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 spatial 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:

$\lambda _{J}^{2}\phi _{{xx}}-\omega _{p}^{{-2}}\phi _{{tt}}-\sin(\phi )=\omega _{c}^{{-1}}\phi _{t}-j/j_{c},$

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:

$\phi _{{xx}}-\phi _{{tt}}-\sin(\phi )=\alpha \phi _{t}-\gamma ,$

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. $\phi (x,t)=A\exp[i(kx-\omega t)]$
• Soliton (aka fluxon, Josephson vortex):^[1]

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

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 $\omega _{p}^{{-1}}$.

References

1. M. Tinkham, Introduction to superconductivity, 2nd ed., Dover New York (1996).