Dynamics of coupled parametric oscillators beyond coupled Ising spins
Coupled parametric oscillators have been recently employed as simulators of artificial Ising networks, with the potential to efficiently solve computationally hard minimization problems. We report on a detailed study of two coupled degenerate parametric oscillators, exploring the entire phase diagram, in terms of pump power, phase and coupling strength, both analytically and experimentally in a radio-frequency (RF) experiment. In addition to a regime where the oscillators act as coupled spin-1/2 degrees of freedom, we predict and observe a wide range of parameters in the vicinity of the oscillation threshold where the spin-1/2 description does not apply. In this regime, the oscillators never synchronize, but rather show persistent, full-scale, coherent beats, whose frequency reflects the coupling strength. Our comprehensive study can be used as the the building block of a coherent Ising machines that combine dissipative and conservative couplings.
Introduction. The history of parametric oscillators probably traces back to the XIX century with studies by Michael Faraday on the crispations of wine glasses Faraday et al. (1831). In modern physics, the optical parametric oscillator (OPO) is widely known due to its applications in classical and quantum optics. Below the oscillation threshold, the OPO generates squeezed vacuum Yurke (1984); Collett and Gardiner (1984); Wu et al. (1987); Lvovsky (2015), with applications in precise metrology Caves (1981); Harry (2010); Aasi and LIGO collaborators (2013); Steinlechner et al. (2013), micro- and nano-electromechanical systems Lifshitz and Cross (2003); Kenig et al. (2009a, b, 2011), quantum information Ralph (1999a); Furusawa et al. (1998); Braunstein et al. (2000); Ciattoni et al. (2018) and quantum communications Ralph (1999b); Shaked et al. (2018). Above threshold, an OPO provides efficient energy conversion between optical frequencies, rendering it the primary source of coherent light at wavelengths that are not laser accessible.
The working mechanism of a degenerate parametric oscillator is the well known period doubling instability Strogatz (2007). In contrast to the lasing instability, the gain in a parametric oscillator depends strongly on the phase of the oscillation, relying on the coherent nonlinear coupling between the pump field (at frequency ) and the oscillation (at exactly ) to amplify a single quadrature component of the oscillation field while attenuating the other quadrature component. The phase of the amplified quadrature can acquire two distinct values, which give rise to two solutions that are relatively shifted by one period of the pump. Each solution breaks the time-translational symmetry of the pump 111The symmetry that connects even to odd periods has two elements and when applied twice it comes back to its own. This situation corresponds to the group defined by the set (0,1) with the operation and is commonly refereed to as ., and it is the simplest example of a classical discrete (Floquet) time crystal Khemani et al. (2016); Else et al. (2016); von Keyserlingk et al. (2016); Yao et al. (2017); Choi et al. (2017); Sacha and Zakrzewski (2017); Yao et al. (2018); O’Sullivan et al. (2018); Yao and Nayak (2018); Gambetta et al. (2019).
Borrowing the common terminology from condensed matter systems, one can refer to a parametric oscillator as a classical two-level system (spin-1/2, or Ising spin). Based on this analogy, it has been recently suggested that coupled parametric oscillators can be used to simulate chains or networks of Ising spins Wang et al. (2013); Inagaki et al. (2016); Yamamoto et al. (2017); King et al. (2018); Hamerly et al. (2018a, b); Cervera-Lierta (2018). The basic idea relies on the inherent mode competition and positive feedback within the oscillators to find the most efficient coupled-mode oscillation, which reflects the ground-state configuration of the corresponding Ising model (under certain assumptions). The idea is to implement a set of optical parametric oscillators to realize many independent spin-1/2 systems, where the coupling of the field between the oscillators will reflect the coupling between spins, giving rise to a network of coupled spins. This coherent Ising machine (CIM) can simulate the spin dynamics and aims at calculating the ground state of the corresponding Ising model, thereby solving a complicated minimization problems that cannot be solved on a classical computer.
In this letter, we consider the simplest case of two coupled degenerate parametric oscillators with the most general coupling, including both energy-preserving and energy-dissipating coupling terms. We show that the Ising-type interaction arises from the dissipative terms, which give preference to either in-phase (“ferromagnetic”) or anti-phase (“anti-ferromagnetic”) synchronization of the oscillators. Energy-preserving couplings induce unique coherent dynamics, where the oscillators do not synchronize near threshold, but rather display coherent everlasting beats for a wide range of the parameter space. These oscillations may have implications for the operation of CIMs, as we discuss below.
