Understanding and minimizing the sources of frequency noise in nanomechanical resonators is crucial for many sensing applications. In this work, we report an ultracoherent perimeter-mode nanomechanical resonator co-integrated with an on-chip optical cavity. This device combines low thermomechanical force noise and low detector noise, allowing us to study its intrinsic frequency fluctuations in detail. We find that the fluctuations of two mechanical modes are strongly correlated; this holds not only for thermal drifts but also for intrinsic frequency flicker noise. Moreover, we demonstrate the generation of a signal at the frequency difference between the two modes directly on chip via nonlinear optomechanical transduction. This "difference signal" has vastly reduced intrinsic frequency fluctuations and can be used for frequency tracking with high precision, as we establish in a proof-of-principle experiment.

The response of nonlinear resonators to multifrequency driving reveals rich dynamics beyond conventional single-tone theory. We study a Duffing resonator under bichromatic excitation and identify a competition between the two drives, governed by their detuning and relative amplitudes. In the slow-beating regime, where the tones are closely spaced, the secondary tone acts as a modulation that induces dynamical phase transitions between coexisting stationary states. We introduce the cycle-averaged amplitude as an order parameter and map the resulting phase diagram as a function of the drive detuning and amplitude ratio, capturing the pronounced asymmetry observed for blue versus red detuning between the drives in experiment. We link the onset of these transitions to the resonance properties around the nonlinear stationary mode of the system. We apply a two-tone ansatz to this system, for the first time, to provide a comprehensive map of stationary states. Our results provide a framework for controlling driven nonlinear systems, enabling state manipulation, and sensing in nanomechanical, optical, and superconducting circuit platforms.

Mechanical resonators operating in the megahertz range have become a versatile platform for fundamental and applied quantum research. Their exceptional properties, such as low mass and high quality factor, make them also appealing for force sensing experiments. In this work, we propose a method for detecting, and ultimately controlling, nuclear spins by coupling them to megahertz resonators via a magnetic field gradient. Dynamical backaction between the sensor and an ensemble of N nuclear spins produces a shift in the sensor's resonance frequency. The mean frequency shift due to the Boltzmann polarization is challenging to measure in nanoscale sample volumes. Here, we show that the fluctuating polarization of the spin ensemble results in a measurable increase of the resonator's frequency variance. On the basis of analytical as well as numerical results, we predict that the variance measurement will allow single nuclear spin detection with existing resonator devices.

Magnetic resonance force microscopy (MRFM) enables three-dimensional imaging of nuclear spin densities in nanoscale objects. On the basis of numerical simulations, we evaluate the performance of strained SiN resonators as force sensors and show that their out-of-plane oscillation direction improves the quality of the reconstructed sample. We further introduce a multislice, compressed-sensing scan protocol that maximizes the information obtained for a given measurement time. Our simulations predict that these scanning protocols and optimized algorithms can shorten the total acquisition time by up to 2 orders of magnitude while maintaining the reconstruction fidelity. Our results demonstrate that combining advanced scanning protocols with state-of-the-art resonators is a promising path toward high-resolution MRFM for volumetric imaging of biological nanostructures.

Noise-activated switching between coexisting stable states is a fundamental mechanism underlying stochastic dynamics in systems ranging from chemical reactions to neural networks. While this phenomenon is well understood for stationary attractors, it remains largely unexplored for limit cycles, whose periodic motion cannot be described by a static potential landscape. Here we experimentally demonstrate activated switching between two coexisting limit-cycle attractors in a driven nonlinear system of coupled resonators. Specifically, we introduce controlled fluctuations to directly observe the rare stochastic transitions between two limit cycles and measure their dependence on noise intensity and driving strength. The measured switching rates are well described by a large-deviation theory, which replaces the conventional activation barrier by the action along the most probable transition path. Our results extend the concept of activated dynamics from stationary to limit-cycle attractors and establish a framework for modeling stochastic transitions between limit cycles in driven-dissipative systems.

