For diffusive stochastic dynamics, the probability to observe any individual trajectory is vanishingly small, making it unclear how to experimentally validate theoretical results for ratios of path probabilities. We provide the missing link between theory and experiment by establishing a protocol to extract ratios of path probabilities from measured time series. For experiments on a single colloidal particle in a microchannel, we extract both ratios of path probabilities and the most probable path for a barrier crossing, and find excellent agreement with independently calculated predictions based on the Onsager-Machlup stochastic action. Our experimental results at room temperature are found to be inconsistent with the low-noise Freidlin-Wentzell stochastic action, and we discuss under which circumstances the latter action is expected to describe the most probable path. Furthermore, while the experimentally accessible ratio of path probabilities is uniquely determined, the formal path-integral action is known to depend on the time-discretization scheme used for deriving it; we reconcile these two seemingly contradictory facts by careful analysis of the time-slicing derivation of the path integral. Our experimental protocol enables us to probe probability distributions on path space and allows us to relate theoretical single-trajectory results to measurement.

The realization of quantum networks critically depends on establishing efficient, coherent light-matter interfaces. Optically active spins in diamond have emerged as promising quantum nodes based on their spin-selective optical transitions, long-lived spin ground states, and potential for integration with nanophotonics. Tin-vacancy (${\mathrm{SnV}}^{-}$) centers in diamond are of particular interest because they exhibit narrow-linewidth emission in nanostructures and possess long spin coherence times at temperatures above 1 K. However, a nanophotonic interface for ${\mathrm{SnV}}^{-}$ centers has not yet been realized. Here, we report cavity enhancement of the emission of ${\mathrm{SnV}}^{-}$ centers in diamond. We integrate ${\mathrm{SnV}}^{-}$ centers into one-dimensional photonic crystal resonators and observe a 40-fold increase in emission intensity. The Purcell factor of the coupled system is 25, resulting in a channeling of the majority of photons (90%) into the cavity mode. Our results pave the way for the creation of efficient, scalable spin-photon interfaces based on ${\mathrm{SnV}}^{-}$ centers in diamond.

We present a statistical approximate solution of the bound, nonhierarchical three-body problem, and extend it to a general analysis of encounters between hard binary systems and single stars. Any such encounter terminates when one of the three stars is ejected to infinity, leaving behind a remnant binary; the problem with binary-single-star scattering consists of finding the probability distribution of the orbital parameters of the remnant binary as a function of the total energy and the total angular momentum. Here, we model the encounter as a series of close, nonhierarchical, triple approaches, interspersed with hierarchical phases, in which the system consists of an inner binary and a star that orbits it; this series of approaches turns the evolution of the entire encounter to a random walk between consecutive hierarchical phases. We use the solution of the bound, nonhierarchical three-body problem to find the walker’s transition probabilities, which we generalize to situations in which tidal interactions are important. Besides tides, any dissipative process may be incorporated into the random-walk model, as it is completely general. Our approximate solution can reproduce the results of the extensive body of past numerical simulations and can account for different environments and different dissipative effects. Therefore, this model can effectively replace the need for direct few-body integrations for the study of binary-single encounters in any environment. Furthermore, it allows for a simply inclusion of dissipative forces typically not accounted for in full $N$-body integration schemes.

We study random quantum circuits and their rate of producing bipartite entanglement, specifically with respect to the choice of 2-qubit gates and the order (protocol) in which these are applied. The problem is mapped to a Markovian process, and we prove that there are large spectral equivalence classes—different configurations have the same spectrum. Optimal gates and the protocol that generate entanglement with the fastest theoretically possible rate are identified. Relaxation towards the asymptotic thermal entanglement proceeds via a series of phase transitions in the local relaxation rate, which is a consequence of non-Hermiticity. In particular, non-Hermiticity can cause the rate to be either faster or, even more interestingly, slower than predicted by the matrix eigenvalue gap. This result is caused by expansion coefficients that grow exponentially with system size, resulting in a “phantom” eigenvalue and is due to nonorthogonality of non-Hermitian eigenvectors. We numerically demonstrate that the phenomenon also occurs in random circuits with nonoptimal generic gates, random U(4) gates, and also without spatial or temporal randomness, suggesting that it could be of wide importance in other non-Hermitian settings, including correlations.

