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 [math]\displaystyle{ \lambda_J }[/math]. This definition is not strict.

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

Simple model: the sine-Gordon equation

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

[math]\displaystyle{ \lambda_J^2\phi_{xx}-\omega_p^{-2}\phi_{tt}-\sin(\phi) =\omega_c^{-1}\phi_t - j/j_c, }[/math]

where subscripts [math]\displaystyle{ x }[/math] and [math]\displaystyle{ t }[/math] denote partial derivatives with respect to [math]\displaystyle{ x }[/math] and [math]\displaystyle{ t }[/math], [math]\displaystyle{ \lambda_J }[/math] is the Josephson penetration depth, [math]\displaystyle{ \omega_p }[/math] is the Josephson plasma frequency, [math]\displaystyle{ \omega_c }[/math] is the so-called characteristic frequency and [math]\displaystyle{ j/j_c }[/math] is the bias current density [math]\displaystyle{ j }[/math] normalized to the critical current density [math]\displaystyle{ j_c }[/math]. In the above equation, the r.h.s. is considered as perturbation.

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

[math]\displaystyle{ \phi_{xx}-\phi_{tt}-\sin(\phi)=\alpha\phi_t - \gamma, }[/math]

where spatial coordinate is normalized to the Josephson penetration depth [math]\displaystyle{ \lambda_J }[/math] and time is normalized to the inverse plasma frequency [math]\displaystyle{ \omega_p^{-1} }[/math]. The parameter [math]\displaystyle{ \alpha=1/\sqrt{\beta_c} }[/math] is the dimensionless damping parameter ([math]\displaystyle{ \beta_c }[/math] is McCumber-Stewart parameter), and, finally, [math]\displaystyle{ \gamma=j/j_c }[/math] is a normalized bias current.

Important solutions

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

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

Here [math]\displaystyle{ x }[/math], [math]\displaystyle{ t }[/math] and [math]\displaystyle{ u=v/c_0 }[/math] are the normalized coordinate, normalized time and normalized velocity. The physical velocity [math]\displaystyle{ v }[/math] is normalized to the so-called Swihart velocity [math]\displaystyle{ c_0=\lambda_J\omega_p }[/math], which represent a typical unit of velocity and equal to the unit of space [math]\displaystyle{ \lambda_J }[/math] divided by unit of time [math]\displaystyle{ \omega_p^{-1} }[/math].