What is the state of the light emitted by an esemble of independent single photon emitters?

Question

Upon proper excitation, single molecules (e.g. fluorescent dyes, quantum dots, etc.) spontaneously emit single photons, namely Fock states $|1\rangle$. These states have no statistical uncertainity over the photon number. However, experimental measure of fluorescence (say, an image of fluorescence microscopy) is well-known to be affected by shot-noise, namely the photons collected by the detector follow a Poisson distribution. This distribution is the same of that of a coherent state $|\alpha\rangle$. Why is this happening? What is causing the transition from number-state to an (almost) coherent-state in an ensemble of single-photon emitters?

Answer 1

Upon proper excitation, single molecules (e.g. fluorescent dyes, quantum dots, etc.) spontaneously emit single photons, namely Fock states |1⟩.

Most substances have a quantum yield that is much smaller than unity, so it would probably be more accurate to describe the light emitted as a density matrix in which, say, 90% of the probability is zero photons, and 10% is the one-photon state. This can be described by a binomial distribution $B(1,p)$, where $p=0.1$. Because $p\ll 1$, this distribution is actually well approximated by a Poisson distribution with mean $p$, although the approximation is not exact. (It can't be exact, because the Poisson distribution will have some probability for $n\ge 2$, in fact for arbitrarily large $n$, which is impossible based on conservation of energy.)

These states have no statistical uncertainity over the photon number.

So this is not quite true. The mean is $p$, and the standard deviation is $\sqrt{p(1-p)}$.

However, experimental measure of fluorescence (say, an image of fluorescence microscopy) is well-known to be affected by shot-noise, namely the photons collected by the detector follow a Poisson distribution.

Experimentally, your detector has some efficiency, which is also probably much less than one. This reduces $p$ by an additional factor. But at the same time, the detector counts photons for a certain time window, and you can see photons from lots of different excitation events. The sum of a large number of low-$p$ binomials is well approximated by a Poisson distribution.

This distribution is the same of that of a coherent state |α⟩. Why is this happening? What is causing the transition from number-state to an (almost) coherent-state in an ensemble of single-photon emitters?

But there is no close logical connection here. The different molecules are all emitting photons independently, so their radiation is not coherent. Poisson distributions arise in lots of real-world situations (insurance claims, etc.), most of which have nothing to do with coherent photon states.