Hmm. I just found the paper at:
https://sci-hub.tw/https://doi.org/10.1038/s41586-019-1287-z

They initiate quantum jumps with a Rabi drive. I do not know exactly what it is, but I found Rabi cycle.

In physics, the Rabi cycle (or Rabi flop) is the cyclic behaviour of a two-level quantum system in the presence of an oscillatory driving field. A great variety of physical processes belonging to the areas of quantum computing, condensed matter, atomic and molecular physics, and nuclear and particle physics can be conveniently studied in terms of two-level quantum mechanical systems, and exhibit Rabi flopping when coupled to an oscillatory driving field. The effect is important in quantum optics, magnetic resonance and quantum computing, and is named after Isidor Isaac Rabi.

A two-level system has two possible levels, and if they are not degenerate (i.e. not equal energy), the system can become "excited" when it absorbs a quantum of energy. When an atom (or some other two-level system) is illuminated by a coherent beam of photons, it will cyclically absorb photons and re-emit them by stimulated emission. One such cycle is called a Rabi cycle and the inverse of its duration the Rabi frequency of the photon beam. The effect can be modeled using the Jaynes–Cummings model and the Bloch vector formalism.

The rabi drive seems an optical pump

"Optical pumping is also used to cyclically pump electrons bound within an atom or molecule to a well-defined quantum state."

Now my interpretation:

They are feeding atom with two certain frequencies, that switches the atom from the Ground-state into one of the two nuclear resonance states (Dark and Bright). They can detect when an atom is in the Dark state or Bright state or Ground state. They measure only the B-state via the resonance circuit.

In figure 2 of the paper you can clearly see how the atom switch between states. There is a clear catch-time when the atom switches to the B-state. By switching off one the 2 rabi-drives, they can see how the atom reacts while its state is changing.

They seem to be able to tell when the state is changing, by measuring the exact amount of light that the cavity is passing through.

From the paper:

Quantum jumps between |G〉 and |D〉 are induced by a weak Rabi drive ΩDG —although this drive can eventually be turned off during the jump, as explained later...
Frequency landscape of atom and cavity responses, overlaid with the control tones shown as vertical arrows. The cavity pull χ of the atom is nearly identical for |G〉 and |D〉, but markedly distinct for |B〉.

When the atom is in |B〉, the resonance frequency of the LC circuit shifts to a lower frequency than when the atom is in |G〉 or |D〉 (effect schematically represented by switch). Therefore, the probe tone performs a |B〉/not-|B〉 measurement on the atom and is blind to any superposition of |G〉 and |D〉.

Whereas the jump starts at a random time and can be prematurely interrupted by a click, the deterministic nature of the uninterrupted flight comes as a surprise given the quantum fluctuations in the heterodyne record I rec during the jump—an island of predictability in a sea of uncertainty. Now when they know how long it takes for light to be released, they can see how long it takes for the absorbed light to change the state of the atom.