Long Josephson junction simulator

A live solution of the sine-Gordon equation for a long junction. Press, drag and release on the phase plot to launch a fluxon, antifluxon, breather or plasma wave at the velocity you drag, choose what each end of the line does, and paint the line impedance and plasma frequency along its length.

Time ω_p0t

Net fluxons

Energy / E₀

Mean voltage / (ħω_p0/2e)

Pendulum row pendula on a torsion wire, as in Fig. 3 of the review; each hangs at the local phase φ, straight down at φ = 0

Phase φ(x) modulo 2π a fluxon shows as a jump from π to −π; press, drag sideways and release to launch the selected excitation

Space–time map field φ_x; newest at the top

Line impedance Z(x)/Z₀ drag up or down to paint

Plasma frequency ω_p(x)/ω_p0 drag up or down to paint

Run and drive

Simulation speed

Ends of the line

Left end

Right end

Periodic joins both ends into a ring.

Applied field at open ends, φ_x

Length L/λ_J0

Bias γ = j/j_c0

Shunt loss α

Place by dragging

Press on the phase plot or pendulum row, drag sideways to set the velocity u = v/c̄ and, for a breather or plasma wave, up or down to set its size, then release. The simulation pauses and previews while you drag.

Line profile

Preset

Brush width / λ_J0

The equation being solved

φ_tt + α φ_t = Z ∂_x(Z⁻¹ ∂_xφ) − ω_p² sin φ + γ Z

Position is in Josephson lengths λ_J0 and time in units of 1/ω_p0, both at the reference values Z₀ and ω_p0. Z(x) and ω_p(x) are the local line impedance and plasma frequency relative to that reference. The Swihart velocity c̄ stays uniform, so the local Josephson length is c̄/ω_p. The energy is in units of E₀, the Josephson energy stored in one λ_J0 of the reference line, so a fluxon at rest on it has energy 8. A fluxon's rest energy scales as 8ω_p/Z: fluxons are pushed towards low plasma frequency and high impedance, and an impedance step reflects plasma waves.

The line is discretized at λ_J0/20 and stepped with a time step of 0.02/ω_p0; the shunt loss is treated implicitly.

What each end does

Open (applied field). The bare edge of a junction. No current leaves the electrodes, and the field at the edge is held at the applied value φ_x = H, which is how a magnetic field enters a long junction.
Short circuit. The two electrodes are joined at the end by a superconducting short, so the voltage there is zero and the phase at the end is held at a whole multiple of 2π.
Periodic (ring). The line closes on itself, as in an annular junction. Fluxons placed on the ring are trapped and counted.
Absorbing (infinite line). A hidden lossy buffer, 30 λ_J0 long, sits beyond the window. Waves and fluxons that leave do not come back, as if the line went on forever. Fluxons that leave stop inside the buffer.