Understanding the Effect of Applied Magnetic Field on SQUIDs in Circuit QED

[maximus123]

Hello everyone, here is my problem

Consider the SQUID shown in the figure (figure is attached to this post if it helps). The loop is made of a
superconducting metal and the superconducting phase must be single-valued
at every point. This means that the total phase difference around the loop
must be an integer multiple of 2$\pi$. i.e

$\Delta\phi_{tot}=2n\pi\\$
and

$\\\Delta\phi_{tot}=\Delta\phi_{a}+\Delta\phi_{b}+\Delta\phi_{I}+ \Delta \phi_{B}\\$​

$\Delta\phi_{a}$ and $\Delta\phi_{b}$ are the phase differences across the junctions a and b, while $\Delta\phi_{I}$ and $\Delta\phi_{B}$ are the phase differences due to the circulating current I and applied magnetic field B respectively. However in circuit QED, the SQUIDs usually have a
small intrinsic inductance. Thus the phase difference $\Delta\phi_{I}$ is negligible.

Explain why $\Delta\phi_{B}$ can be written as

$\Delta\phi_{B}=2\pi\frac{\Phi}{\Phi_{0}}$​

Where $\Phi$ is the flux due to the applied magnetic field and $\Phi_{0}$ is the flux quantum where

$\Phi_{0}=\frac{h}{2e}$​

I'm not sure where to start with this problem, I suppose if the phase difference due to the circulating current is negligible then the total phase difference can be written

$\Delta\phi_{tot}=\Delta\phi_{a}+\Delta\phi_{b}+ \Delta \phi_{B}$​

In our notes we also have it that

$V=\frac{\Phi_{0}}{2\pi}\dot{\delta}$​

Where $\delta$ is the difference between the superconducting phase in the
two electrodes. Which seems like it must be relevant due to the similarity with the intended final result

Any help on how to go about this problem would be greatly appreciated

 

[tman12321]

You must relate the phase to the vector potential A, where B = curl(A). This will include a factor of the flux quantum and 2 pi. Then you can do a line integral of A around the loop to get the total flux. Argue that the sum of the finite phase differences across the links must be zero mod 2 pi. You will find that the sum of the gauge-invariant phase differences is what you're looking for. A good reference is Tinkham's book on superconductivity, specifically chapter six section four. Your library should have it, or you can buy it for around $15 I think.

 

[maximus123]

Thank you, this helped.