# In hetero- homodyne detection, what does it mean to operate at the Quantum Shot Noise limit?

I am an electrical engineering by trade, working on the analogue part (Transmitter & Receiver) of a quantum optical communication channel. By this, I mean I have not much experience on things quantum.

There are, however, some notions or concepts that I would like to understand, but no matter how much I read articles I don't seem to be able to get a clear answer.

My question would be:

• What does it mean for a hetero- or homodyne receiver to be operating in the Quantum Shot Noise limit?

There are some notions that I have (I don't know if my assumptions are wrong).

• Shot noise is caused by the uneven production/arrival of photons. When the rate of arrival is high enough, we can model this noise as following a Gaussian distribution. If the rate is low (related to the power of the laser, I wager in relation to the Local Oscillator (LO)), we have to model it as a Poisson distribution.
• Dark current of photodetectors can also be modelled as such.
• It is important to operate near the Shot Noise Limit. I don't know exactly why, but some potential hypothesis:
• It limits/allows the elimination of electrical noise in post-processing
• Uncorrelates LO power with signal power, facilitating detection of states
• Simply improves detection by reducing overall noise floor (the more power, the more shot noise)

At the end of the end, what I would like to know is:

• Is the shot noise limit in such a configuration absolute or relative? That is, does it depend on the power level of the LO, or is it a fixed value (beyond physical parameters of the system) that one approaches as the power level of the LO is reduced?
• What would be a formula for determining what the power level of the LO should be for the system to be operating at the Shot Noise Limit?

Apologies if this is too long.

There are good questions here. I don't have time to thoroughly answer all of them.

What does it mean for a hetero- or homodyne receiver to be operating in the Quantum Shot Noise limit?

In general with optical detection "operating at the shot noise limit" means the noise is dominated by optical shot noise. Usually this means optical shot noise > detector electrical Johnson noise. There is electrical shot noise in the detection circuit but this is usually much smaller than the Johnson noise.

Shot noise is caused by the uneven production/arrival of photons. When the rate of arrival is high enough, we can model this noise as following a Gaussian distribution. If the rate is low (related to the power of the laser, I wager in relation to the Local Oscillator (LO)), we have to model it as a Poisson distribution.

We typically always model shot noise by saying: If I detect on average $$n$$ photons then the standard deviation is $$\sqrt{n}$$ photons.

Dark current of photodetectors can also be modelled as such.

Like I said above, there is also electron shot noise. In my experience this is well below the Johnson noise of the photodetector's trans-impedance amplifier so we can ignore it. To be optically shot noise limited the optical shot noise must far exceed electrical shot noise.

It is important to operate near the Shot Noise Limit. I don't know exactly why, but some potential hypothesis:

• It limits/allows the elimination of electrical noise in post-processing
• Uncorrelates LO power with signal power, facilitating detection of states
• Simply improves detection by reducing overall noise floor (the more power, the more shot noise)

Your first two points don't make sense to me. The third point is pretty much correct. Often we are using laser light to measure something and the fundamental physics limit to how well we can measure that thing is the optical shot noise limit. Electrical noise (Johnson noise, electron shot noise) are technical rather than physical or fundamental limits to the measurement sensitivities. So it's a shame if we're limited by these rather than optical shot noise.

Is the shot noise limit in such a configuration absolute or relative? That is, does it depend on the power level of the LO, or is it a fixed value (beyond physical parameters of the system) that one approaches as the power level of the LO is reduced?

The shot noise limit depends on the power of the LO. So you can more easily reach the optical shot noise limit by increasing the power of the LO. This is one of the amazing features of interferometric detection. It makes it so easy to overcome technical noise by just increasing the power in the LO. The problem now becomes one of dynamic range rather than noise. That is, with the LO maxed out so you don't saturate the detection, are the fluctuations in the signal large enough to exceed the measurement sensitivity of the detector so that you can see them. Max power / sensitivity is the detector dynamic range. So with interferometric detection we care about detector dynamic range and don't need to worry so much about e.g. the detector noise floor since we can probably exceed that with a large enough LO. In my opinion this technical discussion about detector specifications is arguably the MAIN technical reason to mean to homo- or heterodyne detection for physics measurements.

What would be a formula for determining what the power level of the LO should be for the system to be operating at the Shot Noise Limit?

PSD from shot noise > PSD from electrical noise sources

Kind of tongue in cheek answer there.. in grad school I spent a lot of time working out these formulas, but I don't have time to reproduce those calculations here right now. The shot noise floor increases linearly with optical power. One way to see if you are shot-noise limited is to scan the power (linearly) falling on your photodetector (probably relying on a linear calibrated power meter) and observe the level of the noise floor. If it is increasing linearly with power then you are shot noise limited. At low enough powers it will stop changing with power. Here you can assume you're limited by the electrical noise from the photodetector, or the noise floor of your spectrum analyzer (in this latter case you don't know if your detector is optically shot noise limited or not).

• This was really helpful, thank you. So from what I'm gathering, one would aim to reduce as much as possible other sources of noise, so that the increase in LO power to reach the shot noise limit would be minimum (and also have as much dynamic range available as possible) Commented Jul 14 at 15:43
• @MrNotSafe4Work yeah, that's pretty much right. The signal is a certain level. We can boost it arbitrarily large with interferometric detection, but at some point we hit the max power/current on the photodetector. It might be worth mentioning: in my opinion, "optical shot noise limited" is not a goal unto itself. Boosting signal-to-noise is a goal unto itself. "Optical shot noise limited" is just a milestone which says you've improved SNR as much as you can on the detection side. You either can't improve things further or you need to modify your physics experiment somehow. Commented Jul 14 at 19:39