Coherent dynamics in frustrated coupled parametric oscillators
Marcello Calvanese Strinati, Igal Aharonovich, Shai Ben-Ami, Emanuele G. Dalla Torre, Leon Bello, Avi Pe'er
CCoherent dynamics in frustrated coupled parametricoscillators
Marcello Calvanese Strinati , Igal Aharonovich ,Shai Ben-Ami , Emanuele G. Dalla Torre , Leon Bello , andAvi Pe’er Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel Department of Physics and BINA Center of Nanotechnology, Bar-Ilan University,52900 Ramat-Gan, Israel
Abstract.
We explore the coherent dynamics in a small network of three coupledparametric oscillators and demonstrate the effect of frustration on the persistentbeating between them. Since a single-mode parametric oscillator represents an analogof a classical Ising spin, networks of coupled parametric oscillators are consideredas simulators of Ising spin models, aiming to efficiently calculate the ground stateof an Ising network - a computationally hard problem. However, the coherentdynamics of coupled parametric oscillators can be considerably richer than that ofIsing spins, depending on the nature of the coupling between them (energy preservingor dissipative), as was recently shown for two coupled parametric oscillators. Inparticular, when the energy-preserving coupling is dominant, the system displayseverlasting coherent beats, transcending the Ising description. Here, we extend thesefindings to three coupled parametric oscillators, focusing in particular on the effect offrustration of the dissipative coupling. We theoretically analyze the dynamics usingcoupled nonlinear Mathieu’s equations, and corroborate our theoretical findings bya numerical simulation that closely mimics the dynamics of the system in an actualexperiment. Our main finding is that frustration drastically modifies the dynamics.While in the absence of frustration the system is analogous to the two-oscillator case,frustration reverses the role of the coupling completely, and beats are found for smallenergy-preserving couplings. This result is specifically relevant to Ising simulators,where residual energy-preserving couplings may lead to coherent beats and preventthe system from converging to the Ising ground state.
1. Introduction
Parametric oscillators are a viable experimental platform to study the physics of timecrystals, i.e., systems that can spontaneously break time translational symmetry [1, 2].The possibility of the existence of such a phase of matter at equilibrium was firstproposed in 2012 by Frank Wilczek and collaborators [3, 4], both for quantum andclassical systems. The original proposal evokes the possibility for a system to break continuous time translational symmetry, in analogy with the formation of space crystalsin condensed matter where space translational symmetry is broken. Shortly after a r X i v : . [ n li n . C D ] M a y oherent dynamics in frustrated coupled parametric oscillators Floquet timecrystals , accounts for the fact that, under certain conditions, a periodically-drivensystem can break the discrete time translational symmetry enforced by the externaldrive [8–18]: Instead of merely following the external drive, the system undergoes aperiodic motion at a frequency that is different from that of the drive (see ref. [1] for areview).The periodically driven single-mode classical parametric oscillator is the canonicalexample of period-doubling instability (see refs. [19, 20] for an introduction), andrepresents the simplest case of a classical Floquet time crystal. Indeed, when excitedabove the amplification threshold, the parametric oscillator oscillates at half thefrequency of the drive and admits only two distinct phase solutions, dubbed “0” and“ π ”, with a relative shift in time by one period of the drive. One of the two solutions ischosen by the system depending on the initial conditions, a phenomenology analogousto a spontaneous breaking of a Z (Ising) symmetry. Because of this, a single degenerateparametric oscillator may be regarded as a classical bit, or an Ising spin, where the twostates “up” or “down” of the spin are given by the two distinct “0” and “ π ” solutions.Exploiting this property, networks of many coupled parametric oscillators have beenproposed as a platform, called coherent Ising machine (CIM) [21], to simulate thebehaviour of a network of many coupled Ising spins. Such a machine, whose experimentalrealization has been reported in refs. [22–24], is envisioned to solve the NP-hard problemof finding the ground state of the classical Ising model [25].In the last years, the analysis of various issues related to the computationalperformance of CIMs has been the focus of a remarkable amount of work [26–32]. Whilethe underlying assumption in CIMs is that a system of coupled parametric oscillatorsbehaves as a set of coupled Ising spins, we pointed out recently that already a pair ofcoupled parametric oscillators may display a much richer dynamics, beyond the Isingdescription, depending on the nature of the coupling (energy-preserving or dissipative)between the oscillators [33, 34]. Specifically, we studied in detail both theoretically andexperimentally in a radio-frequency experiment a pair of coupled parametric oscillators,which is the minimal system to explore nontrivial coupling effects. Our main findingwas that, when driven above the amplification threshold at the parametric resonancecondition, the two oscillators can either display persistent coherent beats when thecoupling is mostly energy preserving, or behave as a CIM [21] when the coupling ismostly dissipative.The existence of such a nontrivial dynamics in just a pair of coupled parametricoscillators opens the question on how the nature of the coupling affects the dynamicsof a larger network composed by more than two parametric oscillators, with potentialimplications in the specific context of CIMs, and also in the broad view of exploitinglarge-scale networks of coupled parametric oscillators to realize classical many-body oherent dynamics in frustrated coupled parametric oscillators
2. Two parametric oscillators
Before moving to the analysis of three-coupled oscillators, let us introduce the relevantnotation and analytical tools, via a review of the simple case of two degenerate coupledparametric oscillators, summarizing the main findings of refs. [33, 34]. oherent dynamics in frustrated coupled parametric oscillators x ( t ) x ( t ) c − c c c A R B R A R B R h g r/g (i)(ii) (iii)(iv) (iv) (i) (ii)(iii) (iv) (00)( ππ ) (00)( ππ ) (0 π )( π h h PA PA
Ising BeatsSub-threshold
Figure 1.
