In superconductivity, a long Josephson junction (LJJ) is a Josephson junction which has one or more dimensions longer than the Josephson penetration depth. This definition is not strict.

In terms of underlying model a short Josephson junction is characterized by the Josephson phase, which is only a function of time, but not of coordinates i.e. the Josephson junction is assumed to be point-like in space. In contrast, in a long Josephson junction the Josephson phase.

Simple model: the sine-Gordon equation

in LJJ is the so-called perturbed sine-Gordon equation. For the case of 1D LJJ it looks like:

is the Josephson penetration depth is the Josephson plasma frequency is the so-called characteristic frequency normalized to the critical current density. In the above equation, the r.h.s. is considered as perturbation.

Usually for theoretical studies one uses normalized sine-Gordon equation: where spatial coordinate is normalized to the Josephson penetration depth is McCumber-Stewart parameter is a normalized bias current.

Important solutions

• Soliton (a.k.a. fluxon, Josephson vortex):

.

References