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 {\displaystyle \lambda _{J}} . This definition is not strict. In terms of underlying model a short Josephson junction is characterized by the Josephson phase ϕ ( t ) {\displaystyle \phi (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 ) {\displaystyle \phi (x,t)} or ϕ ( x , y , t ) {\displaystyle \phi (x,y,t)} .
Simple model: the sine-Gordon equation The simplest and the most frequently used model which describes the dynamics of the Josephson phase ϕ {\displaystyle \phi } in LJJ is the so-called perturbed sine-Gordon equation. For the case of 1D LJJ it looks like:
where subscripts x {\displaystyle x} and t {\displaystyle t} denote partial derivatives with respect to x {\displaystyle x} and t {\displaystyle t} , λ J {\displaystyle \lambda _{J}} is the Josephson penetration depth, ω p {\displaystyle \omega _{p}} is the Josephson plasma frequency, ω c {\displaystyle \omega _{c}} is the so-called characteristic frequency and j / j c {\displaystyle j/j_{c}} is the bias current density j {\displaystyle j} normalized to the critical current density j c {\displaystyle j_{c}} . 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 λ J {\displaystyle \lambda _{J}} and time is normalized to the inverse plasma frequency ω p − 1 {\displaystyle \omega _{p}^{-1}} . The parameter α = 1 / β c {\displaystyle \alpha =1/{\sqrt {\beta _{c}}}} is the dimensionless damping parameter ( β c {\displaystyle \beta _{c}} is McCumber-Stewart parameter), and, finally, γ = j / j c {\displaystyle \gamma =j/j_{c}} is a normalized bias current.
Important solutions Small amplitude plasma waves. ϕ ( x , t ) = A exp [ i ( k x − ω t ) ] {\displaystyle \phi (x,t)=A\exp[i(kx-\omega t)]}
Soliton (a.k.a. fluxon, Josephson vortex):
Here x {\displaystyle x} , t {\displaystyle t} and u = v / c 0 {\displaystyle u=v/c_{0}} are the normalized coordinate, normalized time and normalized velocity. The physical velocity v {\displaystyle v} is normalized to the so-called Swihart velocity c 0 = λ J ω p {\displaystyle c_{0}=\lambda _{J}\omega _{p}} , which represent a typical unit of velocity and equal to the unit of space λ J {\displaystyle \lambda _{J}} divided by unit of time ω p − 1 {\displaystyle \omega _{p}^{-1}} .
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