1. From properties of destruction operator
So, first you have to accept that $a|n\rangle = \sqrt{n}|n-1\rangle$. This is relatively easy to see because the matrix element for absorption of a photon by a two-level system (atom which goes from ground to excited state) is proportional to $\langle n-1|a|n\rangle$ and this must be proportional to the square root of the number of photons in the light mode because the probability for absorption must be proportional to the light intensity. So you need something like $a|n\rangle = \sqrt{n}|n-1\rangle$ (ignoring a possible phase factor).
Then, when you expand the coherent state in number states, $|\alpha\rangle = \sum_n c_n |n\rangle$ and put this into $a|n\rangle = \alpha|n\rangle$, you see that you need $c_n\sqrt{n}=\alpha c_{n-1}$. The result when lowering $c_n|n\rangle$ with $a$ must be the same as a multiplication of $|n-1\rangle$ with $\alpha$. As a consequence, $c_n =(\alpha/\sqrt{n})c_{n-1}$ and you are finished. Iterating this $n$ times yields $c_n = (\alpha^n/\sqrt{n!})c_0$. Normalization gives the value of $c_0$ and then you have $\langle n|\alpha\rangle=c_n$. Now you square the whole stuff and get the Poisson distribution.
So the point is that for large $n$, $\alpha/\sqrt{n}$ will always be smaller than 1. This is why the Poisson distribution decreases in this case. For small $n$, the opposite holds and the Poisson distribution increases.
2. Coherent state in phase space
There's an alternative picture. 976
You know that a single-mode field is like a harmonic oscillator where the mode's quadrature operators play the role of position and momentum of the HO. Now, a coherent state is a wave packet that oscillates in the parabolic potential without changing its shape. There is no dispersion for this wave packet, it coheres (this is where the name coherent state comes from). The energy eigenstates of the HO (which correspond to the number states of the field mode) are static, they don't move. So, to construct a coherent state, you need to use a superposition of number states. And the weighting of number states in the superposition is the square of the probabilities of the Poisson distribution.
This is also not an intuitive physical explanation but it shines a little bit more light on the problem.
3. Coherent state and independent emission events
Another possiblity to get a physical understanding is the independence of the "emission" events. From this, the Poisson distribution is easily understood. What I don't see is the connection between the coherent state $|\alpha\rangle$ and the concept of statistically independent emissions. I think it's even counterintuitive. In the laser, the induced emission events (together with the resonator) create the coherent state. The statistically independent spontaneous emission events disturb the coherent state (phase fluctuations in the laser).
Who can help? 795