Why are there so few ion-trap experiments studying quantum trajectories? The quantum trajectory theory describes the behaviour of a quantum system under continuous monitoring. Initially it is theoretically studied in quantum optics and single atom scenario (e.g., ion trap setups). But as far as I know, experimental studies on quantum trajectories are mainly done in superconducting quantum circuits. So my question is why there are so few experiments using ion traps? Is it because of experimental limitations? Or if there are such ion trap experiments, could you help to list some of them?
 A: The first observations of quantum trajectories were famously carried out using trapped ions, based on a proposal for "electron shelving" by Dehmelt: see the Plenio & Knight review for a historical discussion. The key experimental papers to check out are Sauter et al., Phys. Rev. Lett. 57, 1696 (1986) and Bergquist et al., Phys. Rev. Lett. 57, 1699 (1986) (the latter is free to read). As you can see, these results are several decades old. If more papers are published on quantum trajectories in superconducting circuits nowadays, it is probably just because the technology is much newer and still under development. But electron shelving is used routinely across many platforms --- trapped ions, NV centres, superconducting qubits etc. --- to efficiently measure electronic states of individual (real or artificial) atoms.
The above experiments demonstrate jump-like trajectories corresponding to direct detection of the emitted photons. One can also consider diffusive trajectories that arise from a homodyne-based detection scheme. Such a measurement scheme has been realised to implement feedback cooling of the motion of a trapped ion: see for example Bushev et al., Phys. Rev. Lett. 96, 043003 (2006) and references therein.
