Proliferation of unstable states and their impact on stochastic out-of-equilibrium dynamics in two coupled Kerr parametric oscillators.

Networks of nonlinear parametric resonators are promising candidates as Ising machines for annealing and optimization. These many-body out-of-equilibrium systems host complex phase diagrams of coexisting stationary states. The plethora of states manifest via a series of bifurcations, including bifurcations that proliferate purely unstable solutions. Here we demonstrate that the latter take a fundamental role in the stochastic dynamics of the system. Specifically, they determine the switching paths and the switching rates between stable solutions. We demonstrate experimentally the impact of the added unstable states on noise-activated switching dynamics in a network of two coupled parametric resonators.

Biased Ising Model Using Two Coupled Kerr Parametric Oscillators with External Force.

Networks of coupled Kerr parametric oscillators (KPOs) are a leading physical platform for analog solving of complex optimization problems. These systems are colloquially known as "Ising machines." We experimentally and theoretically study such a network under the influence of an external force. The force breaks the collective phase-parity symmetry of the system and competes with the intrinsic coupling in ordering the network configuration, similar to how a magnetic field biases an interacting spin ensemble. Specifically, we demonstrate how the force can be used to control the system, and highlight the crucial role of the phase and symmetry of the force. Our Letter thereby provides a method to create Ising machines with arbitrary bias, extending even to exotic cases that are impossible to engineer in real spin systems.

Fully Reprogrammable 2D Array of Multistate Molecular Switching Units.

Integration of molecular switching units into complex electronic circuits is considered to be the next step toward the realization of novel logic and memory devices. This paper reports on an ordered 2D network of neighboring ternary switching units represented by triazatruxene (TAT) molecules organized in a honeycomb lattice on a Ag(111) surface. Using low-temperature scanning tunneling microscopy, the bonding configurations of individual TAT molecules can be controlled, realizing up to 12 distinct states per molecule. The switching between those states shows a strong bias dependence ranging from tens of millivolts to volts. The low-bias switching behavior is explored in active units consisting of two and more interacting TAT molecules that are purposefully defined (programmed) by high-bias switching within the honeycomb lattice. Within such a unit the low-bias switching can be triggered and accessed by single-point measurements on a single TAT molecule, demonstrating up to 9 and 19 distinguishable states in a dyad and a tetrad of coupled molecules, respectively. High experimental control over the desired state, owing to bias-dependent hierarchical switching and pronounced switching directionality, as well as full reversibility, make this system particularly appealing, paving the way to design complex molecule-based memory systems.

Tomasch Oscillations as Above-Gap Signature of Topological Superconductivity.

The identification of topological superconductors usually involves searching for in-gap modes that are protected by topology. However, in current experimental settings, the smoking-gun evidence of these in-gap modes is still lacking. In this Letter, we propose to support the distinction between two-dimensional conventional s-wave and topological p-wave superconductors by above-gap transport signatures. Our method utilizes the emergence of Tomasch oscillations of quasiparticles in a junction consisting of a superconductor sandwiched between two metallic leads. We demonstrate that the behavior of the oscillations in conductance as a function of the interface barriers provides a distinctive signature for s-wave and p-wave superconductors. Specifically, the oscillations become weaker as the barrier strength increases in s-wave superconductors, while they become more pronounced in p-wave superconductors, which we prove to be a direct manifestation of the pairing symmetries. Our method can serve as a complimentary probe for identifying some classes of topological superconductors through the above-gap transport.

Dynamical Gauge Fields with Bosonic Codes.

The quantum simulation of dynamical gauge field theories offers the opportunity to study complex high-energy physics with controllable low-energy devices. For quantum computation, bosonic codes promise robust error correction that exploits multiparticle redundancy in bosons. Here, we demonstrate how bosonic codes can be used to simulate dynamical gauge fields. We encode both matter and dynamical gauge fields in a network of resonators that are coupled via three-wave mixing. The mapping to a Z_{2} dynamical lattice gauge theory is established when the gauge resonators operate as Schrödinger cat states. We explore the optimal conditions under which the system preserves the required gauge symmetries. Our findings promote realizing high-energy models using bosonic codes.

