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Long Josephson junctions coupled to a high-Q resonator are studied numerically and compared with recently published approximative results, obtained by using a perturbative approach to the fluxon motion in the junction. The similarities and differences in the two approaches are discussed.

Application of the classic McLaughlin-Scott soliton perturbation theory to a Josephson-junction fluxon subjected to a microwave field that interacts with the fluxon only at the junction boundaries reduces the problem of phase locking of the fluxon oscillation to the study of a two-dimensional functional map. Phase-locked states correspond to fixed points of the map. For junctions of in-line geometry, the existence and stability of such fixed points can be studied analytically. Study of overlap-geometry junctions requires the numerical inversion of a functional equation, but the results are qualitatively very similar. The map predicts significantly different behaviors for locking at odd and even subharmonic frequencies and at superharmonic frequencies. It also gives indications regarding hysteresis in the current-voltage characteristic, the existence of zero-crossing steps, and a description of the locking process in the frequency domain. The principal merit of the map is that it captures much of the experimental phenomenology at a very low computational cost.

The long and intermediate length Josephson tunnel junction oscillator with overlap geometry of linear and circular configuration, is investigated by computational solution of the perturbed sine-Gordon equation model and by experimental measurements. The model predicts the experimental results very well. Line oscillators as well as ring oscillators are treated. For long junctions soliton perturbation methods are developed and turn out to be efficient prediction tools, also providing physical understanding of the dynamics of the oscillator. For intermediate length junctions expansions in terms of linear cavity modes reduce computational costs. The narrow linewidth of the electromagnetic radiation (typically 1 kHz of a line at 10 GHz) is demonstrated experimentally. Corresponding computer simulations requiring a relative accuracy of less than 10 to the -7th power are performed on supercomputer CRAY-1-S. The broadening of linewidth due to external microradiation and internal thermal noise is determined. Keywords include: Cavity mode; Chaos; Fiske steps; Hysteresis; Intermittency, Josephson fluxon oscillator; Josephson ring oscillator; Josephson tunnel junction; Noise rise; Perturbation theory; Radiation line-width; Sine-Gordon equation (perturbed); Solition (fluxon); SQUID; Subharmonics; and Zero field stages.

The dynamical behavior of fluxons propagating in the presence of an applied magnetic field on an overlap-geometry Josephson tunnel junction of length 5λJ having a McCumber βc=5π is studied by numerical integration of the circuit equations of a 50-section lumped RSJ-type (resistive shunted junction) model. Resonant propagating configurations corresponding to the first and third Fiske steps are found. The fundamental frequencies and power levels of the radiation emitted from one end when the junction is biased on the first and third Fiske steps and on the first zero-field step are comparable, but a magnetic field renders the power spectra at the two ends of the junction different.

Numerical integration of the perturbed sine-Gordon equation describing a long overlap-geometry Josephson junction in a magnetic field indicates a branched structure of the first Fiske step. The major portion of the step corresponds to a simply periodic fluxon oscillation whereas the branches are characterized by subharmonic generation. Applied Physics Letters is copyrighted by The American Institute of Physics.

Chaotic intermittency between soliton dynamic states has been found in a perturbed sine-Gordon system in the absence of an external ac driving term. The system is a model of a long Josephson oscillator with constant loss and bias current in an external magnetic field. The results predict the existence of a current step between the first two Fiske steps in the current-voltage characteristic. A simple probability model demonstrates an asymmetry in the statistical nature of the switching in the two directions.

Detailed simulation studies of the dynamics of fluxons in long circular Josephson tunnel junctions under the influence of external microwave radiation and internal thermal noise are presented. The simulation algorithm uses a pseudospectral method well adapted to vector processors (CRAY-1-S), which gives a speed-up factor in computing time of typically 22 in comparison to conventional high-speed computers, and also provides results with a relative accuracy of less than 10-8 thereby making possible the study of the very narrow radiation linewidth of such oscillators. Comparison of calculated linewidths with experimental results shows good agreement.

In long Josephson junctions the motion of fluxons is revealed by the existence of current steps, zero-field steps, in the current-voltage characteristics. In this paper we investigate the stability of the fluxon motion when high values of the current bias are involved. The investigation is carried on by numerical integration of the model equation, the perturbed sine-Gordon equation, simulating junctions of overlap and annular geometry. A detailed description of the mechanism for the switching from the top of the zero-field step for both geometries is reported. Moreover, the effect of the various dissipations and of the junction length on the switching-current value is investigated. A simple boundary model is able to describe, for junctions of overlap geometry, the qualitative dependence of the switching current on the system parameters.

Experimental measurements of current-voltage structure and emitted X-band radiation in applied magnetic field from overlap-geometry Josephson tunnel junctions of normalized length about 2 are compared with numerical simulations obtained with the use of a perturbed sine-Gordon model. The simulations furnish the current and field dependence of the oscillation configuration, from which can be calculated average voltages, frequencies, and power spectra. Simulation and experimental results are in good agreement with regard to the lobe structure of the height of the first zero-field step and/or second Fiske step in magnetic field and the field dependence of the radiation frequency within the various lobes, including details such as hysteresis between lobes. The simulations predict an alternation of the dominant frequency component with increasing field that accounts well for the experimental observations. The usefulness and limitations of cavity-mode analyses, both singlemode and multimode, are evidenced by comparison with the simulation results.

The appearance of zero-field steps (ZFS’s) in the current-voltage characteristics of intermediate-length overlap-geometry Josephson tunnel junctions described by a perturbed sine-Gordon equation (PSGE) is associated with the growth of parametrically excited instabilities of the McCumber background curve (MCB). A linear stability analysis of a McCumber solution of the PSGE in the asymptotic linear region of the MCB and in the absence of magnetic field yields a Hill’s equation which predicts how the number, locations, and widths of the instability regions depend on the junction parameters. A numerical integration of the PSGE in terms of truncated series of time-dependent Fourier spatial modes verifies that the parametrically excited instabilities of the MCB evolve into the fluxon oscillations characteristic of the ZFS’s. An approximate analysis of the Fourier mode equations in the presence of a small magnetic field yields a field-dependent Hill’s equation which predicts that the major effect of such a field is to reduce the widths of the instability regions. Experimental measurements on Nb-NbxOy-Pb junctions of intermediate length, performed at different operating temperatures in order to vary the junction parameters and for various magnetic field values, verify the physical existence of switching from the MCB to the ZFS’s. Good qualitative, and in many cases quantitative, agreement between analytic, numerical, and experimental results is obtained.