# Destroying currents in superconducting rings by vortex tunneling

Consider a superconducting metal ring in which there is a persisting current $I$. I am interested in the failure of this current to remain "persisting" in the ring, although this will occur at unlikely times:

QFT says that it is possible for a vortex line to be created in the region that the ring surrounds, and then passes through the material to the other side. This will decrease the quantized flux, and upon successive such events, $I$ will drop to zero.

It is surely low, but what is the approximate rate/probability of such a process?

I would think it depends on the ring's thickness $\delta$ relative to the coherence length $\xi$ and penetration depth $\lambda$. I would also think it depends on the temperature $T$ relative to the critical temperature $T_0$, since that is the point where there is no difference between the superconductor and the normal metal.

Has this been studied in the literature? Any references are highly appreciated. (The existence of such quantum tunneling processes is known by Witten, but I don't think he studied it, and I'm guessing the analysis is either old or extremely basic.)

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A further investigation led me to the desired reference, which discusses this precise problem:

Hard Superconductivity: Theory of the Motion of Abrikosov Flux Lines
Work of Anderson and Kim, at Bell Labs, around 1964

This is not a full answer to the question, but just to point out existing related studies. This kind of question was considered a lot in the context of Josephson junction, which is basically a superconducting ring but with a weak link (i.e. the junction), where intuitively vortices tunnel through the junction. The simplest model of such a system is just the following Hamiltonian:

$H=\frac{1}{2}E_C(n-n_g)^2+J\cos\phi$.

Here $\phi$ is the phase difference across the junction, $n$ is the conjugate variable, which can be understood as the number of Cooper pairs, and thus quantized (because $\phi$ is $2\pi$ periodic). $n_g$ is the induced charge on the system, controlled by the back gate.

We can think about the Hamiltonian as a quantum mechanical problem of a particle in the cosine potential. When $J$ is much larger than the charging energy $E_C$, the particle is most likely to be found in the minima of the cosine. But there are vortex tunneling processes between the minimums $0, \pm 2\pi, \dots$, the amplitude of which can be calculated using the standard WKB approximation.

Now back to your question. If there is no weak link, then vortex tunneling necessarily has to change the phase of the order parameter everywhere in the ring. Such a process has to depend on the superfluid stiffness as well as simply the length of the ring. To the first approximation, one can model the phase fluctuations on the ring by a Gaussian action, and the cost of a vortex tunneling event in a Gaussian action is roughly log in the system size $L$.

• Thanks for bringing up this related problem; I take it this has negligible effect on the role of SQUIDs. Can you elaborate on your last sentence? I would think the system size $L$ depends on both the ring's thickness $\delta$ and its length $l$. (I initially guessed that the rate of occurrence of vortex tunneling looked like $\text{exp}(-\frac{\delta}{\lambda}\frac{l}{\xi})$, but had no good basis for that guess.) – Chris Gerig Apr 7 '15 at 7:46
• Do you have a reference for details concerning your last paragraph? – Chris Gerig Apr 12 '15 at 19:19