantly altered without reducing the contextuality witness value below the Tsirelson bound, meaning that maximum entanglement is preserved. We also show that two paths may be considered distinguishable even when the path states significantly overlap. Therefore, we have shown that our experimental results are consistent with the distinguishable subsystem assumption down to a separation of less than 100 nm, proving entanglement and the contextual nature of reality on short length scales. This work is the key step in the realization of the modular, robust technique of entangled neutron scattering, which can extract entanglement information from a sample without the knowledge of the microscopic sample Hamiltonian: only semiquantitative knowledge of the correlation lengths of the relevant degrees of freedom and the timescales of the characteristic dynamics is required.