A number of experimental platforms relevant for quantum simulations, ranging from arrays of superconducting circuits hosting correlated states of light to ultracold atoms in optical lattices in the presence of controlled dissipative processes, are described as open quantum many-body systems. Their theoretical understanding is hampered by the exponential scaling of their Hilbert space and by their intrinsic nonequilibrium nature, limiting the applicability of many traditional approaches. In this work, we extend the nonequilibrium bosonic dynamical mean-field theory (DMFT) to Markovian open quantum systems. Within DMFT, a Lindblad master equation describing a lattice of dissipative bosonic particles is mapped onto an impurity problem describing a single site embedded in its Markovian environment and coupled to a self-consistent field and to a non-Markovian bath, where the latter accounts for fluctuations beyond Gutzwiller mean-field theory due to the finite lattice connectivity. We develop a nonperturbative approach to solve this bosonic impurity problem, which dresses the impurity, featuring Markovian dissipative channels, with the non-Markovian bath, in a self-consistent scheme based on a resummation of noncrossing diagrams. As a first application of our approach, we address the steady state of a driven-dissipative Bose-Hubbard model with two-body losses and incoherent pump. We show that DMFT captures hopping-induced dissipative processes, completely missed in Gutzwiller mean-field theory, which crucially determine the properties of the normal phase, including the redistribution of steady-state populations, the suppression of local gain, and the emergence of a stationary quantum-Zeno regime. We argue that these processes compete with coherent hopping to determine the phase transition toward a nonequilibrium superfluid, leading to a strong renormalization of the phase boundary at finite connectivity. We show that this transition occurs as a finite-frequency instability, leading to an oscillating-in-time order parameter, that we connect with a quantum many-body synchronization transition of an array of quantum van der Pol oscillators.

Electrons with large kinetic energy have a superconducting instability for infinitesimal attractive interactions. Quenching the kinetic energy and creating a flat band renders an infinitesimal repulsive interaction the relevant perturbation. Thus, flat-band systems are an ideal platform to study the competition of superconductivity and magnetism and their possible coexistence. Recent advances in the field of twisted bilayer graphene highlight this in the context of two-dimensional materials. Two dimensions, however, put severe restrictions on the stability of the low-temperature phases due to enhanced fluctuations. Only three-dimensional flat bands can solve the conundrum of combining the exotic flat-band phases with stable order existing at high temperatures. Here, we present a way to generate such flat bands through strain engineering in topological nodal-line semimetals. We present analytical and numerical evidence for this scenario and study the competition of the arising superconducting and magnetic orders as a function of externally controlled parameters. We show that the order parameter is rigid because the three-dimensional quantum geometry of the Bloch wave functions leads to a large superfluid stiffness in all three directions. Using density-functional theory and numerical tight-binding calculations, we further apply our theory to strained rhombohedral graphite and CaAgP materials.

We present an optimal protocol for encoding an unknown qubit state into a multiqubit Greenberger-Horne-Zeilinger-like state and, consequently, transferring quantum information in large systems exhibiting power-law ($1/r^{\alpha }$) interactions. For all power-law exponents $\alpha$ between $d$ and $2d+1$, where $d$ is the dimension of the system, the protocol yields a polynomial speed-up for $\alpha >2d$ and a superpolynomial speed-up for $\alpha \le 2d$, compared to the state of the art. For all $\alpha >d$, the protocol saturates the Lieb-Robinson bounds (up to subpolynomial corrections), thereby establishing the optimality of the protocol and the tightness of the bounds in this regime. The protocol has a wide range of applications, including in quantum sensing, quantum computing, and preparation of topologically ordered states. In addition, the protocol provides a lower bound on the gate count in digital simulations of power-law interacting systems.

Constructing systems that exhibit timescales much longer than those of the underlying components, as well as emergent dynamical and collective behavior, is a key goal in fields such as synthetic biology and materials self-assembly. Inspiration often comes from living systems, in which robust global behavior prevails despite the stochasticity of the underlying processes. Here, we present two-dimensional stochastic networks that consist of minimal motifs representing out-of-equilibrium cycles at the molecular scale and support chiral edge currents in configuration space. These currents arise in the topological phase because of the bulk-boundary correspondence and dominate the system dynamics in the steady state, further proving robust to defects or blockages. We demonstrate the topological properties of these networks and their uniquely non-Hermitian features such as exceptional points and vorticity, while characterizing the edge-state localization. As these emergent edge currents are associated with macroscopic timescales and length scales, simply tuning a small number of parameters enables varied dynamical phenomena, including a global clock, dynamical growth and shrinkage, and synchronization. Our construction provides a novel topological formalism for stochastic systems and fresh insights into non-Hermitian physics, paving the way for the prediction of robust dynamical states in new classical and quantum platforms.

