Fundamental noise limit for an ideal photodetector

[Dustgil]

Homework Statement

As the title says, I'm trying to calculate the fundamental noise limit for an ideal photodetector, by specifically looking at the rate of incidence of annihilation of photons (and subsequent excitation of conducting electrons) on the detection surface. Since I'm looking for the theoretical minimum, the quantum efficiency is 100% (taken care of by the ideal nature of the photodetector) and each incident photon excites an electron into functioning as a conducting electron.

Homework Equations

The Attempt at a Solution

So, to describe the generation and annihilation of photons, we need operators. If we have a state described by n particles, then the generation operator acts on that state to create a state with n+1 particles. Likewise, the annihilation operator acts on the n state to produce a state with n-1 particles, UNLESS the n state is the state with zero particles. In that case, the operation produces the n state again, corresponding to the zero state.

This suggests the operators can be represented in the ladder operator framework. Then the properties of the operators should be the same as for all ladder operators. The two operators are then adjoints of each other, and

$$b|\psi_{n}\rangle=\sqrt{n+1}|\psi_{n+1}\rangle$$
$$b^{\dagger}|\psi_{n}\rangle=\sqrt{n}|\psi_{n-1}\rangle$$

where the nth state corresponds to the number of particles, n.

The commutator for ladder operators is 1, for example:

$$[b,b^{\dagger}]|n\rangle=bb^{\dagger}|n\rangle-b^{\dagger}b|n\rangle= [(n+1)-(n)]|n\rangle=|n\rangle$$

Then the uncertainty between the two operators is:

$$\sigma_{b}^{2}\sigma_{b^{\dagger}}^{2}=(\frac{1}{2i}\langle[b,b^{\dagger}]\rangle)^{2}=-\frac{1}{4}$$

THAT doesn't seem right...I was thinking of establishing an minimum possible uncertainty between the creation and annihilation of photons and in that way establish a minimum noise limit for photodetection. Where have I gone wrong? This is a first-semester QM project so I'm hoping I didn't bite off more than I can chew.

 

[blue_leaf77]

How do you define the uncertainty of non-hermitian operators like creation and annihilation operators?