Dynamics of a Josephson array in a resonant cavity

Abstract

We derive dynamical equations for a Josephson array coupled to a resonant cavity by applying the Heisenberg equations of motion to a model Hamiltonian described by us earlier [Phys. Rev. B 63, 144522 (2001); Phys. Rev. B 64, 179902 (E)]. By means of a canonical transformation, we also show that, in the absence of an applied current and dissipation, our model reduces to one described by Shnirman et al [Phys. Rev. Lett. 79, 2371(1997] for coupled qubits, and that it corresponds to a capacitive coupling between the array and the cavity mode. From extensive numerical solutions of the model in one dimension, we find that the array locks into a coherent, periodic state above a critical number of active junctions, that the current-voltage characteristics of the array have self-induced resonant steps (SIRS's), that when Na active junctions are synchronized on a SIRS, the energy emitted into the resonant cavity is quadratic in Na, and that when a fixed number of junctions is biased on a SIRS, the energy is linear in the input power. All these results are in agreement with recent experiments. By choosing the initial conditions carefully, we can drive the array into any of a variety of different integer SIRS's. We tentatively identify terms in the equations of motion which give rise to both the SIRS's and the coherence threshold. We also find higher-order integer SIRS's and fractional SIRS's in some simulations. We conclude that a resonant cavity can produce threshold behavior and SIRS's even in a one-dimensional array with appropriate experimental parameters, and that the experimental data, including the coherent emission, can be understood from classical equations of motion.

Nanomechanical-resonator-induced synchronization in Josephson junction arrays

“…The main criteria for the high detected ac power output was a sharp resonance of the cavity and a coupling mechanism between the cavity and the individual JJs. Subsequent theoretical work by Almaas and Stroud 32,33 resulted in dynamical equations for JJ arrays coupled to a resonant cavity that reproduced well the experimental data of Barbara and co-workers. In this model, Almaas and Stroud showed that the effective coupling resulting in the coherent emission of radiation could be thought of as a capacitive coupling between the charge variables of the junctions and the cavity.…”

“…Our work is based on an approach similar to that of Ref. 32. We construct a Hamiltonian describing a serial array of N underdamped JJs coupled to a dilatational AlN oscillator.…”

Superconducting analogs of quantum optical phenomena: Macroscopic quantum superpositions and squeezing in a superconducting quantum-interference device ring

“…As we emphasize, both are of great interest in the light of current research in the fields of quantum computing and quantum-information processing [10][11][12][13]. Moreover, the results presented in this paper emphasize that the SQUID ring can act as a versatile quantum device and, as we have shown previously, can be used to create correlations across extended, multicomponent, quantum circuit systems [1,2,[30][31][32][33]. These correlations can, for example, be made manifest as quantum entanglements and quantum frequency up/down-conversion [1,2], with each controlled by the external bias flux applied to the SQUID ring.…”

Long Josephson junction in a resonant cavity

“…If no soliton ͑i.e., no fluxon͒ is present, the junction behaves very much like a small Josephson junction. 9 In particular, there are SIRSs just as in a small junction, which occur at the voltages expected for a small junction.…”

“…[7][8][9][10][11] These models generally succeed in reproducing many of the salient features of the experiments: SIRSs at the expected voltage, and an increase in the energy in the cavity proportional to the square of the number of junctions. The models have now been extended to two-dimensional ͑2D͒ junction arrays, 12 where they show that, on a given step, the 2D arrays radiate much more energy into the cavity than the one-dimensional ͑1D͒ arrays.…”