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- Weingarten’s branches from quantum complexityMarch 9, 2022
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## Ground-state cooling by Delic et al. and the potential for dark matter detection

The implacable Aspelmeyer group in Vienna announced a gnarly achievement in November (recently published):

Cooling of a levitated nanoparticle to the motional quantum ground stateUroš Delić, Manuel Reisenbauer, Kahan Dare, David Grass, Vladan Vuletić, Nikolai Kiesel, Markus AspelmeyerGround-state cooling of nanoparticles in laser traps is a very important milestone on the way to producing large spatial superpositions of matter, and I have a long-standing obsession with the possibility of using such superpositions to probe for the existence of new particles and forces like dark matter. In this post, I put this milestone in a bit of context and then and then toss up a speculative plot for the estimated dark-matter sensitivity of a follow-up to Delić et al.’s device.

One way to organize the quantum states of a single continuous degree of freedom, like the center-of-mass position of a nanoparticle, is by their sensitivity to displacements in phase space. This can be formalized as the fidelity between a state and its displacement ,

where has displacement components and in space and momentum, and where is the displacement operator. (The fidelity reduces to the squared overlap when the states are pure.) If the displaced state is highly distinguishable from (has low fidelity with) the undisplaced state, then there are no quantum limitationsAdded: Here, “quantum limitations” is just shorthand for the fundamental limits on our ability to distinguish two quantum states which are each individually localized around distinct points in phase space, but are close enough to have substantial overlap/fidelity. This is to be contrasted with the analogous classical case where points in phase space separated by arbitrarily small distances are perfectly distinguishable given sufficiently accurate measuring equipment.

^{a }on distinguishing the two potential outcomes. This might be mean, e.g., detecting the momentum transfer from a scattering dark-matter particle. States that are hot (large mixedness, smeared over phase space) have low sensitivity to displacements, and sensitivity goes up as the state is cooled, localizing it toward a known location and momentum. However, the sensitivity saturates at a fixed finite value at zero temperature, when the Wigner function has irreducible area in phase space.To increase sensitivity beyond this limit (the standard quantum limit, SQL), we need to move to non-classical states. One possibility is squeezing, producing increased sensitivity in one direction (e.g., position) at the expense of decreased sensitivity in the other (e.g., momentum). Another class of possibilities are “cat states”, i.e., a coherent superposition of two states which are individually roughly classical (localized in phase space) but are distant from each other in phase space. Squeezing or superposing states lets one keep increasing the displacement sensitivity as far as one’s equipment can manage. In a restricted sense, squeezed and superposed states have a “negative effective temperature” with regards to displacement sensitivity. Ground state cooling is a crucial step on the road from a hot messy state to an exquisitely sensitive quantum superposition. Here’s a cartoon I’ve posted previously:

Joyously, Delić et al. not only report cooling a nanoparticle to its ground state, they also ambitiously claim that producing and verifying a spatial superposition of the nanoparticle over length scales similar to its radius may be achieved with some relatively straightforward modifications.Other mechanical modes have been put in their motional ground state before, but these tend to be very difficult to extend to superpositions over large distances. Relative to these, laser-trapped tend to have a simpler path: shut off the laser, allow the nanoparticle wavepacket to expand under free-fall conditions, hit it with something like a (higher-frequency, beam-shaped) laser to simulate a double-slit, wait some more, and then observe it.

^{b }First, here are parameters from the (super-impressive) completed experiment:Now their delicious speculation:

Assuming they can do this, we are looking at a spatial superposition with something in the neighborhood of these properties:

This would be a truly mammoth amount of matter to superpose, beating the current world record — also in Vienna! — by some

fiveorders of magnitude.It’s known that, in a way that can be made precise, big superpositions are sensitive to very small momentum transfers that are otherwise undetectable. In our PRD, Itay Yavin and I looked at some simple models of dark matter to see if any would be identifiable by recently proposed experiments pushing the bounds of superposition size. There were two experiments that would be highly sensitive to a range of parameter space, but which would require many years of technical advances to achieve. (One of them was to operate in space, at a cost of hundreds of millions of euros.) The other, nearer-term experiments could not be sensitive to dark matter except under quite optimistic assumptions in a narrow region of parameter space. In particular, the improved limits on new light scalar mediators from estimated plasma mixing effects in stellar cores by Hardy and Lasenby probably rule out all models that these more tractable experiments might have been sensitive to.

The hypothetical interaction between the dark matter and matter we considered looks like this:

Emboldened by Delić et al., let’s rashly modify one of the sensitivity plots from our paper to get a sense for what we could do with the huge superpositions they suggest are achievable. The solid green curve in the figure below delineates the fraction of the allowed parameter space where dark matter would induce detectable decoherence in such an experiment.

Pretty rad. Producing superpositions of the kind suggested by Delić et al. are likely the most tractable path to begin probing dark matter through decoherence. Of course, there are many caveats:

^{c }If that fraction is, say, 1 in 20 rather than 1 in 160, the sensitivity will be correspondingly reduced. However, since I know of no reason for uncontrolled decoherence from conventional sources to track the sidereal day, I think this number (or something even more aggressive) is reasonable.Once a sidereal decoherence signal is observed, there are several possible methods for identifying whether it has galactic origins, which we discuss in the paper.^{d }The primary reason to be optimistic about future experiments is the strong scaling of the sensitivity with nanoparticle radius due to the coherent scattering enhancement.For much of the parameter space we are considering, the momentum transfer during the matter-dark-matter scattering event is much longer than the size of the nanoparticle, so the dark matter can’t resolve the different nucleons that compose it. The reflected dark-matter waves from each nucleon add together coherently, leading to a cross section that scales quadratically with the number of nucleons in the superposed nanoparticle target.

[^{e }In contrast, the primary sources of decoherence (collisions with ambient gas molecules and emission of blackbody radiation) scale much more slowly with the radius ( and , respectively).I thank Robert Lasenby for discussion.]## Footnotes

(↵ returns to text)

Added: Here, “quantum limitations” is just shorthand for the fundamental limits on our ability to distinguish two quantum states which are each individually localized around distinct points in phase space, but are close enough to have substantial overlap/fidelity. This is to be contrasted with the analogous classical case where points in phase space separated by arbitrarily small distances are perfectly distinguishable given sufficiently accurate measuring equipment.↵quadraticallywith the number of nucleons in the superposed nanoparticle target.↵