We devise a machine learning technique to solve the general problem of inferring network links that have time delays using only time series data of the network nodal states. This task has applications in many fields, e.g., from applied physics, data science, and engineering to neuroscience and biology. Our approach is to first train a type of machine learning system known as reservoir computing to mimic the dynamics of the unknown network. We then use the trained parameters of the reservoir system output layer to deduce an estimate of the unknown network structure. Our technique, by its nature, is noninvasive but is motivated by the widely used invasive network inference method, whereby the responses to active perturbations applied to the network are observed and employed to infer network links (e.g., knocking down genes to infer gene regulatory networks). We test this technique on experimental and simulated data from delay-coupled optoelectronic oscillator networks, with both identical and heterogeneous delays along the links. We show that the technique often yields very good results, particularly if the system does not exhibit synchrony. We also find that the presence of dynamical noise can strikingly enhance the accuracy and ability of our technique, especially in networks that exhibit synchrony.

To fully capitalize on the potential and versatility of resonant inelastic x-ray scattering (RIXS), it is essential to develop the capability to interpret different RIXS contributions through calculations, including the dependence on momentum transfer, from first principles for correlated materials. Toward that objective, we present new methodology for calculating the full RIXS response of a correlated metal in an unbiased fashion. Through comparison of measurements and calculations that tune the incident photon energy over a wide portion of the Fe $L_3$ absorption resonance of the benchmark material ${\mathrm{BaFe}}_2{\mathrm{As}}_2$, we show that the RIXS response in ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ is dominated by the direct-channel contribution, including the Raman-like response below threshold. Calculations are initially performed within the first-principles Bethe-Salpeter equation (BSE) framework, which we then significantly improve by invoking a quasiboson model to describe the secondary excitations within the intermediate state. This enhancement allows the many-electron RIXS signal to be approximated as a convolution of BSE-calculated spectra with effective spectral functions. We construct these spectral functions, also from first principles, by employing the cumulant expansion of the Green’s function and performing a real-time time-dependent density functional theory calculation of the response of the electronic system to the perturbation of the intermediate-state excitation. Importantly, this process allows us to evaluate the indirect RIXS response from first principles, accounting for the full periodicity of the crystal structure and with full dependence on the momentum transfer.

The two main classes of liquid-crystal (LC) phases of rodlike molecules are nematics, where the rods align in the same direction (the nematic director $\mathbf{n}$), and smectics, where the rods not only are aligned but also form layers. The electro-optic effects in LC devices that are a backbone in today’s display industry mainly use the Fréedericksz transition, which is the bulk reorientation of a surface-anchored nematic by an electric field. Conventional (uniaxial) smectics do not present a Fréedericksz transition, because, due to their layered structure, the director reorientation would distort the layers, which would cost too much energy. In a worldwide ongoing effort to extend the variety of LC compounds suitable for applications in the display industry, bent-shaped molecules have recently raised much attention, since they present multiple new LC phases with unusual properties. In this paper, we report on a structural and electro-optic study of the LC phases of a bent-shaped dimer. On cooling from the isotropic liquid, this compound shows a usual nematic ($N$), a twist-bend nematic ($N_{\mathrm{TB}}$), and a biaxial smectic-$A$ phase ($\mathrm{Sm}{A}_b$). Quite surprisingly, contrary to usual smectics, $\mathrm{Sm}{A}_b$ presents a remarkable electro-optic response, with low ($<4\mathrm{V}$) voltage threshold, no reorganization of the smectic layers, and low ($<1\mathrm{ms}$) response time (i.e., 30 times faster than the $N$ phase at higher temperature). We interpret this unexpected electro-optic effect as a Fréedericksz transition affecting the secondary director $\mathbf{m}$ of the $\mathrm{Sm}{A}_b$, and we model it by analogy with the usual Fréedericksz transition of the $\mathbf{n}$ director of the uniaxial $N$ phase. Indeed, a Fréedericksz transition affecting only $\mathbf{m}$ in this biaxial fluid smectic does not alter its layered structure and costs little energy. From the point of view of applications, thanks to its low relaxation time, this “biaxial” Fréedericksz transition could be exploited in electro-optic devices that require fast switching.

