In superconductivity, a long Josephson junction (LJJ) is a Josephson junction which has one or more dimensions longer than the Josephson penetration depth λ J . This definition is not strict.
In terms of underlying model a short Josephson junction is characterized by the Josephson phase ϕ ( t ) , 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., ϕ ( x , t ) or ϕ ( x , y , t ) .
The simplest and the most frequently used model which describes the dynamics of the Josephson phase ϕ in LJJ is the so-called perturbed sine-Gordon equation. For the case of 1D LJJ it looks like:
λ J 2 ϕ x x − ω p − 2 ϕ t t − sin ( ϕ ) = ω c − 1 ϕ t − j / j c , where subscripts x and t denote partial derivatives with respect to x and t , λ J is the Josephson penetration depth, ω p is the Josephson plasma frequency, ω c is the so-called characteristic frequency and j / j c is the bias current density j normalized to the critical current density j c . In the above equation, the r.h.s. is considered as perturbation.
Usually for theoretical studies one uses normalized sine-Gordon equation:
ϕ x x − ϕ t t − sin ( ϕ ) = α ϕ t − γ , where spatial coordinate is normalized to the Josephson penetration depth λ J and time is normalized to the inverse plasma frequency ω p − 1 . The parameter α = 1 / β c is the dimensionless damping parameter ( β c is McCumber-Stewart parameter), and, finally, γ = j / j c is a normalized bias current.
Small amplitude plasma waves. ϕ ( x , t ) = A exp [ i ( k x − ω t ) ] Soliton (aka fluxon, Josephson vortex): ϕ ( x , t ) = 4 arctan exp ( ± x − u t 1 − u 2 ) Here x , t and u = v / c 0 are the normalized coordinate, normalized time and normalized velocity. The physical velocity v is normalized to the so-called Swihart velocity c 0 = λ J ω p , which represent a typical unit of velocity and equal to the unit of space λ J divided by unit of time ω p − 1 .