Strong Parametric Coupling between Two Ultracoherent Membrane Modes.

We demonstrate parametric coupling between two modes of a silicon nitride membrane. We achieve the coupling by applying an oscillating voltage to a sharp metal tip that approaches the membrane surface to within a few 100 nm. When the voltage oscillation frequency is equal to the mode frequency difference, the modes exchange energy periodically and faster than their free energy decay rate. This flexible method can potentially be useful for rapid state control and transfer between modes, and is an important step toward parametric spin sensing experiments with membrane resonators.

Spontaneous Valley Spirals in Magnetically Encapsulated Twisted Bilayer Graphene.

Van der Waals heterostructures provide a rich platform for emergent physics due to their tunable hybridization of layers, orbitals, and spin. Here, we find that twisted bilayer graphene stacked between antialigned ferromagnetic insulators can feature flat electronic bands due to the interplay between twist, exchange proximity, and spin-orbit coupling. These flat bands are nearly degenerate in valley only and are effectively described by a triangular superlattice model. At half filling, we find that interactions induce spontaneous valley correlations that favor spiral order and derive a low-energy valley-Heisenberg model with symmetric and antisymmetric exchange couplings. We also show how electric interlayer bias broadens the bands and tunes these couplings. Our results put forward magnetic van der Waals heterostructures as a platform to explore valley-correlated states.

Publisher Correction: A square-root topological insulator with non-quantized indices realized with photonic Aharonov-Bohm cages.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

A square-root topological insulator with non-quantized indices realized with photonic Aharonov-Bohm cages.

Topological Insulators are a novel state of matter where spectral bands are characterized by quantized topological invariants. This unique quantized nonlocal property commonly manifests through exotic bulk phenomena and corresponding robust boundary effects. In our work we study a system where the spectral bands are associated with non-quantized indices, but nevertheless possess robust boundary states. We present a theoretical analysis, where we show that the square of the Hamiltonian exhibits quantized indices. The findings are experimentally demonstrated by using photonic Aharonov-Bohm cages.

Quantum Transducer Using a Parametric Driven-Dissipative Phase Transition.

We study a dissipative Kerr resonator subject to both single- and two-photon detuned drives. Beyond a critical detuning threshold, the Kerr resonator exhibits a semiclassical first-order dissipative phase transition between two different steady states that are characterized by a π phase switch of the cavity field. This transition is shown to persist deep into the quantum limit of low photon numbers. Remarkably, the detuning frequency at which this transition occurs depends almost linearly on the amplitude of the single-photon drive. Based on this phase-switching feature, we devise a sensitive quantum transducer that translates the observed frequency of the parametric quantum phase transition to the detected single-photon amplitude signal. The effects of noise and temperature on the corresponding sensing protocol are addressed, and a realistic circuit-QED implementation is discussed.

Classical Many-Body Time Crystals.

Discrete time crystals are a many-body state of matter where the extensive system's dynamics are slower than the forces acting on it. Nowadays, there is a growing debate regarding the specific properties required to demonstrate such a many-body state, alongside several experimental realizations. In this work, we provide a simple and pedagogical framework by which to obtain many-body time crystals using parametrically coupled resonators. In our analysis, we use classical period-doubling bifurcation theory and present a clear distinction between single-mode time-translation symmetry breaking and a situation where an extensive number of degrees of freedom undergo the transition. We experimentally demonstrate this paradigm using coupled mechanical oscillators, thus providing a clear route for time crystal realizations in real materials.

Weak Localization and Antilocalization in Nodal-Line Semimetals: Dimensionality and Topological Effects.