Since the discovery of high-temperature superconductivity in cuprates, Josephson junction based phase-sensitive experiments are believed and used to provide the most convincing evidence for determining the pairing symmetry. Regardless of different junction materials and geometries used, quantum tunneling involved in these experiments is essentially a nanoscale process, and thus, actual experimental results are extremely sensitive to atomic details of the junction structures. The situation has led to controversial results as to the nature of the pairing symmetry of cuprates: while in-plane junction experiments generally support $d$-wave pairing symmetry, those based on out-of-plane ($c$-axis) Josephson junctions between two rotated cuprate blocks favor $s$-wave pairing. In this work, we revisit the $c$-axis experiment by fabricating Josephson junctions with atomic-level control in their interface structure. We fabricate over 90 junctions of ultrathin ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+x}$ (BSCCO) flakes by state-of-the-art exfoliation technique and obtain atomically flat junction interfaces in the whole junction regions as characterized by high-resolution transmission electron microscopy. Notably, the resultant uniform junctions at various twist angles all exhibit a single tunneling branch behavior, suggesting that only the first half of a unit cell on both sides of the twisted flakes is involved in the Josephson tunneling process. With such well-defined geometry and structure and the characteristic single tunneling branch, we repeatedly observe Josephson tunneling at a nominal twist angle of 45°, which is against the expectation from a purely $d$-wave pairing scenario. Our results strongly favor the scenario of a persistent $s$-wave order parameter in the junction.

The memory effect in disordered systems is a key physical phenomenon that has been employed for optical imaging, metrology, and communication through opaque media. Under the conventional memory effect, when the incident beam is tilted slightly, the transmitted pattern tilts in the same direction. However, the “memory” is limited in its angular range and tilt direction. Here, we present a general approach to customize the memory effect by introducing an angular memory operator. Its eigenstates possess perfect correlation for tilt angles and directions that can be arbitrarily chosen separately for the incident and transmitted waves, and can be readily realized with wave front shaping. This work reveals the power of wave front shaping in creating any desired memory for applications of classical and quantum waves in complex systems.

Distributed quantum sensing can provide quantum-enhanced sensitivity beyond the shot-noise limit (SNL) for sensing spatially distributed parameters. To date, distributed quantum sensing experiments have mostly been accomplished in laboratory environments without a real space separation for the sensors. In addition, the post-selection is normally assumed to demonstrate the sensitivity advantage over the SNL. Here, we demonstrate distributed quantum sensing with discrete variables in field and show the unconditional violation (without post-selection) of SNL up to 0.916 dB for the field distance of 240 m. The achievement is based on a loophole-free Bell test setup with entangled photon pairs at the averaged heralding efficiency of 73.88%. Moreover, to test quantum sensing in real life, we demonstrate the experiment for long distances (with 10-km fiber), together with the sensing of a completely random parameter. The results represent an important step towards a practical quantum sensing network for widespread applications.

We show experimentally that a dc-biased Josephson junction in series with two microwave resonators emits entangled beams of microwaves leaking out of the resonators. In the absence of a stationary phase reference for characterizing the entanglement of the outgoing beams, we measure second-order coherence functions to prove the entanglement. The experimental results are found in quantitative agreement with theory, proving that the low-frequency noise of the dc bias is the main limitation for the coherence time of the entangled beams. This agreement allows us to evaluate the entropy of entanglement of the resonators, estimate the entanglement flux at their output, and to identify the improvements that could bring this device closer to a useful bright source of entangled microwaves for quantum-technological applications.

The interplay between thermal and quantum fluctuations controls the competition between phases of matter in strongly correlated electron systems. We study finite-temperature properties of the strongly coupled two-dimensional doped Hubbard model using the minimally entangled typical thermal states method on width-four cylinders. We discover that a phase characterized by commensurate short-range antiferromagnetic correlations and no charge ordering occurs at temperatures above the half-filled stripe phase extending to zero temperature. The transition from the antiferromagnetic phase to the stripe phase takes place at temperature $T/t\approx 0.05$ and is accompanied by a steplike feature of the specific heat. We find the single-particle gap to be smallest close to the nodal point at $\mathit{k}=(\pi /2,\pi /2)$ and detect a maximum in the magnetic susceptibility. These features bear a strong resemblance to the pseudogap phase of high-temperature cuprate superconductors. The simulations are verified using a variety of different unbiased numerical methods in the three limiting cases of zero temperature, small lattice sizes, and half filling. Moreover, we compare to and confirm previous determinantal quantum Monte Carlo results on incommensurate spin-density waves at finite doping and temperature.