Schematic representation of the two-parametric-oscillator system. Eachparametric oscillator is described by a classical field x k , with k = 1 ,
2, driven byan external pump h ( t ) = h sin(2 ω t ) injected into a parametric amplifier (PA). Thecoupling between the oscillators is described in the general case by a coupling matrixthat accounts for (i) transmittance coefficients c and c , which renormalize theintrinsic losses of the oscillators, and (ii) coupling coefficients c and c , whichdetermine the rate of energy flow from oscillator 1 to 2, and vice versa . We consider a system of two identical single-mode parametric oscillators, with equalproper frequency ω , driven by an external pump field at frequency 2 ω and withamplitude h , injected into a parametric amplifier (PA) [36] as depicted in figure 1.The field inside each oscillator 1 (or 2) is identified by a classical variable x ( x ).The two oscillators are coupled by a power-splitter coupling [33], which accounts for: (i)transmission coefficients c and c for oscillator 1 and 2, respectively, which renormalizethe intrinsic loss of each oscillator, providing an overall loss rate that we denote by g ,and (ii) coupling coefficients c and c , which give the rate of energy exchange betweenthe two oscillators. In this framework, the fields x and x are coupled according to theequation (see Appendix A) (cid:32) ˙ x ˙ x (cid:33) = ω (cid:32) c − c (cid:33) (cid:32) x x (cid:33) , (1)where the dot denotes the time derivative. The dynamics of the two-oscillator systemis therefore described by a pair of coupled Mathieu’s equations [34] ¨ x + ω [1 + h (1 − β x ) sin(2 ω t )] x + ω g ˙ x − ω c ˙ x = 0¨ x + ω [1 + h (1 − β x ) sin(2 ω t )] x + ω g ˙ x + ω c ˙ x = 0 . (2)Equation (2) also includes a second-order nonlinearity in the amplitude of the pump field(hereafter referred to as “pump-depletion nonlinearity”), whose strength is quantifiedby β . Such a nonlinearity describes the fact that the intensity of the pump field insideeach oscillator is depleted by x and x , and in many experimental contexts capturesthe most relevant nonlinear process (see ref. [34] for a critical discussion). In thegeneral case, the rate of energy flow between the two oscillators can be unbalanced,i.e., c (cid:54) = c , indicating dissipation in the coupling itself. Without loss of generality, oherent dynamics in frustrated coupled parametric oscillators c = r − α and c = r + α , where r ≥ energy-preserving component of the coupling, whereas α ≥ dissipative one.The energy-preserving coupling r induces a coherent exchange of energy betweenthe two oscillators. Its energy-preserving nature follows from the fact that the equationsof motion (2), with β = 0, g = 0, and c = c = r , can be derived from the Hamilton’sequations [19] starting from the Hamiltonian H = p + p m + 12 m ω (cid:20) r h sin(2 ω t ) (cid:21) (cid:0) x + x (cid:1) + ω r p x − p x ) , (3)where p and p are the canonical momentum variables for x and x , respectively. Suchan Hamiltonian is analogous to that of a charged particle (charge q = mω r/
2) movingon a two-dimensional plane identified by the spatial coordinates ( x , x , z = 0), subjectto a vector potential A = ( − x , x , T , where T denotes the transposition (for thedetails of the derivation, see Appendix B).The dissipative coupling α , in contrast, introduces additional loss or gain terms [34],which give rise to the Ising-type coupling between the oscillators, and guides theconvergence of the two-oscillator system to the desired Ising ground state [34], asrequired by a CIM [21]. In the long-time limit, the two oscillators will prefer tolock according to the sign of α : In-phase for “ferromagnetic” coupling ( α > ππ ), or in anti-phase for “anti-ferromagnetic” coupling ( α < π ) or ( π We now review the effect of the interplay between r and α on the long-time dynamics ofthe system. To reach this goal, we employ a perturbative analysis that gives the solutionin terms of the slowly-varying dynamics of the unperturbed oscillators. That techniqueis known as multiple-scale expansion in the context of nonlinear dynamics [37], and asthe slow-varying amplitude approximation in quantum and nonlinear optics. We herereview the main steps, referring the interested reader to ref. [34] for complete description.We take the intra-cavity loss g as a small expansion parameter, and identify thefast-varying time scale as t = 2 π/ω - the period of the fast oscillations at half thepump frequency, whereas the slow-varying time scale is identified by τ = gt . One thenstudies the dynamics only of the slow-varying degrees of freedom, integrating out thefast-varying ones.In the unperturbed case, the two fields x and x are then explicitly expressedby separating the two time scales as x ( t, τ ) = A ( τ ) e iω t + A ∗ ( τ ) e − iω t and x ( t, τ ) = B ( τ ) e iω t + B ∗ ( τ ) e − iω t , for some complex amplitudes A ( τ ) and B ( τ ), in which the long-time dynamics is encoded, and A ∗ ( B ∗ ) is the complex conjugate of A ( B ). One thenstudies the equations of motion (2) in the limit in which all the coupling constants affect oherent dynamics in frustrated coupled parametric oscillators h , r , and α are taken proportional to g , i.e., one defines ˜ h = h/g , ˜ r = r/g , and˜ α = α/g . By defining the dimensionless time as ˜ τ = ω τ , one finds that the slow-varyingcomplex amplitudes A and B obey the following set of coupled first-order differentialequations [34]: ∂A∂ ˜ τ = ˜ h A ∗ − ˜ h β (cid:0) | A | A ∗ − A (cid:1) − A r + ˜ α B∂B∂ ˜ τ = ˜ h B ∗ − ˜ h β (cid:0) | B | B ∗ − B (cid:1) − B − ˜ r − ˜ α A , (4)Equation (4) can be further recast in terms of the real and imaginary parts of thecomplex amplitudes, A = A R + i A I and similarly B = B R + i B I , where A R ( B R ) and A I ( B I ) are, respectively, the real and imaginary parts of A ( B ). The long-time dynamics ofthe amplitudes is determined by the configuration of the fixed points ( A R , A I , B R , B I )of equation (4), and by studying whether they are stable or unstable by looking atthe eigenvalues of the Jacobian matrix J ( A, B ) of the system in equation (4) around aspecific fixed point. Note however that, sufficiently close to the oscillation threshold, theimaginary part of both oscillation amplitudes decays very quickly ( A I = B I = 0) [34] dueto the phase dependent amplification and squeezing in parametric oscillators, allowingto focus the discussion only on the dynamics of the real parts A R and B R .While in general the configuration of the fixed points depends on the form of thenonlinearity, especially far from the amplification threshold, most of the interestingphysics throughout this paper occurs close to the threshold, where nonlinear effects arenegligible and the system is almost linear. The properties of the system at threshold canbe found by studying the spectrum of the Jacobian matrix around the origin A = B = 0.Such an analysis is sufficient to understand if the parametric oscillators converge to asteady-state oscillation, corresponding to reaching the “0” or “ π ” solutions for eachoscillator, or not. It therefore provides informations on whether the overall systemmimics an Ising network, and behaves as a time crystal, or not [34].The general procedure is as follows: For a given set of parameters ˜ h , ˜ r and ˜ α , welook at the eigenvalue of the Jacobian matrix J (0 ,
