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 λ_J. This definition is not strict.

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).

[edit] 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:

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_xx − φ_tt − 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 . The parameter is the dimensionless damping parameter (β_c is McCumber-Stewart parameter), and, finally, γ = j / j_c is a normalized bias current.

[edit] Important solutions

• Small amplitude plasma waves. φ(x,t) = Aexp[i(kx − ωt)]

• Soliton (aka fluxon, Josephson vortex):

Here x, t and u = v / c₀ are the normalized coordinate, normalized time and normalized velocity. The physical velocity v is normalized to the so-called Swihart velocity c₀ = λ_Jω_p, which represent a typical unit of velocity and equal to the unit of space λ_J divided by unit of time .