The nucleation of crystals from liquid melt is often characterized by a competition between different crystalline structures or polymorphs and can result in nuclei with heterogeneous compositions. These mixed-phase nuclei can display nontrivial spatial arrangements, such as layered and onionlike structures, whose composition varies according to the radial distance, and which so far have been explained on the basis of bulk and surface free-energy differences between the competing phases. Here we extend the generality of these nonclassical nucleation processes, showing that layered and onionlike structures can emerge solely based on structural fluctuations even in the absence of free-energy differences. We consider two examples of competing crystalline structures, hcp and fcc forming in hard spheres relevant for repulsive colloids and dense liquids, and the cubic and hexagonal diamond forming in water relevant also for other group 14 elements such as carbon and silicon. We introduce a novel structural order parameter that combined with a neural-network classification scheme allows us to study the properties of the growing nucleus from the early stages of nucleation. We find that small nuclei have distinct size fluctuations and compositions from the nuclei that emerge from the growth stage. The transition between these two regimes is characterized by the formation of onionlike structures, in which the composition changes with the distance from the center of the nucleus, similar to what is seen in the two-step nucleation process.

The physical realization of ${\mathbb{Z}}_2$ topological order as encountered in the paradigmatic toric code has proven to be an elusive goal. We predict that this phase of matter can be realized in a two-dimensional array of Rydberg atoms placed on the ruby lattice, at specific values of the Rydberg blockade radius. First, we show that the blockade model—also known as a “$PXP$” model—realizes a monomer-dimer model on the kagome lattice with a single-site kinetic term. This model can be interpreted as a ${\mathbb{Z}}_2$ gauge theory whose dynamics is generated by monomer fluctuations. We obtain its phase diagram using the numerical density matrix renormalization group method and find a topological quantum liquid (TQL) as evidenced by multiple measures including (i) a continuous transition between two featureless phases, (ii) a topological entanglement entropy of $\ln 2$ as measured in various geometries, (iii) degenerate topological ground states, and (iv) the expected modular matrix from ground state overlap. Next, we show that the TQL persists upon including realistic, algebraically decaying van der Waals interactions $V(r)\sim 1/r^6$ for a choice of lattice parameters. Moreover, we can directly access topological loop operators, including the Fredenhagen-Marcu order parameter. We show how these can be measured experimentally using a dynamic protocol, providing a “smoking gun” experimental signature of the TQL phase. Finally, we show how to trap an emergent anyon and realize different topological boundary conditions, and we discuss the implications for exploring fault-tolerant quantum memories.

The equilibrium contact angle of a droplet resting on a solid substrate can reveal essential properties of the solid’s surface. However, when the motion of a droplet on a surface shows significant hysteresis, it is generally accepted that the solid’s equilibrium properties cannot be determined. Here, we describe a method to measure surface tensions of soft solids with strong wetting hysteresis. With independent knowledge of the surface tension of the wetting fluid and the linear-elastic response of the solid, the solid deformations under the contact line and the contact angle of a single droplet together reveal the difference in surface tension of the solid against the liquid and vapor phases If the solid’s elastic properties are unknown, then this surface tension difference can be determined from the change in substrate deformations with contact angle. These results reveal an alternate equilibrium contact angle, equivalent to the classic form of Young-Dupré, but with surface tensions in place of surface energies. We motivate and apply this approach with experiments on gelatin, a common hydrogel.

We study the stability of quasinormal modes (QNM) in asymptotically flat black hole spacetimes by means of a pseudospectrum analysis. The construction of the Schwarzschild QNM pseudospectrum reveals the following: (i) the stability of the slowest-decaying QNM under perturbations respecting the asymptotic structure, reassessing the instability of the fundamental QNM discussed by Nollert [H. P. Nollert, About the Significance of Quasinormal Modes of Black Holes, Phys. Rev. D 53, 4397 (1996)] as an “infrared” effect; (ii) the instability of all overtones under small-scale (“ultraviolet”) perturbations of sufficiently high frequency, which migrate towards universal QNM branches along pseudospectra boundaries, shedding light on Nollert’s pioneer work and Nollert and Price’s analysis [H. P. Nollert and R. H. Price, Quantifying Excitations of Quasinormal Mode Systems, J. Math. Phys. (N.Y.) 40, 980 (1999)]. Methodologically, a compactified hyperboloidal approach to QNMs is adopted to cast QNMs in terms of the spectral problem of a non-self-adjoint operator. In this setting, spectral (in)stability is naturally addressed through the pseudospectrum notion that we construct numerically via Chebyshev spectral methods and foster in gravitational physics. After illustrating the approach with the Pöschl-Teller potential, we address the Schwarzschild black hole case, where QNM (in)stabilities are physically relevant in the context of black hole spectroscopy in gravitational-wave physics and, conceivably, as probes into fundamental high-frequency spacetime fluctuations at the Planck scale.