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 $lambda\_J$. This definition is not strict.

In terms of underlying model a "short Josephson junction" is characterized by the Josephson phase $phi(t)$, 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 spacial coordinates, i.e., $phi(x,t)$ or $phi(x,y,t)$.

Simple model: the sine-Gordon equation

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

$lambda\_J^2phi\_\{xx\}-omega\_p^\{-2\}phi\_\{tt\}-sin(phi)\; =omega\_c^\{-1\}phi\_t\; -\; j/j\_c,$
where subscripts $x$ and $t$ denote partial derivatives with respect to $x$ and $t$, $lambda\_J$ is the Josephson penetration depth, $omega\_p$ is the Josephson plasma frequency, $omega\_c$ 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:
$phi\_\{xx\}-phi\_\{tt\}-sin(phi)=alphaphi\_t\; -\; gamma,$
where spatial coordinate is normalized to the Josephson penetration depth $lambda\_J$ and time is normalized to the inverse plasma frequency $omega\_p^\{-1\}$. The parameter is the dimensionless damping parameter ( is McCumber-Stewart parameter), and, finally, $gamma=j/j\_c$ is a normalized bias current.

Important solutions

* Small amplitude plasma waves. $phi(x,t)=Aexp\; [i(kx-omega\; t)]$

* Soliton (aka fluxon, Josephson vortex):
$phi(x,t)=4arctanexpleft(pmfrac\{x-ut\}\{sqrt\{1-u^2\; ight)$
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 $c\_0=lambda\_Jomega\_p$, which represent a typical unit of velocity and equal to the unit of space $lambda\_J$ divided by unit of time $omega\_p^\{-1\}$.