0) with largest real part ( λ max , whichwe dub “most efficient eigenvalue” from now on). For a pump amplitude below threshold˜ h < ˜ h th , we have Re[ λ max ] <
0, implying that oscillations decay: lim t →∞ x , ( t ) = 0.The value of ˜ h th is defined as the value of ˜ h such that Re[ λ max ] = 0. For ˜ h > ˜ h th , wehave Re[ λ max ] >
0, indicating that parametric amplification sets in, and the amplitudes A and B of the fields x , ( t ) exponentially grow in time, until eventually stabilized bythe pump depletion nonlinearity.Importantly, for ˜ h = ˜ h th , the imaginary part of λ max determines the frequency ofthe beats at threshold ω B = ω g | Im[ λ max ] | between the two oscillators, which modulatethe aforementioned exponential growth of the amplitudes. The beat frequency ω B isthe key observable to describe the behaviour of the system as the oscillators are drivenabove the oscillation threshold: oherent dynamics in frustrated coupled parametric oscillators • When ω B = 0, A and B reach eventually constant values A R and B R , due to thepresence of the nonlinearity. Hence, in the long-time limit, x ( t ) ∼ A R cos( ω t )and similar for x ( t ). This is the situation where the system behaves as a timecrystal, and can simulate Ising spins. Indeed, if A R > x converges to the “0”solution, whereas if A R < x converges to the “ π ” solution, and analogously for x . The two-oscillator system then converges to some of the four possible “Ising”configurations (00), ( ππ ), (0 π ), and ( π • Instead, for ω B > A and B display persistent coherent beats . In such asituation, one has, in the long-time limit, x ( t ) ∼ A R cos( ω B t ) cos( ω t ) and x ( t ) ∼ B R sin( ω B t ) cos( ω t ), for some constant values A R and B R . The presenceof the beats implies that each oscillator x , periodically flips between the “0” and“ π ” solutions, and therefore the system neither obeys the Ising description, nor itbehaves as a time crystal, since the system periodically jumps between all the fourIsing configurations (00), ( ππ ), (0 π ), and ( π Figure 2 shows a concrete calculation of the phase diagram of the system in equation (4).We plot in the left panel the phase diagram in the h/ (2 g ) vs. r/g plane, where differentphases are represented by different colors and correspond to different configurations ofthe fixed points. In the right panels, we show the configuration of the fixed points (blackdots for unstable points and green dots for stable ones) of each phase in the left panel,in the B R vs. A R plane, where the red-arrowed curve represents the flow of B R ( τ ) vs A R ( τ ). This choice is justified by the fact that, A I = B I = 0 for the range of pumpamplitudes ˜ h as in the figure as mentioned before.In the phase diagram of figure 2, we identify the following main phases: (i) The sub-threshold phase, for a pump amplitude ˜ h < ˜ h th , where the origin A = B = 0 is the only stable fixed point of equation (4); (ii) The Ising or CIM phase, for ˜ h > ˜ h th , defined by the presence of two stable fixedpoints, related by the inversion symmetry A ↔ − A and B ↔ − B of equation (4),the origin being unstable. The solutions x ( t ) and x ( t ) display a fast oscillationat half the pump frequency and are phase-locked either at (00) or ( ππ ), for“ferromagnetic” ˜ α >
0, or either at (0 π ) or ( π
0) for “antiferromagnetic” ˜ α < (iii)
The beating phase, for ˜ h > ˜ h th , in which no stable fixed point is found, and thelong-time dynamics is attracted into a stable limit cycle , which is identified by anattractive isolated closed orbit [20] in the space of the amplitudes A and B . Inthis phase, x and x display fast oscillations at half the pump frequency whoseamplitude is modulated (beats) with a frequency that depends on ˜ r , ˜ α , and ˜ h [34]. (iv) A phase with four stable fixed points, corresponding to all the four Isingconfigurations (00), ( ππ ), (0 π ), and ( π oherent dynamics in frustrated coupled parametric oscillators A R B R A R B R (i) (ii)(iii) (iv) (00)( ππ ) (00)( ππ ) (0 π )( π h g r/g (i)(ii) (iii)(iv) (iv) Ising BeatsSub-threshold
Figure 2.
Dynamical phase diagram of two coupled parametric oscillators, describedby equation (4). (Left) Phase diagram in the h/ (2 g ) vs. r/g plane, for ˜ α = 0 .
15, and(Right) configuration of the fixed points in the B R vs. A R plane, where black andgreen dots represent unstable and stable points, respectively. The flow of the slowvarying amplitudes from the solution of equation (4) is marked by red curves. Thephase diagram consists of four main phases, characterized by different configurationsof the fixed points, as shown in right panels: (i) below the oscillation threshold, whereonly the origin A = B = 0 is a stable attractor; (ii) the Ising or CIM region, wherethe system has two stable fixed points, corresponding to the two ground-state Isingsolutions (00) and ( ππ ) (for “ferromagnetic” ˜ α > (iii) a phase in which a stablelimit cycle stabilizes the long-time dynamics, and the system displays coherent beats; (iv) a phase with four stable fixed points, corresponding to both ground-state (00) and( ππ ), and excited-state (0 π ) and ( π
0) configurations. The red dashed line in the phasediagram is the boundary for the oscillation threshold ˜ h th . Other phases with morethan four fixed points [34] are not labelled and not relevant for the present discussion,and additional unstable fixed points different from the origin are not shown for thesake of clarity. decoupled spins. This behaviour matches the one previously discussed in ref. [21].As in phase (ii) , the amplitude of the fast oscillations converge to a constant value,and the system behaves as a time crystal.Slightly above the amplification threshold ˜ h th , the CIM phase is found for ˜ r < ˜ α ( ω B = 0), whereas the beating phase is found for ˜ r > ˜ α ( ω B > r = ˜ α .
3. Three coupled parametric oscillators
The results discussed in the previous section pointed out the existence of a persistentcoherent beating dynamics in coupled parametric oscillators, not considered in thestandard analysis of CIMs. A natural question that arises is how the presence of such oherent dynamics in frustrated coupled parametric oscillators x ( t ) x ( t ) c − c x ( t ) c − c c − c h PA h PA h PA Figure 3.