Furthermore, the coherent transfer of energy between parametric oscillators (beating) is of interest in itself. Coherent dynamics in coupled oscillators (for which beating is a signature mark) is a fundamental principle of wave physics that was widely studied. Such coherent beating however is almost always a transient phenomenon that decays over time due to decoherence, dissipation and non-linear effects. And yet here, the parametric interaction actually acts to preserve this fragile coherence, displaying a steady state of everlasting, full-scale coherent beats.
We realize experimentally a pair of coupled parametric oscillators using parametrically driven RF resonators with a tunable coupling. Our experimental findings agree with a numerical solution of an analytical model that accounts for the coupling strength and phase, periodic drive, gain, losses, nonlinearities, and coupling with energy-preserving and dissipative components. Our main finding is that, depending on the relation between the two components of the coupling, one has two distinct regimes: (i) when the dissipative component of the coupling dominates, the system displays the expected behaviour of coupled spin-1/2 that is the working principle of the two-oscillator CIM Wang et al. (2013) - It can oscillate parametrically only if the pump frequency equals twice the proper oscillators frequency where the oscillation near threshold corresponds to the Ising configuration of lowest energy; (ii) when the energy-preserving component of the coupling dominates, the system displays a richer phenomenology. Differently from the previous case, when the pump frequency is equal to twice the oscillator frequency, as it is usually assumed in the analysis of CIMs, the system exhibits periodic beats that never decay or lose coherence. Only when the pump power is raised further, beyond a higher nonlinear threshold, the oscillators synchronize to the Ising solution. The beating regime represent a trajectory in phase space that visits periodically all the possible spin configurations, which may have implications for the operation of CIMs. This novel regime, in which the system is not amenable to the description of Ising spins, is the main subject of our analysis.
Theoretically, we first study the coupled system by resorting to a linear stability analysis, based on Floquet’s theorem Magnus and Winkler (1979); Chicone (2006); Eckardt and Anisimovas (2015). This approach allows us to describe all the parametric instabilities of the system in the absence of nonlinearities. We then address in detail the case when the system is pumped at twice the oscillator frequency by a multi-scale analysis Kevorkian and Cole (1996), which allows to determine analytically the phase diagram of the coupled OPOs in the presence of nonlinearities. In this case, we find that the system has four distinct major phases of oscillation (see Fig. 1): (i) a stable phase below threshold, where the system does not oscillate (the regime of squeezed noise). (ii) a phase near threshold, where two steady synchronized oscillations are possible, corresponding to the two lower-energy configurations of a pair of coupled Ising spins. This phase exists only with a dissipative component of the coupling. (iii) Further above threshold there is a phase with four possibilities of steady oscillation, behaving as two uncoupled spins, and (iv) an extended region near threshold, where the system shows periodic beats with a non-universal frequency. This beating behavior, which appears only when the coupling includes an energy-preserving component, was not addressed in standard analyses of CIMs, and differs from the usual response of parametric oscillators, whose oscillating frequency is dictated by the pump frequency only. The existence of the beating phase near threshold suggests a different route to the Ising regime of coupling: In addition to the direct transition from sub-threshold squeezed noise to the Ising regime as the pump is increased above threshold (see arrow A on Fig. 1, right panel), the oscillation may also cross first into the coherent beating phase and only then reach the synchronized Ising regime (arrow B), which may be of interest in the dynamical analysis of CIMs.
Theoretical model. We study a system of two degenerate single-mode parametric oscillators, with equal gain and loss terms, coupled via energy-preserving and energy-dissipating terms, in the presence of pump-depletion nonlinearity. We analytically model our system by a set of classical equations of motion:
Here, and represent the fields of the two oscillators, the resonant frequency is parametrically modulated in time as and with being the bare resonant frequency, the pump frequency, the relative phase between the pumps. The quantities represent the normalized pump power, where accounts for the pump depletion nonlinearity when the oscillation is substantial; is the intrinsic loss term, and and represent the strengths of the energy-preserving and energy-dissipating couplings, respectively.
If are sufficiently small, the nonlinearity can be neglected (). This approximation is valid near and below the oscillation threshold and allows us to diagonalize Eq. (1) by introducing the two eigenmodes , where the coefficients are determined by the values of and . The stability analysis of the system can then be carried out by means of a perturbative approach based on Floquet’s theorem. We discuss here the main results, referring the interested reader to Ref. Calvanese Strinati et al. (2018) for a detailed discussion.
