Noise in dynamical systems is usually considered a nuisance. But in certain nonlinear systems, including electronic circuits and biological sensory apparatus, the presence of noise can in fact enhance the detection of weak signals. This phenomenon, called stochastic resonance, may find useful application in physical, technological and biomedical contexts.
We show that a current-biased series array of nonidentical Josephson junctions undergoes two transitions as a function of the spread of natural frequencies. One transition corresponds to the onset of partial synchronization, and the other corresponds to complete phase locking. In the limit of weak coupling and disorder, the system can be mapped onto an exactly solvable model introduced by Kuramoto and the transition points can be accurately predicted. PACS numbers: 05.45.+b, 74.40.+k, 74.50.+r Populations of coupled nonlinear oscillators can spontaneously synchronize to a common frequency, despite differences in their natural frequencies. This remarkable phenomenon, known as collective synchronization, has been observed in many physical and biological systems, including relaxation oscillator circuits, networks of neurons and cardiac pacemaker cells, chorusing crickets, and fireflies that flash in unison [1,2].In a pioneering study, Winfree [3] developed a mathematical framework for studying large populations of limitcycle oscillators, and he showed that the onset of synchronization is analogous to a thermodynamic phase transition. This observation was refined by Kuramoto [4] who proposed and analyzed an exactly solvable mean-field model of coupled oscillators with distributed natural frequencies. The Kuramoto model has stimulated a great deal of theoretical work [5][6][7][8][9][10][11][12], thanks to its analytical tractability, but it has not been used to describe any experimental system.In this Letter we show that a series array of Josephson junctions provides a physical realization of the Kuramoto model. This connection allows us to give the first analytical treatment of mutual synchronization in a Josephson array for the realistic case where the junctions are nonidentical. We find that the array displays two transitions: the first corresponds to the onset of dynamical order, while the second coincides with total phase locking and the quenching of fluctuations. We calculate that both of these transitions are experimentally accessible with existing technology.Consider a series array of N junctions, biased with a constant current I B and subject to a load with inductance L, resistance R, and capacitance C. For resistively shunted junctions with negligible capacitance, the governing circuit equations arē h 2er j
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