Smallest uncertainty ever achieved in position measurement in QM?

Question

The Heisenberg uncertainty principle states that

$$\Delta x\Delta y\geq\hbar/2$$

Since the magnitude of $\hbar$ is $10^{-34}$ we could measure both $x$ and $p$ with an uncertainty magnitude of $10^{-17}$, but we could also measure $x$ down to an uncertainty magnitude of $10^{-25}$, causing $\Delta p$ to be bounded by $10^{-9}$. For all I know, we could also have $\Delta x$ about $10^{-100}$, at least in principle, thus causing $\Delta p$ to be no better than $10^{66}$, that is, the values the momentum can take are wildly spread out and nearly-completely uninformative.

Now the question: what is the smallest $\Delta x$ (or $\Delta p$) ever achieved in experiments? Is there any theoretical lower bound, except for $0$?

Answer 1

There are theoretical limits, they happen in regimes where classical (as in non-relativistic) quantum mechanics, respective special relativistic quantum mechanics break down.

The first is Zitterbewegung, an effect of relativistic quantum mechanics, which predicts that the position of a particle jitters with a frequency of $\frac{2mc^2}h$ with an amplitude of the order of the Compton wavelength of the particle.

When the momentum uncertainty becomes to big there will be quantum gravitational effects (semi-classically the particle will have so much momentum uncertainty that it can reach energies where it forms a black hole). Since we do not know the theory of quantum gravitation, we do, however not know exactly what happens if we approach this limit. Intuitively, this limit is reached at an uncertainty of the order of the Planck length.

On the momentum side, the problem is, that the measurement apparatus must grow with the precision of the measurement, here we again run in a limit with classical quantum mechanics (where no limit is set on the propagation of effects) and in a relativistic setting we will get problems with synchronizing the measurement apparatus as it grows (I know of no detailed analysis of this, hints are welcome). The latest time we run into trouble is when our measurement apparatus reaches the size of the observable universe.