# How can there only be "11 phonons" in the mirrors of LIGO interferometers?

LIGO is an incredibly sensitive detector of small changes in space due to the passing of gravitational waves and uses some very high-level mathematics and physics and experimental techniques to drive its noise level low enough to make this happen.

Ars Technica's refers to the new paper in Science Approaching the motional ground state of a 10-kg object (readable in arXiv) and says:

Chilling feedback

Fortunately, the delay involved here turned out to dampen the system, rather than changing its frequency. (This is technically true for only a single mode, or frequency range, of the pendulum's swing.) Over time, as the system was constantly tweaked, the effect was to bleed energy off the system, effectively cooling it. By the end of a period of operation, the researchers estimate that its effective temperature was only 77 nano-Kelvin, or very close to absolute zero.

The researchers also put that in terms of phonons, a quantum unit of vibration. At the end of the process, there were likely 11 phonons in the 40 kilogram mirror. That's not the quantum ground state, which would involve emptying the system of phonons. But it's quite close and could potentially already be useful for studying quantum phenomena on large objects; if not, it wouldn't take much improvement to get it there.

The most exciting prospect the authors see is that the motion of the pendulum is also dependent upon gravitational effects, which we've not been able to reconcile with quantum mechanics. The new work, they suggest, "hints at the tantalizing prospect of studying gravitational decoherence on massive quantum systems." And, compared to a grain of sand, 40 kg certainly qualifies as massive.

I am confused by the use of phonons here. This is about the motional ground state of the mirrors and I'm thinking that the state is the spatial configuration of the four mirrors, the relative motion of either their centers of mass or front reflecting surfaces along their optical axes is what's been cooled to 77 nano-kelvin is it not?

The temperature of the bulk glass from which the mirrors are fabricated is not that cold, right?

So in the glass there are likely to be a heck of a lot more than 11 phonons, aren't there?

Where are these 11 phonons and how are they defined?

They are looking at one very specific degree freedom of the individual mirrors. The collective oscillatory motion the mirror in the direction of the laser beam. When isolated from the full equations of motion for the mirror, the effective equation of motion for this degree of freedom is a harmonic oscillator with an effective mass of 10 kg (remember the actual mass of the mirror is 40 kg) and frequency ~ 2.7 Hz.

When treating the mirror as a quantum system, the energy levels of this oscillator (and all the other degrees of freedom of the mirror) are quantized. We can thus count the number of excitations of this particular mode. Such quanta are called phonons.

Because this mode in particular would interfere with the detection of gravitational waves, its the subject of various schemes to dampen the mode, i.e. to ensure that there is as little energy as possible in this particular mechanical mode. The point of the LIGO paper is that this dampening effort is so effective that the number of phonons in this particular mode is reduced to about 10 during operation. This means that this particular macroscopic mechanical degree of freedom is brought very close to its quantum mechanical ground state. This should be compared to the natural occupation number of this mode at room temperature based on the equipartition theorem, which is $$\sim 10^{12}$$ phonons.

However, it should be stressed it is only this particular mode that dampened to this extreme. The mirror as a whole is still at room temperature, and all the other mechanical modes are still populated by trillions of phonons.

• Thank you very much for your answer! This may be best asked as a new SE question or two, or it may be obvious to everyone except me; though they treat the mirror as a quantum system, 1) is this degree of freedom really one? 2) Did they demonstrate quantization of energy levels in some way, or is it that we simply believe that there's no reason for macroscopic masses not to behave just as quantum-mechanically as small ones do?
– uhoh
Jun 22 at 14:02

The confusing language is due to wave-particle duality. At higher frequencies and larger distances, the particle model is more accurate: Ray-tracers can track photons as they reflect, refract and scatter (but not diffract) and photons arrive in discrete events. But at low frequencies the wave model is more accurate: individual photons are less important and may not be detectable at all and diffraction and interference are strong effects. Thus one is better off solving Maxwell's equations with boundary conditions for mirrors and dielectrics. They are talking about the lowest frequency there is in the system so it is confusing to use the "particle" terminology.