Activation processes govern noise-induced switching between long-lived states. In an equilibrium double well, the thermally activated switching rate exhibits a prefactor with a nonmonotonic dependence on environmental coupling, a foundational crossover known as Kramers turnover. Here, we demonstrate a Kramers turnover analogue in a Kerr parametric oscillator, a driven-dissipative nonlinear system featuring two stable phase states. First, we analytically establish turnover physics in this out-of-equilibrium setting. There, the strong physical correlation between the activation barrier and intrinsic damping fundamentally obscures the underlying turnover physics. To overcome this limitation, we rescale the rotating-frame dynamics and introduce a tunable effective friction controlled entirely by the parametric drive. This rescaling comes at the cost of a concurrent rescaling of the effective temperature. Exploiting this simultaneous scaling, we leverage the effective temperature to extract the turnover directly from temperature-dependent observations. Subsequently, measuring noise-induced phase slips in a micro-electromechanical device, we observe a distinct crossover in the prefactor's temperature dependence. Our results unambiguously isolate the out-of-equilibrium turnover regime and highlight that the competition between dissipation and fluctuations profoundly shapes activation dynamics also beyond equilibrium.

Coupled Kerr parametric oscillators (KPOs) are a promising resource for classical and quantum analog computation, for example to find the ground state of Ising Hamiltonians. Yet, the state space of strongly coupled KPO networks is very involved. As such, their phase diagram sometimes features either too few or too many states, including some that cannot be mapped to Ising spin configurations. This complexity makes it challenging to find and meet the conditions under which an analog optimization algorithm can be successful. Here, we demonstrate how to use three strongly coupled KPOs as a simulator for an Ising Hamiltonian, and estimate its ground state using a Boltzmann sampling measurement. While fully classical, our Letter is directly relevant for quantum systems operating on coherent states.

Nonlinear dynamics are studied in diverse fields as climate models, avalanches, nanomechanical sensors, optical frequency converters, and electrical quantum amplifiers. A widely studied nonlinear model is the so-called Duffing (or Kerr) resonator, which features a quartic potential term. Two hallmark properties of this model are (i) a shift of a system's resonance frequency as a function of the driving strength, and (ii) monostable or bistable responses, depending on the drive strength and detuning from resonance. Together, these two properties can lead to dynamical phase transitions when several drives are applied simultaneously. Here, we report an experimental and theoretical study of a driven-dissipative nonlinear system with two detuned drives. We observe distinct response regimes characterized by the system's ability to follow the system's time-dependent vector-flow topology. Our work provides an example for understanding dynamical phase transitions in out-of-equilibrium nonlinear systems.

The separation of physical forces acting on the tip of a magnetic force microscope (MFM) is essential for correct magnetic imaging. Electrostatic forces can be modulated by varying the tip-sample potential and minimized to map the local Kelvin potential. However, distinguishing magnetic forces from van der Waals forces typically requires two measurements with opposite tip magnetizations under otherwise identical measurement conditions. Here, we present an inverted magnetic force microscope where the sample is mounted on a flat cantilever for force sensing, and the magnetic tip is attached to a miniaturized electromagnet that periodically flips the tip magnetization. This setup enables the extraction of magnetic tip-sample interactions from the sidebands occurring at the switching rate in the cantilever oscillation spectrum. Our method achieves the separation of magnetic signals from other force contributions in a single-scan mode. Future iterations of this setup may incorporate membrane, trampoline, or string resonators with ultra-high quality factors, potentially improving measurement sensitivity by up to three orders of magnitude compared to the state-of-the-art MFM systems using cantilevers.

Networks of nonlinear resonators offer a promising platform for analog computing and the emulation of complex systems. However, realizing such networks remains challenging, as it requires resonators with high quality factors, individual frequency tunability, and strong inter-resonator coupling. In this work, we present a system that meets all these criteria. Our system is based on metallized silicon nitride membranes that are coupled via their common substrate and controlled capacitively via electrodes. We demonstrate individual frequency tuning and strong parametric driving of each membrane. Notably, we tune membrane frequencies through avoided crossings and demonstrate tunability of the coupled membrane's parametric response. This platform provides a scalable and controllable setting for exploring collective phenomena, dynamical phase transitions, nonlinear topology, and analog computing.

