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Flow of the nonlinear Mathieu's equation with pump-depletion nonlinearity [Eq. (A5)] in the AI vs. AR plane. Blue arrows represent the lines of the flow, black and green dots represent unstable and stable fixed point, respectively. We show the flow for four prototype cases: (a) for β = 0 and˜hand˜ and˜h < 2, (b) β > 0 and˜hand˜ and˜h < 2, (c) β > 0 and 2 < ˜ h < 4, and (d) β > 0 and˜hand˜ and˜h > 4.
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Periodically driven parametric oscillators offer a convenient way to simulate classical Ising spins. When many parametric oscillators are coupled dissipatively, they can be analogous to networks of Ising spins, forming an effective coherent Ising machine (CIM) that efficiently solves computationally hard optimization problems. In the companion pape...
Citations
... On the application side, CNMPs are one implementation of phase-based Ising systems 12 , which are intensely studied to solve computationally hard problems. ...
We demonstrate that parametrically excited eigenmodes in nearby nanomagnets can be coupled to each other. Both positive (in-phase) and negative (anti-phase) couplings can be realized by a combination of appropriately chosen geometry and excitation field frequency. The oscillations are sufficiently stable against thermal fluctuations. The phase relation between field-coupled nanomagnets shows a hysteretic behavior with the phase relation being locked over a wide frequency range. We envision that this computational study lays the groundwork to use field-coupled nanomagnets as parametrons as building blocks of logic devices, neuromorphic systems or Ising machines.
... strongly [28] and weakly coupled KPOs [29]. Previous experimental and theoretical studies of KPObased Ising simulators with bilinear coupling considered [19,26,[28][29][30][31][32][33][34]. For example, the state "down-up" shown in Fig. 1(b) can be identically replaced by "up-down", as the individual spin levels are degenerate. ...
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 work 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.
... Practically, the difficulty to implement the spin coupling interactions with the proposed hardware has become the main factor limiting scalability and performance for unconventional Ising simulators [27,28]. Also complete characterization of possible stable phases is necessary for estimating the Ising description and plays a key role to understand the working principle in spin systems [29][30][31][32][33]. ...
Recently, spatial photonic Ising machines (SPIMs) have demonstrated the abilities to compute the Ising Hamiltonian of large-scale spin systems, with the advantages of ultrafast speed and high power efficiency. However, such optical computations have been limited to specific Ising models with fully connected couplings. Here we develop a wavelength-division multiplexing SPIM to enable programmable spin couplings and external magnetic fields as well for general Ising models. We experimentally demonstrate such a wavelength-division multiplexing SPIM with a single spatial light modulator, where the gauge transformation is implemented to eliminate the impact of pixel alignment. To show the programmable capability of general spin coupling interactions, we explore three spin systems: $\pm J$ models, Sherrington-Kirkpatrick models, and only locally connected ${{J}_{1}}\texttt{-}{{J}_{2}}$ models and observe the phase transitions between the spin-glass, ferromagnetic and paramagnetic phases. These results show that the wavelength-division multiplexing approach has great programmable flexibility of spin couplings and external magnetic fields, which provides the opportunities to solve general combinatorial optimization problems with large-scale and on-demand SPIM.
... Although recent works have suggested exploiting this bi-stability to realize a coherent Ising machine (CIM) [1][2][3][4][5][6][7][8][9], we find the unique dynamics of coupled parametric oscillators far richer, and we point out that only under certain, rather narrow conditions can a network of coupled parametric oscillators help solving the Ising problem [10][11][12][13][14]. These works show that a steady state solution to a network of coupled parametric oscillators can be coherent, everlasting beats of power, as also manifested in this work. ...
... This result of permanent beats also contradicted the standard intuition of coupled active oscillators, which expects pump saturation to quickly eliminate coherent multi-mode dynamics due to the competition for pump resources between the participating modes. Indeed, even in [10,11] the persistent beating regime appeared only near the oscillation threshold, where pump saturation is low, and raising the pump-power further resulted in collapsing of both POs to a synchronized oscillation at a single frequency. Here we show that when both POs share a single pump, the pump saturation does not affect the oscillators' dynamics. ...
... Our experiment employed a new, frequency-domain approach to coupling parametric oscillators that used the different frequency modes of a single cavity as independent POs, which were coupled arbitrarily with an FPGA. Compared to the standard space [10,11,13,14] or time [1][2][3][4][7][8][9] coupling methods, this frequency approach expands the research of coupled POs in two aspects: First, the implementation of several POs within a single cavity that share a single pump allows to examine the role of pump competition and depletion, as we discussed. In addition, our frequency approach reveals the role of non-degenerate (two-mode) POs, that can now be considered on equal footing with degenerate POs, exploring the implications of the additional phase freedom for non-degenerate POs (the envelope phase) on the coupled dynamics. ...