Schematic representation of the three-parametric-oscillator system. Similarto figure 1, the parametric oscillators are described by a classical field x k ( k = 1 , , h ( t ) = h sin(2 ω t ). The coupling between theoscillators, describing the connectivity of the system, includes all the mutual couplings c jk for j, k = 1 , ,
3. The transmittance coefficients c jj for j = 1 , , a dynamics affects a more structured network with more than two coupled parametricoscillators. Here, we begin to address this question by extending the previous discussionto the case of three degenerate coupled parametric oscillators, which is the simplestconfiguration where one can systematically study the role of connectivity. Specifically,our main focus is to study how frustration of the dissipative coupling (which reflects theunderlying Ising model) affects the coherent dynamics of the system. A schematic representation of a three-oscillator system is shown in figure 3. Now, thecoupling matrix c between the oscillators, according to equation (1), is ˙ x ˙ x ˙ x = c c − c c − c − c x x x . (5)The system is now described by a set of three coupled nonlinear Mathieu’s equations,which we write in a compact form for the sake of simplicity ( j, k = 1 , , x j + ω (cid:2) h (cid:0) − β x j (cid:1) sin(2 ω t ) (cid:3) x j + ω g ˙ x j − ω (cid:88) k (cid:54) = j sgn( k − j ) c jk ˙ x k = 0 , (6)where sgn( · ) denotes the sign function. From equation (6), one obtains the correspondingmultiple-scale equations for the slow-varying amplitudes of the fields { x j } as insection 2.3 [equation (4)]. Here, we renormalize each element of the coupling matrix as˜ c jk = c jk /g , and to ease the notation, we use the symbols { X j ( τ ) } of the amplitudes,such that x j ( t, τ ) = X j ( τ ) e iω t + X ∗ j ( τ ) e − iω t . The equations for the complex amplitudes oherent dynamics in frustrated coupled parametric oscillators j, k = 1 , , ∂X j ∂ ˜ τ = ˜ h X ∗ j − X j − ˜ hβ (cid:0) | X j | X ∗ j − X j (cid:1) + 12 (cid:88) k (cid:54) = j sgn( k − j )˜ c jk X k . (7)As in section 2, we decompose the coupling matrix c jk in equations (5)-(7) in termsof an antisymmetric and symmetric part, respectively r jk and α jk . Physically, theantisymmetric part corresponds to the energy-preserving coupling, and the symmetricpart corresponds to the dissipative one. Due to this increase of parameter space withrespect to the case in section 2 (the coupling matrix now has in general six independentcomponents), we focus on a specific choice of the coupling matrix, with the ambition tohighlight the role of frustration in the dissipative components of the coupling matrix.We choose r jk = r for all j and k , and introduce two different dissipative couplings: α = η and α = α = α , so that the coupling matrix in equation (5) reads ˙ x ˙ x ˙ x = g r + ˜ η ˜ r + ˜ α − ˜ r + ˜ η r + ˜ α − ˜ r + ˜ α − ˜ r + ˜ α x x x . (8)In the rest of the paper, our goal is to study the physics of the system near thresholdas a function of the coupling parameters ˜ r , ˜ α , and ˜ η , where as before all quadraturesare real ( X ,I = X ,I = X ,I = 0). Before discussing this general case, we first focuson two main configurations of interest for the coupling matrix in equation (8). Namely,assuming ˜ r, ˜ α ≥
0: (i) the non-frustrated case, for ˜ η = ˜ α , and (ii) the fully-frustrated case, for ˜ η = − ˜ α . The reason why we focus on these two fine-tuned cases is because theyare, on one hand, easily analytically tractable, and on the other hand, they capture thedramatic effect of frustration in the dissipative coupling. We discuss the general caselater in section 3.4. First, we analytically discuss the threshold properties of the non-frustrated network,for ˜ η = ˜ α in equation (8). The most efficient eigenvalue λ max can be found by explicitinspection of the Jacobian matrix J ( X , X , X ) around the origin X = X = X = 0.We analyze separately the cases of ˜ r > ˜ α and ˜ r < ˜ α .For ˜ r > ˜ α , we have λ max = − / h/ − e iπ/ F (˜ r, ˜ α ) + e − i π/ G (˜ r, ˜ α ), where F (˜ r, ˜ α ) = ˜ r − ˜ α G (˜ r, ˜ α ) G (˜ r, ˜ α ) = 12 (cid:2)(cid:0) ˜ r − ˜ α (cid:1) (˜ r + ˜ α ) (cid:3) / . (9)Since λ max is complex, beats are found. From the imaginary parts of λ max , one can findthe expression of the beat frequency: ω B = gω √ (cid:0) ˜ r − ˜ α (cid:1) / (cid:104) (˜ r − ˜ α ) / + (˜ r + ˜ α ) / (cid:105) . (10)In particular, for ˜ r approaching ˜ α , the frequency of the beats reduces towards zero withthe critical behaviour of ω B ∼ (˜ r − ˜ α ) / , differently from the critical exponent of 1 / oherent dynamics in frustrated coupled parametric oscillators X ,R X ,R X ,R (000)( πππ ) ˜ η = ˜ α ˜ η = − ˜ α (0 ππ )(0 π π π π ) (000)( πππ ) X ,R X ,R X ,R Figure 4.
Stability diagram of three parametric oscillators in the ( X ,R , X ,R , X ,R )space. Green dots represent stable configurations of oscillations for ˜ r = 0 and ˜ α > X ,R = 0plane. (Left panel) In the non-frustrated network (˜ η = ˜ α ), the two possible phase-locked configurations are (000) and ( πππ ), which correspond to the ferromagneticIsing solutions. The system converges to one of them as long as ˜ r < ˜ α ( ω B = 0 for˜ r < ˜ α ). (Right panel) In the frustrated case (˜ η = − ˜ α ) there are six possible phase-locked configurations: (0 ππ ), (0 π π π π ), and ( πππ ), which correspondto the six degenerate Ising ground states. In contrast to the non-frustrated case, anyinfinitesimal ˜ r > ω B > < ˜ r < ˜ α ). When ˜ r < ˜ α , we find that λ max = − / h/ F ( ˜ α, − ˜ r ) + G ( ˜ α, − ˜ r ). Now, λ max isreal, and above the oscillation threshold, parametric amplification occurs without beats, ω B = 0. In this case, the phase-locked steady-state oscillations correspond to the two“ferromagnetic” configurations (000) or ( πππ ), as shown in the left panel figure 4 (in thefigure specifically for ˜ r = 0). As one may expect, the behaviour of the non-frustratednetwork is qualitatively the same as the behaviour of two coupled oscillators (section 2). We now move to the case of the fully-frustrated network, i.e., ˜ η = − ˜ α in equation (8).By proceeding as before, we find that, for ˜ r > ˜ α , the most efficient eigenvalue is λ max = − / h/ F (˜ r, − ˜ α ) − G (˜ r, − ˜ α ). Now, in stark contrast to the non-frustratedcase, λ max is real, and above the oscillation threshold parametric amplification occurswithout beats, ω B = 0.For ˜ r < ˜ α , instead, λ max reads as λ max = − / h/ e i π/ F ( ˜ α, ˜ r ) + e − i π/ G ( ˜ α, ˜ r ).Therefore, the parametric oscillation occurs with beats, where the beat frequency is ω B = gω √ (cid:0) ˜ α − ˜ r (cid:1) / (cid:104) − ( ˜ α − ˜ r ) / + ( ˜ α + ˜ r ) / (cid:105) . (11)One can see by inspection that, when ˜ r < ˜ α , the presence of the limit cycle at thresholdmakes the system periodically flip between six possible phase-locked configurations, oherent dynamics in frustrated coupled parametric oscillators ω B /αrα η/α (cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:1)(cid:2)(cid:4)(cid:1)(cid:1)(cid:2)(cid:5)(cid:4)(cid:6)(cid:2)(cid:1)(cid:1)(cid:6)(cid:2)(cid:3)(cid:4) ω B /α (cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:1)(cid:2)(cid:4)(cid:1)(cid:1)(cid:2)(cid:5)(cid:4)(cid:6)(cid:2)(cid:1)(cid:1)(cid:6)(cid:2)(cid:3)(cid:4) (a) (b) η/α Theory Simulated experiment
IsingBeatsBeatsNonIsing
Figure 5.
