Physics: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 $\varphi (t)$, 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., $\varphi (x,t)$ or $\varphi (x,y,t)$.

Simple model: the sine-Gordon equation

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

${\lambda }_J^2{\varphi }_{xx}-{\omega }_p^{-2}{\varphi }_{tt}-\sin (\varphi )={\omega }_c^{-1}{\varphi }_t-j/j_c,$

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:

${\varphi }_{xx}-{\varphi }_{tt}-\sin (\varphi )=\alpha {\varphi }_t-\gamma ,$

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 $\alpha =1/\sqrt{{\beta }_c}$ is the dimensionless damping parameter (${\beta }_c$ is McCumber-Stewart parameter), and, finally, $\gamma =j/j_c$ is a normalized bias current.

Important solutions

• Small amplitude plasma waves. $\varphi (x,t)=A\exp [i(kx-\omega t)]$
• Soliton (aka fluxon, Josephson vortex):

$\varphi (x,t)=4\arctan \exp (\pm \frac{x-ut}{\sqrt{1-u^2}})$

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 $c_0={\lambda }_J{\omega }_p$, which represent a typical unit of velocity and equal to the unit of space ${\lambda }_J$ divided by unit of time ${\omega }_p^{-1}$.