Tuesday, March 30, 2021

Amazingly good harmonic oscillators

One way that we judge the "quality" of a harmonic oscillator by how long it takes to ring down.  A truly perfect, lossless harmonic oscillator would ring forever, so that's the limiting ideal.  If you ding a tuning fork, it will oscillate about 1000 times before its energy falls by a factor of around \(\exp(-2\pi) \approx 1/535\).  That means that its quality factor, \(Q\), is about 1000.  (An ideal, lossless harmonic oscillator would have \(Q = \infty\).   In contrast, if you ding the side of a coffee mug, the sound dies out almost immediately - it doesn't seem bell-like at all, because it has a much lower \(Q\), something like 10-50.  The quality is limited by damping, and in a mechanical system this is the lossy frictional process that, in the simplest treatment, acts on the moving parts of the oscillator with a force proportional to the speed of the motion.  That damping can be from air resistance, or in the case of the coffee mug example, it's dominated by "internal friction".

So, how good of a mechanical oscillator can we make?  This paper on the arxiv last night shows a truly remarkable (to me, anyway) example, where \(Q \sim 10^{8}\) in vacuum.  The oscillators in question are nanofabricated (drumhead-like) membranes of silicon nitride, with resonant frequencies of about 300 kHz.  To put this in perspective, if a typical 1 kHz tuning fork had the same product of \(Q\) and frequency, it would take \(3 \times 10^{10}\) seconds, or 950 years, for its energy content to ring down by that 1/535 factor.  The product of \(Q\) and frequency is so high, it should be possible to do quantum mechanics experiments with these resonators at room temperature.  
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That's impressive, but it's even more so if you know a bit about internal friction in most solids, especially amorphous ones like silicon nitride.  If you made a similar design out of ordinary silicon dioxide glass, it would have a \(Q\) at room temperature of maybe 1000.  About 15 years ago, it was discovered that there is something special about silicon nitride, so that when it is stretched into a state of high tensile stress, its internal friction falls dramatically.  This actually shows a failure of the widely used tunneling two-level system model for glasses.  The investigators in the present work have taken this to a new extreme, and it could really pave the way for some very exciting work in mechanical devices operating in the quantum regime.  

update:  In resonators made from silicon nitride beams with specially engineered clamping geometries, you can do even better.  How about the equivalent of a guitar string that takes 30000 years to ring down?  “Listen to that sustain!


Anonymous said...

Want to know something just as amazing? These membrane resonator designs are often so dead simple that an undergrad can fabricate these in a university cleanroom! Talk about reproducible!

On the other hand, do you have anything to read for why a mechanical sensor with high Q would be so useful, as compared to an optical one?

Douglas Natelson said...

Anon, very cool. As for why one would want mechanical systems like this, it really depends on what people are trying to do. I think it's less about sensing using the high Q, and more about exploiting the quantum properties of a mechanical widget (e.g., https://arxiv.org/abs/1210.3619). The membranes in this paper each contain on the order of 10^14 atoms, and the idea that this could be manipulated coherently as a quantum object is pretty neat and potentially useful.