Phase diagram of the three-oscillator case, showing the colormap plotsof the frequency of the beats ω B /α at threshold in the r/α vs. η/α plane. Thevertical red lines mark the values η = ± α of non-frustrated and fully-frustratednetwork (see sections 3.2 and 3.3), and green lines are calculated boundaries betweenthe phase-locking regions where ω B = 0 (dark blue) and the beating regions where ω B > (a) Analytical value from the theory presented insections 3.4 and Appendix C. The two phase-locking regions are labelled by “Ising”and “Non Ising” depending on whether the systems behaves correctly as a CIM ornot (see section 3.4). (b)
Low-level simulation of the experiment (see section 4).The experimental simulation agrees exceptionally well with the predicted theoreticalbehaviour, apart from small deviations (some noisy regions and small discrepanciesnear the phase boundaries) that we ascribe mostly to the difficulty of estimating theoscillation threshold in the experimental simulation for some values of the systemparameters. namely, (0 ππ ), (0 π π π π ), and ( πππ ), corresponding to the sixdegenerate ground-state configurations of the frustrated Ising model. One of theseconfigurations stabilizes the long-time dynamics only when ˜ r = 0 (see figure 4, rightpanel). The frustration of one of the dissipative components of the coupling has thereforea dramatic effect on the coherent dynamics of the network. In the non-frustrated case,the system at threshold converges to a phase-locked configuration for ˜ r < ˜ α , and displayspersistent beats otherwise. Instead, the behaviour of the fully-frustrated network is reversed with respect to the non-frustrated case: It presents beats for ˜ r < ˜ α , andconverges to phase-locked oscillations otherwise. For fixed ˜ α , the frequency of the beatsincreases linearly from zero for small ˜ r , and goes to zero as ˜ r approaches ˜ α with thesame critical exponent 1 / We now expand the discussion to consider the general case of equation (8), whichinterpolates between the non-frustrated and the frustrated cases, and generalizes the oherent dynamics in frustrated coupled parametric oscillators Label size for LaTeXiT: 22 (000) , ( πππ )( π , ( π π )(0 ππ ) , (0 π π ) , ( ππ Figure 6.
Ising energy levels relative to the ground-state energy as a function of η/α ,from the classical Ising energy that the dissipative components of the coupling reflect: E ising = − ( η/ σ σ − ( α/ σ σ + σ σ ), where σ j = ± π ”) of x j . The three energylevels are shown in different colors for different states: blue for (000) and ( πππ ), redfor ( π π π ), (0 ππ ), and (0 π π ) and ( ππ η/α = −
1, where the six states ( π π π ),(0 ππ ), (0 π πππ ) become degenerate. analysis in sections 3.2 and 3.3. Here, one can find the frequency of the beats at threshold ω B from the imaginary part of the most efficient eigenvalue, for a fixed ˜ α , as a functionof ˜ r and ˜ η , and discern regions in parameter space in which phase-locked oscillationsor beats are observed. We present our findings in figure 5 for both theory [panel (a) ]and low-level simulation of the experiment [panel (b) ], which will be discussed in detailin the next section. To ease the comparison between the analytical prediction and thelow-level simulation results, we express the frequency of the beats in units of α . Thismakes the frequency of the beats ω B /α be a function of r/α multiplied by pure numbersand independent of g (see sections 3.2 and 3.3, and Appendix C).Panel (a) of figure 5 shows ω B /α in the r/α vs. η/α plane, where the red lines markthe special cases of fully frustrated and totally non-frustrated Ising coupling ( η/α = ± ω B = 0, dark blue) from those where persistent beating ismanifested ( ω B >
0, other colors). We see that, for η/α >
0, a region of beats is foundwhen the energy-preserving coupling dominates over the dissipative coupling ( r > α ),and phase-locking is found otherwise, similar to the two-oscillator analysis and to thenon-frustrated case. However, when η/α <
0, a “tooth-shaped” region of beats appearswhen the dissipative coupling dominates ( r < α ), and phase locking is found otherwise.As r is lowered towards r = 0, the width of the tooth region of beats decreases until it oherent dynamics in frustrated coupled parametric oscillators r = 0, at η = − α , i.e., the fully-frustrated point of section 3.3.This peculiar behaviour of beating for small r/α may have important implicationsin the context of CIMs. For example, fixing r and scanning η from positive to negativeallows to interpolate between the ferromagnetic non-frustrated ( η = α ), and fullyfrustrated ( η = − α ) Ising models, where the transition to the fully-frustrated caseoccurs at η/α = −
1. At the transition point, the Ising gap (i.e., the energy differencebetween the ground- and first-excited configurations) closes, causing the multiplicity ofthe ground state to increase (in our case, from two to six, see blue and red curves infigure 6). Our findings suggest that any vanishingly small energy-preserving couplinginduces coherent beating between the oscillators as the Ising gap becomes vanishinglysmall, preventing the system from converging to the Ising ground-state configuration.We therefore conclude that our findings provide a first explicit example where thepresence of an energy-preserving coupling (even if small) represents a source of errorfor CIMs.In addition, we find that, while the three-oscillator systems correctly behaves as aCIM at the fully-frustrated point and in the phase-locking “Ising” region in figure 5, inthe other phase-locking (“Non Ising”) region, the system does not yield the expectedIsing behaviour. Indeed, the Ising model predicts a four-fold degenerate ground-statewhen η < − α (see red curve in figure 6). However, in the “Non Ising” region in figure 5,the oscillator system slightly above the threshold converges only to two fixed points. Inparticular, we find the following behavour: • For r = 0 and r = α , the two fixed points are found on the X ,R = 0 plane,implying that lim t →∞ x ( t ) = 0, and the other two oscillators converge to (0 π ) or( π • For 0 < r < α , the two fixed points correspond to the states (0 ππ ) and ( π • For r > α , the two fixed points correspond to the states (0 π
0) and ( π π ). Thesetwo configurations, for η < − α , are two of the four ground states, as before, but for − α < η <
0, they correspond to two of the four excited states of the Ising model.This finding hints that frustration may cause phase-locked oscillation at threshold thathowever transcend the Ising description. A deeper analysis on these points requiresfurther analysis of larger spin models, which is beyond the scope of the presentmanuscript, and it is left for future work.
