Quantum graininess

Everybody knows about the Heisenberg Uncertainty Principle. How much they know about it varies. Most know it as a limit to the knowledge that we can have about quantum systems. Some know it more specifically as a limit to the knowledge we can have about the value of two conjugate variables. Some can derive the uncertainty principle from other basic rules of quantum mechanics.

Last year as I first began to learn about quantum optics I was extremely surprised to learn that phase and number can be considered to be conjugate variables in a loose sense. If you know photon number perfectly you can do so only by destroying photon phase.

Anyway, the HUP shows up all over the place and becomes a problem any time you want to perform a measurement. I suppose that the question that I want to pose now is simple: when you have a nonlinear effect that manifests itself as an effect on the phase of a system that phase will have an uncertainty that corresponds to your knowledge of the number state of that system. If the phase is classically proportional to the strength of your nonlinear interaction then you might map that uncertainty onto the strength of your nonlinear interaction. From this mapping can one state an uncertainty principle for the precision with which the strength of the nonlinear interaction can be specified? It seems like if you could this might open up a way to analyze fundamental limits to your ability to make precision measurements using nonlinear optics.

So how should I get at the problem? The first way you might get at specifying the uncertainty of a quantity is by representing it as an operator and seeing what it commutes with. I think, however, that nonlinear interaction strengths are not likely to be observables but rather parameters that you can estimate using observables. So I guess that the analysis might follow one that you would use for other important but unobservable parameters like phase.