The Superfluid Fountain

To me, Quantum Mechanics is not a lifestyle. Granted, I don’t really know enough about QM, but I honestly believe that Physicists don’t either. The world we live in can be divided into the macroscopic, the microscopic which we can see under a microscope, and the subatomic. And QM seems to reign in this last of the three categories.

But a macroscopic object like a cup of coffee, or a Cat, can ultimately be described using real numbers. OTOH, the subatomic currently requires imaginary and complex numbers to describe. But there is one exception which must be mentioned, in the form of superfluid liquid helium-4. This is the state that liquid helium goes in to, when it’s cooled below 2 Kelvin. In this state it looses all viscosity, and essentially exhibits Quantum properties on the macroscopic scale. I also think that this is the only known practical example. One meaning this has, is that if set in a circular motion through a loop, it will continue to flow through this loop essentially forever, or until somebody intervenes and alters the system that was set up. But there is a deeper observation to make about this.

We’ve all observed the effect of (normal) water, when it rains on a lake. Each water droplet hits the surface of the lake with some kinetic energy, and sets ripples into motion – waves – which radiate locally from the point of impact. But we also know that all the kinetic energy that hits the surface of the lake in the raindrops, eventually gets absorbed by the viscosity of the water.

We need to ask ourselves what would happen, if water was also a superfluid, and had zero viscosity, just like helium-4 below its critical temperature. The raindrops could still hit the surface of the larger pool of superfluid, but where would their kinetic energy escape to?

The immediate answer to this question is that it would not. The surface of the lake, as well as its depths, would continue to become more turbulent, as waves add up and cancel, due to ‘conservation of energy’. It would lead to a ridiculous situation, which actual planets don’t exhibit.

But now the question comes up, of what happens to superfluid helium-4, if it is set into motion according to a fountain, from which droplets form due to non-zero surface tension, and from which those droplet strike the surface, of a pool of superfluid. Would this fountain also continue to flow once set up? The answer is ‘in principle yes’, due to conservation of energy. But then what happens to the waves created by each droplet?

Well those waves form part of a resonant circuit, within our superfluid fountain. They build up in amplitude, but also find their way back to the entry-point of the spout.

In a resonant circuit, if waves arrive slightly phase-shifted after having gone once through a feedback loop, that loop will choose a frequency, at which this phase-shift becomes zero. I.e., this phase-shift eventually cancels out. And if we express this type of system using complex numbers as a notation, the imaginary component corresponds to the sine-function of the phase-shift. A stable resonator will go into any state, in which this imaginary component becomes zero.

So a fountain of superfluid is also possible, in which helium-4 comes back in droplets, instead of always going through a simple loop. Those droplets will resonate in the pool, but in such a way that stable resonance is also set up, and so that kinetic energy is not lost.

If the fountain is constructed in such a way that this cannot happen, then it will either not flow, or not form waves.



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