4. Numerical simulation of the experimental implementation
To corroborate the previous results and confirm our analytical predictions, we conducteda direct numerical simulation of the dynamics of the field inside a parametric-oscillatorcavity with three (or more) modes, aiming to emulate as closely as possible the dynamicsof a future experimental setup. In such an experiment, we intend to couple between oherent dynamics in frustrated coupled parametric oscillators
We consider a multimode cavity where each temporal slot acts as an independentparametric oscillator, dynamically coupled to the other modes. In our simulation, thefield inside the cavity propagates as illustrated in the block diagram in figure 7. At eachround trip inside the cavity, the time signal is partitioned into N = 3 time slots, andparametrized as a three-dimensional vector S = ( S , S , S ) T . In each such interval, thefield is assumed to vary slowly and is amplified independently of the other time slots.This is a reasonable assumption since the parametric gain is an instantaneous processand the pump for each time slot is uncoupled from that of the other slots. Thus, eachtime slot defines a distinct parametrically driven mode that without coupling evolvesindependently from the others. Furthermore, additive noise is fed inside the cavity ateach round trip through an output coupler device to simulate thermal noise or vacuumfluctuations. During the first round trip, before being injected into the parametricamplifier, the signal inside the cavity consists of noise alone.A round trip inside the cavity is identified by the following steps: First, the pumpfield and the signal are injected into the parametric amplifier. Importantly, since ourgoal is to probe the linear, near threshold, properties of the system, the pump intensityis set slightly above the oscillation threshold (section 3). At the output of the parametricamplifier, the residual pump field is blocked (dark grey parallel lines in figure 7) andthe signal is injected into a coupler, which splits the field according to transmission andreflection coefficients T c = 1 / R c = (cid:112) − T . The transmitted signal T c S is sentinto a coupling mechanism, which implements the coupling matrix c of equation (8). Ina future experiment, such a coupling mechanism will be implemented by an FPGA. Afterthe coupling, the coupled signal F = T c c S and the reflected signal R c S are combinedon another coupling device, again with transmission and reflection coefficients T c and R c . At this step, the reflected signal T c R c S + R c F is blocked (contributing to theoverall cavity losses), and the transmitted signal R S + T c F is fed back into the cavity.Last, the output coupler with transmission and reflection coefficients T out = 1 / R out = (cid:112) − T allows to couple out the field T out ( R S + T c F ) from the cavity andanalyze it both it in time and frequency, and the remaining field R out ( R S + T c F ), fedwith noise, is input together with the pump field into the parametric amplifier to startthe next round trip. oherent dynamics in frustrated coupled parametric oscillators Output Modecoupling c Pump Parametricampli fi erNoiseSignal SignalPump ( R c , T c )( R c , T c )( R out , T out ) S T c S R c S R S + T c F T c R c S + R c F F
Figure 7.
Schematic of the simulated experiment. At each round trip, the fieldinside the cavity is fed with noise and injected together with the pump field into aparametric amplifier. After the parametric amplification, part of the signal is sentinto the coupling mechanism that couples between the time slots, and injected backinto the cavity. At the end of the round trip, part of the signal is extracted from thecavity and measured. The green backslashed boxes denote a coupler device, identifiedby reflection and transmission coefficients (
R, T ), whose values are chosen differentlydepending on the simulation step [( R c , T c ) for mode coupling, or ( R out , T out ) for outputcoupling and noise feeding]. For a given set of parameters, we run the simulation over a sufficient number ofround trips n rt in order for the oscillation to reach a steady state. We then examinethe steady-state dynamics to obtain the slow-varying amplitude of all oscillators andnumerically extract the frequency of the beats. The numerical frequency of the beats is measured by computing the fast Fouriertransform (FFT) of the signal at the output to identify the frequency componentwith largest amplitude, for different values of r/α and/or η/α . In the numerics, thefrequencies obtained from the FFT are given in units of 1 /τ sim , where the total simulationtime is τ sim = n rt τ rt , and round-trip time τ rt in our numerical context is an arbitrarytime scale. In order to quantitatively compare the numerical results with the analyticalprediction, we express the frequency of the beats in units of α (section 3.4) and use τ rt = 0 . r/α for both thenumerical simulation of the experiment and the theoretical analysis. We comparethe results in the special cases of the non-frustrated network ( η = α ) and the fully-frustrated one ( η = − α ). As evident, the numerical results in both cases agree wellwith the theoretical curves, with some slight deviations especially in the fully-frustratedcases, which we ascribe to the difficulty of correctly estimating the oscillation thresholdand therefore choosing the proper value of h , due to both noise and nonlinearities.Indeed, it has been shown that pump-depletion nonlinearity tends to lower the beating oherent dynamics in frustrated coupled parametric oscillators r/α r/αω B α η = α η = − α (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:7)(cid:2)(cid:1) (cid:7)(cid:2)(cid:3) (cid:7)(cid:2)(cid:4)(cid:1)(cid:2)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:8)(cid:1)(cid:2)(cid:7)(cid:1)(cid:1)(cid:2)(cid:7)(cid:8)(cid:1)(cid:2)(cid:3)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:4)(cid:2)(cid:1) (cid:4)(cid:2)(cid:3) (cid:5)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:3) Figure 8.
Frequency of the beats at threshold ω B /α as a function of r/α , (Left)in the non-frustrated network ( η = α ), and (Right) in the fully-frustrated network( η = − α ). The numerical data (red points) obtained from the simulation of theexperiment with τ rt = 0 . frequency (divergence of the period of the beats), eventually inducing phase-locking as h is increased above the oscillation threshold [33].Last, in figure 5, panel (b) , we evaluate the beating frequency ω B /α as a functionof r/α and η/α , in order to verify the theoretical phase diagram in panel (a) . Theanalytical phase boundary, marked by the green line, which is the same for both panelsof figure 5, is superimposed to the numerical phase diagram to ease comparison betweentheory and simulated experiment. As evident, our numerical data agree exceptionallywell with the analytical prediction also in the interpolating case.
