What does a mechanical Josephson Effect look like?

Question

This question is related to my research work in physics concerning the brittle-ductile-transition (BDT) in solid state mechanics, which ended quite some time ago and is summarized in my thesis from 2013, and though I have quit any serious research work, yet one particular question has bothered me since then from time to time and as there is this place to ask it, I finally decided to do so:

At the end of some years of looking at the brittle-ductile-transition I came to the conclusion, that it is essentially like the transition from a superconductor of the first kind (brittle) to a superconductor of the second kind (ductile), it is just almost never thought of in this way in the mechanical community; people in the superconducting community seem to think much more often about rigid bodies as analogs of superconductors, and I believe they are spot on and then some. So:

a) A solid state (rigid body) is a superconductor for mechanical forces, the corresponding Meisner effect is the exclusion of (relative) mass transport within the solid state.

b) Dislocations in crystalline solids are related to mass transport in the solid as the vortices in superconductors of the second kind are related to magnetic fields penetrating the superconductor.

c) Mechanical effects in plasticity like hardening etc. can also be interpreted in this framework.

d) Abstract superconductivity is not particularly mysterious, as we know it from our everyday-surroundings, just in mechanics, not in electricity.

Yet, while I am prepared to defend this (much more than) analogy, I never did any serious number crunching, I have long since changed careers and I could never wrap my head around the obvious nontrivial question of the title, so:

I) If one believes the above analogy to hold some water, how would a mechanical Josephson effect look like?

II) More specifically: Could it have been observed, and if so, in which kinds of experiments (think of length scales etc.), has it maybe been observed, with or without recognition as such, say in AFM?

My thesis does not contain much on the analogy and nothing about the question, as this was only an afterthought in those days, but the question never left me; the doi of the thesis is here if somebody is interested in the BDT, for the question in a narrow sense it is certainly not required.

Answer 1

The analogy is interesting but inexact, in a way that leads to significant differences.

Entropy

Rigid bodies can't spin forever in the universe. Even if you rotate an object in space, things will happen that will slow the rotation. It will absorb and emit particles, and these particles will carry away angular momentum. As long as it is above 0 K this is true. Superconductors can maintain a supercurrent state that doesn't interact at all with entropy from the environment (to the best of our knowledge). This is possible because the supercurrent occupies the ground state of a quantized system and thermal fluctuations need to overcome an energy barrier to influence the state.

Quantization

There's a few ways to make the mechanical analogy better. One is to imagine an ideal inelastic non-interacting rigid body at 0 K, but there's a number of unphysical assumptions here, so it's not very helpful analogy for understanding how real world systems behave.

Another way is to quantize it. Imagine a spinning rigid body where the angular momentum is quantized. So that to increase or decrease the spin of the body, there is an energy threshold. If you then put the body in an environment that's cold, it can indeed spin forever.

Instead of spinning, you can also consider oscillating systems (like mass-spring systems). Turns out people have made systems like this, and the vibrational states are quantized. See this paper for example.

This turns out to be much harder to achieve than superconductivity - it requires lower temperatures and is only possible for smaller systems. For example, in that paper, device is just 15 μm in size and has to be cooled to 0.015 K. Intuitively, it makes sense that it's harder than achieving superconductivity, because Cooper pairs are very light and have relatively large binding energy for their mass. Mechanical systems are much heavier in comparison and their energy spectrum contains many more states.

In this kind of analogy, you could form a Josephson junction by having two coupled oscillator systems, for instance.

Superfluids

Consider flux pinning. A very simplified explanation of flux pinning is that it's the supercurrent forming tiny vortices in the superconductor. The external magnetic field lines pass through the centers of these vortices. While the vortices stay fixed in place, the bulk supercurrent flows around them. It would be really difficult to imagine this having an analogy to a bulk rigid system. The defect isn't allowed to move with the crystal. The system would somehow have to ‘flow’ around the crystal defect. So you need to relax the assumption of rigidity and instead imagine it as something that can flow with zero viscosity e.g. a superfluid. This helps the analogy in other ways too, for example it explains how you can maintain the supercurrent when the superconductor is moved or rotated, which wouldn't be possible with a spinning rigid system.

Superfluids turn out to be really good analogies to superconductors, and in fact a lot of the physics really is the same. And it turns out you can observe Josephson junction effects in superfluids, for example in this paper.

Let us suppose that superfluid 4He is forced through a small aperture at very low temperatures. The superfluid will accelerate as it flows through the aperture until its velocity reaches a critical value $v_c$ at which a quantized vortex is formed. This object consists of a central region inside which the wavefunction becomes zero (and hence is no longer in the superfluid state) and around which supercurrent flows. The term ‘topological defect’ is often used to describe such objects in superfluids, superconductors and other systems. Because the phase of the wavefunction is well-defined around the vortex and varies from 0 all the way to $2\pi$, a moving vortex can lead to a changing phase between two points lying across the direction of vortex motion. The formation of a vortex and its subsequent motion leads to a quantized decrease $\Delta v_s$ in the superfluid velocity (which depends on the phase), a process known as phase slippage. The flow of the superfluid across the aperture should therefore exhibit a characteristic sawtooth variation with time.