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 ${\displaystyle \lambda _{J}}$. This definition is not strict.

In terms of underlying model a short Josephson junction is characterized by the Josephson phase ${\displaystyle \phi (t)}$, 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., ${\displaystyle \phi (x,t)}$ or ${\displaystyle \phi (x,y,t)}$.

Simple model: the sine-Gordon equation

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

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

where subscripts ${\displaystyle x}$ and ${\displaystyle t}$ denote partial derivatives with respect to ${\displaystyle x}$ and ${\displaystyle t}$, ${\displaystyle \lambda _{J}}$ is the Josephson penetration depth, ${\displaystyle \omega _{p}}$ is the Josephson plasma frequency, ${\displaystyle \omega _{c}}$ is the so-called characteristic frequency and ${\displaystyle j/j_{c}}$ is the bias current density ${\displaystyle j}$ normalized to the critical current density ${\displaystyle j_{c}}$. In the above equation, the r.h.s. is considered as perturbation.

Usually for theoretical studies one uses normalized sine-Gordon equation:

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

where spatial coordinate is normalized to the Josephson penetration depth ${\displaystyle \lambda _{J}}$ and time is normalized to the inverse plasma frequency ${\displaystyle \omega _{p}^{-1}}$. The parameter ${\displaystyle \alpha =1/{\sqrt {\beta _{c}}}}$ is the dimensionless damping parameter (${\displaystyle \beta _{c}}$ is McCumber-Stewart parameter), and, finally, ${\displaystyle \gamma =j/j_{c}}$ is a normalized bias current.

Important solutions

• Small amplitude plasma waves. ${\displaystyle \phi (x,t)=A\exp[i(kx-\omega t)]}$
• Soliton (aka fluxon, Josephson vortex):^[1]

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

Here ${\displaystyle x}$, ${\displaystyle t}$ and ${\displaystyle u=v/c_{0}}$ are the normalized coordinate, normalized time and normalized velocity. The physical velocity ${\displaystyle v}$ is normalized to the so-called Swihart velocity ${\displaystyle c_{0}=\lambda _{J}\omega _{p}}$, which represent a typical unit of velocity and equal to the unit of space ${\displaystyle \lambda _{J}}$ divided by unit of time ${\displaystyle \omega _{p}^{-1}}$.^[2]

References