Synchronization Transitions in a Disordered Josephson Series Array
Kurt Wiesenfeld · Pere Colet · Steven H. Strogatz
Summary and citation · read the original at the source
En una página
Put a hundred Josephson junctions in a series array, bias them with a current, and every junction oscillates at its own frequency because no two junctions are ever identical. What this paper shows is that the array does not have to stay a crowd. Wiesenfeld, Colet and Strogatz demonstrate that the circuit equations for such an array, in the limit of weak coupling and small spread, map exactly onto Kuramoto's mean-field model of coupled oscillators, which had until then been an elegant piece of mathematics with no experimental system attached to it. The mapping is not an analogy: the all-to-all coupling that the Kuramoto model assumes falls out of the load equation of the circuit itself. With the mapping in hand they predict two distinct transitions as the spread of critical currents is reduced, first a partial locking in which a growing fraction of junctions share one frequency, then complete phase locking, and they work out how each one would show up in the voltage spectrum of a real array.
Por qué importa aquíThis site's coherence material keeps making the same claim in different hardware: a population of independent oscillators can be made to act as one, and the acting-as-one is the useful part. This is the paper that turns that claim into a circuit with numbers, a predicted onset, and a measurement you could make on a bench. It also supplies the bridge between the Josephson physics of chapter 11 and the synchronisation language used for collective coherence elsewhere on the site, because it shows the two are the same equations.
Qué afirma
01A current-biased series array of nonidentical Josephson junctions, loaded with an inductance, resistance and capacitance, undergoes two distinct transitions as the spread of natural frequencies is reduced: an onset of partial frequency locking, and at smaller spread a second transition into complete phase locking.Abstract; Figure 1 and the three dynamical regimes described with it
Published and peer-reviewed02In the limit of weak coupling and weak disorder the array's circuit equations reduce, by averaging, to the Kuramoto model of coupled phase oscillators, with the coupling strength and the phase shift given explicitly in terms of the circuit elements. The all-to-all coupling that model assumes is not imposed as a mean-field approximation here; it arises from the circuit analysis of the shared load.Equations 1 and 2; equations 6 to 9; the remark on all-to-all coupling following equation 2
Published and peer-reviewed03The paper states that before this work the Kuramoto model had not been used to describe any experimental system, and that the mapping gives the first analytical treatment of mutual synchronization in a Josephson array for the realistic case of nonidentical junctions.Introductory paragraphs, second and third
Published and peer-reviewed04The two transitions have different observable signatures in the total voltage across the load, which matters because the number of locked junctions cannot be measured directly. The onset of order appears as the birth of a narrow spectral line at the locking frequency, whose amplitude is proportional to the Kuramoto order parameter; the complete-locking transition instead shows up as the quenching of the broadband low-frequency part of the output, and is invisible in the line amplitude.Figure 2 and the discussion of the three regimes; equation 13 and the paragraph around it
Published and peer-reviewed05The analytic prediction tracks the simulation. For the parameters of the main run, a hundred junctions with a 50 ohm load, 25 picohenry inductance, 0.04 picofarad capacitance, mean critical current 0.5 milliamperes and mean junction resistance 0.5 ohms, the equivalent Kuramoto coupling is 0.0601 with the cosine of the phase shift at 0.3878, and the predicted locked fraction agrees with direct numerical integration of the circuit equations. Adding Johnson noise at 4 kelvin only raises a flat noise floor.Figure 1 caption and the paragraph reporting the Kuramoto parameters; the Johnson-noise remark following Figure 2
Published and peer-reviewed06What to watch, and it is the paper's own closing argument: both transitions are within reach of existing technology if the bias current is used as the control parameter rather than the disorder, and the power delivered to the load at the locking frequency for the simulated design is about 30 nanowatts per junction, which the authors say should be enough to detect with on-chip measurements. They also name the open question, whether two-dimensional arrays can be related to the Kuramoto model in the same way.Figure 3 and the five experimental considerations preceding it; closing paragraph
What to watch
La puerta de entrada
https://doi.org/10.1103/PhysRevLett.76.404SOURCE READ IN FULL, TEXT NOT REPRODUCED. The four-page paper was read on 2026-09-11 from the author-hosted copy on Steven Strogatz's own publications page, which is a legitimate way to read it and is not a licence to republish it. The footer of the paper's first page carries the copyright line of the American Physical Society for 1996 and no open licence appears anywhere on it, so no sentence of the paper is carried here. The summary and the claims are written in this site's own words from the paper read in full; each locator names the equation, figure or paragraph the statement comes from, and the simulation parameters are quoted as figures because they are the settings of a run, not expression. The two figures are not reproduced.
Cómo citarlo
Kurt Wiesenfeld, Pere Colet, Steven H. Strogatz (1996) Synchronization Transitions in a Disordered Josephson Series Array. doi:10.1103/PhysRevLett.76.404
Dónde encaja en el currículo
Control de la gravedad y superconductoresEl vacío como fluido cuántico