# Shot noise and single photon detection

I am looking into the noise considerations of a single photon detector, specifically an avalanche detector. I am wondering if it makes sense to think about shot noise when considering a single photon.

I believe that since we only wish to detect one photon then it just doesn't make sense to discuss shot noise.

Thoughts?

• isn't there shot noise in the avalanche? – JEB Dec 4 '18 at 2:17

There is still shot noise in the following sense.

You have some experiment. That experiment is emitting light towards you're single photon counter. Say you record a the signal coming from the single photon counter for 100 ms. You will get some streak of "clicks" indicating photons have been detected.

Now repeat the experiment a thousand times.

You now have a 1000 time traces. Now chunk the time window into 1 ms bins. You can now ask for any given time bin (for example from $$38 ms) what is the average number of photons detected in that window, averaged over all trials of the experiment? You can also ask what is the standard deviation of the photon number detected in that window.

You will find that the mean photon count in that window is proportional to the average photon flux during that time window, $$\bar{N}$$, during that time and the standard deviation of the photon count during that time window will be proportional to the square root of the photon flux, $$\sqrt{\bar{N}}$$ during that time window. Note that this will be true even if $$\bar{N}<1$$.

We can define continuous and single photon detection modes. Every detector has some bandwidth $$f_{BW}$$. It is natural to consider the time having to do with the inverse of the bandwidth: $$\Delta t = \frac{1}{f_{BW}}$$. $$\Delta t$$ can be thought of as the minimal resolvable time window for the detector. That is the detector "averages" the field incident upon it over a time $$\Delta t$$. So if there is a field incident on the detector with a flux of $$10^9$$ photons per second and the bandwidth of the detector is $$1 GHz = 10^{-9} s$$ then the detector will detect one photon per time window $$\Delta t$$. If many photons are falling on the detector per one time bandwidth window then we can say the detector is in a continuous detection mode. If less than one photon falls on the detector per detection window then we can say it is in a single photon counting mode.

In the continuous mode for a single trace you will see a continuous photocurrent which has fluctuations. In the single photon mode you will see a stream of pulses with fluctuating spacings in time.

In the continuous mode shot noise is manifested as a variance of the signal level. In the single photon mode shot noise is manifested as random timings of the pulses. The idea of shot noise applies to both of them in as much as it is a statement about what happens if you consider the statistics of many realizations of the experiment that created the photosignal.

It is important when it comes to the source because you cannot predict when exactly the single photos will come. And I believe the avalanche diode also has a probability that a dark current electron will avalanche so timing can be tricky in your experiment. Also cosmic rays can upset things and they are also shot noise based.

Shot noise is defined here as the noise that can be modeled through a Poisson distribution and that is ultimately determined by the fact that the fundamental quantity you are measuring is quantized. This noise term, therefore, is relevant when we are dealing with photons, that are quanta of the electromagnetic field. It will thus be related to the fact that every time a photon strikes on your detector you will have a certain probability $$p$$ of detecting it and, conversely, a probability $$1-p$$ of not detecting it. The noise will thus be related, roughly speaking, to the fact that we have obtained, at the output of the detector, a signal when the photon was not present or that we have not obtained any signal when the photon was present.

The technique you are probably referring to, when it comes to the detection of single photons, is called single photon counting. In this technique, the detection comes down to hard limits in terms of noise, that are related to the quantum nature of light, thus leading to the so called quantum noise. Shot noise is a type of quantum noise and it is, therefore, unavoidable and fundamental in this kind of process.