Phonons = Mechanical disturbances in the mirror material, just like "there are photons" is the same as saying "there are electromagnetic disturbances".

11 phonons = the 11'th excited state of a quantum harmonic oscillator. This is amazing for an object heavier than a cat, although it is only an estimate. They haven't "seen" the quantization the way we can in a hydrogen lamp.

If the spacetime metric does not enter a superposition this means there would be two half-strength "gravity wells", one for each position of the mirror (although we are talking about distances that make protons look huge). This may force a quantum collapse and take the item out of quantum superposition if it exceeds the Planck mass. It seems unlikely that gravity would play by such different rules at low energies, but if it does it would be revolutionary to theoretical physics.

Non-thermal air-conditioning. The mirrors are at ~300K and would feel warm to the touch. There are huge numbers of phonons overall, on the order of 10^27 and the mirrors are constantly exchanging thermal phoTons with their environment as well as collisions with residual gas molecules. But the bulk motion in the direction of the laser beam is what matters to us as a quantum system.

An active feedback loop has pulled energy from the bulk translational mode of the mirror. This has accomplished what a 77nK Helium bath would do without the enormous difficulty of chilling a large object to 77nK.

I give more details how they achieve such a low number on the physics stack.

• Thank you for stopping by! I appreciate the very clear explanation and "...although it is only an estimate. They haven't "seen" the quantization..." which nicely addresses my comment elsewhere as well as touching on what if it didn't behave in the way assumed. I can't exactly say Welcome to Astronomy SE! since you've been here almost as long as the virgo cluster!
– uhoh
Jun 28 at 1:37
• Two good answers means I'll award two bounties, SE forces the 2nd to be double the first so higher one goes to lower rep (my standard algorithm).
– uhoh
Jun 28 at 1:37

Gravitational waves would even be detectable (in principle) if the mirrors had a relative velocity. This relative velocity (kinetic energy) is not related to temperature. Phonons can have an influence on the relative distance of the mirrors. If this distance is steady (no acceleration) then an accurate measurement can be made.

The 11 phonons correspond to the 70 nanoKelvin of the mirrors. One phonon can already vary the distance between the mirrors. But 11 is too little to do this significantly.

So, the only objects for which the ground state is considered are the mirrors themselves (their lattice structure on which the phonons can move). You can read here that the mirrors are kept at a fixed distance. The only thing that can "do damage" to the measurements are the spatial variations of the mirrors themselves. Now, phonons moving in the mirror are tantamount to spatial distortions, although these are small, but there is a high precision required. So to make these distortions as small as possible it's best to make the temperature of the mirrors as small as possible. Eleven phonons correspond to a pretty low temperature (70 nanoKelvin!). I think it's a modern day miracle that this is achieved.

• @uhoh The lattice of the mirrors is quantized. On this lattice, the phonons exist. Any motion inside the mirrors will reduce their ability to reflect radiation as precisely as possible. The mirrors themselves are kept at a fixed distance so it's only the "motion" due to phonons that can do "damage" to the measurements. See ligo.caltech.edu/page/faq Jun 18 at 13:53
• @uhoh There you go! Hope this is better! Jun 18 at 14:50
• ...and here, a proposed future upgrade will replace the 40 kg fused silica mirrors with 150 kg silicon mirrors cooled to 124 K where their coefficient of thermal expansion crosses zero, but right now their bulk temperature is (and must be) in equilibrium with the rest of the room. They don't sit within nanokelvin refrigerators/cryostats; in fact I am not sure if even tiny solid objects have ever been cooled that low. Nuclear spins, small numbers of separate rubidium atoms floating in laser traps perhaps.
– uhoh
Jun 20 at 6:23
• I'm sure it's an estimate and at any moment there will be statistical variation, but you are now just re-asking my original question!
– uhoh
Jun 20 at 7:11
• If you don't believe the paper I've cited and quoted in the question, then perhaps you should not have answered the question explaining it?
– uhoh
Jun 21 at 2:31