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 $λ J$ . This definition is not strict.

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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 $t$ , $λ J$ is the Josephson penetration depth, $ω p$ is the Josephson plasma frequency, $ω c$ 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:

φ x x - φ t t - sin(φ) = αφ t - γ,

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

Important solutions

Small amplitude plasma waves. $φ(x, t) = A exp[i (k x - ω t)]$

Soliton (aka fluxon, Josephson vortex):

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

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 $c 0 = λ J ω p$ , which represent a typical unit of velocity and equal to the unit of space $λ J$ divided by unit of time $\omega_p^{-1}$ .