When the system is governed by the energy-dissipating coupling, , it has only one parametric resonance at . The two eigenmodes have different thresholds . Therefore, by increasing above the lower threshold, one can selectively excite , and for higher , also . The two modes are excited independently and oscillate with the same frequency (), with an exponentially growing amplitude on top of such oscillations: . This is the standard case for CIMs.
In contrast, when , the system displays a richer phase diagram and can show three distinct parametric resonance at and , depending on the relative phase of the pumps . Specifically, when , only the resonance at can be excited, whereas for only the resonances at exist (for a generic , all three resonances are found). For the resonance at , both eigenmodes are excited simultaneously, leading to full scale beats above the threshold : . When , the modes are independently excited (at , respectively), and parametric amplification occurs without beats. We therefore see that, when , at the system undergoes a transition from a CIM to a coherent beating behaviour.
Intuitively, the beating phase can be understood from a simple reasoning of coupled harmonic oscillators: Coupling two oscillators of frequency in an energy-preserving manner splits the oscillation modes into a symmetric mode at a lower frequency and an anti-symmetric mode at (where reflects the coupling strength). As opposed to laser oscillators that can oscillate on any frequency or phase, parametric oscillators are restricted in phase and frequency by the pump. Thus, when pumped at , parametric oscillators cannot oscillate on any single coupled mode, since its frequency is now shifted away from the pump. Nevertheless, the system can show simultaneous oscillations of both modes, which now act as non-degenerate signal and idler of the combined oscillator. As a consequence, energy-preserving coupling of parametric oscillators generates tunable two-mode squeezed vacuum and parametric light, where the frequency separation between the modes can be tuned by varying the strength of the coupling without any change to the internal geometry/structure of the individual oscillators.
Let us now expand the analysis further above the threshold beyond the linear Floquet analysis, by incorporating the nonlinearity . For brevity, we focus on pumping at the degenerate frequency , where the system displays richer physics. To describe analytically the oscillation field in this nonlinear regime, we resort to a multiple-scale perturbative expansion Kevorkian and Cole (1996), which is known in non-linear optics as the slow-varying envelope approximation. In the following, we specify to the case of .
The fast time scale of the oscillator is associated with the carrier frequency , where the loss is the small expansion parameter of the theory, allowing to identify the slow time scale as . We therefore write and , where and are the complex amplitudes of and , respectively. Normalizing the pump and the coupling in Eq. (1) with respect to the loss as , and , and defining the dimensionless time as , the long-time dynamics is captured by the set of ODEs Calvanese Strinati et al. (2018):
Such equations are equivalent to the slow-varying amplitude equations of nonlinear optics.
We can now calculate the phase diagram of Eq. (2) in the vs. plane (see Fig. 1), where in this case, using tools of nonlinear dynamics Strogatz (2007) to determine the number of fixed points and their stability. Below the threshold , a unique stable fixed point exists at (the origin). Above the threshold () the origin becomes unstable and two situations are encountered (assuming ): For , one finds two stable fixed points that correspond to the correct ground states of the two Ising spins - the oscillators synchronize together with a fixed amplitude and phase. For larger , one finds two additional stable points that correspond to the two other Ising configurations, as discussed in the analysis of CIMs Wang et al. (2013). For , one first finds a stable limit cycle, which manifests itself as beats in the time evolution of and , and only for larger , the phase with two or four stable fixed points is met. If the CIM phase does not exist at all. For , the width of the limit cycle region gradually decreases, eventually vanishing at (see Calvanese Strinati et al. (2018) for details).
Experimental methods. Since the dynamics described here is coherent and purely classical, it is suitable to realize the system of coupled parametric oscillators in a radio-frequency (RF) set-up. In addition to its technical simplicity, the RF experiment offers the great advantage of observing the oscillation directly in time (on an oscilloscope), which is not possible in an optical realization. Furthermore, although an RF parametric amplifier cannot perform quantum squeezing, it can still realize semiclassical squeezing, where the classical thermal noise is squeezed by the phase-dependent gain (to be reported in a future publication).