New materials such as nodal-line semimetals offer a unique setting for novel transport phenomena. Here, we calculate the quantum correction to conductivity in a disordered nodal-line semimetal. The torus-shaped Fermi surface and encircled π Berry flux carried by the nodal loop result in a fascinating interplay between the effective dimensionality of electron diffusion and band topology, which depends on the scattering range of the impurity potential relative to the size of the nodal loop. For a short-range impurity potential, backscattering is dominated by the interference paths that do not encircle the nodal loop, yielding a 3D weak localization effect. In contrast, for a long-range impurity potential, the electrons effectively diffuse in various 2D planes and the backscattering is dominated by the interference paths that encircle the nodal loop. The latter leads to weak antilocalization with a 2D scaling law. Our results are consistent with symmetry consideration, where the two regimes correspond to the orthogonal and symplectic classes, respectively. Furthermore, we present weak-field magnetoconductivity calculations at low temperatures for realistic experimental parameters and predict that clear scaling signatures ∝sqrt[B] and ∝-lnB, respectively. The crossover between the 3D weak localization and 2D weak antilocalization can be probed by tuning the Fermi energy, giving a unique transport signature of the nodal-line semimetal.

Tunneling into a Finite Luttinger Liquid Coupled to Noisy Capacitive Leads.

Tunneling spectroscopy of one-dimensional interacting wires can be profoundly sensitive to the boundary conditions of the wire. Here, we analyze the tunneling spectroscopy of a wire coupled to capacitive metallic leads. Strikingly, with increasing many-body interactions in the wire, the impact of the boundary noise becomes more prominent. This interplay allows for a smooth crossover from standard 1D tunneling signatures into a regime where the tunneling is dominated by the fluctuations at the leads. This regime is characterized by an elevated zero-bias tunneling alongside a universal power-law decay at high energies. Furthermore, local tunneling measurements in this regime show a unique spatial dependence that marks the formation of plasmonic standing waves in the wire. Our result offers a tunable method by which to control the boundary effects and measure the interaction strength (Luttinger parameter) within the wire.

Rapid Flipping of Parametric Phase States.

We experimentally demonstrate flipping the phase state of a parametron within a single period of its oscillation. A parametron is a binary logic element based on a driven nonlinear resonator. It features two stable phase states that define an artificial spin. The most basic operation performed on a parametron is a bit flip between these two states. Thus far, this operation involved changing the energetic population of the resonator and therefore required a number of oscillations on the order of the quality factor Q. Our technique takes a radically different approach and relies on rapid control of the underlying potential. Our work represents a paradigm shift for phase-encoded logic operations by boosting the speed of a parametron bit flip to its ultimate limit.

Proposal for Detecting Nodal-Line Semimetal Surface States with Resonant Spin-Flipped Reflection.

Topological nodal-line semimetals are predicted to exhibit unique drumheadlike surface states (DSSs). Yet, direct detection of such states remains a challenge. Here, we propose spin-resolved transport in a junction between a normal metal and a spin-orbit coupled nodal-line semimetal as the mechanism for their detection. Specifically, we find that in such an interface the DSSs induce resonant spin-flipped reflection. This effect can be probed by both vertical spin transport and lateral charge transport between antiparallel magnetic terminals. In the tunneling limit of the junction, both spin and charge conductances exhibit a resonant peak around zero energy, providing unique evidence of the DSSs. This signature is robust to both dispersive DSSs and interface disorder. Based on numerical calculations, we show that the scheme can be implemented in the topological semimetal HgCr_{2}Se_{4}.

Quantum interference of topological states of light.

Topological insulators are materials that have a gapped bulk energy spectrum but contain protected in-gap states appearing at their surface. These states exhibit remarkable properties such as unidirectional propagation and robustness to noise that offer an opportunity to improve the performance and scalability of quantum technologies. For quantum applications, it is essential that the topological states are indistinguishable. We report high-visibility quantum interference of single-photon topological states in an integrated photonic circuit. Two topological boundary states, initially at opposite edges of a coupled waveguide array, are brought into proximity, where they interfere and undergo a beamsplitter operation. We observe Hong-Ou-Mandel interference with 93.1 ± 2.8% visibility, a hallmark nonclassical effect that is at the heart of linear optics-based quantum computation. Our work shows that it is feasible to generate and control highly indistinguishable single-photon topological states, opening pathways to enhanced photonic quantum technology with topological properties, and to study quantum effects in topological materials.

Cavity-Mediated Coherent Coupling between Distant Quantum Dots.

