Synchronization of coupled rotators: Josephson junction ladders and the locally coupled Kuramoto model

BRYAN DANIELS, S. T.M. Dissanayake, B. R. Trees

Research output: Contribution to journalArticle

Abstract

We show that the resistively shunted junction (RSJ) equations describing a ladder array of overdamped, critical-current disordered Josephson junctions that are current biased along the rungs of the ladder can be mapped onto a Kuramoto model with nearest neighbor, sinusoidal couplings. This result is obtained by an averaging method, in which the fast dynamics of the RSJ equations are integrated out, leaving the dynamics which describe the time scale over which neighboring junctions along the rungs of the ladder phase and frequency synchronize. We quantify the degree of frequency synchronization of the rung junctions by calculating the standard deviation of their time-averaged voltages, [Formula presented] and the phase synchronization is quantified by calculating the time average of the modulus of the Kuramoto order parameter, [Formula presented] We test the results of our averaging process by comparing the values of [Formula presented] and [Formula presented] for the original RSJ equations and our averaged equations. We find excellent agreement for dc bias currents of [Formula presented] where [Formula presented] is the average critical current of the rung junctions, and critical current disorders of up to 10%. We also study the effects of thermal noise on the synchronization properties of the overdamped ladder. Finally, we find that including the effects of junction capacitance can lead to a discontinuous synchronization transition as the strength of the coupling between neighboring junctions is smoothly varied.

Original languageEnglish (US)
Number of pages1
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Volume67
Issue number2
DOIs
StatePublished - Jan 1 2003
Externally publishedYes

Fingerprint

Kuramoto Model
Josephson Junction
Coupled Model
ladders
Josephson junctions
synchronism
Synchronization
critical current
Phase Synchronization
frequency synchronization
Averaging Method
Time-average
Capacitance
Order Parameter
Standard deviation
Biased
Averaging
Disorder
Nearest Neighbor
thermal noise

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Physics and Astronomy(all)

Cite this

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title = "Synchronization of coupled rotators: Josephson junction ladders and the locally coupled Kuramoto model",
abstract = "We show that the resistively shunted junction (RSJ) equations describing a ladder array of overdamped, critical-current disordered Josephson junctions that are current biased along the rungs of the ladder can be mapped onto a Kuramoto model with nearest neighbor, sinusoidal couplings. This result is obtained by an averaging method, in which the fast dynamics of the RSJ equations are integrated out, leaving the dynamics which describe the time scale over which neighboring junctions along the rungs of the ladder phase and frequency synchronize. We quantify the degree of frequency synchronization of the rung junctions by calculating the standard deviation of their time-averaged voltages, [Formula presented] and the phase synchronization is quantified by calculating the time average of the modulus of the Kuramoto order parameter, [Formula presented] We test the results of our averaging process by comparing the values of [Formula presented] and [Formula presented] for the original RSJ equations and our averaged equations. We find excellent agreement for dc bias currents of [Formula presented] where [Formula presented] is the average critical current of the rung junctions, and critical current disorders of up to 10{\%}. We also study the effects of thermal noise on the synchronization properties of the overdamped ladder. Finally, we find that including the effects of junction capacitance can lead to a discontinuous synchronization transition as the strength of the coupling between neighboring junctions is smoothly varied.",
author = "BRYAN DANIELS and Dissanayake, {S. T.M.} and Trees, {B. R.}",
year = "2003",
month = "1",
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doi = "10.1103/PhysRevE.67.026216",
language = "English (US)",
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AU - DANIELS, BRYAN

AU - Dissanayake, S. T.M.

AU - Trees, B. R.

PY - 2003/1/1

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N2 - We show that the resistively shunted junction (RSJ) equations describing a ladder array of overdamped, critical-current disordered Josephson junctions that are current biased along the rungs of the ladder can be mapped onto a Kuramoto model with nearest neighbor, sinusoidal couplings. This result is obtained by an averaging method, in which the fast dynamics of the RSJ equations are integrated out, leaving the dynamics which describe the time scale over which neighboring junctions along the rungs of the ladder phase and frequency synchronize. We quantify the degree of frequency synchronization of the rung junctions by calculating the standard deviation of their time-averaged voltages, [Formula presented] and the phase synchronization is quantified by calculating the time average of the modulus of the Kuramoto order parameter, [Formula presented] We test the results of our averaging process by comparing the values of [Formula presented] and [Formula presented] for the original RSJ equations and our averaged equations. We find excellent agreement for dc bias currents of [Formula presented] where [Formula presented] is the average critical current of the rung junctions, and critical current disorders of up to 10%. We also study the effects of thermal noise on the synchronization properties of the overdamped ladder. Finally, we find that including the effects of junction capacitance can lead to a discontinuous synchronization transition as the strength of the coupling between neighboring junctions is smoothly varied.

AB - We show that the resistively shunted junction (RSJ) equations describing a ladder array of overdamped, critical-current disordered Josephson junctions that are current biased along the rungs of the ladder can be mapped onto a Kuramoto model with nearest neighbor, sinusoidal couplings. This result is obtained by an averaging method, in which the fast dynamics of the RSJ equations are integrated out, leaving the dynamics which describe the time scale over which neighboring junctions along the rungs of the ladder phase and frequency synchronize. We quantify the degree of frequency synchronization of the rung junctions by calculating the standard deviation of their time-averaged voltages, [Formula presented] and the phase synchronization is quantified by calculating the time average of the modulus of the Kuramoto order parameter, [Formula presented] We test the results of our averaging process by comparing the values of [Formula presented] and [Formula presented] for the original RSJ equations and our averaged equations. We find excellent agreement for dc bias currents of [Formula presented] where [Formula presented] is the average critical current of the rung junctions, and critical current disorders of up to 10%. We also study the effects of thermal noise on the synchronization properties of the overdamped ladder. Finally, we find that including the effects of junction capacitance can lead to a discontinuous synchronization transition as the strength of the coupling between neighboring junctions is smoothly varied.

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