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# 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 λJ. This definition is not strict.

In terms of underlying model a short Josephson junction is characterized by the Josephson phase φ(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., φ(x,t) or φ(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 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, λJ is the Josephson penetration depth, ωp is the Josephson plasma frequency, ωc is the so-called characteristic frequency and j / jc is the bias current density j normalized to the critical current density jc. In the above equation, the r.h.s. is considered as perturbation.

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

φxx − φtt − sin(φ) = αφt − γ,

where spatial coordinate is normalized to the Josephson penetration depth λJ and time is normalized to the inverse plasma frequency $\omega_p^{-1}$. The parameter $\alpha=1/\sqrt{\beta_c}$ is the dimensionless damping parameter (βc is McCumber-Stewart parameter), and, finally, γ = j / jc is a normalized bias current.

## Important solutions

• Small amplitude plasma waves. φ(x,t) = Aexp[i(kx − ωt)]
$\phi(x,t)=4\arctan\exp\left(\pm\frac{x-ut}{\sqrt{1-u^2}}\right)$

Here x, t and u = v / c0 are the normalized coordinate, normalized time and normalized velocity. The physical velocity v is normalized to the so-called Swihart velocity c0 = λJωp, which represent a typical unit of velocity and equal to the unit of space λJ divided by unit of time $\omega_p^{-1}$.