5. Conclusions
We analyzed the behaviour of three coupled degenerate parametric oscillators - theminimal case to study nontrivial coupling and connectivity effects. By extendingour previous work on two coupled parametric oscillators, we modelled the system asthree coupled Mathieu’s equations, where the coupling between any two oscillatorsis comprised of both energy-preserving and dissipative components. We focused inparticular on two main cases of connectivity, namely, for frustrated and non-frustrateddissipative coupling. Our theoretical predictions, obtained by linearizing the effectiveequations of motion, were confirmed by a direct numerical simulation in time of thedynamics inside a parametric oscillator cavity, as it would be implemented in an actualexperiment. The good agreement between the results obtained by these two differentapproaches strengthens the fact that the coupled nonlinear Mathieu’s equations capturethe relevant dynamics of coupled parametric oscillators.Our main finding was that frustration of the dissipative component of the couplinghas a dramatic effect on the coherent dynamics of the system. While in the non-frustrated case the system phase locks once the dissipative coupling exceeds theenergy-preserving one, and behaves as a CIM, in qualitative agreement with the oherent dynamics in frustrated coupled parametric oscillators
Acknowledgements
We thank Itzhack Dana for fruitful discussions. A. P. acknowledges support from theIsrael Science Foundation (ISF) Grants No. 44/14 and U.S.-Israel Binational ScienceFoundation (BSF) Grant No. 2017743. M. C. S. acknowledges support from the ISFGrants No. 231/14, 1452/14, and 993/19, and BSF Grants No. 2016130 and 2018726.
Appendix A. Derivation of the power-splitter coupling
In this appendix, we explicit the origin of the power-splitter coupling as in equation (1).In an actual physical implementation, the parametric oscillators are realized by anonlinear cavity. The fields x and x inside each cavity propagate with a characteristicround-trip time τ rt = D/v , which depends on the linear dimension D of the cavity, andon the field propagation velocity v inside the cavity.The fields after n + 1 round-trip times, t n +1 relate to the fields after n round-trip oherent dynamics in frustrated coupled parametric oscillators t n = n τ rt , for n = 0 , , , . . . , via the splitter matrix as (cid:32) x ( t n +1 ) x ( t n +1 ) (cid:33) = (cid:32) c c − c c (cid:33) (cid:32) x ( t n ) x ( t n ) (cid:33) . (A.1)We consider c = c ≡ c , where 0 ≤ c ≤ x ( t n +1 ) = c x ( t n ) + c x ( t n ) x ( t n +1 ) = c x ( t n ) − c x ( t n ) , (A.2)and by rewriting c x , = x , − (1 − c ) x , , equation (A.2) becomes x ( t n +1 ) = x ( t n ) − (1 − c ) x ( t n ) + c x ( t n ) x ( t n +1 ) = x ( t n ) − (1 − c ) x ( t n ) − c x ( t n ) . (A.3)Since x , ( t n +1 ) − x , ( t n ) ∝ ˙ x , ( t ) /ω , equation (A.3) can be rewritten as ˙ x = − ω (1 − c ) x + ω c x ˙ x = − ω (1 − c ) x − ω c x . (A.4)Without loss of generality, we consider c >
0. The terms proportional to 1 − c inequation (A.4) can be seen as loss terms that can be absorbed into the definition of g ,the intrinsic loss of the cavities. Therefore, by taking the time derivative on both sides ofequation (A.4), and by including this coupling in the equations of motion, equation (2)is obtained. Appendix B. Hamiltonian for the power-splitter coupling
In this appendix, we report the derivation of the equations of motion (2) (with β = 0and g = 0) from the Hamilton’s equations, in order to show that the coupling with c = c = r is indeed energy preserving. First, one takes the two oscillatorsfields, x and x , and their conjugate momentum variables, p and p , and defines thevectors of canonical coordinates p = ( p , p ) T and x = ( x , x ) T , where T denotes thetransposition. The Hamiltonian of the system is analogous to that of a particle of charge q in a two-dimensional plane, in a vector potential along the z -axis (i.e., perpendicularto the plane), given by A = ( − x , x , T , so that the corresponding effective magneticfield is B = ∇ × A = ˆ z ( ∂ x A x − ∂ x A x ) = ˆ z H = 12 m ( p − q A ) + 12 mω [1 + h sin(2 ω t )] x = p m + 12 mω [1 + h sin(2 ω t ) x − qm p · A + q m A , (B.1)or explicitly in terms of the canonical variables x , x , and p , p H = p + p m + 12 m ω [1+ h sin(2 ω t )]( x + x )+ q m ( x + x )+ qm ( p x − p x ) . (B.2) oherent dynamics in frustrated coupled parametric oscillators { ˙ x } variables are˙ x = ∂H∂p = p m + qm x ˙ x = ∂H∂p = p m − qm x , (B.3)and the Hamilton’s equations for the { ˙ p } variables are˙ p = − ∂H∂x = − mω [1 + h sin(2 ω t )] x − q m x + qm p ˙ p = − ∂H∂x = − mω [1 + h sin(2 ω t )] x − q m x − qm p . (B.4)From equation (B.3), by deriving both sides with respect to time, one has˙ p = m ¨ x − q ˙ x ˙ p = m ¨ x + q ˙ x . (B.5)By substituting ˙ p and ˙ p in the left-hand sides of equation (B.5) with the expressionsin equation (B.4), one has m ¨ x − q ˙ x = − mω [1 + h sin(2 ω t )] x − q m x + qm p m ¨ x + q ˙ x = − mω [1 + h sin(2 ω t )] x − q m x − qm p , (B.6)but, from equation (B.3), one has p /m = ˙ x − ( q/m ) x and p /m = ˙ x + ( q/m ) x that,when substituted in the right-hand side of equation (B.6), yields m ¨ x − q ˙ x = − mω [1 + h sin(2 ω t )] x + q ˙ x m ¨ x + q ˙ x = − mω [1 + h sin(2 ω t )] x − q ˙ x , (B.7)from which one obtains the equations of motion¨ x + ω [1 + h sin(2 ω t )] x − (2 q/m ) ˙ x = 0¨ x + ω [1 + h sin(2 ω t )] x + (2 q/m ) ˙ x = 0 , (B.8)which are indeed the equations of motion in equation (2) with c = c = r , β = 0, and g = 0, where ω r = 2 q/m . Appendix C. Jacobian matrix spectrum in the interpolating case
In this appendix, we report the expressions of the eigenvalues of the Jacobian matrixaround the origin in the interpolating case discussed in section 3.4 (see also figure 5).By generalizing the functions in equation (9), one has G (˜ r, ˜ α, ˜ η ) = 12 ˜ η (cid:0) ˜ r − ˜ α (cid:1) + (cid:115)(cid:18) ˜ r − α + ˜ η (cid:19) + η (˜ r − ˜ α ) / (C.1) F (˜ r, ˜ α, ˜ η ) = ˜ r − (2 ˜ α + ˜ η ) / G (˜ r, ˜ α, ˜ η ) , (C.2) oherent dynamics in frustrated coupled parametric oscillators r/α r/αω B α η = α η = − α (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:7)(cid:2)(cid:1) (cid:7)(cid:2)(cid:3) (cid:7)(cid:2)(cid:4)(cid:1)(cid:2)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:8)(cid:1)(cid:2)(cid:7)(cid:1)(cid:1)(cid:2)(cid:7)(cid:8)(cid:1)(cid:2)(cid:3)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:4)(cid:2)(cid:1) (cid:4)(cid:2)(cid:3) (cid:5)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:3) x τ rt = 0 . τ rt = 0 . τ rt = 0 . x τ rt = 0 . τ rt = 0 . τ rt = 0 . Figure D1.
