Nanomechanical-resonator-induced synchronization in Josephson junction arrays

Abstract

We show that a serial array of N nonuniform, underdamped Josephson junctions coupled piezoelectrically to a nanoelectromechanical ͑NEM͒ oscillator results in phase locking of the junctions. Our approach is based on a semiclassical solution to a set of coupled differential equations that were generated by the Heisenberg operator equations, which in turn are based on a model Hamiltonian that includes the following effects: the charging and Josephson energies of the junctions, dissipation in the junctions, the effect of a dc bias current, an undamped simple harmonic oscillator ͑representing the NEM͒, and an interaction energy ͑due to the piezoelectric effect͒ between the NEM and the junctions. Phase locking of the junctions is signaled by a step in the current-voltage ͑I-V͒ curve. We find the phase-locked states are ͑neutrally͒ stable at the bottom and top of the step but not for bias currents in the middle of the step. Using harmonic balance, we are able to calculate an analytical expression for the location of the resonance step, v step , in the I-V curve. Because of the multistability of the underdamped junctions, it is possible, with a judicious choice of initial conditions and bias current, to set a desired number N a ഛ N of junctions on the resonance step, with N − N a junctions in the zero-voltage state. We are also able to show that, when N a junctions are in the phase-locked configuration, the time-averaged energy of the NEM oscillator scales like N a 2 .

The self-actuating InAs nanowire-based nanoelectromechanical Josephson junction

“…It was proposed to use the nanomechanical resonator to store the Josephson junction phase qubit states, 12,13 and to build a coherent source of microwave radiation based on an array of Josephson junctions. 14 In addition, a vibrationmodulated Josephson junction has been predicted to exhibit self-induced Shapiro steps in the I − V characteristics at a frequency of mechanical resonance, 15 and the weak-link made of a doubly-clamped nanowire can be used to pump the supercurrent into nanowire vibrations. 16 In all these aspects NEMS act similarly to an external microwave resonator coupled to a Josephson junction.…”

“…The piezoelectric coupling of the Josephson oscillations to the acoustic mode of a crystal resonator has been theoretically studied for the case of an array of Josephson junctions. 14 From this point of view our experimental realisation can be seen as a special case of a single weakly coupled junction. Other possible coupling mechanisms such as the magnetomotive force, 16 and modulation of the junction dimensions 15 are ruled out since the resonance is observed at B = 0, and the junction length L is fixed.…”

“…11,13,18 Also, our experiment demonstrates the possibility of building a nano-scale source of coherent microwave radiation based on an array of Josephson junctions coupled to a high-Q piezoelectric resonator. 14 METHODS -The high-quality InAs nanowires used in this work have been synthesised by the Molecular-Beam Epitaxy in Riber 32 solid source system by the Au-assisted Vapour-Liquid-Solid (VLS) method. The nanowires were grown along 111 direction on a (011)oriented InAs substrate at 400 o C and the group V/III ratio of 100.…”

Renormalization group approach to oscillator synchronization

“…In summary, the effect of the strong coupling decimation step is to replace the pair of oscillators with phases θ n and θ n+1 by a single oscillator with phase Θ, mass M and frequency Ω given by Eqs. ( 6), (7), and (8).…”

“…In contrast, with proper connectivity and coupling strength, the population can display the same frequency as order forms. This collective behavior has been observed in diverse systems including neural networks [2,3,4], Josephson junctions [5,6,7], lasers [8], electronic circuits [9], and a range of biological systems, including sleep cycles [10], animal gaits [11] and biological rhythms in general [12].…”