Long Josephson junction
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In terms of underlying model a short Josephson junction is characterized by the Josephson phase , 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., or .
Simple model: the sine-Gordon equation
The simplest and the most frequently used model which describes the dynamics of the Josephson phase in LJJ is the so-called perturbed sine-Gordon equation. For the case of 1D LJJ it looks like:
where subscripts and denote partial derivatives with respect to and , is the Josephson penetration depth, is the Josephson plasma frequency, is the so-called characteristic frequency and is the bias current density normalized to the critical current density . In the above equation, the r.h.s. is considered as perturbation.
Usually for theoretical studies one uses normalized sine-Gordon equation:
where spatial coordinate is normalized to the Josephson penetration depth and time is normalized to the inverse plasma frequency . The parameter is the dimensionless damping parameter ( is McCumber-Stewart parameter), and, finally, is a normalized bias current.
Here , and are the normalized coordinate, normalized time and normalized velocity. The physical velocity is normalized to the so-called Swihart velocity , which represent a typical unit of velocity and equal to the unit of space divided by unit of time .
- M. Tinkham, Introduction to superconductivity, 2nd ed., Dover New York (1996).