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>The researchers say this could eventually be rectified with atomic clocks that fit into portable devices, which are in development but are currently too expensive for widespread use. The other components used are already able to be miniaturized. The experiment also only tracked the researcher’s path after the fact, but future work will investigate doing so in real-time.

What would be an eta for an atomic clock inside an Iphone?

I would guess never. The iPhone derives a lot of it's value from communicating with servers all the time. High-accuracy timekeeping just isn't valuable.
I find this quite surprising. We use chip-scale atomic clocks in our equipment, and while expensive, it hardly seems out-of-reach for the type of customers who might be interested in underground surveying (mines, for example).
Microchip sells SA65, which is 40x35mm and retails for about $3000.
I'd be moderately surprised if it was necessary: atomic clocks are only really king of time stability over longer time periods (days/months). If this just needs to stay in sync over the course of a few hours (after syncing outside with GPS, for example), then it's much more plausible to do it with a high-precision crystal or silicon MEMS oscillator (SiTime has already become a big player in the phone market because of their low power consumption for the precision they offer, and they have some really impressive higher-end offerings that wouldn't currently fit but perhaps could be made to).
Requires connection to a base station (or 4). If you're already underground (or underwater) then real-time connection to that is going to be tricky.
It doesn’t neeed to be real-time though: for some applications it’s fine to just record then establish after the fact where exactly the receiver traveled.
This is really cool--going to send this to my friend who is an experimental submarine pilot up in Newfoundland. Seems like something he could use.