Graphene nanomechanical resonators are well suited for the study of parametric oscillations. Their large frequency tunability and their pronounced nonlinearities enable an efficient modulation of their resonant frequencies. Here, we present measurements of the response of few-layer-graphene nanomechanical resonators, each driven by a large parametric pump at frequency 2 omega and a weak external drive at omega, where omega is set near the mechanical resonant frequency omega 0. The pump actuates the resonator beyond the threshold for large-amplitude parametric oscillations, while the drive breaks the symmetry between the parametric phase states. By increasing and decreasing a gate voltage to detune omega 0 in the presence of the pump and the drive, we observe a double hysteresis in the response. The double hysteresis reveals the existence of two possible large-amplitude vibrational states whose phase difference is nearly pi radians. We deterministically prepare the resonator in either one of these states by cycling the gate voltage. We measure the stationary occupation probabilities of the two states in the presence of a white Gaussian force noise, and find that they strongly depend on the amplitude and on the phase of the external drive. The phase states of parametric oscillations with broken amplitude symmetry can be mapped to biased bimodal degrees of freedom, such as Ising spins in an external magnetic field. Therefore, they hold promise as units of binary information.

In topology, averaging over local geometrical details reveals robust global features. These are crucial in physics for understanding quantized bulk transport and exotic boundary effects of linear wave propagation in (meta-)materials. Beyond linear Hamiltonian systems, topological physics strives to characterize open (non-Hermitian) and interacting systems. Here, we establish a framework for the topological classification of driven-dissipative nonlinear systems by defining a graph index for their Floquet semiclassical equations of motion. Our index builds upon the topology of vector flows and encodes the particle-hole nature of excitations around all out-of-equilibrium stationary states. Thus, we uncover the topology of nonlinear resonator's dynamics under external and parametric forcing. Our framework sheds light on the topology of driven-dissipative phases, including under- to overdamped responses and symmetry-broken phases linked to population inversion. We therefore expose the pervasive link between topology and nonlinear dynamics, with broad implications for interacting topological insulators, topological solitons, neuromorphic networks, and bosonic codes.

Networks of coupled Kerr parametric oscillators (KPOs) hold promise for the realization of neuromorphic and quantum computation. Yet, their rich bifurcation structure remains poorly understood. Here, we employ secular perturbation theory to map the stability regions of these networks and identify the regime where the system can be mapped to an Ising model. Starting with two coupled KPOs, we show how the bifurcations arise from the competition between the global parametric drive and linear coupling between the KPOs. We then extend this framework to larger networks with all-to-all equal coupling, deriving analytical expressions for the full cascade of bifurcation transitions. In the thermodynamic limit, we find that these transitions become uniformly spaced, leading to a highly regular structure. Our results reveal the precise bounds under which KPO networks have an Ising-like solution space, and thus provide crucial guidance for their experimental implementation.

Nanoscale magnetic resonance imaging (NanoMRI) is an active area of applied research with potential applications in structural biology and quantum engineering. The success of this technological vision hinges on improving the instrument's sensitivity and functionality. A particular challenge is the optimization of the magnetic field gradient required for spatial encoding and of the radio frequency field used for spin control, in analogy to the components used in clinical MRI. In this work, we present the fabrication and characterization of a magnet-in-microstrip device that yields a compact form factor for both elements. We find that our design leads to a number of advantages, among them a 4-fold increase of the magnetic field gradient compared to those achieved with traditional fabrication methods. Our results can be useful for boosting the efficiency of a variety of different experimental arrangements and detection principles in the field of NanoMRI.

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.