Scalable architectures for quantum information technologies require one to selectively couple long-distance qubits while suppressing environmental noise and cross talk. In semiconductor materials, the coherent coupling of a single spin on a quantum dot to a cavity hosting fermionic modes offers a new solution to this technological challenge. Here, we demonstrate coherent coupling between two spatially separated quantum dots using an electronic cavity design that takes advantage of whispering-gallery modes in a two-dimensional electron gas. The cavity-mediated, long-distance coupling effectively minimizes undesirable direct cross talk between the dots and defines a scalable architecture for all-electronic semiconductor-based quantum information processing.

Exploring 4D quantum Hall physics with a 2D topological charge pump.

The discovery of topological states of matter has greatly improved our understanding of phase transitions in physical systems. Instead of being described by local order parameters, topological phases are described by global topological invariants and are therefore robust against perturbations. A prominent example is the two-dimensional (2D) integer quantum Hall effect: it is characterized by the first Chern number, which manifests in the quantized Hall response that is induced by an external electric field. Generalizing the quantum Hall effect to four-dimensional (4D) systems leads to the appearance of an additional quantized Hall response, but one that is nonlinear and described by a 4D topological invariant-the second Chern number. Here we report the observation of a bulk response with intrinsic 4D topology and demonstrate its quantization by measuring the associated second Chern number. By implementing a 2D topological charge pump using ultracold bosonic atoms in an angled optical superlattice, we realize a dynamical version of the 4D integer quantum Hall effect. Using a small cloud of atoms as a local probe, we fully characterize the nonlinear response of the system via in situ imaging and site-resolved band mapping. Our findings pave the way to experimentally probing higher-dimensional quantum Hall systems, in which additional strongly correlated topological phases, exotic collective excitations and boundary phenomena such as isolated Weyl fermions are predicted.

Photonic topological boundary pumping as a probe of 4D quantum Hall physics.

When a two-dimensional (2D) electron gas is placed in a perpendicular magnetic field, its in-plane transverse conductance becomes quantized; this is known as the quantum Hall effect. It arises from the non-trivial topology of the electronic band structure of the system, where an integer topological invariant (the first Chern number) leads to quantized Hall conductance. It has been shown theoretically that the quantum Hall effect can be generalized to four spatial dimensions, but so far this has not been realized experimentally because experimental systems are limited to three spatial dimensions. Here we use tunable 2D arrays of photonic waveguides to realize a dynamically generated four-dimensional (4D) quantum Hall system experimentally. The inter-waveguide separation in the array is constructed in such a way that the propagation of light through the device samples over momenta in two additional synthetic dimensions, thus realizing a 2D topological pump. As a result, the band structure has 4D topological invariants (known as second Chern numbers) that support a quantized bulk Hall response with 4D symmetry. In a finite-sized system, the 4D topological bulk response is carried by localized edge modes that cross the sample when the synthetic momenta are modulated. We observe this crossing directly through photon pumping of our system from edge to edge and corner to corner. These crossings are equivalent to charge pumping across a 4D system from one three-dimensional hypersurface to the spatially opposite one and from one 2D hyperedge to another. Our results provide a platform for the study of higher-dimensional topological physics.

Parametric Symmetry Breaking in a Nonlinear Resonator.

Much of the physical world around us can be described in terms of harmonic oscillators in thermodynamic equilibrium. At the same time, the far-from-equilibrium behavior of oscillators is important in many aspects of modern physics. Here, we investigate a resonating system subject to a fundamental interplay between intrinsic nonlinearities and a combination of several driving forces. We have constructed a controllable and robust realization of such a system using a macroscopic doubly clamped string. We experimentally observe a hitherto unseen double hysteresis in both the amplitude and the phase of the resonator's response function and present a theoretical model that is in excellent agreement with the experiment. Our work unveils that the double hysteresis is a manifestation of an out-of-equilibrium symmetry breaking between parametric phase states. Such a fundamental phenomenon, in the most ubiquitous building block of nature, paves the way for the investigation of new dynamical phases of matter in parametrically driven many-body systems and motivates applications ranging from ultrasensitive force detection to low-energy computing memory units.