Long Josephson junction


In superconductivity, a long Josephson junction 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 can be a function of one or two spatial coordinates, i.e., or.

Simple model: the sine-Gordon equation

The simplest and the most frequently used model which describes the dynamics of the Josephson phase in LJJ is the so-called perturbed sine-Gordon equation. For the case of 1D LJJ it looks like:

where subscripts and denote partial derivatives with respect to and, is the Josephson penetration depth, is the Josephson plasma frequency, is the so-called characteristic frequency and is the bias current density 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 and time is normalized to the inverse plasma frequency. The parameter is the dimensionless damping parameter, and, finally, is a normalized bias current.

Important solutions


Here, and are the normalized coordinate, normalized time and normalized velocity. The physical velocity is normalized to the so-called Swihart velocity, which represent a typical unit of velocity and equal to the unit of space divided by unit of time.