Funny that you pick the one experiment that was used to show the discrete nature of light. Conrad's next post

: How do I make a very low noise voltage reference that can source enough current to power a UV light.
Not sure if moving the problem around is going to make it any easier to solve. Never mind the noise that will likely be in whatever light source you use.
Some other thoughts (which are probably not practical):
Johnson noise is proportional to output impedance. So if you increase the output voltage, you lower the relative Johnson noise (obviously will not work for series combinations like many battery cells). Of course now you need to figure out what to do with that high voltage. Resistive dividers are probably out

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What if you had a low(ish) noise voltage reference that is connected to a cap via a switch. Every say 10s, the switch would close for say 100 ms to charge the cap. This will likely cause a small step in voltage (due to voltage reference noise). The cap should be large enough to stay pretty much at the same voltage for that 10s. This can probably be a better quality cap than if it had to maintain the same voltage for minutes. If you sample at 100 NPLC, then you could get around 5 samples per step (obviously auto-zero disabled).
Repeat this a couple of hundred times, and calculate variance by averaging the variance calculated for each individual step over all steps. Because the expectation value of their means would be very close (on the level of the noise from the voltage reference, maybe 1 ppm?), you do not have to worry about standardizing the variances.This is probably one of those devil (charge injection has horns) is in the details problems, but it might work. Of course the relatively simple hardware and complicated analysis might not be everybody's cup of tea. And I am not sure if the short discharge time reduces of exacerbates the imperfections in the cap and the rest of the setup.