Static solitons of the sine-Gordon equation and equilibrium vortex structure in Josephson junctions

Abstract

The problem of vortex structure in a single Josephson junction in an external magnetic field, in the absence of transport currents, is reconsidered from a new mathematical point of view. In particular, we derive a complete set of exact analytical solutions representing all the stationary points (minima and saddle-points) of the relevant Gibbs free-energy functional. The type of these solutions is determined by explicit evaluation of the second variation of the Gibbs free-energy functional. The stable (physical) solutions minimizing the Gibbs free-energy functional form an infinite set and are labelled by a topological number Nv = 0, 1, 2, . . .. Mathematically, they can be interpreted as nontrivial "vacuum" (Nv = 0) and static topological solitons (Nv = 1, 2, . . .) of the sine-Gordon equation for the phase difference in a finite spatial interval: solutions of this kind were not considered in previous literature. Physically, they represent the Meissner state (Nv = 0) and Josephson vortices (Nv = 1, 2, . . .). Major properties of the new physical solutions are thoroughly discussed. An exact, closed-form analytical expression for the Gibbs free energy is derived and analyzed numerically. Unstable (saddle-point) solutions are also classified and discussed.

Coherent diffraction of thermal currents in Josephson tunnel junctions

“…33 By contrast, the phase slips discussed above have an energetic origin and they occur when the system passes from one energetically stable configuration to another one. 34 This transition takes place when the magnetic flux crosses one of the critical points and therefore can be experimentally induced by changing the magnetic flux. The different origin of the slips is exemplified by the fact that the fluctuation-induced phase slips are always of 2π while in the present case we have slips 094506-3 of π .…”

Phase-coherent solitonic Josephson heat oscillator

“…Instead, for a magnetic field above the critical value, i.e., for H(t) > 2, solitons in the form of fluxons penetrate the LJJ from its ends. However, in this case at a specific value of the magnetic field several solutions, describing distinct configurations with different amount of solitons, may concurrently exist 51,[53][54][55] . The dynamical approach is essential to describe the JJ state when multiple solutions are available.…”

Thermal flux-flow regime in long Josephson tunnel junctions

“…2(b) we show the behaviour of both the number of solitons (left vertical scale, blue line) and the mean voltage drop (right vertical scale, red line) by quasi-adiabatically, i.e., very slowly, increasing the bias current, at a fixed external magnetic field H = 1.75. We observe that at a low bias current the system is in the Meissner state [72], that is the fluxon-free state, corresponding to N = 0. Instead, when I b 0.156 a flux-flow regime is established.…”

“…The bias-current conditions giving a flux-flow regime can be grasped by studying the number of solitons and the mean voltage drop across the junction. The number of solitons N excited along the junction can be roughly evaluated through the quantity [72]…”