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 spatial coordinates, i.e.,
ϕ
(
x
,
t
)
or
ϕ
(
x
,
y
,
t
)
.
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:
λ
J
2
ϕ
x
x
−
ω
p
−
2
ϕ
t
t
−
sin
(
ϕ
)
=
ω
c
−
1
ϕ
t
−
j
/
j
c
,
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
x
x
−
ϕ
t
t
−
sin
(
ϕ
)
=
α
ϕ
t
−
γ
,
where spatial coordinate is normalized to the Josephson penetration depth
λ
J
and time is normalized to the inverse plasma frequency
ω
p
−
1
. The parameter
α
=
1
/
β
c
is the dimensionless damping parameter (
β
c
is McCumber-Stewart parameter), and, finally,
γ
=
j
/
j
c
is a normalized bias current.
Small amplitude plasma waves.
ϕ
(
x
,
t
)
=
A
exp
[
i
(
k
x
−
ω
t
)
]
Soliton (aka fluxon, Josephson vortex):
ϕ
(
x
,
t
)
=
4
arctan
exp
(
±
x
−
u
t
1
−
u
2
)
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-called Swihart velocity
c
0
=
λ
J
ω
p
, which represent a typical unit of velocity and equal to the unit of space
λ
J
divided by unit of time
ω
p
−
1
.
