Long Josephson junction

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 can be a function of one or two spatial coordinates, i.e.,

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

denote partial derivatives with respect to

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

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

is the dimensionless damping parameter (

is McCumber-Stewart parameter), and, finally,

is a normalized bias current.

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