Comment by maho

The simple explanation: The positrons fly around the trap center. In the z-direction, this is a simple "up and down" oscillation. There are electrodes in the top and bottom of the trap called the "end-caps". The end-caps are connected to each other with a large resistor (technically, a coil forming an LC resonator with the parasitic capacitance, but on resonance it looks like a large resistor).

When the positrons are at the top, they attract negative charges into the top end-cap. This is called an "image charge". When the positrons are close to the bottom end-cap, the image charge into bottom end-cap. This forms a current, the "image current". The more positrons you have, the larger the image current.

Since the image current (on the order of femtoamperes) has to flow through the large resistor, it is turned into a voltage that can be amplified and measured by cryogenic amplifiers. But it has a second effect: The voltage "back-acts" on the positrons, damping their motion. If you have more positrons in the trap, this damping gets stronger, because each positron experiences the back-action of all other positrons.

In the paper, you can see pictures of the driven axial motion, where you take an RF excitation and sweep the frequency over the resonance, recording the positrons' amplitude as a response. The more positrons you have, the "wider" this resonance, because of the damping effect.

As you reduce the number of positrons by doing a "dip" (making the trap really shallow, hoping that some, but not all, positrons escape), you can see the width getting smaller and smaller in discrete steps. That's how you know you are dealing with individual particles.

I haven't worked on Gabrielse's setup, but I have worked on other, very similar (better, ha!) experiments 10 years ago. I don't see what's so special about this trap, but putting antimatter into it always makes for a nice headline.