Long Josephson junction
Encyclopedia
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 
.
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 
.
Superconductivity
Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
, a long Josephson junction (LJJ) is a Josephson junction which has one or more dimensions longer than the Josephson penetration depth
Josephson penetration depth
In superconductivity, Josephson penetration depth characterizes the typical length on which an externally-applied magnetic field penetrates into the long Josephson junction...
. This definition is not strict.In terms of underlying model a short Josephson junction is characterized by the Josephson phase
Josephson phase
In superconductivity, the Josephson phase is the difference of the phases of the quantum mechanical wave function in two superconducting electrodes forming a Josephson junction....
, 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 phaseJosephson phase
In superconductivity, the Josephson phase is the difference of the phases of the quantum mechanical wave function in two superconducting electrodes forming a Josephson junction....
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 equationSine-Gordon equation
The sine–Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally considered in the nineteenth century in the course of study of surfaces of constant negative...
. For the case of 1D LJJ it looks like:
where subscripts
and denote partial derivatives with respect to and , is the Josephson penetration depthJosephson penetration depth
In superconductivity, Josephson penetration depth characterizes the typical length on which an externally-applied magnetic field penetrates into the long Josephson junction...
,
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
Josephson penetration depth
In superconductivity, Josephson penetration depth characterizes the typical length on which an externally-applied magnetic field penetrates into the long Josephson junction...
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
- Small amplitude plasma waves.
- Soliton (aka fluxonFluxonIn physics, a fluxon is a quantum of electromagnetic flux. The term may have any of several related meanings.- Superconductivity :In the context of superconductivity, a fluxon is a small whisker of normal phase surrounded by superconducting phase. Supercurrents circulate around its center...
, Josephson vortexJosephson vortexIn superconductivity, a Josephson vortex is a vortex of supercurrent in a long Josephson junction . The supercurrents circulate around the vortex center which is situated inside the Josephson barrier...
):
Here
, and are the normalized coordinate, normalized time and normalized velocity. The physical velocity is normalized to the so-called Swihart velocitySwihart velocity
Originally, in superconducting transmission lines, the Swihart velocity is the propagation velocity of electromagnetic waves along transmission line....
, which represent a typical unit of velocity and equal to the unit of space divided by unit of time .