I was preparing a short write-up / test procedure for the custom HV coil driver, when I thought of something I can't resolve. I'm probably missing some really basic physics here - but why do we not account for the shot noise from DC current flowing through the series resistor? For a 4kohm resistor, the Johnson current noise is ~2pA/rtHz. This is the target we were trying to beat with our custom designed HV bias circuit. But if there is a 1 mA DC current flowing through this resistor, the shot noise of this current is 18pA/rtHz, which is ~9 times larger than the Johnson noise of the same resistor. One could question the applicability of this formula to calculate the shot noise of a DC current through a wire-wound resistor - e.g. maybe the electron transport is not really "ballistic", and so the assumption that the electrons transported through it are independent and non-interacting isn't valid. There are some modified formulae for the shot noise through a metal resistor, which evaluates to 10pA/rtHz for the same 4kohm resistor, which is still ~5x the Johnson noise.
In the case of the HV coil driver circuit, the passive filtering stage I added at the output to filter out the excess PA95 noise unwittingly helps us - the pole at ~0.7 Hz filters the shot noise (but not the Johnson noise) such that at ~10 Hz, the Johnson noise does indeed dominate the total contribution. So, for this circuit, I think we don't have to worry about some un-budgeted noise. However, I am concerned about the fast actuation path - we were all along assuming that this path would be dominated by the Johnson noise of the 4kohm series resistor. But if we need even 1mA of current to null some DC DARM drift, then we'd have the shot noise contribution become comparable, or even dominant?
I looked through the iLIGO literature, where single-stage suspensions were being used, e.g. Rana's manifesto, but I cannot find any mention of shot noise due to DC current, so probably there is a simple explanation why - but it eludes me, at least for the moment. The iLIGO coil drivers did not have a passive filter at the output of the coil driver circuit (at least, not till this work), and there isn't any feedback gain for the DARM loop at >100 Hz (where we hope to measure squeezing) to significantly squash this noise.
Attachment #1 shows schematic topologies of the iLIGO and proposed 40m configs. It may be that I have completely misunderstood the iLIGO config and what I've drawn there is wrong. Since we are mainly interested in the noise from the resistor, I've assumed everything upstream of the final op-amp is noiseless (equivalently, we assume we can sufficiently pre-filter these noises).
Attachment #2 shows the relative magnitudes of shot noise due to a DC current, and thermal noise of the series resistor, as a function of frequency, for a few representative currents, for the slow bias path assuming a 0.7Hz corner from the 4kohm/3uF RC filter at the output of the PA95.
Some lit review suggests that it's actually pretty hard to measure shot noise in a resistor - so I'm guessing that's what it is, the mean free path of electrons is short compared to the length of the resistor such that the assumption that electrons arrive independently and randomly isn't valid. So Ohm's law dictates and that's what sets the current noise. See, for example, pg 432 of Horowitz and Hill.