en>Rcsprinter123: Disambiguated: fingers → finger, Schecter → Schecter Guitar Research using Dab solver

2013-10-19T23:37:39Z

Disambiguated: fingers → finger, Schecter → Schecter Guitar Research using Dab solver

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{{Refimprove|date=November 2013}}
In [[superconductivity]], a '''long Josephson junction''' (LJJ) is a [[Josephson junction]] which has one or more dimensions longer than the [[Josephson penetration depth]] <math>\lambda_J</math>. This definition is not strict.

In terms of underlying model a ''short Josephson junction'' is characterized by the [[Josephson phase]] <math>\phi(t)</math>, 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., <math>\phi(x,t)</math> or <math>\phi(x,y,t)</math>.

==Simple model: the sine-Gordon equation==
The simplest and the most frequently used model which describes the dynamics of the Josephson phase <math>\phi</math> in LJJ is the so-called perturbed [[sine-Gordon equation]]. For the case of 1D LJJ it looks like:

<center><math>
\lambda_J^2\phi_{xx}-\omega_p^{-2}\phi_{tt}-\sin(\phi)
=\omega_c^{-1}\phi_t - j/j_c,
</math></center>
where subscripts <math>x</math> and <math>t</math> denote partial derivatives with respect to <math>x</math> and <math>t</math>, <math>\lambda_J</math> is the [[Josephson penetration depth]], <math>\omega_p</math> is the [[Josephson plasma frequency]], <math>\omega_c</math> is the so-called [[Josephson characteristic frequency|characteristic frequency]] and <math>j/j_c</math> is the bias current density <math>j</math> normalized to the critical current density <math>j_c</math>. In the above equation, the r.h.s. is considered as perturbation.

Usually for theoretical studies one uses normalized sine-Gordon equation:
<center><math>
\phi_{xx}-\phi_{tt}-\sin(\phi)=\alpha\phi_t - \gamma,
</math></center>
where spatial coordinate is normalized to the [[Josephson penetration depth]] <math>\lambda_J</math> and time is normalized to the inverse plasma frequency <math>\omega_p^{-1}</math>. The parameter <math>\alpha=1/\sqrt{\beta_c}</math> is the dimensionless damping parameter (<math>\beta_c</math> is [[McCumber-Stewart parameter]]), and, finally, <math>\gamma=j/j_c</math> is a normalized bias current.

==Important solutions==
* Small amplitude plasma waves. <math>\phi(x,t)=A\exp[i(kx-\omega t)]</math>

* Soliton (aka [[fluxon]], [[Josephson vortex]]):<ref>M. Tinkham, Introduction to superconductivity, 2nd ed., Dover New York (1996).</ref>
<center><math>
\phi(x,t)=4\arctan\exp\left(\pm\frac{x-ut}{\sqrt{1-u^2}}\right)
</math></center>
Here <math>x</math>, <math>t</math> and <math>u=v/c_0</math> are the normalized coordinate, normalized time and normalized velocity. The physical velocity <math>v</math> is normalized to the so-called [[Swihart velocity]] <math>c_0=\lambda_J\omega_p</math>, which represent a typical unit of velocity and equal to the unit of space <math>\lambda_J</math> divided by unit of time <math>\omega_p^{-1}</math>.

==References==
{{reflist}}

{{DEFAULTSORT:Long Josephson Junction}}
[[Category:Superconductivity]]
[[Category:Josephson effect]]

Scale length (string instruments) - Revision history

en>Rcsprinter123: Disambiguated: fingers → finger, Schecter → Schecter Guitar Research using Dab solver