The coherent dynamics in networks of coupled oscillators is of great interest in wave-physics since the coupling produces various dynamical effects, such as coherent energy exchange (beats) between the oscillators. However, it is common wisdom that these coherent dynamics are transients that quickly decay in active oscillators (e.g. lasers) since pump saturation causes mode competition that results, for homogeneous gain, in the prevalence of the single winning mode. We observe that pump saturation in coupled parametric oscillators counter-intuitively encourages the multi-mode dynamics of beating and indefinitely preserves it, despite the existence of mode competition. We explore in detail the coherent dynamics of a pair of coupled parametric oscillators with a shared pump and arbitrary coupling in a radio frequency (RF) experiment, as well as in simulation. Specifically, we realize two parametric oscillators as different frequency-modes of a single RF cavity and couple them arbitrarily using a digital high-bandwidth FPGA. We observe persistent coherent beats that are maintained at any pump level, even high above the threshold. The simulation highlights how the interplay of pump depletion between the two oscillators prevents them from synchronizing, even when the oscillation is deeply saturated.
... X μ + ðqÀ1ÞD X ðqÞ μ (see Fig. 2 for a pictorial representation with N = D = 2). The coupling matrix can in general be written as the sum of a symmetric and antisymmetric part, identifying the dissipative and energy-preserving part of the coupling, respectively 50,51 . A dissipative coupling is commonly considered when using POs for optimiziation 59 , while the energy-preserving coupling proposed in 30 inducing persistent coherent beats between POs has been recently used to realize photonic spiking neurons 60 . ...
... We show in panels b, d (blue color) the PO relative energy deviation ΔE/|E GS |, where ΔE = E PO − E GS , as a function of the pump amplitude deviation from threshold Δh = h − h th . The pump amplitude h varies from the analytical threshold to a numerically determined value, above which the PO amplitudes acquire a nonzero imaginary part 50 . The PO energy E PO is found from Eq. (3) by determining the hyperspins from the PO steadystate amplitudes σ ! ...
From condensed matter to quantum chromodynamics, multidimensional spins are a fundamental paradigm, with a pivotal role in combinatorial optimization and machine learning. Machines formed by coupled parametric oscillators can simulate spin models, but only for Ising or low-dimensional spins. Currently, machines implementing arbitrary dimensions remain a challenge. Here, we introduce and validate a hyperspin machine to simulate multidimensional continuous spin models. We realize high-dimensional spins by pumping groups of parametric oscillators, and show that the hyperspin machine finds to a very good approximation the ground state of complex graphs. The hyperspin machine can interpolate between different dimensions by tuning the coupling topology, a strategy that we call “dimensional annealing”. When interpolating between the XY and the Ising model, the dimensional annealing substantially increases the success probability compared to conventional Ising simulators. Hyperspin machines are a new computational model for combinatorial optimization. They can be realized by off-the-shelf hardware for ultrafast, large-scale applications in classical and quantum computing, condensed-matter physics, and fundamental studies.
... This analogy leads to the idea of using networks of coupled KPOs to build noisy intermediate-scale quantum (NISQ) machines. 16,17 These machines can simulate the dynamics of mathematical problems that overwhelm traditional computers, such as the ground state of an Ising Hamiltonian, [18][19][20][21][22][23][24][25][26][27] or of other complex systems that can be mapped onto the same framework. [28][29][30][31][32] An important quantity for many applications of TLSs is their lifetime s. 33 It is the typical time spent on a level before the interaction with an environment induces a (seemingly) spontaneous "jump" from one state to the other. ...
... The two responses belong to stable attractors (1 and 2) with opposite phases, i.e., v 1 ¼ Àv 2 (and u 1 ¼ Àu 2 ). 27,[50][51][52] To study switching between the phase states of our KPO, we apply white electrical noise n characterized by a standard deviation r V (over a bandwidth of 30 MHz) that causes the state of the resonator to fluctuate around its initial solution. If the fluctuations are large enough, they will occasionally carry the resonator across the threshold in the middle between the phase states. ...
