Long Josephson junction

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): \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}.{{cite journal

References

References

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