Many experimental techniques aim at determining the Hamiltonian of a given system. The Hamiltonian describes the system's evolution in the absence of dissipation, and is often central to control or interpret an experiment. Here, we theoretically propose and experimentally demonstrate a method for Hamiltonian reconstruction from measurements over a large area of phase space, overcoming the main limitation of previous techniques. A crucial ingredient for our method is the presence of dissipation, which enables sampling of the Hamiltonian through ringdown-type measurements. We apply the method to a driven-dissipative system -- a parametric oscillator -- observed in a rotating frame, and reconstruct the (quasi-)Hamiltonian of the system. Furthermore, we demonstrate that our method provides direct experimental access to the so-called symplectic norm of the stationary states of the system, which is tied to the particle- or hole-like nature of excitations of these states. In this way, we establish a method to unveil qualitative differences between the fluctuations around stabilized minima and maxima of the nonlinear out-of-equilibrium stationary states. Our method constitutes a versatile approach to characterize a wide class of driven-dissipative systems.

The combination of a strong pump and a weak probe has been widely applied to investigate both optical and nanomechanical devices. Such pump-probe measurements allows for the exploration of nonlinear dynamics, driven by the large pump tone, by measuring the system response to a probe tone. In contrast, here we report on the dynamics of a mechanical Duffing resonator driven with a combination of two large tones at different frequencies. Our results indicate the presence of various distinct regimes with very different dynamics. We systematically investigate the impact of the relative strength and detuning between the two drives on the dynamical response. This provides an illustrative example of dynamical phase transitions in out-of-equilibrium systems.

The new generation of strained-silicon-nitride resonators harbors great promise for scanning force microscopy, especially when combined with the extensive toolbox of cavity optomechanics. However, accessing a mechanical resonator inside an optical cavity with a scanning tip is challenging. Here we experimentally demonstrate a cavity-based scanning force microscope based on a silicon-nitridemembrane sensor. We overcome geometric constraints by making use of the extended nature of the mechanical resonator normal modes, which allows us to spatially separate the scanning and readout sites of the membrane. Our microscope is geared towards low-temperature applications in the zeptonewton regime, such as nanoscale nuclear-spin detection and imaging.

Many experimental techniques aim at determining the energy landscape of a system given and compare it to a model Hamiltonian. This landscape governs the system's evolution in the absence of dissipation. Here, we theoretically propose and experimentally demonstrate a method to measure the energy landscape of a system without knowing its functional form. A crucial ingredient for our method is the presence of dissipation, which enables sampling of the landscape over a large area of phase space through ringdown-type measurements, overcoming the main limitation of previous techniques. We apply the method to a driven-dissipative system–a parametric oscillator–observed in a rotating frame. We first measure the phase-space flow dynamics of the system via ringdown measurements, unveiling its attractors and separatrices. With these measurements, we reconstruct the (quasi-)energy landscape of the system. Furthermore, we demonstrate that our method provides direct experimental access to the so-called symplectic norm of the stationary states of the system, which is tied to the particle- or holelike nature of excitations of these states. In this way, we establish a method to identify qualitative differences between the fluctuations around stabilized minima and maxima of the nonlinear out-of-equilibrium stationary states. Our method constitutes a versatile approach to characterize a wide class of driven-dissipative systems.

The field of nanoscale magnetic resonance imaging (NanoMRI) was started 30 years ago. It was motivated by the desire to image single molecules and molecular assemblies, such as proteins and virus particles, with near-atomic spatial resolution and on a length scale of 100 nm. Over the years, the NanoMRI field has also expanded to include the goal of useful high-resolution nuclear magnetic resonance (NMR) spectroscopy of molecules under ambient conditions, including samples up to the micron-scale. The realization of these goals requires the development of spin detection techniques that are many orders of magnitude more sensitive than conventional NMR and MRI, capable of detecting and controlling nanoscale ensembles of spins. Over the years, a number of different technical approaches to NanoMRI have emerged, each possessing a distinct set of capabilities for basic and applied areas of science. The goal of this roadmap article is to report the current state of the art in NanoMRI technologies, outline the areas where they are poised to have impact, identify the challenges that lie ahead, and propose methods to meet these challenges. This roadmap also shows how developments in NanoMRI techniques can lead to breakthroughs in emerging quantum science and technology applications.