The Kerr Parametric Oscillator (KPO) is a nonlinear resonator system that is often described as a synthetic two-level system. In the presence of noise, the system switches between two states via a fluctuating trajectory in phase space, instead of following a straight path. The presence of such fluctuating trajectories makes it hard to establish a precise count or even a useful definition, of the “lifetime” of the state. Addressing this issue, we compare several rate counting methods that allow to estimate a lifetime for the levels. In particular, we establish that a peak in the Allan variance of fluctuations can also be used to determine the levels' lifetime. Our work provides a basis for characterizing KPO networks for simulated annealing where an accurate determination of the state lifetime is of fundamental importance.
... We unify a variety of existing methods for the analysis of time-dependent nonlinear ODEs into an integrated framework and take advantage of open-source Julia libraries to achieve ease of use, flexibility, and high performance [ Fig. 1c]. Our package is readily applicable to a range of active fields where nonlinear harmonically-driven systems appear, such as modal analysis in structural dynamics [10,33], electric circuits [11,34,35], nonlinear optics [12,[36][37][38][39][40][41][42], optomechanics [22,[43][44][45], micro-and nanomechanics [18,[46][47][48][49][50][51][52][53][54][55][56][57], oscillator networks [58][59][60][61][62][63], Ising machines [64][65][66][67][68][69], and many-body light-matter systems [70][71][72][73][74][75][76]. ...
... x j (t) = F cos(ωt) , i = 1, 2, ..., N . (17) Similar systems have been explored in the context of combinatorial optimisation machines based on the mapping of effective spins to parametron networks [64][65][66][67][68][69]. The displacement of each oscillator, x i , is expanded using the harmonic oscillating at ω, giving the harmonic ansatz ...
... Its modular design paves the way for future methodological extensions, including detection of Hopf bifurcations [107], limit cycles and chaos [108], higher-order Krylov-Bogoliubov averaging method [109], as well as interfaces with existing dedicated libraries to treat nonlinear spatially-extended or quantum systems [110,111]. Its usage can assist a breadth of fields, where nonlinear harmonically-driven systems appear, such as modal analysis in structural dynamics [10,33], electric circuits [11,34,35], nonlinear optics [12,[36][37][38][39][40][41][42], optomechanics [22,[43][44][45], micro-and nanomechanics [18,46,[51][52][53][54][55][56][57], oscillator networks [58][59][60][61][62][63], Ising machines [64][65][66][67][68][69], and many-body light-matter systems [70][71][72][73][74][75][76]. ...
HarmonicBalance.jl is a publicly available Julia package designed to simplify and solve systems of periodic time-dependent nonlinear ordinary differential equations. Time dependence of the system parameters is treated with the harmonic balance method, which approximates the system’s behaviour as a set of harmonic terms with slowly-varying amplitudes. Under this approximation, the set of all possible steady-state responses follows from the solution of a polynomial system. In HarmonicBalance.jl, we combine harmonic balance with contemporary implementations of symbolic algebra and the homotopy continuation method to numerically determine all steady-state solutions and their associated fluctuation dynamics. For the exploration of involved steady-state topologies, we provide a simple graphical user interface, allowing for arbitrary solution observables and phase diagrams. HarmonicBalance.jl is a free software available at https://github.com/NonlinearOscillations/HarmonicBalance.jl.
... Parametric resonators are more naturally treated in this case in terms of the natural amplitudes x (refs. 47,48 ) or employing quadratures in a generalized rotating frame 49 . For modulation frequencies resonant with Δω ⟨ij⟩ , Σω ⟨ij⟩ , equation (7) is exactly time-independent. ...
Imposing chirality on a physical system engenders unconventional energy flow and responses, such as the Aharonov–Bohm effect1 and the topological quantum Hall phase for electrons in a symmetry-breaking magnetic field. Recently, great interest has arisen in combining that principle with broken Hermiticity to explore novel topological phases and applications2–16. Here we report phononic states with unique symmetries and dynamics that are formed when combining the controlled breaking of time-reversal symmetry with non-Hermitian dynamics. Both of these are induced through time-modulated radiation pressure forces in small nano-optomechanical networks. We observe chiral energy flow among mechanical resonators in a synthetic dimension and Aharonov–Bohm tuning of their eigenmodes. Introducing particle-non-conserving squeezing interactions, we observe a non-Hermitian Aharonov–Bohm effect in ring-shaped networks in which mechanical quasiparticles experience parametric gain. The resulting complex mode spectra indicate flux-tuning of squeezing, exceptional points, instabilities and unidirectional phononic amplification. This rich phenomenology points the way to exploring new non-Hermitian topological bosonic phases and applications in sensing and transport that exploit spatiotemporal symmetry breaking. Time-reversal symmetry breaking is combined with non-Hermitian dynamics in an optomechanical system with squeezing interactions to produce chirality in the system, and a non-Hermitian Aharonov–Bohm effect is observed.