The coupled parametric oscillators are realized with two ring RF resonators (see Fig. 2) of 50 cm long coaxial cables with repetition rate of roughly 85 MHz. Each resonator includes: (i) an RF frequency mixer (Mini-Circuits ZX05-10-S+) pumped at 170 MHz by an RF synthesizer acting as the nonlinear parametric amplifier (paramp), (ii) a broadband (regular) low-noise amplifier with gain of approximately 15 dB (Mini-Circuits ZX60-P105LN), which compensates for the losses of the cavity, (iii) a -15 dB coupler for the resonator output (Mini-Circuits ZFDC-15-5), and (iv) a tunable attenuator to electronically tune the overall gain of the oscillator. The coupling between the parametric oscillators is achieved using a fixed power splitter and a couple of tunable attenuators to control the effective coupling. The oscillators are pumped by two phase-locked synthesizers, which allows us to control the relative phase of the pumps.
Since we aim primarily to demonstrate the properties of the beating phase (limit cycle) with energy-preserving coupling, we focus experimentally on and monitor the field emitted from the parametric oscillators for various values of the pump power with respect to the oscillation threshold and various coupling strengths , determined by the beat frequency at threshold. Our results are shown in Fig. 3, top panels. The left plots show the experimental results, while the right panels show the theoretical solution, obtained by numerically solving Eq. (1), and overlapped by the oscillation envelopes , (orange and magenta), computed by solving the slow-varying Eq. (2). For pumping above threshold, both oscillators demonstrate regular, a nearly sinusoidal beating envelope over a carrier frequency at half the pump frequency, which matches the cavity resonance at 87 MHz [Fig. 3(a),(b)]. As we further increase the pump power, the period of the beats increases and their shape becomes more elongated and complex [pear shaped, see Fig. 3(c),(d)], until finally diverging at the transition to a synchronized steady-state [Fig. 3(e),(f)].
In Fig. 3, bottom panels, we show the flow of Eq. (2) as vs. , where the three panels refer to the three different cases in the top panels: (a)-(b) slightly above the oscillation threshold, where all fixed points are unstable and the limit cycle is nearly a perfect circle around the origin, corresponding to perfect beats, (c)-(d) just before synchronization, where the limit cycle becomes sharper and beats assume an asymmetric shape, and (e)-(f) after synchronization, where stable attractors around the origin stabilize the dynamics.
From the observed behavior of the field inside the cavities, we can obtain a phase diagram to be compared to the theoretical behaviour discussed before (see Fig. 4). For a given set of value of and , we superimpose the experimental points on the theoretical phase diagram by marking red dots when beats are observed, or by blue crosses when synchronization is observed. In order to compare theory and experiment, we use as a fit parameter. Close to the synchronization threshold, the system is very sensitive to noise, and the observed behaviour alternates between beats and syncronization. This fact limits the experimental precision in the estimation of the transition line from beats to synchronization.
Conclusion and discussion. We reported a detailed study of two coupled degenerate parametric oscillators, explored in an RF experiment, analytically and numerically. A single parametric oscillator is the prototype example of a discrete time crystal, which spontaneously breaks the symmetry associated with the time-periodicity of the pump. From a physical perspective, this means that a single parametric oscillator is always synchronized with half the frequency of the pump. Naively, one would expect this observation to hold true when several parametric oscillators are coupled together.
Our study reveals a richer phase diagram that includes a new phase, where the amplitudes of the amplified quadratures follow a limit cycle. In this phase, which is found when the coupling has significant energy-preserving component, the oscillators perform coherent beats that never decay or lose coherence. This type of behavior was not previously considered in the study of coupled parametric oscillators, and has interesting implications to CIMs. Specifically, it suggests a different route to the Ising synchronized state, where instead of a direct transition from sub-threshold squeezed noise to the Ising regime, the oscillation will cross first into the coherent beating phase and only then reach the synchronized Ising regime.
An interesting question for future work involves the role of the suppressed quadratures. Two coupled parametric oscillators are described by four unconstrained degrees of freedom (without any conserved quantity). As a consequence, this system is not constrained by the KAM theorem and can show a chaotic behavior. In addition, our theoretical analysis neglected the role of noise forces that can restore the ergodicity of the system. In our experiment we indeed observed that close to threshold, the system shows a complicate and seemingly random behavior, which we plan to study in the future.
Acknowledgements. We are grateful to J. Avron, I. Bonamassa, C. Conti, N. Davidson, I. Gershenzon, D. A. Kessler, R. Lifshitz, Y. Michael and C. Tradonsky for fruitful discussions. A. P. acknowledges support from ISF grant No. 46/14. M. C. S. acknowledges support from the ISF, grants No. 231/14 and 1452/14.
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