- Long Josephson junction
In
superconductivity , a long Josephson junction (LJJ) is aJosephson junction which has one or more dimensions longer than theJosephson penetration depth lambda_J. This definition is not strict.In terms of underlying model a "short Josephson junction" is characterized by the
Josephson phase phi(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 theJosephson phase can be a function of one or two spacial coordinates, i.e., phi(x,t) or phi(x,y,t).Simple model: the sine-Gordon equation
The simplest and the most frequently used model which describes the dynamics of the Josephson phase phi in LJJ is the so-called perturbed
sine-Gordon equation . For the case of 1D LJJ it looks like:
lambda_J^2phi_{xx}-omega_p^{-2}phi_{tt}-sin(phi) =omega_c^{-1}phi_t - j/j_c,
where subscripts x and t denote partial derivatives with respect to x and t, lambda_J is theJosephson penetration depth , omega_p is theJosephson plasma frequency , omega_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:
phi_{xx}-phi_{tt}-sin(phi)=alphaphi_t - gamma,
where spatial coordinate is normalized to theJosephson penetration depth lambda_J and time is normalized to the inverse plasma frequency omega_p^{-1}. The parameter alpha=1/sqrt{eta_c} is the dimensionless damping parameter (eta_c isMcCumber-Stewart parameter ), and, finally, gamma=j/j_c is a normalized bias current.Important solutions
* Small amplitude plasma waves. phi(x,t)=Aexp [i(kx-omega t)]
* Soliton (aka
fluxon ,Josephson vortex ):
phi(x,t)=4arctanexpleft(pmfrac{x-ut}{sqrt{1-u^2 ight)
Here x, t and u=v/c_0 are the normalized coordinate, normalized time and normalized velocity. The physical velocity v is normalized to the so-calledSwihart velocity c_0=lambda_Jomega_p, which represent a typical unit of velocity and equal to the unit of space lambda_J divided by unit of time omega_p^{-1}.
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