# Long Josephson junction

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In superconductivity, a **long Josephson junction** (LJJ) is a Josephson junction which has one or more dimensions longer than the Josephson penetration depth . This definition is not strict.

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[edit]

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.

### Important solutions[edit]

- Small amplitude plasma waves.
- Soliton (aka fluxon, Josephson vortex):
^{[1]}

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 .^{[2]}

## References[edit]

**^**M. Tinkham, Introduction to superconductivity, 2nd ed., Dover New York (1996).**^**J. C. Swihart (1961). "Field Solution for a Thin-Film Superconducting Strip Transmission Line".*J. Appl. Phys*.**32**(3): 461–469. Bibcode:1961JAP....32..461S. doi:10.1063/1.1736025.