Beat frequency at threshold ω B as a function of r/α , (Left panel) in thenon-frustrated case and (Right panel) fully-frustrated case, as in figure 8. The datafrom the simulated experiment for three different values of τ rt as in the legends (purplecircles for τ rt = 0 . τ rt = 0 . τ rt = 0 . and then the eigenvalues of the Jacobian that can have a positive real part can bewritten as λ (˜ r, ˜ α, ˜ η ) = −
12 + ˜ h F (˜ r, ˜ α, ˜ η ) − G (˜ r, ˜ α, ˜ η ) (C.3) λ (˜ r, ˜ α, ˜ η ) = −
12 + ˜ h − e iπ/ F (˜ r, ˜ α, ˜ η ) + e − iπ/ G (˜ r, ˜ α, ˜ η ) (C.4) λ (˜ r, ˜ α, ˜ η ) = −
12 + ˜ h − e − iπ/ F (˜ r, ˜ α, ˜ η ) + e iπ/ G (˜ r, ˜ α, ˜ η ) . (C.5)By finding the most efficient eigenvalue λ max (˜ r, ˜ α, ˜ η ) according to the usual condition(the eigenvalue with largest real part), the frequency of the beats at threshold reads ω B (˜ r, ˜ α, ˜ η ) = ω g | Im[ λ max (˜ r, ˜ α, ˜ η )] | . Appendix D. Additional details on the choice of the round-trip time
In this appendix, we provide some details on the choice of τ rt discussed in figure 8. Infigure D1, we show the comparison between the theoretical expressions of the frequencyof the beats at threshold [equations (10) and (11)] ω B /α , as a function of r/α , andthe data from the simulated experiment for different values of the round-trip time τ rt ,as in the legends. The effect of changing τ rt is to renormalize the frequency units forthe simulated experiment. Because of the excellent agreement between theory and datafrom the simulated experiment in the non-frustrated case for τ rt = 0 . τ rt was chosen to quantitatively match the theoretical phase diagram in figure 5. Indeed,as evident, for this τ rt , theory and data are essentially overlapped. References [1] Sacha K and Zakrzewski J 2017
Rep. Prog. Phys. oherent dynamics in frustrated coupled parametric oscillators [2] Khemani V, Moessner R and Sondhi S L 2019 arXiv:1910.10745 [3] Wilczek F 2012 Phys. Rev. Lett. (16) 160401[4] Shapere A and Wilczek F 2012
Phys. Rev. Lett. (16) 160402[5] Bruno P 2013
Phys. Rev. Lett. (7) 070402[6] Nozi`eres P 2013
EPL (Europhysics Letters)
Phys. Rev. Lett. (25) 251603[8] Sacha K 2015
Phys. Rev. A (3) 033617[9] Khemani V, Lazarides A, Moessner R and Sondhi S L 2016 Phys. Rev. Lett. (25) 250401[10] Else D V, Bauer B and Nayak C 2016
Phys. Rev. Lett. (9) 090402[11] von Keyserlingk C W, Khemani V and Sondhi S L 2016
Phys. Rev. B (8) 085112[12] Khemani V, von Keyserlingk C W and Sondhi S L 2017 Phys. Rev. B (11) 115127[13] Yao N Y, Potter A C, Potirniche I D and Vishwanath A 2017 Phys. Rev. Lett. (3) 030401[14] Else D V, Bauer B and Nayak C 2017
Phys. Rev. X (1) 011026[15] Yao Y N, Nayak C, Balents L and Zaletel P M 2018 arXiv:1801.02628 [16] O’Sullivan J, Lunt O, Zollitsch C W, Thewalt M L W, Morton J J L and Pal A 2018 arXiv:1807.09884 [17] Yao N Y and Nayak C 2018 Phys. Today No. 9, 40–47[18] Gambetta F M, Carollo F, Marcuzzi M, Garrahan J P and Lesanovsky I 2019
Phys. Rev. Lett. (1) 015701[19] Landau L D and Lifshitz E M 1982
Mechanics (Elsevier Science, Amsterdam)[20] Strogatz S H 2007
Nonlinear Dynamics And Chaos
Studies in nonlinearity (Perseus Books,Reading) ISBN 9788187169857[21] Wang Z, Marandi A, Wen K, Byer R L and Yamamoto Y 2013
Phys. Rev. A (6) 063853[22] Inagaki T, Inaba K, Hamerly R, Inoue K, Yamamoto Y and Takesue H 2016 Nature Photonics njp Quantum Information Nat. Commun. J. Phys. A arXiv:1806.08422 [27] Hamerly R, Inagaki T, McMahon P L, Venturelli D, Marandi A, Onodera T, Ng E, Rieffel E,Fejer M M, Utsunomiya S, Takesue H and Yamamoto Y 2018 Quantum vs. optical annealing:Benchmarking the opo ising machine and d-wave Proceedings of the Conference on Lasers andElectro-Optics (Optical Society of America, San Jose) p FTu4A.2[28] Hamerly R, Inagaki T, McMahon P L, Venturelli D, Marandi A, Onodera T, Ng E, Langrock C,Inaba K, Honjo T, Enbutsu K, Umeki T, Kasahara R, Utsunomiya S, Kako S, KawarabayashiK, Byer R L, Fejer M M, Mabuchi H, Englund D, Rieffel E, Takesue H and Yamamoto Y 2019
Sci. Adv. eaau0823[29] Cervera-Lierta A 2018 Quantum
114 ISSN 2521-327X[30] Pierangeli D, Marcucci G and Conti C 2019
Phys. Rev. Lett. (21) 213902[31] Tiunov E S, Ulanov A E and Lvovsky A I 2019
Opt. Express Unconventional Computation and Natural Computation (Springer,Cham) pp 232–256.[33] Bello L, Calvanese Strinati M, Dalla Torre E G and Pe’er A 2019
Phys. Rev. Lett. (8) 083901[34] Calvanese Strinati M, Bello L, Pe’er A and Dalla Torre E G 2019
Phys. Rev. A (2) 023835[35] Heugel T L, Oscity M, Eichler A, Zilberberg O and Chitra R 2019
Phys. Rev. Lett. (12) 124301[36] Boyd R 2008
Nonlinear Optics (Elsevier Science)[37] Kevorkian J and Cole J 1996