Advanced Solid State Physics

Figure 3: Superconducting ring with a capacitor parallel to a Josephson junction

3.1.2 Quantization of the Magnetic Flux

The next example is the quantization of magnetic flux in a superconducting ring with a Josephson
junction and a capacitor parallel to it (see fig. 3). The equivalent to the equations of motion is here
the equation for current conservation:

CΦ + ̈ ICsin

( 2 πΦ

Φ 0

)

=−

Φ−Φe

L 1

What are the terms of this equation? The chargeQin the capacitor with capacityCist given by

Q=C·V

withV as the voltage applied. Derivate this equation after the timetgives the currentI:

Q ̇=I=C·V ̇

With Faraday’s lawΦ = ̇ V (Φas the magnetic flux) this gives the first term for the current through
the capacitor

I=C·Φ ̈.

The currentIin a Josephson junction is given by

I=Ic·sin

(

2 πΦ

Φ 0

)

with the flux quantumΦ 0 , which is the second term. The third term comes from the relationship
between the fluxΦin an inductor with inductivityL 1 and a currentI:

Φ =L 1 ·I

With the applied fluxΦeand the flux in the ring this results in

I=

Φ−Φe

L 1

,

which is the third term.

In this example it isn’t as easy as before to construct a Lagrangian. But luckily we are quite smart
and so the right Lagrangian is

L(Φ,Φ) = ̇

CΦ ̇^2

2

−

(Φ−Φe)^2

2 L 1

+Ejcos

(

2 πΦ

Φ 0

)

.