... Such simulators have exciting potential applications in quantum computation. 20 In addition, the behavior of coupled parametric oscillators has been used to study the response of microelectromechanical systems (MEMs). 21 These examples provide only a small glimpse of the wide range of applications that these coupled oscillator systems have across a variety of scientific disciplines. ...
... [22][23][24] Although most real-world applications involve oscillators with differing phases or differing frequencies, most of the analysis that appears in the literature have ignored both of these essential features while also ignoring the effects of coupling strength, especially at frequencies away from parametric resonance. 20 Ignoring frequencies away from parametric resonance frequencies is natural if one is interested in studying phase-locking or if one has energy harvesting applications in mind. [25][26][27][28] However, real-world systems typically operate over a broad range of parametric, modulation frequencies, and many engineering and biological applications including coupled oscillator systems are designed to operate away from parametric resonance to avoid excessive vibration, noise, and accelerate fatigue. ...
... 40 However, these asymptotic methods fail to capture the full range of the parameters in question by restricting attention to regions of phase space for which the modulation frequency is the same or an integer multiple of the resonance frequency. 20,40 Moreover, as we demonstrate and explain in Sec. V, the effects of the oscillator's coupling term can make the higher order terms non-negligible. ...
Coupled, parametric oscillators are often studied in applied biology, physics, fluids, and many other disciplines. In this paper, we study a parametrically driven, coupled oscillator system where the individual oscillators are subjected to varying frequency and phase with a focus on the influence of the damping and coupling parameters away from parametric resonance frequencies. In particular, we study the long-term statistics of the oscillator system’s trajectories and stability. We present a novel, robust, and computationally efficient method, which has come to be known as an auxiliary function method for long-time averages, and we pair this method with classical, perturbative-asymptotic analysis to corroborate the results of this auxiliary function method. These paired methods are then used to compute the regions of stability for a coupled oscillator system. The objective is to explore the influence of higher order, coupling effects on the stability region across a broad range of modulation frequencies, including frequencies away from parametric resonances. We show that both simplified and more general asymptotic methods can be dangerously un-conservative in predicting the true regions of stability due to high order effects caused by coupling parameters. The differences between the true stability region and the approximate stability region can occur at physically relevant parameter values in regions away from parametric resonance. As an alternative to asymptotic methods, we show that the auxiliary function method for long-time averages is an efficient and robust means of computing true regions of stability across all possible initial conditions.
... The pump feeds the D oscillators, with saturation value h(1 − βI), where β is a saturation coefficient and I is the total PO energy. We describe the dynamics by D coupled Math-ieu's equations [30,50,51] x j +ω 2 ...
... The dynamics of the complex amplitudes X j is found from Eq. (1) by a multiple-scale expansion [50,52], as detailed in the supplementary information (SI). For a range of h values above threshold, the dynamics amplifies the amplitude real parts, and suppresses the imaginary parts. ...
... Furthermore, C denotes the coupling matrix, whose offdiagonal element C jl quantifies the coupling strength between any two POs x j and x l . The coupling matrix can in general be written as the sum of a symmetric and antisymmetric part, identifying the dissipative and energy-preserving part of the coupling, respectively [50,51]. Dissipative couplings are commonly considered when using POs for optimiziation [59], while energy-preserving couplings inducing persistent coherent beats between POs [30] have been recently proposed to realize photonic spiking neurons [60]. ...
From condensed matter to quantum chromodynamics, multidimensional spins are a fundamental paradigm, with a pivotal role in combinatorial optimization and machine learning. Machines formed by coupled parametric oscillators can simulate spin models, but only for Ising or low-dimensional spins. Currently, machines implementing arbitrary dimensions remain a challenge. Here, we introduce and validate a hyperspin machine to simulate multidimensional continuous spin models. We realize high-dimensional spins by pumping groups of parametric oscillators, and study NP-hard graphs of hyperspins. The hyperspin machine can interpolate between different dimensions by tuning the coupling topology, a strategy that we call "dimensional annealing". When interpolating between the XY and the Ising model, the dimensional annealing impressively increases the success probability compared to conventional Ising simulators. Hyperspin machines are a new computational model for combinatorial optimization. They can be realized by off-the-shelf hardware for ultrafast, large-scale applications in classical and quantum computing, condensed-matter physics, and fundamental studies.
