This award supports a theoretical project to study the phenomenon of synchronization in collections of disparate oscillators, with a special focus on the application to arrays of nanomechanical oscillators.

Synchronization is the formation of collective states of coherent motion of oscillators with diverse intrinsic frequencies, through the interaction between the elements and their nonlinear behavior. It represents a particularly interesting and practically important collective behavior of a driven, non-equilibrium system. A focus of the research will be the synchronization of large arrays of coupled nonlinear mechanical devices at the scale of tens to a hundred nanometers with quantitatively understood and controllable coupling and nonlinearity. These nanomechanical systems possess a unique combination of properties, including small size, high frequencies, and easily accessible and controllable nonlinearities, which make them extremely well-suited for realizing an experimental system of large coupled oscillator arrays, to test preexisting theory and to provoke further extensions. This project is a renewed theoretical study of synchronization exploiting ideas and methods from statistical mechanics, dynamical systems theory, and pattern formation theory, motivated by the prospect of experimental realization in nanomechanical systems.

The project will have two main thrusts. The first is a quantitative analysis of the systems of a few nanomechanical oscillators that will likely be the first target of experiment. This effort will help the design of experimental systems, suggest protocols for experimental measurements, and will test the results as they become available against the theoretical models. The second thrust will be a study of the basic theory of large arrays of oscillators, extending the understanding particularly in directions relevant to future experimental work on large arrays of nanomechanical oscillators. A variety of systems will be investigated, including ones with short range, power-law long range, and all-to-all coupling, using techniques such as real space renormalization group methods, expansions analogous to spin-wave methods in magnets, and numerical simulations. The role of spatial dimension, the importance of topological defects, and effects of noise driven fluctuations, will also be investigated.

The proposed study of synchronization in nanomechanical oscillators has important technological applications, will have a broad impact on diverse areas of basic science, and provides unique educational opportunities, particularly for training graduate students.


This award supports a theoretical project to study the phenomenon of synchronization in collections of disparate oscillators, with a special focus on the application to arrays of nanomechanical oscillators.

This research project involves the study of systems of oscillators, each with different intrinsic frequencies. Oscillators of particular interest are many tiny vibrating beams some ten thousand to a hundred thousand times smaller than the diameter of a human hair, the nanometer scale. Synchronization occurs when the oscillators reach a state where their motion is in lock-step. This is a consequence of the interactions among the oscillators or vibrating beams. Synchronization occurs in many different contexts in nature and in artificially fabricated systems.

Large arrays of coupled mechanical devices at the scale of ten to a hundred nanometers lie on the forefront of lithographic fabrication technology, provide a unique laboratory to test theory, and have many potential technological applications such as exquisitely precise clocks, and sensitive detectors even down to the single molecule level. The collective behavior of disparate oscillators is also important in wide areas of basic science, including the dynamics of neurons in the brain and muscle cells in the heart, and the coherent grouping of lasers to make high power sources. A deeper understanding of the general phenomenon of synchronization gained from the careful study of nanomechanical systems will impact these and other areas of research.

This award provides unique educational opportunities, particularly for training graduate students. It represents fundamental research and contributes to the intellectual foundation of future device and nanoscale device technologies.

Project Report

Oscillators - devices with an output signal that vary periodically in time with a period set by internal dynamics rather than an external drive - play a crucial role as clocks and frequency references in technology and living systems. These devices convert energy from a steady input source (such as a battery) into sustained periodic motion, counteracting the inevitable dissipative forces. They fall in the class of driven, nonequilibrium physical systems. Mathematically, an oscillator is represented by a limit cycle, a closed curve in the phase space of dynamical variables, and has a perfectly periodic signal. In the real world, small uncontrolled perturbations that we call noise, maybe resulting from fundamental Brownian molecular motion or from more prosaic causes such as mechanical vibrations, cause deviations from the perfectly periodic dynamics of the limit cycle, and degrade frequency or timing precision of the device. In this project we have investigated the collective dynamics of oscillators and related driven nonequilibrium systems, in both classical and quantum regimes. The basic motivation of the research is that coupled oscillators may synchronize to all oscillate with exactly the same frequency, even though, due to fabrication limitations, each one would have a different oscillation frequency. In the synchronized state it was widely expected on the basis of averaging over the independent noise forces acting each oscillators that the effect of the noise would be reduced by a factor of the number N of synchronized oscillators, and so the frequency and timing imprecision reduced by a factor of 1/N. We have investigated this question by carefully formulating the effect of the noise on the collective phase variable of the synchronized oscillators, reducing the problem from the difficult calculation of stochastic nonlinear evolution equations to the much simpler calculation of the linear stability analysis of the synchronized oscillators in the absence of noise. Using this approach we have verified that for a commonly used model of synchronization, in which each oscillator phase tends to relax in a completely dissipative way to the average phase of the oscillators to which it is coupled, the reduction of the degradation due to noise is indeed the naively anticipated factor of N. In this simple model the synchronized state is rather featureless, with each phase only differing from a uniform average by a small amount needed to pull its frequency to the common synchronized value. However, this model is nongeneric, since the relaxation of each phase will typically have a reactive oscillatory component, as well as the purely dissipative exponential decay. In this more general model, the synchronized state shows nontrivial structures reminiscent of the patterns formed in many other driven nonequilibrium systems: the synchronized state consists of phase waves propagating out from a source, which, for example in a two dimensional array of oscillators, may take the form of roughly circular waves propagating out from a core region formed by a collection of higher frequency oscillators (targets), or of a topologically conserved spiral - see figure. For such states we find that the phase noise reduction is not by the factor of the total number of oscillators, but only by the smaller number of oscillators in the core region that acts as the source of the waves, and we drive precise results for calculating the size of the core region for target and spiral sources. The efficacy of improving the timing and frequency precision is correspondingly reduced. A major consequence of our results is that in practical applications of synchronization, the pattern formation aspects must be carefully studied. To facilitate the experimental investigation in nanomechanical systems, in collaboration with the NEMS experimental group we have developed a prototype for arrays of oscillators, with each oscillator constructed from a nanomechanical resonator with motion sustained by an electronic feedback loop. A novel feature of our system is that the coupling between oscillators is also formed from electronic feedback. This allows flexible coupling, with both reactive and dissipative components. A key advantage is that all the parameters of the system are known quantitatively and may be tuned, and we have used a system of two coupled oscillators to verify the predictions of synchronization theory quantitatively. In addition we have proposed, and predicted results for, experimental systems using cold trapped atomes to extend the study to smaller scales and into the strongly quantum regime. One type of system is an array of trapped atoms excited by a laser to a highly excited "Rydberg" state. In these states, the electronic wave function extends over much larger distances than the atomic size in the ground state, facilitating the study of interacting arrays. We find a rich variety of spatiotemporal dynamics, which can be measured experimentally as collective dynamics in the atomic fluorescence. Another system studied is a collection of coupled optical cavities to investigate synchronization in the quantum limit.

National Science Foundation (NSF)
Division of Materials Research (DMR)
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Daryl W. Hess
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California Institute of Technology
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